Split (1)

train · ~317k rows (showing the first 75.7k)

id string	text string	source string	created timestamp[s]	added string	metadata dict
0902.0430	Singular Bott-Chern Classes and the Arithmetic Grothendieck Riemann Roch Theorem for Closed Immersions --- José I. Burgos Gil111Partially supported by Grant DGI MTM2006-14234-C02-01. and Răzvan Liţcanu222Partially supported by CNCSIS Grant 1338/2007 and PN II Grant ID_2228 (502/2009) Received: May 13, 2009 Communicated by Peter Schneider Abstract. We study the singular Bott-Chern classes introduced by Bismut, Gillet and Soulé. Singular Bott-Chern classes are the main ingredient to define direct images for closed immersions in arithmetic $K$-theory. In this paper we give an axiomatic definition of a theory of singular Bott-Chern classes, study their properties, and classify all possible theories of this kind. We identify the theory defined by Bismut, Gillet and Soulé as the only one that satisfies the additional condition of being homogeneous. We include a proof of the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions that generalizes a result of Bismut, Gillet and Soulé and was already proved by Zha. This result can be combined with the arithmetic Grothendieck-Riemann-Roch theorem for submersions to extend this theorem to arbitrary projective morphisms. As a byproduct of this study we obtain two results of independent interest. First, we prove a Poincaré lemma for the complex of currents with fixed wave front set, and second we prove that certain direct images of Bott-Chern classes are closed. 2010 Mathematics Subject Classification: 14G40 32U40 Keywords and Phrases: Arakelov Geometry, Closed immersions, Bott-Chern classes, Arithmetic Riemann-Roch theorem, currents, wave front sets. ###### Contents 1. 0 Introduction 2. 1 Characteristic classes in analytic Deligne cohomology 3. 2 Bott-Chern classes 4. 3 Direct images of Bott-Chern classes 5. 4 Cohomology of currents and wave front sets 6. 5 Deformation of resolutions 7. 6 Singular Bott-Chern classes 8. 7 Classification of theories of singular Bott-Chern classes 9. 8 Transitivity and projection formula 10. 9 Homogeneous singular Bott-Chern classes 11. 10 The arithmetic Riemann-Roch theorem for regular closed immersions ## 0 Introduction Chern-Weil theory associates to each hermitian vector bundle a family of closed characteristic forms that represent the characteristic classes of the vector bundle. The characteristic classes are compatible with exact sequences. But this is not true for the characteristic forms. The Bott-Chern classes measure the lack of compatibility of the characteristic forms with exact sequences. The Grothendieck-Riemann-Roch theorem gives a formula that relates direct images and characteristic classes. In general this formula is not valid for the characteristic forms. The singular Bott-Chern classes measure, in a functorial way, the failure of an exact Grothendieck-Riemann-Roch theorem for closed immersions at the level of characteristic forms. In the same spirit, the analytic torsion forms measure the failure of an exact Grothendieck- Riemann-Roch theorem for submersions at the level of characteristic forms. Hence singular Bott-Chern classes and analytic torsion forms are analogous objects, the first for closed immersions and the second for submersions. Let us give a more precise description of Bott-Chern classes and singular Bott-Chern classes. Let $X$ be a complex manifold and let $\varphi$ be a symmetric power series in $r$ variables with real coefficients. Let $\overline{E}=(E,h)$ be a rank $r$ holomorphic vector bundle provided with a hermitian metric. Using Chern-Weil theory, we can associate to $\overline{E}$ a differential form $\varphi(\overline{E})=\varphi(-K)$, where $K$ is the curvature tensor of $E$ viewed as a matrix of 2-forms. The differential form $\varphi(\overline{E})$ is closed and is a sum of components of bidegree $(p,p)$ for $p\geq 0$. If $\overline{\xi}\colon 0\longrightarrow\overline{E}^{\prime}\longrightarrow\overline{E}\longrightarrow\overline{E}^{\prime\prime}\longrightarrow 0$ is a short exact sequence of holomorphic vector bundles provided with hermitian metrics, then the differential forms $\varphi(\overline{E})$ and $\varphi(\overline{E}^{\prime}\oplus\overline{E}^{\prime})$ may be different, but they represent the same cohomology class. The Bott-Chern form associated to $\overline{\xi}$ is a solution of the differential equation $-2\partial\bar{\partial}\varphi(\overline{\xi})=\varphi(\overline{E}^{\prime}\oplus\overline{E}^{\prime})-\varphi(\overline{E})$ (0.1) obtained in a functorial way. The class of a Bott-Chern form modulo the image of $\partial$ and $\overline{\partial}$ is called a Bott-Chern class and is denoted by $\widetilde{\varphi}(\overline{\xi})$. There are three ways of defining the Bott-Chern classes. The first one is the original definition of Bott and Chern [7]. It is based on a deformation between the connection associated to $\overline{E}$ and the connection associated to $\overline{E}^{\prime}\oplus\overline{E}^{\prime\prime}$. This deformation is parameterized by a real variable. In [17] Gillet and Soulé introduced a second definition of Bott-Chern classes that is based on a deformation between $\overline{E}$ and $\overline{E}^{\prime}\oplus\overline{E}^{\prime\prime}$ parameterized by a projective line. This second definition is used in [4] to prove that the Bott- Chern classes are characterized by three properties 1. (i) The differential equation (0.1). 2. (ii) Functoriality (i.e. compatibility with pull-backs via holomorphic maps). 3. (iii) The vanishing of the Bott-Chern class of a orthogonally split exact sequence. In [4] Bismut, Gillet and Soulé have a third definition of Bott-Chern classes based on the theory of superconnections. This definition is useful to link Bott-Chern classes with analytic torsion forms. The definition of Bott-Chern classes can be generalized to any bounded exact sequence of hermitian vector bundles (see section 2 for details). Let $\overline{\xi}\colon 0\longrightarrow(E_{n},h_{n})\longrightarrow\dots\longrightarrow(E_{1},h_{1})\longrightarrow(E_{0},h_{0})\longrightarrow 0$ be a bounded acyclic complex of hermitian vector bundles; by this we mean a bounded acyclic complex of vector bundles, where each vector bundle is equipped with an arbitrarily chosen hermitian metric. Let $r=\sum_{i\text{ even}}\operatorname{rk}(E_{i})=\sum_{i\text{ odd}}\operatorname{rk}(E_{i}).$ As before, let $\varphi$ be a symmetric power series in $r$ variables. A Bott- Chern class associated to $\overline{\xi}$ satisfies the differential equation $-2\partial\bar{\partial}\widetilde{\varphi}(\overline{\xi})=\varphi(\bigoplus_{k}\overline{E}_{2k})-\varphi(\bigoplus_{k}\overline{E}_{2k+1}).$ In particular, let “$\operatorname{ch}$” denote the power series associated to the Chern character class. The Chern character class has the advantage of being additive for direct sums. Then, the Bott-Chern class associated to the long exact sequence $\overline{\xi}$ and to the Chern character class satisfies the differential equation $-2\partial\bar{\partial}\widetilde{\operatorname{ch}}(\overline{\xi})=-\sum_{k=0}^{n}(-1)^{i}\operatorname{ch}(\overline{E}_{k}).$ Let now $i\colon Y\longrightarrow X$ be a closed immersion of complex manifolds. Let $\overline{F}$ be a holomorphic vector bundle on $Y$ provided with a hermitian metric. Let $\overline{N}$ be the normal bundle to $Y$ in $X$ provided also with a hermitian metric. Let $0\longrightarrow\overline{E}_{n}\longrightarrow\overline{E}_{n-1}\longrightarrow\dots\longrightarrow\overline{E}_{0}\longrightarrow i_{\ast}F\longrightarrow 0$ be a resolution of the coherent sheaf $i_{\ast}F$ by locally free sheaves, provided with hermitian metrics (following Zha [32] we shall call such a sequence a metric on the coherent sheaf $i_{\ast}F$). Let $\operatorname{Td}$ denote the Todd characteristic class. Then the Grothendieck-Riemann-Roch theorem for the closed immersion $i$ implies that the current $i_{\ast}(\operatorname{Td}(\overline{N})^{-1}\operatorname{ch}(\overline{F}))$ and the differential form $\sum_{k}(-1)^{k}\operatorname{ch}(\overline{E}_{k})$ represent the same class in cohomology. We denote $\overline{\xi}$ the data consisting in the closed embedding $i$, the hermitian bundle $\overline{N}$, the hermitian bundle $\overline{F}$ and the resolution $\overline{E}_{\ast}\longrightarrow i_{\ast}F$. In the paper [5], Bismut, Gillet and Soulé introduced a current associated to the above situation. These currents are called singular Bott-Chern currents and denoted in [5] by $T(\overline{\xi})$. When the hermitian metrics satisfy a certain technical condition (condition A of Bismut) then the singular Bott- Chern current $T(\overline{\xi})$ satisfies the differential equation $-2\partial\bar{\partial}T(\overline{\xi})=i_{\ast}(\operatorname{Td}(\overline{N})^{-1}\operatorname{ch}(\overline{F}))-\sum_{i=0}^{n}(-1)^{i}\operatorname{ch}(\overline{E}_{i}).$ These singular Bott-Chern currents are among the main ingredients of the proof of Gillet and Soulé’s arithmetic Riemann-Roch theorem. In fact it is the main ingredient of the arithmetic Riemann-Roch theorem for closed immersions [6]. This definition of singular Bott-Chern classes is based on the formalism of superconnections, like the third definition of ordinary Bott-Chern classes. In his thesis [32], Zha gave another definition of singular Bott-Chern currents and used it to give a proof of a different version of the arithmetic Riemann-Roch theorem. This second definition is analogous to Bott and Chern’s original definition. Nevertheless there is no explicit comparison between the two definitions of singular Bott-Chern currents. One of the purposes of this note is to give a third construction of singular Bott-Chern currents, in fact of their classes modulo the image of $\partial$ and $\overline{\partial}$, which could be seen as analogous to the second definition of Bott-Chern classes. Moreover we will use this third construction to give an axiomatic definition of a theory of singular Bott-Chern classes. A theory of singular Bott-Chern classes is an assignment that, to each data $\overline{\xi}$ as above, associates a class of currents $T(\overline{\xi})$, that satisfies the analogue of conditions (i), (ii) and (iii). The main technical point of this axiomatic definition is that the conditions analogous to (i), (ii) and (iii) above are not enough to characterize the singular Bott- Chern classes. Thus we are led to the problem of classifying the possible theories of Bott-Chern classes, which is the other purpose of this paper. We fix a theory $T$ of singular Bott-Chern classes. Let $Y$ be a complex manifold and let $\overline{N}$ and $\overline{F}$ be two hermitian holomorphic vector bundles on $Y$. We write $P=\mathbb{P}(N\oplus 1)$ for the projective completion of $N$. Let $s\colon Y\longrightarrow P$ be the inclusion as the zero section and let $\pi_{P}\colon P\longrightarrow Y$ be the projection. Let $\overline{K}_{\ast}$ be the Koszul resolution of $s_{\ast}\mathcal{O}_{Y}$ endowed with the metric induced by $\overline{N}$. Then we have a resolution by hermitian vector bundles $K(\overline{F},\overline{N})\colon\overline{K}_{\ast}\otimes\pi_{P}^{\ast}\overline{F}\longrightarrow s_{\ast}F.$ To these data we associate a singular Bott-Chern class $T(K(\overline{F},\overline{N}))$. It turns out that the current $\frac{1}{(2\pi i)^{\operatorname{rk}N}}\int_{\pi_{P}}T(K(\overline{F},\overline{N}))=(\pi_{P})_{\ast}T(K(\overline{F},\overline{N}))$ is closed (see section 3 for general properties of the Bott-Chern classes that imply this property) and determines a characteristic class $C_{T}(F,N)$ on $Y$ for the vector bundles $N$ and $F$. Conversely, any arbitrary characteristic class for pairs of vector bundles can be obtained in this way. This allows us to classify the possible theories of singular Bott-Chern classes: ###### Claim (theorem 7.1). The assignment that sends a singular Bott-Chern class $T$ to the characteristic class $C_{T}$ is a bijection between the set of theories of singular Bott-Chern classes and the set of characteristic classes. The next objective of this note is to study the properties of the different theories of singular Bott-Chern classes and of the corresponding characteristic classes. We mention, in the first place, that for the functoriality condition to make sense, we have to study the wave front sets of the currents representing the singular Bott-Chern classes. In particular we use a Poincaré Lemma for currents with fixed wave front set. This result implies that, in each singular Bott-Chern class, we can find a representative with controlled wave front set that can be pulled back with respect certain morphisms. We also investigate how different properties of the singular Bott-Chern classes $T$ are reflected in properties of the characteristic classes $C_{T}$. We thus characterize the compatibility of the singular Bott-Chern classes with the projection formula, by the property of $C_{T}$ of being compatible with the projection formula. We also relate the compatibility of the singular Bott- Chern classes with the composition of successive closed immersions to an additivity property of the associated characteristic class. Furthermore, we show that we can add a natural fourth axiom to the conditions analogue to (i), (ii) and (iii), namely the condition of being homogeneous (see section 9 for the precise definition). ###### Claim (theorem 9.11). There exists a unique homogeneous theory of singular Bott-Chern classes. Thanks to this axiomatic characterization, we prove that this theory agrees with the theories of singular Bott-Chern classes introduced by Bismut, Gillet and Soulé [6], and by Zha [32]. In particular this provides us a comparison between the two definitions. We will also characterize the characteristic class $C_{T^{h}}$ for the theory of homogeneous singular Bott-Chern classes. The last objective of this paper is to give a proof of the arithmetic Riemann- Roch theorem for closed immersions. A version of this theorem was proved by Bismut, Gillet and Soulé and by Zha. Next we will discuss the contents of the different sections of this paper. In section §1 we recall the properties of characteristic classes in analytic Deligne cohomology. A characteristic class is just a functorial assignment that associates a cohomology class to each vector bundle. The main result of this section is that any characteristic class is given by a power series on the Chern classes, with appropriate coefficients. In section §2 we recall the theory of Bott-Chern forms and its main properties. The contents of this section are standard although the presentation is slightly different to the ones published in the literature. In section §3 we study certain direct images of Bott-Chern forms. The main result of this section is that, even if the Bott-Chern classes are not closed, certain direct images of Bott-Chern classes are closed. This result generalizes previous results of Bismut, Gillet and Soulé and of Mourougane. This result is used to prove that the class $C_{T}$ mentioned previously is indeed a cohomology class, but it can be of independent interest because it implies that several identities in characteristic classes are valid at the level of differential forms. In section §4 we study the cohomology of the complex of currents with a fixed wave front set. The main result of this section is a Poincaré lemma for currents of this kind. This implies in particular a $\partial\bar{\partial}$-lemma. The results of this section are necessary to state the functorial properties of singular Bott-Chern classes. In section §5 we recall the deformation of resolutions, that is a generalization of the deformation to the normal cone, and we also recall the construction of the Koszul resolution. These are the main geometric tools used to study singular Bott-Chern classes. Sections §6 to §9 are devoted to the definition and study of the theories of singular Bott-Chern classes. Section §6 contains the definition and first properties. Section §7 is devoted to the classification theorem of such theories. In section §8 we study how properties of the theory of singular Bott-Chern classes and of the associated characteristic class are related. And in section §9 we define the theory of homogeneous singular Bott-Chern classes and we prove that it agrees with the theories defined by Bismut, Gillet and Soulé and by Zha. Finally in section §10 we define arithmetic $K$-groups associated to a $\mathcal{D}_{\log}$-arithmetic variety $(\mathcal{X},\mathcal{C})$ (in the sense of [13]) and push-forward maps for closed immersions of metrized arithmetic varieties, at the level of the arithmetic $K$-groups. After studying the compatibility of these maps with the projection formula and with the push-forward map at the level of currents, we prove a general Riemann-Roch theorem for closed immersions (theorem 10.28) that compares the direct images in the arithmetic $K$-groups with the direct images in the arithmetic Chow groups. This theorem is compatible, if we choose the theory of homogeneous singular Bott-Chern classes, with the arithmetic Riemann-Roch theorem for closed immersions proved by Bismut, Gillet and Soulé [6] and it agrees with the theorem proved by Zha [32]. Theorem 10.28, together with the arithmetic Grothendieck-Riemann-Roch theorem for submersions proved in [16], can be used to obtain an arithmetic Grothendieck-Riemann-Roch theorem for projective morphisms of regular arithmetic varieties. _Acknowledgements_ : This project was started during the Special Year on Arakelov Theory and Shimura Varieties held at the CRM (Bellaterra, Spain). We would like to thank the CRM for his hospitality during that year. We would also like to thank the University of Barcelona and the University Alexandru Ioan Cuza of Iaşi for their hospitality during several visits that allowed us to finish the project. We would also like to thank K. Köhler, J. Kramer, U. Kühn, V. Maillot, D. Rossler, and J. Wildeshaus with whom we have had many discussions on the subject of this paper. Our special thanks to G. Freixas and Shun Tang for their careful reading of the paper and for suggesting some simplifications of the proofs. Finally we would like to thank the referee for his excellent work. ## 1 Characteristic classes in analytic Deligne cohomology A characteristic class for complex vector bundles is a functorial assignment which, to each complex continuous vector bundle on a paracompact topological space $X$, assigns a class in a suitable cohomology theory of $X$. For example, if the cohomology theory is singular cohomology, it is well known that each characteristic class can be expressed as a power series in the Chern classes. This can be seen for instance, showing that continuous complex vector bundles on a paracompact space $X$ can be classified by homotopy classes of maps from $X$ to the classifying space $BGL_{\infty}(\mathbb{C})$ and that the cohomology of $BGL_{\infty}(\mathbb{C})$ is generated by the Chern classes (see for instance [28]). The aim of this section is to show that a similar result is true if we restrict the class of spaces to the class of quasi-projective smooth complex manifolds, the class of maps to the class of algebraic maps and the class of vector bundles to the class of algebraic vector bundles and we choose analytic Deligne cohomology as our cohomology theory. This result and the techniques used to prove it are standard. We will use the splitting principle to reduce to the case of line bundles and will then use the projective spaces as a model of the classifying space $BGL_{1}(\mathbb{C})$. In this section we also recall the definition of Chern classes in analytic Deligne cohomology and we fix some notations that will be used through the paper. ###### Definition 1.1. Let $X$ be a complex manifold. For each integer $p$, _the analytic real Deligne complex_ of $X$ is $\mathbb{R}_{X,\mathcal{D}}(p)=(\underline{\mathbb{R}}(p)\longrightarrow\mathcal{O}_{X}\longrightarrow\Omega^{1}_{X}\longrightarrow\dots\longrightarrow\Omega^{p-1}_{X})\\\ \cong s(\underline{\mathbb{R}}(p)\oplus F^{p}\Omega_{X}^{\ast}\longrightarrow\Omega^{\ast}_{X}),$ where $\underline{\mathbb{R}}(p)$ is the constant sheaf $(2\pi i)^{p}\underline{\mathbb{R}}\subseteq\underline{\mathbb{C}}$. The _analytic real Deligne cohomology of $X$_, denoted $H^{\ast}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(p))$, is the hyper-cohomology of the above complex. Analytic Deligne cohomology satisfies the following result. ###### Theorem 1.2. The assignment $X\longmapsto H^{\ast}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(\ast))=\bigoplus_{p}H^{\ast}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(p))$ is a contravariant functor between the category of complex manifolds and holomorphic maps and the category of unitary bigraded rings that are graded commutative (with respect to the first degree) and associative. Moreover there exists a functorial map $c\colon\operatorname{Pic}(X)=H^{1}(X,\mathcal{O}^{\ast}_{X})\longrightarrow H^{2}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(1))$ and, for each closed immersion of complex manifolds $i\colon Y\longrightarrow X$ of codimension $p$, there exists a morphism $i_{\ast}\colon H^{}_{\mathcal{D}^{\text{{\rm an}}}}(Y,\mathbb{R}())\longrightarrow H^{+2p}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(+p))$ satisfying the properties 1. A1 Let $X$ be a complex manifold and let $E$ be a holomorphic vector bundle of rank $r$. Let $\mathbb{P}(E)$ be the associated projective bundle and let $\mathcal{O}(-1)$ the tautological line bundle. The map $\pi^{\ast}\colon H^{\ast}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(\ast))\longrightarrow H^{\ast}_{\mathcal{D}^{\text{{\rm an}}}}(\mathbb{P}(E),\mathbb{R}(\ast))$ induced by the projection $\pi\colon\mathbb{P}(E)\longrightarrow X$ gives to the second ring a structure of left module over the first. Then the elements $c(\operatorname{cl}(\mathcal{O}(-1)))^{i}$, $i=0,\dots,r-1$ form a basis of this module. 2. A2 If $X$ is a complex manifold, $L$ a line bundle, $s$ a holomorphic section of $L$ that is transverse to the zero section, $Y$ is the zero locus of $s$ and $i\colon Y\longrightarrow X$ the inclusion, then $c(\operatorname{cl}(L))=i_{\ast}(1_{Y}).$ 3. A3 If $j\colon Z\longrightarrow Y$ and $i\colon Y\longrightarrow X$ are closed immersions of complex manifolds then $(ij)_{\ast}=i_{\ast}j_{\ast}$. 4. A4 If $i\colon Y\longrightarrow X$ is a closed immersion of complex manifolds then, for every $a\in H^{\ast}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(\ast))$ and $b\in H^{\ast}_{\mathcal{D}^{\text{{\rm an}}}}(Y,\mathbb{R}(\ast))$ $i_{\ast}(bi^{\ast}a)=(i_{\ast}b)a.$ ###### Proof. The functoriality is clear. The product structure is described, for instance, in [15]. The morphism $c$ is defined by the morphism in the derived category $\mathcal{O}^{\ast}_{X}[1]\overset{\cong}{\longleftarrow}s(\underline{\mathbb{Z}}(1)\rightarrow\mathcal{O}_{X})\longrightarrow s(\underline{\mathbb{R}}(1)\rightarrow\mathcal{O}_{X})=\mathbb{R}_{\mathcal{D}}(1).$ The morphism $i_{\ast}$ can be constructed by resolving the sheaves $\mathbb{R}_{\mathcal{D}}(p)$ by means of currents (see [26] for a related construction). Properties A3 and A4 follow easily from this construction. By abuse of notation, we will denote by $c_{1}(\mathcal{O}(-1))$ the first Chern class of $\mathcal{O}(-1)$ with the algebro-geometric twist, in any of the groups $H^{2}(\mathbb{P}(E),\underline{\mathbb{R}}(1))$, $H^{2}(\mathbb{P}(E),\underline{\mathbb{C}})$, $H^{1}(\mathbb{P}(E),\Omega^{1}_{\mathbb{P}(E)})$. Then, we have sheaf isomorphisms (see for instance [22] for a related result), $\displaystyle\bigoplus_{i=0}^{r-1}\underline{\mathbb{R}}_{X}(p-i)[-2i]$ $\displaystyle\longrightarrow R\pi_{\ast}\underline{\mathbb{R}}_{\mathbb{P}(E)}(p)$ $\displaystyle\bigoplus_{i=0}^{r-1}\Omega^{\ast}_{X}[-2i]$ $\displaystyle\longrightarrow R\pi_{\ast}\Omega^{\ast}_{\mathbb{P}(E)}$ $\displaystyle\bigoplus_{i=0}^{r-1}F^{p-i}\Omega^{\ast}_{X}[-2i]$ $\displaystyle\longrightarrow R\pi_{\ast}F^{p}\Omega^{\ast}_{\mathbb{P}(E)}$ given, all of them, by $(a_{0},\dots,a_{r-1})\longmapsto\sum a_{i}c_{1}(\mathcal{O}(-1))^{i}$. Hence we obtain a sheaf isomorphism $\bigoplus_{i=0}^{r-1}\mathbb{R}_{X,\mathcal{D}}(p-i)[-2i]\longrightarrow R\pi_{\ast}\mathbb{R}_{\mathbb{P}(E),\mathcal{D}}(p)$ from which property A1 follows. Finally property A2 in this context is given by the Poincare-Lelong formula (see [13] proposition 5.64). ∎ ###### Notation 1.3. For the convenience of the reader, we gather here together several notations and conventions regarding the differential forms, currents and Deligne cohomology that will be used through the paper. Throughout this paper we will use consistently the algebro-geometric twist. In particular the Chern classes $c_{i}$, $i=0,\dots$ in Betti cohomology will live in $c_{i}\in H^{2i}(X,\mathbb{R}(i))$; hence our normalizations differ from the ones in [18] where real forms and currents are used. Moreover we will use the following notations. We will denote by $\mathscr{E}^{\ast}_{X}$ the sheaf of Dolbeault algebras of differential forms on $X$ and by $\mathscr{D}^{\ast}_{X}$ the sheaf of Dolbeault complexes of currents on $X$ (see [13] §5.4 for the structure of Dolbeault complex of $\mathscr{D}^{\ast}_{X}$). We will denote by $E^{\ast}(X)$ and by $D^{\ast}(X)$ the complexes of global sections of $\mathscr{E}^{\ast}_{X}$ and $\mathscr{D}^{\ast}_{X}$ respectively. Following [9] and [13] definition 5.10, we denote by $(\mathcal{D}^{\ast}(\underline{\phantom{A}},\ast),\operatorname{d}_{\mathcal{D}})$ the functor that associates to a Dolbeault complex its corresponding Deligne complex. For shorthand, we will denote $\displaystyle\mathcal{D}^{\ast}(X,p)$ $\displaystyle=\mathcal{D}^{\ast}(E^{\ast}(X),p),$ $\displaystyle\mathcal{D}^{\ast}_{D}(X,p)$ $\displaystyle=\mathcal{D}^{\ast}(D^{\ast}(X),p).$ To keep track of the algebro-geometric twist we will use the conventions of [13] §5.4 regarding the current associated to a locally integrable differential form $[\omega](\eta)=\frac{1}{(2\pi i)^{\dim X}}\int_{X}\eta\land\omega$ and the current associated with a subvariety $Y$ $\delta_{Y}(\eta)=\frac{1}{(2\pi i)^{\dim Y}}\int_{Y}\eta.$ With these conventions, we have a bigraded morphism $\mathcal{D}^{\ast}(X,\ast)\to\mathcal{D}^{\ast}_{D}(X,\ast)$ and, if $Y$ has codimension $p$, the current $\delta_{Y}$ belongs to $\mathcal{D}^{2p}_{D}(X,p)$. Then $\mathcal{D}^{\ast}(X,p)$ and $\mathcal{D}_{D}^{\ast}(X,p)$ are the complex of global sections of an acyclic resolution of $\mathbb{R}_{X,\mathcal{D}}(p)$. Therefore $H^{\ast}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(p))=H^{\ast}(\mathcal{D}(X,p))=H^{\ast}(\mathcal{D}_{D}(X,p)).$ If $f:X\to Y$ is a proper smooth morphism of complex manifolds of relative dimension $e$, then the integral along the fibre morphism $f_{\ast}:\mathcal{D}^{k}(X,p)\longrightarrow\mathcal{D}^{k-2e}(X,p-e)$ is given by $f_{\ast}\omega=\frac{1}{(2\pi i)^{e}}\int_{f}\omega.$ (1.4) If $(\mathcal{D}^{\ast}(\ast),\operatorname{d}_{\mathcal{D}})$ is a Deligne complex associated to a Dolbeault complex, we will write $\widetilde{\mathcal{D}}^{k}(X,p):=\mathcal{D}^{k}(X,p)/\operatorname{d}_{\mathcal{D}}\mathcal{D}^{k-1}(X,p).$ Finally, following [13] 5.14 we denote by $\bullet$ the product in the Deligne complex that induces the usual product in Deligne cohomology. Note that, if $\omega\in\bigoplus_{p}\mathcal{D}^{2p}(X,p)$, then for any $\eta\in\mathcal{D}^{\ast}(X,\ast)$ we have $\omega\bullet\eta=\eta\bullet\omega=\eta\land\omega$. Sometimes, in this case we will just write $\eta\omega:=\eta\bullet\omega$. We denote by $\ast$ the complex manifold consisting on one single point. Then $H^{n}_{\mathcal{D^{\text{{\rm an}}}}}(\ast,p)=\begin{cases}\mathbb{R}(p):=(2\pi i)^{p}\mathbb{R},&\text{ if }n=0,\ p\leq 0,\\\ \mathbb{R}(p-1):=(2\pi i)^{p-1}\mathbb{R},&\text{ if }n=1,\ p>0.\\\ \\{0\\},&\text{otherwise.}\end{cases}$ The product structure in this case is the bigraded product that is given by complex number multiplication when the degrees allow the product to be non zero. We will denote by $\mathbb{D}$ this ring. This is the base ring for analytic Deligne cohomology. Note that, in particular, $H^{1}_{\mathcal{D^{\text{{\rm an}}}}}(\ast,1)=\mathbb{R}=\mathbb{C}/\mathbb{R}(1)$. We will denote by ${\bf 1}_{1}$ the image of $1$ in $H^{1}_{\mathcal{D^{\text{{\rm an}}}}}(\ast,1)$. Following [23], theorem 1.2 implies the existence of a theory of Chern classes for holomorphic vector bundles in analytic Deligne cohomology. That is, to every vector bundle $E$, we can associate a collection of Chern classes $c_{i}(E)\in H^{2i}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(i))$, $i\geq 1$ in a functorial way. We want to see that all possible characteristic classes in analytic Deligne cohomology can be derived from the Chern classes. ###### Definition 1.5. Let $n\geq 1$ be an integer and let $r_{1}\geq 1,\dots,r_{n}\geq 1$ be a collection of integers. A _theory of characteristic classes for $n$-tuples of vector bundles of rank $r_{1},\dots,r_{n}$_ is an assignment that, to each $n$-tuple of isomorphism classes of vector bundles $(E_{1},\dots,E_{n})$ over a complex manifold $X$, with $\operatorname{rk}(E_{i})=r_{i}$, assigns a class $\operatorname{cl}(E_{1},\dots,E_{n})\in\bigoplus_{k,p}H^{k}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(p))$ in a functorial way. That is, for every morphism $f\colon X\longrightarrow Y$ of complex manifolds, the equality $f^{\ast}(\operatorname{cl}(E_{1},\dots,E_{n}))=\operatorname{cl}(f^{\ast}E_{1},\dots,f^{\ast}E_{n})$ holds The first consequence of the functoriality and certain homotopy property of analytic Deligne cohomology classes is the following. ###### Proposition 1.6. Let $\operatorname{cl}$ be a theory of characteristic classes for $n$-tuples of vector bundles of rank $r_{1},\dots,r_{n}$. Let $X$ be a complex manifold and let $(E_{1},\dots,E_{n})$ be a $n$-tuple of vector bundles over $X$ with $\operatorname{rk}(E_{i})=r_{i}$ for all $i$. Let $1\leq j\leq n$ and let $0\longrightarrow E^{\prime}_{j}\longrightarrow E_{j}\longrightarrow E^{\prime\prime}_{j}\longrightarrow 0,$ be a short exact sequence. Then the equality $\operatorname{cl}(E_{1},\dots,E_{j},\dots,E_{n})=\operatorname{cl}(E_{1},\dots,E^{\prime}_{j}\oplus E^{\prime\prime}_{j},\dots,E_{n})$ holds. ###### Proof. Let $\iota_{0},\iota_{\infty}\colon X\longrightarrow X\times\mathbb{P}^{1}$ be the inclusion as the fiber over $0$ and the fiber over $\infty$ respectively. Then there exists a vector bundle $\widetilde{E}_{j}$ on $X\times\mathbb{P}^{1}$ (see for instance [19] (1.2.3.1) or definition 2.5 below) such that $\iota^{\ast}_{0}\widetilde{E}_{j}\cong E_{j}$ and $\iota^{\ast}_{\infty}\widetilde{E}_{j}\cong E^{\prime}_{j}\oplus E^{\prime\prime}_{j}$. Let $p_{1}\colon X\times\mathbb{P}^{1}\longrightarrow X$ be the first projection. Let $\omega\in\bigoplus_{k,p}\mathcal{D}^{k}(X,p)$ be any $\operatorname{d}_{\mathcal{D}}$-closed form that represents $\operatorname{cl}(p_{1}^{\ast}E_{1},\dots,\widetilde{E}_{j},\dots,p_{1}^{\ast}E_{n})$. Then, by functoriality we know that $\iota_{0}^{\ast}\omega$ represents $\operatorname{cl}(E_{1},\dots,E_{j},\dots,E_{n})$ and $\iota_{\infty}^{\ast}\omega$ represents $\operatorname{cl}(E_{1},\dots,E^{\prime}_{j}\oplus E^{\prime\prime}_{j},\dots,E_{n})$. We write $\beta=\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log t\bar{t}\bullet\omega,$ where $t$ is the absolute coordinate of $\mathbb{P}^{1}$. Then $\operatorname{d}_{\mathcal{D}}\beta=\iota_{\infty}^{\ast}\omega-\iota^{\ast}_{0}\omega$ which implies the result. ∎ A standard method to produce characteristic classes for vector bundles is to choose hermitian metrics on the vector bundles and to construct closed differential forms out of them. The following result shows that functoriality implies that the cohomology classes represented by these forms are independent from the hermitian metrics and therefore are characteristic classes. When working with hermitian vector bundles we will use the convention that, if $E$ denotes the vector bundle, then $\overline{E}=(E,h)$ will denote the vector bundle together with the hermitian metric. ###### Proposition 1.7. Let $n\geq 1$ be an integer and let $r_{1}\geq 1,\dots,r_{n}\geq 1$ be a collection of integers. Let $\operatorname{cl}$ be an assignment that, to each $n$-tuple $(\overline{E}_{1},\dots,\overline{E}_{n})=((E_{1},h_{1}),\dots,(E_{n},h_{n}))$ of isometry classes of hermitian vector bundles of rank $r_{1},\dots,r_{n}$ over a complex manifold $X$, associates a cohomology class $\operatorname{cl}(\overline{E}_{1},\dots,\overline{E}_{n})\in\bigoplus_{k,p}H_{\mathcal{D}}^{k}(X,\mathbb{R}(p))$ such that, for each morphism $f:Y\to X$, $\operatorname{cl}(f^{\ast}\overline{E}_{1},\dots,f^{\ast}\overline{E}_{n})=f^{\ast}\operatorname{cl}(\overline{E}_{1},\dots,\overline{E}_{n}).$ Then the cohomology class $\operatorname{cl}(\overline{E}_{1},\dots,\overline{E}_{n})$ is independent from the hermitian metrics. Therefore it is a well defined characteristic class. ###### Proof. Let $1\leq j\leq n$ be an integer and let $\overline{E}^{\prime}_{j}=(E_{j},h^{\prime}_{j})$ be the vector bundle underlying $\overline{E}_{j}$ with a different choice of metric. Let $\iota_{0}$, $\iota_{\infty}$ and $p_{1}$ be as in the proof of proposition 1.6. Then we can choose a hermitian metric $h$ on $p_{1}^{\ast}E_{j}$, such that $\iota_{0}^{\ast}(p_{1}^{\ast}E_{j},h)=\overline{E}_{j}$ and $\iota_{\infty}^{\ast}(p_{1}^{\ast}E_{j},h)=\overline{E}_{j}^{\prime}$. Let $\omega$ be any smooth closed differential form on $X\times\mathbb{P}^{1}$ that represents $\operatorname{cl}(p_{1}^{\ast}\overline{E}_{1},\dots,(p_{1}^{\ast}E_{1},h),\dots,p_{1}^{\ast}\overline{E}_{n}).$ Then, $\beta=\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log t\bar{t}\bullet\omega$ satisfies $\operatorname{d}_{\mathcal{D}}\beta=\iota_{\infty}^{\ast}\omega-\iota^{\ast}_{0}\omega$ which implies the result. ∎ We are interested in vector bundles that can be extended to a projective variety. Therefore we will restrict ourselves to the algebraic category. So, by a complex algebraic manifold we will mean the complex manifold associated to a smooth quasi-projective variety over $\mathbb{C}$. When working with an algebraic manifold, by a vector bundle we will mean the holomorphic vector bundle associated to an algebraic vector bundle. We will denote by $\mathbb{D}[[x_{1},\dots,x_{r}]]$ the ring of commutative formal power series. That is, the unknowns $x_{1},\dots,x_{r}$ commute with each other and with $\mathbb{D}$. We turn it into a commutative bigraded ring by declaring that the unknowns $x_{i}$ have bidegree $(2,1)$. The symmetric group in $r$ elements, $\mathfrak{S}_{r}$ acts on $\mathbb{D}[[x_{1},\dots,x_{r}]]$. The subalgebra of invariant elements is generated over $\mathbb{D}$ by the elementary symmetric functions. The main result of this section is the following ###### Theorem 1.8. Let $\operatorname{cl}$ be a theory of characteristic classes for $n$-tuples of vector bundles of rank $r_{1},\dots,r_{n}$. Then, there is a power series $\varphi\in\mathbb{D}[[x_{1},\dots,x_{r}]]$ in $r=r_{1}+\dots+r_{n}$ variables with coefficients in the ring $\mathbb{D}$, such that, for each complex algebraic manifold $X$ and each $n$-tuple of algebraic vector bundles $(E_{1},\dots,E_{n})$ over $X$ with $\operatorname{rk}(E_{i})=r_{i}$ this equality holds: $\operatorname{cl}(E_{1},\dots,E_{n})=\varphi(c_{1}(E_{1}),\dots,c_{r_{1}}(E_{1}),\dots,c_{1}(E_{n}),\dots,c_{r_{n}}(E_{n})).$ (1.9) Conversely, any power series $\varphi$ as before determines a theory of characteristic classes for $n$-tuples of vector bundles of rank $r_{1},\dots,r_{n}$, by equation (1.9). ###### Proof. The second statement is obvious from the properties of Chern classes. Since we are assuming $X$ quasi-projective, given $n$ algebraic vector bundles $E_{1},\dots,E_{n}$ on $X$, there is a smooth projective compactification $\widetilde{X}$ and vector bundles $\widetilde{E}_{1},\dots,\widetilde{E}_{n}$ on $\widetilde{X}$, such that $E_{i}=\widetilde{E}_{i}\|_{X}$ (see for instance [14] proposition 2.2), we are reduced to the case when $X$ is projective. In this case, analytic Deligne cohomology agrees with ordinary Deligne cohomology. Let us assume first that $r_{1}=\dots=r_{n}=1$ and that we have a characteristic class $\operatorname{cl}$ for $n$ line bundles. Then, for each $n$-tuple of positive integers $m_{1},\dots,m_{n}$ we consider the space $\mathbb{P}^{m_{1},\dots,m_{n}}=\mathbb{P}^{m_{1}}_{\mathbb{C}}\times\dots\times\mathbb{P}^{m_{n}}_{\mathbb{C}}$ and we denote by $p_{i}$ the projection over the $i$-th factor. Then $\left.\bigoplus_{k,p}H^{k}_{\mathcal{D}}(\mathbb{P}^{m_{1},\dots,m_{n}},\mathbb{R}(p))=\mathbb{D}[x_{1},\dots,x_{n}]\right/(x_{1}^{m_{1}},\dots,x_{n}^{m_{n}})$ is a quotient of the polynomial ring generated by the classes $x_{i}=c_{1}(p_{i}^{\ast}\mathcal{O}(1))$ with coefficients in the ring $\mathbb{D}$. Therefore, there is a polynomial $\varphi_{m_{1},\dots,m_{n}}$ in $n$ variables such that $\operatorname{cl}(p_{1}^{\ast}\mathcal{O}(1),\dots,p_{1}^{\ast}\mathcal{O}(1))=\varphi_{m_{1},\dots,m_{n}}(x_{1},\dots,x_{n}).$ If $m_{1}\leq m^{\prime}_{1}$, …, $m_{n}\leq m_{n}^{\prime}$ then, by functoriality, the polynomial $\varphi_{m_{1},\dots,m_{n}}$ is the truncation of the polynomial $\varphi_{m^{\prime}_{1},\dots,m^{\prime}_{n}}$. Therefore there is a power series in $n$ variables, $\varphi$ such that $\varphi_{m_{1},\dots,m_{n}}$ is the truncation of $\varphi$ in the appropriate quotient of the polynomial ring. Let $L_{1},\dots,L_{n}$ be line bundles on a projective algebraic manifold that are generated by global sections. Then they determine a morphism $f\colon X\longrightarrow\mathbb{P}^{m_{1},\dots,m_{n}}$ such that $L_{i}=f^{\ast}p^{\ast}_{i}\mathcal{O}(1)$. Therefore, again by functoriality, we obtain $\operatorname{cl}(L_{1},\dots,L_{n})=\varphi(c_{1}(L_{1}),\dots,c_{1}(L_{n})).$ From the class $\operatorname{cl}$ we can define a new characteristic class for $n+1$ line bundles by the formula $\operatorname{cl}^{\prime}(L_{1},\dots,L_{n},M)=\operatorname{cl}(L_{1}\otimes M^{\vee},\dots,L_{n}\otimes M^{\vee}).$ When $L_{1},\dots,L_{n}$ and $M$ are generated by global sections we have that there is a power series $\psi$ such that $\operatorname{cl}^{\prime}(L_{1},\dots,L_{n},M)=\psi(c_{1}(L_{1}),\dots,c_{1}(L_{n}),c_{1}(M)).$ Moreover, when the line bundles $L_{i}\otimes M^{\vee}$ are also generated by global sections the following holds $\displaystyle\psi(c_{1}(L_{1}),\dots,c_{1}(L_{n}),c_{1}(M))$ $\displaystyle=\varphi(c_{1}(L_{1}\otimes M^{\vee}),\dots,c_{1}(L_{n}\otimes M^{\vee}))$ $\displaystyle=\varphi(c_{1}(L_{1})-c_{1}(M),\dots,c_{1}(L_{n})-c_{1}(M)).$ Considering the system of spaces $\mathbb{P}^{m_{1},\dots,m_{n},m_{n+1}}$ with line bundles $L_{i}=p_{i}^{\ast}\mathcal{O}(1)\otimes p_{n+1}^{\ast}\mathcal{O}(1),\ i=1,\dots,n,\quad M=p_{n+1}^{\ast}\mathcal{O}(1),$ we see that there is an identity of power series $\varphi(x_{1}-y,\dots,x_{n}-y)=\psi(x_{1},\dots,x_{n},y).$ Now let $X$ be a projective complex manifold and let $L_{1},\dots,L_{n}$ be arbitrary line bundles. Then there is a line bundle $M$ such that $M$ and $L_{i}^{\prime}=L_{i}\otimes M$, $i=1,\dots,n$ are generated by global sections. Then we have $\displaystyle\operatorname{cl}(L_{1},\dots,L_{n})$ $\displaystyle=\operatorname{cl}(L^{\prime}_{1}\otimes M^{\vee},\dots,L^{\prime}_{n}\otimes M^{\vee})$ $\displaystyle=\operatorname{cl}^{\prime}(L^{\prime}_{1},\dots,L^{\prime}_{n},M)$ $\displaystyle=\psi(c_{1}(L^{\prime}_{1}),\dots,c_{1}(L^{\prime}_{n}),c_{1}(M))$ $\displaystyle=\varphi((c_{1}(L^{\prime}_{1})-c_{1}(M),\dots,c_{1}(L^{\prime}_{n})-c_{1}(M)))$ $\displaystyle=\varphi(c_{1}(L_{1}),\dots,c_{1}(L_{n})).$ The case of arbitrary rank vector bundles follows from the case of rank one vector bundles by proposition 1.6 and the splitting principle. We next recall the argument. Given a projective complex manifold $X$ and vector bundles $E_{1},\dots,E_{n}$ of rank $r_{1},\dots,r_{n}$, we can find a proper morphism $\pi\colon\widetilde{X}\longrightarrow X$, with $\widetilde{X}$ a complex projective manifold, and such that the induced morphism $\pi^{\ast}\colon H^{\ast}_{\mathcal{D}}(X,\mathbb{R}(\ast))\longrightarrow H^{\ast}_{\mathcal{D}}(\widetilde{X},\mathbb{R}(\ast))$ is injective and every bundle $\pi^{\ast}(E_{i})$ admits a holomorphic filtration $0=K_{i,0}\subset K_{i,1}\subset\dots\subset K_{i,r_{i}-1}\subset K_{i,r_{i}}=\pi^{\ast}(E_{i}),$ with $L_{i,j}=K_{i,j}/K_{i,j-1}$ a line bundle. If $\operatorname{cl}$ is a characteristic class for $n$-tuples of vector bundles of rank $r_{1},\dots,r_{n}$, we define a characteristic class for $r_{1}+\dots+r_{n}$-tuples of line bundles by the formula $\operatorname{cl}^{\prime}(L_{1,1},\dots,L_{1,r_{1}},\dots,L_{n,1},\dots,L_{n,r_{n}})=\\\ \operatorname{cl}(L_{1,1}\oplus\dots\oplus L_{1,r_{1}},\dots,L_{n,1}\oplus\dots\oplus,L_{n,r_{n}}).$ By the case of line bundles we know that there is a power series in $r_{1}+\dots+r_{n}$ variables $\psi$ such that $\operatorname{cl}^{\prime}(L_{1,1},\dots,L_{1,r_{1}},\dots,L_{n,1},\dots,L_{n,r_{n}})=\psi(c_{1}(L_{1,1}),\dots,c_{1}(L_{n,r_{n}})).$ Since the class $\operatorname{cl}^{\prime}$ is symmetric under the group $\mathfrak{S}_{r_{1}}\times\dots\times\mathfrak{S}_{r_{n}}$, the same is true for the power series $\psi$. Therefore $\psi$ can be written in terms of symmetric elementary functions. That is, there is another power series in $r_{1}+\dots+r_{n}$ variables $\varphi$, such that $\psi(x_{1,1},\dots,x_{n,r_{n}})=\varphi(s_{1}(x_{1,1},\dots,x_{1,r_{1}}),\dots,s_{r_{1}}(x_{1,1},\dots,x_{1,r_{1}}),\dots{\\\ }\dots,s_{1}(x_{n,1},\dots,x_{n,r_{n}}),\dots,s_{r_{n}}(x_{n,1},\dots,x_{n,r_{n}})),$ where $s_{i}$ is the $i$-th elementary symmetric function of the appropriate number of variables. Then $\displaystyle\pi^{\ast}(\operatorname{cl}(E_{1},\dots,E_{n}))$ $\displaystyle=\operatorname{cl}(\pi^{\ast}E_{1},\dots,\pi^{\ast}E_{n}))$ $\displaystyle=\operatorname{cl}^{\prime}(L_{1,1},\dots,L_{n,r_{n}})$ $\displaystyle=\psi(c_{1}(L_{1,1}),\dots,c_{1}(L_{n,r_{n}}))$ $\displaystyle=\varphi(c_{1}(\pi^{\ast}E_{1}),\dots,c_{r_{1}}(\pi^{\ast}E_{1}),\dots,c_{1}(\pi^{\ast}E_{n}),\dots,c_{r_{n}}(\pi^{\ast}E_{n}))$ $\displaystyle=\pi^{\ast}\varphi(c_{1}(E_{1}),\dots,c_{r_{1}}(E_{1}),\dots,c_{1}(E_{n}),\dots,c_{r_{n}}(E_{n})).$ Therefore, the result follows from the injectivity of $\pi^{\ast}$. ∎ ###### Remark 1.10. It would be interesting to know if the functoriality of a characteristic class in enough to imply that it is a power series in the Chern classes for arbitrary complex manifolds and holomorphic vector bundles. ## 2 Bott-Chern classes The aim of this section is to recall the theory of Bott-Chern classes. For more details we refer the reader to [7], [4], [19], [31], [14], [10] and [12]. Note however that the theory we present here is equivalent, although not identical, to the different versions that appear in the literature. Let $X$ be a complex manifold and let $\overline{E}=(E,h)$ be a rank $r$ holomorphic vector bundle provided with a hermitian metric. Let $\phi\in\mathbb{D}[[x_{1},\dots,x_{r}]]$ be a formal power series in $r$ variables that is symmetric under the action of $\mathfrak{S}_{r}$. Let $s_{i}$, $i=1,\dots,r$ be the elementary symmetric functions in $r$ variables. Then $\phi(x_{1},\dots,x_{r})=\varphi(s_{1},\dots,s_{r})$ for certain power series $\varphi$. By Chern-Weil theory we can obtain a representative of the class $\phi(E):=\varphi(c_{1}(E),\dots,c_{r}(E))\in\bigoplus_{k,p}H^{k}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(p))$ as follows. We denote also by $\phi$ the invariant power series in $r\times r$ matrices defined by $\phi$. Let $K$ be the curvature matrix of the hermitian holomorphic connection of $(E,h)$. The entries of $K$ in a particular trivialization of $E$ are local sections of $\mathcal{D}^{2}(X,1)$. Then we write $\phi(E,h)=\phi(-K)\in\bigoplus_{k,p}\mathcal{D}^{k}(X,p).$ The form $\phi(E,h)$ is well defined, closed, and it represents the class $\phi(E)$. Now let $\overline{E}_{\ast}=(\dots\overset{f_{n+1}}{\longrightarrow}\overline{E}_{n}\overset{f_{n}}{\longrightarrow}\overline{E}_{n-1}\overset{f_{n-1}}{\longrightarrow}\dots)$ be a bounded acyclic complex of hermitian vector bundles; by this we mean a bounded acyclic complex of vector bundles, where each vector bundle is equipped with an arbitrarily chosen hermitian metric. Write $r=\sum_{i\text{ even}}\operatorname{rk}(E_{i})=\sum_{i\text{ odd}}\operatorname{rk}(E_{i}).$ and let $\phi$ be a symmetric power series in $r$ variables. As before, we can define the Chern forms $\phi(\bigoplus_{i\text{ even}}(E_{i},h_{i}))\text{ and }\phi(\bigoplus_{i\text{ odd}}(E_{i},h_{i})),$ that represent the Chern classes $\phi(\bigoplus_{i\text{ even}}E_{i})$ and $\phi(\bigoplus_{i\text{ odd}}E_{i})$. The Chern classes are compatible with respect to exact sequences, that is, $\phi(\bigoplus_{i\text{ even}}E_{i})=\phi(\bigoplus_{i\text{ odd}}E_{i}).$ But, in general, this is not true for the Chern forms. This lack of compatibility with exact sequences on the level of Chern forms is measured by the Bott-Chern classes. ###### Definition 2.1. Let $\overline{E}_{\ast}=(\dots\overset{f_{n+1}}{\longrightarrow}\overline{E}_{n}\overset{f_{n}}{\longrightarrow}\overline{E}_{n-1}\overset{f_{n-1}}{\longrightarrow}\dots)$ be an acyclic complex of hermitian vector bundles, we will say that $\overline{E}_{\ast}$ is an _orthogonally split complex_ of vector bundles if, for any integer $n$, the exact sequence $0\longrightarrow\operatorname{Ker}f_{n}\ \longrightarrow\overline{E}_{n}\longrightarrow\operatorname{Ker}f_{n-1}\longrightarrow 0$ is split, there is a splitting section $s_{n}\colon\operatorname{Ker}f_{n-1}\to E_{n}$ such that $\overline{E}_{n}$ is the orthogonal direct sum of $\operatorname{Ker}f_{n}$ and $\operatorname{Im}s_{n}$ and the metrics induced in the subbundle $\operatorname{Ker}f_{n-1}$ by the inclusion $\operatorname{Ker}f_{n-1}\subset\overline{E}_{n-1}$ and by the section $s_{n}$ agree. ###### Notation 2.2. Let $(x:y)$ be homogeneous coordinates of $\mathbb{P}^{1}$ and let $t=x/y$ be the absolute coordinate. In order to make certain choices of metrics in a functorial way, we fix once and for all a partition of unity $\\{\sigma_{0},\sigma_{\infty}\\}$, over $\mathbb{P}^{1}$ subordinated to the open cover of $\mathbb{P}^{1}$ given by the open subsets $\left\\{\\{\|y\|>1/2\|x\|\\},\\{\|x\|>1/2\|y\|\\}\right\\}$. As usual we will write $\infty=(1:0)$, $0=(0:1)$. The fundamental result of the theory of Bott-Chern classes is the following theorem (see [7], [4], [19]). ###### Theorem 2.3. There is a unique way to attach to each bounded exact complex $\overline{E}_{\ast}$ as above, a class $\widetilde{\phi}(\overline{E}_{\ast})$ in $\bigoplus_{k}\widetilde{\mathcal{D}}^{2k-1}(X,k)=\bigoplus_{k}\mathcal{D}^{2k-1}(X,k)/\operatorname{Im}(\operatorname{d}_{\mathcal{D}})$ satisfying the following properties 1. (i) (Differential equation) $\operatorname{d}_{\mathcal{D}}\widetilde{\phi}(\overline{E}_{\ast})=\phi(\bigoplus_{i\text{ even}}(E_{i},h_{i}))-\phi(\bigoplus_{i\text{ odd}}(E_{i},h_{i})).$ (2.4) 2. (ii) (Functoriality) $f^{\ast}\widetilde{\phi}(\overline{E}_{\ast})=\widetilde{\phi}(f^{\ast}\overline{E}_{\ast})$, for every holomorphic map $f\colon X^{\prime}\longrightarrow X$. 3. (iii) (Normalization) If $\overline{E}_{\ast}$ is orthogonally split, then $\widetilde{\phi}(\overline{E}_{\ast})=0$. ###### Proof. We first recall how to prove the uniqueness. Let $\overline{K}_{i}=(K_{i},g_{i})$, where $K_{i}=\operatorname{Ker}f_{i}$ and $g_{i}$ is the metric induced by the inclusion $K_{i}\subset E_{i}$. Consider the complex manifold $X\times\mathbb{P}^{1}$ with projections $p_{1}$ and $p_{2}$. For every vector bundle $F$ on $X$ we will denote $F(i)=p_{1}^{\ast}F\otimes p_{2}^{\ast}\mathcal{O}_{\mathbb{P}^{1}}(i)$. Let $\widetilde{C}_{\ast}=\widetilde{C}(E_{\ast})_{\ast}$ be the complex of vector bundles on $X\times\mathbb{P}^{1}$ given by $\widetilde{C}_{i}=E_{i}(i)\oplus E_{i-1}(i-1)$ with differential $d(s,t)=(t,0)$. Let $\widetilde{D}_{\ast}=\widetilde{D}(E_{\ast})_{\ast}$ be the complex of vector bundles with $\widetilde{D}_{i}=E_{i-1}(i)\oplus E_{i-2}(i-1)$ and differential $d(s,t)=(t,0)$. Using notation 2.2 we define the map $\psi\colon\widetilde{C}(E_{\ast})_{i}\longrightarrow\widetilde{D}(E_{\ast})_{i}$ given by $\psi(s,t)=(f_{i}(s)-t\otimes y,f_{i-1}(t))$. It is a morphism of complexes. ###### Definition 2.5. The _first transgression exact sequence_ of $E_{\ast}$ is given by $\operatorname{tr}_{1}(E_{\ast})_{\ast}=\operatorname{Ker}\psi.$ On $X\times\mathbb{A}^{1}$, the map $p_{1}^{\ast}E_{i}\longrightarrow\widetilde{C}(E_{\ast})_{i}$ given by $s\longmapsto(s\otimes y^{i},f_{i}(s)\otimes y^{i-1})$ induces an isomorphism of complexes $p_{1}^{\ast}E_{\ast}\longrightarrow\operatorname{tr}_{1}(E_{\ast})_{\ast}\|_{X\times\mathbb{A}^{1}},$ (2.6) and in particular isomorphisms $\operatorname{tr}_{1}(E_{\ast})_{i}\|_{X\times\\{0\\}}\cong E_{i}.$ (2.7) Moreover, we have isomorphisms $\operatorname{tr}_{1}(E_{\ast})_{i}\|_{X\times\\{\infty\\}}\cong K_{i}\oplus K_{i-1}.$ (2.8) ###### Definition 2.9. We will denote by $\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}$ the complex $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ provided with any hermitian metric such that the isomorphisms (2.7) and (2.8) are isometries. If we need a functorial choice of metric, we proceed as follows. On $X\times(\mathbb{P}^{1}\setminus\\{0\\})$ we consider the metric induced by $\widetilde{C}$ on $\operatorname{tr}_{1}(E_{\ast})_{\ast}$. On $X\times(\mathbb{P}^{1}\setminus\\{\infty\\})$ we consider the metric induced by the isomorphism (2.6). We glue both metrics by means of the partition of unity of notation 2.2. In particular, we have that $\operatorname{tr}_{1}(\overline{E}_{\ast})\|_{X\times\\{\infty\\}}$ is orthogonally split. We assume that there exists a theory of Bott-Chern classes satisfying the above properties. Thus, there exists a class of differential forms $\widetilde{\phi}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})$ with the following properties. By (i) this class satisfies $\operatorname{d}_{\mathcal{D}}\widetilde{\phi}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})=\phi(\bigoplus_{i\text{ even}}\operatorname{tr}_{1}(\overline{E}_{\ast})_{i}))-\phi(\bigoplus_{i\text{ odd}}\operatorname{tr}_{1}(\overline{E}_{\ast})_{i}).$ By (ii), it satisfies $\widetilde{\phi}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\mid_{X\times\\{0\\}}=\widetilde{\phi}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}\mid_{X\times\\{0\\}})=\widetilde{\phi}(\overline{E}_{\ast}).$ Finally, by (ii) and (iii) it satisfies $\widetilde{\phi}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\mid_{X\times\\{\infty\\}}=\widetilde{\phi}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}\mid_{X\times\\{\infty\\}})=0.$ Let $\phi(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})$ be any representative of the class $\widetilde{\phi}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})$. Then, in the group $\bigoplus_{k}\widetilde{\mathcal{D}}^{2k-1}(X,k)$, we have $\displaystyle 0$ $\displaystyle=\operatorname{d}_{\mathcal{D}}\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log(t\bar{t})\bullet\phi(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})$ $\displaystyle=\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\left(\operatorname{d}_{\mathcal{D}}\frac{-1}{2}\log(t\bar{t})\bullet\phi(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})-\frac{-1}{2}\log(t\bar{t})\bullet\operatorname{d}_{\mathcal{D}}\phi(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\right)$ $\displaystyle=\widetilde{\phi}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\|_{X\times\\{\infty\\}}-\widetilde{\phi}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\|_{X\times\\{0\\}}$ $\displaystyle\phantom{\ }-\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log(t\bar{t})\bullet(\phi(\bigoplus_{i\text{ even}}\operatorname{tr}_{1}(\overline{E}_{\ast})_{i})-\phi(\bigoplus_{i\text{ odd}}\operatorname{tr}_{1}(\overline{E}_{\ast})_{i}))$ $\displaystyle=-\widetilde{\phi}(\overline{E}_{\ast})-\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log(t\bar{t})\bullet(\phi(\bigoplus_{i\text{ even}}\operatorname{tr}_{1}(\overline{E}_{\ast})_{i})-\phi(\bigoplus_{i\text{ odd}}\operatorname{tr}_{1}(\overline{E}_{\ast})_{i})).$ Hence, if such a theory exists, it should satisfy the formula $\widetilde{\phi}(\overline{E}_{\ast})=\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log(t\bar{t})\bullet(\phi(\bigoplus_{i\text{ odd}}\operatorname{tr}_{1}(\overline{E}_{\ast})_{i})-\phi(\bigoplus_{i\text{ even}}\operatorname{tr}_{1}(\overline{E}_{\ast})_{i})).$ (2.10) Therefore $\widetilde{\phi}(\overline{E}_{\ast})$ is determined by properties (i), (ii) and (iii). In order to prove the existence of a theory of functorial Bott-Chern forms, we have to see that the right hand side of equation (2.10) is independent from the choice of the metric on $\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}$ and that it satisfies the properties (i), (ii) and (iii). For this the reader can follow the proof of [4] theorem 1.29. ∎ In view of the proof of theorem 2.3, we can define the Bott-Chern classes as follows. ###### Definition 2.11. Let $\overline{E}_{\ast}\colon 0\longrightarrow(E_{n},h_{n})\longrightarrow\dots\longrightarrow(E_{1},h_{1})\longrightarrow(E_{0},h_{0})\longrightarrow 0$ be a bounded acyclic complex of hermitian vector bundles. Let $r=\sum_{i\text{ even}}\operatorname{rk}(E_{i})=\sum_{i\text{ odd}}\operatorname{rk}(E_{i}).$ Let $\phi\in\mathbb{D}[[x_{1},\dots,x_{r}]]^{\mathfrak{S}_{r}}$ be a symmetric power series in $r$ variables. Then the _Bott-Chern class_ associated to $\phi$ and $\overline{E}_{\ast}$ is the element of $\bigoplus_{k,p}\widetilde{\mathcal{D}}^{k}(E_{X},p)$ given by $\widetilde{\phi}(\overline{E_{\ast}})=\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log(t\bar{t})\bullet(\phi(\bigoplus_{i\text{ odd}}\operatorname{tr}_{1}(\overline{E}_{\ast})_{i})-\phi(\bigoplus_{i\text{ even}}\operatorname{tr}_{1}(\overline{E}_{\ast})_{i})).$ The following property is obvious from the definition. ###### Lemma 2.12. Let $\overline{E}_{\ast}$ be an acyclic complex of hermitian vector bundles. Then, for any integer $k$, $\widetilde{\phi}(\overline{E}_{\ast}[k])=(-1)^{k}\widetilde{\phi}(\overline{E}_{\ast}).$ $\square$ Particular cases of Bott-Chern classes are obtained when we consider a single vector bundle with two different hermitian metrics or a short exact sequence of vector bundles. Note however that, in order to fix the sign of the Bott- Chern classes on these cases, one has to choose the degree of the vector bundles involved, for instance as in the next definition. ###### Definition 2.13. Let $E$ be a holomorphic vector bundle of rank $r$, let $h_{0}$ and $h_{1}$ be two hermitian metrics and let $\phi$ be an invariant power series of $r$ variables. We will denote by $\widetilde{\phi}(E,h_{0},h_{1})$ the Bott-Chern class associated to the complex $\overline{\xi}\colon 0\longrightarrow(E,h_{1})\longrightarrow(E,h_{0})\longrightarrow 0,$ where $(E,h_{0})$ sits in degree zero. Therefore, this class satisfies $\operatorname{d}_{\mathcal{D}}\widetilde{\phi}(E,h_{0},h_{1})=\phi(E,h_{0})-\phi(E,h_{1}).$ In fact we can characterize $\widetilde{\phi}(E,h_{0},h_{1})$ axiomatically as follows. ###### Proposition 2.14. Given $\phi$, a symmetric power series in $r$ variables, there is a unique way to attach, to each rank $r$ vector bundle $E$ on a complex manifold $X$ and metrics $h_{0}$ and $h_{1}$, a class $\widetilde{\phi}(E,h_{0},h_{1})$ satisfying 1. (i) $\operatorname{d}_{\mathcal{D}}\widetilde{\phi}(E,h_{0},h_{1})=\phi(E,h_{0})-\phi(E,h_{1})$. 2. (ii) $f^{\ast}\widetilde{\phi}(E,h_{0},h_{1})=\widetilde{\phi}(f^{\ast}(E,h_{0},h_{1}))$ for every holomorphic map $f\colon Y\longrightarrow X$. 3. (iii) $\widetilde{\phi}(E,h,h)=0$. Moreover, if we denote $\widetilde{E}:=\operatorname{tr}_{1}(\overline{\xi})_{1}$, then it satisfies $\widetilde{E}\|_{X\times\\{\infty\\}}\cong(E,h_{0}),\quad\widetilde{E}\|_{X\times\\{0\\}}\cong(E,h_{1})$ and $\widetilde{\phi}(E,h_{0},h_{1})=\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log(t\bar{t})\bullet\phi(\widetilde{E}).$ (2.15) ###### Proof. The axiomatic characterization is proved as in theorem 2.3. In order to prove equation (2.15), if we follow the notations of the proof of theorem 2.3 we have $K_{0}=(E,h_{0})$ and $K_{1}=0$. Therefore $\operatorname{tr}_{1}(\overline{\xi})_{0}=p_{1}^{\ast}(E,h_{0})$, while $\widetilde{E}:=\operatorname{tr}_{1}(\overline{\xi})_{1}$ satisfies $\widetilde{E}\|_{X\times\\{0\\}}=(E,h_{1})$ and $\widetilde{E}\|_{X\times\\{\infty\\}}=(E,h_{0})$. Using the antisymmetry of $\log t\bar{t}$ under the involution $t\mapsto 1/t$ we obtain $\widetilde{\phi}(E,h_{0},h_{1})=\widetilde{\phi}(\overline{\xi})=\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log(t\bar{t})\bullet\phi(\widetilde{E}).$ ∎ We can also treat the case of short exact sequences. If $\overline{\varepsilon}\colon 0\longrightarrow\overline{E}_{2}\longrightarrow\overline{E}_{1}\longrightarrow\overline{E}_{0}\longrightarrow 0$ is a short exact sequence of hermitian vector bundles, by convention, we will assume that $\overline{E}_{0}$ sits in degree zero. This fixs the sign of $\widetilde{\phi}(\overline{\varepsilon})$. ###### Proposition 2.16. Given $\phi$, a symmetric power series in $r$ variables, there is a unique way to attach, to each short exact sequence of hermitian vector bundles on a complex manifold $X$ $\overline{\varepsilon}\colon 0\longrightarrow\overline{E}_{2}\longrightarrow\overline{E}_{1}\longrightarrow\overline{E}_{0}\longrightarrow 0,$ where $\overline{E}_{1}$ has rank $r$, a class $\widetilde{\phi}(\overline{\varepsilon})$ satisfying 1. (i) $\operatorname{d}_{\mathcal{D}}\widetilde{\phi}(\overline{\varepsilon})=\phi(\overline{E}_{0}\oplus\overline{E}_{2})-\phi(\overline{E}_{1})$. 2. (ii) $f^{\ast}\widetilde{\phi}(\overline{\varepsilon})=\widetilde{\phi}(f^{\ast}(\overline{\epsilon}))$ for every holomorphic map $f\colon Y\longrightarrow X$. 3. (iii) $\widetilde{\phi}(\overline{\varepsilon})=0$ whenever $\overline{\varepsilon}$ is orthogonally split. $\square$ The following additivity result of Bott-Chern classes will be useful later. ###### Lemma 2.17. Let $\overline{A}_{\ast,\ast}$ be a bounded exact sequence of bounded exact sequences of hermitian vector bundles. Let $r=\sum_{i,j\text{ even}}\operatorname{rk}(A_{i,j})=\sum_{i,j\text{ odd}}\operatorname{rk}(A_{i,j})=\sum_{\begin{subarray}{c}i\text{ odd}\\\ j\text{ even}\end{subarray}}\operatorname{rk}(A_{i,j})=\sum_{\begin{subarray}{c}i\text{ even}\\\ j\text{ odd}\end{subarray}}\operatorname{rk}(A_{i,j}).$ Let $\phi$ be a symmetric power series in $r$ variables. Then $\widetilde{\phi}(\bigoplus_{k\text{ even}}\overline{A}_{k,\ast})-\widetilde{\phi}(\bigoplus_{k\text{ odd}}\overline{A}_{k,\ast})=\widetilde{\phi}(\bigoplus_{k\text{ even}}\overline{A}_{\ast,k})-\widetilde{\phi}(\bigoplus_{k\text{ odd}}\overline{A}_{\ast,k}).$ ###### Proof. The proof is analogous to the proof of proposition 6.13 and is left to the reader. ∎ ###### Corollary 2.18. Let $\overline{A}_{\ast,\ast}$ be a bounded double complex of hermitian vector bundles with exact rows, let $r=\sum_{i+j\text{ even}}\operatorname{rk}(A_{i,j})=\sum_{i+j\text{ odd}}\operatorname{rk}(A_{i,j})$ and let $\phi$ be a symmetric power series in $r$ variables. Then $\widetilde{\phi}(\operatorname{Tot}\overline{A}_{\ast,\ast})=\widetilde{\phi}(\bigoplus_{k}\overline{A}_{\ast,k}[-k]).$ ###### Proof. Let $k_{0}$ be an integer such that $\overline{A}_{k,l}=0$ for $k<k_{0}$. For any integer $n$ we denote by $\operatorname{Tot}_{n}=\operatorname{Tot}((\overline{A}_{k,l})_{k\geq n})$ the total complex of the exact complex formed by the rows with index greater or equal than $n$. Then $\operatorname{Tot}_{k_{0}}=\operatorname{Tot}(\overline{A}_{\ast,\ast})$. For each $k$ there is an exact sequence of complexes $0\longrightarrow\operatorname{Tot}_{k+1}\longrightarrow\operatorname{Tot}_{k}\oplus\bigoplus_{l<k}\overline{A}_{l,\ast}[-l]\longrightarrow\bigoplus_{l\leq k}\overline{A}_{l,\ast}[-l]\longrightarrow 0,$ which is orthogonally split in each degree. Therefore by lemma 2.17 we obtain $\widetilde{\phi}(\operatorname{Tot}_{k}\oplus\bigoplus_{l<k}\overline{A}_{l,\ast}[-l])=\widetilde{\phi}(\operatorname{Tot}_{k-1}\oplus\bigoplus_{l\leq k}\overline{A}_{l,\ast}[-l]).$ Hence the result follows by induction. ∎ A particularly important characteristic class is the Chern character. This class is additive for exact sequences. Specializing lemma 2.17 and corollary 2.18 to the Chern character we obtain ###### Corollary 2.19. With the hypothesis of lemma 2.17, the following equality holds: $\sum_{k}(-1)^{k}\widetilde{\operatorname{ch}}(\overline{A}_{k,\ast})=\sum_{k}(-1)^{k}\widetilde{\operatorname{ch}}(\overline{A}_{\ast,k})=\widetilde{\operatorname{ch}}(\operatorname{Tot}\overline{A}_{\ast,\ast}).$ $\square$ Our next aim is to extend the Bott-Chern classes associated to the Chern character to metrized coherent sheaves. This extension is due to Zha [32], although it is still unpublished. ###### Definition 2.20. A metrized coherent sheaf $\overline{\mathcal{F}}$ on $X$ is a pair $(\mathcal{F},\overline{E}_{\ast}\to\mathcal{F})$ where $\mathcal{F}$ is a coherent sheaf on $X$ and $0\to\overline{E}_{n}\to\overline{E}_{n-1}\to\dots\to\overline{E}_{0}\to\mathcal{F}\to 0$ is a finite resolution by hermitian vector bundles of the coherent sheaf $\mathcal{F}$. This resolution is also called the metric of $\overline{\mathcal{F}}$. If $\overline{E}$ is a hermitian vector bundle, we will also denote by $\overline{E}$ the metrized coherent sheaf $(E,\overline{E}\overset{\operatorname{id}}{\longrightarrow}E)$. Note that the coherent sheaf $0$ may have non trivial metrics. In fact, any exact sequence of hermitian vector bundles $0\to\overline{A}_{n}\to\dots\to\overline{A}_{0}\to 0\to 0$ can be seen as a metric on $0$. It will be denoted $\overline{0}_{A_{\ast}}$. A metric on $0$ is said to be _orthogonally split_ if the exact sequence is orthogonally split. A morphism of metrized coherent sheaves $\overline{\mathcal{F}}_{1}\to\overline{\mathcal{F}}_{2}$ is just a morphism of sheaves $\mathcal{F}_{1}\to\mathcal{F}_{2}$. A sequence of metrized coherent sheaves $\overline{\varepsilon}\colon\qquad\ldots\longrightarrow\overline{\mathcal{F}}_{n+1}\longrightarrow\overline{\mathcal{F}}_{n}\longrightarrow\overline{\mathcal{F}}_{n-1}\longrightarrow\ldots$ is said to be exact if it is exact as a sequence of coherent sheaves. ###### Definition 2.21. Let $\overline{\mathcal{F}}=(\mathcal{F},\overline{E}_{\ast}\to\mathcal{F})$ be a metrized coherent sheaf. Then the _Chern character form_ associated to $\overline{\mathcal{F}}$ is given by $\operatorname{ch}(\overline{\mathcal{F}})=\sum_{i}(-1)^{i}\operatorname{ch}(\overline{E}_{i}).$ ###### Definition 2.22. _An exact sequence of metrized coherent sheaves with compatible metrics_ is a commutative diagram $\begin{array}[]{ccccccccc}&&\vdots&&\vdots&&\vdots&&\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\overline{E}_{n,1}&\rightarrow&\ldots&\rightarrow&\overline{E}_{0,1}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\overline{E}_{n,0}&\rightarrow&\ldots&\rightarrow&\overline{E}_{0,0}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\mathcal{F}_{n}&\rightarrow&\ldots&\rightarrow&\mathcal{F}_{0}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ &&0&&0&&0&&\end{array}$ (2.23) where all the rows and columns are exact. The columns of this diagram are the individual metrics of each coherent sheaf. We will say that an exact sequence with compatible metrics is _orthogonally split_ if each row of vector bundles is an orthogonally split exact sequence of hermitian vector bundles. As in the case of exact sequences of hermitian vector bundles, the Chern character form is not compatible with exact sequences of metrized coherent sheaves and we can define a secondary Bott-Chern character which measures the lack of compatibility between the metrics. ###### Theorem 2.24. 1. 1) There is a unique way to attach to every finite exact sequence of metrized coherent sheaves with compatible metrics $\overline{\varepsilon}\colon\qquad 0\to\overline{\mathcal{F}}_{n}\to\dots\to\overline{\mathcal{F}}_{0}\to 0$ on a complex manifold $X$ a Bott-Chern secondary character $\widetilde{\operatorname{ch}}(\overline{\varepsilon})\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p)$ such that the following axioms are satisfied: 1. (i) (Differential equation) $\operatorname{d}_{\mathcal{D}}\widetilde{\operatorname{ch}}(\overline{\varepsilon})=\sum_{k}(-1)^{k}\operatorname{ch}(\overline{\mathcal{F}_{k}}).$ 2. (ii) (Functoriality) If $f\colon X^{\prime}\longrightarrow X$ is a morphism of complex manifolds, that is tor-independent from the coherent sheaves $\mathcal{F}_{k}$, then $f^{\ast}(\widetilde{\operatorname{ch}})(\overline{\varepsilon})=\widetilde{\operatorname{ch}}(f^{\ast}\overline{\varepsilon}),$ where the exact sequence $f^{\ast}\overline{\varepsilon}$ exists thanks to the tor-independence. 3. (iii) (Horizontal normalization) If $\overline{\varepsilon}$ is orthogonally split then $\widetilde{\operatorname{ch}}(\overline{\varepsilon})=0.$ 2. 2) There is a unique way to attach to every finite exact sequence of metrized coherent sheaves $\overline{\varepsilon}\colon\qquad 0\to\overline{\mathcal{F}}_{n}\to\dots\to\overline{\mathcal{F}}_{0}\to 0$ on a complex manifold $X$ a Bott-Chern secondary character $\widetilde{\operatorname{ch}}(\overline{\varepsilon})\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p)$ such that the axioms (i), (ii) and (iii) above and the axiom (iv) below are satisfied: 1. (iv) (Vertical normalization) For every bounded complex of hermitian vector bundles $\dots\rightarrow\overline{A}_{k}\rightarrow\dots\rightarrow\overline{A}_{0}\rightarrow 0$ that is orthogonally split, and every bounded complex of metrized coherent sheaves $\overline{\varepsilon}\colon\qquad 0\to\overline{\mathcal{F}}_{n}\to\dots\to\overline{\mathcal{F}}_{0}\to 0$ where the metrics are given by $\overline{E}_{i,\ast}\rightarrow\mathcal{F}_{i}$, if, for some $i_{0}$ we denote $\overline{\mathcal{F}}_{i_{0}}^{\prime}=(\mathcal{F}_{i_{0}},\overline{E}_{i_{0},\ast}\oplus\overline{A}_{\ast}\rightarrow\mathcal{F}_{i_{0}})$ and $\overline{\varepsilon}^{\prime}\colon\qquad 0\to\overline{\mathcal{F}}_{n}\to\dots\to\overline{\mathcal{F}}_{i_{0}}^{\prime}\to\dots\to\overline{\mathcal{F}}_{0}\to 0,$ then $\widetilde{\operatorname{ch}}(\overline{\varepsilon}^{\prime})=\widetilde{\operatorname{ch}}(\overline{\varepsilon})$. ###### Proof. _1)_ The uniqueness is proved using the standard deformation argument. By definition, the metrics of the coherent sheaves form a diagram like (2.23). On $X\times\mathbb{P}^{1}$, for each $j\geq 0$ we consider the exact sequences $\widetilde{E}_{\ast,j}=\operatorname{tr}_{1}(E_{\ast,j})$ associated to the rows of the diagram with the hermitian metrics of definition 2.9. Then, for each $i,j$ there are maps $\operatorname{d}\colon\widetilde{E}_{i,j}\to\widetilde{E}_{i-1,j}$, and $\delta\colon\widetilde{E}_{i,j}\to\widetilde{E}_{i,j-1}$. We denote $\widetilde{\mathcal{F}}_{i}=\operatorname{Coker}(\delta\colon\widetilde{E}_{i,1}\to\widetilde{E}_{i,0}).$ Using the definition of $\operatorname{tr}_{1}$ and diagram chasing one can prove that there is a commutative diagram $\begin{array}[]{ccccccccc}&&\vdots&&\vdots&&\vdots&&\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\widetilde{E}_{n,1}&\rightarrow&\ldots&\rightarrow&\widetilde{E}_{0,1}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\widetilde{E}_{n,0}&\rightarrow&\ldots&\rightarrow&\widetilde{E}_{0,0}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\widetilde{\mathcal{F}}_{n}&\rightarrow&\ldots&\rightarrow&\widetilde{\mathcal{F}}_{0}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ &&0&&0&&0&&\end{array}$ (2.25) where all the rows and columns are exact. In particular this implies that the inclusions $i_{0}\colon X\to X\times\\{0\\}\to X\times\mathbb{P}^{1}$ and $i_{\infty}\colon X\to X\times\\{\infty\\}\to X\times\mathbb{P}^{1}$ are tor- independent from the sheaves $\widetilde{\mathcal{F}}_{i}$. But $i_{0}^{\ast}\widetilde{\mathcal{F}}_{\ast}$ is isometric with $\overline{\mathcal{F}}_{\ast}$ and $i_{\infty}^{\ast}\widetilde{\mathcal{F}}_{\ast}$ is orthogonally split. Hence, by the standard argument, axioms (i), (ii) and (iii) imply that $\widetilde{\operatorname{ch}}(\overline{\varepsilon})=\sum_{j}(-1)^{j}\widetilde{\operatorname{ch}}(\overline{E}_{\ast,j}).$ (2.26) To prove the existence we use equation (2.26) as definition. Then the properties of the Bott-Chern classes of exact sequences of hermitian vector bundles imply that axioms (i), (ii) and (iii) are satisfied. _Proof of 2)_. We first assume that such theory exists. Let $\dots\rightarrow\overline{A}_{k}\rightarrow\dots\rightarrow\overline{A}_{0}\rightarrow 0$ be a bounded complex of hermitian vector bundles, non necessarily orthogonally split, and $\overline{\varepsilon}\colon\qquad 0\to\overline{\mathcal{F}}_{n}\to\dots\to\overline{\mathcal{F}}_{0}\to 0$ a bounded complex of metrized coherent sheaves where the metrics are given by $\overline{E}_{i,\ast}\rightarrow\mathcal{F}_{i}$. As in axiom (iv), for some $i_{0}$ we denote $\overline{\mathcal{F}}_{i_{0}}^{\prime}=(\mathcal{F}_{i_{0}},\overline{E}_{i,\ast}\oplus\overline{A}_{\ast}\rightarrow\mathcal{F}_{i_{0}})$ and $\overline{\varepsilon}^{\prime}\colon\qquad 0\to\overline{\mathcal{F}}_{n}\to\dots\to\overline{\mathcal{F}}_{i_{0}}^{\prime}\to\dots\to\overline{\mathcal{F}}_{0}\to 0.$ By axioms (i), (ii) and (iv), the class $(-1)^{i_{0}}(\widetilde{\operatorname{ch}}(\overline{\varepsilon}^{\prime})-\widetilde{\operatorname{ch}}(\overline{\varepsilon}))$ satisfies the properties that characterize $\widetilde{\operatorname{ch}}(A_{\ast})$. Therefore $\widetilde{\operatorname{ch}}(\overline{\varepsilon}^{\prime})=\widetilde{\operatorname{ch}}(\overline{\varepsilon})+(-1)^{i_{0}}\widetilde{\operatorname{ch}}(A_{\ast})$. Fix again a number $i_{0}$ and assume that there is an exact sequence of resolutions $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{A}_{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{E}_{i_{0},\ast}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{E}_{i_{0},\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{i_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{i_{0}}}$ (2.27) Let now $\overline{\varepsilon}^{\prime}$ denote the exact sequence $\overline{\varepsilon}$ but with the metric $\overline{E}_{i_{0},\ast}^{\prime}$ in the position $i_{0}$. Let $\overline{\eta}_{j}$ denote the $j$-th row of the diagram (2.27). Again using a deformation argument one sees that $\operatorname{\widetilde{ch}}(\overline{\varepsilon}^{\prime})-\operatorname{\widetilde{ch}}(\overline{\varepsilon})=(-1)^{i_{0}}\left(\operatorname{\widetilde{ch}}(\overline{A}_{\ast})-\sum_{j}(-1)^{j}\operatorname{\widetilde{ch}}(\eta_{j})\right).$ (2.28) Choose now a compatible system of metrics $\begin{array}[]{ccccccccc}&&\vdots&&\vdots&&\vdots&&\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\overline{D}_{n,1}&\rightarrow&\ldots&\rightarrow&\overline{D}_{0,1}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\overline{D}_{n,0}&\rightarrow&\ldots&\rightarrow&\overline{D}_{0,0}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\mathcal{F}_{n}&\rightarrow&\ldots&\rightarrow&\mathcal{F}_{0}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ &&0&&0&&0&&\end{array}$ (2.29) we denote by $\overline{\lambda}_{j}$ each row of the above diagram. For each $i$, choose a resolution $\overline{E}^{\prime}_{i,\ast}\longrightarrow\mathcal{F}_{i}$ such that there exist exact sequences of resolutions $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{A}_{i,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{E}_{i,\ast}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{E}_{i,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{i}}$ (2.30) and $\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{B}_{i,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{E}_{i,\ast}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{D}_{i,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{i}}$ (2.31) We denote by $\overline{\eta}_{i,j}$ each row of the diagram (2.30) and by $\overline{\mu}_{i,j}$ each row of the diagram (2.31). Then, by (2.28) and (2.26), we have $\operatorname{\widetilde{ch}}(\overline{\varepsilon})=\sum_{j}(-1)^{j}\operatorname{\widetilde{ch}}(\overline{\lambda}_{j})+\sum_{i}(-1)^{i}(\operatorname{\widetilde{ch}}(\overline{B}_{i,\ast})-\operatorname{\widetilde{ch}}(\overline{A}_{i,\ast}))\\\ +\sum_{i,j}(-1)^{i+j}(\operatorname{\widetilde{ch}}(\overline{\eta}_{i,j})-\operatorname{\widetilde{ch}}(\overline{\mu}_{i,j}))$ (2.32) Thus, $\operatorname{\widetilde{ch}}(\overline{\varepsilon})$ is uniquely determined by axioms (i) to (iv). To prove the existence we use equation (2.32) as definition. We have to show that this definition is independent of the choices of the new resolutions. This independence follows from corollary 2.19. Once we know that the Bott-Chern classes are well defined, it is clear that they satisfy axioms (i), (ii), (iii) and (iv). ∎ ###### Proposition 2.33. (Compatibility with exact squares) If $\begin{array}[]{ccccccccc}&&\vdots&&\vdots&&\vdots&&\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ \dots&\rightarrow&\overline{\mathcal{F}}_{n+1,m+1}&\rightarrow&\overline{\mathcal{F}}_{n+1,m}&\rightarrow&\overline{\mathcal{F}}_{n+1,m-1}&\rightarrow&\dots\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ \dots&\rightarrow&\overline{\mathcal{F}}_{n,m+1}&\rightarrow&\overline{\mathcal{F}}_{n,m}&\rightarrow&\overline{\mathcal{F}}_{n,m-1}&\rightarrow&\dots\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ \dots&\rightarrow&\overline{\mathcal{F}}_{n-1,m+1}&\rightarrow&\overline{\mathcal{F}}_{n-1,m}&\rightarrow&\overline{\mathcal{F}}_{n-1,m-1}&\rightarrow&\dots\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ &&\vdots&&\vdots&&\vdots&&\end{array}$ is a bounded commutative diagram of metrized coherent sheaves, where all the rows …$(\overline{\varepsilon}_{n-1})$, $(\overline{\varepsilon}_{n})$, $(\overline{\varepsilon}_{n+1})$, …and all the columns $(\overline{\eta}_{m-1})$, $(\overline{\eta}_{m})$, $(\overline{\eta}_{m+1})$ are exact, then $\sum_{n}(-1)^{n}\widetilde{\operatorname{ch}}(\overline{\varepsilon}_{n})=\sum_{m}(-1)^{m}\widetilde{\operatorname{ch}}(\overline{\eta}_{m}).$ ###### Proof. This follows from equation (2.32) and corollary 2.19. ∎ We will use the notation of definition 2.13 also in the case of metrized coherent sheaves. It is easy to verify the following result. ###### Proposition 2.34. Let $(\overline{\varepsilon})\qquad\ldots\longrightarrow\overline{E}_{n+1}\longrightarrow\overline{E}_{n}\longrightarrow\overline{E}_{n-1}\longrightarrow\ldots$ be a finite exact sequence of hermitian vector bundles. Then the Bott-Chern classes obtained by theorem 2.24 and by theorem 2.3 agree. $\square$ ###### Proposition 2.35. Let $\overline{\mathcal{F}}=(\mathcal{F},\overline{E}_{\ast}\to\mathcal{F})$ be a metrized coherent sheaf. We consider the exact sequence of metrized coherent sheaves $\overline{\varepsilon}\colon\qquad 0\longrightarrow\overline{E}_{n}\to\dots\to\overline{E}_{0}\to\overline{\mathcal{F}}\to 0,$ where, by abuse of notation, $\overline{E}_{i}=(E_{i},\overline{E}_{i}\overset{=}{\to}E_{i})$. Then $\widetilde{\operatorname{ch}}(\overline{\varepsilon})=0$. ###### Proof. Define $\mathcal{K}_{i}=\operatorname{Ker}(E_{i}\to E_{i-1})$, $i=1,\dots,n$ and $\mathcal{K}_{0}=\operatorname{Ker}(E_{0}\to\mathcal{F})$. Write $\overline{\mathcal{K}_{i}}=(\mathcal{K}_{i},0\to\overline{E}_{n}\to\dots\to\overline{E}_{i+1}\to\mathcal{K}_{i}),\ i=0,\dots,n,$ and $\overline{\mathcal{K}}_{-1}=\overline{\mathcal{F}}$. If we prove that $\widetilde{\operatorname{ch}}(0\to\overline{\mathcal{K}_{i}}\to\overline{E}_{i}\to\overline{\mathcal{K}}_{i-1}\to 0)=0,$ (2.36) then we obtain the result by induction using proposition 2.33. In order to prove equation (2.36) we apply equation (2.32). To this end consider resolutions $\displaystyle\overline{D}_{0,\ast}$ $\displaystyle\longrightarrow\mathcal{K}_{i-1},$ $\displaystyle\qquad\overline{D}_{0,k}$ $\displaystyle=\overline{E}_{k+i}$ $\displaystyle\overline{D}_{1,\ast}$ $\displaystyle\longrightarrow E_{i},$ $\displaystyle\qquad\overline{D}_{1,k}$ $\displaystyle=\overline{E}_{k+i+1}\oplus\overline{E}_{k+i}$ $\displaystyle\overline{D}_{2,\ast}$ $\displaystyle\longrightarrow\mathcal{K}_{i},$ $\displaystyle\qquad\overline{D}_{2,k}$ $\displaystyle=\overline{E}_{k+i+1}$ with the map $D_{2,k}\overset{\Delta}{\to}D_{1,k}$ given by $s\mapsto(s,\operatorname{d}s)$ and the map $D_{1,k}\overset{\nabla}{\to}D_{0,k}$ given by $(s,t)\mapsto t-\operatorname{d}s$. The differential of the complex $D_{1,k}$ is given by $(s,t)\mapsto(t,0)$. Using equations (2.32) and (2.26) we write the left hand side of equation (2.36) in terms of Bott-Chern classes of vector bundles. All the exact sequences involved are orthogonally split except maybe the sequences $\overline{\lambda}_{k}\colon\qquad 0\to\overline{D}_{2,k}\to\overline{D}_{1,k}\to\overline{D}_{0,k}\to 0.$ But now we consider the diagrams $\textstyle{\overline{E}_{k+i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\scriptstyle{\operatorname{id}}$$\textstyle{\overline{E}_{k+i+1}\oplus\overline{E}_{k+i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\scriptstyle{f}$$\textstyle{\overline{E}_{k+i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{\overline{E}_{k+i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\overline{E}_{k+i+1}\oplus\overline{E}_{k+i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla}$$\textstyle{\overline{E}_{k+i}}$ and $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 10.18825pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\&&\crcr}}}\ignorespaces{\hbox{\kern-10.18825pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{E}_{k+i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 16.98245pt\raise 5.95277pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.6639pt\hbox{$\scriptstyle{i_{2}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 34.18825pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 0.0pt\raise-19.12221pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\operatorname{id}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.56665pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 34.18825pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{E}_{k+i+1}\oplus\overline{E}_{k+i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 90.11371pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 107.87465pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 59.03145pt\raise-19.12221pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 59.03145pt\raise-29.41109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 107.87465pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{E}_{k+i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 121.64069pt\raise-19.12221pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\operatorname{id}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 121.64069pt\raise-29.56665pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-10.18825pt\raise-38.24442pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{E}_{k+i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 16.98245pt\raise-32.29164pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.6639pt\hbox{$\scriptstyle{i_{2}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 34.18825pt\raise-38.24442pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 34.18825pt\raise-38.24442pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{E}_{k+i+1}\oplus\overline{E}_{k+i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 90.11371pt\raise-33.05692pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 107.87465pt\raise-38.24442pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 107.87465pt\raise-38.24442pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{E}_{k+i+1}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$ where $i_{i}$, $i_{2}$ are the natural inclusions, $p_{1}$ and $p_{2}$ are the projections and $f(s,t)=(s,t+f(s))$. These diagrams and corollary 2.19 imply that $\widetilde{\operatorname{ch}}(\overline{\lambda}_{k})=0$. ∎ ###### Remark 2.37. In [32], Zha shows that the Bott-Chern classes associated to exact sequences of metrized coherent sheaves are characterized by proposition 2.34, proposition 2.35 and proposition 2.33. We prefer the characterization in terms of the differential equation, the functoriality and the normalization, because it relies on natural extensions of the corresponding axioms that define the Bott-Chern classes for exact sequences of hermitian vector bundles. Moreover, this approach will be used in a subsequent paper where we will study singular Bott-Chern classes associated to arbitrary proper morphisms. The following generalization of proposition 2.35 will be useful later. Let $\varepsilon\colon 0\rightarrow\mathcal{G}_{n}\rightarrow\mathcal{G}_{n-1}\rightarrow\dots\rightarrow\mathcal{G}_{0}\rightarrow\mathcal{F}\rightarrow 0$ be a finite resolution of a coherent sheaf by coherent sheaves. Assume that we have a commutative diagram $\begin{array}[]{ccccccccccc}&&\vdots&&\vdots&&\vdots&&&&\\\ &&\downarrow&&\downarrow&&\downarrow&&&&\\\ &&\overline{E}_{1,n}&\rightarrow&\ldots&\rightarrow&\overline{E}_{1,0}&&&&\\\ &&\downarrow&&\downarrow&&\downarrow&&&&\\\ &&\overline{E}_{0,n}&\rightarrow&\ldots&\rightarrow&\overline{E}_{0,0}&&&&\\\ &&\downarrow&&\downarrow&&\downarrow&&&&\\\ 0&\rightarrow&\overline{\mathcal{G}}_{n}&\rightarrow&\ldots&\rightarrow&\overline{\mathcal{G}}_{0}&\rightarrow&\mathcal{F}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&&&\\\ &&0&&0&&0&&\end{array}$ where the columns are exact, the rows are complexes and the $\overline{E}_{i,j}$ are hermitian vector bundles. The columns of this diagram define metrized coherent sheaves $\overline{\mathcal{G}}_{i}$. Let $\overline{\mathcal{F}}$ be the metrized coherent sheaf defined by the resolution $\operatorname{Tot}(\overline{E}_{\ast,\ast})\longrightarrow\mathcal{F}$. ###### Proposition 2.38. With the notations above, let $\overline{\varepsilon}$ be the exact sequence of metrized coherent sheaves $\overline{\varepsilon}\colon 0\rightarrow\overline{\mathcal{G}}_{n}\rightarrow\overline{\mathcal{G}}_{n-1}\rightarrow\dots\rightarrow\overline{\mathcal{G}}_{0}\rightarrow\overline{\mathcal{F}}\rightarrow 0$ Then $\widetilde{\operatorname{ch}}(\overline{\varepsilon})=0$. ###### Proof. For each $k$, let $\operatorname{Tot}_{k}=\operatorname{Tot}((E_{\ast,j})_{j\geq k})$. There are inclusions $\operatorname{Tot}_{k}\longrightarrow\operatorname{Tot}_{k-1}$. Let $\overline{D}_{\ast,j}=s(\operatorname{Tot}_{j+1}\to\operatorname{Tot}_{j})$ with the hermitian metric induced by $\overline{E}_{\ast,\ast}$. There are exact sequences of complexes $0\longrightarrow\overline{E}_{\ast,j}\longrightarrow\overline{D}_{\ast,j}\longrightarrow s(\operatorname{Tot}_{j+1}\to\operatorname{Tot}_{j+1})\longrightarrow 0$ (2.39) that are orthogonally split at each degree. The third complex is orthogonally split. Therefore, if we denote by $h_{E}$ and $h_{D}$ the metric structures of $\mathcal{G}_{j}$ induced respectively by the first and second column of diagram (2.39), then $\widetilde{\operatorname{ch}}(\mathcal{G}_{j},h_{E},h_{D})=0.$ (2.40) There is a commutative diagram of resolutions $\begin{array}[]{ccccccccccc}&&\vdots&&\vdots&&\vdots&&\vdots&&\\\ &&\downarrow&&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\overline{D}_{1,n}&\rightarrow&\ldots&\rightarrow&\overline{D}_{1,0}&\rightarrow&(\operatorname{Tot}_{0})_{1}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\overline{D}_{0,n}&\rightarrow&\ldots&\rightarrow&\overline{D}_{0,0}&\rightarrow&(\operatorname{Tot}_{0})_{0}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\mathcal{G}_{n}&\rightarrow&\ldots&\rightarrow&\mathcal{G}_{0}&\rightarrow&\mathcal{F}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\downarrow&&\\\ &&0&&0&&0&&0&&\end{array}$ where the rows of degree greater or equal than zero are orthogonally split. Hence the result follows from equation (2.26), equation (2.40) and proposition 2.33. ∎ ###### Remark 2.41. We have only defined the Bott-Chern classes associated to the Chern character. Everything applies without change to any additive characteristic class. The reader will find no difficulty to adapt the previous results to any multiplicative characteristic class like the Todd genus or the total Chern class. ## 3 Direct images of Bott-Chern classes The aim of this section is to show that certain direct images of Bott-Chern classes are closed. This result is a generalization of results of Bismut, Gillet and Soulé [6] page 325 and of Mourougane [29] proposition 6. The fact that these direct images of Bott-Chern classes are closed implies that certain relations between characteristic classes are true at the level of differential forms (see corollary 3.7 and corollary 3.8). In the first part of this section we deal with differential geometry. Thus all the varieties will be differentiable manifolds. Let $G_{1}$ be a Lie group and let $\pi\colon N_{2}\longrightarrow M_{2}$ be a principal bundle with structure group $G_{2}$ and connection $\omega_{2}$. Assume that there is a left action of $G_{1}$ over $N_{2}$ that commutes with the right action of $G_{2}$ and such that the connection $\omega_{2}$ is $G_{1}$-invariant. Let $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ be the Lie algebras of $G_{1}$ and $G_{2}$. Every element $\gamma\in\mathfrak{g}_{1}$ defines a tangent vector field $\gamma^{\ast}$ over $N_{2}$ given by $\gamma^{\ast}_{p}=\left.\frac{d}{dt}\right\|_{t=0}\exp(t\gamma)p.$ Let $(\gamma^{\ast})^{V}$ be the vertical component of $\gamma^{\ast}$ with respect to the connection $\omega_{2}$. For every point $p\in N_{2}$, we denote by $\varphi(\gamma,p)\in\mathfrak{g}_{2}$ the element characterized by $(\gamma^{\ast})_{p}^{V}=\varphi(\gamma,p)_{p}^{\ast}$, where $\varphi(\gamma,p)^{\ast}$ is the fundamental vector field associated to $\varphi(\gamma,p)$. The commutativity of the actions of $G_{1}$ and $G_{2}$ and the invariance of the connection $\omega_{2}$ implies that, for $g\in G_{1}$ and $\gamma\in\mathfrak{g}_{1}$, the following equalities hold $\displaystyle L_{g\ast}(\gamma^{\ast})$ $\displaystyle=(\operatorname{ad}(g)\gamma^{\ast}),$ (3.1) $\displaystyle L_{g\ast}(\gamma^{\ast})^{V}$ $\displaystyle=(\operatorname{ad}(g)\gamma^{\ast})^{V},$ (3.2) $\displaystyle\varphi(\operatorname{ad}(g)\gamma,p)$ $\displaystyle=\varphi(\gamma,g^{-1}p).$ (3.3) Let $\mathcal{G}_{2}$ be the vector bundle over $M_{2}$ associated to $N_{2}$ and the adjoint representation of $G_{2}$. That is, $\mathcal{G}_{2}=N_{2}\times\mathfrak{g}_{2}\left/\big{\langle}(pg,v)\sim(p,\operatorname{ad}(g)v)\big{\rangle}\right..$ Thus, we can identify smooth sections of $\mathcal{G}_{2}$ with $\mathfrak{g}_{2}$-valued functions on $N_{2}$ that are invariant under the action of $G_{2}$. In this way, $\varphi(\gamma,p)$ determines a section $\varphi(\gamma)\in C^{\infty}(N_{2},\mathfrak{g}_{2})^{G_{2}}=C^{\infty}(M_{2},\mathcal{G}_{2}).$ Equation (3.3) implies that, for $g\in G_{1}$ and $\gamma\in\mathfrak{g}_{1}$, $\varphi(\operatorname{ad}(g)\gamma)=L_{g^{-1}}^{\ast}\varphi(\gamma).$ We denote by $\Omega^{\omega_{2}}$ the curvature of the connection $\omega_{2}$. Let $P$ be an invariant function on $\mathfrak{g}_{2}$, then $P(\Omega^{\omega_{2}}+\varphi(\gamma))$ is a well defined differential form on $M_{2}$. ###### Proposition 3.4. Let $P$ be an invariant function on $\mathfrak{g}_{2}$ and let $\mu$ be a current on $M_{2}$ invariant under the action of $G_{1}$. Then $\mu(P(\Omega^{\omega_{2}}+\varphi(\gamma)))$ is an invariant function on $\mathfrak{g}_{1}$. ###### Proof. Let $g\in G_{1}$. Then, $\displaystyle\mu(P(\Omega^{\omega_{2}}+\varphi(\operatorname{ad}(g)\gamma)))$ $\displaystyle=\mu(P(\Omega^{\omega_{2}}+L_{g^{-1}}^{\ast}\varphi(\gamma)))$ $\displaystyle=\mu(P(L_{g^{-1}}^{\ast}\Omega^{\omega_{2}}+L_{g^{-1}}^{\ast}\varphi(\gamma)))$ $\displaystyle=L_{g^{-1}\ast}(\mu)(P(\Omega^{\omega_{2}}+\varphi(\gamma)))$ $\displaystyle=\mu(P(\Omega^{\omega_{2}}+\varphi(\gamma)))$ ∎ Let now $N_{1}\longrightarrow M_{1}$ be a principal bundle with structure group $G_{1}$ and provided with a connection $\omega_{1}$. Then we can form the diagram $\begin{CD}N_{1}\times N_{2}@>{\pi_{1}}>{}>N_{1}\underset{G_{1}}{\times}N_{2}\\\ @V{}V{\pi^{\prime}}V@V{}V{\pi}V\\\ N_{1}\times M_{2}@>{\pi_{2}}>{}>N_{1}\underset{G_{1}}{\times}M_{2}\\\ @V{}V{q}V\\\ M_{1}\end{CD}$ Then $\pi$ is a principal bundle with structure group $G_{2}$. The connections $\omega_{1}$ and $\omega_{2}$ induce a connection on the principal bundle $\pi$. The subbundle of horizontal vectors with respect to this connection is given by $\pi_{1\ast}(T^{H}N_{1}\oplus T^{H}N_{2})$. We will denote this connection by $\omega_{1,2}$. We are interested in computing the curvature $\omega_{1,2}$. In fact, all the maps in the above diagram are fiber bundles provided with a connection. When applicable, given a vector field $U$ in any of these spaces, we will denote by $U^{H,1}$ the horizontal lifting to $N_{1}\times N_{2}$, by $U^{H,2}$ the horizontal lifting to $N_{1}\underset{G_{1}}{\times}N_{2}$ and by $U^{H,3}$ the horizontal lifting to $N_{1}\underset{G_{1}}{\times}M_{2}$. The tangent space $T(N_{1}\times N_{2})$ can be decomposed as direct sum in the following ways $\displaystyle T(N_{1}\times N_{2})$ $\displaystyle=T^{H}N_{1}\oplus T^{V}N_{1}\oplus T^{H}N_{2}\oplus T^{V}N_{2}$ $\displaystyle=T^{H}N_{1}\oplus T^{V}N_{1}\oplus T^{H}N_{2}\oplus\operatorname{Ker}\pi_{1\ast},$ (3.5) For every point $(x,y)\in N_{1}\times N_{2}$ we have that $(\operatorname{Ker}\pi_{1\ast})_{(x,y)}\subset T^{V}_{x}N_{1}\oplus T_{y}N_{2}$. Moreover, there is an isomorphism $\mathfrak{g}_{1}\longrightarrow(\operatorname{Ker}\pi_{1\ast})_{(x,y)}$ that sends an element $\gamma\in\mathfrak{g}_{1}$ to the element $(\gamma^{\ast}_{x},-\gamma^{\ast}_{y})\in T^{V}_{x}N_{1}\oplus T_{y}N_{2}$. The tangent space to $N_{1}\underset{G_{1}}{\times}M_{2}$ can be decomposed as the sum of the subbundle of vertical vectors with respect to $q$ and the subbundle of horizontal vectors defined by the connection $\omega_{1}$. The horizontal lifting to $N_{1}\times N_{2}$ of a vertical vector lies in $T^{H}N_{2}$ and the horizontal lifting of a horizontal vector lies in $T^{H}N_{1}$. Let $U$, $V$ be two vector fields on $M_{1}$ and let $U^{H,3}$, $V^{H,3}$ be the horizontal liftings to $N_{1}\underset{G_{1}}{\times}M_{2}$. Then $\displaystyle\Omega^{\omega_{1,2}}(U^{H,3},$ $\displaystyle V^{H,3})=[U^{H,3},V^{H,3}]^{H,2}-[U^{H,2},V^{H,2}]$ $\displaystyle=\pi_{1\ast}([U^{H,3},V^{H,3}]^{H,1}-[U^{H,1},V^{H,1}])$ $\displaystyle=\pi_{1\ast}([U^{H,3},V^{H,3}]^{H,1}-[U,V]^{H,1}+[U,V]^{H,1}-[U^{H,1},V^{H,1}])$ $\displaystyle=\pi_{1,\ast}([U^{H,3},V^{H,3}]^{H,1}-[U,V]^{H,1}+\Omega^{\omega_{1}}(U,V)).$ But, we have $\displaystyle\Omega^{\omega_{1,2}}(U^{H,3},V^{H,3})$ $\displaystyle\in T^{V}N_{2},$ $\displaystyle\Omega^{\omega_{1}}(U,V)$ $\displaystyle\in T^{V}N_{1},$ $\displaystyle[U^{H,3},V^{H,3}]^{H,1}-[U,V]^{H,1}$ $\displaystyle\in T^{H}N_{2}.$ Therefore, by the direct sum decomposition (3.5) we obtain that $\Omega^{\omega_{1,2}}(U^{H,3},V^{H,3})=((\pi_{1\ast}\Omega^{\omega_{1}}(U,V)))^{V},$ where the vertical part is taken with respect to the fib re bundle $\pi$. If $U$ is a horizontal vector field over $N_{1}\underset{G_{1}}{\times}M_{2}$ and $V$ is a vertical vector field, a similar argument shows that $\Omega^{\omega_{1,2}}(U,V)=0$. Finally, if $U$ and $V$ are vector fields on $M_{2}$, they determine vertical vector fields on $N_{1}\underset{G_{1}}{\times}M_{2}$. Then the horizontal liftings $U^{H,1}$ and $V^{H,1}$ are induced by horizontal liftings of $U$ and $V$ to $N_{2}$. Therefore, reasoning as before we see that $\Omega^{\omega_{1,2}}(U,V)=\Omega^{\omega_{2}}(U,V).$ ###### Proposition 3.6. Let $G_{1}$ and $G_{2}$ be Lie groups, with Lie algebras $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$. For $i=1,2$, let $N_{i}\longrightarrow M_{i}$ be a principal bundle with structure group $G_{i}$, provided with a connection $\omega_{i}$. Assume that there is a left action of $G_{1}$ over $N_{2}$ that commutes with the right action of $G_{2}$ and that the connection $\omega_{2}$ is invariant under the $G_{1}$-action. We form the $G_{2}$-principal bundle $\pi\colon N_{1}\underset{G_{1}}{\times}N_{2}\longrightarrow N_{1}\underset{G_{1}}{\times}M_{2}$ with the induced connection $\omega_{1,2}$ and curvature $\Omega^{\omega_{1,2}}$. Let $P$ be any invariant function on $\mathfrak{g}_{2}$. Thus $P(\Omega^{\omega_{1,2}})$ is a well defined closed differential form on $N_{1}\underset{G_{1}}{\times}M_{2}$. Let $\mu$ be a current on $M_{2}$ invariant under the $G_{1}$-action. Being $G_{1}$ invariant, the current $\mu$ induces a current on $N_{1}\underset{G_{1}}{\times}M_{2}$, that we denote also by $\mu$. Let $q\colon N_{1}\underset{G_{1}}{\times}M_{2}\longrightarrow M_{1}$ be the projection. Then $q_{\ast}(P(\Omega^{\omega_{1,2}})\land\mu)$ is a closed differential form on $M_{1}$. ###### Proof. Let $U\subset M_{1}$ be a trivializing open subset for $N_{1}$ and choose a trivialization of $N_{1}\mid_{U}\cong U\times G_{1}$. With this trivialization, we can identify $\Omega^{\omega_{1}}\mid_{U}$ with a 2-form on $U$ with values in $\mathfrak{g}_{1}$. For $\gamma\in\mathfrak{g}_{1}$, we denote by $\psi_{\mu}(\gamma)=\mu(P(\Omega^{\omega_{2}}+\varphi(\gamma)))$ the invariant function provided by proposition 3.4. Then $q_{\ast}(P(\Omega^{\omega_{1,2}})\land\mu)=\psi_{\mu}(\Omega^{\omega_{1}}).$ Therefore, the result follows from the usual Chern-Weil theory. ∎ We go back now to complex geometry and analytic real Deligne cohomology and to the notations 1.3, in particular (1.4). ###### Corollary 3.7. Let $X$ be a complex manifold and let $\overline{E}=(E,h^{E})$ be a rank $r$ hermitian holomorphic vector bundle on $X$. Let $\pi\colon\mathbb{P}(E)\longrightarrow X$ be the associated projective bundle. On $\mathbb{P}(E)$ we consider the tautological exact sequence $\overline{\xi}\colon 0\longrightarrow\overline{\mathcal{O}(-1)}\longrightarrow\pi^{\ast}\overline{E}\longrightarrow\overline{Q}\longrightarrow 0$ where all the vector bundles have the induced metric. Let $P_{1}$, $P_{2}$ and $P_{3}$ be invariant power series in $1$, $r-1$ and $r$ variables respectively with coefficients in $\mathbb{D}$. Let $P_{1}(\overline{\mathcal{O}(-1)})$ and $P_{2}(\overline{Q})$ be the associated Chern forms and let $\widetilde{P}_{3}(\overline{\xi})$ the associated Bott-Chern class. Then $\pi_{\ast}(P_{1}(\overline{\mathcal{O}(-1)})\bullet P_{2}(\overline{Q})\bullet\widetilde{P}_{3}(\overline{\xi}))\in\bigoplus_{k}\widetilde{\mathcal{D}}^{2k-1}(X,k)$ is closed. Hence it defines a class in analytic real Deligne cohomology. This class does not depend on the hermitian metric of $E$. ###### Proof. We consider $\mathbb{C}^{r}$ with the standard hermitian metric. On the space $\mathbb{P}(\mathbb{C}^{r})$ we have the tautological exact sequence $0\longrightarrow\mathcal{O}_{\mathbb{P}(\mathbb{C}^{r})}(-1)\overset{f}{\longrightarrow}\mathbb{C}^{r}\longrightarrow Q\longrightarrow 0.$ Let $(x:y)$ be homogeneous coordinates on $\mathbb{P}^{1}$ and let $t=x/y$ be the absolute coordinate. Let $p_{1}$ and $p_{2}$ be the two projections of $M_{2}=\mathbb{P}(\mathbb{C}^{r})\times\mathbb{P}^{1}$. Let $\widetilde{E}$ be the cokernel of the map $\begin{matrix}p_{1}^{\ast}\mathcal{O}_{\mathbb{P}(\mathbb{C}^{r})}(-1)&\longrightarrow&p_{1}^{\ast}\mathcal{O}_{\mathbb{P}(\mathbb{C}^{r})}(-1)\otimes p_{2}^{\ast}\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus p_{1}^{\ast}\mathbb{C}^{r}\otimes p_{2}^{\ast}\mathcal{O}_{\mathbb{P}^{1}}(1)\\\ s&\longmapsto&s\otimes y+f(s)\otimes x\end{matrix}$ with the metric induced by the standard metric of $\mathbb{C}^{r}$ and the Fubini-Study metric of $\mathcal{O}_{\mathbb{P}(1)}(1)$. Let $N_{2}$ be the principal bundle over $M_{2}$ formed by the triples $(e_{1},e_{2},e_{3})$, where $e_{1}$, $e_{2}$ and $e_{3}$ are unitary frames of $p_{1}^{\ast}\mathcal{O}_{\mathbb{P}(\mathbb{C}^{r})}(-1)$, $p_{1}^{\ast}Q$ and $\widetilde{E}$ respectively. The structure group of this principal bundle is $G_{2}=U(1)\times U(r-1)\times U(r)$. Let $\omega_{2}$ be the connection induced by the hermitian holomorphic connections on the vector bundles $p_{1}^{\ast}\mathcal{O}_{\mathbb{P}(\mathbb{C}^{r})}(-1)$, $p_{1}^{\ast}Q$ and $\widetilde{E}$. Now we denote $M_{1}=X$, and let $N_{1}$ be the bundle of unitary frames of $\overline{E}$. This is a principal bundle over $M_{1}$ with structure group $G_{1}=U(r)$. The group $G_{1}$ acts on the left on $N_{2}$. This action commutes with the right action of $G_{2}$ and the connection $\omega_{2}$ is invariant under this action. Let $\mu=[-\log(\|t\|)]$ be the current on $M_{2}$ associated to the locally integrable function $-\log(\|t\|)$. This current is invariant under the action of $G_{1}$ because this group acts trivially on the factor $\mathbb{P}^{1}$. The invariant power series $P_{1}$, $P_{2}$ and $P_{3}$ determine an invariant function $P$ on $\mathfrak{g}_{2}$, the Lie algebra of $G_{2}$. Let $\omega_{1}$ be the connection induced in $N_{1}$ by the holomorphic hermitian connection on $\overline{E}$. As before let $\omega_{1,2}$ be the connection on $N_{1}\underset{G_{1}}{\times}N_{2}$ induced by $\omega_{1}$ and $\omega_{2}$ and let $q\colon N_{1}\underset{G_{1}}{\times}M_{2}\longrightarrow M_{1}$ be the projection. Observe that $N_{1}\underset{G_{1}}{\times}M_{2}=\mathbb{P}(E)\times\mathbb{P}^{1}$ and $q=\pi\circ p_{1}$. By the projection formula and the definition of Bott-Chern classes we have $\pi_{\ast}(P_{1}(\overline{\mathcal{O}(-1)})\land P_{2}(\overline{Q})\land\widetilde{P}_{3}(\overline{\xi}))=q_{\ast}(\mu\bullet P(\Omega^{\omega_{1,2}})),$ Therefore the fact that it is closed follows from 3.6. Since, for fixed $P_{1}$, $P_{2}$ and $P_{3}$, the construction is functorial on $(X.\overline{E})$, the fact that the class in analytic real Deligne cohomology does not depend on the choice of the hermitian metric follows from proposition 1.7. ∎ ###### Corollary 3.8. Let $\overline{E}=(E,h^{E})$ be a hermitian holomorphic vector bundle on a complex manifold $X$. We consider the projective bundle $\pi\colon\mathbb{P}(E\oplus\mathbb{C})\longrightarrow X$. Let $\overline{Q}$ be the universal quotient bundle on the space $\mathbb{P}(E\oplus\mathbb{C})$ with the induced metric. Then the following equality of differential forms holds $\pi_{\ast}\sum_{i}(-1)^{i}\operatorname{ch}(\bigwedge^{i}\overline{Q}^{\vee})=\pi_{\ast}(c_{r}(\overline{Q})\operatorname{Td}^{-1}(\overline{Q}))=\operatorname{Td}^{-1}(\overline{E}).$ ###### Proof. Let $\overline{\xi}$ be the tautological exact sequence with induced metrics. We first prove that $\pi_{\ast}(c_{r}(\overline{Q})\operatorname{Td}(\overline{\mathcal{O}(-1)}))=1.$ We can write $\operatorname{Td}(\overline{\mathcal{O}(-1)})=1+c_{1}(\overline{\mathcal{O}(-1)})\phi(\overline{\mathcal{O}(-1)})$ for certain power series $\phi$. Since $c_{r+1}(\overline{E}\oplus\mathbb{C})=0$ we have $c_{r}(\overline{Q})c_{1}(\overline{\mathcal{O}(-1)})=\operatorname{d}_{\mathcal{D}}\widetilde{c}_{r+1}(\overline{\xi}).$ Therefore, by corollary 3.7, we have $\displaystyle\pi_{\ast}(c_{r}(\overline{Q})\operatorname{Td}(\overline{\mathcal{O}(-1)}))$ $\displaystyle=\pi_{\ast}(c_{r}(\overline{Q}))+\pi_{\ast}(c_{r}(\overline{Q})c_{1}(\overline{\mathcal{O}(-1)})\phi(\overline{\mathcal{O}(-1)}))$ $\displaystyle=1+\operatorname{d}_{\mathcal{D}}\pi_{\ast}(\widetilde{c}_{r+1}(\overline{\xi})\phi(\overline{\mathcal{O}(-1)}))$ $\displaystyle=1.$ Then the corollary follows from corollary 3.7 by using the identity $\pi_{\ast}(c_{r}(\overline{Q})\operatorname{Td}^{-1}(\overline{Q}))=\pi_{\ast}(c_{r}(\overline{Q})\operatorname{Td}(\overline{\mathcal{O}(-1)})\pi^{\ast}\operatorname{Td}^{-1}(\overline{E}))\\\ +\operatorname{d}_{\mathcal{D}}\pi_{\ast}(c_{r}(\overline{Q})\operatorname{Td}(\overline{\mathcal{O}(-1)})\widetilde{\operatorname{Td}^{-1}}(\overline{\xi})).$ ∎ The following generalization of corollary 3.7 provides many relations between integrals of Bott-Chern classes and is left to the reader. ###### Corollary 3.9. Let $X$ be a complex manifold and let $\overline{E}=(E,h^{E})$ be a rank $r$ hermitian holomorphic vector bundle on $X$. Let $\pi\colon\mathbb{P}(E)\longrightarrow X$ be the associated projective bundle. On $\mathbb{P}(E)$ we consider the tautological exact sequence $\overline{\xi}\colon 0\longrightarrow\overline{\mathcal{O}(-1)}\longrightarrow\pi^{\ast}\overline{E}\longrightarrow\overline{Q}\longrightarrow 0$ where all the vector bundles have the induced metric. Let $P_{1}$ and $P_{2}$ be invariant power series in $1$ and $r-1$ variables respectively with coefficients in $\mathbb{D}$ and let $P_{3},\dots,P_{k}$ be invariant power series in $r$ variables with coefficients in $\mathbb{D}$. Let $P_{1}(\overline{\mathcal{O}(-1)})$ and $P_{2}(\overline{Q})$ be the associated Chern forms and let $\widetilde{P}_{3}(\overline{\xi}),\dots,\widetilde{P}_{k}(\overline{\xi})$ be the associated Bott-Chern classes. Then $\pi_{\ast}(P_{1}(\overline{\mathcal{O}(-1)})\bullet P_{2}(\overline{Q})\bullet\widetilde{P}_{3}(\overline{\xi})\bullet\dots\bullet\widetilde{P}_{k}(\overline{\xi}))$ is a closed differential form on $X$ for any choice of the ordering in computing the non associative product under the integral. ## 4 Cohomology of currents and wave front sets The aim of this section is to prove the Poincaré lemma for the complex of currents with fixed wave front set. This implies in particular a certain $\partial\bar{\partial}$-lemma (corollary 4.7) that will allow us to control the singularities of singular Bott-Chern classes. Let $X$ be a complex manifold of dimension $n$. Following notation 1.3 recall that there is a canonical isomorphism $H^{\ast}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(p))\cong H^{\ast}(\mathcal{D}^{\ast}_{D}(X,p)).$ A current $\eta$ can be viewed as a generalized section of a vector bundle and, as such, has a wave front set that is denoted by $\operatorname{WF}(\eta)$. The theory of wave front sets of distributions is developed in [25] chap. VIII. For the theory of wave front sets of generalized sections, the reader can consult [24] chap. VI. Although we will work with currents and hence with generalized sections of vector bundles, we will follow [25]. The wave front set of $\eta$ is a closed conical subset of the cotangent bundle of $X$ minus the zero section $T^{\ast}X_{0}=T^{\ast}X\setminus\\{0\\}$. This set describes the points and directions of the singularities of $\eta$ and it allows us to define certain products and inverse images of currents. Let $S\subset T^{\ast}X_{0}$ be a closed conical subset, we will denote by $\mathscr{D}^{\ast}_{X,S}$ the subsheaf of currents whose wave front set is contained in $S$. We will denote by $D^{\ast}(X,S)$ its complex of global sections. For every open set $U\subset X$ there is an appropriate notion of convergence in $\mathscr{D}^{\ast}_{X,S}(U)$ (see [25] VIII Definition 8.2.2). All references to continuity below are with respect to this notion of convergence. We next summarize the basic properties of wave front sets. ###### Proposition 4.1. Let $u$ be a generalized section of a vector bundle and let $P$ be a differential operator with smooth coefficients. Then $\operatorname{WF}(Pu)\subseteq\operatorname{WF}(u).$ ###### Proof. This is [25] VIII (8.1.11). ∎ ###### Corollary 4.2. The sheaf $\mathscr{D}^{\ast}_{X,S}$ is closed under $\partial$ and $\bar{\partial}$. Therefore it is a sheaf of Dolbeault complexes. Let $f\colon X\longrightarrow Y$ be a morphism of complex manifolds. The _set of normal directions_ of $f$ is $N_{f}=\\{(f(x),v)\in T^{\ast}Y\mid df(x)^{t}v=0\\}.$ This set measures the singularities of $f$. For instance, if $f$ is a smooth map then $N_{f}=0$ whereas, if $f$ is a closed immersion, $N_{f}$ is the conormal bundle of $f(X)$. Let $S\subset T^{\ast}Y_{0}$ be a closed conical subset. We will say that $f$ is transverse to $S$ if $N_{f}\cap S=\emptyset$. We will denote $f^{\ast}S=\\{(x,df(x)^{t}v)\in T^{\ast}X_{0}\mid(f(x),v)\in S\\}.$ ###### Theorem 4.3. Let $f\colon X\longrightarrow Y$ be a morphism of complex manifolds that is transverse to $S$. Then there exists one and only one extension of the pull- back morphism $f^{\ast}\colon\mathscr{E}^{\ast}_{Y}\longrightarrow\mathscr{E}^{\ast}_{X}$ to a continuous morphism $f^{\ast}\colon\mathscr{D}^{\ast}_{Y,S}\longrightarrow\mathscr{D}^{\ast}_{X,f^{\ast}S}.$ In particular there is a continuous morphism of complexes $D^{\ast}(Y,S)\longrightarrow D^{\ast}(X,f^{\ast}S).$ ###### Proof. This follows from [25] theorem 8.2.4. ∎ We now recall the effect of correspondences on the wave front sets. Let $K\in D^{\ast}(X\times Y)$, and let $S$ be a conical subset of $T^{\ast}Y_{0}$. We will write $\displaystyle\operatorname{WF}(K)_{X}$ $\displaystyle=\\{(x,\xi)\in T^{\ast}X_{0}\mid\exists y\in Y,(x,y,\xi,0)\in\operatorname{WF}(K)\\}$ $\displaystyle\operatorname{WF}^{\prime}(K)_{Y}$ $\displaystyle=\\{(y,\eta)\in T^{\ast}Y_{0}\mid\exists x\in X,(x,y,0,-\eta)\in\operatorname{WF}(K)\\}$ $\displaystyle\operatorname{WF}^{\prime}(K)\circ S$ $\displaystyle=\\{(x,\xi)\in T^{\ast}X_{0}\mid\exists(y,\eta)\in S,(x,y,\xi,-\eta)\in\operatorname{WF}(K)\\}.$ ###### Theorem 4.4. The image of the correspondence map $\begin{matrix}E^{\ast}_{c}(Y)&\longrightarrow&D^{\ast}(X)\\\ \eta&\longmapsto&p_{1\ast}(K\land p_{2}^{\ast}(\eta))\end{matrix}$ is contained in $D^{\ast}(X,WF(K)_{X})$. Moreover, if $S\cap\operatorname{WF}^{\prime}(K)_{Y}=\emptyset$, then there exists one and only one extension to a continuous map $D^{\ast}_{c}(Y,S)\longrightarrow D^{\ast}(X,S^{\prime}),$ where $S^{\prime}=\operatorname{WF}(K)_{X}\cup\operatorname{WF}^{\prime}(K)\circ S$. ###### Proof. This is [25] theorem 8.2.13. ∎ We are now in a position to state and prove the Poincaré lemma for currents with fixed wave front set. As usual, we will denote by $F$ the Hodge filtration of any Dolbeault complex. ###### Theorem 4.5 (Poincaré lemma). Let $S$ be any conical subset of $T^{\ast}X_{0}$. Then the natural morphism $\iota\colon(E^{\ast}(X),F)\longrightarrow(D^{\ast}(X,S),F)$ is a filtered quasi-isomorphism. ###### Proof. Let $K$ be the Bochner-Martinelli integral operator on $\mathbb{C}^{n}\times\mathbb{C}^{n}$. It is the operator $\begin{matrix}E_{c}^{p,q}(\mathbb{C}^{n})&\longrightarrow&E^{p,q-1}(\mathbb{C}^{n})\\\ \varphi&\longmapsto&\int_{w\in\mathbb{C}^{n}}k(z,w)\land\varphi(w),\end{matrix}$ where $k$ is the Bochner-Martinelli kernel ([21] pag. 383). Thus $k$ is a differential form on $\mathbb{C}^{n}\times\mathbb{C}^{n}$ with singularities only along the diagonal. Using the explicit description of $k$ in [21], it can be seen that $WF(k)=N^{\ast}\Delta_{0}$, the conormal bundle of the diagonal. By theorem 4.4, the operator $K$ defines a continuous linear map from $\Gamma_{c}(\mathbb{C}^{n},\mathscr{D}^{\ast}_{\mathbb{C}^{n},S})$ to $\Gamma(\mathbb{C}^{n},\mathscr{D}^{\ast}_{\mathbb{C}^{n},S})$. This is the key fact that allows us to adapt the proof of the Poincaré Lemma for arbitrary currents to the case of currents with fixed wave front set. We will prove that the sheaf inclusion $(\mathscr{E}_{X},F)\longrightarrow(\mathscr{D}_{X,S},F)$ is a filtered quasi-isomorphism. Then the theorem will follow from the fact that both are fine sheaves. The previous statement is equivalent to the fact that, for any integer $p\geq 0$, the inclusion $\iota\colon\mathscr{E}^{p,}_{X}\longrightarrow\mathscr{D}^{p,}_{X,S}$ is a quasi-isomorphism. Let $x\in X$, since exactness can be checked at the level of stalks, we need to show that $\iota_{x}\colon\mathscr{E}^{p,}_{X,x}\longrightarrow\mathscr{D}^{p,}_{X,S,x}$ is a quasi-isomorphism. let $U$ be a coordinate neighborhood around $x$ and let $x\in V\subset U$ be a relatively compact open subset. Let $\rho\in C_{c}^{\infty}(U)$ be a function with compact support such that $\rho\mid_{V}=1$. We define an operator $K\rho\colon\mathscr{D}^{p,q}_{X,S}(U)\longrightarrow\mathscr{D}^{p,q-1}_{X,S}(V).$ If $T\in\mathscr{D}^{p,q}_{X,S}(U)$ and $\varphi\in E_{c}^{\ast}(V)$ is a test form, then $K\rho(T)(\varphi)=(-1)^{p+q}T(\rho K(\varphi)).$ Hence, using that $\bar{\partial}K(\varphi)+K(\bar{\partial}\varphi)=\varphi$, and that $\varphi=\rho\varphi$, we have $(\bar{\partial}K\rho T+K\rho\bar{\partial}T+T)(\varphi)=-T(\bar{\partial}(\rho)\land K(\varphi)).$ Observe that, even if the support of $\varphi$ is contained in $V$, the support of $K(\varphi)$ can be $\mathbb{C}^{n}$; therefore the right hand side of the above equation may be non zero. We compute $\displaystyle T(\bar{\partial}(\rho)\land K(\varphi))$ $\displaystyle=T\left(\bar{\partial}(\rho)\land\int_{w\in\mathbb{C}^{n}}k(w,z)\land\varphi(w)\right)$ $\displaystyle=T\left(\int_{w\in\mathbb{C}^{n}}\bar{\partial}(\rho)\land k(w,z)\land\varphi(w)\right).$ Since $\operatorname{supp}(\varphi)\subset V$ and $\bar{\partial}(\rho)\|_{V}\equiv 0$, we can find a number $\epsilon>0$ such that, if $\\|z-w\\|<\epsilon$, then $\bar{\partial}(\rho)\land k(w,z)\land\varphi(w)=0$. Since the singularities of $k(w,z)$ are concentrated on the diagonal, it follows that the differential form $\bar{\partial}(\rho)\land k(w,z)\land\varphi(w)$ is smooth. Therefore, the current in $V$ given by $\varphi\longmapsto T\left(\int_{w\in\mathbb{C}^{n}}\bar{\partial}(\rho)\land k(w,z)\land\varphi(w)\right),$ is the current associated to the smooth differential form $T_{z}\left(\bar{\partial}(\rho)\land k(w,z)\right)$, where the subindex $z$ means that $T$ only acts on the $z$ variable, being $w\in V$ a parameter. This smooth form will be denoted by $\Psi(T)$. Summing up, we have shown that, for any current $T\in\mathscr{D}^{p,q}_{X,S}(U)$ there exists a smooth differential form $\Psi(T)\in\mathscr{E}^{p,q}_{X}(V)$ such that $T\mid_{V}=-\bar{\partial}K\rho T-K\rho\bar{\partial}T-\Psi(T).$ Observe that we can not say that $\Psi$ is a quasi-inverse of $\iota_{x}$ because it depends on the choice of $\rho$ and it is not possible to choose a single $\rho$ that can be applied to all $T$. Hence it is not a well defined operator at the level of stalks. Let now $T\in\mathscr{D}^{p,}_{X,S,x}$ be closed. It is defined in some neighborhood of $x$, say $U^{\prime}$. Applying the above procedure we find a smooth differential form $\Psi(T)$ defined on a relatively compact subset of $U^{\prime}$, say $V^{\prime}$, that is cohomologous to $T$. Hence the map induced by $\iota_{x}$ in cohomology is surjective. Let $\omega\in\mathscr{E}^{p,}_{X,x}$ be closed and such that $\iota_{x}\omega=\bar{\partial}T$ for some $T\in\mathscr{D}^{p,-1}_{X,S,x}$. We may assume that $\omega$ and $T$ are defined is some neighborhood $U^{\prime\prime}$ of $x$. Then, on some relatively compact subset $V^{\prime\prime}\subset U^{\prime\prime}$, we have $\omega\mid_{V^{\prime\prime}}=\bar{\partial}T\mid_{V^{\prime\prime}}=-\bar{\partial}K\rho\omega-\bar{\partial}\Psi(T).$ Since $K\rho\omega$ and $\Psi(T)$ are smooth differential forms we conclude that the map induced by $\iota_{x}$ in cohomology is injective. ∎ We will denote by $\mathcal{D}^{\ast}_{D}(X,S,p)$ the Deligne complex associated to $D^{\ast}(X,S)$. The following two results are direct consequences of theorem 4.5. ###### Corollary 4.6. The inclusion $\mathcal{D}^{\ast}_{D}(X,S,p)\longrightarrow\mathcal{D}^{\ast}_{D}(X,p)$ induces an isomorphism $H^{\ast}(\mathcal{D}^{\ast}_{D}(X,S,p))\cong H^{\ast}_{\mathcal{D}^{\text{{\rm an}}}}(X,\mathbb{R}(p)).$ ###### Corollary 4.7. 1. (i) Let $\eta\in\mathcal{D}^{n}_{D}(X,p)$ be a current such that $\operatorname{d}_{\mathcal{D}}\eta\in\mathcal{D}^{n+1}_{D}(X,S,p),$ then there is a current $a\in\mathcal{D}^{n-1}_{D}(X,p)$ such that $\eta+\operatorname{d}_{\mathcal{D}}a\in\mathcal{D}^{n}_{D}(X,S,p)$. 2. (ii) Let $\eta\in\mathcal{D}^{n}_{D}(X,S,p)$ be a current such that there is a current $a\in\mathcal{D}^{n-1}_{D}(X,p)$ with $\eta=\operatorname{d}_{\mathcal{D}}a$, then there is a current $b\in\mathcal{D}^{n-1}_{D}(X,S,p)$ such that $\eta=\operatorname{d}_{\mathcal{D}}b$. $\square$ ## 5 Deformation of resolutions In this section we will recall the deformation of resolutions based on the Grassmannian graph construction of [1]. We will also recall the Koszul resolution associated to a section of a vector bundle. The main theme is that given a bounded complex $E_{\ast}$ of locally free sheaves (with some properties) on a complex manifold $X$, one can construct a bounded complex $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ over a certain manifold $W$. This new manifold has a birational map $\pi\colon W\longrightarrow X\times\mathbb{P}^{1}$, that is an isomorphism over $X\times\mathbb{P}^{1}\setminus\\{\infty\\}$. The complex $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ agrees with the original complex over $X\times\\{0\\}$ and is particularly simple over $\pi^{-1}(X\times\\{\infty\\})$. Thus $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ is a deformation of the original complex to a simpler one. The two examples we are interested in are: first, when the original complex is exact, then $W$ agrees with $X\times\mathbb{P}^{1}$ and $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ was defined in 2.5. Its restriction to $\pi^{-1}(X\times\\{\infty\\})$ is split; second, when $i\colon Y\longrightarrow X$ is a closed immersion of complex manifolds, and $E_{\ast}$ is a bounded resolution of $i_{\ast}\mathcal{O}_{Y}$, then $W$ agrees with the deformation to the normal cone of $Y$ and the restriction of $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ to $\pi^{-1}(X\times\\{\infty\\})$ is an extension of a Koszul resolution by a split complex. Note that, if we allow singularities, then the Grassmannian graph construction is much more general. The deformation of resolutions is based on the Grassmannian graph construction of [1], and, in the form that we present here, has been developed in [6] and [20]. In order to fix notations we first recall the deformation to the normal cone and the Koszul resolution associated to the zero section of a vector bundle. Let $Y\hookrightarrow X$ be a closed immersion of complex manifolds, with $Y$ of pure codimension $n$. In the sequel we will use notation 2.2. Let $W=W_{Y/X}$ be the blow-up of $X\times\mathbb{P}^{1}$ along $Y\times\\{\infty\\}$. Since $Y$ and $X\times\mathbb{P}^{1}$ are manifolds, $W$ is also a manifold. The map $\pi\colon W\longrightarrow X\times\mathbb{P}^{1}$ is an isomorphism away from $Y\times\\{\infty\\}$; we will write $P$ for the exceptional divisor of the blow-up. Then $P=\mathbb{P}(N_{Y/X}\otimes N^{-1}_{\infty/\mathbb{P}^{1}}\oplus\mathbb{C}).$ Thus $P$ can be seen as the projective completion of the vector bundle $N_{Y/X}\otimes N^{-1}_{\infty/\mathbb{P}^{1}}$. Note that $N_{\infty/\mathbb{P}^{1}}$ is trivial although not canonically trivial. Nevertheless we can choose to trivialize it by means of the section $y\in\mathcal{O}_{\mathbb{P}^{1}}(1)$. Sometimes we will tacitly assume this trivialization and omit $N_{\infty/\mathbb{P}^{1}}$ from the formulae. The map $q_{W}\colon W\longrightarrow\mathbb{P}^{1}$, obtained by composing $\pi$ with the projection $q\colon X\times\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1}$, is flat and, for $t\in\mathbb{P}^{1}$, we have $q_{W}^{-1}(t)\cong\begin{cases}X\times\\{t\\},&$\text{ if } $t\not=\infty,\\\ P\cup\widetilde{X},&\text{ if }t=\infty,\end{cases}$ where $\widetilde{X}$ is the blow-up of $X$ along $Y$, and $P\cap\widetilde{X}$ is, at the same time, the divisor at $\infty$ of $P$ and the exceptional divisor of $\widetilde{X}$. Following [6] we will use the following notations $\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi_{P}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Y\times\\{\infty\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{\infty}}$$\textstyle{X\times\mathbb{P}^{1}}$ $\begin{array}[]{cr}i\colon Y\longrightarrow X,&\\\ W_{\infty}=\pi^{-1}(\infty)=P\cup\widetilde{X},&\\\ q\colon X\times\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1},&\text{the projection,}\\\ p\colon X\times\mathbb{P}^{1}\longrightarrow X,&\text{the projection,}\\\ q_{W}=q\circ\pi&\\\ p_{W}=p\circ\pi&\\\ q_{Y}\colon Y\times\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1},&\text{the projection,}\\\ p_{Y}\colon Y\times\mathbb{P}^{1}\longrightarrow Y,&\text{the projection,}\\\ j\colon Y\times\mathbb{P}^{1}\longrightarrow W&\text{the induced map,}\\\ j_{\infty}\colon Y\times\\{\infty\\}\longrightarrow P.&\\\ \end{array}$ Given any map $g\colon Z\longrightarrow X\times\mathbb{P}^{1}$, we will denote $p_{Z}=p\circ g$ and $q_{Z}=q\circ g$. For instance $p_{P}=p\circ\pi\circ f=p_{W}\circ f=i\circ\pi_{P}$, where, in the last equality, we are identifying $Y$ with $Y\times\\{\infty\\}$. We next recall the construction of the Koszul resolution. Let $Y$ be a complex manifold and let $N$ be a rank $n$ vector bundle. Let $P=\mathbb{P}(N\oplus\mathbb{C})$ be the projective bundle of lines in $N\oplus\mathbb{C}$. It is obtained by completing $N$ with the divisor at infinity. Let $\pi_{P}\colon P\longrightarrow Y$ be the projection and let $s\colon Y\longrightarrow P$ be the zero section. On $P$ there is a tautological short exact sequence $0\longrightarrow\mathcal{O}(-1)\longrightarrow\pi_{P}^{\ast}(N\oplus\mathbb{C})\longrightarrow Q\longrightarrow 0.$ (5.1) The above exact sequence and the inclusion $\mathbb{C}\longrightarrow\pi_{P}^{\ast}(N\oplus\mathbb{C})$ induce a section $\sigma\colon\mathcal{O}_{P}\longrightarrow Q$ that vanishes along the zero section $s(Y)$. By duality we obtain a morphism $Q^{\vee}\longrightarrow\mathcal{O}_{P}$ that induces a long exact sequence $0\longrightarrow\bigwedge^{n}Q^{\vee}\longrightarrow\dots\longrightarrow\bigwedge^{1}Q^{\vee}\longrightarrow\mathcal{O}_{P}\longrightarrow s_{\ast}\mathcal{O}_{Y}\longrightarrow 0.$ If $F$ is another vector bundle over $Y$, we obtain an exact sequence, $0\longrightarrow\bigwedge^{n}Q^{\vee}\otimes\pi_{P}^{\ast}F\longrightarrow\dots\longrightarrow\bigwedge^{1}Q^{\vee}\otimes\pi_{P}^{\ast}F\longrightarrow\pi_{P}^{\ast}F\longrightarrow s_{\ast}F\longrightarrow 0.$ (5.2) ###### Definition 5.3. The _Koszul resolution_ of $s_{\ast}(F)$ is the resolution (5.2). The complex $0\longrightarrow\bigwedge^{n}Q^{\vee}\otimes\pi_{P}^{\ast}F\longrightarrow\dots\longrightarrow\bigwedge^{1}Q^{\vee}\otimes\pi_{P}^{\ast}F\longrightarrow\pi_{P}^{\ast}F\longrightarrow 0$ will be denoted by $K(F,N)$. When $\overline{N}$ is a hermitian vector bundle, the exact sequence (5.1) induces a hermitian metric on $Q$. If, moreover, $\overline{F}$ is also a hermitian vector bundle, all the vector bundles that appear in the Koszul resolution have an induced hermitian metric. We will denote by $K(\overline{F},\overline{N})$ the corresponding complex of hermitian vector bundles. In particular, we shall write $K(\overline{\mathcal{O}_{Y}},\overline{N})$ if $F=\mathcal{O}_{Y}$ is endowed with the trivial metric $\\|1\\|=1$, unless expressly stated otherwise. We finish this section by recalling the results about deformation of resolutions that will be used in the sequel. For more details see [1] II.1, [6] Section 4 (c) and [20] Section 1. ###### Theorem 5.4. Let $i:Y\hookrightarrow X$ be a closed immersion of complex manifolds, where $Y$ may be empty. Let $U=X\setminus Y$. Let $F$ be a vector bundle over $Y$ and $E_{\ast}\longrightarrow i_{\ast}F\longrightarrow 0$ be a resolution of $i_{\ast}F$. Then there exists a complex manifold $W=W(E_{\ast})$, called the Grassmannian graph construction, with a birational map $\pi\colon W\longrightarrow X\times\mathbb{P}^{1}$ and a complex of vector bundles, $\operatorname{tr}_{1}(E_{\ast})_{\ast}$, over $W$ such that 1. (i) The map $\pi$ is an isomorphism away from $Y\times\\{\infty\\}$. The restriction of $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ to $X\times(\mathbb{P}^{1}\setminus\\{\infty\\})$ is isomorphic to $p_{W}^{\ast}E_{\ast}$ restricted to $X\times(\mathbb{P}^{1}\setminus\\{\infty\\})$. Moreover, If $\widetilde{X}$ is the Zariski closure of $U\times\\{\infty\\}$ inside $W$, the restriction of $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ to $\widetilde{X}$ is split acyclic. In particular, if $Y$ is empty or $F$ is the zero vector bundle, hence $E_{\ast}$ is acyclic in the whole $X$, then $W=X\times\mathbb{P}^{1}$ and $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ is the first transgression exact sequence introduced in 2.5. 2. (ii) When $Y$ is non-empty and $F$ is a non-zero vector bundle over $Y$, then $W(E_{\ast})$ agrees with $W_{Y/X}$, the deformation to the normal cone of $Y$. Moreover, there is an exact sequence of resolutions on $P$ $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\\\&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{A_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 69.80002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 37.65001pt\raise-31.78888pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 69.80002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{tr}_{1}(E_{\ast})_{\ast}\mid_{P}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 138.337pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 92.06851pt\raise-30.73332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 138.337pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{K(F,N_{Y/X}\otimes N^{-1}_{\infty/\mathbb{P}^{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 241.14465pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 177.74083pt\raise-30.73332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 241.14465pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-3.0pt\raise-41.23332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 32.15001pt\raise-41.23332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 74.72464pt\raise-41.23332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 74.72464pt\raise-41.23332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{(j_{\infty})_{\ast}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 129.18243pt\raise-36.94926pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.28406pt\hbox{$\scriptstyle{=}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 160.39696pt\raise-41.23332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 160.39696pt\raise-41.23332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{(j_{\infty})_{\ast}F}$}}}}}}}{\hbox{\kern 243.64465pt\raise-41.23332pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces,$ where $A_{\ast}$ is split acyclic and $K(F,N_{Y/X}\otimes N^{-1}_{\infty/\mathbb{P}^{1}})$ is the Koszul resolution. 3. (iii) Let $f\colon X^{\prime}\longrightarrow X$ be a morphism of complex manifolds and assume that we are in one of the following cases: 1. (a) The map $f$ is smooth. 2. (b) The map $f$ is arbitrary and $E_{\ast}$ is acyclic. 3. (c) $f$ is transverse to $Y$. Then $E^{\prime}_{\ast}:=f^{\ast}(E_{\ast})$ is exact over $f^{-1}(U)$, $W^{\prime}:=W(E^{\prime}_{\ast})=W\underset{X}{\times}X^{\prime},$ with $f_{W}\colon W^{\prime}\longrightarrow W$ the induced map, and we have $f_{W}^{\ast}(\operatorname{tr}_{1}(E_{\ast})_{\ast})=\operatorname{tr}_{1}(f^{\ast}(E_{\ast}))_{\ast}$. 4. (iv) If the vector bundles $E_{i}$ are provided with hermitian metrics, then one can choose a hermitian metric on $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ such that its restriction to $X\times\\{0\\}$ is isometric to $E_{\ast}$ and the restriction to $U\times\\{\infty\\}$ is orthogonally split. We will denote by $\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}$ the complex $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ with such a choice of hermitian metrics. Moreover, this choice of metrics can be made functorial. That is, if $f$ is a map as in item (iii), then $f_{W}^{\ast}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})=\operatorname{tr}_{1}(f^{\ast}(\overline{E}_{\ast}))_{\ast}$ ###### Proof. The case when $E_{\ast}$ is acyclic has already been treated. For the case when $Y$ is non-empty and $F$ is non zero, we first recall the construction of the Grassmannian graph of an arbitrary complex from [20], which is more general than what we need here. If $E$ is a vector bundle over $X$ we will denote by $E(i)$ the vector bundle over $X\times\mathbb{P}^{1}$ given by $E(i)=p^{\ast}E\otimes q^{\ast}\mathcal{O}(i)$. Let $\widetilde{C}_{\ast}$ be the complex of locally free sheaves given by $\widetilde{C}_{i}=E_{i}(i)\oplus E_{i-1}(i-1)$ with differential given by $\operatorname{d}(a,b)=(b,0)$. On $X\times(\mathbb{P}^{1}\setminus\\{\infty\\})$ we consider, for each $i$, the inclusion of vector bundles $\gamma_{i}\colon E_{i}\hookrightarrow\widetilde{C}_{i}$ given by $s\longmapsto(s\otimes y^{i},\operatorname{d}s\otimes y^{i-1})$. Let $G$ be the product of the Grassmann bundles $Gr(n_{i},\widetilde{C}_{i})$ that parametrize rank $n_{i}=\operatorname{rk}E_{i}$ subbundles of $\widetilde{C}_{i}$ over $X\times\mathbb{P}^{1}$. The inclusion $\gamma_{\ast}\colon\bigoplus E_{i}\longrightarrow\bigoplus\widetilde{C}_{i}$ induces a section $s$ of $G$ over $X\times\mathbb{A}^{1}$. Then $W(E_{\ast})$ is defined to be the closure of $s(X\times\mathbb{A}^{1})$ in $G$. Since the projection from $G$ to $X\times\mathbb{P}^{1}$ is proper, the same is true for the induced map $\pi\colon W\longrightarrow X\times\mathbb{P}^{1}$. For each $i$, the induced map $W\longrightarrow Gr(n_{i},\widetilde{C}_{i})$ defines a subbundle $\operatorname{tr}_{1}(E_{\ast})_{i}$ of $\pi^{\ast}\widetilde{C}_{i}$. This subbundle agrees with $E_{i}$ over $X\times\mathbb{A}^{1}$. The differential of $\widetilde{C}_{\ast}$ induces a differential on $\operatorname{tr}_{1}(E_{\ast})_{\ast}$. Assume now that the bundles $E_{i}$ are provided with hermitian metrics. Using the Fubini-Study metric of $\mathcal{O}(1)$ we obtain induced metrics on $\widetilde{C}_{i}$. Over $\pi^{-1}(X\times(\mathbb{P}^{1}\setminus\\{\infty\\}))$ we induce a metric on $\operatorname{tr}_{1}(E_{\ast})_{i}$ by means of the identification with $E_{i}$. Over $\pi^{-1}(X\times(\mathbb{P}^{1}\setminus\\{0\\}))$ we consider on $\operatorname{tr}_{1}(E_{\ast})_{i}$ the metric induced by $\widetilde{C}_{i}$. We glue together both metrics with the partition of unity $\\{\sigma_{0},\sigma_{\infty}\\}$ of notation 2.2. In the case we are interested there is a more explicit description of $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ given in [6] Section 4 (c). Namely, $\operatorname{tr}_{1}(E_{\ast})_{i}$ is the kernel of the morphism $\phi\colon p_{W}^{\ast}\widetilde{C}_{i}=p_{W}^{\ast}E_{i}(i)\oplus p_{W}^{\ast}E_{i-1}(i-1)\longrightarrow p_{W}^{\ast}E_{i-1}(i)\oplus p_{W}^{\ast}E_{i-2}(i-1)$ (5.5) given by $\phi(s,t)=(\operatorname{d}s-t\otimes y,\operatorname{d}t)$. The only statements that are not explicitly proved in [6] Section 4 (c) or [20] Section 1 are the functoriality when $f$ is not smooth and the properties of the explicit choice of metrics. If the complex $E_{\ast}$ is acyclic, then the same is true for $E^{\prime}_{\ast}=f^{\ast}E_{\ast}$. In this case $W=X\times\mathbb{P}^{1}$ and $W^{\prime}=X^{\prime}\times\mathbb{P}^{1}$. Then the functoriality follows from the definition of $\operatorname{tr}_{1}(E_{\ast})_{\ast}$. Assume now that we are in case (iii)c. We can form the Cartesian square $\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\scriptstyle{g}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{X}$ where $i^{\prime}$ is also a closed immersion of complex manifolds. Then we have that $E^{\prime}_{\ast}$ is a resolution of $i^{\prime}_{\ast}g^{\ast}F$. Hence $W^{\prime}=W(E^{\prime}_{\ast})$ is the deformation to the normal cone of $Y^{\prime}$ and therefore $W^{\prime}=W\underset{X}{\times}X^{\prime}$. Again the functoriality of $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ can be checked using the explicit construction of [20] Section 1 that we have recalled above. ∎ ###### Remark 5.6. 1. (i) The definition of $\operatorname{tr}_{1}(E_{\ast})$ can be extended to any bounded chain complex over a integral scheme (see [20]). 2. (ii) There is a sign difference in the definition of the inclusion $\gamma$ used in [20] and the one used in [6]. We have followed the signs of the first reference. ## 6 Singular Bott-Chern classes Throughout this section we will use notation 1.3. In particular we will write $\displaystyle\widetilde{\mathcal{D}}^{n}_{D}(X,p)$ $\displaystyle=\left.\mathcal{D}^{n}_{D}(X,p)\right/\operatorname{d}_{\mathcal{D}}\mathcal{D}^{n-1}_{D}(X,p),$ $\displaystyle\widetilde{\mathcal{D}}^{n}_{D}(X,S,p)$ $\displaystyle=\left.\mathcal{D}^{n}_{D}(X,S,p)\right/\operatorname{d}_{\mathcal{D}}\mathcal{D}^{n-1}_{D}(X,S,p).$ A particularly important current is $W_{1}\in\mathcal{D}^{1}_{D}(\mathbb{P}^{1},1)$ given by $W_{1}=[\frac{-1}{2}\log\\|t\\|^{2}].$ (6.1) With the above convention, this means that $W_{1}(\eta)=\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{-1}{2}\log\\|t\\|^{2}\bullet\eta.$ (6.2) By the Poincaré-Lelong equation $\operatorname{d}_{\mathcal{D}}W_{1}=\delta_{\infty}-\delta_{0}.$ (6.3) Note that the current $W_{1}$ was used in the construction of Bott-Chern classes (definition 2.11) and will also have a role in the definition of singular Bott-Chern classes. Before defining singular Bott-Chern classes we need to define the objects that give rise to them. ###### Definition 6.4. Let $i\colon Y\longrightarrow X$ be a closed immersion of complex manifolds. Let $N$ be the normal bundle of $Y$ and let $h_{N}$ be a hermitian metric on $N$. We denote $\overline{N}=(N,h_{N})$. Let $r_{N}$ be the rank of $N$, that agrees with the codimension of $Y$ in $X$. Let $\overline{F}=(F,h_{F})$ be a hermitian vector bundle on $Y$ of rank $r_{F}$. Let $\overline{E}_{\ast}\to i_{\ast}F$ be a metric on the coherent sheaf $i_{\ast}F$. The four-tuple $\overline{\xi}=(i,\overline{N},\overline{F},\overline{E}_{\ast}).$ (6.5) is called a _hermitian embedded vector bundle_. The number $r_{F}$ will be called the _rank_ of $\overline{\xi}$ and the number $r_{N}$ will be called the _codimension_ of $\overline{\xi}$. By convention, any exact complex of hermitian vector bundles on $X$ will be considered a hermitian embedded vector bundle of any rank and codimension. Obviously, to any hermitian embedded vector bundle we can associate the metrized coherent sheaf $(i_{\ast}F,\overline{E}_{\ast}\to i_{\ast}F)$. ###### Definition 6.6. A _singular Bott-Chern class_ for a hermitian embedded vector bundle $\overline{\xi}$ is a class $\widetilde{\eta}\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,p)$ such that $\operatorname{d}_{\mathcal{D}}\eta=\sum_{i=0}^{n}(-1)^{i}[\operatorname{ch}(\overline{E}_{i})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})])$ (6.7) for any current $\eta\in\tilde{\eta}$. The existence of this class is guaranteed by the Grothendieck-Riemann-Roch theorem, which implies that the two currents in the right hand side of equation (6.7) are cohomologous. Even if we have defined singular Bott-Chern classes as classes of currents with arbitrary singularities, it is an important observation that in each singular Bott-Chern class we can find representatives with controlled singularities. Let $N^{\ast}_{Y,0}$ be the conormal bundle of $Y$ with the zero section deleted. It is a closed conical subset of $T^{\ast}_{0}(X)$. Since the current $\sum_{i=0}^{n}(-1)^{i}[\operatorname{ch}(\overline{E}_{i})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})])\\\ =\sum_{i=0}^{n}(-1)^{i}[\operatorname{ch}(\overline{E}_{i})]-\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})\delta_{Y}$ belongs to $\mathcal{D}^{\ast}_{D}(X,N^{\ast}_{Y,0},p)$, by corollary 4.7, we obtain ###### Proposition 6.8. Let $\overline{\xi}=(i,\overline{N},\overline{F},\overline{E}_{\ast})$ be a hermitian embedded vector bundle as before. Then any singular Bott-Chern class for $\overline{\xi}$ belongs to the subset $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,N^{\ast}_{Y,0},p)\subset\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,p).$ $\square$ This result will allow us to define inverse images of singular Bott-Chern classes for certain maps. Let $f\colon X^{\prime}\longrightarrow X$ be a morphism of complex manifolds that is transverse to $Y$. We form the Cartesian square $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.78389pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\crcr}}}\ignorespaces{\hbox{\kern-7.78389pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 15.28851pt\raise 5.86389pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.86389pt\hbox{$\scriptstyle{i^{\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.78389pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 0.0pt\raise-18.80554pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.77777pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 31.78389pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 40.0886pt\raise-18.80554pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 40.0886pt\raise-27.77777pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-7.01389pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 15.8385pt\raise-32.30275pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.30833pt\hbox{$\scriptstyle{i}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.55388pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 32.55388pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X}$}}}}}}}\ignorespaces}}}}\ignorespaces.$ Observe that, by the transversality hypothesis, the normal bundle to $Y^{\prime}$ on $X^{\prime}$ is the inverse image of the normal bundle to $Y$ on $X$ and $f^{\ast}E_{\ast}$ is a resolution of $i^{\prime}_{\ast}g^{\ast}F$. Thus we write $f^{\ast}\overline{\xi}=(i^{\prime},f^{\ast}\overline{N},g^{\ast}\overline{F},f^{\ast}\overline{E}_{\ast})$, which is a hermitian embedded vector bundle. By proposition 6.8, given any singular Bott-Chern class $\widetilde{\eta}$ for $\xi$, we can find a representative $\eta\in\bigoplus_{p}\mathcal{D}^{2p-1}_{D}(X,N^{\ast}_{Y,0},p)$. By theorem 4.3, there is a well defined current $f^{\ast}\eta$ and it is a singular Bott- Chern current for $f^{\ast}\xi$. Therefore we can define $f^{\ast}(\widetilde{\eta})=\widetilde{f^{\ast}(\eta)}$. Again by theorem 4.3, this class does not depend on the choice of the representative $\eta$. Our next objective is to study the possible definitions of functorial singular Bott-Chern classes. ###### Definition 6.9. Let $r_{F}$ and $r_{N}$ be two integers. A _theory of singular Bott-Chern classes of rank $r_{F}$ and codimension $r_{N}$_ is an assignment which, to each hermitian embedded vector bundle $\overline{\xi}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast})$ of rank $r_{F}$ and codimension $r_{N}$, assigns a class of currents $T(\overline{\xi})\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,p)$ satisfying the following properties 1. (i) (Differential equation) The following equality holds $\operatorname{d}_{\mathcal{D}}T(\overline{\xi})=\sum_{i}(-1)^{i}[\operatorname{ch}(\overline{E}_{i})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})]).$ (6.10) 2. (ii) (Functoriality) For every morphism $f\colon X^{\prime}\longrightarrow X$ of complex manifolds that is transverse to $Y$, then $f^{\ast}T(\overline{\xi})=T(f^{\ast}\overline{\xi}).$ 3. (iii) (Normalization) Let $\overline{A}=(A_{\ast},g_{\ast})$ be a non-negatively graded orthogonally split complex of vector bundles. Write $\overline{\xi}\oplus\overline{A}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast}\oplus\overline{A}_{\ast})$. Then $T(\overline{\xi})=T(\overline{\xi}\oplus\overline{A})$. Moreover, if $X=\operatorname{Spec}\mathbb{C}$ is one point, $Y=\emptyset$ and $\overline{E}_{\ast}=0$, then $T(\overline{\xi})=0$. A _theory of singular Bott-Chern classes_ is an assignment as before, for all positive integers $r_{F}$ and $r_{M}$. When the inclusion $i$ and the bundles $F$ and $N$ are clear from the context, we will denote $T(\overline{\xi})$ by $T(\overline{E}_{\ast})$. Sometimes we will have to restrict ourselves to complex algebraic manifolds and algebraic vector bundles. In this case we will talk of _theory of singular Bott-Chern classes for algebraic vector bundles_. ###### Remark 6.11. 1. (i) Recall that the case when $Y=\emptyset$ and $\overline{E}_{\ast}$ is any bounded exact sequence of hermitian vector bundles is considered a hermitian embedded vector bundle of arbitrary rank. In this case, the properties above imply that $T(\overline{\xi})=[\widetilde{\operatorname{ch}}(\overline{E}_{\ast})],$ where $\widetilde{\operatorname{ch}}$ is the Bott-Chern class associated to the Chern character. That is, for acyclic complexes, any theory of singular Bott-Chern classes agrees with the Bott-Chern classes associated to the Chern character. 2. (ii) If the map $f$ is transverse to $Y$, then either $f^{-1}(Y)$ is empty or it has the same codimension as $Y$. Moreover, it is clear that $f^{\ast}F$ has the same rank as $F$. Therefore, the properties of singular Bott-Chern classes do not mix rank or codimension. This is why we have defined singular Bott- Chern classes for a particular rank and codimension. 3. (iii) By contrast with the case of Bott-Chern classes, the properties above are not enough to characterize singular Bott-Chern classes. For the rest of this section we will assume the existence of a theory of singular Bott-Chern classes and we will obtain some consequences of the definition. We start with the compatibility of singular Bott-Chern classes with exact sequences and Bott-Chern classes. Let $\overline{\chi}\colon 0\longrightarrow\overline{F}_{n}\longrightarrow\dots\longrightarrow\overline{F}_{1}\longrightarrow\overline{F}_{0}\longrightarrow 0$ (6.12) be a bounded exact sequence of hermitian vector bundles on $Y$. For $j=0,\dots,n$, let $\overline{E}_{j,\ast}\longrightarrow i_{\ast}F_{j}$ be a resolution, and assume that they fit in a commutative diagram $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\\\&&&&&\crcr}}}\ignorespaces{\hbox{\kern-5.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 31.8545pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 31.8545pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{E}_{n,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 76.9259pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 41.21295pt\raise-29.61334pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 76.9259pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 116.78043pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 116.78043pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{E}_{1,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 163.64502pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 125.8582pt\raise-29.61334pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 163.64502pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{E}_{0,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 208.15509pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 172.7228pt\raise-29.61334pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 208.15509pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern-5.5pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 29.5pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 29.5pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{i_{\ast}F_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 76.9259pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 76.9259pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 114.4259pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 114.4259pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{i_{\ast}F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 161.2905pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 161.2905pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{i_{\ast}F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 208.15509pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 208.15509pt\raise-39.44666pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces,$ with exact rows. We write $\overline{\xi_{j}}=(i\colon Y\longrightarrow X,\overline{N},\overline{F}_{j},\overline{E}_{j,\ast})$. For each $k$, we denote by $\overline{\eta}_{k}$ the exact sequence $0\longrightarrow\overline{E}_{n,k}\longrightarrow\dots\longrightarrow\overline{E}_{1,k}\longrightarrow\overline{E}_{0,k}\longrightarrow 0.$ ###### Proposition 6.13. With the above notations, the following equation holds: $T(\bigoplus_{j\text{ even}}\overline{\xi}_{j})-T(\bigoplus_{j\text{ odd}}\overline{\xi}_{j})=\sum_{k}(-1)^{k}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\widetilde{\operatorname{ch}}(\overline{\chi})]).$ Here the direct sum of hermitian embedded vector bundles, involving the same embedding and the same hermitian normal bundle, is defined in the obvious manner. ###### Proof. We consider the construction of theorem 5.4 for each of the exact sequences $\overline{\eta}_{k}$ and the exact sequence $\overline{\chi}$. For each $k$, we have $W_{X}:=W(\overline{\eta}_{k})=X\times\mathbb{P}^{1}$ and we denote $W_{Y}:=W(\overline{\chi})=Y\times\mathbb{P}^{1}$. On $W_{Y}$ we consider the transgression exact sequence $\operatorname{tr}_{1}(\overline{\chi})_{\ast}$ and on $W_{X}$ we consider the transgression exact sequences $\operatorname{tr}_{1}(\overline{\eta_{k}})_{\ast}$. We denote by $j\colon W_{Y}\longrightarrow W_{X}$ the induced morphism. Then there is an exact sequence (of exact sequences) $\dots\longrightarrow\operatorname{tr}_{1}(\overline{\eta}_{1})_{\ast}\longrightarrow\operatorname{tr}_{1}(\overline{\eta}_{0})_{\ast}\longrightarrow j_{\ast}\operatorname{tr}_{1}(\overline{\chi})_{\ast}\longrightarrow 0.$ We denote $\displaystyle\operatorname{tr}_{1}(\overline{\chi})_{+}$ $\displaystyle=\bigoplus_{j\text{ even}}\operatorname{tr}_{1}(\overline{\chi})_{j},\quad$ $\displaystyle\operatorname{tr}_{1}(\overline{\chi})_{-}$ $\displaystyle=\bigoplus_{j\text{ odd}}\operatorname{tr}_{1}(\overline{\chi})_{j},$ $\displaystyle\operatorname{tr}_{1}(\overline{\eta}_{k})_{+}$ $\displaystyle=\bigoplus_{j\text{ even}}\operatorname{tr}_{1}(\overline{\eta}_{k})_{j},\quad$ $\displaystyle\operatorname{tr}_{1}(\overline{\eta}_{k})_{-}$ $\displaystyle=\bigoplus_{j\text{ odd}}\operatorname{tr}_{1}(\overline{\eta}_{k})_{j},$ and $\displaystyle\operatorname{tr}_{1}(\overline{\xi})_{+}$ $\displaystyle=(j\colon W_{Y}\longrightarrow W_{X},p_{Y}^{\ast}\overline{N},\operatorname{tr}_{1}(\overline{\chi})_{+},\operatorname{tr}_{1}(\overline{\eta}_{\ast})_{+}),$ $\displaystyle\operatorname{tr}_{1}(\overline{\xi})_{-}$ $\displaystyle=(j\colon W_{Y}\longrightarrow W_{X},p_{Y}^{\ast}\overline{N},\operatorname{tr}_{1}(\overline{\chi})_{-},\operatorname{tr}_{1}(\overline{\eta}_{\ast})_{-}),$ where here $p_{Y}\colon W_{Y}\longrightarrow Y$ denotes the projection. We consider the current on $X\times\mathbb{P}^{1}$ given by $W_{1}\bullet\left(T(\operatorname{tr}_{1}(\overline{\xi})_{+})-T(\operatorname{tr}_{1}(\overline{\xi})_{-})\right)$. This current is well defined because the wave front set of $W_{1}$ is the conormal bundle of $(X\times\\{0\\})\cup(X\times\\{\infty\\})$, whereas the wave front set of $T(\operatorname{tr}_{1}(\overline{\xi})_{\pm})$ is the conormal bundle of $Y\times\mathbb{P}^{1}$. By the functoriality of the transgression exact sequences, we obtain that $\operatorname{tr}_{1}(\overline{\xi})_{+}\mid_{X\times\\{0\\}}=\bigoplus_{j\text{ even}}\overline{\xi}_{j},\quad\operatorname{tr}_{1}(\overline{\xi})_{-}\mid_{X\times\\{0\\}}=\bigoplus_{j\text{ odd}}\overline{\xi}_{j}.$ Moreover, using the fact that, for any bounded acyclic complex of hermitian vector bundles $\overline{E}_{\ast}$, the exact sequence $\operatorname{tr}_{1}(\overline{E}_{\ast})\mid_{X\times\\{\infty\\}}$ is orthogonally split, we have an isometry $\operatorname{tr}_{1}(\overline{\xi})_{+}\mid_{X\times\\{\infty\\}}\cong\operatorname{tr}_{1}(\overline{\xi})_{-}\mid_{X\times\\{\infty\\}}.$ We now denote by $p_{X}\colon W_{X}\longrightarrow X$ the projection. Using the properties that define a theory of singular Bott-Chern classes, in the group $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,N^{\ast}_{Y,0},p)$, the following holds $\displaystyle 0$ $\displaystyle=\operatorname{d}_{\mathcal{D}}(p_{X})_{\ast}\left(W_{1}\bullet T(\operatorname{tr}_{1}(\overline{\xi})_{+})-W_{1}\bullet T(\operatorname{tr}_{1}(\overline{\xi})_{-})\right)$ $\displaystyle=\left(T(\operatorname{tr}_{1}(\overline{\xi})_{+})-T(\operatorname{tr}_{1}(\overline{\xi})_{-})\right)\mid_{X\times\\{\infty\\}}-\left(T(\operatorname{tr}_{1}(\overline{\xi})_{+})-T(\operatorname{tr}_{1}(\overline{\xi})_{-})\right)\mid_{X\times\\{0\\}}$ $\displaystyle-(p_{X})_{\ast}\sum_{k}(-1)^{k}W_{1}\bullet\left(\operatorname{ch}(\operatorname{tr}_{1}(\overline{\eta}_{k})_{+})-\operatorname{ch}(\operatorname{tr}_{1}(\overline{\eta}_{k})_{-})\right)$ $\displaystyle+(p_{X})_{\ast}\left(W_{1}\bullet j_{\ast}\left[\operatorname{Td}^{-1}(p_{Y}^{\ast}\overline{N})\operatorname{ch}(\operatorname{tr}_{1}(\overline{\chi})_{+})-\operatorname{Td}^{-1}(p_{Y}^{\ast}\overline{N})\operatorname{ch}(\operatorname{tr}_{1}(\overline{\chi})_{-})\right]\right)$ $\displaystyle=-T(\bigoplus_{j\text{ even}}\overline{\xi}_{j})+T(\bigoplus_{j\text{ odd}}\overline{\xi}_{j})+\sum(-1)^{k}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]-i_{\ast}[\operatorname{Td}^{-1}(\overline{N})\bullet\widetilde{\operatorname{ch}}(\overline{\chi})],$ which implies the proposition. ∎ The following result is a consequence of proposition 6.13 and theorem 2.24. ###### Corollary 6.14. Let $Y\longrightarrow X$ be a closed immersion of complex manifolds. Let $\overline{\chi}$ be an exact sequence of hermitian vector bundles on $Y$ as (6.12). For each $j$, let $\xi_{j}=(i\colon Y\longrightarrow X,\overline{N},\overline{F}_{j},\overline{E}_{j,\ast})$ be a hermitian embedded vector bundle. We denote by $\overline{\varepsilon}$ the induced exact sequence of metrized coherent sheaves. Then $T(\bigoplus_{j\text{ even}}\overline{\xi}_{j})-T(\bigoplus_{j\text{ odd}}\overline{\xi}_{j})=[\widetilde{\operatorname{ch}}(\overline{\varepsilon})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\widetilde{\operatorname{ch}}(\overline{\chi})]).$ $\square$ We now study the effect of changing the metric of the normal bundle $N$. ###### Proposition 6.15. Let $\overline{\xi}_{0}=(i,\overline{N}_{0},\overline{F},\overline{E}_{\ast})$ be a hermitian embedded vector bundle, where $\overline{N}_{0}=(N,h_{0})$. Let $h_{1}$ be another metric in the vector bundle $N$ and write $\overline{N}_{1}=(N,h_{1})$, $\overline{\xi}_{1}=(i,\overline{N}_{1},\overline{F},\overline{E}_{\ast})$. Then $T(\overline{\xi}_{0})-T(\overline{\xi}_{1})=-i_{\ast}[\widetilde{\operatorname{Td}^{-1}}(N,h_{0},h_{1})\operatorname{ch}(\overline{F})].$ ###### Proof. The proof is completely analogous to the proof of proposition 6.13. ∎ We now study the case when $Y$ is the zero section of a completed vector bundle. Let $\overline{F}$ and $\overline{N}$ be hermitian vector bundles over $Y$. We denote $P=\mathbb{P}(N\oplus\mathbb{C})$, the projective bundle of lines in $N\oplus\mathcal{O}_{Y}$. Let $s\colon Y\longrightarrow P$ denote the zero section and let $\pi_{P}\colon P\longrightarrow Y$ denote the projection. Let $K(\overline{F},\overline{N})$ be the Koszul resolution of definition 5.3. We will use the notations before this definition. The following result is due to Bismut, Gillet and Soulé for the particular choice of singular Bott-Chern classes defined in [6]. ###### Theorem 6.16. Let $T$ be a theory of singular Bott-Chern classes of rank $r_{F}$ and codimension $r_{N}$. Let $Y$ be a complex manifold and let $\overline{F}$ and $\overline{N}$ be hermitian vector bundles of rank $r_{F}$ and $r_{N}$ respectively. Then the current $(\pi_{P})_{\ast}(T(K(\overline{F},\overline{N})))$ is closed. Moreover the cohomology class that it represents does not depend on the metric of $N$ and $F$ and determines a characteristic class for pairs of vector bundles of rank $r_{F}$ and $r_{N}$. We denote this class by $C_{T}(F,N)$. ###### Proof. We have that $\displaystyle\operatorname{d}_{\mathcal{D}}(\pi_{P})_{\ast}$ $\displaystyle(T(K(\overline{F},\overline{N})))$ $\displaystyle=(\pi_{P})_{\ast}(\operatorname{d}_{\mathcal{D}}T(K(\overline{F},\overline{N})))$ $\displaystyle=(\pi_{P})_{\ast}\left(\sum_{k=0}^{r}(-1)^{k}[\operatorname{ch}(\bigwedge^{k}\overline{Q}^{\vee})\pi_{P}^{\ast}\operatorname{ch}(\overline{F})]-s_{\ast}[\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})]\right)$ $\displaystyle=\left((\pi_{P})_{\ast}[c_{r}(\overline{Q})\operatorname{Td}^{-1}(\overline{Q})]-[\operatorname{Td}^{-1}(\overline{N})]\right)\operatorname{ch}(\overline{F})).$ Therefore, the fact that the current $(\pi_{P})_{\ast}(T(K(\overline{F},\overline{N})))$ is closed follows from corollary 3.8. The fact that this class is functorial on $(Y,\overline{N},\overline{F})$ is clear from the construction Thus, the fact that it does not depend on the hermitian metrics of $N$ and $F$ follows from proposition 1.7. ∎ ###### Remark 6.17. By theorem 1.8 we know that, if we restrict ourselves to the algebraic category, $C_{T}(F,N)$ is given by a power series on the Chern classes with coefficients in $\mathbb{D}$. By degree reasons $C_{T}(F,N)\in\bigoplus_{p}H_{\mathcal{D}^{\text{{\rm an}}}}^{2p-1}(Y,\mathbb{R}(p)).$ Let ${\bf 1}_{1}\in H^{1}_{\mathcal{D}}(\ast,\mathbb{R}(1))$ be the element determined by the constant function with value 1 in $\mathcal{D}^{1}(\ast,1)$. Then $C_{T}(F,N)/{\bf 1}_{1}$ is a power series in the Chern classes of $N$ and $F$ with real coefficients. ## 7 Classification of theories of singular Bott-Chern classes The aim of this section is to give a complete classification of the possible theories of singular Bott-Chern classes. This classification is given in terms of the characteristic class $C_{T}$ introduced in the previous section. ###### Theorem 7.1. Let $r_{F}$ and $r_{N}$ be two positive integers. Let $C$ be a characteristic class for pairs of vector bundles of rank $r_{F}$ and $r_{N}$. Then there exists a unique theory $T_{C}$ of singular Bott-Chern classes of rank $r_{F}$ and codimension $r_{N}$ such that $C_{T_{C}}=C$. ###### Proof. We first prove the uniqueness. Assume that $T$ is a theory of singular Bott- Chern classes such that $C_{T}=C$. Let $\overline{\xi}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast})$ be a hermitian embedded vector bundle as in section 6. Let $W$ be the deformation to the normal cone of $Y$. We will use all the notations of section 5. In particular, we will denote by $p_{\widetilde{X}}\colon\widetilde{X}\longrightarrow X$ and $p_{P}\colon P\longrightarrow X$ the morphisms induced by restricting $p_{W}$. Recall that $p_{P}$ can be factored as $P\overset{\pi_{P}}{\longrightarrow}Y\overset{i}{\longrightarrow}X.$ The normal vector bundle to the inclusion $j\colon Y\times\mathbb{P}^{1}\longrightarrow W$ is isomorphic to $p_{Y}^{\ast}N\otimes q_{Y}^{\ast}\mathcal{O}(-1)$. We provide it with the hermitian metric induced by the metric of $N$ and the Fubini-Study metric of $\mathcal{O}(-1)$ and we denote it by $\overline{N}^{\prime}$. By theorem 5.4 we have a complex of hermitian vector bundles, $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ such that the restriction $\operatorname{tr}_{1}(E_{\ast})_{\ast}\|_{X\times\\{0\\}}$ is isometric to $E_{\ast}$, the restriction $\operatorname{tr}_{1}(E_{\ast})_{\ast}\|_{\widetilde{X}}$ is orthogonally split and there is an exact sequence on $P$ $0\longrightarrow A_{\ast}\longrightarrow\operatorname{tr}_{1}(E_{\ast})_{\ast}\|_{P}\longrightarrow K(F,N)\longrightarrow 0,$ where $A_{\ast}$ is split acyclic and $K(F,N)$ is the Koszul resolution. Recall that we have trivialized $N^{-1}_{\infty/\mathbb{P}^{1}}$ by means of the section $y$ of $\mathcal{O}_{\mathbb{P}^{1}}(1)$. We choose a hermitian metric in every bundle of $A_{\ast}$ such that it becomes orthogonally split. For each $k$ we will denote by $\overline{\eta}_{k}$ the exact sequence of hermitian vector bundles $0\longrightarrow\overline{A}_{k}\longrightarrow\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}\|_{P}\longrightarrow K(\overline{F},\overline{N})_{k}\longrightarrow 0.$ (7.2) Observe that the current $W_{1}$ is defined as the current associated to a locally integrable differential form. The pull-back of this form to $W$ is also locally integrable. Therefore it defines a current on $W$ that we also denote by $W_{1}$. Moreover, since the wave front sets of $W_{1}$ and of $T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})$ are disjoint, there is a well defined current $W_{1}\bullet T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})$. Then, using the properties of singular Bott-Chern classes in definition 6.9, the equality $\displaystyle 0$ $\displaystyle=\operatorname{d}_{\mathcal{D}}(p_{W})_{\ast}\left(W_{1}\bullet T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\right)$ $\displaystyle=(p_{\widetilde{X}})_{\ast}(T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\|_{\widetilde{X}})+(p_{P})_{\ast}(T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\|_{P})-T(\overline{\xi})$ $\displaystyle-(p_{W})_{\ast}\left(W_{1}\bullet\left(\sum_{k}(-1)^{k}\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})-(j_{\ast}(\operatorname{ch}(p_{Y}^{\ast}\overline{F})\operatorname{Td}^{-1}(\overline{N}^{\prime}))\right)\right)$ holds in the group $\bigoplus_{k}\widetilde{\mathcal{D}}^{2k-1}(X,k)$. By properties 6.9(ii) and 6.9(iii), $T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\|_{\widetilde{X}}=T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}\|_{\widetilde{X}})=0$. By proposition 6.13 we have $T(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}\|_{P})=T(K(\overline{F},\overline{N}))-\sum_{k}(-1)^{k}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})].$ Moreover, we have $(p_{P})_{\ast}(T(K(\overline{F},\overline{N})))=i_{\ast}(\pi_{P})_{\ast}(T(K(\overline{F},\overline{N})))=i_{\ast}C_{T}(F,N).$ By the definition of $N^{\prime}$ and the choice of its metric, there are two differential forms $a,b$ on $Y$, such that $\operatorname{ch}(p_{Y}^{\ast}\overline{F})\operatorname{Td}^{-1}(\overline{N}^{\prime})=p_{Y}^{\ast}(a)+p_{Y}^{\ast}(b)\land q_{Y}^{\ast}(c_{1}(\mathcal{O}(-1))).$ We denote $\omega=-c_{1}(\mathcal{O}(-1))$. By the properties of the Fubini- Study metric, $\omega$ is invariant under the involution of $\mathbb{P}^{1}$ that sends $t$ to $1/t$. Then $(p_{W})_{\ast}\left(W_{1}\bullet(j_{\ast}(\operatorname{ch}(p_{Y}^{\ast}\overline{F})\operatorname{Td}^{-1}(\overline{N}^{\prime}))\right)=i_{\ast}(p_{Y})_{\ast}(W_{1}\bullet(p_{Y}^{\ast}a+p_{Y}^{\ast}b\omega))=0$ because the current $W_{1}$ changes sign under the involution $t\longmapsto 1/t$. Summing up, we have obtained the equation $T(\overline{\xi})=-(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k})\right)\\\ -\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]+i_{\ast}C_{T}(F,N).$ (7.3) Hence the singular Bott-Chern class is characterized by the properties of definition 6.9 and the characteristic class $C_{T}$. In order to prove the existence of a theory of singular Bott-Chern classes, we use equation (7.3) to define a class $T_{C}(\xi)$ as follows. ###### Definition 7.4. Let $C$ be a characteristic class for pairs of vector bundles of rank $r_{F}$ and $r_{N}$ as in theorem 7.1. Let $\overline{\xi}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast})$ be as in definition 6.9. Let $\overline{A}_{\ast}$, $\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}$ and $\overline{\eta}_{\ast}$ be as in (7.2). Then we define $T_{C}(\overline{\xi})=-(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k})\right)\\\ -\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]+i_{\ast}C(F,N).$ (7.5) We have to prove that this definition does not depend on the choice of the metric of $\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}$ or the metric of $\overline{A}_{\ast}$, that $T_{C}$ satisfies the properties of definition 6.9 and that the characteristic class $C_{T_{C}}$ agrees with $C$. First we prove the independence from the metrics. We denote by $h_{k}$ the hermitian metric on $\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}$ and by $g_{k}$ the hermitian metric on $A_{k}$. Let $h^{\prime}_{k}$ and $g^{\prime}_{k}$ be another choice of metrics satisfying also that $(A_{\ast},g^{\prime}_{\ast})$ is orthogonally split, that $(\operatorname{tr}_{1}(E_{\ast})_{k},h^{\prime}_{k})\|_{X\times\\{0\\}}$ is isometric to $\overline{E}_{k}$ and that $(\operatorname{tr}_{1}(E_{\ast})_{k},h^{\prime}_{k})\|_{\widetilde{X}}$ is orthogonally split. We denote by $\overline{\eta}^{\prime}_{k}$ the exact sequence $\eta_{k}$ provided with the metrics $g^{\prime}$ and $h^{\prime}$. Then, in the group $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p)$, we have $\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]-\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}^{\prime}_{k})]=\\\ \sum_{k}(-1)^{k}(p_{P})_{\ast}\left[\widetilde{\operatorname{ch}}(A_{k},g_{k},g^{\prime}_{k})\right]-\sum_{k}(-1)^{k}(p_{P})_{\ast}\left[\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k}\|_{P},h_{k},h^{\prime}_{k})\right].$ (7.6) Observe that the first term of the right hand side vanishes due to the hypothesis of $A_{\ast}$ being orthogonally split for both metrics. Moreover, we also have, $(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k})\right)-\\\ (p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(E_{\ast})_{k},h^{\prime}_{k})\right)=\\\ (p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{d}_{\mathcal{D}}\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k},h^{\prime}_{k})\right).$ (7.7) But, in the group $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p)$, $(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{d}_{\mathcal{D}}\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k},h^{\prime}_{k})\right)=\\\ \sum_{k}(-1)^{k}(p_{\widetilde{X}})_{\ast}[\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k},h^{\prime}_{k})]\|_{\widetilde{X}}\\\ +\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k},h^{\prime}_{k})]\|_{P})\\\ -\sum_{k}(-1)^{k}[\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(E_{\ast})_{k},h_{k},h^{\prime}_{k})]\|_{X\times\\{0\\}}.$ (7.8) The last term of the right hand side vanishes because the metrics $h_{k}$ and $h^{\prime}_{k}$ agree when restricted to $X\times\\{0\\}$ and the first term vanishes by the hypothesis that $\operatorname{tr}_{1}(E_{\ast})_{\ast}\|_{\widetilde{X}}$ is orthogonally split with both metrics. Combining equations (7.6), (7.7) and (7.8) we obtain that the right hand side of equation (7.5) does not depend on the choice of metrics. We next prove the property (i) of definition 6.9. We compute $\operatorname{d}_{\mathcal{D}}T_{C}(\overline{\xi})=-\sum_{k}(-1)^{k}\left((p_{\widetilde{X}})_{\ast}\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}\|_{\widetilde{X}})+(p_{P})_{\ast}\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}\|_{P})\right)\\\ +\sum_{k}(-1)^{k}\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}\|_{X\times\\{0\\}})\\\ -\sum_{k}(-1)^{k}(p_{P})_{\ast}\left(\operatorname{ch}(\overline{A}_{k})+\operatorname{ch}(K(\overline{F},\overline{N})_{k})-\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}\|_{P})\right).$ Using that $\overline{A}_{\ast}$ and that $\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}\|_{\widetilde{X}}$ are orthogonally split and corollary 3.8 we obtain $\displaystyle\operatorname{d}_{\mathcal{D}}T_{C}(\overline{\xi})$ $\displaystyle=\sum_{k}(-1)^{k}\operatorname{ch}(\overline{E}_{k})-\sum_{k}(-1)^{k}(p_{P})_{\ast}\operatorname{ch}(K(\overline{F},\overline{N})_{k})$ $\displaystyle=\sum_{k}(-1)^{k}[\operatorname{ch}(\overline{E}_{k})]-(p_{P})_{\ast}[c_{r}(\overline{Q})\operatorname{Td}^{-1}(\overline{Q})]$ $\displaystyle=\sum_{k}(-1)^{k}[\operatorname{ch}(\overline{E}_{k})]-i_{\ast}[\operatorname{ch}(\overline{F})\operatorname{Td}^{-1}(\overline{N})].$ We now prove the normalization property. We consider first the case when $Y=\emptyset$ and $\overline{E}_{\ast}$ is a non-negatively graded orthogonally split complex. We denote by $\overline{K}_{i}=\operatorname{Ker}(\operatorname{d}_{i}\colon E_{i}\longrightarrow E_{i-1})$ with the induced metric. By hypothesis there are isometries $\overline{E}_{i}=\overline{K}_{i}\oplus\overline{K}_{i-1}.$ Under these isometries, the differential is $\operatorname{d}(s,t)=(t,0)$. Following the explicit construction of $\operatorname{tr}_{1}(E_{\ast})$ given in [20], recalled in definition 2.5, we see that $\operatorname{tr}_{1}(E_{\ast})_{i}=p^{\ast}K_{i}\otimes q^{\ast}\mathcal{O}(i)\oplus p^{\ast}K_{i-1}\otimes q^{\ast}\mathcal{O}(i-1)=K_{i}(i)\oplus K_{i-1}(i-1).$ Moreover, we can induce a metric on $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ satisfying the hypothesis of definition 2.9 by means of the metric of the bundles $K_{i}$ and the Fubini-Study metric on the bundles $\mathcal{O}(i)$. It is clear that the second and third terms of the right hand side of equation (7.3) are zero. For the first term we have $\displaystyle\sum_{k}(-1)^{k}$ $\displaystyle(p_{W})_{\ast}W_{1}\bullet\left(\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k})\right)$ $\displaystyle=(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\overline{K}_{k}(k)\overset{\perp}{\oplus}\overline{K}_{k-1}(k-1))\right)$ $\displaystyle=(p_{W})_{\ast}\left(W_{1}\bullet(a+b\land\omega)\right),$ where $\omega$ is the Fubini-Study $(1,1)$-form on $\mathbb{P}^{1}$ and $a,b$ are inverse images of differential forms on $X$. Therefore we obtain that $T_{C}(\overline{E}_{\ast})=0$. Now let $\overline{\xi}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast})$ and let $\overline{B}_{\ast}$ be a non-negatively graded orthogonally split complex of vector bundles. By [20] section 1.1, we have that $W(E_{\ast}\oplus B_{\ast})=W(E_{\ast})$ and that $\operatorname{tr}_{1}(E_{\ast}\oplus B_{\ast})=\operatorname{tr}_{1}(E_{\ast})\oplus\pi^{\ast}\operatorname{tr}_{1}(B_{\ast}).$ In order to compute $T_{C}(\overline{\xi})$, we have to consider the exact sequences of hermitian vector bundles over $P$ $\overline{\eta}_{k}\colon 0\longrightarrow\overline{A}_{k}\longrightarrow\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}\|_{P}\longrightarrow K(\overline{F},\overline{N})_{k}\longrightarrow 0,$ whereas, in order to compute $T_{C}(\overline{\xi}\oplus\overline{B}_{\ast})$, we consider the sequences $\overline{\eta}^{\prime}_{k}\colon\\\ 0\longrightarrow\overline{A}_{k}\oplus\pi^{\ast}(\operatorname{tr}_{1}(\overline{B})_{k})\|_{P}\longrightarrow\operatorname{tr}_{1}(\overline{E}_{\ast})_{k}\oplus\pi^{\ast}(\operatorname{tr}_{1}(\overline{B})_{k})\|_{P}\longrightarrow K(\overline{F},\overline{N})_{k}\longrightarrow 0.$ By the additivity of Bott-Chern classes, we have that $\widetilde{\operatorname{ch}}(\overline{\eta}_{k})=\widetilde{\operatorname{ch}}(\overline{\eta}^{\prime}_{k})$. Therefore $\displaystyle T_{C}(\overline{\xi}\oplus\bar{B}_{\ast})-T_{C}(\overline{\xi})$ $\displaystyle=-(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast}\oplus\overline{B}_{\ast})_{k})\right)$ $\displaystyle\phantom{AAA}+(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{k})\right)$ $\displaystyle=-(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{B}_{\ast})_{k})\right)$ $\displaystyle=0.$ The proof of the functoriality is left to the reader. Finally we prove that $C_{T_{C}}=C$. Let $Y$ be a complex manifold and let $\overline{F}$ and $\overline{N}$ be two hermitian vector bundles. We write $X=\mathbb{P}(N\oplus\mathbb{C})$. Let $i\colon Y\longrightarrow X$ be the inclusion given by the zero section and let $\pi_{X}\colon X\longrightarrow Y$ be the projection. On $X$ we have the tautological exact sequence $0\longrightarrow\mathcal{O}(-1)\longrightarrow\pi_{X}^{\ast}(N\oplus\mathbb{C})\longrightarrow Q\longrightarrow 0$ and the Koszul resolution, denoted $K(\overline{F},\overline{N})$. We denote $\overline{\xi}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},K(\overline{F},\overline{N})).$ Using the definition of $T_{C}$, that is, equation (7.5), and the fact that $T_{C}$ satisfies the properties of definition 6.9, hence equation (7.3) is satisfied, we obtain that $i_{\ast}C(F,N)=i_{\ast}C_{T_{C}}(F,N)$ Applying $(\pi_{X})_{\ast}$ we obtain that $C(F,N)=C_{T_{C}}(F,N)$ which finishes the proof of theorem 7.1. ∎ ## 8 Transitivity and projection formula We now investigate how different properties of the characteristic class $C_{T}$ are reflected in the corresponding theory of singular Bott-Chern classes. ###### Proposition 8.1. Let $i\colon Y\hookrightarrow X$ be a closed immersion of complex manifolds. Let $\overline{F}$ be a hermitian vector bundle on $Y$ and $\overline{G}$ a hermitian vector bundle on $X$. Let $\overline{N}$ denote the normal bundle to $Y$ provided with a hermitian metric. Let $\overline{E}_{\ast}$ be a finite resolution of $i_{\ast}F$ by hermitian vector bundles. We denote $\overline{\xi}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast})$ and $\overline{\xi}\otimes\overline{G}=(i\colon Y\longrightarrow X,\overline{N},\overline{F}\otimes i^{\ast}\overline{G},\overline{E}_{\ast}\otimes\overline{G})$. Then $T(\overline{\xi}\otimes\overline{G})-T(\overline{\xi})\bullet\operatorname{ch}(\overline{G})=i_{\ast}(C_{T}(F\otimes i^{\ast}G,N))-i_{\ast}(C_{T}(F,N))\bullet\operatorname{ch}(\overline{G}).$ ###### Proof. Since the construction of $\operatorname{tr}_{1}(E_{\ast})_{\ast}$ is local on $X$ and $Y$ and compatible with finite sums, we have that $W(E_{\ast})=W(E_{\ast}\otimes G),\qquad\operatorname{tr}_{1}(\overline{E}_{\ast}\otimes\overline{G})_{\ast}=\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast}\otimes p_{W}^{\ast}\overline{G}.$ We first compute $(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast}\otimes\overline{G})_{\ast})\right)\\\ =(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})p^{\ast}_{W}\operatorname{ch}(\overline{G})\right)\\\ =(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E}_{\ast})_{\ast})\right)\operatorname{ch}(\overline{G}).$ (8.2) The Koszul resolution of $i_{\ast}(F\otimes i^{\ast}G)$ is given by $K(F\otimes i^{\ast}G,N)=K(F,N)\otimes p_{P}^{\ast}G.$ For each $k\geq 0$, we will denote by $\overline{\eta}_{k}\otimes p_{P}^{\ast}\overline{G}$ the exact sequence $0\longrightarrow\overline{A}_{k}\otimes p_{P}^{\ast}\overline{G}\longrightarrow\operatorname{tr}_{1}(\overline{E}_{\ast}\otimes\overline{G})_{k}\|_{P}\longrightarrow K(\overline{F},\overline{N})_{k}\otimes p_{P}^{\ast}\overline{G}\longrightarrow 0.$ Then, we have $(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k}\otimes p_{P}^{\ast}\overline{G})]=(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})\bullet p_{P}^{\ast}\operatorname{ch}(\overline{G})]=(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]\bullet\operatorname{ch}(\overline{G})$ (8.3) Thus the proposition follows from equation (8.2), equation (8.3) and formula (7.3). ∎ ###### Definition 8.4. We will say that a theory of singular Bott-Chern classes is _compatible with the projection formula_ if, whenever we are in the situation of proposition 8.1, the following equality holds: $T(\overline{\xi}\otimes\overline{G})=T(\overline{\xi})\bullet\operatorname{ch}(\overline{G}).$ We will say that a characteristic class $C$ (of pairs of vector bundles) is _compatible with the projection formula_ if it satisfies $C(F,N)=C(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F).$ ###### Corollary 8.5. A theory of singular Bott-Chern classes $T$ is compatible with the projection formula if and only if it is the case for the associated characteristic class $C_{T}$. ###### Proof. Assume that $C_{T}$ is compatible with the projection formula and that we are in the situation of proposition 8.1. Then $\displaystyle i_{\ast}C_{T}(F\otimes i^{\ast}G,N))$ $\displaystyle=i_{\ast}(C_{T}(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F\otimes i^{\ast}G))$ $\displaystyle=i_{\ast}(C_{T}(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F)i^{\ast}\operatorname{ch}(G))$ $\displaystyle=i_{\ast}(C_{T}(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F))\operatorname{ch}(G)$ $\displaystyle=i_{\ast}(C_{T}(F,N))\bullet\operatorname{ch}(G).$ Thus, by proposition 8.1, $T$ is compatible with the projection formula. Assume that $T$ is compatible with the projection formula. Let $\overline{F}$ and $\overline{N}$ be hermitian vector bundles over a complex manifold $Y$. Let $s\colon Y\hookrightarrow P:=\mathbb{P}(N\oplus\mathbb{C})$ be the zero section and let $\pi\colon P\longrightarrow Y$ be the projection. Then $\displaystyle C_{T}(F,N)$ $\displaystyle=\pi_{\ast}(T(K(\overline{F},\overline{N})))$ $\displaystyle=\pi_{\ast}(T(K(\overline{\mathcal{O}}_{Y},\overline{N})\otimes\pi^{\ast}\overline{F}))$ $\displaystyle=\pi_{\ast}(T(K(\overline{\mathcal{O}}_{Y},\overline{N}))\bullet\pi^{\ast}\operatorname{ch}(F))$ $\displaystyle=\pi_{\ast}(T(K(\overline{\mathcal{O}}_{Y},\overline{N})))\bullet\operatorname{ch}(F)$ $\displaystyle=C_{T}(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F).$ ∎ We will next investigate the relationship between singular Bott-Chern classes and compositions of closed immersions. Thus, let $\textstyle{Y\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{Y/X}}$$\scriptstyle{i_{Y/M}}$$\textstyle{\,X\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{X/M}}$$\textstyle{\,M}$ be a composition of closed immersions. Assume that the normal bundles $N_{Y/X}$, $N_{X/M}$ and $N_{Y/M}$ are provided with hermitian metrics. We will denote by $\overline{\varepsilon}$ the exact sequence $\overline{\varepsilon}\colon 0\rightarrow\overline{N}_{Y/X}\rightarrow\overline{N}_{Y/M}\rightarrow i_{Y/X}^{\ast}\overline{N}_{X/M}\rightarrow 0.$ (8.6) Let $P_{X/M}=\mathbb{P}(N_{X/M}\oplus\mathbb{C})$ be the projective completion of the normal cone to $X$ in $M$. Then there is an isomorphism $N_{Y/P_{X/M}}\cong N_{Y/X}\oplus i^{\ast}_{Y/X}N_{X/M}.$ (8.7) We denote by $\overline{N}_{Y/P_{X/M}}$ the vector bundle on the left hand side with the hermitian metric induced by the isomorphism (8.7). Let $\overline{F}$ be a hermitian vector bundle over $Y$, let $\overline{E}_{\ast}\longrightarrow(i_{Y/X})_{\ast}F$ be a resolution by hermitian vector bundles. Let $\overline{E}^{\prime}_{\ast,\ast}$ be a complex of complexes of vector bundles over $M$, such that, for each $k\geq 0$, $\overline{E}^{\prime}_{k,\ast}\longrightarrow(i_{X/M})_{\ast}E_{k}$ is a resolution, and there is a commutative diagram of resolutions $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\\\&&&&\crcr}}}\ignorespaces{\hbox{\kern-6.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 48.6873pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 48.6873pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{E^{\prime}_{k+1,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 120.4824pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 56.43645pt\raise-30.71109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 120.4824pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{E^{\prime}_{k,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 191.03307pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 128.23155pt\raise-30.71109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 191.03307pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{E^{\prime}_{k-1,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 247.22421pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 198.78221pt\raise-30.71109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 247.22421pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots}$}}}}}}}{\hbox{\kern-6.75pt\raise-41.21109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 30.75pt\raise-41.21109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 30.75pt\raise-41.21109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{(i_{X/M})_{\ast}E_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 106.1229pt\raise-41.21109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 106.1229pt\raise-41.21109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{(i_{X/M})_{\ast}E_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 174.34021pt\raise-41.21109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 174.34021pt\raise-41.21109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{(i_{X/M})_{\ast}E_{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 247.22421pt\raise-41.21109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 247.22421pt\raise-41.21109pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots}$}}}}}}}\ignorespaces}}}}\ignorespaces.$ It follows that we have a resolution $\operatorname{Tot}(\overline{E}^{\prime}_{\ast,\ast})\longrightarrow(i_{Y/M})_{\ast}F$ of $(i_{Y/M})_{\ast}F$ by hermitian vector bundles. ###### Notation 8.8. We will denote $\displaystyle\overline{\xi}_{Y\hookrightarrow X}$ $\displaystyle=(i_{Y/X},\overline{N}_{Y/X},\overline{F},\overline{E}_{\ast}),$ $\displaystyle\overline{\xi}_{Y\hookrightarrow M}$ $\displaystyle=(i_{Y/M},\overline{N}_{Y/M},\overline{F},\operatorname{Tot}(\overline{E}^{\prime}_{\ast,\ast})),$ $\displaystyle\overline{\xi}_{X\hookrightarrow M,k}$ $\displaystyle=(i_{X/M},\overline{N}_{X/M},\overline{E}_{k},\overline{E}^{\prime}_{k,\ast}).$ We will also denote by $\overline{\xi}_{Y\hookrightarrow P_{X/M}}$ the hermitian embedded vector bundle $\left(Y\hookrightarrow P_{X/M},\overline{N}_{Y/P_{X/M}},\overline{F},\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}_{\ast}\otimes K(\mathcal{O}_{X},\overline{N}_{X/M}))\right).$ Let $T$ be a theory of singular Bott-Chern classes, and let $C_{T}$ be its associated characteristic class. Our aim now is to relate $T(\overline{\xi}_{Y\hookrightarrow X})$, $T(\overline{\xi}_{Y\hookrightarrow M})$ and $T(\overline{\xi}_{X\hookrightarrow M,k})$. Let $W_{X}$ be the deformation to the normal cone of $X$ in $M$. As before we denote by $j_{X}\colon X\times\mathbb{P}^{1}\longrightarrow W_{X}$ the inclusion. We denote by $W$ the deformation to the normal cone of $j_{X}(Y\times\mathbb{P}^{1})$ in $W_{X}$. $M$ $W_{Y}$ $P_{Y/M}$$\widetilde{W}_{X}$$P_{Y\times\mathbb{P}^{1}}$ $\widetilde{M}_{X}\times\mathbb{P}^{1}$ $W_{Y/P}$ $W_{X}$$\widetilde{M}_{X}\times\\{\infty\\}$$\widetilde{P}_{X/M}$$P_{Y/P_{X/M}}$$W$$\mathbb{P}^{1}\times\mathbb{P}^{1}$$(0,\infty)$$(0,0)$$(\infty,0)$$(\infty,\infty)$$P_{X/M}$$\widetilde{M}_{X}\times\\{0\\}$$\widetilde{M}_{Y}$ Figure 1: Double deformation This double deformation is represented in figure 1. There is a proper map $q_{W}\colon W\longrightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}$. The fibers of $q_{W}$ over the corners of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ are as follows: $\displaystyle q_{W}^{-1}(0,0)$ $\displaystyle=M,$ $\displaystyle q_{W}^{-1}(\infty,0)$ $\displaystyle=\widetilde{M}_{X}\times\\{0\\}\cup P_{X/M},$ $\displaystyle q_{W}^{-1}(0,\infty)$ $\displaystyle=\widetilde{M}_{Y}\cup P_{Y/M},$ $\displaystyle q_{W}^{-1}(\infty,\infty)$ $\displaystyle=\widetilde{M}_{X}\times\\{\infty\\}\cup\widetilde{P}_{X/M}\cup P_{Y/P_{X/M}},$ where $\widetilde{M}_{X}$ and $\widetilde{M}_{Y}$ are the blow-up of $M$ along $X$ and $Y$ respectively, $P_{Y/M}=\mathbb{P}(N_{Y/M}\oplus\mathbb{C})$ is the projective completion of the normal cone to $Y$ in $M$, $P_{Y/P_{X/M}}$ of the normal cone to $Y$ in $P_{X/M}$ and $\widetilde{P}_{X/M}$ is the blow-up of $P_{X/M}$ along $Y$. The preimages by $\pi$ of the different faces of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ are as follows: $\displaystyle q_{W}^{-1}(\mathbb{P}^{1}\times\\{0\\})$ $\displaystyle=W_{X},$ $\displaystyle q_{W}^{-1}(\\{0\\}\times\mathbb{P}^{1})$ $\displaystyle=W_{Y},$ $\displaystyle q_{W}^{-1}(\mathbb{P}^{1}\times\\{\infty\\})$ $\displaystyle=\widetilde{W}_{X}\cup P_{Y\times\mathbb{P}^{1}},$ $\displaystyle q_{W}^{-1}(\\{\infty\\}\times\mathbb{P}^{1})$ $\displaystyle=\widetilde{M}_{X}\times\mathbb{P}^{1}\cup W_{Y/P},$ where $W_{Y}$ is the deformation to the normal cone of $Y$ in $M$, the component $\widetilde{W}_{X}$ is the blow-up of $W_{X}$ along $j_{X}(Y\times\mathbb{P}^{1})$, while $P_{Y\times\mathbb{P}^{1}}=\mathbb{P}(N_{Y\times\mathbb{P}^{1}/W_{X}}\oplus\mathbb{C})$ is the projective completion of the normal cone to $j_{X}(Y\times\mathbb{P}^{1})$ in $W_{X}$ and $W_{Y/P}$ is the deformation to the normal cone of $Y$ inside $P_{X/M}$. All the above subvarieties will be called boundary components of $W$. We will use the following notations for the different maps. $\displaystyle p_{X}\colon X\times\mathbb{P}^{1}\longrightarrow X$ $\displaystyle p_{Y}\colon Y\times\mathbb{P}^{1}\longrightarrow Y$ $\displaystyle p_{Y\times\mathbb{P}^{1}}\colon Y\times\mathbb{P}^{1}\times\mathbb{P}^{1}\longrightarrow Y\times\mathbb{P}^{1}$ $\displaystyle p_{\widetilde{M}_{X}\times\mathbb{P}^{1}}\colon\widetilde{M}_{X}\times\mathbb{P}^{1}\longrightarrow M$ $\displaystyle p_{W_{Y/P}}\colon W_{Y/P}\longrightarrow M$ $\displaystyle p_{W_{Y}}\colon W_{Y}\longrightarrow M$ $\displaystyle p_{W_{X}}\colon W_{X}\longrightarrow M$ $\displaystyle p_{P_{Y\times\mathbb{P}^{1}}}\colon P_{Y\times\mathbb{P}^{1}}\longrightarrow M$ $\displaystyle p_{\widetilde{W}_{X}}\colon\widetilde{W}_{X}\longrightarrow M$ $\displaystyle p_{P_{Y/P_{X/M}}}\colon P_{Y/P_{X/M}}\longrightarrow M$ $\displaystyle p_{P_{X/M}}\colon P_{X/M}\longrightarrow M$ $\displaystyle p_{\widetilde{P}_{X/M}}\colon\widetilde{P}_{X/M}\longrightarrow M$ $\displaystyle p_{P_{Y/M}}\colon P_{Y/M}\longrightarrow M$ $\displaystyle p_{W}\colon W\longrightarrow M$ $\displaystyle j_{Y}\colon Y\times\mathbb{P}^{1}\longrightarrow W_{Y}$ $\displaystyle j^{\prime}_{Y}\colon Y\times\mathbb{P}^{1}\longrightarrow W_{X}$ $\displaystyle j_{Y\times\mathbb{P}^{1}}\colon Y\times\mathbb{P}^{1}\times\mathbb{P}^{1}\longrightarrow W$ $\displaystyle i_{Y/P_{X/M}}\colon Y\longrightarrow P_{X/M}$ $\displaystyle\pi_{P_{X/M}}\colon P_{X/M}\longrightarrow X$ $\displaystyle\pi_{P_{Y/M}}\colon P_{Y/M}\longrightarrow Y$ $\displaystyle\pi_{P_{Y/P}}\colon P_{Y/P_{X/M}}\longrightarrow Y$ $\displaystyle\pi_{P_{Y\times\mathbb{P}^{1}}}\colon P_{Y\times\mathbb{P}^{1}}\longrightarrow Y\times\mathbb{P}^{1}$ $\displaystyle\pi_{\widetilde{M}_{X}}\colon\widetilde{M}_{X}\longrightarrow M$ $\displaystyle\pi_{\widetilde{M}_{Y}}\colon\widetilde{M}_{Y}\longrightarrow M$ Note that the map $p_{\widetilde{M}_{X}\times\mathbb{P}^{1}}$ factors through the blow-up $\widetilde{M}_{X}\longrightarrow M$ and the map $p_{\widetilde{W}_{X}}$ factors through the blow-up $\widetilde{M}_{Y}\longrightarrow M$, whereas the maps $p_{W_{Y/P}}$, $p_{P_{X/M}}$ and $p_{\widetilde{P}_{X/M}}$ factor through the inclusion $X\hookrightarrow M$ and the maps $p_{P_{Y\times\mathbb{P}^{1}}}$, $p_{P_{Y/M}}$ and $p_{P_{Y/P_{X/M}}}$ factor through the inclusion $Y\hookrightarrow M$. The normal bundle to $X\times\mathbb{P}^{1}$ in $W_{X}$ is isomorphic to $p_{X}^{\ast}N_{X/M}\otimes q_{X}^{\ast}\mathcal{O}(-1)$ and we consider on it the metric induced by the metric on $\overline{N}_{X/M}$ and the Fubini-Study metric on $\mathcal{O}(-1)$. We denote it by $\overline{N}_{X\times\mathbb{P}^{1}/W_{X}}$. The normal bundle to $Y\times\mathbb{P}^{1}$ in $W_{X}$ satisfies $\displaystyle N_{Y\times\mathbb{P}^{1}/W_{X}}\|_{Y\times\\{0\\}}$ $\displaystyle\cong N_{Y/M}$ $\displaystyle N_{Y\times\mathbb{P}^{1}/W_{X}}\|_{Y\times\\{\infty\\}}$ $\displaystyle\cong N_{Y/X}\oplus i^{\ast}_{Y/X}N_{X/M}.$ On $N_{Y\times\mathbb{P}^{1}/W_{X}}$ we choose a hermitian metric such that the above isomorphisms are isometries. Finally, on the normal bundle to $Y\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ in $W$, we define a metric using the same procedure as the definition of the metric of $\overline{N}_{X\times\mathbb{P}^{1}/W_{X}}$. On $W_{X}$ we obtain a sequence of resolutions $\operatorname{tr}_{1}(\overline{E}^{\prime})_{n,\ast}\longrightarrow(j_{X})_{\ast}p_{X}^{\ast}E_{n}$. They form a complex of complexes $\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast}$ and the associated total complex $\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast})$ provides us with a resolution $\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast})_{\ast}\longrightarrow(j^{\prime}_{Y})_{\ast}p_{Y}^{\ast}F.$ (8.9) The restriction of $\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast})$ to $M$ is $\operatorname{Tot}(\overline{E}^{\prime}_{\ast,\ast})$. The restriction of each complex $\operatorname{tr}_{1}(\overline{E}^{\prime})_{n,\ast}$ to $\widetilde{M}_{X}\times\\{0\\}$ is orthogonally split. Therefore the restriction of $\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime}))$ to $\widetilde{M}_{X}\times\\{0\\}$ is the total complex of a complex of orthogonally split complexes. So it is acyclic although not necessarily orthogonally split. The restriction of each complex $\operatorname{tr}_{1}(\overline{E}^{\prime})_{n,\ast}$ to $P_{X/M}$ fits in an exact sequence $0\longrightarrow\overline{A}_{n,\ast}\longrightarrow\operatorname{tr}_{1}(\overline{E}^{\prime})_{n,\ast}\|_{P_{X/M}}\longrightarrow\pi_{P_{X/M}}^{\ast}\overline{E}_{n}\otimes K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{\ast}\longrightarrow 0.$ These exact sequences glue together giving a commutative diagram $\textstyle{\operatorname{Tot}(\overline{A}_{\ast,\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast}\|_{P_{X/M}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}_{\ast}\otimes K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{\ast})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(i_{Y/P_{X/M}})_{\ast}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(i_{Y/P_{X/M}})_{\ast}F}$ where the rows are short exact sequences. Even if the complexes $(\overline{A}_{n})_{\ast}$ are orthogonally split, this is not necessarily the case for $\operatorname{Tot}(\overline{A}_{\ast,\ast})$. To ease the notation we will denote $\overline{A}_{\ast}=\operatorname{Tot}(\overline{A}_{\ast,\ast})$. Applying theorem 5.4 to the resolution (8.9), we obtain a complex of hermitian vector bundles $\widetilde{E}^{\prime}_{\ast}=\operatorname{tr}_{1}(\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast}))$ which is a resolution of the coherent sheaf $(j_{Y\times\mathbb{P}^{1}})_{\ast}p_{Y\times\mathbb{P}^{1}}^{\ast}p_{Y}^{\ast}F$. We now study the restriction of $\widetilde{E}^{\prime}_{\ast}$ to each of the boundary components of $W$. • The restriction of $\widetilde{E}^{\prime}_{\ast}$ to $W_{X}$ is just $\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime}))$ which has already been described. For each $k\geq 0$, we will denote by $\eta^{1}_{k}$ the short exact sequence of hermitian vector bundles on $P_{X/M}$ $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.04584pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-7.04584pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{A}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 7.04584pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 31.04584pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 31.04584pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{Tot}(\operatorname{tr}_{1}(\overline{E}^{\prime})_{\ast,\ast}\|_{P_{X/M}})_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 133.93396pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 133.93396pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}\otimes K(\mathcal{O}_{X},\overline{N}_{X/M}))_{k}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$ whereas, for each $n,k\geq 0$ we will denote by $\eta^{1}_{n,k}$ the short exact sequence $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 9.50427pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-9.50427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{A}_{n,k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 9.50427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 33.50427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 33.50427pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\operatorname{tr}_{1}(\overline{E}^{\prime})_{n,k}\|_{P_{X/M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 111.09811pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 111.09811pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\pi_{P_{X/M}}^{\ast}\overline{E}_{n}\otimes K(\mathcal{O}_{X},\overline{N}_{X/M})_{k}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$ * • Its restriction to $W_{Y}$ is $\operatorname{tr}_{1}(\operatorname{Tot}(\overline{E}^{\prime}))$. It is a resolution of $(j_{Y})_{\ast}p_{Y}^{\ast}F$. Its restriction to $\widetilde{M}_{Y}$ is orthogonally split, whereas its restriction to $P_{Y/M}$ fits in an exact sequence $0\longrightarrow\overline{B}_{\ast}\longrightarrow\operatorname{tr}_{1}(\operatorname{Tot}(\overline{E}^{\prime}))_{\ast}\|_{P_{Y/M}}\longrightarrow\pi_{P_{Y/M}}^{\ast}\overline{F}\otimes K(\overline{\mathcal{O}}_{Y},\overline{N}_{Y/M})\longrightarrow 0.$ For each $k\geq 0$ we will denote by $\eta^{2}_{k}$ the degree $k$ piece of the above exact sequence. * • Its restriction to $\widetilde{M}_{X}\times\mathbb{P}^{1}$ is an acyclic complex, such that its further restriction to $\widetilde{M}_{X}\times\\{0\\}$ is acyclic and its restriction to $\widetilde{M}_{X}\times\\{\infty\\}$ is orthogonally split. * • Its restriction to $W_{Y/P}$ fits in a short exact sequence $0\rightarrow\operatorname{tr}_{1}(\overline{A}_{\ast})\rightarrow\widetilde{E}^{\prime}_{\ast}\|_{W_{Y/P}}\rightarrow\operatorname{tr}_{1}(\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}\otimes K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})))\rightarrow 0.$ For each $k\geq 0$, we will denote by $\mu^{1}_{k}$ the exact sequence of hermitian vector bundles over $W_{Y/P}$ given by the piece of degree $k$ of this exact sequence. The three terms of the above exact sequence become orthogonally split when restricted to $\widetilde{P}_{X/M}$. By contrast, when restricted to $P_{Y/P_{X/M}}$ they fit in a commutative diagram $\textstyle{\overline{C}^{1}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{C}^{2}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{C}^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{tr}_{1}(\overline{A})_{\ast}\|_{P_{Y/P_{X/M}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{E}^{\prime}_{\ast}\|_{P_{Y/P_{X/M}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{D}^{2}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{D}^{1}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{D}^{1}_{\ast}}$ where the complexes $\overline{C}^{i}_{\ast}$ are orthogonally split, and $\displaystyle\overline{D}^{1}_{\ast}$ $\displaystyle=\pi^{\ast}_{P_{Y/P}}\overline{F}\otimes K(\overline{\mathcal{O}}_{Y},\overline{N}_{Y/P_{X/M}}),$ $\displaystyle\overline{D}^{2}_{\ast}$ $\displaystyle=\operatorname{tr}_{1}(\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}\otimes K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})))\|_{P_{Y/P_{X/M}}}.$ For each $k\geq 0$, we will denote by $\eta^{3}_{k}$ the exact sequence corresponding to the piece of degree $k$ of the second row of the above diagram, by $\eta^{4}_{k}$ that of the second column and by $\eta^{5}_{k}$ that of the third column. Notice that the map in the third row is an isometry. We assume that the metric on $C^{1}_{\ast}$ is chosen in such a way that the first column is an isometry. Since the complexes $\overline{C}^{i}_{\ast}$ are orthogonally split, by lemma 2.17 we obtain $\sum_{k}(-1)^{k}\left(\widetilde{\operatorname{ch}}(\eta^{3}_{k})-\widetilde{\operatorname{ch}}(\eta^{4}_{k})+\widetilde{\operatorname{ch}}(\eta^{5}_{k})\right)=0.$ (8.10) Note that the restriction of $\mu^{1}_{k}$ to $P_{X/M}$ agrees with $\eta^{1}_{k}$, whereas its restriction to $P_{Y/P_{X/M}}$ agrees with $\eta^{3}_{k}$. * • Its restriction to $\widetilde{W}_{X}$ is orthogonally split. * • Finally its restriction to $P_{Y\times\mathbb{P}^{1}}$ fits in an exact sequence $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.90001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\overline{D}_{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 6.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 30.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 30.90001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\widetilde{E}^{\prime}_{\ast}\|_{P_{Y\times\mathbb{P}^{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 83.66197pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 83.66197pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\pi^{\ast}_{P_{Y\times\mathbb{P}^{1}}}p^{\ast}_{Y\times\mathbb{P}^{1}}\overline{F}\otimes K(\mathcal{O}_{Y\times\mathbb{P}^{1}},\overline{N}_{Y\times\mathbb{P}^{1}/W_{X}})}$}}}}}}}\ignorespaces}}}}\ignorespaces,$ where $\overline{D}_{\ast}$ is orthogonally split. For each $k\geq 0$ we will denote by $\mu^{2}_{k}$ the piece of degree $k$ of this exact sequence. Note that the restriction of $\mu^{2}_{k}$ to $P_{Y/M}$ agrees with $\eta^{2}_{k}$ and the restriction of $\mu^{2}_{k}$ to $P_{Y/P_{X/M}}$ agrees with $\eta^{4}_{k}$. On $\mathbb{P}^{1}\times\mathbb{P}^{1}$ we denote the two projections by $p_{1}$ and $p_{2}$. Since the currents $p_{1}^{\ast}W_{1}$ and $p_{2}^{\ast}W_{1}$ have disjoint wave front sets we can define the current $W_{2}=p_{1}^{\ast}W_{1}\bullet p_{2}^{\ast}W_{1}\in\mathcal{D}^{2}_{D}(\mathbb{P}^{1}\times\mathbb{P}^{1},2)$ which satisfies $\operatorname{d}_{\mathcal{D}}W_{2}=(\delta_{\\{\infty\\}\times\mathbb{P}^{1}}-\delta_{\\{0\\}\times\mathbb{P}^{1}})\bullet p_{2}^{\ast}W_{1}-p_{1}^{\ast}W_{1}\bullet(\delta_{\mathbb{P}^{1}\times\\{\infty\\}}-\delta_{\mathbb{P}^{1}\times\\{0\\}}).$ (8.11) The key point in order to study the compatibility of singular Bott-Chern classes and composition of closed immersions is that, in the group $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(M,p)$, we have $\operatorname{d}_{\mathcal{D}}(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{2}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k})\right)=0.$ We compute this class using the equation (8.11). It can be decomposed as follows. $\displaystyle\operatorname{d}_{\mathcal{D}}(p_{W})_{\ast}$ $\displaystyle\left(\sum_{k}(-1)^{k}W_{2}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k})\right)=$ $\displaystyle(p_{\widetilde{M}_{X}\times\mathbb{P}^{1}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}\|_{\widetilde{M}_{X}\times\mathbb{P}^{1}})\right)$ (a) $\displaystyle+(p_{W_{Y/P}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}\|_{W_{Y/P}})\right)$ (b) $\displaystyle-(p_{W_{Y}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}\|_{W_{Y}})\right)$ (c) $\displaystyle-(p_{\widetilde{W}_{X}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}\|_{\widetilde{W}_{X}})\right)$ (d) $\displaystyle-(p_{P_{Y\times\mathbb{P}^{1}}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}\|_{P_{Y\times\mathbb{P}^{1}}})\right)$ (e) $\displaystyle+(p_{W_{X}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}\|_{W_{X}})\right)$ (f) $\displaystyle\phantom{AAAA}=\colon I_{a}+I_{b}-I_{c}-I_{d}-I_{e}+I_{f}$ We compute each of the above terms. (a) Since the restriction $\widetilde{E}^{\prime}\|_{\widetilde{M}_{X}\times\\{\infty\\}}$ is orthogonally split, we have $I_{a}=-(\pi_{\widetilde{M}_{X}})_{\ast}\widetilde{\operatorname{ch}}(\widetilde{E}^{\prime}\|_{\widetilde{M}_{X}\times\\{0\\}}).$ But, using lemma 2.17 and the fact, for each $k$, the complexes $\operatorname{tr}_{1}(\overline{E}^{\prime})_{k,\ast}\|_{\widetilde{M}_{X}}$ are orthogonally split, we obtain that $I_{a}=0$. (b) We compute $\displaystyle I_{b}=$ $\displaystyle(p_{W_{Y/P}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}\|_{W_{Y/P}})\right)$ $\displaystyle=$ $\displaystyle(p_{W_{Y/P}})_{\ast}\left(W_{1}\bullet\sum_{k}(-1)^{k}(-\operatorname{d}_{\mathcal{D}}\widetilde{\operatorname{ch}}(\mu^{1}_{k})+\operatorname{ch}(\operatorname{tr}_{1}(\overline{A}_{\ast})_{k})\right.$ $\displaystyle\left.\phantom{(p_{W_{Y/P}})_{\ast}\sum_{k}(-1)^{k}}+\operatorname{ch}(\operatorname{tr}_{1}(\operatorname{Tot}(\pi_{P_{X/M}}^{\ast}\overline{E}\otimes K(\mathcal{O}_{X},\overline{N}_{X/M})))_{k}))\right)$ $\displaystyle=$ $\displaystyle\sum_{k}(-1)^{k}(-(p_{P_{Y/P_{X/M}}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{3}_{k})-(p_{\widetilde{P}_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\mu^{1}_{k}\|_{\widetilde{P}_{X/M}})+(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{k}))$ $\displaystyle-\widetilde{\operatorname{ch}}(\overline{A})$ $\displaystyle-(i_{X/M})_{\ast}(\pi_{P_{X/M}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow P_{X/M}})+(i_{Y/M})_{\ast}C_{T}(F,N_{Y/P_{X/M}})$ $\displaystyle-\sum_{k}(-1)^{k}(p_{P_{Y/P_{X/M}}})\widetilde{\operatorname{ch}}(\eta^{5}_{k}),$ where $\xi_{Y\hookrightarrow P_{X/M}}$ is as in notation 8.8. By corollary 2.19 and the fact that the exact sequences $\overline{A}_{k,\ast}$ are orthogonally split, the term $\widetilde{\operatorname{ch}}(\overline{A})$ vanishes. Also by corollary 2.19 we can see that $\sum_{k}(-1)^{k}(p_{\widetilde{P}_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\mu^{1}_{k}\|_{\widetilde{P}_{X/M}})$ vanishes. Therefore we conclude $\displaystyle I_{b}=$ $\displaystyle\sum_{k}(-1)^{k}(-(p_{P_{Y/P_{X/M}}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{3}_{k})+(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{k}))-(p_{P_{Y/P_{X/M}}})\widetilde{\operatorname{ch}}(\eta^{5}_{k})$ $\displaystyle-(i_{X/M})_{\ast}(\pi_{P_{X/M}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow P_{X/M}})+(i_{Y/M})_{\ast}C_{T}(F,N_{Y/P_{X/M}}).$ (c) By the definition of singular Bott-Chern forms we have $I_{c}=-T(\overline{\xi}_{Y\hookrightarrow M})+(i_{Y/M})_{\ast}C_{T}(F,N_{Y/M})-\sum_{k}(-1)^{k}(p_{P_{Y/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{2}_{k}),$ (d) Since the restriction of $\widetilde{E}^{\prime}_{\ast}$ to $\widetilde{W}_{X}$ is orthogonally split, we have $I_{d}=0$. (e) We compute $\displaystyle I_{e}=$ $\displaystyle(p_{P_{Y\times\mathbb{P}^{1}}})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\widetilde{E}^{\prime}_{k}\|_{P_{Y\times\mathbb{P}^{1}}})\right)$ $\displaystyle=$ $\displaystyle(p_{P_{Y\times\mathbb{P}^{1}}})_{\ast}\left(W_{1}\bullet\sum_{k}(-1)^{k}\big{(}-\operatorname{d}_{\mathcal{D}}\widetilde{ch}(\mu^{2}_{k})+\operatorname{ch}(\overline{D}_{k})\right.$ $\displaystyle\left.\phantom{(p_{P_{Y\times\mathbb{P}^{1}}})_{\ast}\sum_{k}}+\operatorname{ch}(\pi^{\ast}_{P_{Y\times\mathbb{P}^{1}}}p^{\ast}_{Y}\overline{F}\otimes K(\mathcal{O}_{Y\times\mathbb{P}^{1}},\overline{N}_{Y\times\mathbb{P}^{1}/W_{X}})_{k})\big{)}\right).$ The term $\sum(-1)^{k}\operatorname{ch}(\overline{D}_{k})$ vanishes because the complex $D_{\ast}$ is orthogonally split. We have $\sum_{k}(-1)^{k}(p_{P_{Y\times\mathbb{P}^{1}}})_{\ast}(W_{1}\bullet\operatorname{ch}(\pi^{\ast}_{P_{Y\times\mathbb{P}^{1}}}p^{\ast}_{Y}\overline{F}\otimes K(\overline{\mathcal{O}}_{Y\times\mathbb{P}^{1}},\overline{N}_{Y\times\mathbb{P}^{1}/W_{X}})_{k}))\\\ =(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet(p_{Y})_{\ast}\left(W_{1}\bullet\pi^{\ast}_{P_{Y\times\mathbb{P}^{1}}}\sum_{k}(-1)^{k}\operatorname{ch}(K(\overline{\mathcal{O}}_{Y\times\mathbb{P}^{1}},\overline{N}_{Y\times\mathbb{P}^{1}/W_{X}})_{k})\right)\\\ =(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet(p_{Y})_{\ast}\left(W_{1}\bullet\operatorname{Td}^{-1}(\overline{N}_{Y\times\mathbb{P}^{1}/W_{X}})\right)\\\ =(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N}),$ (8.12) where $\overline{\varepsilon}_{N}$ is the exact sequence (8.6). Therefore we obtain $I_{e}=-\sum_{k}(-1)^{k}(p_{P_{Y/P_{X/M}}})_{\ast}\widetilde{\operatorname{ch}}(\eta_{k}^{4})+\sum_{k}(-1)^{k}(p_{P_{Y/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta_{k}^{2})\\\ +(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N}).$ (f) Finally we have $\displaystyle I_{f}=$ $\displaystyle-\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow M,k})+\sum_{k}(-1)^{k}(i_{X/M})_{\ast}C_{T}(E_{k},N_{X/M})$ $\displaystyle-\sum_{k,l}(-1)^{k+l}(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{k,l}).$ By corollary 2.19 we have that $\sum_{m,l}(-1)^{m+l}(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{m,l})=\sum_{k}(-1)^{k}(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{k}).$ Thus $\displaystyle I_{f}=$ $\displaystyle-\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow M,k})+\sum_{k}(-1)^{k}(i_{X/M})_{\ast}C_{T}(E_{k},N_{X/M})$ $\displaystyle-\sum_{k}(-1)^{k}(p_{P_{X/M}})_{\ast}\widetilde{\operatorname{ch}}(\eta^{1}_{k}).$ Summing up all the terms we have computed, and taking into account equation (8.10) and the fact that $C_{T}(F,N_{Y/M})=C_{T}(F,N_{Y/P_{X/M}})$ we have obtained the following partial result. ###### Lemma 8.13. Let $i_{Y/M}=i_{X/M}\circ i_{Y/X}$ be a composition of closed immersions of complex manifolds. Let $T$ be a theory of singular Bott-Chern classes with $C_{T}$ its associated characteristic class. Let $\overline{\xi}_{Y\hookrightarrow M}$, $\overline{\xi}_{X\hookrightarrow M,k}$ and $\overline{\xi}_{Y\hookrightarrow P_{X/M}}$ be as in notation 8.8, and let $\overline{\varepsilon}$ be as in (8.6). Then, in the group $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(M,p)$, the equation $T(\overline{\xi}_{Y\hookrightarrow M})=\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow M,k})-\sum_{k}(-1)^{k}(i_{X/M})_{\ast}C_{T}(E_{k},N_{X/M})\\\ +(i_{X/M})_{\ast}(\pi_{P_{X/M}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow P_{X/M}})+(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N})$ (8.14) holds. In order to compute the third term of the right hand side of equation (8.14) we consider the following situation $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 25.03004pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\crcr}}}\ignorespaces{\hbox{\kern-25.03004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{Y\times_{X}P_{X/M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 26.61172pt\raise 5.98889pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.62779pt\hbox{$\scriptstyle{j}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 49.03004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-15.99019pt\raise-19.29166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern-4.55803pt\raise-28.74915pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern 49.03004pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{P_{X/M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 46.51659pt\raise-19.29166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 57.94876pt\raise-28.74915pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern-7.01389pt\raise-38.58331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 6.0pt\raise-19.29166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{s}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 3.36191pt\raise-6.49973pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 27.04759pt\raise-33.27498pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.30833pt\hbox{$\scriptstyle{i}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 54.97206pt\raise-38.58331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 54.97206pt\raise-38.58331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 68.50677pt\raise-19.29166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{$\scriptstyle{s}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern 65.86868pt\raise-6.49973pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}\ignorespaces.$ To ease the notation, we denote $P_{X/M}$ by $P$, $Y\underset{X}{\times}P_{X/M}$ by $X^{\prime}$ and we denote by $P^{\prime}$ the projective completion of the normal cone to $X^{\prime}$ in $P$ and by $\pi_{P^{\prime}}\colon P^{\prime}\longrightarrow X^{\prime}$, $\pi_{X^{\prime}/Y}\colon X^{\prime}\longrightarrow Y$ and $\pi_{P^{\prime}/Y}\colon P^{\prime}\longrightarrow Y$ the projections. Observe that $X$ and $X^{\prime}$ intersect transversely along $Y$. Moreover, $N_{Y/X^{\prime}}=i^{\ast}_{Y/X}N_{X/M}$, $N_{X^{\prime}/P}=\pi^{\ast}_{X^{\prime}/Y}N_{Y/X}$ and $N_{Y/P}=N_{Y/X}\oplus N_{Y/X^{\prime}}$. We use these identifications to define metrics on $N_{Y/X^{\prime}}$, $N_{X^{\prime}/P}$ and $N_{Y/P}$. Therefore the exact sequence $0\longrightarrow\overline{N}_{Y/X^{\prime}}\longrightarrow\overline{N}_{Y/P}\longrightarrow i^{\ast}_{Y/X^{\prime}}\overline{N}_{X^{\prime}/P}\longrightarrow 0$ is orthogonally split. We apply the previous lemma to the composition of closed inclusions $Y\hookrightarrow X^{\prime}\hookrightarrow P,$ the vector bundle $\overline{F}$ over $Y$ and the resolutions $\displaystyle\pi^{\ast}\overline{F}\otimes j^{\ast}K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{\ast}\longrightarrow s_{\ast}F$ $\displaystyle\pi^{\ast}\overline{E}_{\ast}\otimes K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{k}\longrightarrow j_{\ast}(\pi^{\ast}F\otimes j^{\ast}K(\mathcal{O}_{X},N_{X/M})_{k}).$ We denote by $\overline{\xi}_{Y\hookrightarrow P}$ and $\overline{\xi}_{X^{\prime}\hookrightarrow P,k}$ the hermitian embedded vector bundles corresponding to the above resolutions. If $i_{Y/P^{\prime}}\colon Y\hookrightarrow P^{\prime}$ is the induced inclusion, we denote by $\overline{\xi}_{Y\hookrightarrow P^{\prime}}$ the hermitian embedded vector bundle $\left(i_{Y/P^{\prime}},\overline{N}_{Y/P^{\prime}},\overline{F},\operatorname{Tot}(\pi^{\ast}_{P^{\prime}}j^{\ast}K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})\otimes K(\overline{\mathcal{O}}_{X^{\prime}},\overline{N}_{X^{\prime}/P})\otimes(\pi_{P^{\prime}/Y})^{\ast}\overline{F})\right).$ Note that the hermitian embedded vector bundle $\overline{\xi}_{Y\hookrightarrow P}$ agrees with the hermitian embedded vector bundle denoted $\overline{\xi}_{Y\hookrightarrow P_{X/M}}$ in lemma 8.13. Moreover, we have that $\overline{\xi}_{X^{\prime}\hookrightarrow P,k}=\pi^{\ast}\overline{\xi}_{Y\hookrightarrow X}\otimes K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{k}.$ Applying lemma 8.13, we obtain $T(\overline{\xi}_{Y\hookrightarrow P_{X/M}})=\sum_{k}(-1)^{k}T(\overline{\xi}_{X^{\prime}\hookrightarrow P_{X/M},k})\\\ -\sum_{k}(-1)^{k}j_{\ast}C_{T}(\pi^{\ast}F\otimes j^{\ast}K(\mathcal{O}_{X},N_{X/M})_{k},N_{X^{\prime}/P})\\\ +j_{\ast}(\pi_{P^{\prime}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow P^{\prime}})$ (8.15) By proposition 8.1, $\sum_{k}(-1)^{k}T(\overline{\xi}_{X^{\prime}\hookrightarrow P_{X/M},k})=\sum_{k}(-1)^{k}T(\pi^{\ast}\overline{\xi}_{Y\hookrightarrow X}\otimes K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{k})\\\ =T(\pi^{\ast}\overline{\xi}_{Y\hookrightarrow X})\bullet\sum_{k}(-1)^{k}\operatorname{ch}(K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})_{k})\\\ +\sum_{k}(-1)^{k}j_{\ast}C_{T}(\pi^{\ast}F\otimes j^{\ast}K(\mathcal{O}_{X},N_{X/M})_{k},N_{X^{\prime}/P})\\\ -\sum_{k}(-1)^{k}j_{\ast}C_{T}(\pi^{\ast}F,N_{X^{\prime}/P})\bullet\operatorname{ch}(K(\mathcal{O}_{X},N_{X/M})_{k})$ (8.16) We now want to compute the term $(i_{X/M})_{\ast}(\pi_{P_{X/M}})_{\ast}j_{\ast}(\pi_{P^{\prime}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow P^{\prime}})$. Observe that we can identify $P^{\prime}=\mathbb{P}(i_{Y/X}^{\ast}N_{X/M}\oplus\mathbb{C})\underset{Y}{\times}\mathbb{P}(s^{\ast}N_{X^{\prime}/P}\oplus\mathbb{C}),$ where $s^{\ast}N_{X^{\prime}/P}$ is canonically isomorphic to $N_{Y/X}$. Moreover $(i_{X/M})_{\ast}(\pi_{P_{X/M}})_{\ast}j_{\ast}(\pi_{P^{\prime}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow P^{\prime}})=(i_{Y/M})_{\ast}(\pi_{P^{\prime}/Y})_{\ast}T(\overline{\xi}_{Y\hookrightarrow P^{\prime}}).$ ###### Definition 8.17. We denote $C_{T}^{\operatorname{ad}}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})=(\pi_{P^{\prime}/Y})_{\ast}T(\overline{\xi}_{Y\hookrightarrow P^{\prime}})$ and we define $\rho(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})=C_{T}(F,N_{Y/M})-C_{T}^{\operatorname{ad}}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M}).$ (8.18) ###### Lemma 8.19. The current $C_{T}^{\operatorname{ad}}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})$ is closed and defines a characteristic class of triples of vector bundles. Therefore $\rho$ is also a characteristic class. Moreover the class $\rho$ does not depend on the theory of singular Bott-Chern classes $T$. ###### Proof. The fact that $C_{T}^{\operatorname{ad}}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})$ is closed and determines a characteristic class is proved as in 6.16. The independence of $\rho$ from to $T$ is seen as follows. We denote by $\overline{K}^{\prime}_{\ast}$ the complex $\operatorname{Tot}(\pi^{\ast}_{P^{\prime}}j^{\ast}K(\overline{\mathcal{O}}_{X},\overline{N}_{X/M})\otimes K(\overline{\mathcal{O}}_{X^{\prime}},\overline{N}_{X^{\prime}/P}))\otimes(\pi_{P^{\prime}/Y})^{\ast}\overline{F}.$ This complex is a resolution of $(i_{Y/P^{\prime}})_{\ast}\overline{F}$ Let $W$ be the blow-up of $P^{\prime}\times\mathbb{P}^{1}$ along $Y\times\infty$, and let $\operatorname{tr}_{1}(\overline{K}^{\prime})_{\ast}$ be the deformation of complexes on $W$ given by theorem 5.4. Just by looking at the rank of the different vector bundles we see that the restriction of $\operatorname{tr}_{1}(\overline{K}^{\prime})_{\ast}$ to $P_{Y/P^{\prime}}$, the exceptional divisor of this blow-up, is isomorphic (although not necessarily isometric) to the Koszul complex $K(\overline{F},\overline{N}_{X/M})_{\ast}$. Then, by equation (7.3) $T(\overline{\xi}_{Y\hookrightarrow P^{\prime}})-(i_{Y/P^{\prime}})_{\ast}C_{T}(F,N_{Y/M})=\\\ -(p_{W})_{\ast}\left(W_{1}\bullet\sum_{k}(-1)^{k}\operatorname{ch}(\operatorname{tr}_{1}(\overline{K}^{\prime})_{k})\right)\\\ -\sum_{k}(-1)^{k}(p_{P})_{\ast}\widetilde{\operatorname{ch}}(\operatorname{tr}_{1}(\overline{K}^{\prime})_{k}\|_{P_{Y/P^{\prime}}},K(\overline{F},\overline{N}_{X/M})_{k}).$ Since the right hand side of this equation does not depend on the theory $T$, the result is proved. ∎ Using equations (8.15), (8.16), lemma 8.19 and the projection formula, we obtain $\displaystyle(\pi_{P_{X/M}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow P_{X/M}})=$ $\displaystyle\left(T(\overline{\xi}_{Y\hookrightarrow X})-(i_{Y/X})_{\ast}C_{T}(F,N_{Y/X})\right)$ $\displaystyle\phantom{AA}\bullet(\pi_{P_{X/M}})_{\ast}\sum_{k}(-1)^{k}\operatorname{ch}(K(\mathcal{O}_{X},\overline{N}_{X/M})_{k})$ $\displaystyle+(\pi_{P_{X/M}})_{\ast}j_{\ast}(\pi_{P^{\prime}})_{\ast}T(\overline{\xi}_{Y\hookrightarrow P^{\prime}})$ $\displaystyle=$ $\displaystyle\left(T(\overline{\xi}_{Y\hookrightarrow X})-(i_{Y/X})_{\ast}C_{T}(F,N_{Y/X})\right)\bullet\operatorname{Td}^{-1}(\overline{N}_{X/M})$ $\displaystyle+(i_{Y/X})_{\ast}C^{\operatorname{ad}}_{T}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})$ $\displaystyle=$ $\displaystyle\left(T(\overline{\xi}_{Y\hookrightarrow X})-(i_{Y/X})_{\ast}C_{T}(F,N_{Y/X})\right)\bullet\operatorname{Td}^{-1}(\overline{N}_{X/M})$ $\displaystyle+(i_{Y/X})_{\ast}C_{T}(F,N_{Y/M})-\rho(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M}).$ (8.20) Joining this equation and lemma 8.13 we obtain the main relationship between singular Bott-Chern classes and composition of closed immersions. ###### Proposition 8.21. Let $i_{Y/M}=i_{X/M}\circ i_{Y/X}$ be a composition of closed immersions of complex manifolds. Let $T$ be a theory of singular Bott-Chern classes with $C_{T}$ its associated characteristic class. Let $\overline{\xi}_{Y\hookrightarrow M}$, $\overline{\xi}_{X\hookrightarrow M,k}$ and $\overline{\xi}_{Y\hookrightarrow P_{X/M}}$ be as in notation 8.8 and let $\overline{\varepsilon}$ be as in (8.6). Then, in the group $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(M,p)$, we have the equation $T(\overline{\xi}_{Y\hookrightarrow M})=\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow M,k})+(i_{X/M})_{\ast}(T(\overline{\xi}_{Y\hookrightarrow X})\bullet\operatorname{Td}^{-1}(\overline{N}_{X/M}))\\\ +(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N})\\\ +(i_{Y/M})_{\ast}C^{\operatorname{ad}}_{T}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})\\\ -(i_{X/M})_{\ast}((i_{Y/X})_{\ast}C_{T}(F,N_{Y/X})\bullet\operatorname{Td}^{-1}(N_{X/M}))\\\ -(i_{X/M})_{\ast}\sum_{k}(-1)^{k}C_{T}(E_{k},N_{X/M})$ We can simplify the formula of proposition 8.21 if we assume that our theory of singular Bott-Chern classes is compatible with the projection formula. ###### Corollary 8.22. With the hypothesis of proposition 8.21, assume furthermore that $T$ is compatible with the projection formula. Then $T(\overline{\xi}_{Y\hookrightarrow M})=\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow M,k})+(i_{X/M})_{\ast}(T(\overline{\xi}_{Y\hookrightarrow X})\bullet\operatorname{Td}^{-1}(\overline{N}_{X/M}))\\\ +(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N})\\\ +(i_{Y/M})_{\ast}\left[C^{\operatorname{ad}}_{T}(F,N_{Y/X},i_{Y/X}^{\ast}N_{X/M})-C_{T}(F,N_{Y/X})\bullet\operatorname{Td}^{-1}(i_{Y/X}^{\ast}N_{X/M}))\right.\\\ \left.-C_{T}(F,i_{Y/X}^{\ast}N_{X/M})\bullet\operatorname{Td}^{-1}(N_{Y/X})\right]$ ###### Proof. Since $T$ is compatible with the projection formula, then $C_{T}$ is also. Therefore, using the Grothendieck-Riemann-Roch theorem for closed immersions at the level of analytic Deligne cohomology classes, we have $\displaystyle\sum_{k}(-1)^{k}C_{T}(E_{k},$ $\displaystyle N_{X/M})=C_{T}(\mathcal{O}_{X},N_{X/M})\bullet\sum_{k}(-1)^{k}\operatorname{ch}(E_{k})$ $\displaystyle=C_{T}(\mathcal{O}_{X},N_{X/M})\bullet(i_{Y/X})_{\ast}(\operatorname{ch}(F)\bullet\operatorname{Td}^{-1}(N_{Y/X}))$ $\displaystyle=(i_{Y/X})_{\ast}(i_{Y/X}^{\ast}C_{T}(\mathcal{O}_{X},N_{X/M})\bullet\operatorname{ch}(F)\bullet\operatorname{Td}^{-1}(N_{Y/X}))$ $\displaystyle=(i_{Y/X})_{\ast}(C_{T}(F,i_{Y/X}^{\ast}N_{X/M})\bullet\operatorname{Td}^{-1}(N_{Y/X})),$ which implies the result. ∎ ###### Definition 8.23. Let $T$ be a theory of singular Bott-Chern classes. We will say that $T$ is _transitive_ if the equation $T(\overline{\xi}_{Y\hookrightarrow M})=\sum_{k}(-1)^{k}T(\overline{\xi}_{X\hookrightarrow M,k})+(i_{X/M})_{\ast}(T(\overline{\xi}_{Y\hookrightarrow X})\bullet\operatorname{Td}^{-1}(\overline{N}_{X/M}))\\\ +(i_{Y/M})_{\ast}\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\varepsilon}_{N})$ (8.24) holds. When equation (8.24) is satisfied for a particular choice of complex immersions and resolutions, we say that the theory $T$ is _transitive with respect to this particular choice_. We now introduce an abstract version of definition 8.17. ###### Definition 8.25. Given any characteristic class $C$ of pairs of vector bundles, we will denote $C^{\rho}(F,N_{1},N_{2}):=C(F,N_{1}\oplus N_{2})-\rho(F,N_{1},N_{2}),$ where $\rho$ is the characteristic class of definition 8.17. Note that, when $T$ is a theory of singular Bott-Chern classes we have $C^{\rho}_{T}(F,N_{1},N_{2})=C^{\operatorname{ad}}_{T}(F,N_{1},N_{2}).$ ###### Definition 8.26. We will say that a characteristic class $C$ (of pairs of vector bundles) is _$\rho$ -Todd additive_ (in the second variable) if it satisfies $C(F,N_{1}\oplus N_{2})=C(F,N_{1})\bullet\operatorname{Td}^{-1}(N_{2})+C(F,N_{2})\bullet\operatorname{Td}^{-1}(N_{1})+\rho(F,N_{1},N_{2})$ or, equivalently, $C^{\rho}(F,N_{1},N_{2})=C(F,N_{1})\bullet\operatorname{Td}^{-1}(N_{2})+C(F,N_{2})\bullet\operatorname{Td}^{-1}(N_{1}).$ A direct consequence of corollary 8.22 is ###### Corollary 8.27. Let $T$ be a theory of singular Bott-Chern classes that is compatible with the projection formula. Then it is transitive if and only if the associated characteristic class $C_{T}$ is $\rho$-Todd additive. Since we are mainly interested in singular Bott-Chern classes that are transitive and compatible with the projection formula, we will study characteristic classes that are compatible with the projection formula and $\rho$-Todd-additive in the second variable. Since we want to express any characteristic class in terms of a power series we will restrict ourselves to the algebraic category. ###### Proposition 8.28. Let $C$ be a class that is compatible with the projection formula and $\rho$-Todd additive in the second variable. Then $C$ determines a power series $\phi_{C}(x)$ given by $C(\mathcal{O}_{Y},L)=\phi_{C}(c_{1}(L)),$ (8.29) for every complex algebraic manifold $Y$ and algebraic line bundle $Y$. Conversely, given any power series in one variable $\phi(x)$, there exists a unique characteristic class for algebraic vector bundles that is compatible with the projection formula and $\rho$-Todd additive in the second variable such that equation (8.29) holds. ###### Proof. This result follows directly from the splitting principle and theorem 1.8. ∎ ###### Remark 8.30. The utility of corollary 8.27 and proposition 8.28 is limited by the fact that we do not know an explicit formula for the class $\rho(\mathcal{O}_{Y},N_{1},N_{2})$. This class is related with the arithmetic difference between $\mathbb{P}_{Y}(N_{1}\oplus N_{2}\oplus\mathbb{C})$ and $\mathbb{P}_{Y}(N_{1}\oplus\mathbb{C})\underset{Y}{\times}\mathbb{P}_{Y}(N_{2}\oplus\mathbb{C})$, the second space being simpler than the first. The main ingredients needed to compute this class are the Bott-Chern classes of the tautological exact sequence. Therefore the work of Mourougane [29] might be useful for computing this class. Recall that an additive genus is a characteristic class for algebraic vector bundles $S$ such that $S(N_{1}\oplus N_{2})=S(N_{1})+S(N_{2}).$ Let $\phi(x)=\sum_{i=0}^{\infty}a_{i}x^{i}$ be a power series in one variable. There is a one to one correspondence between additive genus and power series characterized by the condition that $S(L)=\phi(c_{1}(L))$, for each line bundle $L$. Since the class $\rho$ does not depend on the theory $T$ it cancels out when considering the difference between two different theories of singular Bott- Chern classes. ###### Proposition 8.31. Let $C_{1}$ and $C_{2}$ be two characteristic classes for pairs of algebraic vector bundles that are compatible with the projection formula and $\rho$-Todd-additive in the second variable. Then there is a unique additive genus $S_{12}$ such that $C_{1}(F,N)-C_{2}(F,N)=\operatorname{ch}(F)\bullet\operatorname{Td}(N)^{-1}\bullet S_{12}(N).$ (8.32) We can summarize the results of this section in the following theorem. ###### Theorem 8.33. There is a one to one correspondence between theories of singular Bott-Chern classes for complex algebraic manifolds that are transitive and compatible with the projection formula, and formal power series $\phi(x)\in\mathbb{R}[[x]]$. To each theory of singular Bott-Chern classes corresponds the power series $\phi$ such that $C_{T}(\mathcal{O}_{Y},L)={\bf 1}_{1}\bullet\phi(c_{1}(L)),$ (8.34) for every complex algebraic manifold $Y$ and every algebraic line bundle $L$. To each power series $\phi$ it corresponds a unique class $C$, compatible with the projection formula and $\rho$-Todd-additive in the second variable, characterized by equation (8.34) and a theory of singular Bott-Chern given by definition 7.4. Even if we do not know the exact value of the class $\rho$ another consequence of corollary 8.27 is that, in order to prove the transitivity of a theory of singular Bott-Chern classes it is enough to check it for a particular class of compositions. ###### Corollary 8.35. Let $T$ be a theory of singular Bott-Chern classes compatible with the projection formula. Then $T$ is transitive if and only if for any compact complex manifold $Y$ and vector bundles $N_{1}$, $N_{2}$, the theory $T$ is transitive with respect to the composition of inclusions $Y\hookrightarrow\mathbb{P}_{Y}(N_{1}\oplus\mathbb{C})\hookrightarrow\mathbb{P}_{Y}(N_{1}\oplus\mathbb{C})\times_{Y}\mathbb{P}_{Y}(N_{2}\oplus\mathbb{C})$ and the Koszul resolutions. $\square$ We can make the previous corollary a little more explicit. Let $\pi_{1}$ and $\pi_{2}$ be the projections from $P:=\mathbb{P}_{Y}(N_{1}\oplus\mathbb{C})\times_{Y}\mathbb{P}_{Y}(N_{2}\oplus\mathbb{C})$ to $P_{1}:=\mathbb{P}_{Y}(N_{1}\oplus\mathbb{C})$ and $P_{2}:=\mathbb{P}_{Y}(N_{2}\oplus\mathbb{C})$ respectively. Let $\overline{K}_{1}=K(\overline{\mathcal{O}}_{Y},\overline{N}_{1})$ and $\overline{K}_{2}=K(\overline{\mathcal{O}}_{Y},\overline{N}_{2})$ be the Koszul resolutions in $P_{1}$ and $P_{2}$ respectively. Then, $\overline{K}=\pi_{1}^{\ast}K_{1}\otimes\pi_{2}^{\ast}K_{2}$ is a resolution of $\mathcal{O}_{Y}$ in $P$. Then the theory $T$ is transitive in this case if $T(\overline{K})=\pi_{2}^{\ast}T(\overline{K}_{2})\bullet\pi_{1}^{\ast}(c_{r_{1}}(\overline{Q}_{1})\bullet\operatorname{Td}^{-1}(\overline{Q}_{1}))+(i_{1})_{\ast}(T(\overline{K}_{1})\bullet p_{1}^{\ast}\operatorname{Td}^{-1}(\overline{N}_{2})),$ where $r_{1}$ is the rank of $N_{1}$, $\overline{Q}_{1}$ is the tautological quotient bundle in $P_{1}$ with the induced metric, $i_{1}\colon P_{1}\longrightarrow P$ is the inclusion and $p_{1}\colon P_{1}\longrightarrow Y$ is the projection. The singular Bott-Chern classes that we have defined depend on the choice of a hermitian metric on the normal bundle and behave well with respect inverse images. Nevertheless, when one is interested in covariant functorial properties and, in particular, in a composition of closed immersions, it might be interesting to consider a variant of singular Bott-Chern classes that depend on the choice of metrics on the tangent bundles to $Y$ and $X$. ###### Notation 8.36. Let $\overline{\xi}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast}\to i_{\ast}F)$ be a hermitian embedded vector bundle. Let $\overline{T}_{X}$ and $\overline{T}_{Y}$ be the tangent bundles to $X$ and $Y$ provided with hermitian metrics. As usual we write $\operatorname{Td}(Y)=\operatorname{Td}(\overline{T}_{Y})$ and $\operatorname{Td}(X)=\operatorname{Td}(\overline{T}_{X})$. We put $\overline{\xi}_{c}=(i\colon Y\longrightarrow X,\overline{T}_{X},\overline{T}_{Y},\overline{F},\overline{E}_{\ast}\to i_{\ast}F).$ By abuse of notation we will also say that $\overline{\xi}_{c}$ is a hermitian embedded vector bundle. In this situation we denote by $\overline{\xi}_{N_{Y/X}}$ the exact sequence of hermitian vector bundles $\overline{\xi}_{N_{Y/X}}\colon 0\longrightarrow\overline{T}_{Y}\longrightarrow i^{\ast}\overline{T}_{X}\longrightarrow\overline{N}_{Y/X}\longrightarrow 0.$ If there is no danger of confusion we will denote $\overline{N}=\overline{N}_{Y/X}$ and therefore $\overline{\xi}_{N}=\overline{\xi}_{N_{Y/X}}$. ###### Definition 8.37. Let $T$ be a theory of singular Bott-Chern classes. Then the _covariant singular Bott-Chern class_ associated to $T$ is given by $T_{c}(\overline{\xi}_{c})=T(\overline{\xi})+i_{\ast}(\operatorname{ch}(\overline{F})\bullet\widetilde{\operatorname{Td}^{-1}}(\overline{\xi}_{N_{Y/X}})\operatorname{Td}(Y))$ (8.38) ###### Proposition 8.39. The covariant singular Bott-Chern classes satisfy the following properties 1. (i) The class $T_{c}(\overline{\xi}_{c})$ does not depend on the choice of the metric on $N_{Y/X}$. 2. (ii) The differential equation $\operatorname{d}_{\mathcal{D}}T_{c}(\overline{\xi}_{c})=\sum_{k}(-1)^{k}\operatorname{ch}(\overline{E}_{k})-i_{\ast}(\operatorname{ch}(\overline{F})\bullet\operatorname{Td}(Y))\bullet\operatorname{Td}^{-1}(X)$ (8.40) holds. 3. (iii) If the theory $T$ is compatible with the projection formula, then $T_{c}(\overline{\xi}_{c}\otimes\overline{G})=T_{c}(\overline{\xi}_{c})\bullet\operatorname{ch}(\overline{G}).$ 4. (iv) If, moreover, the theory $T$ is transitive, then, using notation 8.8 adapted to the current setting, we have $T_{c}(\overline{\xi}_{Y\hookrightarrow M,c})=\sum_{k}(-1)^{k}T_{c}(\overline{\xi}_{X\hookrightarrow M,k,c})\\\ +(i_{X/M})_{\ast}(T_{c}(\overline{\xi}_{Y\hookrightarrow X,c})\bullet\operatorname{Td}(X))\bullet\operatorname{Td}^{-1}(M).$ (8.41) 5. (v) With the hypothesis of corollary 6.14, we have $T_{c}(\bigoplus_{j\text{ even}}\overline{\xi}_{j,c})-T_{c}(\bigoplus_{j\text{ odd}}\overline{\xi}_{j,c})=[\widetilde{\operatorname{ch}}(\overline{\varepsilon})]-i_{\ast}([\widetilde{\operatorname{ch}}(\overline{\chi})\bullet\operatorname{Td}(Y)])\bullet\operatorname{Td}^{-1}(X).$ (8.42) ###### Proof. All the statements follow from straightforward computations. ∎ ## 9 Homogeneous singular Bott-Chern classes In this section we will show that, by adding a natural fourth axiom to definition 6.9, we obtain a unique theory of singular Bott-Chern classes that we call homogeneous singular Bott-Chern classes, and we will compare it with the classes previously defined by Bismut, Gillet and Soulé and by Zha. In the paper [6], Bismut, Gillet and Soulé introduced a theory of singular Bott-Chern classes that is the main ingredient in their construction of direct images for closed immersions. Strictly speaking, the construction of [6] only produces a theory of singular Bott-Chern classes in the sense of this paper when the metrics involved satisfy a technical condition, called Condition (A) of Bismut. Nevertheless, there is a unique way to extend the definition of [6] from metrics satisfying Bismut’s condition (A) to general metrics in such a way that one obtains a theory of singular Bott-Chern classes in the sense of this paper. In his thesis [32], Zha gave another definition of singular Bott-Chern classes, and he also used them to define direct images for closed immersions in Arakelov theory. We will recall the construction of both theories of singular Bott-Chern classes and we will show that they agree with the theory of homogeneous singular Bott-Chern classes. We warn the reader that the normalizations we use differ from the normalizations in [6] and [32]. The main difference is that we insist on using the algebro-geometric twist in cohomology, whereas in the other two papers the authors use cohomology with real coefficients. Let $r_{F}$ and $r_{N}$ be two positive integers. Let $Y$ be a complex manifold and let $\overline{F}$ and $\overline{N}$ be two hermitian vector bundles of rank $r_{F}$ and $r_{N}$ respectively. Let $P=\mathbb{P}(N\oplus\mathbb{C})$ and let $s$ be the zero section. We will follow the notations of definition 5.3. Then $T(K(\overline{F},\overline{N}))$ satisfies the differential equation $\operatorname{d}_{\mathcal{D}}T(K(\overline{F},\overline{N}))=c_{r_{N}}(\overline{Q})\operatorname{Td}^{-1}(\overline{Q})\operatorname{ch}(\pi_{P}^{\ast}\overline{F})-s_{\ast}(\operatorname{ch}(\overline{F})\operatorname{Td}^{-1}(\overline{N})).$ Therefore, the class $\widetilde{e}_{T}(\overline{F},\overline{N}):=T(K(\overline{F},\overline{N}))\bullet\operatorname{Td}(\overline{Q})\bullet\operatorname{ch}^{-1}(\pi_{P}^{\ast}\overline{F})$ satisfies the simpler equation $\operatorname{d}_{\mathcal{D}}\widetilde{e}_{T}(\overline{F},\overline{N})=[c_{r_{N}}(\overline{Q})]-\delta_{Y}.$ (9.1) Observe that the right hand side of this equation belongs to $\mathcal{D}^{2r_{N}}_{D}(P,r_{N})$. Thus it seems natural to introduce the following definition. ###### Definition 9.2. Let $T$ be a theory of singular Bott-Chern classes of rank $r_{F}>0$ and codimension $r_{N}$. Then the class $\widetilde{e}_{T}(\overline{F},\overline{N})=T(K(\overline{F},\overline{N}))\bullet\operatorname{Td}(\overline{Q})\bullet\operatorname{ch}^{-1}(\pi_{P}^{\ast}\overline{F})$ is called the _Euler-Green class associated to_ $T$. The class $T(K(\overline{F},\overline{N}))$ is said to be _homogeneous_ if $\widetilde{e}_{T}(\overline{F},\overline{N})\in\widetilde{\mathcal{D}}^{2r_{N}-1}_{D}(P,r_{N}).$ A theory of singular Bott-Chern classes of rank $0$ is said to be _homogeneous_ if it agrees with the theory of Bott-Chern classes associated to the Chern character. Finally, a theory of singular Bott-Chern classes is said to be _homogeneous_ if its restrictions to all ranks and codimensions are homogeneous. The main interest of the above definition is the following result. ###### Theorem 9.3. Given two positive integers $r_{F}$ and $r_{N}$ there exists a unique theory of homogeneous singular Bott-Chern classes of rank $r_{F}$ and codimension $r_{N}$. ###### Proof. The proof of this result is based on the theory of Euler-Green classes. Let $P=\mathbb{P}(N\oplus\mathbb{C})$ be as before, and let $s$ denote the zero section of $P$. Let $D_{\infty}$ be the subvariety of $P$ that parametrizes the lines contained in $N$. Then $D_{\infty}=\mathbb{P}(N)$. ###### Lemma 9.4. There exists a unique class $\widetilde{e}(P,\overline{Q},s)\in\mathcal{D}^{2r_{N}-1}_{D}(P,r_{N})$ such that 1. (i) It satisfies $\operatorname{d}_{\mathcal{D}}\widetilde{e}(P,\overline{Q},s)=[c_{r_{N}}(\overline{Q})]-\delta_{Y}.$ (9.5) 2. (ii) The restriction $\widetilde{e}(P,\overline{Q},s)\|_{D_{\infty}}=0.$ ###### Proof. We first show the uniqueness. Assume that $\widetilde{e}$ and $\widetilde{e}^{\prime}$ are two classes that satisfy the hypothesis of the theorem. Then $\widetilde{e}^{\prime}-\widetilde{e}$ is closed. Hence it determines a cohomology class in $H^{2r_{N}-1}_{\mathcal{D}^{\text{{\rm an}}}}(P,r_{N})$. Since, by theorem 1.2, the restriction $H^{2r_{N}-1}_{\mathcal{D}^{\text{{\rm an}}}}(P,r_{N})\longrightarrow H^{2r_{N}-1}_{\mathcal{D}^{\text{{\rm an}}}}(D_{\infty},r_{N})$ (9.6) is an isomorphism, condition (ii) implies that $\widetilde{e}^{\prime}=\widetilde{e}$. Now we prove the existence. Since $Y$ is the zero locus of the section $s$, that is transversal to the zero section of $Q$, we know that the currents $[c_{r_{N}}]$ and $\delta_{Y}$ are cohomologous. Therefore there exists an element $\widetilde{a}\in\widetilde{\mathcal{D}}^{2r_{N}-1}_{D}(P,r_{N})$ such that $\operatorname{d}_{\mathcal{D}}\widetilde{a}=[c_{r_{N}}(\overline{Q})]-\delta_{Y}$. Since $\overline{Q}$ restricted to $D_{\infty}$ splits as an orthogonal direct sum $\overline{Q}\|_{D_{\infty}}=\overline{S}\oplus\overline{\mathbb{C}}$ (9.7) where the metric on the factor $\mathbb{C}$ is trivial, and the section $s$ restricts to the constant section $1$, we obtain that $([c_{r_{N}}(\overline{Q})]-\delta_{Y})\|_{D_{\infty}}=0$. Therefore $\widetilde{a}$ determines a class in $H^{2r_{N}-1}_{\mathcal{D}^{\text{{\rm an}}}}(P,r_{N})$. Using again that (9.6) is an isomorphism, we find an element $\widetilde{b}\in H^{2r_{N}-1}_{\mathcal{D}^{\text{{\rm an}}}}(P,r_{N})$, such that $\widetilde{e}=\widetilde{a}-\widetilde{b}$ satisfies the conditions of the lemma. ∎ We continue with the proof of theorem 9.3. We first prove the uniqueness. Let $T$ be a theory of homogeneous singular Bott-Chern classes. The splitting (9.7) implies easily that the restriction of the Koszul resolution $K(\overline{F},\overline{N})$ to $D_{\infty}$ is orthogonally split. By the functoriality of singular Bott-Chern classes, $T(K(\overline{F},\overline{N}))\|_{D_{\infty}}=0$. Thus the class $\widetilde{e}_{T}(\overline{F},\overline{N}):=T(K(\overline{F},\overline{N}))\bullet\operatorname{Td}(\overline{Q})\bullet\operatorname{ch}^{-1}(\pi_{P}^{\ast}\overline{F})\in\widetilde{\mathcal{D}}^{2r_{N}-1}_{D}(P,r_{N})$ satisfies the two conditions of lemma 9.4. Therefore $\widetilde{e}_{T}(\overline{F},\overline{N})=\widetilde{e}(P,\overline{Q},s)$ and $T(K(\overline{F},\overline{N}))=\widetilde{e}(P,\overline{Q},s)\bullet\operatorname{Td}^{-1}(\overline{Q})\bullet\operatorname{ch}(\pi_{P}^{\ast}\overline{F}),$ (9.8) where the right hand side does not depend on the theory $T$. In consequence we have that $C_{T}(F,N)=(\pi_{P})_{\ast}T(K(\overline{F},\overline{N}))$ (9.9) does not depend on the theory $T$. Thus by the uniqueness in theorem 7.1 we obtain the uniqueness here. For the existence we observe ###### Lemma 9.10. The current $C(F,N)=(\pi_{P})_{\ast}(\widetilde{e}(P,\overline{Q},s)\bullet\operatorname{Td}^{-1}(\overline{Q}))\bullet\operatorname{ch}(\overline{F})$ is a characteristic class for pairs of vector bundles of rank $r_{F}$ and $r_{N}$. ###### Proof. We first compute, using equation (9.5) and corollary 3.8, $\displaystyle\operatorname{d}_{\mathcal{D}}C(F,N)$ $\displaystyle=(\pi_{P})_{\ast}\left(\operatorname{d}_{\mathcal{D}}\widetilde{e}(P,\overline{Q},s)\bullet\operatorname{Td}^{-1}(\overline{Q})\right)\bullet\operatorname{ch}(\overline{F})$ $\displaystyle=(\pi_{P})_{\ast}\left(([c_{r_{N}}(\overline{Q})]-\delta_{Y})\bullet\operatorname{Td}^{-1}(\overline{Q})\right)\bullet\operatorname{ch}(\overline{F})$ $\displaystyle=(\pi_{P})_{\ast}\left(c_{r_{N}}(\overline{Q})\bullet\operatorname{Td}^{-1}(\overline{Q})\right)\bullet\operatorname{ch}(\overline{F})-\operatorname{Td}^{-1}(\overline{N})\bullet\operatorname{ch}(\overline{F})$ $\displaystyle=0.$ Thus $C(F,N)$ determines a cohomology class. This class is functorial by construction. By proposition 1.7 this class does not depend on the metric and defines a characteristic class. ∎ By the existence in theorem 7.1 we obtain a theory of singular Bott-Chern classes $T_{C}$ that is easily seen to be homogeneous. ∎ A reformulation of theorem 9.3 is ###### Theorem 9.11. There exists a unique way to associate to each hermitian embedded vector bundle $\overline{\xi}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast})$ a class of currents $T^{h}(\overline{\xi})\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,N^{\ast}_{Y,0},p)$ that we call homogeneous singular Bott-Chern class, satisfying the following properties 1. (i) (Differential equation) The equality $\operatorname{d}_{\mathcal{D}}T^{h}(\overline{\xi})=\sum_{i}(-1)^{i}[\operatorname{ch}(\overline{E}_{i})]-i_{\ast}([\operatorname{Td}^{-1}(\overline{N})\operatorname{ch}(\overline{F})])$ (9.12) holds. 2. (ii) (Functoriality) For every morphism $f\colon X^{\prime}\longrightarrow X$ of complex manifolds that is transverse to $Y$, $f^{\ast}T^{h}(\overline{\xi})=T^{h}(f^{\ast}\overline{\xi}).$ 3. (iii) (Normalization) Let $\overline{A}=(A_{\ast},g_{\ast})$ be a non-negatively graded orthogonally split complex of vector bundles. Write $\overline{\xi}\oplus\overline{A}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast}\oplus\overline{A}_{\ast})$. Then $T^{h}(\overline{\xi})=T^{h}(\overline{\xi}\oplus\overline{A})$. Moreover, if $X=\operatorname{Spec}\mathbb{C}$ is one point, $Y=\emptyset$ and $\overline{E}_{\ast}=0$, then $T^{h}(\overline{\xi})=0$. 4. (iv) (Homogeneity) If $r_{F}=\operatorname{rk}(F)>0$ and $r_{N}=\operatorname{rk}(N)>0$, then, with the notations of definition 9.2, $T^{h}(K(\overline{F},\overline{N}))\bullet\operatorname{Td}(\overline{Q})\bullet\operatorname{ch}^{-1}(\pi_{P}^{\ast}\overline{F})\in\widetilde{\mathcal{D}}^{2r_{N}-1}_{D}(P,r_{N}).$ $\square$ The class $\widetilde{e}(P,\overline{Q},s)$ of lemma 9.4 is a particular case of the Euler-Green classes introduced by Bismut, Gillet and Soulé in [6]. The basic properties of the Euler-Green classes are summarized in the following results. ###### Proposition 9.13. Let $X$ be a complex manifold, let $\overline{E}$ be a hermitian holomorphic vector bundle of rank $r$ and let $s$ be a holomorphic section of $E$ that is transverse to the zero section. Denote by $Y$ the zero locus of $s$. There is a unique way to assign to each $(X,\overline{E},s)$ as before a class of currents $\widetilde{e}(X,\overline{E},s)\in\widetilde{\mathcal{D}}^{2r-1}_{D}(X,N^{\ast}_{Y,0},r)$ satisfying the following properties 1. (i) (Differential equation) $\operatorname{d}_{\mathcal{D}}\widetilde{e}(X,\overline{E},s)=c_{r}(\overline{E})-\delta_{Y}.$ (9.14) 2. (ii) (Functoriality) If $f\colon X^{\prime}\longrightarrow X$ is a morphism transverse to $Y$ then $\widetilde{e}(X^{\prime},f^{\ast}\overline{E},f^{\ast}s)=f^{\ast}\widetilde{e}(X,\overline{E},s).$ (9.15) 3. (iii) (Multiplicativity) Let $\overline{E}_{1}$ and $\overline{E}_{2}$ be hermitian holomorphic vector bundles, and let $s_{1}$ and $s_{2}$ be holomorphic sections of $\overline{E}_{1}$ and $\overline{E}_{2}$ respectively that are transverse to the zero section and with zero locus $Y_{1}$ and $Y_{2}$. We write $\overline{E}=\overline{E}_{1}\oplus\overline{E}_{2}$ and $s=s_{1}\oplus s_{2}$. Assume that $s$ is transverse to the zero section; hence $Y_{1}$ and $Y_{2}$ meet transversely. With this hypothesis we have $\widetilde{e}(X,\overline{E},s)=\widetilde{e}(X,\overline{E}_{1},s_{1})\land c_{r_{2}}(\overline{E}_{2})+\delta_{Y_{1}}\land\widetilde{e}(X,\overline{E}_{2},s_{2})\\\ =\widetilde{e}(X,\overline{E}_{1},s_{1})\land\delta_{Y_{2}}+c_{r_{1}}(\overline{E}_{1})\land\widetilde{e}(X,\overline{E}_{2},s_{2}).$ 4. (iv) (Line bundles) If $\overline{L}$ is a hermitian line bundle and $s$ is a section of $L$, then $\widetilde{e}(X,\overline{L},s)=-\log\\|s\\|.$ (9.16) ###### Proof. Bismut, Gillet and Soulé prove the existence by constructing explicitly an Euler-Green current in the total space of $E$ and pulling it back to $X$ by the section $s$. For the uniqueness, first we see that properties (i) and (ii) imply that, if $h_{0}$ and $h_{1}$ are two hermitian metrics in $E$, then $\widetilde{e}(X,(E,h_{0}),s)-\widetilde{e}(X,(E,h_{1}),s)=\widetilde{c}_{r}(E,h_{0},h_{1}).$ (9.17) We now consider $\pi\colon P=\mathbb{P}(E\oplus\mathbb{C})\longrightarrow X$, with the tautological exact sequence $0\longrightarrow\mathcal{O}(-1)\longrightarrow\pi^{\ast}E\oplus\mathbb{C}\longrightarrow Q\longrightarrow 0$ On $Q$ we consider the metric induced by the metric of $\overline{E}$ and the trivial metric on the factor $\mathbb{C}$, and let $s_{Q}$ the section of $Q$ induced by the section $1$ of $\mathbb{C}$. Let $D_{\infty}$ be as in lemma 9.4. Then properties (ii) to (iv) imply that $\widetilde{e}(P,\overline{Q},s_{Q})\|_{D_{\infty}}=0$. Hence by lemma 9.4 $\widetilde{e}$ is uniquely determined. Finally, let $f\colon X\longrightarrow P$ be the map given by $x\longmapsto(s(x):-1)$. Then $f^{\ast}Q\cong E$, although they are not necessarily isometric, and $f^{\ast}s_{Q}=s$. Therefore, the functoriality and equation (9.17) determine $\widetilde{e}(X,\overline{E},s)$. To prove the existence, we use lemma 9.4, functoriality and equation (9.17) to define the Euler-Green classes. It is easy to show that they are well defined and satisfy properties (i) to (iv). ∎ Equation (9.8) relating homogeneous singular Bott-Chern classes and Euler- Green classes in a particular case can be generalized to arbitrary vector bundles. ###### Proposition 9.18. Let $X$ be a complex manifold, $\overline{E}$ a hermitian vector bundle over $X$, $s$ a section of $E$ transversal to the zero section and $i\colon Y\longrightarrow X$ the zero locus of $s$. Let $K(\overline{E})$ be the Koszul resolution of $i_{\ast}\mathcal{O}_{Y}$ determined by $\overline{E}$ and $s$. We can identify $N_{Y/X}$ with $i^{\ast}E$. We denote by $\overline{N}_{Y/X}$ the vector bundle with the metric induced by the above identification. Then $T^{h}(i,\overline{\mathcal{O}}_{Y},\overline{N}_{Y/X},K(\overline{E}))=\widetilde{e}(X,\overline{E},s)\bullet\operatorname{Td}^{-1}(\overline{E}).$ ###### Proof. Let $P=\mathbb{P}(E\oplus\mathbb{C})$. We follow the notation of proposition 9.13. We denote by $h_{0}$ the original metric of $\overline{E}$ and by $h_{1}$ the metric induced by the isomorphism $E\cong f^{\ast}Q$. Observe that $h_{0}$ and $h_{1}$ agree when restricted to $Y$, because the preimage of $\overline{Q}$ by the zero section agrees with $\overline{E}$. Hence there is an isometry $\overline{N}_{Y/X}\cong i^{\ast}f^{\ast}\overline{Q}$. We denote $T^{h}(K(\overline{E}))=T^{h}(i,\overline{\mathcal{O}}_{Y},\overline{N}_{Y/X},K(\overline{E}))$. Then we have $\displaystyle T^{h}(K(\overline{E}))$ $\displaystyle=f^{\ast}T^{h}(K(\overline{\mathcal{O}_{X}},\overline{E}))+\sum_{i}(-1)^{i}\widetilde{\operatorname{ch}}(\bigwedge^{i}E^{\vee},h_{0},h_{1})$ $\displaystyle=f^{\ast}(\widetilde{e}(P,\overline{Q},s_{Q})\bullet\operatorname{Td}^{-1}(\overline{Q}))+\widetilde{c}_{r}(E,h_{0},h_{1})\bullet\operatorname{Td}^{-1}(E,h_{1})$ $\displaystyle\phantom{AAAA}+c_{r}(E,h_{0})\bullet\widetilde{\operatorname{Td}^{-1}}(E,h_{0},h_{1})$ $\displaystyle=\widetilde{e}(X,\overline{E},s)\bullet\operatorname{Td}^{-1}(E,h_{1})-\widetilde{c}_{r}(E,h_{0},h_{1})\bullet\operatorname{Td}^{-1}(E,h_{1})$ $\displaystyle\phantom{AAAA}+\widetilde{c}_{r}(E,h_{0},h_{1})\bullet\operatorname{Td}^{-1}(E,h_{1})+c_{r}(E,h_{0})\bullet\widetilde{\operatorname{Td}^{-1}}(E,h_{0},h_{1})$ $\displaystyle=\widetilde{e}(X,\overline{E},s)\bullet\operatorname{Td}^{-1}(E,h_{0})-\widetilde{e}(X,\overline{E},s)\bullet\operatorname{d}_{\mathcal{D}}\widetilde{\operatorname{Td}^{-1}}(E,h_{0},h_{1})$ $\displaystyle\phantom{AAAA}+c_{r}(E,h_{0})\bullet\widetilde{\operatorname{Td}^{-1}}(E,h_{0},h_{1})$ $\displaystyle=\widetilde{e}(X,\overline{E},s)\bullet\operatorname{Td}^{-1}(E,h_{0})-\operatorname{d}_{\mathcal{D}}\widetilde{e}(X,\overline{E},s)\bullet\widetilde{\operatorname{Td}^{-1}}(E,h_{0},h_{1})$ $\displaystyle\phantom{AAAA}+c_{r}(E,h_{0})\bullet\widetilde{\operatorname{Td}^{-1}}(E,h_{0},h_{1})$ $\displaystyle=\widetilde{e}(X,\overline{E},s)\bullet\operatorname{Td}^{-1}(E,h_{0})+i_{\ast}\widetilde{\operatorname{Td}^{-1}}(E,h_{0},h_{1})\|_{Y}$ $\displaystyle=\widetilde{e}(X,\overline{E},s)\bullet\operatorname{Td}^{-1}(\overline{E}),$ which concludes the proof. ∎ ###### Theorem 9.19. The theory of homogeneous singular Bott-Chern classes is compatible with the projection formula and transitive. ###### Proof. We have $\displaystyle C_{T^{h}}(F,N)$ $\displaystyle=(\pi_{P})_{\ast}T^{h}(K(\overline{F},\overline{N}))$ $\displaystyle=(\pi_{P})_{\ast}(\widetilde{e}(P,\overline{Q},s)\bullet\operatorname{Td}^{-1}(\overline{Q})\bullet\operatorname{ch}(\pi_{P}^{\ast}\overline{F}))$ $\displaystyle=(\pi_{P})_{\ast}(\widetilde{e}(P,\overline{Q},s)\bullet\operatorname{Td}^{-1}(\overline{Q}))\bullet\operatorname{ch}(\overline{F})$ $\displaystyle=C_{T^{h}}(\mathcal{O}_{Y},N)\bullet\operatorname{ch}(F).$ Thus $C_{T^{h}}$ is compatible with the projection formula. We now prove the transitivity. Let $Y$, $N_{1}$ and $N_{2}$ be as in corollary 8.35. We follow the notation after this corollary. Then applying proposition 9.18 we obtain $T^{h}(\overline{K})=\widetilde{e}(P,\pi_{1}^{\ast}\overline{Q}_{1}\oplus\pi_{2}^{\ast}\overline{Q}_{2},s_{1}+s_{2})\bullet\operatorname{Td}^{-1}(\pi_{1}^{\ast}\overline{Q}_{1}\oplus\pi_{2}^{\ast}\overline{Q}_{2}),$ (9.20) where $s_{i}$ denote the tautological section of $\overline{Q}_{i}$ or its preimage by $\pi_{i}$. Then, by proposition 9.13 (iii), taking into account that $Y_{1}=P_{2}$, $T^{h}(\overline{K})=\pi_{1}^{\ast}(c_{r_{1}}(\overline{Q}_{1})\operatorname{Td}^{-1}(\overline{Q}_{1}))\bullet\pi_{2}^{\ast}(\widetilde{e}(P_{2},\overline{Q}_{2},s_{2})\operatorname{Td}^{-1}(\overline{Q}_{2}))\\\ +(i_{1})_{\ast}(\widetilde{e}(P_{1},\overline{Q}_{1},s_{1})\operatorname{Td}^{-1}(\overline{Q}_{1})\bullet p_{1}^{\ast}\operatorname{Td}^{-1}(\overline{N}_{2})).$ (9.21) Applying again proposition 9.18 we obtain $T^{h}(\overline{K})=\pi_{1}^{\ast}(c_{r_{1}}(\overline{Q}_{1})\operatorname{Td}^{-1}(\overline{Q}_{1}))\bullet\pi_{2}^{\ast}(T^{h}(\overline{K}_{2}))+(i_{1})_{\ast}(T^{h}(\overline{K}_{1})\bullet p_{1}^{\ast}\operatorname{Td}^{-1}(\overline{N}_{2})).$ (9.22) Thus, by corollary 8.35 the theory of homogeneous singular Bott-Chern classes is transitive. ∎ We next recall the construction of singular Bott-Chern classes of Bismut, Gillet and Soulé. Let $i\colon Y\longrightarrow X$ be a closed immersion of complex manifolds and let $\overline{\xi}=(i,\overline{N},\overline{F},\overline{E}_{\ast})$ be a hermitian embedded vector bundle. We consider the associated complex of sheaves $0\to E_{n}\overset{v}{\to}\dots\overset{v}{\to}E_{0}\to 0,$ where we denote by $v$ the differential of this complex. This complex is exact for all $x\in X\setminus Y$. The cohomology sheaves of this complex are holomorphic vector bundles on $Y$ which we denote by $H_{n}=\mathcal{H}_{n}(E_{\ast}\|_{Y}),\quad H=\bigoplus_{n}H_{n}.$ For each $x\in Y$ and $U\in T_{x}X$ we denote by $\partial_{U}v(x)$ the derivative of the map $v$ calculated in any holomorphic trivialization of $E$ near $x$. Then $\partial_{U}v(x)$ acts on $H_{x}$. Moreover, this action only depends on the class $y$ of $U$ in $N_{x}$. We denote it by $\partial_{y}v(x)$. Moreover $(\partial_{y}v(x))^{2}=0$; therefore the pull- back of $H$ to the total space of $N$ together with $\partial_{y}v$ is a complex that we denote by $(H,\partial_{y}v)$. On the total space of $N$, the interior multiplication by $y\in N$ turns $\bigwedge N^{\vee}$ into a Koszul complex. By abuse of notation we denote also by $\iota_{y}$ the operator $\iota_{y}\otimes 1$ acting on $\bigwedge N^{\vee}\otimes F$. There is a canonical isomorphism between the complexes $(H,\partial_{y}v)$ and $(\bigwedge N^{\vee}\otimes F,\iota_{y})$. An explicit description of this isomorphism can be found in [3] §1. Let $v^{\ast}$ be the adjoint of the operator $v$ with respect to the metrics of $\overline{E}_{\ast}$. Then we have an identification of vector bundles over $Y$ $H_{k}=\\{f\in E_{k}\mid vf=v^{\ast}f=0\\}.$ This identification induces a hermitian metric on $H_{k}$, and hence on $H$. Note that the metrics on $N$ and $F$ also induce a hermitian metric on $\bigwedge N^{\vee}\otimes F$. ###### Definition 9.23. We say that $\overline{\xi}=(i,\overline{N},\overline{F},\overline{E}_{\ast})$ satisfies Bismut assumption (A) if the canonical isomorphism between $(H,\partial_{y}v)$ and $(\bigwedge N^{\vee}\otimes F,\iota_{y})$ is an isometry. ###### Proposition 9.24. Let $\overline{\xi}=(i,\overline{N},\overline{F},\overline{E}_{\ast})$ be as before, with $\overline{N}=(N,h_{N})$ and $\overline{F}=(F,h_{F})$. Then there exist metrics $h^{\prime}_{E_{k}}$ over $E_{k}$ such that the hermitian embedded vector bundle $\overline{\xi}^{\prime}=(i,\overline{N},\overline{F},(E_{\ast},h^{\prime}_{E_{\ast}}))$ satisfies Bismut assumption (A). ###### Proof. This is [3] proposition 1.6. ∎ Let $\nabla^{E}$ be the canonical hermitian holomorphic connection on $E$ and let $V=v+v^{\ast}$. Then $A_{u}=\nabla^{E}+\sqrt{u}V$ is a superconnection on $E$. Let $\nabla^{H}$ be the canonical hermitian connection on $H$. Then $B=\nabla^{H}+\partial_{y}v+(\partial_{y}v)^{\ast}$ is a superconnection on $H$. Let $N_{H}$ be the number operator on the complex $(E,v)$, that is, $N_{H}$ acts on $E_{k}$ by multiplication by $k$, and let $\operatorname{Tr}_{s}$ denote the supertrace. Recall that here we are using the symbol $\left[\ \right]$ to denote the current associated to a locally integrable differential form and the symbol $\delta_{Y}$ to denote the current integration along a subvariety, both with the normalizations of notation 1.3. For $0<Re(s)\leq 1/2$ let $\zeta_{E}(s)$ be the current on $X$ given by the formula $\zeta_{E}(s)=\frac{1}{\Gamma(s)}\left.\int_{0}^{\infty}u^{s-1}\right\\{\left[\operatorname{Tr}_{s}\left(N_{H}\exp(-A_{u}^{2})\right)\right]\\\ -\left.i_{\ast}\left[\int_{N}\operatorname{Tr}_{s}\left(N_{H}\exp(-B^{2})\right)\right]\right\\}\operatorname{d}u.$ (9.25) This current is well defined and extends to a current that depends holomorphically on $s$ near $0$. ###### Definition 9.26. Assume that $\overline{\xi}=(i,\overline{N},\overline{F},\overline{E}_{\ast})$ satisfies Bismut assumption (A). Then we denote $T^{BGS}(\overline{\xi})=-\frac{1}{2}\zeta^{\prime}_{E}(0).$ By abuse of notation we will denote also by $T^{BGS}(\overline{\xi})$ its class in $\widetilde{\bigoplus}_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,p).$ Let now $\overline{\xi}=(i,\overline{N},\overline{F},(E_{\ast},h_{E_{\ast}}))$ be general and let $\overline{\xi}^{\prime}=(i,\overline{N},\overline{F},(E_{\ast},h^{\prime}_{E_{\ast}}))$ be any hermitian embedded vector bundle satisfying assumption (A) provided by proposition 9.24. Then we denote $T^{BGS}(\overline{\xi})=T^{BGS}(\overline{\xi}^{\prime})+\sum_{i}(-1)^{i}\widetilde{\operatorname{ch}}(E_{i},h_{E_{i}},h^{\prime}_{E_{i}}),$ where $\widetilde{\operatorname{ch}}(E_{i},h_{E_{i}},h^{\prime}_{E_{i}})$ is as in definition 2.13. ###### Remark 9.27. This definition only agrees (up to a normalization factor) with the definition in [6] for hermitian embedded vector bundles that satisfy assumption (A). ###### Theorem 9.28. The assignment that, to each hermitian embedded vector bundle $\overline{\xi}$, associates the current $T^{BGS}(\overline{\xi})$, is a theory of singular Bott-Chern classes that agrees with $T^{h}$. ###### Proof. First we have to show that, when $\overline{\xi}$ does not satisfy assumption (A) then $T^{BGS}(\overline{\xi})$ is well defined. Assume that $\overline{\xi}^{\prime\prime}=(i,\overline{N},\overline{F},(E_{\ast},h^{\prime}_{E_{\ast}}))$ is another choice of hermitian embedded vector bundle satisfying assumption (A). By lemma 2.17 we have that $\widetilde{\operatorname{ch}}(E_{i},h_{i},h^{\prime}_{i})+\widetilde{\operatorname{ch}}(E_{i},h^{\prime}_{i},h^{\prime\prime}_{i})+\widetilde{\operatorname{ch}}(E_{i},h^{\prime\prime}_{i},h_{i})=0.$ By [6] theorem 2.5 we have that $T^{BGS}(\overline{\xi}^{\prime})-T^{BGS}(\overline{\xi}^{\prime\prime})=\sum_{i}(-1)^{i}\widetilde{\operatorname{ch}}(E_{i},h^{\prime}_{E_{i}},h^{\prime\prime}_{E_{i}}).$ Summing up we obtain that $T^{BGS}(\overline{\xi})$ is well defined. If the hermitian embedded vector bundle $\overline{\xi}$ satisfies Bismut assumption (A) then, by [6] theorem 1.9, $T^{BGS}(\overline{\xi})$ satisfies equation (6.10). If $\overline{\xi}$ does not satisfy assumption (A) then, combining [6] theorem 1.9 and equation (2.4), we also obtain that $T^{BGS}(\overline{\xi})$ satisfies equation (6.10). The functoriality property is [6] theorem 1.10. In order to prove the normalization property, let $\overline{\xi}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast})$ be a hermitian embedded vector bundle that satisfies assumption (A) and let $\overline{A}$ be a non-negatively graded orthogonally split complex of vector bundles on $X$. Observe that $\overline{A}$ is also a (trivial) hermitian embedded vector bundle. Then $\overline{A}$ and $\overline{\xi}\oplus\overline{A}$ also satisfy assumption (A). By [6] theorem 2.9 $T^{BGS}(\overline{\xi}\oplus\overline{A})=T^{BGS}(\overline{\xi})+T^{BGS}(\overline{A}).$ But by [5] remark 2.3, $T^{BGS}(\overline{A})$ agrees with the Bott-Chern class associated to the Chern character and the exact complex $\overline{A}$. Since $A$ is orthogonally split we have $T^{BGS}(\overline{A})=0$. Now the case when $\xi$ does not satisfy assumption (A) follows from the definition. By [6] theorem 3.17, with the hypothesis of proposition 9.18, we have that $\displaystyle T^{BGS}(i,\overline{\mathcal{O}}_{Y},\overline{N}_{Y/X},K(\overline{E}))$ $\displaystyle=\widetilde{e}(X,\overline{E},s)\bullet\operatorname{Td}^{-1}(\overline{E})$ $\displaystyle=T^{h}(i,\overline{\mathcal{O}}_{Y},\overline{N}_{Y/X},K(\overline{E})).$ From this it follows that $C_{T^{BGS}}=C_{T^{h}}$ and by theorem 7.1, $T^{BGS}=T^{h}$. ∎ We now recall Zha’s construction. Note that, in order to obtain a theory of singular Bott-Chern classes, we have changed the normalization convention from the one used by Zha. Note also that Zha does not define explicitly a singular Bott-Chern class, but such a definition is implicit in his definition of direct images for closed immersions. Let $Y$ be a complex manifold and let $\overline{N}=(N,h)$ be a hermitian vector bundle. We denote $P=\mathbb{P}(N\oplus\mathbb{C})$. Let $\pi\colon P\longrightarrow Y$ denote the projection and let $\iota\colon Y\longrightarrow P$ denote the inclusion as the zero section. On $P$ we consider the tautological exact sequence $0\longrightarrow\mathcal{O}(-1)\longrightarrow\pi^{\ast}N\oplus\mathcal{O}_{P}\longrightarrow Q\longrightarrow 0.$ Let $h_{1}$ denote the hermitian metric on $Q^{\vee}$ induced by the metric of $N$ and the trivial metric on $\mathcal{O}_{P}$ and let $h_{0}$ denote the semi-definite hermitian form on $Q^{\vee}$ induced by the map $Q^{\vee}\longrightarrow\mathcal{O}_{P}$ obtained from the above exact sequence and the trivial metric on $\mathcal{O}_{P}$. Let $h_{t}=(1-t^{2})h_{0}+t^{2}h_{1}$. It is a hermitian metric on $Q^{\vee}$. We will denote $\overline{Q}_{t}^{\vee}=(Q^{\vee},h_{t})$. Let $\nabla_{t}$ be the associated hermitian holomorphic connection and let $N_{t}$ denote the endomorphism defined by $\frac{\operatorname{d}}{\operatorname{d}t}\left<v,w\right>_{t}=\left<N_{t}v,w\right>.$ For each $n\geq 1$, let $\operatorname{Det}$ denote the alternate $n$-linear form on the space of $n$ by $n$ matrices such that $\det(A)=\operatorname{Det}(A,\dots,A).$ We denote $\det(B;A)=\operatorname{Det}(B,A,\dots,A)$. Zha introduced the differential form $\widetilde{e}_{Z}(\overline{Q}^{\vee})=\frac{-1}{2}\lim_{s\rightarrow 0}\int_{s}^{1}\det(N_{t},\nabla_{t}^{2})\operatorname{d}t$ (9.29) which is a smooth form on $P\setminus\iota(Y)$, locally integrable on $P$. Hence it defines a current, also denoted by $\widetilde{e}_{Z}(\overline{Q}^{\vee})$ on $P$. The important property of this current is that it satisfies $\operatorname{d}_{\mathcal{D}}\overline{e}_{Z}(Q^{\vee})=c_{n}(\overline{Q}_{1})-\delta_{Y}.$ (9.30) In [32], Zha denotes by $C(\overline{Q}^{\vee})$ a form that differs from $\widetilde{e}_{Z}$ by the normalization factor and the sign. We denote it by $\widetilde{e}_{Z}$ because it agrees with the Euler-Green current introduced in [6]. ###### Proposition 9.31. The equality $\widetilde{e}_{Z}(Q^{\vee})=\widetilde{e}(P,\overline{Q}_{1},s_{Q})$ holds. ###### Proof. With the notations of lemma 9.4, both classes satisfy equation (9.30) and their restriction to $D_{\infty}$ is zero. By lemma 9.4 they agree. ∎ ###### Definition 9.32. Let $\overline{\xi}=(i\colon Y\longrightarrow X,\overline{N},\overline{F},\overline{E}_{\ast})$ be as in definition 6.9. Let $\overline{A}_{\ast}$, $\operatorname{tr}_{1}(\overline{E})_{\ast}$ and $\overline{\eta}_{\ast}$ be as in (7.2). Then we define $T^{Z}(\overline{\xi})=-(p_{W})_{\ast}\left(\sum_{k}(-1)^{k}W_{1}\bullet\operatorname{ch}(\operatorname{tr}_{1}(\overline{E})_{k})\right)\\\ -\sum_{k}(-1)^{k}(p_{P})_{\ast}[\widetilde{\operatorname{ch}}(\overline{\eta}_{k})]+(p_{P})_{\ast}(\operatorname{ch}(\pi_{p}^{\ast}\overline{F})\operatorname{Td}^{-1}(\overline{Q}_{1})\widetilde{e}_{Z}(\overline{Q}_{1}^{\vee})).$ (9.33) It follows directly from the definition that $T^{Z}$ is the theory of singular Bott-Chern classes associated to the class $C_{Z}(F,N)=(p_{P})_{\ast}(\operatorname{ch}(\pi_{p}^{\ast}\overline{F})\operatorname{Td}^{-1}(\overline{Q}_{1})\widetilde{e}_{Z}(\overline{Q}_{1}^{\vee})).$ (9.34) ###### Theorem 9.35. The theory of singular Bott-Chern classes $T^{Z}$ agrees with the theory of homogeneous singular Bott-Chern classes $T^{h}$. ###### Proof. The result follows directly from theorem 7.1, equation (9.34) and proposition 9.18. ∎ Next we want to use 8.33 to give another characterization of $T^{h}$. To this end we only need to compute the characteristic class $C_{T^{h}}(\mathcal{O}_{Y},L)$ for a line bundle $L$ as a power series in $c_{1}(L)$. ###### Theorem 9.36. The theory of homogeneous singular Bott-Chern classes of algebraic vector bundles is the unique theory of singular Bott-Chern classes of algebraic vector bundles that is compatible with the projection formula and transitive and that satisfies $C_{T^{h}}(\mathcal{O}_{Y},L)={\bf 1}_{1}\bullet\phi(c_{1}(L)),$ where $\phi$ is the power series $\phi(x)=\frac{1}{2}\sum_{n=0}^{\infty}\frac{(-1)^{n+1}H_{n+1}}{(n+2)!}x^{n},$ and where $H_{n}=1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n}$, $n\geq 1$ are the harmonic numbers. We already know that $T^{h}$ is compatible with the projection formula and transitive. Thus it only remains to compute the power series $\phi$. Let $\overline{L}=(L,h_{L})$ be a hermitian line bundle over a complex manifold $Y$. Let $z$ be a system of holomorphic coordinates of $Y$. Let $e$ be a local section of $L$ and let $h(z)=h(e_{z},e_{z})$. Let $P=\mathbb{P}(L\oplus\mathbb{C})$, with $\pi\colon P\longrightarrow Y$ the projection and $\iota\colon Y\longrightarrow P$ the zero section. We choose homogeneous coordinates on $P$ given by $(z,(x:y))$, here $(x:y)$ represents the line of $L_{z}\oplus\mathbb{C}$ generated by $xe(z)+y\mathbf{1}$, where $\mathbf{1}$ is a generator of $\mathbb{C}$ of norm 1. On the open set $y\not=0$ we will use the absolute coordinate $t=x/y$. Let $0\longrightarrow\mathcal{O}(-1)\longrightarrow\pi^{\ast}(L\oplus\mathbb{C})\longrightarrow Q\longrightarrow 0$ be the tautological exact sequence. The section $s=\\{\mathbf{1}\\}$ is a global section of $Q$ that vanishes along the zero section. Moreover we have $\\|s\\|^{2}_{(z,(x:y))}=\frac{x\bar{x}h(z)}{y\bar{y}+x\bar{x}h(z)}=\frac{t\bar{t}h}{1+t\bar{t}h}.$ Then (recall that we are using the algebro-geometric normalization) $\displaystyle c_{1}(\overline{Q})$ $\displaystyle=\partial\bar{\partial}\log\\|s\\|^{2}$ (9.37) $\displaystyle=\partial\bar{\partial}\log\frac{t\bar{t}h}{1+t\bar{t}h}$ (9.38) $\displaystyle=\partial\left(\frac{1+t\bar{t}h}{t\bar{t}h}\frac{t\bar{\partial}(\bar{t}h)(1+t\bar{t}h)-t^{2}\bar{t}h\bar{\partial}(\bar{t}h)}{(1+t\bar{t}h)^{2}}\right)$ (9.39) $\displaystyle=\partial\left(\frac{t\bar{\partial}(\bar{t}h)}{t\bar{t}h(1+t\bar{t}h)}\right)$ (9.40) $\displaystyle=\partial\left(\frac{\bar{\partial}(\bar{t}h)}{\bar{t}h}\right)\frac{1}{1+t\bar{t}h}-\frac{\bar{t}\partial(ht)\land\bar{\partial}(\bar{t}h)}{\bar{t}h(1+t\bar{t}h)^{2}}$ (9.41) $\displaystyle=\frac{\pi^{\ast}c_{1}(\overline{L})}{1+t\bar{t}h}-\frac{\partial(th)\land\bar{\partial}(\bar{t}h)}{h(1+t\bar{t}h)^{2}}.$ (9.42) We now consider the Koszul resolution $\overline{K}\colon 0\longrightarrow Q^{\vee}\overset{s}{\longrightarrow}\mathcal{O}_{p}\longrightarrow\iota_{\ast}\mathcal{O}_{X}\longrightarrow 0.$ We denote by $T^{h}(\overline{K})$ the singular Bott-Chern class associated to this Koszul complex. Then, by proposition 9.13 and proposition 9.18, $T^{h}(\overline{K})=-\frac{1}{2}\operatorname{Td}^{-1}(\overline{Q})\log\\|s\\|^{2}.$ In order to compute $\pi_{\ast}T^{h}(\overline{K})$ we have to compute first $\pi_{\ast}c_{1}(\overline{Q})^{n}\log\\|s\\|^{2}$. But $c_{1}(\overline{Q})^{n}=\frac{\pi^{\ast}c_{1}(\overline{L})^{n}}{(1+t\bar{t}h)^{n}}-n\left(\frac{\pi^{\ast}c_{1}(\overline{L})}{(1+t\bar{t}h)}\right)^{n-1}\frac{\partial(th)\land\bar{\partial}(\bar{t}h)}{h(1+t\bar{t}h)^{2}}.$ Therefore $\displaystyle\pi_{\ast}c_{1}(\overline{Q})^{n}\log\\|s\\|^{2}$ $\displaystyle=-nc_{1}(\overline{L})^{n-1}\frac{1}{2\pi i}\int_{\mathbb{P}^{1}}\frac{\partial(th)\land\bar{\partial}(\bar{t}h)}{h(1+t\bar{t}h)^{n+1}}\log\frac{t\bar{t}h}{1+t\bar{t}h}$ $\displaystyle=-nc_{1}(\overline{L})^{n-1}\frac{1}{2\pi i}\int_{0}^{2\pi}\int_{0}^{\infty}\log\frac{r^{2}}{1+r^{2}}\frac{-2ir\operatorname{d}\theta\operatorname{d}r}{(1+r^{2})^{n+1}}$ $\displaystyle=nc_{1}(\overline{L})^{n-1}\int_{0}^{1}\log(1-w)w^{n-1}\operatorname{d}w$ $\displaystyle=-c_{1}(\overline{L})^{n-1}H_{n},$ where $H_{n}$, $n\geq 1$ are the harmonic numbers. Since $Td^{-1}(\overline{Q})=\frac{1-\exp(-c_{1}(\overline{Q}))}{c_{1}(\overline{Q})}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+1)!}c_{1}(\overline{Q})^{n},$ we obtain $\displaystyle C_{T^{h}}(\mathcal{O}_{Y},L)=\pi_{\ast}T^{h}(\overline{K})=\frac{1}{2}\sum_{n=0}^{\infty}\frac{(-1)^{n+1}H_{n+1}}{(n+2)!}c_{1}(\overline{L})^{n}{\bf 1}_{1}.$ Then, a reformulation of proposition 8.31 is ###### Corollary 9.43. Let $T$ be a theory of singular Bott-Chern classes for algebraic vector bundles that is compatible with the projection formula and transitive. Then there is a unique additive genus $S_{T}$ such that $C_{T}(F,N)-C_{T^{h}}(F,N)=\operatorname{ch}(F)\bullet\operatorname{Td}(N)^{-1}\bullet S_{T}(N).$ (9.44) Conversely, any additive genus determines a theory of singular Bott-Chern classes by the formula (9.44). ## 10 The arithmetic Riemann-Roch theorem for regular closed immersions In this section we recall the definition of arithmetic Chow groups and arithmetic $K$-groups. We see that each choice of an additive theory of singular Bott-Chern classes allows us to define direct images for closed immersions in arithmetic $K$-theory. Once the direct images for closed immersions are defined, we prove the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions. A version of this theorem was proved earlier by Bismut, Gillet and Soulé [6] when there is a commutative diagram $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\crcr}}}\ignorespaces{\hbox{\kern-6.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 14.5442pt\raise 5.30833pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.30833pt\hbox{$\scriptstyle{i}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 18.66052pt\raise-12.30556pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.44444pt\raise-30.89012pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 37.5pt\raise-18.41666pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 37.5pt\raise-27.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 31.44444pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathcal{Z}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$ where $i$ is a closed immersion and $f$ and $g$ are smooth over $\mathbb{C}$. The version of this theorem given in this paper is due to Zha [32], but still unpublished. The theorem of Bismut, Gillet and Soulé compares $g_{\ast}\operatorname{\widehat{ch}}(i_{\ast}\overline{E})$ with $f_{\ast}\operatorname{\widehat{ch}}(\overline{E})$, whereas the theorem of Zha compares directly $\operatorname{\widehat{ch}}(i_{\ast}\overline{E})$ with $i_{\ast}\operatorname{\widehat{ch}}(\overline{E})$. The main difference between the theorem of Bismut, Gillet and Soulé and that of Zha is the kind of arithmetic Chow groups they use. In the first case these groups are only covariant for proper morphisms that are smooth over $\mathbb{C}$; thus the Grothendieck-Riemann-Roch can only be stated for a diagram as above, while in the second case a version of these groups that are covariant for arbitrary proper morphisms is used. Since each choice of a theory of singular Bott-Chern classes gives rise to a different definition of direct images for closed immersions, the arithmetic Grothendieck-Riemann-Roch theorem will have a correction term that depends on the theory of singular Bott-Chern classes used. In the particular case of the homogeneous singular Bott-Chern classes, which are the theories used by Bismut, Gillet and Soulé and by Zha, this correction term vanishes and we obtain the simplest formula. In this case the arithmetic Grothendieck-Riemann- Roch theorem is formally identical to the classical one. Let $(A,\Sigma,F_{\infty})$ be an arithmetic ring [18]. Since we will allow the arithmetic varieties to be non regular and we will use Chow groups indexed by dimension, following [20] we will assume that the ring $A$ is equidimensional and Jacobson. Let $F$ be the field of fractions of A. An arithmetic variety $\mathcal{X}$ is a scheme flat and quasi-projective over $A$ such that $\mathcal{X}_{F}=\mathcal{X}\times\operatorname{Spec}F$ is smooth. Then $X:=\mathcal{X}_{\Sigma}$ is a complex algebraic manifold, which is endowed with an anti-holomorphic automorphism $F_{\infty}$. One also associates to $\mathcal{X}$ the real variety $X_{\mathbb{R}}=(X,F_{\infty})$. Following [13], to each regular arithmetic variety we can associate different kinds of arithmetic Chow groups. Concerning arithmetic Chow groups, we shall use the terminology and notation in op. cit. §4 and §6. Let $\mathcal{D}_{\log}$ be the Deligne complex of sheaves defined in [13] section 5.3; we refer to op. cit. for the precise definition and properties. A $\mathcal{D}_{\log}$-arithmetic variety is a pair $(\mathcal{X},\mathcal{C})$ consisting of an arithmetic variety $\mathcal{X}$ and a complex of sheaves $\mathcal{C}$ on $X_{\mathbb{R}}$ which is a $\mathcal{D}_{\log}$-complex (see op. cit. section 3.1). We are interested in the following $\mathcal{D}_{\log}$-complexes of sheaves: 1. (i) The Deligne complex $\mathcal{D}_{{\text{\rm l,a}},X}$ of differential forms on $X$ with logarithmic and arbitrary singularities. That is, for every Zariski open subset $U$ of $X$, we write $E^{\ast}_{{\text{\rm l,a}},X}(U)=\lim_{\begin{subarray}{c}\longrightarrow\\\ \overline{U}\end{subarray}}\Gamma(\overline{U},\mathscr{E}^{\ast}_{\overline{U}}(\log B)),$ where the limit is taken over all diagrams $\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\iota}}$$\scriptstyle{\iota}$$\textstyle{\overline{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{X}$ such that $\overline{\iota}$ is an open immersion, $\beta$ is a proper morphism, $B=\overline{U}\setminus U$, is a normal crossing divisor and $\mathscr{E}^{\ast}_{\overline{U}}(\log B)$ denotes the sheaf of smooth differential forms on $U$ with logarithmic singularities along $B$ introduced in [8] . For any Zariski open subset $U\subseteq X$, we put $\mathcal{D}^{\ast}_{{\text{\rm l,a}},X}(U,p)=(\mathcal{D}^{\ast}_{{\text{\rm l,a}},X}(U,p),\operatorname{d}_{\mathcal{D}})=(\mathcal{D}^{\ast}(E_{{\text{\rm l,a}},X}(U),p),\operatorname{d}_{\mathcal{D}}).$ If $U$ is now a Zariski open subset of $X_{\mathbb{R}}$, then we write $\mathcal{D}^{\ast}_{{\text{\rm l,a}},X}(U,p)=(\mathcal{D}^{\ast}_{{\text{\rm l,a}},X}(U,p),\operatorname{d}_{\mathcal{D}})=(\mathcal{D}^{\ast}_{{\text{\rm l,a}},X}(U_{\mathbb{C}},p)^{\sigma},\operatorname{d}_{\mathcal{D}}),$ where $\sigma$ is the involution $\sigma(\eta)=\overline{F_{\infty}^{\ast}\eta}$ as in [13] notation 5.65. Note that the sections of $\mathcal{D}^{\ast}_{{\text{\rm l,a}},X}$ over an open set $U\subset X$ are differential forms on $U$ with logarithmic singularities along $X\setminus U$ and arbitrary singularities along $\overline{X}\setminus X$, where $\overline{X}$ is an arbitrary compactification of $X$. Therefore the complex of global sections satisfy $\mathcal{D}^{\ast}_{{\text{\rm l,a}},X}(X,)=\mathcal{D}^{\ast}(X,\ast),$ where the right hand side complex has been introduced in section §1. The complex $\mathcal{D}^{\ast}_{{\text{\rm l,a}},X}$ is a particular case of the construction of [12] section 3.6. 2. (ii) The Deligne complex $\mathcal{D}_{\text{{\rm cur}},X}$ of currents on $X$. This is the complex introduced in [13] definition 6.30. When $\mathcal{X}$ is regular, applying the theory of [13] we can define the arithmetic Chow groups $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})$ and $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$. These groups satisfy the following properties 1. (i) There are natural morphisms $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})\longrightarrow\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$ and, when applicable, all properties below will be compatible with these morphisms. 2. (ii) There is a product structure that turns $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})_{{\mathbb{Q}}}$ into an associative and commutative algebra. Moreover, it turns $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})_{{\mathbb{Q}}}$ into a $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})_{{\mathbb{Q}}}$-module. 3. (iii) If $f\colon\mathcal{Y}\longrightarrow\mathcal{X}$ is a map of regular arithmetic varieties, there are pull-back morphisms $f^{\ast}\colon\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})\longrightarrow\operatorname{\widehat{CH}}^{\ast}(\mathcal{Y},\mathcal{D}_{{\text{\rm l,a}},Y}).$ If moreover, $f$ is smooth over $F$, there are pull-back morphisms $f^{\ast}\colon\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})\longrightarrow\operatorname{\widehat{CH}}^{\ast}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y}).$ The inverse image is compatible with the product structure. 4. (iv) If $f\colon\mathcal{Y}\longrightarrow\mathcal{X}$ is a proper map of regular arithmetic varieties of relative dimension $d$, there are push-forward morphisms $f_{\ast}\colon\operatorname{\widehat{CH}}^{\ast}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})\longrightarrow\operatorname{\widehat{CH}}^{\ast-d}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X}).$ If moreover, $f$ is smooth over $F$, there are push-forward morphisms $f_{\ast}\colon\operatorname{\widehat{CH}}^{\ast}(\mathcal{Y},\mathcal{D}_{{\text{\rm l,a}},Y})\longrightarrow\operatorname{\widehat{CH}}^{\ast-d}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X}).$ The push-forward morphism satisfies the projection formula and is compatible with base change. 5. (v) The groups $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})$ are naturally isomorphic to the groups defined by Gillet and Soulé in [18] (see [12] theorem 3.33). When $X$ is generically projective, the groups $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$ are isomorphic to analogous groups introduced by Kawaguchi and Moriwaki [27] and are very similar to the weak arithmetic Chow groups introduced by Zha (see [11]). 6. (vi) There are well-defined maps $\displaystyle\zeta$ $\displaystyle\colon\operatorname{\widehat{CH}}^{p}(\mathcal{X},{\mathcal{C}})\longrightarrow\operatorname{CH}^{p}(\mathcal{X}),$ $\displaystyle\operatorname{a}$ $\displaystyle\colon\widetilde{{\mathcal{C}}}^{2p-1}(X_{\mathbb{R}},p)\longrightarrow\operatorname{\widehat{CH}}^{p}(\mathcal{X},{\mathcal{C}}),$ $\displaystyle\omega$ $\displaystyle\colon\operatorname{\widehat{CH}}^{p}(\mathcal{X},{\mathcal{C}})\longrightarrow{\rm Z}{\mathcal{C}}^{2p}(X_{\mathbb{R}},p),$ where ${\mathcal{C}}$ is either $\mathcal{D}_{{\text{\rm l,a}},X}$ or $\mathcal{D}_{\text{{\rm cur}},X}$. For the precise definition of these maps see [13] notation 4.12. When $\mathcal{X}$ is not necessarily regular, following [20] and combining with the definition of [13] we can define the arithmetic Chow groups indexed by dimension $\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})$ and $\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$ (see [12] section 5.3). They have the following properties (see [20]). 1. (i) If $\mathcal{X}$ is regular and equidimensional of dimension $n$ then there are isomorphisms $\displaystyle\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})$ $\displaystyle\cong\operatorname{\widehat{CH}}^{n-\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X}),$ $\displaystyle\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$ $\displaystyle\cong\operatorname{\widehat{CH}}^{n-\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X}).$ 2. (ii) If $f\colon\mathcal{Y}\longrightarrow\mathcal{X}$ is a proper map between arithmetic varieties then there is a push-forward map $f_{\ast}\colon\operatorname{\widehat{CH}}_{\ast}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})\longrightarrow\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X}).$ If $f$ is smooth over $F$ then there is a push-forward map $f_{\ast}\colon\operatorname{\widehat{CH}}_{\ast}(\mathcal{Y},\mathcal{D}_{{\text{\rm l,a}},Y})\longrightarrow\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X}).$ 3. (iii) If $f\colon\mathcal{Y}\longrightarrow\mathcal{X}$ is a flat map or, more generally, a local complete intersection (l.c.i) map of relative dimension $d$, there are pull-back morphisms $f^{\ast}\colon\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})\longrightarrow\operatorname{\widehat{CH}}_{\ast+d}(\mathcal{Y},\mathcal{D}_{{\text{\rm l,a}},Y}).$ If moreover, $f$ is smooth over $F$, there are pull-back morphisms $f^{\ast}\colon\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})\longrightarrow\operatorname{\widehat{CH}}_{\ast+d}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y}).$ 4. (iv) If $f\colon\mathcal{Y}\longrightarrow\mathcal{X}$ is a morphism of arithmetic varieties with $\mathcal{X}$ regular, then there is a cap product $\operatorname{\widehat{CH}}^{p}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})\otimes\operatorname{\widehat{CH}}_{d}(\mathcal{Y},\mathcal{D}_{{\text{\rm l,a}},Y})\longrightarrow\operatorname{\widehat{CH}}_{d-p}(\mathcal{Y},\mathcal{D}_{{\text{\rm l,a}},Y})_{{\mathbb{Q}}},$ and a similar cap-product with the groups $\operatorname{\widehat{CH}}_{d}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})$. This product is denoted by $y\otimes x\mapsto y._{f}x$, For more properties of these groups see [20]. We will define now the arithmetic $K$-groups in this context. As a matter of convention, in the sequel we will use slanted letters to denote a object defined over $A$ and the same letter in roman type for the corresponding object defined over $\mathbb{C}$. For instance we will denote a vector bundle over $\mathcal{X}$ by $\mathcal{E}$ and the corresponding vector bundle over $X$ by $E$. ###### Definition 10.1. A _hermitian vector bundle_ on an arithmetic variety $\mathcal{X}$, $\overline{\mathcal{E}}$, is a locally free sheaf $\mathcal{E}$ with a hermitian metric $h_{E}$ on the vector bundle $E$ induced on $X$, that is invariant under $F_{\infty}$. A sequence of hermitian vector bundles on $\mathcal{X}$ $(\overline{\varepsilon})\qquad\ldots\longrightarrow\overline{\mathcal{E}}_{n+1}\longrightarrow\overline{\mathcal{E}}_{n}\longrightarrow\overline{\mathcal{E}}_{n-1}\longrightarrow\ldots$ is said to be exact if it is exact as a sequence of vector bundles. A _metrized coherent sheaf_ is a pair $\overline{\mathcal{F}}=(\mathcal{F},\overline{E}_{\ast}\to F)$, where $\mathcal{F}$ is a coherent sheaf on $\mathcal{X}$ and $\overline{E}_{\ast}\to F$ is a resolution of the coherent sheaf $F=\mathcal{F}_{{\mathbb{C}}}$ by hermitian vector bundles, that is defined over $\mathbb{R}$, hence is invariant under $F_{\infty}$. We assume that the hermitian metrics are also invariant under $F_{\infty}$. Recall that to every hermitian vector bundle we can associate a collection of Chern forms, denoted by $c_{p}$. Moreover, the invariance of the hermitian metric under $F_{\infty}$ implies that the Chern forms will be invariant under the involution $\sigma$. Thus $c_{p}(\overline{\mathcal{E}})\in\mathcal{D}^{2p}_{{\text{\rm l,a}},X}(X_{\mathbb{R}},p)=\mathcal{D}^{2p}(X,p)^{\sigma}.$ We will denote also by $c_{p}(\overline{\mathcal{E}})$ its image in $\mathcal{D}^{2p}_{\text{{\rm cur}},X}(X_{\mathbb{R}},p)$. In particular we have defined the Chern character $\operatorname{ch}(\overline{\mathcal{E}})$ in either of the groups $\bigoplus_{p}\mathcal{D}^{2p}_{{\text{\rm l,a}},X}(X_{\mathbb{R}},p)$ or $\bigoplus_{p}\mathcal{D}^{2p}_{\text{{\rm cur}},X}(X_{\mathbb{R}},p)$. Moreover, to each finite exact sequence $(\overline{\varepsilon})$ of hermitian vector bundles on $\mathcal{X}$ we can attach a secondary Bott-Chern class $\widetilde{\operatorname{ch}}(\overline{\varepsilon})$. Again, the fact that the sequence is defined over $A$ and the invariance of the metrics with respect to $F_{\infty}$ imply that $\widetilde{\operatorname{ch}}(\overline{\varepsilon})\in\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{{\text{\rm l,a}},X}(X_{\mathbb{R}},p)=\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p)^{\sigma}.$ We will denote also by $\widetilde{\operatorname{ch}}(\overline{\varepsilon})$ its image in $\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{\text{{\rm cur}},X}(X_{\mathbb{R}},p)$. The Bott-Chern classes associated to exact sequences of metrized coherent sheaves enjoy the same properties. ###### Definition 10.2. Let $\mathcal{X}$ be an arithmetic variety and let $\mathcal{C}^{\ast}(\ast)$ be one of the two $\mathcal{D}_{\log}$-complexes $\mathcal{D}_{{\text{\rm l,a}},X}$ or $\mathcal{D}_{\text{{\rm cur}},X}$. The arithmetic $K$-group associated to the $\mathcal{D}_{\log}$-arithmetic variety $(\mathcal{X},\mathcal{C})$ is the abelian group $\widehat{K}(\mathcal{X},\mathcal{C})$ generated by pairs $(\overline{\mathcal{E}},\eta)$, where $\overline{\mathcal{E}}$ is a hermitian vector bundle on $\mathcal{X}$ and $\eta\in\bigoplus_{p\geq 0}\widetilde{\mathcal{C}}^{2p-1}(X_{\mathbb{R}},p)$, modulo relations $(\overline{\mathcal{E}}_{1},\eta_{1})+(\overline{\mathcal{E}}_{2},\eta_{2})=(\overline{\mathcal{E}},\tilde{\operatorname{ch}}(\overline{\varepsilon})+\eta_{1}+\eta_{2})$ (10.3) for each short exact sequence $(\overline{\varepsilon})\qquad 0\longrightarrow\overline{\mathcal{E}}_{1}\longrightarrow\overline{\mathcal{E}}\longrightarrow\overline{\mathcal{E}}_{2}\longrightarrow 0\ .$ The arithmetic $K^{\prime}$-group associated to the $\mathcal{D}_{\log}$-arithmetic variety $(\mathcal{X},\mathcal{C})$ is the abelian group $\widehat{K}^{\prime}(\mathcal{X},\mathcal{C})$ generated by pairs $(\overline{\mathcal{F}},\eta)$, where $\overline{\mathcal{F}}$ is a metrized coherent sheaf on $\mathcal{X}$ and $\eta\in\bigoplus_{p\geq 0}\widetilde{\mathcal{C}}^{2p-1}(X_{\mathbb{R}},p)$, modulo relations $(\overline{\mathcal{F}}_{1},\eta_{1})+(\overline{\mathcal{F}}_{2},\eta_{2})=(\overline{\mathcal{F}},\tilde{\operatorname{ch}}(\overline{\varepsilon})+\eta_{1}+\eta_{2})$ (10.4) for each short exact sequence of metrized coherent sheaves $(\overline{\varepsilon})\qquad 0\longrightarrow\overline{\mathcal{F}}_{1}\longrightarrow\overline{\mathcal{F}}\longrightarrow\overline{\mathcal{F}}_{2}\longrightarrow 0\ .$ We now give some properties of the arithmetic $K$-groups. As their proofs are similar, in the essential points, to those of analogous statements in, for example, [18] in the regular case and [20] in the singular case, we omit them. 1. (i) We have natural morphisms $\widehat{K}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})\longrightarrow\widehat{K}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})\text{ and }\widehat{K}^{\prime}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})\longrightarrow\widehat{K}^{\prime}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X}).$ When applicable, all properties below will be compatible with these morphisms. 2. (ii) $\widehat{K}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})$ is a ring. The product structure is given by $(\overline{\mathcal{F}}_{1},\eta_{1})\cdot(\overline{\mathcal{F}}_{2},\eta_{2})=(\overline{\mathcal{F}}_{1}\otimes\overline{\mathcal{F}}_{2},\operatorname{ch}(\overline{\mathcal{F}}_{1})\bullet\eta_{2}+\eta_{1}\bullet\operatorname{ch}(\overline{\mathcal{F}}_{2})+\operatorname{d}_{\mathcal{D}}\eta_{1}\bullet\eta_{2})$ (10.5) 3. (iii) $\widehat{K}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$ is a $\widehat{K}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})$-module. 4. (iv) There are natural maps $\widehat{K}(\mathcal{X},\mathcal{C})\longrightarrow\widehat{K}^{\prime}(\mathcal{X},\mathcal{C})$ that, when $\mathcal{X}$ is regular, are isomorphisms. 5. (v) The groups $\widehat{K}^{\prime}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})$ and $\widehat{K}^{\prime}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$ are $\widehat{K}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})$-modules. 6. (vi) There are natural maps $\omega\colon\widehat{K}^{\prime}(\mathcal{X},\mathcal{C})\longrightarrow\bigoplus_{p}Z\mathcal{C}^{2p}(p)$ that send the class of a pair $(\overline{\mathcal{F}},\eta)$ with $\overline{\mathcal{F}}=(\mathcal{F},\overline{E}_{\ast}\to\mathcal{F}_{{\mathbb{C}}})$ to the form (or current) $\omega(\overline{\mathcal{F}},\eta)=\sum_{i}(-1)^{i}\operatorname{ch}(\overline{E}_{i})+\operatorname{d}_{\mathcal{D}}\eta.$ 7. (vii) When $\mathcal{X}$ is regular, there exists a Chern character, $\widehat{\operatorname{ch}}\colon\widehat{K}(\mathcal{X},\mathcal{C})_{{\mathbb{Q}}}\longrightarrow\bigoplus_{p}\widehat{\operatorname{CH}}^{p}(\mathcal{X},\mathcal{C})_{{\mathbb{Q}}},$ that is an isomorphism. Moreover, if $\mathcal{C}=\mathcal{D}_{{\text{\rm l,a}},X}$ this isomorphism is compatible with the product structure. If $\mathcal{X}$ is not regular, there is a biadditive pairing $\widehat{K}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})\otimes\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})\longrightarrow\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})_{{\mathbb{Q}}},$ and a similar pairing with the groups $\operatorname{\widehat{CH}}_{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$, which is denoted in both cases by $\alpha\otimes x\mapsto\widehat{\operatorname{ch}}(\alpha)\cap x$. For the properties of this product see [20] pg. 496. 8. (viii) If $\mathcal{Y}$ and $\mathcal{X}$ are arithmetic varieties and $f\colon\mathcal{Y}\to\mathcal{X}$ is a morphism of arithmetic varieties, $f$ induces a morphism of rings: $f^{}\colon\widehat{K}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})\rightarrow\widehat{K}(\mathcal{Y},\mathcal{D}_{{\text{\rm l,a}},Y}).$ When $f$ is flat, the inverse image is also defined for the groups $\widehat{K}^{\prime}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})$. Moreover, if $f_{\mathbb{C}}$ is smooth, the inverse image can be defined for the groups $\widehat{K}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$ and, when in addition $f$ is flat, for the groups $\widehat{K}^{\prime}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$. In what follows we will be interested in direct images for closed immersions. Since the direct images in arithmetic $K$-theory will depend on the choice of a metric, we have the following ###### Definition 10.6. A metrized arithmetic variety is a pair $(\mathcal{X},h_{X})$ consisting of an arithmetic variety $\mathcal{X}$ and a hermitian metric on the complex tangent bundle $T_{X}$ that is invariant under $F_{\infty}$. Let $(\mathcal{X},h_{X})$ and $(\mathcal{Y},h_{Y})$ be metrized arithmetic varieties and let $i\colon\mathcal{Y}\longrightarrow\mathcal{X}$ be a closed immersion. Over the complex numbers, we are in the situation of notation 8.36. In particular we have a canonical exact sequence of hermitian vector bundles $\overline{\xi}_{N}\colon 0\longrightarrow\overline{T}_{Y}\longrightarrow i^{}\overline{T}_{X}\longrightarrow\overline{N}_{Y/X}\longrightarrow 0$ (10.7) where the tangent bundles $T_{Y}$, $T_{X}$ are endowed with the hermitian metrics $h_{Y}$, $h_{X}$ respectively and the normal bundle $N_{Y/X}$ is endowed with an arbitrary hermitian metric $h_{N}$. We will follow the conventions of notation 8.36. We next define push-forward maps, via a closed immersion, for the elements of the arithmetic $K$-group of a metrized arithmetic variety. We will define two kinds of push-forward maps. One will depend only on a metric on the complex normal bundle $N_{Y/X}$. By contrast, the second will depend on the choice of metrics on the complex tangent bundles $T_{X}$ and $T_{Y}$. The second definition allows us to see $K^{\prime}(\underline{\phantom{A}},\mathcal{D}_{\text{{\rm cur}},Y})$ as a functor from the category whose objects are metrized arithmetic varieties and whose morphisms are closed immersions to the category of abelian groups. As we deal with hermitian vector bundles and metrized coherent sheaves, both definitions will involve the choice of a theory of singular Bott-Chern classes. In order for the push forward to be well defined in $K$-theory we need a minimal additivity property for the singular Bott-Chern classes. ###### Definition 10.8. A theory of singular Bott-Chern classes $T$ is called _additive_ if for any closed embedding of complex manifolds $i\colon Y\hookrightarrow X$ and any hermitian embedded vector bundles $\overline{\xi}_{1}=(i,\overline{N},\overline{F}_{1},\overline{E}_{1,\ast})$, $\overline{\xi}_{2}=(i,\overline{N},\overline{F}_{2},\overline{E}_{2,\ast})$ the equation $T(\overline{\xi}_{1}\oplus\overline{\xi}_{2})=T(\overline{\xi}_{1})+T(\overline{\xi}_{2})$ is satisfied. Let $C$ be a characteristic class for pairs of vector bundles. We say that it is _additive_ (in the first variable) if $C(F_{1}\oplus F_{2},N)=C(F_{1},N)+C(F_{2},N)$ for any vector bundles $F_{1},F_{2},N$ on a complex manifold $X$. The following statement follows directly from equation 7.5: ###### Proposition 10.9. A theory of singular Bott-Chern classes $T$ is additive if and only if the corresponding characteristic class $C_{T}$ is additive in the first variable. Note that a theory of singular Bott-Chern classes consists in joining theories of singular Bott-Chern classes in arbitrary rank and codimension (definition 6.9). The property of being additive gives a compatibility condition for these theories, by respect to the hermitian vector bundles $\overline{F}$ (with the notation used in definition 6.9). Note also that if a theory of singular Bott- Chern classes is compatible with the projection formula then it is additive. ###### Definition 10.10. Let $T$ be an additive theory of singular Bott-Chern classes, and let $T_{c}$ be the associated covariant class as in definition 8.37. Let $i\colon(\mathcal{Y},h_{Y})\longrightarrow(\mathcal{X},h_{X})$ be a closed immersion of metrized arithmetic varieties and let $\overline{N}=\overline{N}_{Y/X}=(N_{Y/X},h_{N})$ be a choice of a hermitian metric on the complex normal bundle. The _push-forward maps_ $i^{T_{c}}_{\ast},i^{T}_{\ast}\colon\widehat{K}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})\longrightarrow\widehat{K}^{\prime}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$ are defined by $\displaystyle i^{T_{c}}_{\ast}(\overline{\mathcal{F}},\eta)$ $\displaystyle=[((i_{\ast}\mathcal{F},\overline{E}_{\ast}\to(i_{\ast}\mathcal{F})_{{\mathbb{C}}}),0)]-[(0,T_{c}(\overline{\xi}_{c}))]$ $\displaystyle\phantom{AA}+[(0,i_{\ast}(\eta\operatorname{Td}(Y)i^{}\operatorname{Td}^{-1}(X)))]$ (10.11) $\displaystyle i^{T}_{\ast}(\overline{\mathcal{F}},\eta)$ $\displaystyle=[((i_{\ast}\mathcal{F},\overline{E}_{\ast}\to(i_{\ast}\mathcal{F})_{{\mathbb{C}}}),0)]-[(0,T(\overline{\xi}))]$ $\displaystyle\phantom{AA}+[(0,i_{\ast}(\eta\operatorname{Td}^{-1}(\overline{N}_{Y/X})))].$ (10.12) Here $0\rightarrow\overline{E}_{n}\rightarrow\ldots\rightarrow\overline{E}_{1}\rightarrow\overline{E}_{0}\rightarrow(i_{}\mathcal{F})_{\mathbb{C}}\rightarrow 0$ is a finite resolution of the coherent sheaf $(i_{}\mathcal{F})_{{\mathbb{C}}}$ by hermitian vector bundles, $\overline{\xi}=(i,\overline{N}_{X/Y},\overline{\mathcal{F}}_{\mathbb{C}},\overline{E}_{})$ is the induced hermitian embedded vector bundle on $X$, and $\overline{\xi}_{c}=(i,\overline{T}_{X},\overline{T}_{Y},\overline{\mathcal{F}}_{\mathbb{C}},\overline{E}_{})$ as in definition 8.37. We can extend this definition to push-forward maps $i^{T_{c}}_{\ast},i^{T}_{\ast}\colon\widehat{K}^{\prime}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})\longrightarrow\widehat{K}^{\prime}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$ by the rule $\displaystyle i^{T_{c}}_{\ast}(\overline{\mathcal{F}},\eta)$ $\displaystyle=[((i_{\ast}\mathcal{F},\operatorname{Tot}(\overline{E}_{\ast,\ast})\to(i_{\ast}\mathcal{F})_{{\mathbb{C}}}),0)]-\sum_{i}(-1)^{i}[(0,T_{c}(\overline{\xi}_{i,c}))]$ $\displaystyle\phantom{AAA}+[(0,i_{\ast}(\eta\operatorname{Td}(Y)i^{}\operatorname{Td}^{-1}(X)))],$ (10.13) $\displaystyle i^{T}_{\ast}(\overline{\mathcal{F}},\eta)$ $\displaystyle=[((i_{\ast}\mathcal{F},\operatorname{Tot}(\overline{E}_{\ast,\ast})\to(i_{\ast}\mathcal{F})_{{\mathbb{C}}}),0)]-\sum_{i}(-1)^{i}[(0,T(\overline{\xi}_{i}))]$ $\displaystyle\phantom{AAA}+[(0,i_{\ast}(\eta\operatorname{Td}^{-1}(\overline{N}_{Y/X})))],$ (10.14) where $0\to\overline{E}_{n}\to\dots\to\overline{E}_{0}\to\mathcal{F}_{{\mathbb{C}}}\to 0$ is a resolution of $\mathcal{F}_{{\mathbb{C}}}$ by hermitian vector bundles, $\overline{E}_{\ast,\ast}$ is a complex of complexes of vector bundles over $X$, such that, for each $i\geq 0$, $\overline{E}_{i,\ast}\to i_{\ast}E_{i}$ is also a resolution by hermitian vector bundles and $\overline{\xi}_{i}=(i,\overline{N}_{X/Y},\overline{E}_{i},\overline{E}_{i,})$ is the induced hermitian embedded vector bundle and $\overline{\xi}_{i,c}$ is as in definition 8.37. We suppose that there is a commutative diagram of resolutions $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&\\\&&&&\crcr}}}\ignorespaces{\hbox{\kern-6.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 31.69478pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 31.69478pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{E_{k+1,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 86.14545pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 45.97533pt\raise-29.8889pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 86.14545pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{E_{k,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 133.44055pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 96.8482pt\raise-29.8889pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 133.44055pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{E_{k-1,\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 184.45752pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 146.47664pt\raise-29.8889pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 184.45752pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots}$}}}}}}}{\hbox{\kern-6.75pt\raise-39.72221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 30.75pt\raise-39.72221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 30.75pt\raise-39.72221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{i_{\ast}E_{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 85.20065pt\raise-39.72221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 85.20065pt\raise-39.72221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{i_{\ast}E_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 132.49576pt\raise-39.72221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 132.49576pt\raise-39.72221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{i_{\ast}E_{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 184.45752pt\raise-39.72221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 184.45752pt\raise-39.72221pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots}$}}}}}}}\ignorespaces}}}}\ignorespaces.$ hence a resolution $\operatorname{Tot}(\overline{E}_{\ast,\ast})\longrightarrow(i_{\ast}\mathcal{F})_{{\mathbb{C}}}$ by hermitian vector bundles. Note that, whenever the push-forward $i^{T}_{\ast}$ appears, we will assume that we have chosen a metric on $N_{Y/X}$. The two push-forward maps are related by the equation $i^{T_{c}}_{\ast}(\overline{\mathcal{F}},\eta)=i^{T}_{\ast}(\overline{\mathcal{F}},\eta)-\left[\left(0,i_{\ast}\left(\omega(\overline{\mathcal{F}},\eta)\widetilde{\operatorname{Td}^{-1}}(\overline{\xi}_{N})\operatorname{Td}(Y)\right)\right)\right],$ (10.15) where $\overline{\xi}_{N}$ is the exact sequence (10.7). ###### Proposition 10.16. The push-forward maps $i^{T}_{\ast}$, $i^{T_{c}}_{\ast}$ are well defined. That is, they do not depend on the choice of a representative of a class in $\widehat{K}$, nor on the choice of metrics on the coherent sheaf $(i_{\ast}\mathcal{F})_{{\mathbb{C}}}$. The first one does not depend on the choice of metrics on $T_{X}$ nor on $T_{Y}$, whereas the second one does not depend on the choice of a metric on the normal bundle $N_{Y/X}$. Moreover, if $i$ is a regular closed immersion or $\mathcal{X}$ is a regular arithmetic variety, then $i^{T_{c}}_{\ast}$ and $i^{T}_{\ast}$ can be lifted to maps $i^{T_{c}}_{\ast},i^{T}_{\ast}\colon\widehat{K}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})\longrightarrow\widehat{K}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},Y}).$ ###### Proof. The fact that $i^{T}_{\ast}$ only depends on the metric on $\overline{N}$ and not on the metrics on $T_{X}$ and $T_{Y}$ and that for $i^{T_{c}}_{\ast}$ is the opposite, follows directly from the definition in the first case and from proposition 8.39 in the second. We will only prove the other statements for $i^{T_{c}}_{\ast}$, as the other case is analogous. We first prove the independence from the metric chosen on the coherent sheaf $(i_{\ast}\mathcal{F})_{{\mathbb{C}}}$. If $\overline{E}_{\ast}\to(i_{\ast}\mathcal{F})_{{\mathbb{C}}}$, $\overline{E}^{\prime}_{\ast}\to(i_{\ast}\mathcal{F})_{{\mathbb{C}}}$ are two such metrics, inducing the hermitian embedded vector bundles $\overline{\xi}$ respectively $\overline{\xi}^{\prime}$, then, using corollary 6.14 $T_{c}(\overline{\xi}_{c}^{\prime})-T_{c}(\overline{\xi}_{c})=T(\overline{\xi}^{\prime})-T(\overline{\xi})=\widetilde{\operatorname{ch}}(\overline{\varepsilon}),$ where $\overline{\varepsilon}$ is the exact complex of hermitian embedded vector bundles $\overline{\varepsilon}\colon 0\longrightarrow\overline{\xi}\longrightarrow\overline{\xi}^{\prime}\longrightarrow 0,$ where $\overline{\xi}^{\prime}$ sits in degree zero. Therefore, by equation 10.4, $[((i_{\ast}\mathcal{F},\overline{E}_{\ast}\to(i_{\ast}\mathcal{F})_{{\mathbb{C}}}),0)]-[(0,T_{c}(\overline{\xi_{c}}))]\\\ =[((i_{\ast}\mathcal{F},\overline{E}^{\prime}_{\ast}\to(i_{\ast}\mathcal{F})_{{\mathbb{C}}}),0)]-[(0,T_{c}(\overline{\xi}_{c}^{\prime}))].$ Since the last term of equation 10.11 does not depend on the metric on $(i_{\ast}\mathcal{F})_{{\mathbb{C}}}$, we obtain that $i^{T_{c}}_{\ast}$ does not depend on this metric. For proving that the push-forward map $i^{T_{c}}_{\ast}$ is well defined it remains to show the independence from the choice of a representative of a class in $\widehat{K}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})$. We consider an exact sequence of hermitian vector bundles on $\mathcal{Y}$ $\overline{\varepsilon}\colon 0\longrightarrow\overline{\mathcal{F}}_{1}\longrightarrow\overline{\mathcal{F}}\longrightarrow\overline{\mathcal{F}}_{2}\longrightarrow 0$ and two classes $\eta_{1},\eta_{2}\in\bigoplus_{p\geq 0}\widetilde{\mathcal{D}}_{\text{{\rm cur}}}^{2p-1}(Y,p)$. We also denote $\overline{\varepsilon}$ the induced exact sequence of hermitian vector bundles on $Y$. We have to prove $i^{T_{c}}_{\ast}([(\overline{\mathcal{F}},\eta_{1}+\eta_{2}+\widetilde{\operatorname{ch}}(\overline{\varepsilon})])=i^{T_{c}}_{\ast}([(\overline{\mathcal{F}_{1}},\eta_{1})])+i^{T_{c}}_{\ast}([(\overline{\mathcal{F}_{2}},\eta_{2})]).$ (10.17) Since it is clear that $i^{T_{c}}_{\ast}(0,\eta_{1}+\eta_{2})=i^{T_{c}}_{\ast}(0,\eta_{1})+i^{T_{c}}_{\ast}(0,\eta_{2})$, we are led to prove $i^{T_{c}}_{\ast}([(\overline{\mathcal{F}},\widetilde{\operatorname{ch}}(\overline{\varepsilon})])=i^{T_{c}}_{\ast}([(\overline{\mathcal{F}_{1}},0)])+i^{T_{c}}_{\ast}([(\overline{\mathcal{F}_{2}},0)]).$ (10.18) We choose metrics on the coherent sheaves $(i_{\ast}\mathcal{F}_{1})_{{\mathbb{C}}}$, $(i_{\ast}\mathcal{F}_{2})_{{\mathbb{C}}}$ and $(i_{\ast}\mathcal{F})_{{\mathbb{C}}}$ respectively: $\overline{E}_{1,\ast}\longrightarrow(i_{\ast}\mathcal{F}_{1})_{{\mathbb{C}}}\ ,\ \overline{E}_{2,\ast}\longrightarrow(i_{\ast}\mathcal{F}_{2})_{{\mathbb{C}}}\ ,\ \overline{E}_{\ast}\longrightarrow(i_{\ast}\mathcal{F})_{{\mathbb{C}}}.$ We denote $\overline{\xi}_{1}$, $\overline{\xi}_{2}$, $\overline{\xi}$ the induced hermitian embedded vector bundles. We obtain an exact sequence of metrized coherent sheaves on $\mathcal{X}$: $\overline{\nu}\colon 0\longrightarrow\overline{i_{\ast}\mathcal{F}_{1}}\longrightarrow\overline{i_{\ast}\mathcal{F}}\longrightarrow\overline{i_{\ast}\mathcal{F}_{2}}\longrightarrow 0.$ Then, using the fact that the theory $T$ is additive and equation (8.42) we have $T_{c}(\overline{\xi}_{1,c})+T_{c}(\overline{\xi}_{2,c})-T_{c}(\overline{\xi}_{c})=[\widetilde{\operatorname{ch}}(\overline{\nu})]-i_{\ast}([\widetilde{\operatorname{ch}}(\overline{\varepsilon})\bullet\operatorname{Td}(Y)])\bullet\operatorname{Td}^{-1}(X).$ (10.19) Moreover, by the relation (10.4), $[(\overline{i_{\ast}\mathcal{F}_{1}},0)]+[(\overline{i_{\ast}\mathcal{F}_{2}},0)]=[(\overline{i_{\ast}\mathcal{F}},\widetilde{\operatorname{ch}}(\overline{\nu}))].$ (10.20) Hence, we compute, $\displaystyle i^{T_{c}}_{\ast}([(\overline{\mathcal{F}},\widetilde{\operatorname{ch}}(\overline{\varepsilon})])$ $\displaystyle-i^{T_{c}}_{\ast}([(\overline{\mathcal{F}_{1}},0)])-i^{T_{c}}_{\ast}([(\overline{\mathcal{F}_{2}},0)])$ $\displaystyle=[(i_{\ast}\overline{\mathcal{F}},0)]-[(i_{\ast}\overline{\mathcal{F}_{1}},0)]-[(i_{\ast}\overline{\mathcal{F}_{2}},0)]$ $\displaystyle\phantom{A}-[(0,T_{c}(\overline{\xi}_{c}))]+[(0,T_{c}(\overline{\xi_{1,c}}))]+[(0,T_{c}(\overline{\xi_{2,c}}))]$ $\displaystyle\phantom{A}+[(0,i_{\ast}([\widetilde{\operatorname{ch}}(\overline{\varepsilon})]\bullet\operatorname{Td}(Y)\bullet i^{\ast}\operatorname{Td}^{-1}(X)))]$ $\displaystyle=-[(0,i_{\ast}([\widetilde{\operatorname{ch}}(\overline{\varepsilon})]\bullet\operatorname{Td}(Y)\bullet i^{\ast}\operatorname{Td}^{-1}(X))))]$ $\displaystyle\phantom{AA}+[(0,i_{\ast}([\widetilde{\operatorname{ch}}(\overline{\varepsilon})]\bullet\operatorname{Td}(Y)\bullet i^{\ast}\operatorname{Td}^{-1}(X))))]$ $\displaystyle=0.$ The proof that $i^{T_{c}}_{\ast}$ for metrized coherent sheaves is well defined is similar. The proof of its independence from choice of a metric on $N_{Y/X}$ or from the choice of the resolutions and metrics in $X$ is the same as before. Now let $0\longrightarrow\overline{\mathcal{F}}^{\prime}\longrightarrow\overline{\mathcal{F}}\longrightarrow\overline{\mathcal{F}}^{\prime\prime}\longrightarrow 0$ be a short exact sequence of metrized coherent sheaves on $\mathcal{Y}$. This means that we have resolutions $\overline{E}^{\prime}_{\ast}\to\mathcal{F}^{\prime}_{\mathbb{C}}$, $\overline{E}_{\ast}\to\mathcal{F}_{\mathbb{C}}$ and $\overline{E}^{\prime\prime}_{\ast}\to\mathcal{F}^{\prime\prime}_{\mathbb{C}}$. Using theorem 2.24 we can suppose that there is a commutative diagram of resolutions $\begin{array}[h]{ccccccccc}0&\rightarrow&\overline{E}^{\prime}_{\ast}&\rightarrow&\overline{E}_{\ast}&\rightarrow&\overline{E}^{\prime\prime}_{\ast}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&\mathcal{F}^{\prime}_{\mathbb{C}}&\rightarrow&\mathcal{F}_{\mathbb{C}}&\rightarrow&\mathcal{F}^{\prime\prime}_{\mathbb{C}}&\rightarrow&0,\end{array}$ (10.21) with exact rows. Moreover, we can assume that the complexes of complexes $\overline{E}^{\prime}_{\ast,\ast}$, $\overline{E}_{\ast,\ast}$, $\overline{E}^{\prime\prime}_{\ast,\ast}$ used in definition 10.10 are chosen compatible with diagram (10.21). Thus we obtain a commutative diagram $\begin{array}[h]{ccccccccc}0&\rightarrow&\operatorname{Tot}\overline{E}^{\prime}_{\ast,\ast}&\rightarrow&\operatorname{Tot}\overline{E}_{\ast,\ast}&\rightarrow&\operatorname{Tot}\overline{E}^{\prime\prime}_{\ast,\ast}&\rightarrow&0\\\ &&\downarrow&&\downarrow&&\downarrow&&\\\ 0&\rightarrow&i_{\ast}\mathcal{F}^{\prime}_{\mathbb{C}}&\rightarrow&i_{\ast}\mathcal{F}_{\mathbb{C}}&\rightarrow&i_{\ast}\mathcal{F}^{\prime\prime}_{\mathbb{C}}&\rightarrow&0.\end{array}$ (10.22) We denote by $\overline{\nu}$ the exact sequence of metrized coherent sheaves on $X$ defined by diagram (10.22). We denote $\overline{\chi}_{i}$ the exact sequence of hermitian vector bundles on $Y$ $\overline{\chi}_{i}\colon 0\longrightarrow\overline{E}^{\prime}_{i}\longrightarrow\overline{E}_{i}\longrightarrow\overline{E}^{\prime\prime}_{i}\longrightarrow 0,$ and by $\overline{\varepsilon}$ the exact sequence of metrized coherent sheaves on $X$ $\overline{\varepsilon}_{i}\colon 0\longrightarrow\overline{i_{\ast}E}^{\prime}_{i}\longrightarrow\overline{i_{\ast}E}_{i}\longrightarrow\overline{i_{\ast}E}^{\prime\prime}_{i}\longrightarrow 0.$ Moreover, let $\overline{\xi}_{i}$, $\overline{\xi}^{\prime}_{i}$ and $\overline{\xi}^{\prime\prime}_{i}$ denote the hermitian embedded vector bundles defined by the above resolutions and $\overline{E}_{i}$, $\overline{E}^{\prime}_{i}$ and $\overline{E}^{\prime\prime}_{i}$ respectively and let $\overline{\xi}_{i,c}$, $\overline{\xi}^{\prime}_{i,c}$ and $\overline{\xi}^{\prime\prime}_{i,c}$ be as in definition 8.37. Then, using proposition 2.38 and equation (8.42) we obtain $\displaystyle\widetilde{\operatorname{ch}}(\overline{\nu})$ $\displaystyle=\sum_{i}(-1)^{i}\widetilde{\operatorname{ch}}(\overline{\varepsilon})$ $\displaystyle=\sum_{i}(-1)^{i}(T_{c}(\overline{\xi}^{\prime}_{i,c})+T_{c}(\overline{\xi}^{\prime\prime}_{i,c})-T_{c}(\overline{\xi}_{i,c}))$ (10.23) $\displaystyle\phantom{AA}+\sum_{i}(-1)^{i}i_{\ast}(\widetilde{\operatorname{ch}}(\overline{\chi}_{i})\bullet\operatorname{Td}(Y))\bullet\operatorname{Td}^{-1}(X)$ Now the proof follows as before, but using equation (10.23) instead of equation (10.19). If $\mathcal{X}$ is a regular arithmetic variety, the lifting property follows from the isomorphism between the $\widehat{K}$-groups and the $\widehat{K}^{\prime}$-groups. Suppose now that $i\colon\mathcal{Y}\longrightarrow\mathcal{X}$ is a regular closed immersion and let $[\overline{\mathcal{F}},\eta]\in\widehat{K}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})$. Then it follows from [2] III that the coherent sheaf $i_{\ast}\mathcal{F}$ can be resolved $0\longrightarrow\mathcal{E}_{n}\longrightarrow\ldots\longrightarrow\mathcal{E}_{0}\longrightarrow i_{\ast}\mathcal{F}\longrightarrow 0$ with $\mathcal{E}_{i}$ locally free sheaves on $\mathcal{X}$. Moreover we endow the vector bundles $E_{i}$ induced on $X$ with hermitian metrics and so we obtain a metric on the coherent sheaf $i_{\ast}\mathcal{F}$ and the corresponding hermitian embedded vector bundle $\overline{\xi}$. Using the independence from the resolutions and on the metrics we see that the equation 10.11 defines an element in $\widehat{K}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$. ∎ ###### Proposition 10.24. For any element $\alpha\in\widehat{K}^{\prime}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})$ we have $\displaystyle\omega(i^{T_{c}}_{\ast}(\alpha))\operatorname{Td}(X)$ $\displaystyle=i_{\ast}(\omega(\alpha)\operatorname{Td}(Y))$ (10.25) $\displaystyle\omega(i^{T}_{\ast}(\alpha))$ $\displaystyle=i_{\ast}(\omega(\alpha)\operatorname{Td}^{-1}(N_{Y/X}))$ (10.26) ###### Proof. We will prove the statement only for $i_{\ast}^{T_{c}}$. We consider first a class of the form $[\overline{\mathcal{F}},0]$. Using equation (8.38) we obtain, after choosing a metric $\overline{E}_{i}\longrightarrow(i_{\ast}\mathcal{F})_{{\mathbb{C}}}$, and considering the induced hermitian embedded vector bundle $\overline{\xi_{c}}$: $\displaystyle\omega(i^{T_{c}}_{\ast}([\overline{\mathcal{F}},0]))\operatorname{Td}(X)$ $\displaystyle=\left(\sum(-1)^{i}\operatorname{ch}(\overline{E_{i}})-\operatorname{d}_{\mathcal{D}}T_{c}(\overline{\xi_{c}})\right)\operatorname{Td}(X)$ $\displaystyle=i_{\ast}(\operatorname{ch}(\overline{F})\bullet\operatorname{Td}(Y)\bullet i^{\ast}\operatorname{Td}^{-1}(X)i^{\ast}(\operatorname{Td}(X)))$ $\displaystyle=i_{\ast}(\operatorname{ch}(\overline{F})\bullet\operatorname{Td}(Y))$ $\displaystyle=i_{\ast}(\omega([\overline{\mathcal{F}},0])\operatorname{Td}(Y))$ Taking now a class of the form $[0,\eta]$ we obtain: $\displaystyle\omega(i^{T_{c}}_{\ast}([0,\eta]))\operatorname{Td}(X)$ $\displaystyle=\operatorname{d}_{\mathcal{D}}\left(i_{\ast}(\eta\operatorname{Td}(Y)i^{}\operatorname{Td}^{-1}(X))\right)\operatorname{Td}(X)$ $\displaystyle=i_{\ast}\operatorname{d}_{\mathcal{D}}(\eta\operatorname{Td}(Y))$ $\displaystyle=i_{\ast}(\omega([0,\eta])\operatorname{Td}(Y))$ and hence the equality 10.25 is proved. ∎ The next proposition explains the terminology “compatible with the projection formula” and “transitive” that we used for theories of singular Bott-Chern classes. The second statement is the main reason to introduce the push-forward $i_{\ast}^{T_{c}}$. ###### Proposition 10.27. If the theory of singular Bott-Chern classes is compatible with the projection formula, we have that, for $\alpha\in\widehat{K}^{\prime}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})$ and $\beta\in\widehat{K}(\mathcal{X},\mathcal{D}_{{\text{\rm l,a}},X})$ the following equalities hold $\displaystyle i^{T_{c}}_{\ast}(\alpha i^{\ast}\beta)$ $\displaystyle=i^{T_{c}}_{\ast}(\alpha)\beta,$ $\displaystyle i^{T}_{\ast}(\alpha i^{\ast}\beta)$ $\displaystyle=i^{T}_{\ast}(\alpha)\beta.$ If moreover the theory of singular Bott-Chern classes is transitive and $j\colon(\mathcal{Z},h_{Z})\longrightarrow(\mathcal{Y},h_{Y})$ is another closed immersion of metrized arithmetic varieties, then $(i\circ j)_{\ast}^{T_{c}}=i_{\ast}^{T_{c}}\circ j_{\ast}^{T_{c}}.$ ###### Proof. We prove first the projection formula. For simplicity we will treat the case when $\alpha\in\widehat{K}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})$. Let $\alpha=(\overline{\mathcal{F}},\eta)$, let $\overline{\xi_{c}}=(i,\overline{T}_{X},\overline{T}_{Y},\overline{\mathcal{F}}_{\mathbb{C}},\overline{E}_{\ast})$ be a hermitian embedded vector bundle and let $\beta=(\overline{\mathcal{E}},\chi)$. Using equations (10.11) and (10.5), we obtain $\displaystyle i^{T_{c}}_{\ast}(\alpha i^{\ast}\beta)-i^{T_{c}}_{\ast}(\alpha)\beta$ $\displaystyle=-\sum_{i}(-1)^{i}\operatorname{ch}(\overline{E}_{i})\bullet\chi+\operatorname{d}_{\mathcal{D}}(T_{c}(\overline{\xi}_{c}))\bullet\chi$ $\displaystyle\phantom{AA}+i_{\ast}(\operatorname{ch}((\overline{\mathcal{F}})_{\mathbb{C}})\bullet\operatorname{Td}(Y)))\bullet\operatorname{Td}^{-1}(X)\bullet\chi$ $\displaystyle\phantom{AA}+T_{c}(\overline{\xi}_{c})\bullet\operatorname{ch}(\overline{\mathcal{E}}_{\mathbb{C}})-T_{c}(\overline{\xi}_{c}\otimes\overline{\mathcal{E}}_{\mathbb{C}})$ $\displaystyle=T_{c}(\overline{\xi}_{c}\otimes\overline{\mathcal{E}}_{\mathbb{C}})-T_{c}(\overline{\xi}_{c})\bullet\operatorname{ch}(\overline{\mathcal{E}}_{\mathbb{C}}).$ Therefore, if $T$ is compatible with the projection formula, then the projection formula holds. The fact that, if moreover $T$ is transitive then $(i\circ j)_{\ast}^{T_{c}}=i_{\ast}^{T_{c}}\circ j_{\ast}^{T_{c}}$ follows directly from the definition and equation (8.41). ∎ If $i\colon\mathcal{Y}\longrightarrow\mathcal{X}$ is a regular closed immersion between arithmetic varieties, then the normal cone $\mathcal{N}_{\mathcal{Y}/\mathcal{X}}$ is a locally free sheaf. The choice of a hermitian metric on $N_{Y/X}$ determines a hermitian vector bundle $\overline{\mathcal{N}}_{\mathcal{Y}/\mathcal{X}}$. If now $i\colon(\mathcal{Y},h_{Y})\longrightarrow(\mathcal{X},h_{X})$ is a closed immersion between regular metrized arithmetic varieties, then the tangent bundles ${\mathcal{T}}_{\mathcal{Y}}$ and ${\mathcal{T}}_{\mathcal{X}}$ are virtual vector bundles. Since over $\mathbb{C}$ they define vector bundles, we can provide them with hermitian metrics and denote the hermitian virtual vector bundles by $\overline{\mathcal{T}}_{\mathcal{X}}$ and $\overline{\mathcal{T}}_{\mathcal{Y}}$. There are well defined clases $\widehat{\operatorname{Td}}(\mathcal{Y})=\widehat{\operatorname{Td}}(\overline{\mathcal{T}}_{\mathcal{Y}})$ and $\widehat{\operatorname{Td}}(\mathcal{X})=\widehat{\operatorname{Td}}(\overline{\mathcal{T}}_{\mathcal{X}})$. The arithmetic Grothendieck-Riemann-Roch theorem for closed immersions compares the direct images in the arithmetic $K$-groups with the direct images in the arithmetic Chow groups. ###### Theorem 10.28 ([6], [32]). Let $T$ be a theory of singular Bott-Chern classes and let $S_{T}$ be the additive genus of corollary 9.43. 1. (i) Let $i\colon\mathcal{Y}\longrightarrow\mathcal{X}$ be a regular closed immersion between arithmetic varieties. Assume that we have chosen a hermitian metric on the complex bundle $N_{Y/X}$. Then, for any $\alpha=(\overline{\mathcal{F}},\eta)\in\widehat{K}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})$ the equation $\widehat{\operatorname{ch}}(i^{T}_{\ast}(\alpha))=i_{\ast}(\widehat{\operatorname{ch}}(\alpha)\widehat{\operatorname{Td}}^{-1}(\overline{\mathcal{N}}_{\mathcal{Y}/\mathcal{X}}))-\operatorname{a}(i_{\ast}(\operatorname{ch}(\mathcal{F}_{{\mathbb{C}}})\operatorname{Td}^{-1}(N_{Y/X})S_{T}(N))$ (10.29) holds. 2. (ii) Let $i\colon(\mathcal{Y},h_{Y})\longrightarrow(\mathcal{X},h_{X})$ be a closed immersion between regular metrized arithmetic varieties. Then, for any $\alpha=(\overline{\mathcal{F}},\eta)\in\widehat{K}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})$ the equation $\widehat{\operatorname{ch}}(i^{T_{c}}_{\ast}(\alpha))\widehat{\operatorname{Td}}(\mathcal{X})=i_{\ast}(\widehat{\operatorname{ch}}(\alpha)\widehat{\operatorname{Td}}(\mathcal{Y}))-\operatorname{a}(i_{\ast}(\operatorname{ch}(\mathcal{F}_{{\mathbb{C}}})\operatorname{Td}(Y)S_{T}(N)))$ (10.30) holds. ###### Proof. The proof follows the classical pattern of the deformation to the normal cone as in [6] and [32]. Let $\mathcal{W}$ be the deformation to the normal cone to $\mathcal{Y}$ in $\mathcal{X}$. We will follow the notation of section 5. Since $i$ is a regular closed immersion, there is a finite resolution by locally free sheaves $0\to\mathcal{E}_{n}\to\dots\to\mathcal{E}_{1}\to\mathcal{E}_{0}\to i_{\ast}\mathcal{F}\to 0.$ We choose hermitian metrics on the complex bundles $E_{i}=(\mathcal{E}_{i})_{\mathbb{C}}$. The immersion $j\colon\mathcal{Y}\times\mathbb{P}^{1}\longrightarrow\mathcal{W}$ is also a regular immersion. The construction of theorem 5.4 is valid over the arithmetic ring $A$. Therefore we have a resolution by hermitian vector bundles $0\to\widetilde{\mathcal{G}}_{n}\to\dots\to\widetilde{\mathcal{G}}_{1}\to\widetilde{\mathcal{G}}_{0}\to i_{\ast}\mathcal{F}\to 0.$ such that its restriction to $\mathcal{X}\times\\{0\\}$ is isometric to $\mathcal{E}_{\ast}$. Its restriction to $\widetilde{\mathcal{X}}$ is orthogonally split, and its restriction to $\mathcal{P}=\mathbb{P}(\mathcal{N}_{\mathcal{Y}/\mathcal{X}}\oplus\mathcal{O}_{\mathcal{Y}})$ fits in a short exact sequence $0\longrightarrow\overline{\mathcal{A}}_{\ast}\longrightarrow\widetilde{\mathcal{E}}_{\ast}\|_{\mathcal{P}}\longrightarrow K(\overline{\mathcal{F}},\overline{\mathcal{N}}_{\mathcal{Y}/\mathcal{X}})\longrightarrow 0,$ where $\overline{\mathcal{A}}_{\ast}$ is orthogonally split and $K(\overline{\mathcal{F}},\overline{\mathcal{N}}_{\mathcal{Y}/\mathcal{X}})$ is the Koszul resolution. We denote by $\overline{\eta}_{k}$ the piece of degree $k$ of this exact sequence. Let $t$ be the absolute coordinate of $\mathbb{P}^{1}$. It defines a rational function in $\mathcal{W}$ and $\widehat{\operatorname{div}}(t)=(\mathcal{X}_{0}+\mathcal{P}+\widetilde{\mathcal{X}},(0,-\frac{1}{2}\log t\overline{t}))$ The key point of the proof of the theorem is that, in the group $\operatorname{\widehat{CH}}^{\ast}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},X})$, we have $(p_{\mathcal{W}})_{\ast}(\widehat{\operatorname{ch}}(\widetilde{\mathcal{E}}_{\ast})\widehat{\operatorname{div}}(t))=0.$ Using the definition of the product in the arithmetic Chow rings we obtain $(p_{\mathcal{W}})_{\ast}(\widehat{\operatorname{ch}}(\widetilde{\mathcal{E}}_{\ast})\widehat{\operatorname{div}}(t))=\widehat{\operatorname{ch}}(\overline{\mathcal{E}}_{\ast})-(p_{\widetilde{\mathcal{X}}})_{\ast}\widehat{\operatorname{ch}}(\widetilde{\mathcal{E}}_{\ast}\|_{\widetilde{\mathcal{X}}})-(p_{\widetilde{\mathcal{P}}})_{\ast}\widehat{\operatorname{ch}}(\widetilde{\mathcal{E}}_{\ast}\|_{\mathcal{P}})\\\ +\operatorname{a}((p_{W})_{\ast}(\operatorname{ch}((\widetilde{\mathcal{E}}_{\ast})_{\mathbb{C}})\bullet W_{1})).$ (10.31) But we have $\displaystyle\widehat{\operatorname{ch}}(\overline{\mathcal{E}}_{\ast})$ $\displaystyle=\widehat{\operatorname{ch}}(i^{T}_{\ast}(\overline{\mathcal{F}}))+\operatorname{a}(T(\overline{\xi})),$ (10.32) $\displaystyle(p_{\widetilde{\mathcal{X}}})_{\ast}\widehat{\operatorname{ch}}(\widetilde{\mathcal{E}}_{\ast}\|_{\widetilde{\mathcal{X}}})$ $\displaystyle=0,$ (10.33) $\displaystyle(p_{\widetilde{\mathcal{P}}})_{\ast}\widehat{\operatorname{ch}}(\widetilde{\mathcal{E}}_{\ast}\|_{\mathcal{P}})$ $\displaystyle=i_{\ast}(\pi_{\mathcal{P}})_{\ast}(\widehat{\operatorname{ch}}(K(\overline{\mathcal{F}},\overline{\mathcal{N}}_{\mathcal{Y}/\mathcal{X}}))-\sum_{k}(-1)^{k}\operatorname{a}(\widetilde{\operatorname{ch}}(\overline{\eta}_{k}))).$ (10.34) Moreover, by equation (7.3), $\operatorname{a}((p_{W})_{\ast}(\operatorname{ch}((\widetilde{\mathcal{E}}_{\ast})_{\mathbb{C}})\bullet W_{1}))=-\operatorname{a}(T(\overline{\xi}))-\sum_{k}(-1)^{k}\operatorname{a}(\widetilde{\operatorname{ch}}(\overline{\eta}_{k})))\\\ +\operatorname{a}(i_{\ast}C_{T}(\mathcal{F}_{\mathbb{C}},\mathcal{N}_{\mathbb{C}})).$ (10.35) Thus we are led to compute $i_{\ast}(\pi_{\mathcal{P}})_{\ast}\widehat{\operatorname{ch}}(K(\overline{\mathcal{F}},\overline{\mathcal{N}}_{\mathcal{Y}/\mathcal{X}}))$. This is done in the following two lemmas. ###### Lemma 10.36. Let $\mathcal{Y}$ be an arithmetic variety, $\overline{\mathcal{N}}$ a rank $r$ hermitian vector bundle over $\mathcal{Y}$ and denote $\mathcal{P}=\mathbb{P}^{1}(\mathcal{N}\oplus\mathcal{O}_{\mathcal{Y}})$, and $\overline{\mathcal{Q}}$ the tautological quotient bundle. Let $\mathcal{Y}_{0}$ be the cycle defined by the zero section of $\mathcal{P}$. Then $\widehat{c}_{r}(\overline{\mathcal{Q}})=(\mathcal{Y}_{0},(c_{r}(\overline{\mathcal{Q}}_{\mathbb{C}}),\widetilde{e}(\mathcal{P}_{\mathbb{C}},\overline{\mathcal{Q}}_{\mathbb{C}},s))),$ (10.37) where $\widetilde{e}(\mathcal{P}_{\mathbb{C}},\overline{\mathcal{Q}}_{\mathbb{C}},s)$ is the Euler-Green current of lemma 9.4. ###### Proof. We know that $\widehat{c}_{r}(\overline{\mathcal{Q}})=(\mathcal{Y}_{0},(c_{r}(\overline{\mathcal{Q}}_{\mathbb{C}}),\widetilde{e}))$ for certain Green current $\widetilde{e}$. By definition this Green current satisfies $\operatorname{d}_{\mathcal{D}}\widetilde{e}=c_{r}(\overline{\mathcal{Q}}_{\mathbb{C}})-\delta_{\mathcal{Y}_{\mathbb{C}}}.$ Moreover, since the restriction of $\overline{\mathcal{Q}}_{\mathbb{C}}$ to $D_{\infty}$ has a global section of constant norm we have that $\widetilde{e}\|_{D_{\infty}}=0$. Therefore, by lemma 9.4, $\widetilde{e}=\widetilde{e}(\mathcal{P}_{\mathbb{C}},\overline{\mathcal{Q}}_{\mathbb{C}},s).$ ∎ ###### Lemma 10.38. The following equality hold: $(\pi_{\mathcal{P}})_{\ast}\widehat{\operatorname{ch}}(K(\overline{\mathcal{F}},\overline{\mathcal{N}})_{\ast})=\\\ \widehat{\operatorname{ch}}(\overline{\mathcal{F}})\widehat{\operatorname{Td}^{-1}}(\overline{\mathcal{N}})+\operatorname{a}(C_{T}(\overline{\mathcal{F}},\overline{\mathcal{N}})-\operatorname{ch}(\mathcal{F}_{{\mathbb{C}}})\operatorname{Td}^{-1}(N_{Y/X})S_{T}(N)).$ (10.39) ###### Proof. We just compute, using lemma 10.36, $\displaystyle(\pi_{\mathcal{P}})_{\ast}\widehat{\operatorname{ch}}(K(\overline{\mathcal{F}}$ $\displaystyle,\overline{\mathcal{N}})_{\ast})=(\pi_{\mathcal{P}})_{\ast}\sum_{k}(-1)^{k}\widehat{\operatorname{ch}}(\bigwedge^{k}\overline{\mathcal{Q}}^{\vee})\widehat{\operatorname{ch}}(\pi_{\mathcal{P}}^{\ast}\overline{\mathcal{F}})$ $\displaystyle=(\pi_{\mathcal{P}})_{\ast}(\widehat{c}_{r}(\overline{\mathcal{Q}})\widehat{\operatorname{Td}^{-1}}(\overline{\mathcal{Q}}))\widehat{\operatorname{ch}}(\overline{\mathcal{F}})$ $\displaystyle=\widehat{\operatorname{Td}^{-1}}(\overline{\mathcal{N}})\widehat{\operatorname{ch}}(\overline{\mathcal{F}})+\operatorname{a}((\pi_{P})_{\ast}(\widetilde{e}\operatorname{Td}^{-1}(\overline{Q}))\operatorname{ch}(\overline{F}))$ $\displaystyle=\widehat{\operatorname{Td}^{-1}}(\overline{\mathcal{N}})\widehat{\operatorname{ch}}(\overline{\mathcal{F}})+\operatorname{a}((\pi_{P})_{\ast}(T^{h}(K(\overline{F},\overline{N})))\operatorname{ch}(\overline{F}))$ $\displaystyle=\widehat{\operatorname{Td}^{-1}}(\overline{\mathcal{N}})\widehat{\operatorname{ch}}(\overline{\mathcal{F}})+\operatorname{a}(C_{T^{h}}(F,N))$ $\displaystyle=\widehat{\operatorname{Td}^{-1}}(\overline{\mathcal{N}})\widehat{\operatorname{ch}}(\overline{\mathcal{F}})+C_{T}(F,N)-\operatorname{a}(\operatorname{Td}^{-1}(N)\operatorname{ch}(F)S_{T}(N)).$ ∎ The equation (10.29) follows by combining equations (10.31), (10.32), (10.33), (10.34), (10.35) and (10.39). The equation (10.30) follows from equation (10.29) by a straightforward computation. ∎ Since $T$ is homogeneous if and only if $S_{T}=0$, in view of this result, the theory of homogeneous singular Bott-Chern classes is characterized for being the unique theory of singular Bott-Chern classes that provides an exact arithmetic Grothendieck-Riemann-Roch theorem for closed immersions. By contrast, if one uses a theory of singular Bott-Chern classes that is not homogeneous, there is an analogy between the genus $S_{T}$ and the $R$-genus that appears in the arithmetic Grothendieck-Riemann-Roch theorem for submersions. Since there is a unique theory of homogeneous singular Bott-Chern classes, the following definition is natural. ###### Definition 10.40. Let $i\colon(\mathcal{Y},h_{Y})\longrightarrow(\mathcal{X},h_{X})$ be a closed immersion of metrized arithmetic varieties, the _push-forward map_ $i_{\ast}\colon\widehat{K}^{\prime}(\mathcal{Y},\mathcal{D}_{\text{{\rm cur}},Y})\longrightarrow\widehat{K}^{\prime}(\mathcal{X},\mathcal{D}_{\text{{\rm cur}},Y})$ is defined as $i_{\ast}=i^{T_{c}^{h}}_{\ast}$. ###### Corollary 10.41. The push-forward map makes $\widehat{K}^{\prime}(\underline{\phantom{\mathcal{Y}}},\mathcal{D}_{\text{{\rm cur}},Y})$ and $\widehat{K}(\underline{\phantom{\mathcal{Y}}},\mathcal{D}_{\text{{\rm cur}},Y})$ functors from the category of regular metrized arithmetic varieties and closed immersions to the category of abelian groups. ###### Corollary 10.42. Let $i\colon(\mathcal{Y},h_{Y})\longrightarrow(\mathcal{X},h_{X})$ be a closed immersion of regular metrized arithmetic varieties, then $\widehat{\operatorname{ch}}(i^{T}_{\ast}(\alpha))\widehat{\operatorname{Td}}(\mathcal{X})=i_{\ast}(\widehat{\operatorname{ch}}(\alpha)\widehat{\operatorname{Td}}(\mathcal{Y})).$ (10.43) ###### Remark 10.44. Combining theorem 10.28 with [16] we can obtain an arithmetic Grothendieck- Riemann-Roch theorem for projective morphisms of regular arithmetic varieties. In a forthcoming paper we will show that the higher torsion forms used to define the direct images for submersions can also be characterized axiomatically. ## References [1] P. Baum, W. Fulton, and R. MacPherson, _Riemann-Roch for singular varieties_ , Publ. Math. Inst. Hautes Etud. Sci. 45 (1975), 101–145. * [2] P. Berthelot, A. Grothendieck, and L. Illusie, _Théorie des intersections et théorème de Riemann-Roch_ , Lecture Notes in Math., vol. 225, Springer-Verlag, 1971. * [3] J. M. Bismut, _Superconnection currents and complex immersions_ , Invent. Math. 90 (1990), 59–113. * [4] J.-M. Bismut, H. Gillet, and C. Soulé, _Analytic torsion and holomorphic determinant bundles I: Bott-chern forms and analytic torsion_ , Comm. Math. Phys. 115 (1988), 49–78. * [5] , _Bott-Chern currents and complex immersions_ , Duke Math. J. 60 (1990), no. 1, 255–284. MR MR1047123 (91d:58239) * [6] , _Complex immersions and Arakelov geometry_ , The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 249–331. MR MR1086887 (92a:14019) * [7] R. Bott and S.S. Chern, _Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections_ , Acta Math. 114 (1968), 71–112. * [8] J. I. Burgos Gil, _A ${C}^{\infty}$-logarithmic Dolbeault complex_, Compositio Math. 92 (1994), 61–86. * [9] , _Arithmetic Chow rings and Deligne-Beilinson cohomology_ , J. Alg. Geom. 6 (1997), 335–377. * [10] , _Hermitian vector bundles and characteristic classes_ , The arithmetic and geometry of algebraic cycles (Banff 1998), CRM Proc. Lecture Notes, vol. 24, AMS, 2000, pp. 155–182. * [11] , _Semipurity of tempered Deligne cohomology_ , Collect. Math. 59 (2008), no. 1, 79–102. MR MR2384539 * [12] J. I. Burgos Gil, J. Kramer, and U. Kühn, _Arithmetic characteristic classes of automorphic vector bundles_ , Documenta Math. 10 (2005), 619–716. * [13] , _Cohomological arithmetic Chow rings_ , J. Inst. Math. Jussieu 6 (2007), no. 1, 1–172. MR MR2285241 * [14] J. I. Burgos Gil and S. Wang, _Higher Bott-Chern forms and Beilinson’s regulator_ , Invent. Math. 132 (1998), 261–305. * [15] H. Esnault and E. Viehweg, _Deligne-Beilinson cohomology_ , in Rapoport et al. [30], pp. 43–91. * [16] H. Gillet, D. Rössler, and C Soulé, _An arithmetic Riemann-Roch theorem in higher degrees_ , Ann. Inst. Fourier 58 (2008), no. 6, 2169–2189. * [17] H. Gillet and C. Soulé, _Direct images of Hermitian holomorphic bundles_ , Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 2, 209–212. MR MR854556 (88d:58107) * [18] , _Arithmetic intersection theory_ , Publ. Math. IHES 72 (1990), 94–174. * [19] , _Characteristic classes for algebraic vector bundles with hermitian metric I, II_ , Annals of Math. 131 (1990), 163–203,205–238. * [20] , _An arithmetic Riemann-Roch theorem_ , Invent. Math. 110 (1992), 473–543. * [21] P. Griffiths and J. Harris, _Principles of algebraic geometry_ , John Wiley & Sons, Inc., 1994. * [22] M. Gros, _Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique_ , Mém. Soc. Math. France (N.S.) (1985), no. 21, 87. MR MR844488 (87m:14021) * [23] A. Grothendieck, _La théorie des classes de Chern_ , Bull. Soc. Math. France 86 (1958), 137–154. * [24] V. Guillemin and S. Sternberg, _Geometric asymptotics_ , American Mathematical Society, Providence, R.I., 1977, Mathematical Surveys, No. 14. MR MR0516965 (58 #24404) * [25] L. Hörmander, _The analysis of linear partial differential operators. I_ , second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990, Distribution theory and Fourier analysis. MR MR1065993 (91m:35001a) * [26] U. Jannsen, _Deligne homology, Hodge- $D$-conjecture and motives_, in Rapoport et al. [30], pp. 305–372. * [27] S. Kawaguchi and A. Moriwaki, _Inequalities for semistable families of arithmetic varieties_ , J. Math. Kyoto Univ. 41 (2001), no. 1, 97–182. MR MR1844863 (2002f:14036) * [28] J.W. Milnor and J.S. Stasheff, _Characteristic classes_ , Annals of Math. Studies, vol. 76, Princeton University Press, Princeton, New Jersey, 1974. * [29] Ch. Mourougane, _Computations of Bott-Chern classes on $\mathbb{P}(E)$_, Duke Mathematical Journal 124 (2004), 389–420. * [30] M. Rapoport, N. Schappacher, and P. Schneider (eds.), _Beilinson’s conjectures on special values of ${L}$-functions_, Perspectives in Math., vol. 4, Academic Press, 1988. * [31] C. Soulé, D. Abramovich, J.-F. Burnol, and J. Kramer, _Lectures on Arakelov Geometry_ , Cambridge Studies in Advanced Math., vol. 33, Cambridge University Press, 1992. * [32] Y. Zha, _A general arithmetic Riemann-Roch theorem_ , Ph.D. thesis, University of Chicago, 1998. José I. Burgos Gil Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) burgos@icmat.es, jiburgosgil@gmail.com Temporary Address: Centre de Recerca Matemática CRM UAB Science Faculty 08193 Bellaterra Barcelona, Spain Răzvan Liţcanu University Al. I. Cuza Faculty of Mathematics Bd. Carol I, 11 700506 Iaşi Romania litcanu@uaic.ro	arxiv-papers	2009-02-03T14:34:53	2024-09-04T02:49:00.384750	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "J. I. Burgos Gil and R. Litcanu", "submitter": "Jos\\'e Ignacio Burgos Gil", "url": "https://arxiv.org/abs/0902.0430" }
0902.0605	# Gravitational Stability of Vortices in Bose-Einstein Condensate Dark Matter Mark N Brook1 and Peter Coles2 1 School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD, UK 2 Cardiff School of Physics and Astronomy, Cardiff University, Queens Buildings, 5 The Parade, Cardiff, CF24 3AA, UK ppxmb3@nottingham.ac.uk ###### Abstract We investigate a simple model for a galactic halo under the assumption that it is dominated by a dark matter component in the form of a Bose-Einstein condensate involving an ultra-light scalar particle. In particular we discuss the possibility if the dark matter is in superfluid state then a rotating galactic halo might contain quantised vortices which would be low-energy analogues of cosmic strings. Using known solutions for the density profiles of such vortices we compute the self-gravitational interactions in such halos and place bounds on the parameters describing such models, such as the mass of the particles involved. ###### pacs: 03.75.Nt, 11.27.+d, 95.35.+d ## 1 Introduction In the standard model of galaxy formation, the visible component of a galaxy is supposed to be embedded in an invisible halo of non-baryonic matter [1, 2]. This dark component is further supposed to be cold, meaning that it is usually assumed to consist of very heavy particles with very low thermal velocities. However, it has been known for some time that Cold Dark Matter (CDM) models have certain problems in reproducing observable properties of galaxies, among them being the predicted presence of central density cusps and the overabundance of small scale structure [3, 4, 5]. In the light of these issues, some authors (e.g. [6]) have suggested that the Dark Matter could instead consist of ultralight particles possessing a de Broglie wavelength sufficiently large that quantum-mechanical effects might manifest themselves on astrophysically interesting scales. Such models would naturally predict smoother and less centrally concentrated galaxy haloes owing than in the CDM case. Advocating a particular version of this idea, Silverman & Mallett [7] suggested a symmetry breaking mechanism for the production of such a particle, based upon a real-valued scalar field. Although in this case the symmetry breaking mechanism provides a nice example of particle production in a universe with a cosmological constant, symmetry breaking with a real scalar field generically produces a catastrophic domain wall problem [8], and this example would seem to be no exception [9] so this is probably not a viable scenario. However, these papers consider the possibility that the Dark Matter component resides in a Bose-Einstein Condensate (BEC). The dynamics and possible observational consequences of a Cosmological fluid with such properties has been investigated [10], using techniques developed in the field of condensed matter physics. The equation describing a BEC is known to condensed matter theorists as the Gross-Pitaevskii (GP) equation, but is probably more familiar to cosmologists as the nonlinear Schrödinger equation (NLSE). In condensed matter theory, the term Bose-Einstein Condensate is usually applied to a dilute bosonic gas confined by an external potential, the bosons occupying the lowest available quantum state. Typically, in the limit of large particle number, the density distribution of the condensate is taken to be described by a macroscopic wave-function that is considered to be a quantum field. This field is manipulated by the Gross-Pitaevskii equation, or nonlinear Schrödinger equation, rather than working with the usual creation and annihilation operators of quantum mechanics. The density distribution of the condensate can be represented by a macroscopic wave-function of the same form as the ground state wave-function of a single particle. The momentum distribution of the condensate is obtained by taking the Fourier transform of this wave-function. In an experimental setup, the occurrence of a Bose- Einstein condensate is confirmed by a sharp peak in the momentum space distribution of the gas of particles. More speculatively, the concept of a BEC can also be applied to such hypothetical particles as axions or ghosts. In this context, the axion field, for example, is coherent and has relatively small spatial gradients. The gradient energy can be interpreted as particle momenta, which will be the same and small for each particle, hence giving a sharp peak in the momentum space distribution as in the case of the more familiar BEC described above. In quantum field theory, a condensate corresponds to a non-zero expectation value for some operator in the vacuum and, in the limit of large quantum number, this condensate can be considered to be a classical field. This is a good model for the condensate of Cooper pairs in a superconductor, or for helium atoms in a superfluid [11]. The usual, linear Schrödinger equation, coupled to the Poisson equation can be used to model many phenomena in Cosmology. As well as modelling a quantum mechanical system, as in [6], it has also been used as a classical wave equation to model structure formation. It has been shown that using the Condensed Matter concept of a Madelung transformation to yield the Euler and Continuity equations from the Schrödinger equation, applies as well as to a Cosmological fluid as it does to fluids in the laboratory [12, 13, 14, 15, 16, 17]. Silverman & Mallett [7] also considered the rotation of a galactic-scale dark matter halo. Using a phenomenological description taken directly from condensed matter, they concluded that a galactic halo should be threaded by a lattice of quantised vortices, as a consequence of the rotation of that galaxy. Indeed from studies of rotating BECs and quantum turbulence [18, 19], it would seem to be difficult to prevent such vortices from forming. The galaxy velocity rotation curve produced by these authors reproduces the approximate form of observed rotation curves. A similar conclusion was reached in Yu and Morgan [20]. This paper considered stationary cylindrical solutions of a complex $\phi^{4}$ scalar field model, coupled to gravity. These solutions are Nielson-Olesen vortices, also known as local U(1) Cosmic Strings [8]. To describe the motion of these vortices in the galaxy, Yu and Morgan’s procedure was to calculate the motion of one vortex according to a gradient in the phase induced by the surrounding vortices. There are many models using the Schrödinger-Poisson, or the relativistic Einstein-Klein-Gordon, system to describe slightly different physical processes. A non-exhaustive list includes scalar field dark matter [21, 22], boson stars [23], Oscillatons [24]; condensate stars [25], repulsive dark matter [26] and fluid dark matter [27, 28], as well as the fuzzy dark matter and classical fluid approaches that we have already mentioned, and the more established theories such as the Abelian-Higgs model in field theory, and the Landau-Ginzberg model in condensed matter. We will not attempt a thorough review of each model here, except to say that it is sometimes difficult to explicitly distinguish between them. The effects of the interaction of gravity with a coherent state of matter, such as a BEC, have certainly been considered [29, 30], and prompted the question of whether it is actually possible for DM to be in a coherent quantum state, if the only interaction with visible matter is gravitational. Penrose has also used the Schrödinger-Poisson system during his ‘Quantum State reduction’ research program [31]. In this paper we seek to determine some of the properties of a quantised vortex residing in a galactic-scale Bose-Einstein Condensate dark matter. In particular, we will place bounds on the parameters that are used to describe such a vortex. For the purposes of this paper we presume that the DM does indeed consist of a BEC, formed at an earlier stage of Cosmological history and described by the coupled nonlinear Schrödinger-Poisson system, and that vortices are present in this cosmological fluid. In Section 2 we introduce the basic formalism for describing a BEC using the Gross-Pitaevskii (nonlinear Schrödinger) equation, and vortices within it. In Section 3 we discuss coupling the NLSE to the Poisson equation. In Sections 4 and 5 we look at some of the properties of a vortex as a result of gravitational coupling. We present some results in Section 6 and a discussion in Section 7. An appendix contains some of the approximations we have used in our work, and is referenced in the main body of the paper. ## 2 Setup For our discussion, we use some of the conventions and proceedures set out by Berloff & Roberts [32], and Pethick & Smith [11]. The nonlinear Schrödinger equation is written in the form $i\hbar\Psi_{t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+\Psi\int\|\Psi(x^{\prime},t)\|^{2}V(\|x-x^{\prime}\|)dx^{\prime},$ (1) where $m$ is the mass of a particle in the BEC, and $V(\|x-x^{\prime}\|)$ is the interaction potential between bosons. The potential is simplified for a weakly interacting Bose system by replacing $V(\|x-x^{\prime}\|)$ with a $\delta$-function repulsive potential of strength $V_{0}$, giving $i\hbar\Psi_{t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+V_{0}\|\Psi\|^{2}\Psi.$ (2) Defining a state that is independent of time to be the ‘laboratory frame’, $\Psi=\exp(iE_{\upsilon}/\hbar)$, it is then possible to consider deviations from that state by considering the evolution of $\psi$, where $\psi=\Psi\exp(iE_{\upsilon}t/\hbar)$. Here, $E_{\upsilon}$ is the chemical potential of a boson, in the sense that it is the increase in ground state energy when one boson is added to the system. The nonlinear Schrödinger equation used for subsequent analysis is then $i\hbar\psi_{t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+V_{0}\|\psi\|^{2}\psi- E_{\upsilon}\psi.$ (3) Multiplying equation (3) by $\phi^{}$ and subtracting the complex conjugate of the resulting equation we obtain $\frac{\partial\|\psi\|^{2}}{\partial t}={\bf\nabla}.\left[\frac{\hbar}{2mi}(\psi^{}{\bf\nabla}\psi-\psi{\bf\nabla}{\psi^{}})\right].$ (4) We notice that this is of the form of a continuity equation. $\frac{\partial\|\psi\|^{2}}{\partial t}+{\bf\nabla}.(\|\psi\|^{2}{\bf v}).$ (5) We identify $\|\psi\|^{2}$ as the number density $n$, and the related momentum density is given by ${\bf j}=\frac{\hbar}{2i}(\psi^{}{\bf\nabla}\psi-\psi{\bf\nabla}{\psi^{}}),$ (6) which is equivalent to ${\bf j}=mn{\bf v}.$ (7) This defines for us the mass density, as $\rho=mn=m\|\psi\|^{2}$, and the velocity ${\bf v}=\frac{\hbar}{2mi}\frac{(\psi^{}{\bf\nabla}\psi-\psi{\bf\nabla}{\psi^{}})}{\|\psi\|^{2}}.$ (8) As suggested in the introduction, we can make a ‘Madelung transformation’ $\psi=\alpha\exp\left(i\phi_{\omega}\right),$ (9) and, from equation (8), we obtain an expression for the velocity of the condensate ${\bf v}=\frac{\hbar}{m}{\bf\nabla}\phi_{\omega}.$ (10) Here, $\phi_{\omega}$ is the velocity potential. Substituting the Madelung transformation, and taking real and imaginary parts yields the fluid equations: the continuity equation $\frac{\partial\left(\alpha^{2}\right)}{\partial t}+\frac{\hbar}{m}\nabla.(\alpha^{2}\nabla\phi_{\omega})=0;$ (11) and the (integrated) Euler equation: $\hbar\frac{\partial\phi_{\omega}}{\partial t}=\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\alpha}{\alpha}-\frac{1}{2}m{\bf v}^{2}-V_{0}\alpha^{2}+E_{\upsilon}.$ (12) Often, the identification ${\phi_{\omega}}^{\prime}=\frac{\hbar}{m}\phi_{\omega}$ (13) is used, to maintain contact with the more familiar form of the fluid equations: $\frac{\partial\left(\alpha^{2}\right)}{\partial t}+\nabla.(\alpha^{2}\nabla{\phi_{\omega}}^{\prime})=0,$ (14) $\frac{\partial{\phi_{\omega}}^{\prime}}{\partial t}=\frac{\hbar^{2}}{2m^{2}}\frac{\nabla^{2}\alpha}{\alpha}-\frac{(\nabla{\phi_{\omega}}^{\prime})^{2}}{2}-\frac{V_{0}}{m}\alpha^{2}+\frac{E_{\upsilon}}{m}.$ (15) Here the quantum nature of the fluid is evident only in the first term on the right hand side of the second equation, which is often known as the quantum pressure term, although dimensionally it is a chemical potential. This term is relevant only on small scales, where quantum effects become important, such as in a vortex core, or where the condensate meets a boundary. This identification rather hides the quantum nature of the fluid with respect to the fluid velocity, which will become particularly relevant when we start talking about vortices in the next section. By assuming that the condensate reaches a stationary equilibrium state at a distance far from any disturbance, equation (3) gives us the relation $\psi_{\infty}=\left(\frac{E_{\upsilon}}{V_{0}}\right)^{\frac{1}{2}}.$ (16) When the condensate wave-function reaches a boundary, such as the wall of a container, or the core of a vortex is being considered, we can define a distance over which the wave-function changes from zero to its bulk value, or where quantum effects become important [32, 11]. $a_{0}=\frac{\hbar}{(2mE_{\upsilon})^{\frac{1}{2}}}$ (17) This is known as the coherence length, or healing length, as it is the distance over which the wave-function requires ‘healing’. ### 2.1 Vortices We have already seen that the velocity of the condensate is given by ${\bf v}=\frac{\hbar}{m}{\bf\nabla}\phi_{\omega}.$ (18) One would expect then, that the condensate would be irrotational, as ${\bf\nabla}\times({\bf\nabla}f)=0$ (19) for any scalar, $f$. This restricts the motion of the condensate much more than a classical fluid. The circulation around any contour then, should also be zero. By Stokes’ theorm $\Gamma=\oint_{l}{\bf v}.d{\bf l}=\int_{A}({\bf\nabla}\times{\bf v}.d{\bf A}=0$ (20) This condition, known as the Landau state, was first derived in an analysis of superfluid HeII [36], and suggests that rotation of such a condensate should be impossible. Experiments by Osbourne [37] indicated that the condensate did indeed experience rotation. Feynman [38], building on the independent work of Onsager [39], suggested that rotation and hence non-zero circulation could be explained by assuming that the condensate is threaded by a lattice of parallel vortex lines. It is possible to have circulation surrounding a region from which the condensate is excluded, and in this case, this would be the vortex core. To see this, we note that the condensate wave-function must be single valued, and so around any closed contour, the change in the phase of the wave- function $\Delta\phi$ must be a multiple of 2$\pi$. $\Delta\phi_{\omega}=\oint{\bf\nabla}\phi_{\omega}.d{\bf l}=2\pi l$ (21) where $l$ is an integer. We immediately see that the circulation is quantised in units of $h/m$. $\Gamma=\oint{\bf v}.d{\bf l}=\frac{\hbar}{m}2\pi l=l\frac{h}{m}$ (22) To obtain vortex solutions, we work in cylindrical coordinates $(r,\chi,z)$, and look for a static solution of the nonlinear Schrödinger equation, equation (3). To satisfy the requirement of single-valuedness, the condensate wave- function must vary as $\exp(in\chi)$, with $n$ integer. We make the vortex ansatz $\psi=R(r)\exp(in\chi).$ (23) It is interesting to note the similarity between this procedure, and that used in obtaining Nielson-Olesen vortices, or Cosmic Strings, in the Abelian-Higgs model [8]. This was mentioned in Section 1, and will be useful shortly for obtaining equation (27), as shown in A.1. We can obtain an expression for the velocity of a vortex by substituting the vortex ansatz (23) into equation (8) ${\bf v}_{\omega}=\frac{\hbar n}{r}\frac{1}{m}\bf{\hat{\chi}},$ (24) and we note again the discrete nature of the allowed values of velocity. From now on we will consider only $n=1$ vortices. Vortices with $n>1$ are generally expected to be unstable, from energy considerations (see for example Chapter 9.2.2 of [11]), and will break up into several $n=1$ vortices to make up a vortex lattice, as described above. We can note further that Cosmic Strings with winding numbers $n>1$ are also unstable to perturbations [8]. Such defects break down to several $n=1$ configurations in both a Condensed Matter environment, and a High Energy Field Theoretic one. Feynman initially introduced quantised vortices as a purely theoretical tool with which to explain the rotation of the condensate, but the experimental verification of the quantisation of rotational velocities (e.g. by [40]) demonstrated that these vortices were indeed real. The density profile of a vortex ($\rho(r)=m\|R(r)\|^{2}$) is defined by the vortex equation, which results from substituting the vortex ansatz into equation (3) $-\frac{\hbar^{2}}{2mE_{\upsilon}}{\Bigg{[}}\frac{d^{2}R(r)}{dr^{2}}+\frac{1}{r}\frac{dR(r)}{dr}-\frac{1}{r^{2}}R(r){\Bigg{]}}+\frac{V_{0}}{E_{\upsilon}}{R(r)}^{3}-R(r)=0$ (25) From equation (16) we see that the density far from the vortex is given by $\rho_{\infty}=mR_{\infty}=m\frac{E_{\upsilon}}{V_{0}}.$ (26) Analytic solutions of this equation are not known so it must be solved numerically. For our anaylses we will use the approximation $R(r)\simeq\left(\frac{E}{V_{0}}\right)^{1/2}\left[1-\exp(-r/a_{0})\right],$ (27) as discussed in A.1. ## 3 Self-gravity of a BEC Vortex In considering Bose-Einstein condensates on scales relevant to structure formation in the universe, we must necessarily include gravitational effects. BECs are typically sufficiently dilute that the mass densities are not very large, and so a Newtonian approximation is sufficient. Gravitational effects can be added to the BEC by including a term in the nonlinear Schrödinger equation that couples to the Poisson equation. We then have a pair of equations modelling a gravitationally coupled fluid. $i\hbar\psi_{t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+V_{0}\|\psi\|^{2}\psi- E_{\upsilon}\psi+m\phi_{G}\psi$ (28) $\nabla^{2}\phi_{G}=4\pi G\rho=4\pi Gm\|\psi\|^{2}.$ (29) ### 3.1 Vortices in Gravitationally Coupled BECs To obtain vortex solutions, we again work in cylindrical coordinates $(r,\chi,z)$, and substitute the vortex ansatz $\psi=R(r)exp(i\chi)$ into equations (28) and (29). The system of equations describing a gravitationally coupled BEC fluid become $-\frac{\hbar^{2}}{2mE_{\upsilon}}{\Bigg{[}}\frac{d^{2}R(r)}{dr^{2}}+\frac{1}{r}\frac{dR(r)}{dr}-\frac{1}{r^{2}}R(r){\Bigg{]}}+\frac{V_{0}}{E_{\upsilon}}{R(r)}^{3}-R(r)+m\phi_{G}(r)=0$ (30) $\nabla^{2}\phi_{G}(r)=\frac{d^{2}\phi_{G}(r)}{dr^{2}}+\frac{1}{r}\frac{d\phi_{G(r)}}{dr}=4\pi GmR(r)^{2}$ (31) Ideally, we would like to find a solution describing the function $R(r)$ in this system, so we can compare the density profile of a quantum vortex, to that of one that is gravitationally coupled. However, finding a full simultaneous solution to these coupled equations is difficult. Firstly, because the nonlinear Schrödinger equation itself is not soluble analytically. Secondly, because the vortex density tends to a constant, so the Newtonian potential tends to diverge, and thirdly because these equations do not define the vortex velocity, which would be providing the centripetal force to withstand the gravitational collapse. In other words, all the variables required to provide a fully simultaneous static solution are not defined within these two equations. ## 4 Vortex Stability in Gravitationally Coupled BECs Rather than solving the coupled equations (28) and (29) directly, we can make some arguments regarding the stability of a gravitationally coupled BEC vortex, and consequently give some bounds on the parameters that describe it. Our analysis is based upon consideration of the circular velocity of a BEC vortex, $v_{\omega}(r)$, and the radial velocity induced from gravitational attraction, $v_{\rm G}(r)$. $v_{\omega}(r)$ is the velocity that the vortex density distribution is moving at, for a particular r, while $v_{\rm G}(r)$ would be the velocity experienced by a test particle orbiting that density distribution, at a distance r. To sustain a vortex, $v_{\omega}(r)$ must at least be greater than $v_{\rm G}(r)$, otherwise the quantum-mechanical forces at work in the vortex are not sufficiently strong to hold itself up against gravitational collapse. That is, the vortex is spinning too slowly to provide enough centripetal force to balance the gravitational force. For stability, we therefore have the bound, $v_{\omega}(r)\geq v_{\rm G}(r)$ (32) ### 4.1 Gravitational Field of a Cylindrically Symmetric System To obtain $v_{\rm G}(r)$, we turn to Gauss’s Law to determine the gravitational field of a cylindrically symmetric mass distribution, and hence obtain the radial gravitational velocity of a test particle moving in the field of that system. Gauss’s law is $\oint{\bf g}\cdot d{\bf A}=-4\pi GM_{\rm encl}$ (33) The density, $\rho(r)=m\|R(r)\|^{2}$, is already determined in terms of the cylindrical r co-ordinate, as it is a solution of the vortex equation. The mass enclosed is the density pervading a cylinder of radius r and length L. $M_{\rm encl}=L\int_{0}^{r}2\pi r\rho(r)dr$ (34) The left-hand side of Gauss’s law, in cylindrical co-ordinates, is $\int grd\chi dz,$ (35) where the integral over the z co-ordinate is again L, the length of the vortex. Gauss’s law, then, gives us $gr2\pi L=-4\pi G2\pi L\int_{0}^{r}\rho(r)rdr$ (36) giving $g=-\frac{4\pi Gm}{r}\int_{0}^{r}\|R(r)\|^{2}rdr.$ (37) The sign is negative as we have chosen an outward-pointing surface normal in our formulation of Gauss’s Law, equation (33), which indicates that the gravitational flux will always be towards the origin. This leads to the slightly counter-intuitive conclusion that a hole (the vortex) in a constant mass density background would seem to produce a gravitational force towards it, but this is really a manifestation of the (extremely) thick shell condition. Viewed another way, this static configuration will want to act to collapse in, and close the hole. It is this force that is ‘unopposed’ in equations (30) and (31). This need not concern us further, as it is the magnitude that is required for our argument. The magnitude of the induced centripetal force is $g=\frac{{v_{\rm G}}^{2}}{r}$ (38) and the gravitational circular velocity profile $v_{\rm G}$ is given by ${{v_{G}}(r)}^{2}=4\pi G\int_{0}^{r}\rho(r)rdr=4\pi Gm\int_{0}^{r}\|R(r)\|^{2}rdr.$ (39) ## 5 Bounds on Parameters We now have expressions for $v_{\rm G}(r)$ and $v_{\omega}(r)$, equations (24) and (39), to go in the bound given by equation (32). In Figure 1 we plot, as an example, $v_{\omega}(r)$ and $v_{\rm G}(r)$ and the density profile for comparison. For this example, we have used values of m $=$ 3.56 $\times$ 10-59kg (2 $\times$ 10-23 eV), Eυ $=$ 2.5 $\times$ 10-49 J (1.56 $\times$ 10-30eV) and $V_{0}$ $=$ 4.45 $\times$ 10-84 Jm3 (3.7 $\times$ 10-45 eV-2) as explained in Appendix A.2. Figure 1: Velocity Profiles for vG (green) and $v_{\omega}$ (blue). Density profile plotted schematically for comparison (red). The bound on stability, $v_{\omega}(r)\geq v_{\rm G}(r)$, will always be violated at some point, as outside the vortex core $v_{\omega}(r)\sim 1/r$ and $v_{\rm G}(r)\sim r$. We must specify what might be an acceptable value of r for $v_{\omega}(r)$ and $v_{\rm G}(r)$ to cross. For a vortex to exist, the density profile should be fully established. We take this to mean that the density has essentially reached its background level. From the scaled density profile in discussed in A.1, and plotted in Figure (4), we see that the density reaches its background level at a value of about ten times the healing length. Using equation (27) in (39), equations (39) and (24) in (32), and substituting for $E_{\upsilon}$ from equation (17) we obtain $\frac{\sqrt{2\pi}}{2}\left(\frac{G\hbar^{2}}{V_{0}{a_{0}}^{2}}\left[2r^{2}+8r{a_{0}}e^{-\frac{r}{{a_{0}}}}+8{a_{0}}^{2}e^{-\frac{r}{{a_{0}}}}-2r{a_{0}}e^{-\frac{2r}{{a_{0}}}}-{a_{0}}^{2}e^{-\frac{2r}{{a_{0}}}}\right]\right)^{\frac{1}{2}}\leq\frac{\hbar}{mr}.$ (40) We will fix the healing length $a_{0}$, and plot $V_{0}$ against $m$ (fixing $a_{0}$ and $m$ fixes $E_{\upsilon}$, from equation (17)) to give an allowed range of parameter values. We will do this for various values of $a_{0}$, and for various values of $r$, which we will take to be an integer number of healing lengths, $r=na_{0}$, with the minimum $n=10$ as outlined above. Equation (40) then becomes $V_{0}\geq\frac{\pi}{2}Gm^{2}n^{2}\left(2n^{2}{a_{0}}^{2}+8n{a_{0}}^{2}e^{-n}+8{a_{0}}^{2}e^{-n}-2n{a_{0}}^{2}e^{-2n}-{a_{0}}^{2}e^{-2n}\right).$ (41) ### 5.1 Other Bounds We can obtain some other bounds to cut off other bits of parameter space. The asymptotic vortex density is given by $\rho_{\infty}=m\left(\frac{E_{\upsilon}}{V_{0}}\right).$ (42) If the vortex exists as a component of a galaxy, then there is a minimum and maximum density that the vortex can have, given by the maximum and minimum known values of mass density within a galaxy: $\rho_{\rm min}\leq\rho_{\infty}\leq\rho_{\rm max}.$ (43) The value of $E_{\upsilon}$ in equation (42) is fixed (as we are fixing the healing length), and so the bound on the density becomes a bound on $V_{0}$. $\frac{\hbar^{2}}{2{a_{0}}^{2}\rho_{max}}\leq V_{0}\leq\frac{\hbar^{2}}{2{a_{0}}^{2}\rho_{min}}.$ (44) Equation (41) gives a lower bound on $V_{0}$, so to obtain an upper bound, we use the second half of the above relation. $V_{0}\leq\frac{\hbar^{2}}{2{a_{0}}^{2}\rho_{min}}.$ (45) Another bound is provided because the vortex velocity should never exceed the speed of light, $v_{\omega}=\frac{\hbar}{mr}\leq c.$ (46) It can be seen from equation (24) that the vortex velocity increases with decreasing radius. This relation breaks down within the vortex core, $a_{0}$, where the vortex velocity diverges. Finding an appropriate description is a topic of some interest in Condensed Matter theory [33]. We evaluate the maximum vortex velocity at a distance of $5a_{0}$ from the origin. i.e. in a regime where we are sure the relation holds. This gives a bound on the mass. $m\geq\frac{\hbar}{5ca_{0}}.$ (47) ### 5.2 Values To see how the restriction on $m$ and $V_{0}$ varies, we can think of a range of healing lengths that cover all possible scales in a galaxy. $\displaystyle 1\times 10^{10}{\rm m}\quad(3.2\times 10^{-10}{\rm kpc},\quad\sim 7\times 10^{-2}{\rm AU})\leq a_{0}$ (48) $\displaystyle a_{0}\leq 1\times 10^{22}{\rm m}\quad(324{\rm kpc})$ (49) This range of scales takes us from sub solar system, to that of the largest known galaxies (e.g. IC 1101 in the Abell 2029 cluster [34]). At fixed $a_{0}$ we will also cover a large range of n; the number of healing lengths where the velocity profiles cross. For the bound given in equation (45), we take the minimum density found within a galaxy to be the cosmological density. This minimum must necessarily be close to the critical density of the universe. $\rho_{\rm min}=\rho_{c}=\frac{3H_{0}^{2}}{8\pi G}.$ (50) With $H_{0}=70$ km s-1 Mpc-1, this gives a value of $\rho_{\rm min}=9.2\times 10^{-27}$ kg m-3. ## 6 Results In Figure (2), we show a region of the $V_{0}-m$ parameter space for the healing length $a_{0}=1\times 10^{16}$ m ($\sim$ 1 pc). The lower bound on $V_{0}$ is given when $v_{\omega}$ and $v_{G}$ cross at a value of ten times the healing length, $n=10$. A vortex could be considered more stable if $v_{\omega}$ and $v_{G}$ cross at a greater number of n, moving us up into the allowed triangular region. However, this can soon reach the minimum density bound on $V_{0}$. A value of $n=10^{6}$ is also plotted, and it is clear that this is outside the bounded region. The lines bounding the region of allowed parameter values are given by equations (41), (47) and (45). Figure 2: Allowed region in $V_{0}-m$ parameter space, for a healing length of $a_{0}=1\times 10^{16}$ m ($\sim$ 1 parsec) Figure (3) shows allowed regions for various healing lengths, all at a value of $n=10$. We see that as we move to smaller values of $a_{0}$, the allowed bounds on $m$ and $V_{0}$ both move up, as expected from equations (45) and (47). More physically, as the mass of the particle is increased, the repulsive potential $V_{0}$ must increase to balance the stronger gravitational force. Figure 3: Allowed regions in $V_{0}-m$ parameter space, with $n=10$. Healing lengths as labelled. ## 7 Discussion In this paper we have used techniques from condensed matter theory in a cosmological setting to place bounds on parameters describing a Dark Matter candidate, on the assumption that the Dark Matter halo consists of a Bose- Einstein Condensate, in which quantised vortices reside. In the case of a laboratory BEC, self-gravitational forces are not important and even in that case analytical progress is limited. Using a simple physical argument, however, we have shown how rough limits on the consistency of such a model can be imposed. Considering a Dark Matter particle of a particular mass, and a vortex of a certain radius, places constraints on the values that the chemical potential, and interaction potential can take. There remain sizeable regions of parameter space in which the model appears to be viable. In future work, it would be interesting to investigate further whether a Dark Matter candidate could reside in a coherent quantum state, if the only interaction was gravitational. A less ambitious undertaking would be to see if the Madelung transformation provides a solution to the problem of defining all the relevant variables, as suggested in Section 3.1. This would give a set of fluid equations that includes the velocity giving rise to the stabilising centripetal force. One problem to be anticipated in such a solution, would be that the velocity in the vortex core would still be ill-defined, as alluded to in Section 5. The system would therefore have to be solved by a more complete numerical method than we have been able to implement so far. ## Acknowledgements Mark Brook acknowledges support from the Science & Technology Facilities Council, and useful comments from Sean Carroll and David Tong. ## Appendix A Approximations ### A.1 Approximations to the Density Profile The numerical solution to the NLSE can be cumbersome to work with, so we provide some discussion of some approximations that can be used. It is possible to scale the the variables $r$ and $R(r)$ in equation (25) to obtain a scale-free equation. Scaling $r$ by the healing length, $r^{\prime}=r/a_{0}$, and $R(r)$ by the steady state value, $R^{\prime}(r^{\prime})=R(r)/R_{\infty}$ we obtain $\frac{d^{2}R^{\prime}(r^{\prime})}{{dr}^{\prime 2}}+\frac{1}{r^{\prime}}\frac{dR^{\prime}(r^{\prime})}{dr^{\prime}}-\frac{1}{{r^{\prime}}^{2}}R^{\prime}(r^{\prime})-{R^{\prime}(r^{\prime})}^{3}+R^{\prime}(r^{\prime})=0.$ (51) Our first idea for an approximation comes from the field of cosmic strings. The method of approximation is detailed in [8]. Looking at the profile of the Higgs field in a Nielson-Olesen vortex we see that it can be written, in a similarly scaled way, as $\frac{d^{2}f^{\prime}(r^{\prime})}{dr^{\prime 2}}+\frac{1}{r^{\prime}}\frac{df^{\prime}(r^{\prime})}{dr^{\prime}}-\frac{1}{r^{\prime 2}}f(r^{\prime})(\alpha(r^{\prime})-1)^{2}-\frac{\lambda}{2}f^{\prime}(r^{\prime})({f^{\prime}(r^{\prime})}^{2}-1)=0$ (52) Here $\alpha$ is a gauge term arising from the coupling to Electromagnetism, and $\lambda$ is determined by the potential term of the theory. It is possible to linearise equation (52) to obtain a modified Bessel function as the first order approximation to f’(r’) - the zeroth order being 1. This happens in the string case, because the gauge contributions serve to cancel one of the terms, leaving the modified Bessel’s equation. The linearised version of equation (51) does not quite reduce to a modified Bessel’s equation, but taking our lead from the cosmic string example, we write $R^{\prime}(r^{\prime})\sim 1-\exp(-r^{\prime}).$ (53) Another approximation, which might seem to be more accurate, was developed by Berloff [35] in a condensed matter context. The Padé approximation has the same asymptotics at $r=0$ and $r=\infty$ as the function one is trying to approximate. The Padé approximation in this case gives $R^{\prime}(r^{\prime})\sim\sqrt{\frac{{r^{\prime}}^{2}(0.3437+0.0286{r^{\prime}}^{2})}{1+0.3333{r^{\prime}}^{2}+0.0286{r^{\prime}}^{4}}}.$ (54) This solution is plotted in Figure 4 along with the numeric solution given by equation (51), and the previous approximation, equation (53). Figure 4: Numeric solution to Equation (51) (blue), the Padè approximation equation (54) (red), and the scaled approximation used in this analysis, equation (53) (green). The Padé approximation is indeed much more accurate in the small and large r regions. However, the Padé approximation has the tendency to overestimate the density in the central region, producing a density function whose derivative is negative in this region. As discussed in the main body of this paper, the gravitational potential is proportional to the density, and so the gravitational force will be proportional to the derivative of the density function. If we chose to use the Padé approximation for our density profile, we could be potentially misled by its behaviour in the central region. We will use the approximation $R(r)=\left(\frac{E_{\upsilon}}{V_{0}}\right)^{\frac{1}{2}}\left[1-\exp[-r/a_{0}]\right).$ (55) ### A.2 Approximations for Parameters Defining the BEC To enable us to obtain actual values for the velocity and density profiles that we are considering, we must provide values for the parameters $m$, $V_{0}$, and $E_{\upsilon}$. The properties of Dark Matter particles are, by their very nature, unknown, so we must make some approximations. We use the analysis in [7] to provide us with some data values. The mass of the Bose Einstein Condensate Dark Matter particle in that paper is 3.56 $\times$ 10-59 kg (2 $\times$ 10-23 eV). Their analysis is based on the mass and angular rotation of the Andromeda galaxy. The mean density is given as 2 $\times$ 10-24kg m-3, and they estimate that the vortex line density in the galaxy would be about 1 vortex per 208 kpc2. This gives a vortex radius of $r_{\omega}\sim$ 2.5 $\times$ 1020 m. We again turn to vortex lattices in condensed matter systems to provide us with some further estimates of vortex properties in a BEC. Taking the distance between two vortices to be twice the vortex radius, we note from experimental observations of vortex lattices in a BEC that the vortex density reaches the normal density at about half the vortex radius; see, for example, Figure 9.3 in [11], taken from [41]. From Figure (4), we also see that the vortex density reaches the normal condensate density at around five healing lengths. This gives us an estimate of ${r_{\omega}}/{2}=5a_{0}$. We then use $r_{\omega}\sim$ 2.5 $\times$ 1020 m, $a_{0}={\hbar}/(2mE_{\upsilon})^{\frac{1}{2}}$, and $\rho_{\infty}=m{E_{\upsilon}}/{V_{0}}$ to give estimates for $E_{\upsilon}$ and $V_{0}$. With these approximations we find values of $E_{\upsilon}$ $=$ 2.5 $\times$ 10-49 J (1.56 $\times$ 10-30eV) and $V_{0}=4.45\times$ 10-84 J m3 (3.7 $\times$ 10-45 eV-2). ## References ## References [1] A. Jenkins et al. [Virgo Consortium Collaboration], Astrophys. J. 499 (1998) 20 [arXiv:astro-ph/9709010]. * [2] P. Coles, Nature 433 (2005) 248. * [3] B. Moore, T. R. Quinn, F. Governato, J. Stadel and G. Lake, Mon. Not. Roy. Astron. Soc. 310, 1147 (1999) [arXiv:astro-ph/9903164]. * [4] J. F. Navarro, C. S. Frenk and S. D. M. White, Astrophys. J. 490, 493 (1997) [arXiv:astro-ph/9611107]. * [5] A. J. Romanowsky et al., Science 301, 1696 (2003) [arXiv:astro-ph/0308518]. * [6] W. Hu, R. Barkana and A. Gruzinov, Phys. Rev. Lett. 85 (2000) 1158 [arXiv:astro-ph/0003365]. * [7] M. P. Silverman and R. L. Mallett, Gen. Rel. Grav. 34 (2002) 633. * [8] A. Vilenkin and E.P.S. Shellard Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge) * [9] M. Brook Ph.D Thesis (in progress) * [10] C. G. Boehmer and T. Harko, JCAP 0706 (2007) 025 [arXiv:0705.4158 [astro-ph]]. * [11] C. J. Pethick and H. Smith Bose-Einstein Condensation in Dilute Gases, Second Edition (Cambridge University Press, Cambridge) * [12] E. A. Spiegel Physica D: Nonlinear Phenomena, 1, Issue 2, 236-240 (1980) * [13] L. M. Widrow and N. Kaiser, ApJ. Lett 416 L71 * [14] P. Coles, Mon. Not. Roy. Astron. Soc. 330 (2002) 421 [arXiv:astro-ph/0110615]. * [15] P. Coles, arXiv:astro-ph/0209576. * [16] P. Coles and K. Spencer, Mon. Not. Roy. Astron. Soc. 342 (2003) 176 [arXiv:astro-ph/0212433]. * [17] C. J. Short and P. Coles, JCAP 0612 (2006) 012 [arXiv:astro-ph/0605012]. * [18] W. F. Vinen, J. Low. Temp. Phys. 121, 367 (2000) * [19] C. Short Ph.D Thesis, University of Nottingham * [20] R. P. Yu and M. J. Morgan, Class. Quant. Grav. 19 (2002) L157. * [21] T. Matos and F. S. Guzman, Class. Quant. Grav. 17, L9 (2000) [arXiv:gr-qc/9810028]. * [22] F. S. Guzman and L. A. Urena-Lopez, Phys. Rev. D 68, 024023 (2003) [arXiv:astro-ph/0303440]. * [23] E. Seidel and W. M. Suen, Phys. Rev. D 42, 384 (1990). * [24] E. Seidel and W. M. Suen, Phys. Rev. Lett. 66, 1659 (1991). * [25] P. O. Mazur and E. Mottola, arXiv:gr-qc/0109035. * [26] J. Goodman, arXiv:astro-ph/0003018. * [27] A. Arbey, J. Lesgourgues and P. Salati, Phys. Rev. D 68 (2003) 023511 [arXiv:astro-ph/0301533]. * [28] P. J. E. Peebles, arXiv:astro-ph/0002495. ApJ, 534, L127 * [29] S. Carroll Cosmic Variance http://blogs.discovermagazine.com/cosmicvariance/2008/10/24/gravity-is-an-important-force/ * [30] S. Carroll Private Communication * [31] I. M. Moroz, R. Penrose and P. Tod, Class. Quant. Grav. 15, 2733 (1998). * [32] P. H. Roberts and N. G. Berloff In “Quantized Vortex Dynamics and Superfluid Turbulence” edited by C. F. Barenghi, R. J. Donnelly and W. F. Vinen, Lecture Notes in Physics, 571, Springer-Verlag (2001). www.damtp.cam.ac.uk/user/ngb23/publications/review.pdf * [33] M. Sadd, G. V. Chester, L. Reatto Phys. Rev. Lett. 79, 2490 (1997) * [34] J. M. Uson, S. P. Boughn and J. R. Kuhn Science 250 539 (1990) * [35] N. Berloff J. Phys. A: Math. Gen. 37 (2004) 11729 [arXiv:/cond-mat/0306596] * [36] L. D. Landau J. Phys. Moscow 5 71 (1941) Reprinted in I. M. Khalatnikov, Introduction to the Theory of Superfluidity pg. 185 (New York, W A Benjamin) * [37] D. V. Osbourne Proc. Phys. Soc. A 63 909 (1950) * [38] R. P. Feynman Progress in Low Temperature Physics ed. C J Gorter Vol 1, Ch. 2 (Amsterdam, North Holland) * [39] L. Onsager Nuovo Cimento 6 suppl. 2 249 (1949) * [40] R. E. Packard and T. M. Sanders Jr. Phys. Rev. A 6 799 (1972) * [41] I. Coddington, P. Engels, V. Schweikhard, and E. A. Cornell, Phys. Rev. Lett. 91, 100402 (2003)	arxiv-papers	2009-02-03T22:04:01	2024-09-04T02:49:00.419708	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "Mark N Brook (1,2), Peter Coles (1,3) ((1) School of Physics &\n Astronomy, University of Nottingham, (2) Institute of Cancer Research,\n London, (3) Maynooth University, Ireland)", "submitter": "Peter Coles", "url": "https://arxiv.org/abs/0902.0605" }
0902.0621	Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions\star\star$\star$This paper is a contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions” (July 21–25, 2008, MPIM, Bonn, Germany). The full collection is available at http://www.emis.de/journals/SIGMA/Elliptic-Integrable-Systems.html Fokko J. VAN DE BULT and Eric M. RAINS F.J. van de Bult and E.M. Rains MC 253-37, California Institute of Technology, 91125, Pasadena, CA, USA vdbult@caltech.edu, rains@caltech.edu Received February 01, 2009; Published online June 10, 2009 We describe a uniform way of obtaining basic hypergeometric functions as limits of the elliptic beta integral. This description gives rise to the construction of a polytope with a different basic hypergeometric function attached to each face of this polytope. We can subsequently obtain various relations, such as transformations and three-term relations, of these functions by considering geometrical properties of this polytope. The most general functions we describe in this way are sums of two very-well-poised ${}_{10}\phi_{9}$’s and their Nassrallah–Rahman type integral representation. elliptic hypergeometric functions, basic hypergeometric functions, transformation formulas 33D15 ## 1 Introduction Hypergeometric functions have played an important role in mathematics, and have been much studied since the time of Euler and Gauß. One of the goals of this research has been to obtain hypergeometric identities, such as evaluation and transformation formulas. Such formulas are of interest due to representation-theoretical interpretations, as well as their use in simplifying sums appearing in combinatorics. In more recent times people have been trying to understand the structure behind these formulas. In particular people have studied the symmetry groups associated to certain hypergeometric functions, or the three terms relations satisfied by them (see [9] and [10]). Another recent development is the advent of elliptic hypergeometric functions. This defines a whole new class of hypergeometric functions, in addition to the ordinary hypergeometric functions and the basic hypergeometric functions. A nice recent overview of this theory is given in [19]. For several of the most important kinds of formulas for classical hypergeometric functions there exist elliptic hypergeometric analogues. It is well known that one obtains basic hypergeometric functions upon taking a limit in these elliptic hypergeometric functions. However a systematic description of all possible limits had not yet been undertaken. In this article we provide such a description of limits, extending work by Stokman and the authors [2]. This description provides some extra insight into elliptic hypergeometric functions, as it indicates what relations for elliptic hypergeometric functions correspond to what kinds of relations for basic hypergeometric functions. Conversely we can now more easily tell for what kind of relations there have not yet been found proper elliptic hypergeometric analogues. More importantly though, this description provides more insight into the structure of basic hypergeometric functions and their relations, in the form of a geometrical description of a large class of functions and relations. All the results for basic hypergeometric functions we obtain can be shown to be limits of previously known relations satisfied by sums of two very-well-poised ${}_{10}\phi_{9}$’s and their Nassrallah–Rahman like integral representation. However, we would have been unable to place them in a geometrical picture as we do in this article without considering these functions as limits of an elliptic hypergeometric function. In this article we focus on the (higher-order) elliptic beta integral [15]. For any $m\in\mathbb{Z}_{\geq 0}$ the function $E^{m}(t)$ is defined for $t\in\mathbb{C}^{2m+6}$ satisfying the balancing condition $\prod_{r=0}^{2m+5}t_{r}=(pq)^{m+1}$ by the formula $E^{m}(t)=\Biggl{(}\prod_{0\leq r<s\leq 2m+5}(t_{r}t_{s};p,q)\Biggr{)}\frac{(p;p)(q;q)}{2}\int_{\mathcal{C}}\frac{\prod\limits_{r=0}^{2m+5}\Gamma(t_{r}z^{\pm 1})}{\Gamma(z^{\pm 2})}\frac{dz}{2\pi iz}.$ Here $\Gamma$ denotes the elliptic gamma function and is defined in Section 2, as are the $(p,q)$-shifted factorials $(x;p,q)$. Two important results for the elliptic beta integral are the existence of an evaluation formula for $E^{0}$ and the fact that $E^{1}$ is invariant under an action of the Weyl group $W(E_{7})$ of type $E_{7}$ [11]. A more thorough discussion of the elliptic beta integral is provided in Section 3. The main result of this paper is the following (see Theorems 5.2–5.4), and its analogues for $m=0$, $m>1$. ###### Theorem 1.1. Let $P$ denote the convex polytope in $\mathbb{R}^{8}$ with vertices $e_{i}+e_{j},\quad 0\leq i<j\leq 7,\qquad\frac{1}{2}\left(\sum_{r=0}^{7}e_{r}\right)-e_{i}-e_{j},\quad 0\leq i<j\leq 7.$ Then for each $\alpha\in P$ the limit $B^{1}_{\alpha}(u)=\lim_{p\to 0}E^{1}\big{(}p^{\alpha_{0}}u_{0},\ldots,p^{\alpha_{7}}u_{7}\big{)}$ exists as a function of $u\in\mathbb{C}^{8}$ satisfying the balancing condition $\prod u_{r}=q^{2}$. Moreover, $B^{1}_{\alpha}$ depends only on the face of the polytope which contains $\alpha$ and is a function of the projection of $\log(u)$ to the space orthogonal to that face. ###### Remark 1.1. The polytope $P$ was studied in an unrelated context in [4], where it was referred to as the “Hesse polytope”, as antipodal pairs of vertices are in natural bijection with the bitangents of a plane quartic curve. As stated the theorem is rather abstract, but for each point in this polytope we have an explicit expression of the limit as either a basic hypergeometric integral, or a basic hypergeometric series, or a product of $q$-shifted factorials (and sometimes several of these options). A graph containing all these functions is presented in Appendix A. We also obtain geometrical descriptions of various relations between these limits $B^{1}_{\alpha}$. Note that the vertices of the polytope are given by the roots satisfying $\rho\cdot u=1$ of the root system $R(E_{8})=\\{u\in\mathbb{Z}^{8}\cup(\mathbb{Z}^{8}+\rho)~{}\|~{}u\cdot u=2\\}$, where $\rho=\\{1/2\\}^{8}$. In particular, the Weyl group $W(E_{7})=\operatorname{Stab}_{W(E_{8})}(\rho)$ acts on the polytope in a natural way, which is consistent with the $W(E_{7})$-symmetry of $E^{1}$. As an immediate corollary of this $W(E_{7})$ invariance we obtain both the symmetries of the limit $B^{1}_{\alpha}$ (determined by the stabilizer in $W(E_{7})$ of the face containing $\alpha$) and transformations relating different limits (determined by the orbits of the face $\alpha$). Special cases of these include many formulas found in Appendix III of Gasper and Rahman [7]. For example, they include Bailey’s four term transformation of very-well-poised ${}_{10}\phi_{9}$’s (as a symmetry of the sum of two ${}_{10}\phi_{9}$’s), the Nassrallah–Rahman integral representation of a very- well-poised ${}_{8}\phi_{7}$ (as a transformation between two different limits) and the expression of a very-well-poised ${}_{8}\phi_{7}$ in terms of the sum of two ${}_{4}\phi_{3}$’s. Three term relations involving the different basic hypergeometric functions can be obtained as limits of $p$-contiguous relations satisfied by $E^{1}$ (and geometrically correspond to triples of points in $P$ differing by roots of $E_{7}$), while the $q$-contiguous relations satisfied by $E^{1}$ reduce to the ($q$-)contiguous relations satisfied by its basic hypergeometric limits. In particular, we see that these two qualitatively different kinds of formulas for basic hypergeometric functions are closely related: indeed, they are different limits of essentially the same elliptic identity! A similar statement can be made for $E^{0}$, which leads to evaluation formulas of its basic hypergeometric limits. Special cases of these include Bailey’s sum for a very-well-poised ${}_{8}\phi_{7}$ and the Askey–Wilson integral evaluation. We would like to remark that a similar analysis can be performed for multivariate integrals. In particular the polytopes we obtain here are the same as the polytopes we get for the multivariate elliptic Selberg integrals (previously called type $I\\!I$ integrals) of [5, 6, 11, 12]. In a future article the authors will also consider the limits of the (bi-)orthogonal functions of [11], generalizing and systematizing the $q$-Askey scheme. The article is organized as follows. We begin with a small section on notations, followed by a review of some of the properties of the elliptic beta integrals. In Section 4 we will describe the explicit limits we consider. In Section 5 we define convex polytopes, each point of which corresponds to a direction in which we can take a limit. Moreover in this section we prove the main theorems of this article, describing some basic properties of these basic hypergeometric limits in terms of geometrical properties of the polytope. In Section 6 we harvest by considering the consequences in the case we know non- trivial transformations of the elliptic beta integral. Section 7 is then devoted to explicitly giving some of these consequences in an example, on the level of ${}_{2\vphantom{1}}\phi_{1\vphantom{2}}$. Section 8 describes some peculiarities specific to the evaluation $(E^{0})$ case. Finally in Section 9 we consider some remaining questions, in particular focusing on what happens for limits outside our polytope. The appendices give a graphical representation of the different limits we obtain and a quick way of determining what kinds of relations these functions satisfy. ## 2 Notation Throughout the article $p$ and $q$ will be complex numbers satisfying $\|p\|,\|q\|<1$, in order to ensure convergence of relevant series and products. Note that $q$ is generally assumed to be fixed, while $p$ may vary. We use the following notations for $q$-shifted factorials and theta functions: $(x;q)=(x;q)_{\infty}=\prod_{j=0}^{\infty}(1-xq^{j}),\qquad(x;q)_{k}=\frac{(x;q)_{\infty}}{(xq^{k};q)_{\infty}},\qquad\theta(x;q)=(x,q/x;q),$ where in the last equation we used the convention that $(a_{1},\ldots,a_{n};q)=\prod_{i=1}^{n}(a_{i};q)$, which we will also apply to gamma functions. Moreover we will use the shorthand $(xz^{\pm 1};q)=(xz,xz^{-1};q)$. Many of the series we obtain as limits are confluent, and in some cases, highly confluent. To simplify the description of such limits, we will use a slightly modified version of the notation for basic hypergeometric series in [7]. In particular we set $\displaystyle{}_{r\vphantom{s}}^{\vphantom{(n)}}\phi_{s\vphantom{r}}^{(n)}\left(\begin{array}[]{c}a_{1},a_{2},\ldots,a_{r}\\\ b_{1},b_{2},\ldots,b_{s}\end{array};q,z\right)=\sum_{k=0}^{\infty}\frac{(a_{1},a_{2},\ldots,a_{r};q)_{k}}{(q,b_{1},b_{2},\ldots,b_{s};q)_{k}}z^{k}\left((-1)^{k}q^{\binom{k}{2}}\right)^{n+s+1-r}.$ In terms of the original ${}_{r}\phi_{s}$ from [7] this is $\displaystyle{}_{r\vphantom{s}}^{\vphantom{(n)}}\phi_{s\vphantom{r}}^{(n)}\left(\begin{array}[]{c}a_{1},a_{2},\ldots,a_{r}\\\ b_{1},b_{2},\ldots,b_{s}\end{array};q,z\right)=\begin{cases}{}_{r\vphantom{s+n}}\phi_{s+n\vphantom{r}}\left(\begin{array}[]{c}a_{1},a_{2},\ldots,a_{r}\\\ b_{1},b_{2},\ldots,b_{s},\underbrace{0,\ldots,0}_{n}\end{array};q,z\right)&\text{ if $n>0$,}\\\ {}_{r\vphantom{s}}\phi_{s\vphantom{r}}\left(\begin{array}[]{c}a_{1},a_{2},\ldots,a_{r}\\\ b_{1},b_{2},\ldots,b_{s}\end{array};q,z\right)&\text{ if $n=0$,}\\\ {}_{r-n\vphantom{s}}\phi_{s\vphantom{r-n}}\left(\begin{array}[]{c}a_{1},a_{2},\ldots,a_{r},\overbrace{0,\ldots,0}^{-n}\\\ b_{1},b_{2},\ldots,b_{s}\end{array};q,z\right)&\text{ if $n<0$.}\end{cases}$ In the case $n=0$ we will of course in general omit the $(0)$, as we then re- obtain the usual definition of ${}_{r}\phi_{s}$. Moreover, when considering specific series, we will often omit the $r$ and $s$ from the notation as they can now be derived by counting the number of parameters. We also extend the definition of very-well-poised series in this way: ${}_{r\vphantom{r-1}}^{\vphantom{(n)}}W_{r-1\vphantom{r}}^{(n)}(a;b_{1},\ldots,b_{r-3};q,z)={}_{r\vphantom{r-1}}^{\vphantom{(n)}}\phi_{r-1\vphantom{r}}^{(n)}\left(\begin{array}[]{c}a,\pm q\sqrt{a},b_{1},\ldots,b_{r-3}\\\ \pm\sqrt{a},aq/b_{1},\ldots,aq/b_{r-3}\end{array};q,z\right).$ Note, however, that this function cannot be obtained simply by setting some parameters to 0 in the usual very-well-poised series. Indeed, setting the parameter $b$ to zero in a very-well-poised series causes the corresponding parameter $aq/b$ to become infinite, making the limit fail. For the basic hypergeometric bilateral series we use the usual notation ${}_{r}\psi_{r}\left(\begin{array}[]{c}a_{1},\ldots,a_{r}\\\ b_{1},\ldots b_{r}\end{array};q,z\right)=\sum_{k\in\mathbb{Z}}\frac{(a_{1},\ldots,a_{r};q)_{k}}{(b_{1},\ldots,b_{r};q)_{k}}z^{k}.$ We define $p,q$-shifted factorials by setting $(z;p,q)=\prod_{j,k\geq 0}(1-p^{j}q^{k}z).$ The elliptic gamma function [14] is defined by $\Gamma(z)=\Gamma(z;p,q)=\frac{(pq/z;p,q)}{(z;p,q)}=\prod_{j,k=0}^{\infty}\frac{1-p^{j+1}q^{k+1}/z}{1-p^{j}q^{k}z}.$ We omit the $p$ and $q$ dependence whenever this does not cause confusion. Note that the elliptic gamma function satisfies the difference equations $\Gamma(qz)=\theta(z;p)\Gamma(z),\qquad\Gamma(pz)=\theta(z;q)\Gamma(z)$ (2.1) and the reflection equation $\Gamma(z)\Gamma(pq/z)=1.$ ## 3 Elliptic beta integrals In this section we introduce the elliptic beta integrals and we recall their relevant properties. As a generalization of Euler’s beta integral evaluation, the elliptic beta integral was introduced by Spiridonov in [15]. An extension by two more parameters was shown to satisfy a transformation formula [16, 11], corresponding to a symmetry with respect to the Weyl group of $E_{7}$. We can generalize the beta integral by adding even more parameters, but unfortunately not much is known about these integrals, beyond some quadratic transformation formulas for $m=2$ [13] and a transformation to a multivariate integral [11]. ###### Definition 3.1. Let $m\in\mathbb{Z}_{\geq 0}$. Define the set $\mathcal{H}_{m}=\\{z\in\mathbb{C}^{2m+6}~{}\|~{}\prod_{i}z_{i}=(pq)^{m+1}\\}/\sim$, where $\sim$ is the equivalence relation induced by $z\sim-z$. For parameters $t\in\mathcal{H}_{m}$ we define the renormalized elliptic beta integral by $E^{m}(t)=\Bigl{(}\prod_{0\leq r<s\leq 2m+5}(t_{r}t_{s};p,q)\Bigr{)}\frac{(p;p)(q;q)}{2}\int_{\mathcal{C}}\frac{\prod\limits_{r=0}^{2m+5}\Gamma(t_{r}z^{\pm 1})}{\Gamma(z^{\pm 2})}\frac{dz}{2\pi iz},$ (3.1) where the integration contour $\mathcal{C}$ circles once around the origin in the positive direction and separates the poles at $z=t_{r}p^{j}q^{k}$ ($0\leq r\leq 2m+5$ and $j,k\in\mathbb{Z}_{\geq 0}$) from the poles at $z=t_{r}^{-1}p^{-j}q^{-k}$ ($0\leq r\leq 2m+5$ and $j,k\in\mathbb{Z}_{\geq 0}$). For parameters $t$ for which such a contour does not exist (i.e. if $t_{r}t_{s}\in p^{\mathbb{Z}_{\leq 0}}q^{\mathbb{Z}_{\leq 0}}$) we define $E^{m}$ to be the analytic continuation of the function to these parameters. Observe that this function is well-defined, in the sense that $E^{m}(t)=E^{m}(-t)$ by a change of integration variable $z\to-z$. We can choose the contour in (3.1) to be the unit circle itself whenever $\|t_{r}\|<1$ for all $r$. If $t_{r}t_{s}=p^{-n_{1}}q^{-n_{2}}$ for some $n_{1},n_{2}\geq 0$, $r\neq s$, then the desired contour fails to exist, but we can obtain the analytic continuation by picking up residues of offending poles before specializing the parameter $t$. In particular the prefactor $\prod_{0\leq r<s\leq 2m+5}(t_{r}t_{s};p,q)$ cancels all the poles of these residues and thus ensures $E^{m}$ is analytic at those points. In this case the integral reduces to a finite sum. Indeed for $t_{0}t_{1}=p^{-n_{1}}q^{-n_{2}}$, we have $\displaystyle E^{m}(t)=(pq/t_{0}t_{1};p,q)\Biggl{(}\prod_{\begin{subarray}{c}0\leq r<s\leq 2m+5\\\ (r,s)\neq(0,1)\end{subarray}}(t_{r}t_{s};p,q)\Biggr{)}\Gamma(pqt_{0}^{2},t_{1}/t_{0})\prod_{r=2}^{2m+5}\Gamma(t_{r}t_{0}^{\pm 1})$ $\displaystyle\phantom{E^{m}(t)=}{}\times\sum_{k=0}^{n_{1}}\prod_{r=0}^{2m+5}\frac{\theta(t_{r}t_{0};q,p)_{k}}{\theta(pqt_{0}/t_{r};q,p)_{k}}\frac{\theta(pqt_{0}^{2};q,p)_{2k}}{\theta(t_{0}^{2};q,p)_{2k}}\sum_{l=0}^{n_{2}}\prod_{r=0}^{2m+5}\frac{\theta(t_{r}t_{0};p,q)_{l}}{\theta(pqt_{0}/t_{r};p,q)_{l}}\frac{\theta(pqt_{0}^{2};p,q)_{2l}}{\theta(t_{0}^{2};p,q)_{2l}},$ where we use the notation $\theta(x;q,p)_{k}=\prod_{r=0}^{k-1}\theta(xp^{r};q)$. There are other singular cases, more difficult to evaluate, but in general $E^{m}(t)$ is analytic on all of ${\mathcal{H}}_{m}$, as follows from [11, Lemma 10.4]. The elliptic beta integral evaluation of [15] is now given by ###### Theorem 3.2. For $t\in\mathcal{H}_{0}$ we have $E^{0}(t)=\prod_{0\leq r<s\leq 5}(pq/t_{r}t_{s};p,q).$ (3.2) Apart from in [15], elementary proofs of this theorem are given in [18] and [11]. Moreover in [11] several multivariate extensions of this result are presented. A second important result is the $E_{7}$ symmetry satisfied by $E^{1}$. Before we can state this in a theorem we first have to introduce the Weyl groups and their actions. ###### Definition 3.3. Let $\rho\in\mathbb{R}^{8}$ be the vector $\rho=(1/2,\ldots,1/2)$. Define the root system $R(E_{8})$ of $E_{8}$ by $R(E_{8})=\\{v\in\mathbb{Z}^{8}\cup(\mathbb{Z}^{8}+\rho)~{}\|~{}v\cdot v=2\\}$. Moreover the root system $R(E_{7})$ of $E_{7}$ is given by $R(E_{7})=\\{v\in R(E_{8})~{}\|~{}v\cdot\rho=0\\}$. Denote by $s_{\alpha}$ the reflection in the hyperplane orthogonal to $\alpha$ (i.e. $s_{\alpha}(\beta)=\beta-(\alpha\cdot\beta)\alpha$ for $\alpha\in R(E_{8})$). The corresponding Weyl group $W(E_{7})$ is the reflection group generated by $\\{s_{\alpha}~{}\|~{}\alpha\in R(E_{7})\\}$. Apart from the natural action of $E_{7}$ on $\mathbb{R}^{8}$, we need the action on $\mathcal{H}_{1}$ given by $wt=\exp(w(\log(t)))$ for $t\in\mathcal{H}_{1}$ (where $\log((t_{0},\ldots,t_{7}))=(\log(t_{0}),\ldots,\log(t_{7}))$ and similarly for $\exp$). Finally we will often meet the $W(E_{7})$ orbit $S$ in $R(E_{8})$ given by $S=\\{v\in R(E_{8})~{}\|~{}s\cdot\rho=1\\}$. Note that the action of $W(E_{7})$ on $\mathcal{H}_{1}$ is well-defined due to the equivalence of $t\sim-t$. Indeed, if we reflect in a root of the form $\rho-e_{i}-e_{j}-e_{k}-e_{l}$ then we have to take square roots of the $t_{j}$, but if we do this consistently (such that $\prod_{j}\sqrt{t_{j}}=pq$), the final result will differ at most by a factor $-1$. A more thorough analysis of this action is given in [2]. Now we can formulate the following theorem describing the transformations satisfied by $E^{1}$ (see [16] and [11], the latter containing also a multivariate extension). ###### Theorem 3.4. The integral $E^{1}$ is invariant under the action of $W(E_{7})$, i.e. for all $w\in W(E_{7})$ and $t\in\mathcal{H}_{1}$ we have $E^{1}(t)=E^{1}(wt)$. In the cited references the transformation has certain products of elliptic gamma functions on one or both sides of the equation, but these factors are precisely canceled by our choice of prefactor. Let us recall the following contiguous relations satisfied by $E^{1}$ [15] (it is shown there for $m=0$, but the proof is identical to that of the $m=1$ case, apart from the use of the Weyl group action). We have rewritten it in a clearly $W(E_{7})$ invariant form. ###### Theorem 3.5. Let us denote $t^{\rho}=\prod_{j}t_{j}^{\rho_{j}}$, and $t\cdot p^{\rho}=(t_{0}p^{\rho_{0}},\ldots,t_{7}p^{\rho_{7}})$. Then if $\alpha,\beta,\gamma\in R(E_{7})$ form an equilateral triangle $($i.e. $\alpha\cdot\beta=\alpha\cdot\gamma=\beta\cdot\gamma=1)$ we have $\displaystyle\prod_{\begin{subarray}{c}\delta\in S\\\ \delta\cdot(\alpha-\beta)=\delta\cdot(\alpha-\gamma)=1\end{subarray}}(t^{\delta}p^{\delta\cdot\beta};q)t^{\gamma}\theta(t^{\beta-\gamma};q)E^{1}(t\cdot p^{\alpha})$ $\displaystyle\qquad\qquad{}+\prod_{\begin{subarray}{c}\delta\in S\\\ \delta\cdot(\beta-\gamma)=\delta\cdot(\beta-\alpha)=1\end{subarray}}(t^{\delta}p^{\delta\cdot\gamma};q)t^{\alpha}\theta(t^{\gamma-\alpha};q)E^{1}(t\cdot p^{\beta})$ $\displaystyle\qquad\qquad{}+\prod_{\begin{subarray}{c}\delta\in S\\\ \delta\cdot(\gamma-\alpha)=\delta\cdot(\gamma-\beta)=1\end{subarray}}(t^{\delta}p^{\delta\cdot\alpha};q)t^{\beta}\theta(t^{\alpha-\beta};q)E^{1}(t\cdot p^{\gamma})=0.$ (3.3) ###### Proof 3.1. Observe that the relation is satisfied by the integrands when $\alpha=e_{1}-e_{0}$, $\beta=e_{2}-e_{0}$ and $\gamma=e_{3}-e_{0}$, where $\\{e_{i}\\}$ form the standard orthonormal basis of $\mathbb{R}^{8}$, due to the fundamental relation $\frac{1}{y}\theta\big{(}wx^{\pm 1},yz^{\pm 1};q\big{)}+\frac{1}{z}\theta\big{(}wy^{\pm 1},zx^{\pm 1};q\big{)}+\frac{1}{x}\theta\big{(}wz^{\pm 1},xy^{\pm 1};q\big{)}=0.$ (3.4) Integrating the identity now proves the contiguous relations for these special $\alpha$, $\beta$ and $\gamma$. As the equation is invariant under the action of $W(E_{7})$, which acts transitively on the set of all equilateral triangles of roots, the result holds for all such triangles. These contiguous relations can be combined to obtain relations of three $E^{1}$’s which differ by shifts along any vector in the root lattice of $E_{7}$ (i.e., the smallest 7-dimensional lattice in $\mathbb{R}^{8}$ containing $R(E_{7})$). In particular the equation relating $E^{1}(t\cdot p^{\alpha})$, $E^{1}(t)$ and $E^{1}(t\cdot p^{-\alpha})$ for $\alpha=e_{1}-e_{0}$ is the elliptic hypergeometric equation studied by Spiridonov in, amongst others, [17]. ## 4 Limits to basic hypergeometric functions In order to obtain basic hypergeometric limits from these integrals we let $p\to 0$. As our parameters can not be chosen independently of $p$ (due to the balancing condition), we have to explicitly describe how they behave as $p\to 0$. Different ways the parameters depend on $p$ require different ways of obtaining the limit. In this section we describe the different limits of interest to us. Using the notation of Theorem 3.5 we see that $u\cdot p^{\alpha}$, for $u$ independent of $p$, is an element of $\mathcal{H}_{m}$ if $\alpha\in\mathbb{R}^{2m+6}$ with $\sum_{r}\alpha_{r}=m+1$, and $u\in\tilde{\mathcal{H}}_{m}=\\{z\in\mathbb{C}^{2m+6}~{}\|~{}\prod_{i}z_{i}=q^{m+1}\\}/\sim$ (where we again have $z\sim-z$). In particular in this section we will describe various conditions on $\alpha$ which ensure that the limit $B^{m}_{\alpha}(u)=\lim_{p\to 0}E^{m}(u\cdot p^{\alpha})$ (4.1) is well-defined, and give explicit expressions for this limit. In particular, for $m=1$ we would like such expressions for $\alpha$ in the entire Hesse polytope as defined in Theorem 1.1. The simplest way to obtain a limit is given by the following proposition. ###### Proposition 4.1. For $\alpha\in\mathbb{R}^{2m+6}$ satisfying $\sum_{r}\alpha_{r}=m+1$ and such that $0\leq\alpha_{r}\leq 1$ for all $r$, the limit in (4.1) exists and we have $B^{m}_{\alpha}(u)=\prod_{\begin{subarray}{c}0\leq r<s\leq 2m+5\\\ \alpha_{r}=\alpha_{s}=0\end{subarray}}(u_{r}u_{s};q)\frac{(q;q)}{2}\int_{\mathcal{C}}(z^{\pm 2};q)\frac{\prod\limits_{r:\alpha_{r}=1}(q/u_{r}z^{\pm 1};q)}{\prod\limits_{r:\alpha_{r}=0}(u_{r}z^{\pm 1};q)}\frac{dz}{2\pi iz},$ where the contour is a deformation of the unit circle which separates the poles at $z=u_{r}q^{n}$ ($\alpha_{r}=0,n\geq 0$) from those at $u_{r}^{-1}q^{-n}$ $(\alpha_{r}=0$, $n\geq 0)$. We want to stress that the limit also exists if the integral above is not well-defined (i.e. when there exists no proper contour, when $u_{r}u_{s}=q^{-n}$ for some $\alpha_{r}=\alpha_{s}=0$). In that case the limit $B^{m}_{\alpha}$ is equal to the analytic continuation of the integral representation to these values of the parameters. ###### Proof 4.1. Observe that we can determine limits of the elliptic gamma function by $\lim_{p\to 0}\Gamma(p^{\gamma}z)=\begin{cases}\frac{1}{(z;q)}&\text{if $\gamma=0$},\\\ 1&\text{if $0<\gamma<1$},\\\ (q/z;q)&\text{if $\gamma=1$}.\end{cases}$ In fact $\Gamma(p^{\gamma}z)$ is well-defined and continuous in $p$ at $p=0$ for $0\leq\gamma\leq 1$. These limits can thus be obtained by just plugging in $p=0$. Similarly observe that $\lim_{p\to 0}(p^{\gamma}z;p,q)=\begin{cases}(z;p,q)&\text{if $\gamma=0$},\\\ 1&\text{if $\gamma>0$}.\end{cases}$ The result now follows from noting that an integration contour which separates the poles at $z=u_{r}q^{n}$ ($\alpha_{r}=0,n\geq 0$) from those at $u_{r}^{-1}q^{-n}$ ($\alpha_{r}=0,n\geq 0$) will also work in the definition of $E_{m}(u\cdot p^{\alpha})$ if $p$ is small enough (as the poles of the integrand created by $u_{r}$’s with $\alpha_{r}>0$ will all converge either to 0 or to infinity; in particular they will remain on the correct side of the contour for small enough $p$). Thus we can just plug in $p=0$ in the integral to obtain the limit. This proof only works when the parameters $u$ are such that there exists a contour for the limiting integral. However, this implies these limits work outside a finite set of co-dimension one divisors. Indeed, on compacta outside these divisors the convergence is uniform. Using the Stieltjes–Vitali theorem we can conclude that the limit also holds on these divisors, and is in fact uniform on compacta of the entire parameter space. Moreover Stieltjes–Vitali tells us that the limit function is analytic in these points as well. A second kind of limit, following [12, § 5], can be obtained by first breaking the symmetry of the integrand. This leads to the following proposition. ###### Proposition 4.2. Let $\alpha\in\mathbb{R}^{2m+6}$ satisfy $\sum_{r}\alpha_{r}=m+1$ and $\alpha_{0}\leq\alpha_{1}\leq\alpha_{2}$. Define $\beta=\alpha_{0}+\alpha_{1}+\alpha_{2}$ and impose the extra conditions $\beta\leq\alpha_{r}\leq-\beta$ for $r=0,1,2$ and $-\beta\leq\alpha_{r}\leq 1+\beta$ for $r\geq 3$. Then the limit in (4.1) exists, and takes one of the following forms: * • If $\alpha_{0}=\alpha_{1}=-\alpha_{2}$ (thus $\beta=\alpha_{0}$), then $\displaystyle B_{\alpha}^{m}(t)=\frac{\prod\limits_{r\geq 3:\alpha_{r}=-\alpha_{0}}(u_{r}u_{0},u_{r}u_{1};q)}{(q/u_{0}u_{2},q/u_{1}u_{2};q)}(u_{0}u_{1};q)^{1_{\\{\alpha_{0}=-1/2\\}}}$ $\displaystyle\phantom{B_{\alpha}^{m}(t)=}{}\times(q;q)\int_{\mathcal{C}}\theta(u_{0}u_{1}u_{2}/z;q)\frac{(q/u_{2}z;q)}{(u_{0}/z,u_{1}/z;q)}$ $\displaystyle\phantom{B_{\alpha}^{m}(t)=}{}\times\frac{\prod\limits_{r\geq 3:\alpha_{r}=1+\alpha_{0}}(qz/u_{r};q)}{\prod\limits_{r\geq 3:\alpha_{r}=-\alpha_{0}}(u_{r}z;q)}\left(\frac{(1-z^{2})(qz/u_{2};q)}{(u_{0}z,u_{1}z;q)}\right)^{1_{\\{\alpha_{0}=-1/2\\}}}\frac{dz}{2\pi iz},$ where the contour separates the downward from the upward pole sequences. Here $1_{\\{\alpha_{0}=-1/2\\}}$ equals 1 if $\alpha_{0}=-1/2$ and 0 otherwise. * • If $\alpha_{0}<\alpha_{1}=-\alpha_{2}$ (again $\beta=\alpha_{0}$), then $\displaystyle B_{\alpha}^{m}(u)=\frac{(q;q)}{(q/u_{1}u_{2};q)}\prod_{\begin{subarray}{c}3\leq r\leq 2m+5\\\ \alpha_{r}=-\alpha_{0}\end{subarray}}(u_{r}u_{0};q)\int_{\mathcal{C}}\theta(u_{0}u_{1}u_{2}/z;q)$ $\displaystyle\phantom{B_{\alpha}^{m}(u)=}{}\times\frac{1}{(u_{0}/z;q)}\frac{\prod\limits_{r\geq 3:\alpha_{r}=1+\alpha_{0}}(qz/u_{r};q)}{\prod\limits_{r\geq 3:\alpha_{r}=-\alpha_{0}}(u_{r}z;q)}\left(\frac{(1-z^{2})}{(u_{0}z;q)}\right)^{1_{\\{\alpha_{0}=-1/2}\\}}\frac{dz}{2\pi iz},$ where the contour separates the downward poles from the upward ones. * • Finally, if $\alpha_{1}<-\alpha_{2}$ (thus $\beta<\alpha_{0}$), then $B^{m}(t)=(q;q)\int_{\mathcal{C}}\theta(u_{0}u_{1}u_{2}/z;q)\frac{\prod\limits_{r:\alpha_{r}=1+\beta}(qz/u_{r};q)}{\prod\limits_{r:\alpha_{r}=-\beta}(u_{r}z;q)}\big{(}1-z^{2}\big{)}^{1_{\\{\beta=-1/2\\}}}\frac{dz}{2\pi iz},$ where the contour excludes the poles but circles the essential singularity at zero. ###### Proof 4.2. In order to obtain these limits we will break the symmetry of the integral. We first rewrite (3.4) in the form $\frac{\theta(s_{0}s_{1}s_{2}/z,s_{0}z,s_{1}z,s_{2}z;q)}{\theta(z^{2},s_{0}s_{1},s_{0}s_{2},s_{1}s_{2};q)}+\big{(}z\leftrightarrow z^{-1}\big{)}=1.$ Since the integrand of $E^{m}$ is invariant under the interchange of $z\to z^{-1}$, we can multiply by the left hand side of the above equation and observe that the integrand splits in two parts, each integrating to the same value. Therefore, the integral itself is equal to twice the integral of either part, and we thus obtain $\displaystyle E^{m}(t)=\prod_{0\leq r<s\leq 2m+5}(t_{r}t_{s};p,q)(p;p)(q;q)$ $\displaystyle\phantom{E^{m}(t)=}{}\times\int_{\mathcal{C}}\frac{\prod\limits_{r=0}^{2m+5}\Gamma(t_{r}z^{\pm 1})}{\Gamma(z^{\pm 2})}\frac{\theta(s_{0}s_{1}s_{2}/z,s_{0}z,s_{1}z,s_{2}z;q)}{\theta(z^{2},s_{0}s_{1},s_{0}s_{2},s_{1}s_{2};q)}\frac{dz}{2\pi iz}.$ (4.2) The poles introduced by the factor $1/\theta(z^{2};q)$ are canceled by zeros of the factor $1/\Gamma(z^{\pm 2})$, as we have $\frac{1}{\Gamma(z^{\pm 2})\theta(z^{2};q)}=\frac{\Gamma(pqz^{2})}{\Gamma(pz^{2})}=\theta\big{(}pz^{2};p\big{)}=\theta\big{(}z^{-2};p\big{)}$ using the difference and reflection equations satisfied by the elliptic gamma functions. This process therefore does not introduce any extra poles to the integrand; we may therefore use the same contour as before. In fact, since some of the original poles might have been cancelled, the constraints on the contour can be correspondingly weakened. Now, specialize $s_{r}=t_{r}$ ($r=0,1,2$) in (4.2) and simplify to obtain $\displaystyle E^{m}(t)=\frac{\prod\limits_{0\leq r<s\leq 2}(pt_{r}t_{s};p,q)\prod\limits_{r=0}^{2}\prod\limits_{s=3}^{2m+5}(t_{r}t_{s};p,q)\prod\limits_{3\leq r<s\leq 2m+5}(t_{r}t_{s};p,q)}{(q/t_{0}t_{1},q/t_{0}t_{2},q/t_{1}t_{2};q)}$ $\displaystyle\phantom{E^{m}(t)=}{}\times(p;p)(q;q)\int_{\mathcal{C}}\theta(z^{-2};p)\theta(t_{0}t_{1}t_{2}/z;q)\prod_{r=0}^{2}\Gamma(pt_{r}z,t_{r}/z)\prod_{r=3}^{2m+5}\Gamma(t_{r}z^{\pm 1})\frac{dz}{2\pi iz}.$ (4.3) Now change the integration variable $z\to zp^{\beta}$. The inequalities $\alpha_{0},\alpha_{1},\alpha_{2}\geq\beta$ and $-\beta\leq\alpha_{r}$, $3\leq r$ ensure that the downward poles remain bounded and the upward poles remain bounded away from 0 as $p\to 0$. There thus (for generic $u_{r}$) exists a contour valid for all sufficiently small $p$. After fixing such a contour, the limit again follows by simply plugging in $p=0$; the constraints on $\alpha$ are necessary and sufficient to ensure that all gamma functions in the integrand have well-defined limits. The two previous limits still do not allow us to take limits for each possible vector in the Hesse polytope (in the $m=1$ case). Indeed (as we will show below) we have covered the polytope, modulo the action of $S_{8}$ to sort the entries $\alpha_{0}\leq\cdots\leq\alpha_{7}$, as long as either $\alpha_{0}\geq 0$ (Proposition 4.1) or $\alpha_{1}+\alpha_{2}\leq 0$ (Proposition 4.2). The remaining limits require a more careful look and are given by the following proposition ###### Proposition 4.3. Let $\alpha\in\mathbb{R}^{2m+6}$ satisfy $\sum_{r}\alpha_{r}=m+1$ and assume $-1/2\leq\alpha_{0}<0$, $1+\alpha_{0}\geq\alpha_{r}\geq\alpha_{0}$ for $r\geq 1$ and for $2\leq k\leq m+3$, $\sum_{r\in I}(\alpha_{r}+\alpha_{0})\geq 2\alpha_{0},\qquad I\subset\\{1,2,\ldots,2m+5\\},\qquad\|I\|=k$ hold. Then the limit in (4.1) exists. * • If $\alpha_{0}=\alpha_{1}=-1/2$ $($thus $\alpha_{2}=\cdots=\alpha_{2m+5}=1/2)$ we have $\displaystyle B^{m}_{\alpha}(u)=\frac{\prod\limits_{r=2}^{2m+5}(u_{r}u_{1},qu_{0}/u_{r};q)}{(qu_{0}^{2},u_{0}u_{1},u_{1}/u_{0};q)}$ $\displaystyle\phantom{B^{m}_{\alpha}(u)=}\times{}_{2m+8\vphantom{2m+7}}W_{2m+7\vphantom{2m+8}}\big{(}u_{0}^{2};u_{0}u_{1},u_{0}u_{2},\ldots,u_{0}u_{2m+5};q,q\big{)}+(u_{0}\leftrightarrow u_{1}).$ * • If $\alpha_{0}=-1/2>\alpha_{1}$, and if $\alpha_{1}+\alpha_{2}=0$ the extra condition $\|u_{1}u_{2}\|<1$, we have with $n=\\#\\{r:\alpha_{r}<1/2\\}-3$ $B^{m}_{\alpha}(u)=\frac{\prod\limits_{r:\alpha_{r}=1/2}(qu_{0}/u_{r};q)}{(qu_{0}^{2};q)}{}_{\vphantom{}}^{\vphantom{(n)}}W_{\vphantom{}}^{(n)}\Bigg{(}u_{0}^{2};u_{0}u_{r}:\alpha_{r}=1/2;q,u_{0}^{n}\prod_{r>0:\alpha_{r}<1/2}u_{r}\Bigg{)},$ where the notation implies we take as parameters $u_{0}u_{r}$ for those $r$ which satisfy $\alpha_{r}=1/2$. * • If $-1/2<\alpha_{0}=\alpha_{1}<0$ then $\displaystyle B^{m}_{\alpha}(u)=\frac{\prod\limits_{\alpha_{r}=-\alpha_{0}}(u_{1}u_{r};q)\prod\limits_{\alpha_{r}=1+\alpha_{0}}(qu_{0}/u_{r};q)}{(u_{1}/u_{0};q)}$ $\displaystyle\phantom{B^{m}_{\alpha}(u)=}{}\times{}_{\vphantom{}}^{\vphantom{(n)}}\phi_{\vphantom{}}^{(n)}\left(\begin{array}[]{c}\quad\;\,\qquad u_{0}u_{r}:\alpha_{r}=-\alpha_{0}\\\ qu_{0}/u_{1},qu_{0}/u_{r}:\alpha_{r}=1+\alpha_{0}\end{array};q,q\right)+(u_{0}\leftrightarrow u_{1}),$ where $n=\\#\\{r:\alpha_{r}=-\alpha_{0}\\}-\\#\\{r:\alpha_{r}=1+\alpha_{0}\\}-2$. * • If $-1<2\alpha_{0}=\sum_{r\geq 1:\alpha_{r}+\alpha_{0}<0}(\alpha_{r}+\alpha_{0})$ and $\alpha_{1}>\alpha_{0}$, and if $\alpha_{1}+\alpha_{2}=0$ the extra condition $\|u_{1}u_{2}\|<1$, we get $B^{m}_{\alpha}(u)=\prod_{r:\alpha_{r}=1+\alpha_{0}}(qu_{0}/u_{r};q){}_{\vphantom{}}^{\vphantom{(n)}}\phi_{\vphantom{}}^{(n)}\left(\begin{array}[]{c}u_{0}u_{r}:\alpha_{r}=-\alpha_{0}\\\ qu_{0}/u_{r}:\alpha_{r}=1+\alpha_{0}\end{array};q,u_{0}^{-2}\prod_{r>0:\alpha_{r}<-\alpha_{0}}(u_{r}u_{0})\right),$ where $n=\\#\\{r:\alpha_{r}<-\alpha_{0}\\}-4-\\#\\{r:\alpha_{r}=1+\alpha_{0}\\}+\\#\\{r:\alpha_{r}=-\alpha_{0}\\}$. * • Finally if $2\alpha_{0}<\sum_{r\geq 1:\alpha_{r}+\alpha_{0}<0}(\alpha_{r}+\alpha_{0})$ we get $B^{m}_{\alpha}(u)=\prod_{r:\alpha_{r}=1+\alpha_{0}}(qu_{0}/u_{r};q).$ ###### Proof 4.3. Note that limits in the cases $\alpha_{0}=\alpha_{1}=-1/2$ and $-1/2<\alpha_{0}=\alpha_{1}\geq-\alpha_{r}$ ($r\geq 2)$ are given in Proposition 4.2. Together with the limits in this proposition we have thus covered all of the possible values for $\alpha$ at least once. Due to the condition $\alpha_{0}<0$, in the integral definition of $E^{m}(u\cdot p^{\alpha})$ there always exist poles which have to be excluded from the contour which go to zero as $p\to 0$, for example $z=u_{0}p^{\alpha_{0}}q^{k}$ for $k\in\mathbb{Z}_{\geq 0}$. Similarly there are poles going to infinity as $p\to 0$ which have to be included. The proof of this proposition in essence consists of first picking up the residues belonging to these poles, and taking the contour of the remaining integral close to the unit circle. Subsequently we take the limit as $p\to 0$ (which involves picking up an increasing number of residues), and show that the sums of these residues converge to one or two basic hypergeometric series, while the remaining integral converges to zero. Proving that we are allowed to interchange sum and limit and that the remaining integral vanishes in the limit consists of a calculation giving upper bounds on the integrand and residues, after which we can use dominated convergence. This calculation is quite tedious and hence omitted. The necessary bounds of the elliptic gamma function can be obtained by using the difference equation (2.1) to ensure the argument of the elliptic gamma function is of the form $\Gamma(p^{\gamma}z)$ for $0\leq\gamma\leq 1$, and using the known asymptotic behavior of the theta functions outside their poles and zeros. This gives a bound on the integrand for a contour which is at least $\epsilon>0$ away from any poles of the integrand, and moreover gives us a summable bound on the residues, thus showing that any residues corresponding to points not of the form $z=t_{0}q^{n}$ must vanish in the limit (here we use $\alpha_{0}<\alpha_{r}$ for $r>0$). However a contour as required does in general not exist for all values of $p$. Therefore choose parameters $u$ in a compact subset $K$ of the complement of the $p$-independent divisors (i.e. such that there are no $p$-independent pole-collisions of the integrand of $E^{m}$). For any $p$ for which we can obtain a contour which stays $\epsilon$ away from any poles of the integrand (for all $u\in K$), we can use our estimates to bound $\|E^{m}-B^{m}_{\alpha}\|$ uniformly for $u\in K$ and $a=\|p\|$, with the bound going to zero as $a\to 0$. As long as $\log(p)$ stays $\epsilon$ away from conditions of the form $u_{r}^{-1}u_{s}^{-1}q^{-n}=p^{l+\alpha_{r}+\alpha_{s}}$ ($l,n\in\mathbb{N}$, $u_{r},u_{s}$ range over the projection of $K$ to the $r$’th and $s$’th coordinate) the poles of the integrand near the unit circle stay $\mathcal{O}(\epsilon)$ away from each other and we can find a desired contour. Moreover this ensures that the residues we pick up are at least $\epsilon$ distance away from any other poles. Note that we only need to consider conditions with $l+\alpha_{r}+\alpha_{s}<0$ as the other condition cannot be satisfied for small enough $p$, this implies there is only a finite set of possible $l$, $r$ and $s$. Hence, if we start with small enough $K$ and $\epsilon$, we can ensure that these excluded values of $p$ form disjoint sets. In particular we can, in the $p$-plane, create a circle around these disjoint sets, and use the maximum principle to show that $E^{m}-B^{m}_{\alpha}$ is bounded in absolute value inside these circles by the maximum of the absolute value on the circle. As the circle consists entirely of $p$’s for which our estimates work, we see that inside the circle the difference is bounded as well (by a bound corresponding to a slightly larger radius). Hence for all values of $p$ with $\|p\|>0$ we find that $\|E^{m}-B^{m}_{\alpha}\|$ is bounded uniformly in $u$ and $a=\|p\|$ with the bound going to zero as $a\to 0$. in particular the limit holds uniformly for $u\in K$. Finally we can use the Stieltjes–Vitali theorem again to show the limit holds for all values of $u$. Note that there is some overlap in the conditions of Proposition 4.2 and Proposition 4.3. Indeed we get two different representations of the same function (one integral and one series) in the case of $\alpha\in\mathbb{R}^{2m+6}$ satisfying $\sum_{r}\alpha_{r}=m+1$, $\alpha_{0}\leq\alpha_{r}\leq-\alpha_{0}$ for $r=1,2$, $\alpha_{1}+\alpha_{2}=0$, $-\alpha_{0}\leq\alpha_{r}\leq 1+\alpha_{0}$ for $r\geq 3$. Moreover, in some special cases we have integral representations of the series in Proposition 4.3, which were not covered in Proposition 4.2. Moreover we sometimes find a second, slightly different, expression for the integrals of Proposition 4.2. Indeed we have ###### Proposition 4.4. For $\alpha\in\mathbb{R}^{2m+6}$ satisfying $\sum_{r}\alpha_{r}=m+1$ and $\alpha_{0}\leq\alpha_{1}\leq\cdots\leq\alpha_{2m+5}$ such that $-1/2\leq\alpha_{0}=\alpha_{1}<0$ and $-\alpha_{0}\leq\alpha_{2}$ and $\alpha_{2m+5}\leq 1+\alpha_{0}$ the limit in (4.1) exists and we have $\displaystyle B^{m}_{\alpha}(u)=\prod_{r\geq 2:\alpha_{r}=-\alpha_{0}}(u_{0}u_{r},u_{1}u_{r};q)(q;q)\int_{\mathcal{C}}\frac{\theta(u_{0}u_{1}w/z,wz;q)}{\theta(u_{0}w,u_{1}w;q)}$ $\displaystyle\phantom{B^{m}_{\alpha}(u)=}{}\times\frac{\prod\limits_{r\geq 2:\alpha_{r}=1+\alpha_{0}}(qz/u_{r};q)}{\prod\limits_{r\geq 2:\alpha_{r}=-\alpha_{0}}(u_{r}z;q)}\frac{1}{(u_{0}/z,u_{1}/z;q)}\left(\frac{1-z^{2}}{(u_{0}z,u_{1}z;q)}\right)^{1_{\\{\alpha_{0}=-1/2}\\}}\frac{dz}{2\pi iz};$ where the contour is a deformation of the unit circle separating the poles in downward sequences from the poles in upward sequences. The theta functions involving the extra parameter $w$ combine to give a $q$-elliptic function of $w$ and in fact the integrals are independent of $w$ (though this is only obvious from the fact that $B_{\alpha}(u)$ does not depend on $w$). In the case $\alpha_{0}=\alpha_{1}=-\alpha_{2}$, which is also treated in Proposition 4.2, we can specialize $w=u_{2}$ to re-obtain the previous integral expression of that limit. ###### Proof 4.4. As in the proof of Proposition 4.2, we start with the symmetry broken version of $E^{m}$, as in (4.2). Now we specialize $s_{0}=t_{0}$, $s_{1}=t_{1}$ and $s_{2}=w$. Thus we get $\displaystyle E^{m}(t)=\frac{(pt_{0}t_{1};p,q)}{(q/t_{0}t_{1};q)}\prod_{r=0,1}\prod_{s=2}^{2m+5}(t_{r}t_{s};p,q)\prod_{2\leq r<s\leq 2m+5}(t_{r}t_{s};p,q)(p;p)(q;q)$ $\displaystyle\phantom{E^{m}(t)=}{}\times\int_{\mathcal{C}}\prod_{r=0}^{1}\Gamma(pt_{r}z,t_{r}/z)\prod_{r=2}^{2m+5}\Gamma(t_{r}z^{\pm 1})\frac{\theta(t_{0}t_{1}w/z,wz;q)}{\theta(t_{0}w,t_{1}w;q)}\theta\big{(}z^{-2};p\big{)}\frac{dz}{2\pi iz}.$ (4.4) Replacing $z\to p^{\alpha_{0}}z$ and $w\to p^{-\alpha_{0}}w$ and using $t_{r}=p^{\alpha_{r}}u_{r}$ we can subsequently plug in $p=0$ as before to obtain the desired limit. ## 5 The polytopes In this section we describe a polytope (for each value of $m$) such that points of the polytope correspond to vectors $\alpha$ with respect to which we can take limits. Moreover we describe how the limiting functions $B_{\alpha}$ depend on geometrical properties of $\alpha$ in the polytope. Let us begin by defining the polytopes. ###### Definition 5.1. For $m\in\mathbb{N}$ we define the vectors $\rho^{(m)}$, $v_{j_{1}j_{2}\cdots j_{m}}^{(m)}$ ($0\leq j_{1}<j_{2}<\cdots<j_{m}\leq 2m+5$) and $w_{ij}^{(m)}$ ($0\leq i<j\leq 2m+5$) by $\rho^{(m)}=\frac{1}{2}\sum_{r=0}^{2m+5}e_{r},\qquad v_{j_{1}j_{2}\cdots j_{m+1}}^{(m)}=\sum_{r=1}^{m+1}e_{j_{r}},\qquad w_{ij}^{(m)}=\rho^{(m)}-e_{i}-e_{j},$ where the $e_{k}$ ($0\leq k\leq 2m+5$) form the standard orthonormal basis of $\mathbb{R}^{2m+6}$. Sometimes we write $v_{S}^{(m)}$ for $S\subset\\{0,1,\ldots,2m+5\\}$ with $\|S\|=m+1$. The polytope $P^{(m)}$ is now defined as the convex hull of the vectors $v_{S}^{(m)}$ ($\|S\|=m+1$) and $w_{ij}^{(m)}$ ($0\leq i<j\leq 2m+5$). In the notation for both vectors and polytopes we often omit the $(m)$ if the value of $m$ is clear from context. We will now state the main results of this section. The proofs follow after we have stated all theorems. The main result of this section will be the following theorem. ###### Theorem 5.2. For $\alpha\in P^{(m)}$ the limit in (4.1) exists and $B_{\alpha}^{m}(u)$ depends only on the (open) face of $P^{(m)}$ which contains $\alpha$, i.e. if $\alpha$ and $\beta$ are contained in the same face of $P^{(m)}$ then $B_{\alpha}^{m}(u)=B_{\beta}^{m}(u)$. Next we have the following iterated limit property. ###### Theorem 5.3. Let $\alpha,\beta\in P^{(m)}$. Then the iterated limit property holds, i.e. $\lim_{x\to 0}B_{\alpha}^{m}(x^{\beta-\alpha}u)=B_{t\alpha+(1-t)\beta}(u)$ for any $0<t<1$. As $t\alpha+(1-t)\beta$ is contained in the same face of $P^{(m)}$ for all values $0<t<1$, we already know that the right hand side does not depend on $t$. The iterated limit property shows that all the functions associated to faces can be obtained as limits of the (basic hypergeometric!) functions associated to vertices of the polytope. There are only two different limits associated to vertices (as there are only two different vertices up to permutation symmetry), so all results follow from identities satisfied by these two functions. Indeed the idea of this article is not so much to show new identities as it is to show how many known identities fit in a uniform geometrical picture. Moreover this picture allows us to simply classify all formulas of certain kinds. As an immediate corollary of the iterated limit property we find the last main theorem of this section. ###### Theorem 5.4. For $\alpha\in P^{(m)}$ the function $B_{\alpha}(u)$ depends only on the space orthogonal to the face containing $\alpha$. To be precise if $\beta$ is in the same (open) face as $\alpha$, then $B_{\alpha}(u)=B_{\alpha}(u\cdot x^{\alpha-\beta}).$ ###### Proof 5.1. Consider the line $v(t)=t\alpha+(1-t)\beta$. As $\alpha$ and $\beta$ are in the same open face there exists $\lambda_{1}>1$ such that $v(\lambda_{1})$ is also in this face. Moreover $\alpha$ is a strictly convex linear combination of $v(\lambda_{1})$ and $\beta$, and $v(\lambda_{1})-\beta=\lambda_{1}(\alpha-\beta)$. Now observe that $B_{\alpha}(u)=\lim_{y\to 0}B_{v(\lambda_{1})}(y^{v(\lambda_{1})-\beta}u)=\lim_{y\to 0}B_{v(\lambda_{1})}(y^{v(\lambda_{1})-\beta}x^{\frac{v(\lambda_{1})-\beta}{\lambda_{1}}}u)=B_{\alpha}(u\cdot x^{\beta-\alpha})$ by the iterated limit property. Here we replaced $y\to yx^{1/\lambda_{1}}$ in the second equality. To prove the first two main theorems, Theorems 5.2 and 5.3, we need to split up $P^{(m)}$ in several (to be precise $1+(2m+6)+\binom{2m+6}{3}$, but essentially only 3) different parts. Let us begin with defining the smaller polytopes. Recall the definition of the vectors $\rho^{(m)}$, $v_{S}^{(m)}$ and $w_{ij}^{(m)}$ from Definition 5.1. ###### Definition 5.5. We define the three convex polytopes $P_{\rm I}^{(m)}$, $P_{\rm II}^{(m)}$ and $P_{\rm III}^{(m)}$ by * • $P_{\rm I}^{(m)}$ is the convex hull of the vectors $v_{S}^{(m)}$ ($S\subset\\{0,1,\ldots,2m+5\\}$); * • $P_{\rm II}^{(m)}$ is the convex hull of the vectors $v_{S}^{(m)}$ ($S\subset\\{1,2,\ldots,2m+5\\}$) and $w_{0j}^{(m)}$ ($1\leq j\leq 2m+5$); * • $P_{\rm III}^{(m)}$ is the convex hull of the vectors $v_{S}^{(m)}$ ($S\subset\\!\\{3,4,\ldots,2m+5\\}$) and $w_{ij}^{(m)}$ ($0\leq i<j\leq 2$). Here we always have $\|S\|=m+1$ (otherwise $v_{S}^{(m)}$ would not make sense). The polytopes $P_{\rm I}^{(m)}$, $P_{\rm II}^{(m)}$ and $P_{\rm III}^{(m)}$ correspond to limits in Propositions 4.1, 4.3, respectively 4.2. The following proposition allows us to prove things about $P^{(m)}$ by proving them for these simpler polytopes. ###### Proposition 5.6. Denote $\sigma(A)=\\{\sigma(a)~{}\|~{}a\in A\\}$ for some permutation $\sigma\in S_{2m+6}$. Then we have $P^{(m)}=P_{\rm I}^{(m)}\cup\bigcup_{\sigma\in S_{2m+6}}\sigma\big{(}P_{\rm II}^{(m)}\big{)}\cup\bigcup_{\sigma\in S_{2m+6}}\sigma\big{(}P_{\rm III}^{(m)}\big{)}.$ (5.1) ###### Proof 5.2. It is sufficient to show that given any set $V$ of vertices of $P^{(m)}$ their convex hull can be written as the union of subsets of the polytopes on the right hand side. If $V$ does not contain one of the following bad sets 1. 1. $\\{w_{ij},v_{S_{1}},v_{S_{2}}\\}$ for $i\in S_{1},j\in S_{2}$; 2. 2. $\\{w_{ij},v_{S}\\}$ for $i,j\in S$; 3. 3. $\\{w_{ij},w_{kl}\\}$; 4. 4. $\\{w_{ij},w_{ik},v_{S}\\}$ for $i\in S$, where $i$, $j$, $k$, $l$ denote different integers, then $V$ is contained in the sets of vertices of $P_{\rm I}^{(m)}$ or one of the permutations of $P_{\rm II}$ or $P_{\rm III}$. This follows from a simple case analysis depending on the number and kind of $w_{ij}$’s in $V$. Given any point $p$ in the (closed) convex hull ${\rm ch}(V)$ of $V$, with $p=\sum_{v\in V}a_{v}v$, we can write ${\rm ch}(V)=\bigcup_{v:a_{v}>0}{\rm ch}((V\backslash\\{v\\})\cup\\{p\\})$. Indeed any point $q$ in ${\rm ch}(V)$ can be written as $q=\sum_{v\in V}b_{v}v=\gamma p+\sum_{v\in V}(b_{v}-a_{v}\gamma)v$, where we can take $\gamma\geq 0$ to be such that $b_{v^{\prime}}=a_{v^{\prime}}\gamma$ for some $v^{\prime}$ with $a_{v^{\prime}}>0$ and $b_{v}\geq a_{v}\gamma$ for all $v\in V$. Now $q$ clearly is a convex linear combination of elements of $(V\backslash\\{v^{\prime}\\})\cup\\{p\\}$. This argument is visualized in Fig. 1. As a generalization we obtain that if $p\in{\rm ch}(W)$ for some set $W$ we have that ${\rm ch}(V)\subset\bigcup_{v:a_{v}>0}{\rm ch}((V\backslash\\{v\\})\cup W)$. (0,0)(100,0)(50,87)(40,40)(0,0)(50,87) (40,40)(100,0) $v_{1}$$v_{2}$$v_{3}$$p$(10,0)(10,10) (20,0)(20,20) (30,0)(30,30) (40,0)(40,40) (50,0)(50,33) (60,0)(60,26) (70,0)(70,20) (80,0)(80,13) (90,0)(90,6) (5,9)(7,7) (10,17)(14,14) (15,26)(22,22) (20,35)(29,29) (25,43)(36,36) (30,52)(41,45) (35,61)(43,56) (40,69)(46,66) (45,78)(48,77) (55,78)(47,71) (60,69)(44,58) (65,61)(41,45) (70,52)(55,43) (75,43)(54,31) (80,35)(63,25) (85,26)(72,19) (90,17)(81,13) (95,9)(90,7) Figure 1: ${\rm ch}(v_{1},v_{2},v_{3})={\rm ch}(v_{1},v_{2},p)\cup{\rm ch}(v_{1},v_{3},p)\cup{\rm ch}(v_{2},v_{3},p)$. Now we can consider a set of vertices $V$ containing a bad configuration, and use the above method to rewrite ${\rm ch}(V)\subset\bigcup_{i}{\rm ch}(V_{i})$, where the $V_{i}$ are sets of vertices of $P^{(m)}$ that do not contain that bad configuration, while not introducing any new bad configurations. Iterating this we end up with ${\rm ch}(V)\subset\bigcup_{i}{\rm ch}(V_{i})$ for some sets $V_{i}$ without bad configurations; in particular ${\rm ch}(V)$ is contained in the right hand side of (5.1). First we consider a bad set of the form $\\{w_{ij},w_{kl}\\}$. Then $p=\frac{1}{2}(w_{ij}+w_{kl})=\frac{1}{2}(v_{T_{1}}+v_{T_{2}})$, where $T_{1}$ and $T_{2}$ are any two sets of size $\|T_{i}\|=m+1$ with $T_{1}\cup T_{2}\cup\\{i,j,k,l\\}=\\{0,1,\ldots,2m+5\\}$. Thus we get $V_{1}=(V\cup\\{v_{T_{1}},v_{T_{2}}\\})\backslash\\{w_{ij}\\}$ and $V_{2}=(V\cup\\{v_{T_{1}},v_{T_{2}}\\})\backslash\\{w_{kl}\\}$, as new sets. In particular the number of $w$’s decreases and we can iterate this until no bad sets of the form $\\{w_{ij},w_{kl}\\}$ exist. For the remaining three bad kind of sets we just indicate the way a strictly convex combination of the vectors in the bad set can be written in terms of better vectors. In each step we assume there are no bad sets of the previous form, to ensure we do not create any new bad sets (at least not of the form currently under consideration or of a form previously considered). 1. 1. For $\\{w_{ij},v_{S}\\}$ with $i,j\in S$ we have $\frac{2}{3}w_{ij}+\frac{1}{3}v_{S}=\frac{1}{3}(v_{T_{1}}+v_{T_{2}}+v_{U})$ where $S\backslash T_{1}=S\backslash T_{2}=\\{i,j\\}$ and $S\cap U=T_{1}\cap U=T_{2}\cap U=\varnothing$ and $T_{1}\cap T_{2}=S\backslash\\{i,j\\}$ (thus $T_{1}$, $T_{2}$ and $U$ cover all the elements of $S$, except $i$ and $j$, twice, and all other points once). 2. 2. For $\\{w_{ij},w_{ik},v_{S}\\}$ with $i\in S$ we have $\frac{1}{3}(w_{ij}+w_{ik}+v_{S})=\frac{1}{3}(v_{T_{1}}+v_{T_{2}}+v_{U})$ for $S\backslash T_{1}=S\backslash T_{2}=\\{i\\}$ and $S\cap U=T_{1}\cap U=T_{2}\cap U=\varnothing$ and $T_{1}\cap T_{2}=S\backslash\\{i\\}$ and $j,k\not\in T_{1},T_{2},U$. 3. 3. For $\\{w_{ij},w_{ik},v_{S}\\}\subset V$, with $i\in S$, then $j,k\not\in S$ and $\frac{1}{3}(w_{ij}+w_{ik}+v_{S})=\frac{1}{3}(v_{T_{1}}+v_{T_{2}}+v_{U})$ for $S\backslash T_{1}=S\backslash T_{2}=\\{i\\}$ and $S\cap U=T_{1}\cap U=T_{2}\cap U=\varnothing$ and $T_{1}\cap T_{2}=S\backslash\\{i\\}$ and $j,k\not\in T_{1},T_{2},U$. ∎ Let us now consider the bounding inequalities related to these polytopes. ###### Proposition 5.7. The polytopes $P^{(m)}$, $P_{\rm I}^{(m)}$, $P_{\rm II}^{(m)}$ and $P_{\rm III}^{(m)}$ are the subsets of the hyperplane $\\{\alpha:\alpha\in\mathbb{R}^{2m+6}\|\sum_{i}\alpha_{i}=m+1\\}$ described by the following bounding inequalities * • For $P^{(m)}$ the bounding inequalities are $\displaystyle-\frac{1}{2}\leq\alpha_{i}\leq 1,\qquad$ $\displaystyle(0\leq i\leq 2m+5),$ $\displaystyle\alpha_{i}\leq 1+\alpha_{j}+\alpha_{k}+\alpha_{l},\qquad$ $\displaystyle(\|\\{i,j,k,l\\}\|=4),$ $\displaystyle\alpha_{i}-\alpha_{j}\leq 1,\qquad$ $\displaystyle(i\neq j),$ $\displaystyle(\|S\|-2)\alpha_{i}+\sum_{j\in S}\alpha_{j}\geq 0,\qquad$ $\displaystyle(i\not\in S,3\leq\|S\|\leq m+3).$ For $m=0$ the equations $\alpha_{r}\leq 1$ and $\alpha_{i}\leq 1+\alpha_{j}+\alpha_{k}+\alpha_{l}$ are valid but not bounding. * • The polytope $P_{\rm I}^{(m)}$ is described by the bounding inequalities $0\leq\alpha_{i}\leq 1,\qquad(0\leq i\leq 2m+5).$ For this polytope too, if $m=0$ the equations $\alpha_{r}\leq 1$ are valid but not bounding. * • The polytope $P_{\rm II}^{(m)}$ is described by the bounding inequalities $\displaystyle-1/2\leq\alpha_{0},$ $\displaystyle\alpha_{r}-\alpha_{0}\leq 1,\qquad$ $\displaystyle(r\geq 1),$ $\displaystyle(\|S\|-2)\alpha_{0}+\sum_{j\in S}\alpha_{j}\geq 0,\qquad$ $\displaystyle(0\not\in S,0\leq\|S\|\leq m+3).$ * • Finally, the polytope $P_{\rm III}^{(m)}$ is described by the bounding inequalities $\displaystyle\alpha_{i}+\alpha_{j}\leq 0,\qquad$ $\displaystyle(0\leq i<j\leq 2),$ $\displaystyle-\alpha_{i}\leq\alpha_{0}+\alpha_{1}+\alpha_{2},\qquad$ $\displaystyle(3\leq i\leq 2m+5),$ $\displaystyle\alpha_{i}-1\leq\alpha_{0}+\alpha_{1}+\alpha_{2},\qquad$ $\displaystyle(3\leq i\leq 2m+5).$ If $m=0$ the equations $\alpha_{i}-1\leq\alpha_{0}+\alpha_{1}+\alpha_{2}$ are valid but not bounding. ###### Proof 5.3. It can be immediately verified that the vertices of the polytopes $P^{(m)}$, $P_{\rm I}^{(m)}$, $P_{\rm II}^{(m)}$ and $P_{\rm III}^{(m)}$ satisfy the relevant inequalities, hence so does any convex linear combination of them. In particular it is clear that the polytopes are contained in the sets defined by these bounding inequalities. Note that the different polytopes have symmetries of $S_{2m+6}$ (for $P^{(m)}$ and $P_{\rm I}^{(m)}$), respectively $S_{1}\times S_{2m+5}$ ($P_{\rm II}^{(m)}$), respectively $S_{3}\times S_{2m+3}$ ($P_{\rm III}^{(m)}$). We only have to find the bounding inequalities of these polytopes intersected with a Weyl chamber of the relevant symmetry group, as all bounding inequalities will be permutations of these. These bounding inequalities can be written in the form $\mu\cdot\alpha\geq 0$ for each $\alpha$ in the polytope; we do not need affine equations as we have $\sum_{r}\alpha_{r}=m+1$. A bounding inequality must attain equality at a codimension 1 space of the vertices of the polytope; in particular if we consider all subsets $V$ of $2m+5$ vertices of the intersection of each polytope with the relevant Weyl chamber and insist on $\mu\cdot v=0$ for each $v\in V$, we find all bounding inequalities (and perhaps some more inequalities). For $P_{\rm I}$, $P_{\rm II}$ and $P_{\rm III}$ we are in the circumstance that there are $2m+6$ vertices for the intersection of the Weyl chamber with the polytope; in particular each set of $2m+5$ vertices corresponds to leaving one vector out. Moreover the sign of $\mu$ is then determined by insisting on $\mu\cdot v>0$ for the remaining vertex $v$. As the equations are all homogeneous the normalization of $\mu$ is irrelevant. Let us consider the case of $P_{\rm II}$. The set of relevant vertices is $\\{v_{S},w_{01},e_{1}-e_{2},\ldots,e_{2m+4}-e_{2m+5}\\}$ for $S=\\{m+6,\ldots,2m+5\\}$. We now have the following options for leaving one vector out. 1. 1. If $\mu\cdot v_{S}^{(m)}>0$ we get $\mu=\rho+(m+1)e_{0}$, thus the equation $\alpha_{0}\geq-1/2$. 2. 2. If $\mu\cdot w_{01}^{(m)}>0$ we get $\mu=-e_{0}$ and the equation $\alpha_{0}\leq 0$. 3. 3. If $\mu\cdot(e_{i}-e_{i+1})>0$ for $1\leq i\leq m+3$ we get $\mu=(i-2)e_{0}+\sum_{r=1}^{i}e_{r}$ and the equation becomes $(i-2)\alpha_{0}+\sum_{r=1}^{i}e_{r}\geq 0$. 4. 4. If $\mu\cdot e_{i}-e_{i+1}>0$ for $m+4\leq i\leq 2m+4$ we get $\mu=(m+2)(2m+5-i)e_{0}+(2m+5-i)\sum_{r=1}^{i}e_{r}+(m+4-i)\sum_{r=i+1}^{2m+5}e_{r}$ and the equation becomes $(\alpha_{0}+1)(2m+5-i)\geq\sum_{r=i+1}^{2m+5}\alpha_{r}$. Note that the equation $\alpha_{0}\leq 0$ is the $\|S\|=0$ case of $(\|S\|-2)\alpha_{0}+\sum_{j\in S}\alpha_{j}\geq 0$. Now the last set of equations all follow from the instance $i=2m+4$, i.e. $\alpha_{0}+1\geq\alpha_{2m+5}$ and the equation $\alpha_{2m+5}\geq\alpha_{r}$. The rest are true bounding inequalities. It is only hard to see that the solutions to $(i-2)\alpha_{0}+\sum_{r=1}^{i}e_{r}=0$ in the set of vertices of the polytope span a codimension one space; however the set $\\{w_{01},\ldots,w_{0i}\\}\cup\\{v_{T}~{}\|~{}T\subset\\{i+1,\ldots,2m+5\\}\\}$ does span a set of codimension one. In a similar way one obtains the bounding inequalities for $P_{\rm I}$ and $P_{\rm III}$, we omit the explicit calculations here. To obtain the bounding inequalities of $P$ itself, we observe that any bounding inequality of $P$ must be a bounding inequality of one of $P_{\rm I}$, $P_{\rm II}$, $P_{\rm III}$ or one of their permutations, as $P$ is the union of those polytopes. Indeed any of these equations which are valid on $P$ are bounding inequalities (as the span of the set of vertices for which equality holds does not reduce in dimension when going from a smaller polytope to $P$). Thus we can find the bounding inequalities for $P$ by checking which of the bounding inequalities of these smaller polytopes are valid on $P$. This we only need to check on the vertices of $P$, which is a straightforward calculation. Note that we could also have obtained the bounding inequalities for $P$ in the same way that we obtained those of $P_{\rm I}$, $P_{\rm II}$ and $P_{\rm III}$. However now we would have to take $2m+5$ vectors from the set $\\{v_{S},w_{01},e_{0}-e_{1},\ldots,e_{2m+4}-e_{2m+5}\\}$, which has $2m+7$ elements. The number of options therefore becomes quite large, thus we prefer to avoid this method. We would like to give special attention to the bounding inequalities of $P^{(1)}$, which is the polytope which interests us most. We can rewrite these bounding inequalities in a clearly $W(E_{7})$ invariant way. ###### Proposition 5.8. The bounding inequalities for $P^{(1)}$ inside the subspace $\alpha\cdot\rho=1$ are given by $\displaystyle\alpha\cdot\delta\leq 1,\qquad$ $\displaystyle(\delta\in R(E_{7})),$ $\displaystyle\alpha\cdot\mu\leq 2,\qquad$ $\displaystyle(\mu\in\Lambda(E_{8}),\mu\cdot\rho=1,\mu\cdot\mu=4)$ for $\alpha\in P^{(1)}$. Here $\Lambda(E_{8})=\mathbb{Z}^{8}\cup(\mathbb{Z}^{8}+\rho)$ is the root lattice of $E_{8}$. ###### Proof 5.4. Up to $S_{8}$ one can classify the roots of $E_{7}$, giving $\delta=e_{i}-e_{j}$ or $\delta=\rho-e_{i}-e_{j}-e_{k}-e_{l}$, which handles the bounding inequalities $\alpha_{i}-\alpha_{j}\leq 1$ and $\alpha_{i}+\alpha_{j}+\alpha_{k}+\alpha_{l}\leq 0$. Similarly we can classify all relevant $\mu\in\Lambda(E_{8})$ as $\mu=2e_{i}$, $\mu=e_{i}+e_{j}+e_{k}-e_{l}$, $\mu=\rho-2e_{i}$ and $\mu=\rho+e_{i}-e_{j}-e_{k}-e_{l}$, the corresponding equations are again directly related to the bounding inequalities of $P^{(1)}$ as given in Proposition 5.7. It is convenient to rewrite the integral limits of Propositions 4.1 and 4.2 in a uniform way which clearly indicates the bounding inequalities for the corresponding polytopes (i.e. $P_{\rm I}$, resp. $P_{\rm III}$). It is much harder to give such a uniform expression for $P_{\rm II}$ (and we need separate expressions for the intersection with $P_{\rm I}$ and $P_{\rm III}$ and the facet $\\{\alpha_{0}=1+\alpha_{2m+5}\\}$), so we omit those. ###### Proposition 5.9. Define vectors $v_{j}=e_{j}$ and $w_{j}=e_{j}-\frac{2}{m+1}\rho$, then the bounding inequalities for $P_{\rm I}^{(m)}$ become $v_{j}\cdot\alpha\geq 0$, $w_{j}\cdot\alpha\geq 0$ $($and the condition $2\rho\cdot\alpha=m+1)$. The limit can be written as $B_{\alpha}^{m}(u)=\prod_{j\neq k:v_{j}\cdot\alpha=v_{k}\cdot\alpha=0}(u^{v_{j}+v_{k}};q)\frac{(q;q)}{2}\int\frac{(z^{\pm 2};q)\prod\limits_{j:w_{j}\cdot\alpha=0}(u^{w_{j}}z^{\pm 1};q)}{\prod\limits_{j:v_{j}\cdot\alpha=0}(u^{v_{j}}z^{\pm 1};q)}\frac{dz}{2\pi iz}.$ ###### Proposition 5.10. Define the vectors $v_{j}=e_{0}+e_{1}+e_{2}-e_{j}$ $(0\leq j\leq 2)$, $w_{j}=e_{0}+e_{1}+e_{2}+e_{j}$ $(3\leq j\leq 2m+5)$, and $x_{j}=e_{0}+e_{1}+e_{2}-e_{j}-\frac{2}{m+1}\rho$ $(3\leq j\leq 2m+5)$, then the bounding inequalities for $P_{\rm III}^{(m)}$ can be written as $v_{j}\cdot\alpha\geq 0$, $w_{j}\cdot\alpha\geq 0$ and $x_{j}\cdot\alpha\geq 0$ $($together with $2\rho\cdot\alpha=m+1)$. Let $y=w_{j}+x_{j}$ $($note $y$ is independent of $j)$ then $\displaystyle B_{\alpha}^{m}(u)=\frac{\prod\limits_{j:w_{j}\cdot\alpha=0}\prod\limits_{k:v_{k}\cdot\alpha=0}(u^{w_{j}+u_{k}};q)}{\prod\limits_{k:v_{k}\cdot\alpha=0}(qu^{v_{k}};q)}\left(\prod_{r\neq s:v_{r}\cdot\alpha=v_{s}\cdot\alpha=0}(qu^{y+v_{r}+v_{s}};q)\right)^{1_{\\{y\cdot\alpha=0\\}}}$ $\displaystyle\phantom{B_{\alpha}^{m}(u)=}{}\times\int\theta(1/z;q)\frac{\prod\limits_{j:x_{j}\cdot\alpha=0}(q^{2}zu^{x_{j}};q)}{\prod\limits_{j:w_{j}\cdot\alpha=0}(zu^{w_{j}};q)}\frac{\prod\limits_{r\neq s:v_{r}\cdot\alpha=v_{s}\cdot\alpha=0}(qu^{v_{r}+v_{s}}/z;q)}{\prod\limits_{r:v_{r}\cdot\alpha=0}(u^{v_{r}}/z;q)}$ $\displaystyle\phantom{B_{\alpha}^{m}(u)=}{}\times\left(\frac{(1-qu^{y}z^{2})\prod\limits_{r\neq s:v_{r}\cdot\alpha=v_{s}\cdot\alpha=0}(q^{2}zu^{y+v_{r}+v_{s}};q)}{\prod\limits_{r:v_{r}\cdot\alpha=0}(qzu^{y+v_{r}};q)}\right)^{1_{\\{y\cdot\alpha=0\\}}}\frac{dz}{2\pi iz}.$ ###### Proof 5.5. These two propositions are just a rewriting of Propositions 4.1 and 4.2. With these expressions the proof of the following proposition becomes fairly straightforward. ###### Proposition 5.11. Let the polytope $Q$ be either $P_{\rm I}^{(m)}$, $P_{\rm II}^{(m)}$ or $P_{\rm III}^{(m)}$. For $\alpha\in Q$ the limit in (4.1) exists and depends only on the face of $Q$ which contains $\alpha$ $($i.e. if $\alpha$ and $\beta$ are contained in the same face of $Q$ then $B_{\alpha}^{m}(u)=B_{\beta}^{m}(u))$. ###### Proof 5.6. Indeed Propositions 5.9, respectively 5.10 give the limits for the vectors $\alpha$ in $P_{\rm I}^{(m)}$, respectively $P_{\rm III}^{(m)}$. For $P_{\rm II}^{(m)}$ the limits are given in Proposition 4.3, except for the cases with $\alpha_{0}=0$ (which is the intersection with $P_{\rm I}^{(m)}$), and $\alpha_{1}+\alpha_{2}=0$ (the intersection with $P_{\rm III}^{(m)}$), or a permutation of such a case. In particular we have obtained limits in those cases as well. For $P_{\rm I}$ and $P_{\rm III}$ the expressions in the previous two propositions immediately show that the limits only depend on which bounding inequalities are strict or not, and hence on the face of the polytope containing $\alpha$. For $P_{\rm II}$ we note that the conditions $\alpha_{0}=0$ and $\alpha_{1}+\alpha_{2}=0$ (governing which proposition to look at) correspond to bounding inequalities. Within Proposition 4.3 we observe that the condition $\alpha_{r}=-\alpha_{0}$ becomes a bounding equation once $2\alpha_{0}=\sum_{r\geq 1:\alpha_{0}+\alpha_{r}<0}\alpha_{r}+\alpha_{0}$ holds (as the difference of the equations with $r\in S$ and with $r\not\in S$). So also for $P_{\rm II}$ the limit only depends on which bounding inequalities are strict and which not. We can now prove the first of the main theorems, the equivalent result for the full polytope $P^{(m)}$. ###### Proof 5.7 (Proof of Theorem 5.2). By the $S_{2m+6}$ symmetry of $E_{m}(t)$ we see that if a limit exists for some $\alpha$, then it also exists for all permutations of $\alpha$. As $P^{(m)}$ is the union of permutations of $P_{\rm I}^{(m)}$, $P_{\rm II}^{(m)}$ and $P_{\rm III}^{(m)}$, by the previous proposition we find that the limit $B_{\alpha}^{m}$ exists for all $\alpha\in P^{(m)}$. We would like to extend the statement about dependence on faces as well. To prove this it would be sufficient to show that all faces of $P^{(m)}$ are in fact a face of one of the polytopes in its decomposition, however this is not true. We do have the following lemma. ###### Lemma 5.12. All faces of $P^{(m)}$ are faces of either $P_{\rm I}^{(m)}$, $\sigma(P_{\rm II}^{(m)})$ or $\sigma(P_{\rm III}^{(m)})$ for some $\sigma\in S_{2m+6}$, except the interior of $P^{(m)}$ and permutations of the facet given by the equality $\alpha_{0}+\alpha_{1}+\alpha_{2}+\alpha_{3}=0$. ###### Proof 5.8. Any face of $P^{(m)}$ can be written as the set of all convex linear combinations of some set $V$ of vertices of $P^{(m)}$. Recall that $V$ is contained in the set of vertices of $P_{\rm I}$, $P_{\rm II}$ or $P_{\rm III}$ (or a permutation thereof), unless it contains one of the four bad sets in the proof of Proposition 5.6. Therefore, except when $V$ contains bad sets, the face determined by $V$ is a face of $P_{\rm I}$, $P_{\rm II}$ or $P_{\rm III}$. We now show that if $V$ contains a bad set, the convex hull of $V$ contains a point in the interior of $P^{(m)}$ or the interior of the facet given by $\alpha_{0}+\alpha_{1}+\alpha_{2}+\alpha_{3}=0$. Hence ${\rm ch}(V)$ is either equal to the interior of $P^{(m)}$, or to the special facet. 1. 1. If $\\{w_{ij},v_{S_{1}},v_{S_{2}}\\}\subset V$ ($i\in S_{1}$, $j\in S_{2}$) then $(w_{ij}+v_{S_{1}}+v_{S_{2}})/3\in{\rm ch}(V)$. This is a point where all elements are $1/6$, $1/2$ or $5/6$, in particular it is a point in the interior of $P_{\rm I}$, and thus of $P$ itself. 2. 2. If $\\{w_{ij},v_{S}\\}\subset V$ ($i,j\in S$), then $(w_{ij}+v_{S})/2\in{\rm ch}(V)$, which is in the interior of $P_{\rm I}$. 3. 3. If $\\{w_{ij},w_{ik},v_{S}\\}\subset V$, ($i\in S$), then $(w_{ij}+w_{ik}+2v_{S})/4\in{\rm ch}(V)$, which is again a point in the interior of $P_{\rm I}$. 4. 4. If $\\{w_{ij},w_{kl}\\}\subset V$, then $(w_{ij}+w_{kl})/2\in{\rm ch}(V)$. All bounding inequalities of $P$ are strict on this point except $\alpha_{i}+\alpha_{j}+\alpha_{k}+\alpha_{l}=0$, thus it is a point in the interior of the corresponding facet. Thus ${\rm ch}(V)$ is either this facet or the interior of $P$. ∎ Now it remains to show that the function $B_{\alpha}^{m}$ is the same for all points in the interior, and all points on the facet given by $\alpha_{0}+\alpha_{1}+\alpha_{2}+\alpha_{3}=0$. This follows from the following lemma. ###### Lemma 5.13. On the facet of $P^{(m)}$ given by $\alpha_{0}+\alpha_{1}+\alpha_{2}+\alpha_{3}=0$ we have $B_{\alpha}^{m}(u)=(u_{0}u_{1}u_{2}u_{3};q)$. Moreover on the interior of $P^{(m)}$ we have $B_{\alpha}^{m}(u)=1$. ###### Proof 5.9. In the $m=0$ case we find that the right hand side of the evaluation formula (3.2) converges for $\alpha$ on the facet to $(q/u_{4}u_{5};q)=(u_{0}u_{1}u_{2}u_{3};q)$, while in the interior of $P^{(m)}$ the limit converges to 1 (as for $m=0$ the condition $\alpha_{r}+\alpha_{s}=1$ is equivalent to the sum of the other four parameters being zero.). Thus for $m=0$ the lemma is true. For $m>0$ we can classify all the faces of $P_{\rm I}^{(m)}$, $P_{\rm II}^{(m)}$ and $P_{\rm III}^{(m)}$ which intersect the given facet and the interior of $P^{(m)}$. The bounding inequality $\alpha_{0}+\alpha_{1}+\alpha_{2}+\alpha_{3}\geq 0$ implies that the only vertices allowed in the closure of this facets are $v_{S}$ for $0,1,2,3\not\in S$ and $w_{ij}$ for $i,j\in\\{0,1,2,3\\}$. We obtain the following set of faces of $P_{\rm I}$, $P_{\rm II}$ and $P_{\rm III}$ in the facet $\alpha_{0}+\alpha_{1}+\alpha_{2}+\alpha_{3}=0$ modulo permutations of the parameters. Polytope \| Vertices \| Relations ---\|---\|--- 0.5ex $P_{\rm I}\cap P_{\rm II}\cap P_{\rm III}$ \| $v_{S}$ ($0,1,2,3\not\in S$) \| $\alpha_{0}=\alpha_{1}=\alpha_{2}=\alpha_{3}=0$ $P_{\rm II}\cap P_{\rm III}$ \| $w_{01}$, $v_{S}$ ($0,1,2,3\not\in S$) \| $\alpha_{0}=\alpha_{1}=-\alpha_{2}=-\alpha_{3}$ \| $w_{01}$, $w_{02}$, $v_{S}$ ($0,1,2,3\not\in S$) \| $\alpha_{0}+\alpha_{3}=0$, $\alpha_{1}=\alpha_{2}=0$ $P_{\rm II}$ \| $w_{01}$, $w_{02}$, $w_{03}$, $v_{S}$ ($0,1,2,3\not\in S$) \| $\alpha_{0}<\alpha_{r}<-\alpha_{0}$ ($r=1,2,3$) $P_{\rm III}$ \| $w_{01}$, $w_{02}$, $w_{12}$, $v_{S}$ ($0,1,2,3\not\in S$) \| $-\alpha_{3}<\alpha_{r}<\alpha_{3}$ ($r=0,1,2$) We omitted the conditions on $\alpha_{r}$ for $r\geq 4$ as they are the same in each case. Indeed the bounding inequalities imply that $-\beta<\alpha_{r}<1+\beta$ for $r\geq 4$, where $\beta$ is the sum of the three smallest parameters. The classification becomes apparent once we realize that the $v_{S}$ part of the vertices of the faces is fixed (they must contain a point $v_{S}$ with $0,1,2,3\not\in S$ to be in the open facet, while they cannot contain any other $v_{S}$). Thus we only have to consider the possibilities for adding some $w_{ij}$’s. In these five faces we can directly check what the limit is, and observe that the corresponding integrals and sums indeed evaluate to the desired $(u_{0}u_{1}u_{2}u_{3};q)$. The required evaluation identities are provided by the $m=0$ cases of these faces. Similarly we can describe all faces of $P_{\rm I}$, $P_{\rm II}$ and $P_{\rm III}$ (modulo permutations) meeting the interior of $P$. Polytope \| Vertices \| Relations ---\|---\|--- 0.5ex $P_{\rm I}\cap P_{\rm II}\cap P_{\rm III}$ \| $v_{S}$, ($0,1,2\not\in S$) \| $\alpha_{0}=\alpha_{1}=\alpha_{2}=0$ $P_{\rm I}\cap P_{\rm II}$ \| $v_{S}$ ($0,1\not\in S$) \| $\alpha_{0}=\alpha_{1}=0<\alpha_{2}<1$ \| $v_{S}$ ($0\not\in S$) \| $\alpha_{0}=0<\alpha_{1},\alpha_{2}<1$ $P_{\rm I}$ \| $v_{S}$ \| $0<\alpha_{0},\alpha_{1},\alpha_{2}<1$ $P_{\rm II}\cap P_{\rm III}$ \| $w_{01}$, $v_{S}$ ($0,1,2\not\in S$) \| $-1/2<\alpha_{0}=\alpha_{1}=-\alpha_{2}$ \| $w_{01}$, $w_{02}$, $v_{S}$ ($0,1,2\not\in S$) \| $-1/2<\alpha_{0}<\alpha_{1}=-\alpha_{2}$ $P_{\rm III}$ \| $w_{01}$, $w_{02}$, $w_{12}$, $v_{S}$ ($0,1,2\not\in S$) \| $\alpha_{r}+\alpha_{s}<0$, ($r,s\in\\{0,1,2\\}$) $P_{\rm II}$ \| $w_{01}$, $v_{S}$ ($0,1,\not\in S$) \| $-1/2<\alpha_{0}=\alpha_{1}>-\alpha_{2}$ \| $w_{01}$, $w_{02}$, $w_{03}$, $v_{S}$ ($0\not\in S$) \| $\alpha_{0}<0$ and $\alpha_{0}<\alpha_{1}>-\alpha_{2}$ For each face we have the extra conditions $-\beta<\alpha_{r}<1+\beta$ for $r\geq 3$. We can check for all these 9 faces that the value on that face equals 1 identically. Again the required evaluations all follow from the $m=0$ case. We conclude that also on the two types of faces of $P^{(m)}$ which are not a face of one of the subpolytopes, the value is the same on the entire face. We can now also prove the second main theorem, the iterated limit property. ###### Proof 5.10 (Proof of Theorem 5.3). It is sufficient to prove this property for the closed polytopes $P_{\rm I}^{(m)}$, $P_{\rm II}^{(m)}$ and $P_{\rm III}^{(m)}$. Observe that for the vector $t\alpha+(1-t)\beta$ precisely those boundary conditions (of the respective polytopes) are strict which are strict for either $\alpha$ or $\beta$. In Propositions 5.9 and 5.10 we have written the limits in $P_{\rm I}$ and $P_{\rm III}$ as an integral of a product of terms $f(u,w)=(u^{w};q)^{1_{\\{w\cdot\alpha=0\\}}}$ where $w\cdot\alpha\geq 0$ is the sum of some bounding inequalities. In particular in the expression for $B_{t\alpha+(1-t)\beta}^{m}$ only those terms remain corresponding to sums of bounding inequalities which are attained in both $\alpha$ and $\beta$. On the other hand, as $(x^{\beta-\alpha}u)^{w}=x^{(\beta-\alpha)\cdot w}u^{w}$ we find that if $w\cdot\alpha=w\cdot\beta=0$ then the term $f_{\alpha}(x^{\beta-\alpha}u,w)$ is constant in $x$ and thus does not change in the limit, while if $w\cdot\alpha>0$, we find that $f_{\alpha}(x^{\beta-\alpha}u,w)=1$ is again independent of $x$, and finally if $w\cdot\alpha=0<w\cdot\beta$, then we have a uniform limit $\lim_{x\to 0}f_{\alpha}(x^{\beta-\alpha}u,w)=\lim_{x\to 0}(x^{\beta\cdot w}u^{w};q)=1=f_{t\alpha+(1-t)\beta}(u,w)$. Thus to prove iterated limits we only have to show we are allowed to interchange limit and integral, which follows from the fact that we integrate over some compact contour, and the fact that the $x$-dependent poles which have to remain inside the contour converge to 0, while the $x$-dependent poles which have to remain outside the contour go to infinity; in particular for $x$ small enough we can take an $x$-independent contour. To prove the result for $P_{\rm II}$ is more complicated as we do not have a uniform description of the limit. For the closed facets determined by $\alpha_{0}=0$, $\alpha_{1}+\alpha_{2}=0$ or $\alpha_{0}=\alpha_{1}$ we have an integral description (see Propositions 4.1, 4.2 and 4.4), and limits within these facets can be treated as for $P_{\rm I}$ and $P_{\rm III}$. For the complement of these facets the limit is given in Proposition 4.3 as a single sum (possibly of only one term), and we can replicate the argument for $P_{\rm I}$ and $P_{\rm III}$ for sums instead of integrals to see that iterated limits within the complement of the facets hold. We are left with showing that limits from one of the three facets to the inside behave correctly. These limits all pass from an integral to a single sum. The simplest argument we found is to simply calculate these limits in all three cases separately (due to the rather uniform expressions of the integrals and single sums, we can handle the different faces in each of the closed facets uniformly). It boils down to picking the residues associated with poles which either go to zero as $x\to 0$, while they should be outside the contour, or go to infinity while they should be inside the contour. Subsequently we bound the integrand around the unit circle to show that the remaining integral vanishes in the limit. Finally we give a bound on the residues and use dominated convergence to show we are allowed to interchange limit and sum. This bound also serves to show that the sum of the residues which are not associated to $u_{0}$ (where we take the first coefficient in $\alpha$ to be the strictly lowest one) vanishes as well. The calculations involved are again tedious, and very similar to the calculations in the proof of Proposition 4.3. ## 6 Transformations: the $\boldsymbol{m=1}$ case In this section we start harvesting the results we can now immediately obtain given this picture of basic hypergeometric functions as faces of the polytope $P^{(1)}$. This was already done for the top two levels (i.e. the functions corresponding to vertices and edges) by Stokman and the authors in [2], though there we did not yet see the polytope. In the next section we give a worked through example (related to ${}_{2}\phi_{1}$) of the abstract results in this section. As a convenient tool to understand the implications of the results mentioned, we refer to Appendix A which contains a list of all the possible functions $B_{\alpha}^{1}$. Recall that we have a basic hypergeometric function attached to each face of the polytope $P^{(1)}$, and that the Weyl group $W(E_{7})$ acts both on $P^{(1)}$ and on sets of parameters $\tilde{\mathcal{H}}_{1}$. As the elliptic hypergeometric function is invariant under this action we immediately obtain ###### Theorem 6.1. Let $w\in W(E_{7})$, $\alpha\in P_{1}$ and $u\in\tilde{\mathcal{H}}_{1}$ then $B_{\alpha}^{1}(u)=B_{w(\alpha)}^{1}(w(u)).$ ###### Proof 6.1. Indeed we have $\displaystyle B_{\alpha}^{1}(u)=\lim_{p\to 0}E^{1}(p^{\alpha}\cdot u)=\lim_{p\to 0}E^{1}(w(p^{\alpha}\cdot u))=\lim_{p\to 0}E^{1}(p^{w(\alpha)}\cdot w(u))=B_{w(\alpha)}^{1}(w(u)).$ ∎ This gives us formulas of two different kinds for the functions $B_{\alpha}^{1}$. First of all we can obtain the symmetries of a function by considering the stabilizer of the corresponding face with respect to $W(E_{7})$. This includes for example Heine’s transformation of a ${}_{2}\phi_{1}$, transformations of non-terminating very-well-poised ${}_{8}\phi_{7}$’s, and Baileys’ four-term relation for very-well-poised ${}_{10}\phi_{9}$’s (as a symmetry of a sum of two ${}_{10}\phi_{9}$’s). The symmetry group of the related function is the stabilizer of a generic point in the face, or equivalently the stabilizer of all the vertices of the face. Indeed if some element of $w$ fixes the face, but non-trivially permutes the vertices of the face, then it can be written as the product of a permutation of the vertices generated by reflections in hyperplanes orthogonal to the face, and a Weyl group element which stabilizes the vertices of the face. However, as the functions $B_{\alpha}^{1}(u)$ only depend on the space orthogonal to the face, the first factor has no effect. Secondly, elements of the Weyl group which send one face to a different face induce transformations relating the two functions associated to these two faces. Examples of this include Nassrallah and Rahman’s integral representation of a very-well-poised ${}_{8}W_{7}$, the expression of this function as a sum of two balanced ${}_{4}\phi_{3}$’s, and a relation relating the sum of two ${}_{3}\phi_{2}$’s with argument $z=q$ to a single ${}_{3}\phi_{2}$ with $z=de/abc$ [7, (III.34)]. Indeed all simplicial faces of $P^{(1)}$ of the same dimension are related by the $W(E_{7})$ symmetry, except for dimension 5. Indeed for dimension 5 there are two orbits: $5$-simplices which bound a 6-simplex and those which do not. In particular in Fig. 2 below, there exist transformation formulas between all functions on the same horizontal level, except for the second-lowest level, where you must distinguish between those faces which are at the boundary of some higher dimensional simplicial face, and those that are not. As two functions between which there exists a transformation formula have the same symmetry group, we have written down the symmetry groups of all the functions on each level on the left hand side. Note that the symmetry group for 5-simplices at the boundary of a 6-simplex is $1$ (i.e. the group with only 1 element), while for the other 5-simplices the symmetry group is $W(A_{1})\cong S_{2}$. We can also consider the limit of the contiguous relations satisfied by $E^{1}$. The $q$-contiguous relations reduce to $q$-contiguous relations. We get a relation for each set of three terms $B_{\alpha}(u\cdot q^{\beta_{i}})$, where the $\beta_{i}$ are projections of points in the root lattice of $E_{7}$ to the space orthogonal to the face containing $\alpha$. More interesting is the limit of a $p$-contiguous relation. In order for us to be able to take a limit we have to find three points on $P_{1}$ whose pairwise differences are roots of $E_{7}$. ###### Proposition 6.2. Let $\alpha,\beta,\gamma\in P^{(1)}$ be such that $\alpha-\beta,\alpha-\gamma,\beta-\gamma\in R(E_{7})$ and form an equilateral triangle $($i.e. $(\alpha-\beta)\cdot(\alpha-\gamma)=1)$, and let $u\in\tilde{\mathcal{H}}_{1}$. Recall $S=\\{v\in R(E_{8})~{}\|~{}v\cdot\rho=1\\}$. Then $\displaystyle\prod_{\begin{subarray}{c}\delta\in S\\\ \delta\cdot(\alpha,\beta,\gamma)=(1,0,0)\end{subarray}}(u^{\delta};q)u^{\gamma}\theta(u^{\beta-\gamma};q)B^{1}_{\alpha}(u)+\prod_{\begin{subarray}{c}\delta\in S\\\ \delta\cdot(\alpha,\beta,\gamma)=(0,1,0)\end{subarray}}(u^{\delta};q)u^{\alpha}\theta(u^{\gamma-\alpha};q)B^{1}_{\beta}(u)$ $\displaystyle\qquad\qquad{}+\prod_{\begin{subarray}{c}\delta\in S\\\ \delta\cdot(\alpha,\beta,\gamma)=(0,0,1)\end{subarray}}(u^{\delta};q)u^{\beta}\theta(u^{\alpha-\beta};q)B^{1}_{\gamma}(u)=0.$ (6.1) Note that (6.1) is written in its most symmetric form. In order to avoid non- integer powers of the constants one should first multiply the entire equation by $u^{-\alpha}$ and use $u^{\rho}=q$. ###### Proof 6.2. Choose $\zeta$ such that $\tilde{\alpha}=\alpha+\zeta$, $\tilde{\beta}=\beta+\zeta$ and $\tilde{\gamma}=\gamma+\zeta$ are all roots of $E_{7}$. This is possible by choosing $\tilde{\alpha}$ such that $\tilde{\alpha}\cdot(\alpha-\beta)=1$ and $\tilde{\alpha}\cdot(\alpha-\gamma)=1$, and we can always find roots satisfying these two conditions. Now observe that $\tilde{\alpha}\cdot\tilde{\beta}\leq 1$ as inner product of two different roots of $E_{7}$, and that $\tilde{\alpha}\cdot\tilde{\beta}=(\alpha+\zeta)\cdot(\beta+\zeta)=(\alpha+\zeta)\cdot(\alpha+\zeta)+(\alpha+\zeta)\cdot(\beta-\alpha)\geq 2-1=1,$ as $\alpha+\zeta\neq-(\beta-\alpha)$ (equality here would imply $\beta+\zeta=0$). Thus $\tilde{\alpha}\cdot\tilde{\beta}=1$, and hence $\tilde{\alpha}$, $\tilde{\beta}$ and $\tilde{\gamma}$ satisfy the conditions of Theorem 3.5. Setting $t=u\cdot p^{\zeta}$ in (3.3) we obtain $\displaystyle\prod_{\begin{subarray}{c}\delta\in S\\\ \delta\cdot(\alpha-\beta)=\delta\cdot(\alpha-\gamma)=1\end{subarray}}(u^{\delta}p^{\delta\cdot\beta};q)u^{\gamma}p^{\gamma\cdot\zeta}\theta(u^{\beta-\gamma}p^{\zeta\cdot(\beta-\gamma)};q)E^{1}(u\cdot p^{\alpha})$ $\displaystyle\qquad\qquad{}+\prod_{\begin{subarray}{c}\delta\in S\\\ \delta\cdot(\beta-\alpha)=\delta\cdot(\beta-\gamma)=1\end{subarray}}(u^{\delta}p^{\delta\cdot\gamma};q)u^{\alpha}p^{\alpha\cdot\zeta}\theta(u^{\gamma-\alpha}p^{\zeta\cdot(\gamma-\alpha)};q)E^{1}(u\cdot p^{\beta})$ $\displaystyle\qquad\qquad{}+\prod_{\begin{subarray}{c}\delta\in S\\\ \delta\cdot(\gamma-\alpha)=\delta\cdot(\gamma-\beta)=1\end{subarray}}(u^{\delta}p^{\delta\cdot\alpha};q)u^{\beta}p^{\beta\cdot\zeta}\theta(u^{\alpha-\beta}p^{\zeta\cdot(\alpha-\beta)};q)E^{1}(u\cdot p^{\gamma})=0.$ (6.2) for $u\in\tilde{\mathcal{H}}_{1}$. Now we prove a lemma ###### Lemma 6.3. Let $\alpha$, $\beta$ and $\gamma$ be as in the Proposition and let $\delta\in S$ satisfy $\delta\cdot(\beta-\gamma)=0$ then $\delta\cdot\beta\geq 0$. Moreover $\zeta\cdot(\beta-\gamma)=0$ for $\zeta$ as in this proof. ###### Proof 6.3. Recall the bounding inequalities for $P^{(1)}$ given in Proposition 5.8. Note that $\mu=\rho-\gamma+\beta-\delta\in\Lambda(E_{8})$ satisfies $\mu\cdot\rho=1$ and $\mu\cdot\mu=4$, thus we get $\beta\cdot(\rho-\gamma+\beta-\delta)\leq 2$ and similarly $\gamma\cdot(\rho+\gamma-\beta-\delta)\leq 2$. Adding these two inequalities and simplifying gives $\delta\cdot(\beta+\gamma)\geq 0$, and as $(\beta-\gamma)\cdot\delta=0$, this implies $\delta\cdot\beta\geq 0$. Now observe that $\zeta\cdot(\beta-\gamma)=(\tilde{\alpha}-\alpha)\cdot(\beta-\gamma)=\tilde{\alpha}\cdot(\tilde{\beta}-\tilde{\gamma})-\alpha\cdot(\beta-\gamma)=-\alpha\cdot(\beta-\gamma),$ where in the last equality we used that $\tilde{\alpha}\cdot\tilde{\beta}=1=\tilde{\alpha}\cdot\tilde{\gamma}$. Thus we need to show that $\alpha\cdot(\beta-\gamma)=0$. By the bounding inequalities we have $\alpha\cdot(\alpha-\beta)\leq 1$, but also $\alpha\cdot(\alpha-\beta)=(\alpha-\beta)\cdot(\alpha-\beta)-\beta\cdot(\beta-\alpha)=2-\beta\cdot(\beta-\alpha)\geq 1.$ Thus we find $\alpha\cdot(\alpha-\beta)=1$. By symmetry we also have $\alpha\cdot(\alpha-\gamma)=1$. Thus it follows that $\alpha\cdot\beta=\alpha\cdot\gamma$, or $\alpha\cdot(\beta-\gamma)=0$. The lemma shows that $p^{\gamma\cdot\zeta}=p^{\alpha\cdot\zeta}=p^{\beta\cdot\zeta}$, so we can divide by this term. Using this lemma we see that we can subsequently take the limit $p\to 0$ in (6.2) directly as the arguments of the $\theta$ functions do not depend on $p$, while the arguments of the $q$-shifted factorials are either independent of $p$ or vanish as $p\to 0$. The relations obtained in this way are three-term relations. By the geometry of the polytope the $\alpha$, $\beta$ and $\gamma$ in the above proposition must be such that the faces they are contained in, are in the same $W(E_{7})$ orbit. In particular we can rewrite our three term relation as a relation between three instances of the same function. Thus we get as examples three- term relations for ${}_{3}\phi_{2}$’s [7, (III.33)]. Moreover we obtain the six-term relations of ${}_{10}\phi_{9}$’s as studied in [8] and [10]. One reason why the $p$-contiguous relations morally should exist on the elliptic level is that the three functions related by $p$-shifts in roots of $E_{7}$ satisfy the same second order $q$-difference equations (after a suitable gauge transformation). In particular we can take the limit of these $q$-difference equations and see that $B_{\alpha}^{1}$, $B_{\beta}^{1}$ and $B_{\gamma}^{1}$ also satisfy the same second order $q$-difference equations. In a very degenerate case, there exist faces for which to a vector $\alpha$ in that face there exists exactly one root $r\in R(E_{7})$ such $\alpha+r\in P^{(1)}$. In particular, while we cannot find a three term relation in this case, we do obtain the second solution of the corresponding $q$-difference equations. In the general case we can obtain the symmetry group of the $q$-difference equations by looking at the stabilizer of the shifted lattice $\alpha+\Lambda(E_{7})$ for a generic point $\alpha$ in the face. This stabilizer, the stabilizer of $\alpha$ under the affine Weyl group, is denoted the affine symmetry group in Fig. 2. ## 7 An extended example: $\boldsymbol{{}_{2}\phi_{1}}$ In this section we consider the simplicial face with vertices $w_{01}$, $w_{02}$, $v_{67}$ and $v_{57}$. The centroid of this face is the point $\alpha=(-1/4,0,0,1/4,1/4,1/2,1/2,3/4)$, and we find using Proposition 4.2 that the limit can be expressed as $\displaystyle B_{\alpha}^{1}(u)=\frac{(q,u_{3}u_{0},u_{4}u_{0};q)}{(q/u_{1}u_{2};q)}\int\theta(u_{0}u_{1}u_{2}/z;q)\frac{(qz/u_{7};q)}{(u_{0}/z,u_{3}z,u_{4}z;q)}\frac{dz}{2\pi iz}$ $\displaystyle\phantom{B_{\alpha}^{1}(u)}{}=(u_{1}u_{2},qu_{0}/u_{7};q){}_{2}\phi_{1}\left(\begin{array}[]{c}u_{0}u_{3},u_{0}u_{4}\\\ qu_{0}/u_{7}\end{array};q,u_{1}u_{2}\right),$ as long as this series converges (this integral expression for a ${}_{2}\phi_{1}$ is not very exciting, as it is related to the series by picking up the residues upon moving the integration contour to zero). The stabilizer group of this face under the $W(E_{7})$ action equals the stabilizer of $\alpha$ (as it should be a permutation of the four vertices of the face). However those reflections in $W(E_{7})$ which non-trivially permute the vertices of the face are in roots which are the difference of two vertices, so they will just induce a shift along a vector in the face; as our functions only depend on the space orthogonal to the face, they act as identity on our function (for example they permute $u_{1}\leftrightarrow u_{2}$ or $u_{5}\leftrightarrow u_{6}$). Thus we are only interested in those elements of $W(E_{7})$ which leave the four vertices of this face invariant. In particular, this includes (and by Coxeter theory, is generated by) the reflections in the hyperplanes orthogonal to the roots $\\{\pm(e_{3}-e_{4}),\pm(\rho-e_{0}-e_{3}-e_{4}-e_{7}),\pm(\rho- e_{0}-e_{3}-e_{5}-e_{6}),\pm(\rho-e_{0}-e_{4}-e_{5}-e_{6})\\}$. These eight roots form the root system of $A_{2}\times A_{1}$, thus the symmetry group of a ${}_{2}\phi_{1}$ is $W(A_{2}\times A_{1})$, or the permutation group $S_{3}\times S_{2}$. For example the reflection $s_{\rho- e_{0}-e_{3}-e_{4}-e_{7}}$ generates the symmetry $u\mapsto(u_{0}/s,u_{1}s,u_{2}s,u_{3}/s,u_{4}/s,u_{5}s,u_{6}s,u_{7}/s)$, with $s=\sqrt{u_{0}u_{3}u_{4}u_{7}/q}$, or $\displaystyle(u_{1}u_{2},qu_{0}/u_{7};q){}_{2}\phi_{1}\left(\begin{array}[]{c}u_{0}u_{3},u_{0}u_{4}\\\ qu_{0}/u_{7}\end{array};q,u_{1}u_{2}\right)$ $\displaystyle\qquad{}=(u_{0}u_{1}u_{2}u_{3}u_{4}u_{7}/q,qu_{0}/u_{7};q){}_{2}\phi_{1}\left(\begin{array}[]{c}q/u_{4}u_{7},q/u_{3}u_{7}\\\ qu_{0}/u_{7}\end{array};q,u_{0}u_{1}u_{2}u_{3}u_{4}u_{7}/q\right),$ or simplifying we get $(z,c;q){}_{2}\phi_{1}\left(\begin{array}[]{c}a,b\\\ c\end{array};q,z\right)=(abz/c,c;q){}_{2}\phi_{1}\left(\begin{array}[]{c}c/b,c/a\\\ c\end{array};q,abz/c\right),$ which is one of Heine’s transformations [7, (III.3)]. Similarly related to $s_{\rho-e_{0}-e_{4}-e_{5}-e_{6}}$ we obtain $(z,c;q){}_{2}\phi_{1}\left(\begin{array}[]{c}a,b\\\ c\end{array};q,z\right)=(c/a,az;q){}_{2}\phi_{1}\left(\begin{array}[]{c}a,abz/c\\\ az\end{array};q,c/a\right),$ another one of Heine’s transformations [7, (III.2)]. Together with the permutation swapping $a$ and $b$ (given by $s_{e_{3}-e_{4}}$) these two transformations generate the entire symmetry group. As for transformations to other functions, there are no less than 6 other faces in the $W(E_{7})$-orbit of the face containing $\alpha$ up to $S_{8}$ symmetry. The related transformations are given by (after some simplification) $\displaystyle(z,c;q){}_{2}\phi_{1}\left(\begin{array}[]{c}a,b\\\ c\end{array};q,z\right)=(bz,c;q){}_{2}\phi_{2}\left(\begin{array}[]{c}b,c/a\\\ bz,c\end{array};q,az\right)$ $\displaystyle\qquad{}=\frac{(a,b,abz/c,q;q)}{2}\int\frac{(y^{\pm 2},\sqrt{cz}y^{\pm 1};q)}{(\sqrt{c/z}y^{\pm 1},a\sqrt{z/c}y^{\pm 1},b\sqrt{z/c}y^{\pm 1};q)}\frac{dy}{2\pi iy}$ $\displaystyle\qquad{}=\frac{(z,c/b,c/a;q)}{(c/ab;q)}{}_{3\vphantom{2}}\phi_{2\vphantom{3}}\left(\begin{array}[]{c}abz/c,a,b\\\ qab/c,0\end{array};q,q\right)+\frac{(a,b,abz/c;q)}{(ab/c;q)}{}_{3}\phi_{2}\left(\begin{array}[]{c}z,c/a,c/b\\\ qc/ab,0\end{array};q,q\right)$ $\displaystyle\qquad{}=\frac{(z,abz/c,c;q)}{(bz/c;q)}{}_{3}\phi_{2}\left(\begin{array}[]{c}c/b,a,0\\\ qc/bz,c\end{array};q,q\right)+\frac{(c/b,a,bz;q)}{(c/bz;q)}{}_{3}\phi_{2}\left(\begin{array}[]{c}z,abz/c,0\\\ qbz/c,bz\end{array};q,q\right)$ $\displaystyle\qquad{}=\frac{(az,bz,c;q)}{(abz;q)}{}_{6\vphantom{5}}^{\vphantom{(2)}}W_{5\vphantom{6}}^{(2)}(\frac{abz}{q};a,b,abz/c;q,cz)$ $\displaystyle\qquad{}=(q;q)\int\theta(z/y;q)\frac{(cy,aby;q)}{(ay,by,cy/z;q)}\frac{dy}{2\pi iy}.$ Let us now consider the three term relations (as limit of $p$-contiguous relations). The points $\beta$ in the polytope with $\alpha-\beta\in\Lambda(E_{7})$ in the root lattice of $E_{7}$ are $\beta$ \| $B_{\beta}^{1}$ ---\|--- 2ex $(-\frac{1}{4},0,0,\frac{1}{4},\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4})=\alpha$ \| $(u_{1}u_{2},qu_{0}/u_{7};q){}_{2}\phi_{1}\left(\begin{array}[]{c}u_{0}u_{3},u_{0}u_{4}\\\ qu_{0}/u_{7}\end{array};q,u_{1}u_{2}\right)$ 2ex $(\frac{3}{4},0,0,\frac{1}{4},\frac{1}{4},\frac{1}{2},\frac{1}{2},-\frac{1}{4})$ \| $(u_{1}u_{2},qu_{7}/u_{0};q){}_{2}\phi_{1}\left(\begin{array}[]{c}u_{7}u_{3},u_{7}u_{4}\\\ qu_{7}/u_{0}\end{array};q,u_{1}u_{2}\right)$ 2ex $(\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4},-\frac{1}{4},0,0,\frac{1}{4})$ \| $(u_{5}u_{6},qu_{4}/u_{3};q){}_{2}\phi_{1}\left(\begin{array}[]{c}u_{0}u_{4},u_{7}u_{4}\\\ qu_{4}/u_{3}\end{array};q,u_{5}u_{6}\right)$ 2ex $(\frac{1}{4},\frac{1}{2},\frac{1}{2},-\frac{1}{4},\frac{3}{4},0,0,\frac{1}{4})$ \| $(u_{5}u_{6},qu_{3}/u_{4};q){}_{2}\phi_{1}\left(\begin{array}[]{c}u_{0}u_{3},u_{7}u_{3}\\\ qu_{3}/u_{4}\end{array};q,u_{5}u_{6}\right)$ Here we can use the balancing condition $\prod_{r}u_{r}=q^{2}$ to rewrite $u_{5}u_{6}$ in terms of the previous parameters. Any three of these four functions now give a three term relation, for example (after simplification) $\displaystyle(bq/c,q/a,c;q)\frac{az}{c}\theta(c/az;q){}_{2}\phi_{1}\left(\begin{array}[]{c}a,b\\\ c\end{array};q,z\right)$ $\displaystyle\qquad{}+(b,c/a,q^{2}/c;q)\frac{q}{c}\theta(az/q;q){}_{2}\phi_{1}\left(\begin{array}[]{c}aq/c,bq/c\\\ q^{2}/c\end{array};q,z\right)$ $\displaystyle\qquad{}+(abz/c,cq/abz,bq/a;q)\theta(q/c;q){}_{2}\phi_{1}\left(\begin{array}[]{c}b,bq/c\\\ bq/a\end{array};q,cq/abz\right)=0.$ The affine symmetry group is now given as the extension of the symmetry group by also allowing elements which permute the four ${}_{2}\phi_{1}$’s amongst themselves. Indeed the index $[W(A_{3}\times A_{1}):W(A_{2}\times A_{1})]=4$. It is also the symmetry group of the $q$-difference equations we discuss next. The $q$-contiguous equations relate three terms of the form $B_{\alpha}^{1}(u\cdot q^{\beta})$ where $\beta$ is in the projection $\Lambda_{\alpha}$ of $\Lambda(E_{7})$ on the space orthogonal to the face containing $\alpha$. In particular the lattice $\Lambda_{\alpha}$ is generated by ($\pi$ denotes the orthogonal projection on $\Lambda_{\alpha}$) $\displaystyle\pi(0,0,0,1,0,-1,0,0)=(1/4,0,0,3/4,-1/4,-1/2,-1/2,1/4),$ $\displaystyle\pi(0,0,0,0,1,-1,0,0)=(1/4,0,0,-1/4,3/4,-1/2,-1/2,1/4),$ $\displaystyle\pi(0,0,0,0,0,1,0,-1)=(1/4,0,0,-1/4,-1/4,1/2,1/2,-3/4),$ $\displaystyle\pi(0,0,1,0,0,-1,0,0)=(0,1/2,1/2,0,0,-1/2,-1/2,0).$ If we simplify ${}_{2}\phi_{1}$ by setting $a=u_{0}u_{3}$, $b=u_{0}u_{4}$, $c=qu_{0}/u_{7}$ and $z=u_{1}u_{2}$, these four vectors correspond to multiplying respectively $a$, $b$, $c$, or $z$ by $q$. In particular we have a relation for any three sets of parameters where $a$, $b$, $c$ and $z$’s differ by an integer power of $q$. For example using shifts $\pi(\rho- e_{2}-e_{4}-e_{5}-e_{6})$, $\pi(e_{1}-e_{5})$ and $\pi(e_{1}-e_{2})$ (corresponding to $a\mapsto aq$, $z\mapsto qz$ and doing nothing), we get $-(1-a){}_{2\vphantom{1}}\phi_{1\vphantom{2}}\left(\begin{array}[]{c}aq,b\\\ c\end{array};q,z\right)-a{}_{2\vphantom{1}}\phi_{1\vphantom{2}}\left(\begin{array}[]{c}a,b\\\ c\end{array};q,qz\right)+{}_{2\vphantom{1}}\phi_{1\vphantom{2}}\left(\begin{array}[]{c}a,b\\\ c\end{array};q,z\right)=0.$ ## 8 Evaluations: the $\boldsymbol{m=0}$ case In the $m=0$ case the general polytope picture is somewhat unsatisfying as a description of the possible limits of the elliptic hypergeometric beta integral evaluation. Indeed there are two issues. First of all the polytope $P^{(0)}$ as described in Section 5 is not the entire polytope for which proper limits exist, indeed we can give a larger polytope for which this is true. Secondly, if we are interested in knowing what the different evaluations on the basic hypergeometric level are, it seems more natural to look at $P_{\rm I}^{(0)}$, $P_{\rm II}^{(0)}$ and $P_{\rm III}^{(0)}$, and consider the faces of these polytopes, instead of looking at $P^{(0)}$. Let us first consider this second issue. In Section 5 we have actually shown that to each face of $P_{\rm I}$, $P_{\rm II}$ and $P_{\rm III}$ there is associated a function, which depends only on the space orthogonal to that face. Moreover the iterated limit property holds in these polytopes. If we therefore want to know what the different limit evaluations are, we only have to write down the faces of these three polytopes and the associated functions with their evaluations. As all faces of these polytopes are simplicial, except for the interior of $P_{\rm II}$, this is a simple combinatorial argument. All faces are listed in the appendix in Fig. 3. For those faces of $P$ which are split in different faces of $P_{\rm I}$, $P_{\rm II}$ and $P_{\rm III}$ (i.e. the interior and the facets given by $\alpha_{r}+\alpha_{s}+\alpha_{t}+\alpha_{u}=0$) the value of the evaluation might be the same on all these different faces of the smaller polytopes, but as the functions are a priori different, we do obtain a different evaluation formula for each of these faces. Now we look at the larger polytope. ###### Definition 8.1. Define the extended polytope $P_{\rm ext}$ to be the polytope given as the convex hull of the vectors $e_{j}$ ($0\leq j\leq 5$) and $f_{j}=\rho^{(0)}-2e_{j}$ ($0\leq j\leq 5$). The bounding inequalities are ###### Proposition 8.2. The bounding inequalities of $P_{\rm ext}$ inside the subspace $2\rho^{(0)}\cdot\alpha=1$ are given by $\displaystyle\alpha_{r}+\alpha_{s}\leq 1,\qquad(0\leq r<s\leq 5).$ ###### Proof 8.1. This follows from a calculation as in Proposition 5.7 (though now we have $\binom{7}{2}$ options as we need to take two vectors from seven). The following proposition follows immediately from the evaluation formula (3.2). ###### Proposition 8.3. For $\alpha\in P_{\rm ext}$ the limit (4.1) exists and $B_{\alpha}^{0}(u)$ is the same for each $\alpha$ in a face of $P_{\rm ext}$ and depends only on $u$ orthogonal to the face containing $\alpha$. Moreover the iterated limit property holds. The question this immediately raises is to what extent one can give series or integrals corresponding to points in $P_{\rm ext}\backslash P^{(0)}$. So far we have not been able to give a good description of these limits, let alone a classification. However we expect this is where we have to look for evaluations of bilateral series. In the next section we consider more generally what we expect. ## 9 Going beyond the polytope As indicated in the previous section there exist proper limits outside the polytopes as described in Section 5. While we only know of the existence of proper limits as in (4.1) outside of $P^{(m)}$ in the case $m=0$, if we let $p$ tend to zero along a geometric progression and rescale the functions we can obtain limits in many more cases. In general these limits will depend on what geometric progression we use for $p$ (unlike in the $m=0$ case). We do not know for which points in $\mathbb{R}^{2m+6}$ we can take limits in this way but it does seem to provide a very rich extra set of functions. In particular, this seems to be where bilateral series reside. For instance, Chen and Fu [3] prove a ${}_{2}\psi_{2}$ transformation as a limit of (in our notation) a transformation of $B^{(1)}_{(w^{(1)}_{01}+w^{(1)}_{02}+v^{(1)}_{67})/3}$, taken in a direction pointing outside the Hesse polytope. As an example we consider $m=1$ and a limit along $\alpha=v_{1}+\epsilon v_{2}$ for small $\epsilon>0$ and $v_{1}$ a vertex of $P^{(1)}$ and $v_{2}$ a root of $E_{7}$ with $v_{1}\cdot v_{2}=0$. Note that all of these limits are related to each other by the Weyl group of $E_{7}$ action, so we expect to obtain transformation formulas relating the functions associated to these vectors. Up to permutations we have the following 6 options: 1. 1. $(0,0,0,0,0,0,1-\epsilon,1+\epsilon)$; 2. 2. $(-\epsilon/2,-\epsilon/2,-\epsilon/2,\epsilon/2,\epsilon/2,\epsilon/2,1-\epsilon/2,1+\epsilon/2)$; 3. 3. $(-\epsilon,0,0,0,0,\epsilon,1,1)$; 4. 4. $(-1/2,-1/2,1/2-\epsilon,1/2,1/2,1/2,1/2,1/2+\epsilon)$; 5. 5. $(-1/2-\epsilon/2,-1/2+\epsilon/2,1/2-\epsilon/2,1/2-\epsilon/2,1/2-\epsilon/2,1/2-\epsilon/2,1/2-\epsilon/2)$; 6. 6. $(-1/2-\epsilon,-1/2+\epsilon,1/2,1/2,1/2,1/2,1/2,1/2)$. For some of these we can obtain limits as integrals. In the rest of this section we suppose $\epsilon=1/N$ for some large integer $N$, and $p=x^{N}q^{kN}$ (where $k$ is allowed to vary). ###### Proposition 9.1. For $\alpha=(0,0,0,0,0,0,1-\epsilon,1+\epsilon)$ we have $\lim_{k\to\infty}E_{1}(p^{\alpha}\cdot u)(xu_{7})^{2k}q^{2\binom{k}{2}}=\prod_{0\leq r<s\leq 5}(u_{r}u_{s};q)\frac{(q;q)}{2}\int\frac{\theta(xu_{7}z^{\pm 1};q)(z^{\pm 2};q)}{\prod\limits_{r=0}^{5}(u_{r}z^{\pm 1};q)}\frac{dz}{2\pi iz}.$ ###### Proof 9.1. We calculate $\displaystyle\lim_{k\to\infty}E_{1}(p^{\alpha}\cdot u)(xu_{7})^{2k}q^{2\binom{k}{2}}$ $\displaystyle\qquad{}=\lim_{k\to\infty}(xu_{7})^{2k}q^{2\binom{k}{2}}\prod_{0\leq r<s\leq 7}(p^{\alpha_{r}+\alpha_{s}}u_{r}u_{s};p,q)\frac{(p;p)(q;q)}{2}$ $\displaystyle\qquad\qquad{}\times\int\frac{\prod\limits_{r=0}^{5}\Gamma(u_{r}z^{\pm 1})\Gamma(p^{1-\epsilon}u_{6}z^{\pm 1},p^{1+\epsilon}u_{7}z^{\pm 1})}{\Gamma(z^{\pm 2})}\frac{dz}{2\pi iz}$ $\displaystyle\qquad{}=\prod_{0\leq r<s\leq 5}(u_{r}u_{s};q)\frac{(q;q)}{2}\lim_{k\to\infty}(xu_{7})^{2k}q^{2\binom{k}{2}}$ $\displaystyle\qquad\qquad{}\times\int\frac{\theta(p^{\epsilon}u_{7}z^{\pm 1};q)\prod\limits_{r=0}^{5}\Gamma(u_{r}z^{\pm 1})\Gamma(p^{1-\epsilon}u_{6}z^{\pm 1},p^{\epsilon}u_{7}z^{\pm 1})}{\Gamma(z^{\pm 2})}\frac{dz}{2\pi iz}$ $\displaystyle\qquad{}=\lim_{k\to\infty}\prod_{0\leq r<s\leq 5}(u_{r}u_{s};q)\frac{(q;q)}{2}$ $\displaystyle\qquad\qquad{}\times\int\frac{\theta(xu_{7}z^{\pm 1};q)\prod\limits_{r=0}^{5}\Gamma(u_{r}z^{\pm 1})\Gamma(p^{1-\epsilon}u_{6}z^{\pm 1},p^{\epsilon}u_{7}z^{\pm 1})}{\Gamma(z^{\pm 2})}\frac{dz}{2\pi iz}$ $\displaystyle\qquad{}=\prod_{0\leq r<s\leq 5}(u_{r}u_{s};q)\frac{(q;q)}{2}\int\frac{\theta(xu_{7}z^{\pm 1};q)(z^{\pm 2};q)}{\prod\limits_{r=0}^{5}(u_{r}z^{\pm 1};q)}\frac{dz}{2\pi iz}.$ Here we used that all the poles are on the right side of the contour as $p\to 0$, so we can interchange limit and integral as before. Moreover we used that $\theta(p^{\epsilon}y;q)=\theta(q^{k}xy;q)=\theta(xy;q)\left(-\frac{1}{xy}\right)^{k}q^{-\binom{k}{2}}$ for $y=u_{7}z$ and $y=u_{7}/z$. The next two limits are analogous. ###### Proposition 9.2. For $\alpha=(-1/2,-1/2,1/2-\epsilon,1/2,1/2,1/2,1/2,1/2+\epsilon)$ we have $\displaystyle\lim_{k\to\infty}E_{1}(p^{\alpha}\cdot u)\left(\frac{qu_{7}x^{2}}{u_{0}u_{1}u_{2}}\right)^{k}q^{2\binom{k}{2}}=(u_{0}u_{1},q;q)\prod_{r=0}^{1}\prod_{s=3}^{6}(u_{r}u_{s};q)$ $\displaystyle\qquad\qquad{}\times\int\frac{(1-z^{2})\theta(u_{0}u_{1}u_{2}/zx,xu_{7}/z;q)(qz/u_{3},qz/u_{4},qz/u_{5},qz/u_{6};q)}{(u_{0}z^{\pm 1},u_{1}z^{\pm 1},u_{3}z,u_{4}z,u_{5}z,u_{6}z;q)}\frac{dz}{2\pi iz}$ and for $\alpha=(-\epsilon/2,-\epsilon/2,-\epsilon/2,\epsilon/2,\epsilon/2,\epsilon/2,1-\epsilon/2,1+\epsilon/2)$ we have $\displaystyle\lim_{k\to\infty}E_{1}(p^{\alpha}\cdot u)\left(\frac{qu_{7}x^{2}}{u_{0}u_{1}u_{2}}\right)^{k}q^{2\binom{k}{2}}=(q;q)\prod_{r=0}^{2}\prod_{s=3}^{5}(u_{r}u_{s};q)$ $\displaystyle\qquad\qquad{}\times\int\frac{\theta(u_{0}u_{1}u_{2}/zx,u_{7}x/z;q)(q/u_{7}z,qz/u_{6};q)}{(u_{0}/z,u_{1}/z,u_{2}/z,u_{3}z,u_{4}z,u_{5}z;q)}\frac{dz}{2\pi iz}.$ ###### Proof 9.2. The first integral is a direct limit in the symmetry broken integral (4.3), while the second limit comes from the symmetric integral with $z\to p^{\epsilon/2}z$. For the other three limits we are unable to describe $B_{\alpha}$ using an integral, but we do have series representations of these limits. As in the case of Proposition 4.3 proofs of these limits involve tedious calculations, so we just give a short sketch. As we have found is quite common for series representations outside the polytope, we obtain bilateral series. For example ###### Proposition 9.3. For $\alpha=w_{01}+\epsilon(e_{1}-e_{0})$ that if $\|u_{0}u_{1}\|<1$ we have $\displaystyle\lim_{k\to\infty}E_{1}(p^{\alpha}\cdot u)x^{2k}q^{2\binom{k}{2}}\left(\frac{q}{u_{0}^{2}}\right)^{k}=\frac{(u_{0}u_{1};q)}{(q;q)}\theta(u_{0}^{2}/x^{2};q)\prod_{r=2}^{7}(qx/u_{r}u_{0},qu_{0}/u_{r}x;q)$ $\displaystyle\qquad\qquad{}\times{}_{8}\psi_{8}\left(\begin{array}[]{c}\pm qu_{0}/x,u_{2}u_{0}/x,\ldots,u_{7}u_{0}/x\\\ \pm u_{0}/x,qu_{0}/u_{2}x,\ldots,qu_{0}/u_{7}x\end{array};q,u_{0}u_{1}\right).$ ###### Proof 9.3. To obtain this limit we look at the symmetry broken integral (4.4) with two $s_{i}$ specialized, and change the integration variable $z\to p^{\epsilon-1/2}z$. Subsequently we pick up the poles at $z=u_{0}p^{-2\epsilon}q^{n}$ (for $n=0$ to $n=2k$) and observe that the remaining integral vanishes in the limit. Moreover the absolute value of the summand in the sum of the residues is maximized near $n=k$ and we can show that we can interchange limit and sum in $\sum_{n=-k}^{k}\operatorname{Res}(z=u_{0}p^{-2\epsilon}q^{n+k})$, giving a bilateral sum. Note that for $\|u_{0}u_{1}\|\geq 1$ we do not have an explicit expression for the limit. The limit does have an analytic extension to the region $\|u_{0}u_{1}\|\geq 1$, but we can only prove this by using the Weyl group symmetry to relate the limit to the previous limits obtained. In the case $\|u_{0}u_{1}\|>1$ we could again use the general method of obtaining a limit: discovering where the integrand or residues are maximized, rescaling properly and interchanging limit and sum/integral. In this case it would lead to $\lim_{k\to\infty}E_{1}(p^{\alpha}\cdot u)x^{2k}q^{2\binom{k}{2}}\left(\frac{q}{u_{0}^{3}u_{1}}\right)^{k}=\frac{\prod\limits_{r=2}^{7}\theta(u_{r}x/u_{0};q)}{\theta(x^{2}/u_{0}^{2};q)}{}_{1\vphantom{0}}\phi_{0\vphantom{1}}\left(\begin{array}[]{c}u_{0}u_{1}\\\ -\end{array};q,\frac{1}{u_{0}u_{1}}\right)$ by picking up the residues at $z=u_{0}p^{-\epsilon}q^{n}$ in the same integral as above but with $z\to p^{-1/2}z$ instead of $z\to p^{\epsilon-1/2}z$. Note that the right hand side vanishes by the evaluation formula for a ${}_{1\vphantom{0}}\phi_{0\vphantom{1}}$. Indeed we can also see that this limit vanishes by applying the Weyl group symmetry before taking the limit and observing a factor $\lim_{k\to\infty}(1/u_{0}u_{1})^{k}=0$ remains after using one of the integral limits above. In particular this shows one has to be careful taking limits in order to obtain something interesting. ###### Proposition 9.4. For $\alpha=(-\epsilon,0,0,0,0,\epsilon,1,1)$ and $\|u_{0}u_{5}\|<1$ we obtain the limit $\displaystyle\lim_{k\to\infty}E^{1}(p^{\alpha}\cdot u)\left(\frac{u_{5}u_{6}u_{7}}{u_{0}}\right)^{k}x^{2k}q^{2\binom{k}{2}}=\theta(u_{0}u_{4}/x,u_{0}u_{1}u_{2}u_{3}/x;q)$ $\displaystyle\qquad\qquad{}\times\frac{(u_{0}u_{5},u_{1}u_{4},u_{2}u_{4},qu_{3}/u_{1},qu_{3}/u_{2},qu_{3}/u_{6},q/u_{3}u_{6},qu_{3}/u_{7},q/u_{3}u_{7};q)}{(q/u_{1}u_{2},qu_{3}^{2},u_{4}/u_{3};q)}$ $\displaystyle\qquad\qquad{}\times{}_{8}W_{7}(u_{3}^{2};u_{1}u_{3},u_{2}u_{3},u_{4}u_{3},u_{6}u_{3},u_{7}u_{3};u_{0}u_{5})+(u_{3}\leftrightarrow u_{4}).$ Moreover for $\alpha=w_{01}+\epsilon(\rho-e_{0}-e_{2}-e_{3}-e_{4})$ $($without convergence conditions$)$ $\displaystyle\lim_{k\to\infty}E^{1}(p^{\alpha}\cdot u)x^{2k}q^{2\binom{k}{2}}\left(\frac{u_{5}u_{6}u_{7}}{u_{0}}\right)^{k}$ $\displaystyle\qquad{}=\frac{(u_{0}u_{1};q)\prod\limits_{r=2}^{4}(qx/u_{0}u_{r},u_{1}u_{r};q)\prod\limits_{r=5}^{7}(qu_{0}/xu_{r},u_{0}u_{r};q)}{(q,u_{0}^{2}/x,xu_{1}/u_{0};q)}$ $\displaystyle\qquad\qquad{}\times{}_{4}\psi_{4}\left(\begin{array}[]{c}u_{0}^{2}/x,u_{0}u_{2}/x,u_{0}u_{3}/x,u_{0}u_{4}/x\\\ qu_{0}/xu_{1},qu_{0}/xu_{5},qu_{0}/xu_{6},qu_{0}/xu_{7}\end{array};q,q\right)$ $\displaystyle\qquad\qquad{}+\frac{\prod\limits_{r=2}^{4}\theta(u_{0}u_{r}/x;q)}{\theta(u_{0}/xu_{1};q)}\prod\limits_{r=5}^{7}(qu_{1}/u_{r},u_{0}u_{r};q){}_{4}\phi_{3}\left(\begin{array}[]{c}u_{0}u_{1},u_{2}u_{1},u_{3}u_{1},u_{4}u_{1}\\\ qu_{1}/u_{5},qu_{1}/u_{6},qu_{1}/u_{7}\end{array};q,q\right).$ ###### Proof 9.4. For the first limit, use the symmetry broken integral (4.3) with three specializations, shift $z\to p^{\epsilon}z$ and pick up the residues at $z=p^{\epsilon}q^{n}/u_{3}$ and $z=p^{\epsilon}q^{n}/u_{4}$. We arbitrarily broke the symmetry between $u_{1}$ and $u_{2}$, with $u_{3}$ and $u_{4}$ here to be able to write this as the sum of only two series. The second limit is obtained by picking up the residues at $z=p^{-3\epsilon/2}u_{0}q^{n}$ (for the ${}_{4}\psi_{4}$) and $z=p^{-\epsilon/2}u_{1}q^{n}$ (for the ${}_{4}\phi_{3}$) in the symmetry broken integral (4.4) with two specializations, with $z\to p^{-1/2+\epsilon}$. Now we have obtained these limits we can obtain relations for these functions as before. In particular we obtain the symmetries of these functions (the symmetry group, the stabilizer of $\alpha$ in $W(E_{7})$, is isomorphic to $W(A_{5})$), and we obtain transformation formulas relating all these six functions in terms of each other. This includes the formula [7, (III.38)] expressing an ${}_{8}\psi_{8}$ in two very-well-poised ${}_{8}\phi_{7}$’s. As mentioned before the big difference between the limits inside the polytope and those outside is that inside we do not need to specialize $p$ to a geometric progression. In particular the functions just obtained do depend non-trivially on the parameter $x$; $x$ is not just a cosmetic factor necessary to calculate the limit as $w$ was in Proposition 4.4. The simplest way to see this is to specialize to an evaluation formula. In the elliptic beta integral $E^{1}$ we reduce to $E^{0}$ if the product of two parameters equals $pq$. Thus we must find a pair $r$, $s$ such that $\alpha_{r}+\alpha_{s}=1$ and set $u_{r}u_{s}=q$. This is not possible for all of the limits, but for example in the case $\alpha=(-\epsilon/2,-\epsilon/2,-\epsilon/2,\epsilon/2,\epsilon/2,\epsilon/2,1-\epsilon/2,1+\epsilon/2)$ we can set $u_{5}u_{6}=q$. The limit for the evaluation formula works in precisely the same way, and we obtain $\displaystyle(q/u_{0}u_{7},q/u_{1}u_{7},q/u_{2}u_{7};q)\theta(q/xu_{3}u_{7},q/xu_{4}u_{7};q)$ $\displaystyle\qquad{}=(q;q)\prod_{r=0}^{2}\prod_{s=3}^{5}(u_{r}u_{s};q)\int\frac{\theta(u_{0}u_{1}u_{2}/zx,u_{7}x/z;q)(q/u_{7}z;q)}{(u_{0}/z,u_{1}/z,u_{2}/z,u_{3}z,u_{4}z;q)}\frac{dz}{2\pi iz}.$ The left hand side is clearly non-trivially dependent on $x$, while the right hand side is the (rescaled) limit for $\alpha$ specialized in $u_{5}u_{6}=q$. ## Appendix A The $\boldsymbol{m=1}$ limits ${}_{3\vphantom{2}}^{\vphantom{(5)}}W_{2\vphantom{3}}^{(5)}(b)$${}_{0\vphantom{0}}^{\vphantom{(1)}}\phi_{0\vphantom{0}}^{(1)}$$SB_{0}^{1}$$NR_{0}^{1}$(395,38)(383, 55) (236,36)(269, 54) (219,38)(204, 55) (131,38)(156, 54) (115,38)(103, 55) (5,38)(16, 54) ${}_{4\vphantom{3}}^{\vphantom{(4)}}W_{3\vphantom{4}}^{(4)}(b)$${}_{0}\phi_{2}(b)$${}_{0}\phi_{1}$${}_{0\vphantom{1}}^{\vphantom{(-1)}}\phi_{1\vphantom{0}}^{(-1)}$${}_{1\vphantom{0}}^{\vphantom{(1)}}\phi_{0\vphantom{1}}^{(1)}$$SB_{1}^{1}$$SB_{2}^{0}$${}_{0\vphantom{1}}^{\vphantom{(-2)}}\phi_{1\vphantom{0}}^{(-2)}$${}_{0\vphantom{2}}^{\vphantom{(-3)}}\phi_{2\vphantom{0}}^{(-3)}(q)+\prime\prime$$NR_{1}^{1}$(375,68)(363, 85) (338,68)(355, 85) (326,68)(317, 85) (289,68)(310, 85) (275,68)(263, 85) (232,68)(213, 85) (230,65)(185, 85) (212,66)(248, 84) (194,68)(179, 85) (175,68)(200, 84) (159,64)(127, 85) (144,63)(243, 86) (129,68)(122, 85) (105,68)(117, 85) (95,68)(83, 85) (65,68)(77, 85) (55,68)(43, 85) (25,68)(36, 84) ${}_{5\vphantom{4}}^{\vphantom{(3)}}W_{4\vphantom{5}}^{(3)}(b)$${}_{1}\phi_{2}(b)$${}_{1}\phi_{1}$$SB_{2}^{1}$${}_{2\vphantom{0}}^{\vphantom{(1)}}\phi_{0\vphantom{2}}^{(1)}$${}_{1\vphantom{1}}^{\vphantom{(-1)}}\phi_{1\vphantom{1}}^{(-1)}$${}_{1\vphantom{2}}^{\vphantom{(-2)}}\phi_{2\vphantom{1}}^{(-2)}(q)+\prime\prime$$NR_{2}^{1}$(355,98)(343, 115) (321,98)(336, 115) (311,98)(303, 114) (268,96)(294, 115) (249,96)(210, 115) (217,98)(237, 115) (207,98)(202, 115) (181,98)(197, 115) (169,98)(154, 115) (130,94)(191, 116) (115,98)(103, 115) (85,98)(97, 115) (75,98)(63, 115) (45,98)(56, 114) ${}_{6\vphantom{5}}^{\vphantom{(2)}}W_{5\vphantom{6}}^{(2)}(b)$${}_{2}\phi_{2}(b)$$SB_{3}^{2}(b)$${}_{2}\phi_{1}$${}_{3\vphantom{1}}^{\vphantom{(1)}}\phi_{1\vphantom{3}}^{(1)}(q)+\prime\prime$${}_{2\vphantom{2}}^{\vphantom{(-1)}}\phi_{2\vphantom{2}}^{(-1)}(q)+\prime\prime$$NR_{3}^{1}$(335,128)(323, 145) (305,128)(317, 145) (281,128)(238, 145) (251,128)(271, 145) (238,128)(228, 145) (207,128)(222, 145) (195,128)(179, 145) (156,128)(171, 145) (144,128)(129, 145) (112,125)(163, 145) (95,128)(83, 145) (65,128)(76, 144) ${}_{7\vphantom{6}}^{\vphantom{(1)}}W_{6\vphantom{7}}^{(1)}(b)$$\widehat{SB}{}^{5}_{5}$${}_{3}\phi_{2}(b)$${}_{3}\phi_{2}(q)+\prime\prime$$NR_{5}^{0}$$NR_{4}^{1}$(315,158)(303, 175) (282,158)(296, 175) (240,156)(291, 176) (220,158)(205, 175) (181,158)(197, 175) (163,155)(112, 175) (119,158)(104, 175) (85,158)(97, 175) ${}_{8}W_{7}(b)$${}_{4}\phi_{3}(qb)+\prime\prime$$NR_{5}^{1}$(295,188)(283, 205) (216,186)(270, 206) (180,188)(133, 205) (105,188)(117, 205) ${}_{10}W_{9}(qb)+\prime\prime$$NR_{6}^{2}(b)$$1$$W(A_{1})$$1$$W(A_{1})$$W(A_{2}\times A_{1})$$W(A_{4})$$W(D_{5})$$W(E_{6})$(-125,35)(-125, 55) (-97,65)(-108, 85) (-123,65)(-112, 85) (-110,95)(-110, 115) (-110,125)(-110, 145) (-110,155)(-110, 175) (-110,185)(-110, 205) SymmetrygroupAffineSymmetrygroup$W(E_{7})$$W(D_{6})$$W(A_{5})$$W(A_{3}\times A_{1})$$W(A_{2})$$W(A_{2})$$W(A_{1})$$1$(-55,35)(-55, 55) (-27,65)(-38, 85) (-53,65)(-42, 85) (-40,95)(-40, 115) (-40,125)(-40, 145) (-40,155)(-40, 175) (-40,185)(-40, 205) Figure 2: The simplicial faces of $P^{(1)}$. Fig. 2 shows the functions associated to the ($S_{8}$-orbits of) simplicial faces of $P^{(1)}$. We connect two faces by an edge if one is a facet of the other; the degenerations of a given function are those connected to it by a downward path in the graph. To simplify the picture, we omitted the non-simplicial faces, i.e., the interior together with the facets cut out by equations of the form $\alpha_{0}+\alpha_{1}+\alpha_{2}+\alpha_{3}=0$ or $\alpha_{7}=1+\alpha_{0}$. Note that these non-simplicial faces all correspond to evaluations. In the scheme we used the following abbreviations for integrals $\displaystyle NR_{a}^{b}=\int\frac{(z^{\pm 2};q)\prod\limits_{r=1}^{b}(w_{r}z^{\pm 1};q)}{\prod\limits_{r=1}^{a}(v_{r}z^{\pm 1};q)}\frac{dz}{2\pi iz},\qquad SB^{b}_{a}=\int\theta(u/z;q)\frac{\prod\limits_{r=1}^{b}(w_{r}z;q)}{\prod\limits_{r=1}^{a}(v_{r}z;q)}(1-z^{2})^{1_{\widehat{SB}}}\frac{dz}{2\pi iz}.$ Here $1_{\widehat{SB}}$ denotes 1 if we consider $\widehat{SB}$ and 0 otherwise. We write ${}_{r\vphantom{s}}\phi_{s\vphantom{r}}+\prime\prime$ or ${}_{r}W_{r-1}+\prime\prime$ to denote the sum of two series with related coefficients (as in Proposition 4.2). After series we write $(q)$ if $z=q$ in the series, $(b)$ if the balancing condition $z\prod a_{r}=\prod b_{r}$ should hold, and $(qb)$ if the series is balanced (i.e. both the above properties hold). Similarly $S_{3}^{2}(b)$ and $NR_{6}^{2}(b)$ indicate that a balancing condition holds amongst their parameters. We want to stress that all functions are entire in their parameters. In particular for the non-confluent series without the condition $z=q$ (which might fail to converge) we have an integral representation which gives an analytic extension to all values of $z$. $SB_{0}^{0}$$NR_{0}^{0}$(382,8)(365, 25) (218,8)(235, 25) (202,8)(185, 25) ${}_{3\vphantom{2}}^{\vphantom{(3)}}W_{2\vphantom{3}}^{(3)}(b)$${}_{0}\phi_{1}(b)$${}_{0}\phi_{0}$$SB_{1}^{0}$${}_{0\vphantom{0}}^{\vphantom{(-1)}}\phi_{0\vphantom{0}}^{(-1)}$${}_{0\vphantom{1}}^{\vphantom{(-2)}}\phi_{1\vphantom{0}}^{(-2)}(q)+\prime\prime$$NR_{1}^{0}$(352,38)(335, 55) (308,38)(325, 55) (292,38)(275, 55) (248,38)(265, 55) (232,38)(215, 55) (188,38)(205, 55) (172,38)(155, 55) (135,35)(195, 55) (112,38)(95, 55) (68,38)(85, 55) (52,38)(35, 55) (8,38)(25, 55) ${}_{4\vphantom{3}}^{\vphantom{(2)}}W_{3\vphantom{4}}^{(2)}(b)$${}_{1}\phi_{1}(b)$$SB_{2}^{1}(b)$${}_{1}\phi_{0}$${}_{1\vphantom{1}}^{\vphantom{(-1)}}\phi_{1\vphantom{1}}^{(-1)}(q)+\prime\prime$$NR_{2}^{0}$(322,68)(305, 85) (278,68)(295, 85) (262,68)(245, 85) (218,68)(235, 85) (202,68)(185, 85) (158,68)(175, 85) (142,68)(125, 85) (105,65)(165, 85) (82,68)(65, 85) (38,68)(55, 85) ${}_{5\vphantom{4}}^{\vphantom{(1)}}W_{4\vphantom{5}}^{(1)}(b)$$\widehat{SB}{}^{3}_{3}$${}_{2}\phi_{1}(b)$${}_{2}\phi_{1}(q)+\prime\prime$$NR_{3}^{0}$(292,98)(275, 115) (248,98)(265, 115) (232,98)(215, 115) (188,98)(205, 115) (172,98)(155, 115) (128,98)(145, 115) (75,95)(135, 115) ${}_{6}W_{5}(b)$${}_{3}\phi_{2}(qb)+\prime\prime$$NR_{4}^{0}$(262,128)(245, 145) (218,128)(235, 145) (202,128)(185, 145) (158,128)(175, 145) ${}_{8}W_{7}(qb)+\prime\prime$$NR_{5}^{1}(b)$ Figure 3: The simplicial faces of $P_{\rm I}^{(0)}$, $P_{\rm II}^{(0)}$ and $P_{\rm III}^{(0)}$. In Fig. 3 we have depicted the functions corresponding to the simplicial faces of $P_{\rm I}^{(0)}$, $P_{\rm II}^{(0)}$, $P_{\rm III}^{(0)}$, i.e. those functions for which we obtain an evaluation formula. The only non-simplicial face of these three polytopes is the interior of $P_{\rm II}$ (on which the function given by the relevant list of functions, Proposition 4.3, equals 1 identically). The faces of $P_{\rm I}$ are those which have $NR_{0}^{0}$ as a limit; i.e., the limits of the form $NR^{}_{}$. Similarly, the faces of $P_{\rm III}$ are those which have $SB_{0}^{0}$ as a limit, and any function not of the form $NR_{0}^{0}$ or $SB$ corresponds to a simplicial face of $P_{\rm II}$. ### Acknowledgements The second author was supported in part by NSF grant DMS-0833464. ## References * [1] * [2] van de Bult F.J., Rains E.M., Stokman J.V., Properties of generalized univariate hypergeometric functions, Comm. Math. Phys. 275 (2007), 37–95, math.CA/0607250. * [3] Chen W.Y.C., Fu A.M., Semi-finite forms of bilateral basic hypergeometric series, Proc. Amer. Math. Soc. 134 (2006), 1719–1725, math.CA/0501242. * [4] Conway J.H., Sloane N.J.A., The cell structures of certain lattices, in Miscellanea Mathematica, Springer, Berlin, 1991, 71–107. * [5] van Diejen J.F., Spiridonov V.P., An elliptic Macdonald–Morris conjecture and multiple modular hypergeometric sums, Math. Res. Lett. 7 (2000), 729–746. * [6] van Diejen J.F., Spiridonov V.P., Elliptic Selberg integrals, Internat. Math. Res. Notices 2001 (2001), no. 20, 1083–1110. * [7] Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and Its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004. * [8] Gupta D.P., Masson D.R., Contiguous relations, continued fractions and orthogonality, Trans. Amer. Math. Soc. 350 (1998), 769–808, math.CA/9511218. * [9] Lievens S., Van der Jeugd J., Invariance groups of three term transformations for basic hypergeometric series, J. Comput. Appl. Math. 197 (2006), 1–14. * [10] Lievens S., Van der Jeugd J., Symmetry groups of Bailey’s transformations for ${}_{10}\phi_{9}$-series, J. Comput. Appl. Math. 206 (2007), 498–519. * [11] Rains E.M., Transformations of elliptic hypergeometric integrals, Ann. Math., to appear, math.QA/0309252. * [12] Rains E.M., Limits of elliptic hypergeometric integrals, Ramanujan J., to appear, math.CA/0607093. * [13] Rains E.M., Elliptic Littlewood identities, arXiv:0806.0871. * [14] Ruijsenaars S.N.M., First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), 1069–1146. * [15] Spiridonov V.P., On the elliptic beta function, Uspekhi Mat. Nauk 56 (2001), no. 1 (337), 181–182 (English transl.: Russian Math. Surveys 56 (2001), no. 1, 185–186). * [16] Spiridonov V.P., Theta hypergeometric integrals, Algebra i Analiz 15 (2003), 161–215 (English transl.: St. Petersburg Math. J. 15 (2004), 929–967), math.CA/0303205. * [17] Spiridonov V.P., Classical elliptic hypergeometric functions and their applications, Rokko Lect. in Math., Vol. 18, Kobe University, 2005, 253–287, math.CA/0511579. * [18] Spiridonov V.P., Short proofs of the elliptic beta integrals, Ramanujan J. 13 (2007), 265–283, math.CA/0408369. * [19] Spiridonov V.P., Essays on the theory of elliptic hypergeometric functions, Uspekhi Mat. Nauk 63 (2008), no. 3, 3–72 (English transl.: Russian Math. Surveys 63 (2008), no. 3, 405–472), arXiv:0805.3135.	arxiv-papers	2009-02-03T21:29:06	2024-09-04T02:49:00.428540	{ "license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/", "authors": "Fokko van de Bult and Eric Rains", "submitter": "Eric M. Rains", "url": "https://arxiv.org/abs/0902.0621" }
0902.0686	# Universal detector efficiency of a mesoscopic capacitor Simon E. Nigg simon.nigg@unige.ch Markus Büttiker Département de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland ###### Abstract We investigate theoretically a novel type of high frequency quantum detector based on the mesoscopic capacitor recently realized by Gabelli et al., [Science 313, 499 (2006)], which consists of a quantum dot connected via a single channel quantum point contact to a single lead. We show that the state of a double quantum dot charge qubit capacitively coupled to this detector can be read out in the GHz frequency regime with near quantum limited efficiency. To leading order, the quantum efficiency is found to be universal owing to the universality of the charge relaxation resistance of the mesoscopic capacitor. ††preprint: The measurement problem is probably one of the oldest topics in quantum physics, which is still of prime interest to researchers nowadays. With the advent of mesoscopic physics, fundamental issues related to Von Neumann’s notion of the instantaneous wave function collapse von Neumann (1932) can now be addressed experimentally. Indeed it has recently become possible to engineer systems in which parts of the measurement device are themselves unambiguously quantum. In the weak coupling regime the dynamics of the wave function collapse itself can be probed and sometimes even reversed Katz et al. (2008); Korotkov and Jordan (2006). Questions such as “how long does it take to acquire the desired information ?” and “how fast does the measurement decoher the state of the measured system ?” become of relevance. This is in particular true in the emergent field of quantum information processing, where one wishes to both manipulate and read-out quantum bits (qubits) with the highest possible efficiencies. An important figure of merit of any quantum detector is its Heisenberg efficiency. Loosely speaking it is the ratio of how fast to how invasive a given detector is. By “fast” we mean how quickly two different states of the measured system can be distinguished from one another and by “invasive” we mean how strong is the back-action of the detector onto the state of the measured system. The Heisenberg uncertainty relation implies that one cannot acquire information about the system faster than one dephases it during the measurement process Korotkov and Averin (2001); Makhlin et al. (2001); Clerk et al. (2003); Pilgram and Büttiker (2002). Hence the Heisenberg efficiency is bounded from above. An important task is thus to find and characterize detectors which reach the maximum allowed Heisenberg efficiency. Several such systems have been described in the literature. In the DC regime Refs. Gurvitz (1997); Aleiner et al. (1997); Clerk and Stone (2004); Averin and Sukhorukov (2005) investigate the quantum point contact (QPC) detector. Refs. Pilgram and Büttiker (2002); Clerk et al. (2003) discuss two terminal scattering detectors capacitively coupled to a double dot charge qubit. In both cases, the average current through the detector functions as a meter, since the electron transmission probability is sensitive to the position of the charge in the qubit. Due to $1/f$ noise DC detectors are generically plagued by a large dephasing rate. To circumvent this, Schoelkopf et al. Schoelkopf et al. (1998) introduced the radio-frequency single-electron transistor (rf-SET). The idea there, is to measure the damping of an oscillator circuit in which the SET is embedded. In this letter we present a novel quantum detector based on the mesoscopic capacitor Büttiker et al. (1993), which consists of a quantum dot connected via a single channel QPC to a single lead. At temperatures low compared with the charging energy, such a system exhibits a universal Büttiker et al. (1993); Gabelli et al. (2006); Nigg et al. (2006) charge relaxation resistance $R_{q}=h/(2e^{2})$. We show that this system embedded in an LC tank circuit with impedance $L$ and capacitance $C$, can be operated as a high frequency detector near the quantum limit despite the presence of intrinsic dissipation. At the resonance frequency $\omega_{0}=1/\sqrt{LC}$ we find to leading order, a universal Heisenberg efficiency $\eta=\frac{L/C}{L/C+R_{q}Z_{0}}\,,$ (1) where $Z_{0}$ is the characteristic impedance of the transmission line connected to the tank circuit. Figure 1: (Color online) Detector and qubit system (a). Equivalent circuit in the adiabatic approximation (b). Incoming photons are reflected ($\mathcal{R}$) and detected or dissipated ($\mathcal{T}$). The system we consider is depicted in Fig. 1 (a). Let us first consider the system without the LC resonator and transmission line. The part being measured; the double dot charge qubit, plus the capacitive coupling term are described by the Hamiltonian $H_{qb}=\frac{1}{2}\left(\epsilon\sigma_{z}+\Delta\sigma_{x}+\kappa\sigma_{z}\hat{N}\right)\quad\text{with}\quad\kappa=\frac{e^{2}}{C_{i}}\,.$ (2) Here $\sigma_{z}=\mathinner{\|{\uparrow}\rangle}\mathinner{\langle{\uparrow}\|}-\mathinner{\|{\downarrow}\rangle}\mathinner{\langle{\downarrow}\|}$ and $\sigma_{x}=\mathinner{\|{\uparrow}\rangle}\mathinner{\langle{\downarrow}\|}+\mathinner{\|{\downarrow}\rangle}\mathinner{\langle{\uparrow}\|}$. In the state $\mathinner{\|{\uparrow}\rangle}$ ($\mathinner{\|{\downarrow}\rangle}$) the excess charge is located on the upper (lower) dot. The energy difference $\epsilon$ and the coupling $\Delta$ between these two states can be tuned by the gate voltages $V_{g1}$ and $V_{g2}$ (see Fig. 1 (a)). $\hat{Q}=e\hat{N}=e\sum_{i}d_{i}^{\dagger}d_{i}$ is the excess charge on the quantum dot (QD) of the mesoscopic capacitor. The latter is described by the Hamiltonian $H_{D}=\sum_{i}\varepsilon_{i}d_{i}^{\dagger}d_{i}+\frac{\hat{Q}^{2}}{2C_{\Sigma}}\,.$ (3) Here the first term describes the unperturbed level spectrum while the second term gives the Coulomb interaction. $C_{\Sigma}=(1/C_{i}+1/C_{2}+1/C_{g})^{-1}$ is the total series capacitance. Finally the QD of the capacitor is coupled to the lead via the tunneling Hamiltonian $H_{T}=\sum_{ik}t_{ik}c_{k}^{\dagger}d_{i}+h.c.$, where $t_{ik}$ is the tunneling matrix element between state $i$ of the dot and state $k$ of the lead and can be tuned with the gate voltage $V_{qpc}$ (see Fig. 1 (a)). The lead, where we neglect the electron electron interaction, is described by $H_{L}=\sum_{k}E_{k}c_{k}^{\dagger}c_{k}$. The entire system is described by the Hamiltonian $H=H_{qb}+H_{D}+H_{L}+H_{T}\,.$ (4) If not for the tunneling term $H_{T}$, which changes the charge on the dot of the capacitor, a qubit prepared in one of the eigenstates of $H_{qb}$ for a given charge $Q$ would remain in this state under the time evolution. Because of $H_{T}$ however, the charge on the dot fluctuates leading to a modulation in time of the level splitting of the qubit. If this modulation is slow enough though, the qubit will remain in an instantaneous eigenstate of $H_{qb}$ at all times. To derive the necessary conditions for this to be true, we follow Johansson et al. (2006), and apply a unitary transformation onto $H$, which diagonalizes $H_{qb}$ in each subspace of fixed $N$. $H^{\prime}={U(\hat{N})}^{\dagger}HU(\hat{N})\,.$ (5) With $\eta_{0}={\rm arccot}\left[\frac{\epsilon}{\Delta}\right]$, the unitary operator to second order in the coupling $\kappa$ is explicitly given by $U(\hat{N})=\hat{a}_{0}U_{0}+\hat{a}_{1}U_{1}$, with $\hat{a}_{0}=1-\frac{\kappa^{2}\Delta^{2}}{{8\Omega_{0}}^{4}}\hat{N}^{2}\quad\text{and}\quad\hat{a}_{1}=\frac{\kappa\Delta}{2\Omega_{0}^{2}}\hat{N}\left(1-\frac{\kappa\epsilon}{\Omega_{0}^{2}}\right)\,,$ (6) where $\Omega_{0}=\sqrt{\epsilon^{2}+\Delta^{2}}$ is the bare Rabi frequency and $U_{0}=\begin{pmatrix}\cos\frac{\eta_{0}}{2}&-\sin\frac{\eta_{0}}{2}\\\ \sin\frac{\eta_{0}}{2}&\cos\frac{\eta_{0}}{2}\end{pmatrix},U_{1}=\begin{pmatrix}\sin\frac{\eta_{0}}{2}&\cos\frac{\eta_{0}}{2}\\\ -\cos\frac{\eta_{0}}{2}&\sin\frac{\eta_{0}}{2}\end{pmatrix}$ (7) Note that ${U_{0}}^{\dagger}U_{0}={U_{1}}^{\dagger}U_{1}=\openone$ while ${U_{0}}^{\dagger}U_{1}=-{U_{1}}^{\dagger}U_{0}=i\sigma_{y}$. Using the fact that $[[H_{T},\hat{N}],\hat{N}]=H_{T}$, we finally obtain $U^{\dagger}H_{T}U={H_{T}}+i\sigma_{y}\frac{\kappa\Delta}{2\Omega_{0}^{2}}[H_{T},\hat{N}]+O(\kappa^{3})\,.$ (8) where we have neglected a small $O(\kappa^{2})$ renormalization of the tunneling amplitudes $t_{ik}$, which is insensitive to the state of the qubit. In the linear response regime, the time scale on which $\mathinner{\langle{\hat{N}(t)}\rangle}$ fluctuates is set by the inverse of the drive frequency $\omega$. Therefore the energy available for making a real transition between the qubit eigenstates, which is given by the second term on the right-hand side of Eq. (8), is proportional to $\hbar\omega\kappa\Delta/(2\Omega_{0}^{2})$. Demanding that this energy be small compared to the level splitting $\Omega_{0}$ of the qubit leads us to the following adiabatic condition on the drive frequency $\hbar\omega\ll\frac{2\Omega_{0}^{3}}{\kappa\Delta}\,.$ (9) Let us briefly discuss this condition. We see that for $\Delta=0$, we can drive the system as fast as we wish provided $\epsilon\not=0$. This simply reflects the fact that for $(\Delta=0)\ll\epsilon$ the two eigenstates of the qubit, which in fact are the charge states in this limit, are decoupled from one another. We also see that the weaker the coupling, the faster we may drive the system without inducing transitions, which is intuitively reasonable. For realistic values of the parameters; $\Delta=\Omega_{0}=5\,\mu{\rm eV}$, $\epsilon=0$ and $\kappa=50\,\eta{\rm eV}$, we find $2\Omega_{0}^{3}/(\kappa\Delta)\gtrsim 1.5\cdot 10^{12}\,{\rm Hz}$, so that even for drive frequencies in the GHz regime we are still safely in the adiabatic regime. In the adiabatic approximation and for weak coupling, i.e. $\kappa\ll\Omega_{0}$, the dynamics of the system is thus appropriately described to second order in $\kappa$ by the purely longitudinal effective Hamiltonian $H_{{\rm eff}}=H_{+}\mathinner{\|{+}\rangle}\mathinner{\langle{+}\|}+H_{-}\mathinner{\|{-}\rangle}\mathinner{\langle{-}\|}$, where $H_{\pm}=\pm\frac{\Omega_{0}}{2}+\sum_{i}\varepsilon_{i}^{\pm}d_{i}^{\dagger}d_{i}+\frac{e^{2}}{2C_{{\rm eff}}^{\pm}}\hat{N}^{2}+H_{L}+{H_{T}}\,.$ (10) Here $\mathinner{\|{\pm}\rangle}$ are the adiabatic eigenstates of $H_{qb}$. The presence of the qubit appears thus as a renormalization of the spectrum of the QD of the detector: $\varepsilon_{i}^{\pm}=\varepsilon_{i}\pm\kappa\epsilon/(2\Omega_{0})$, and a renormalization of the geometric capacitance Duty et al. (2005); Sillanpää et al. (2005) of the dot vis-à-vis the gate $V_{g1}$: $1/C_{{\rm eff}}^{\pm}=1/C_{\Sigma}\pm\kappa^{2}\Delta^{2}/(2\Omega_{0}^{3})$. Formally, the effective Hamiltonian we have just derived is exactly the same as the one of a mesoscopic capacitor with a single level spectrum $\varepsilon_{i}^{\pm}$ and a geometric capacitance $C_{{\rm eff}}^{\pm}$. Within the self-consistent Hartree approximation Büttiker et al. (1993); Brouwer et al. (2005), the linear response of a mesoscopic capacitor to an applied AC voltage is known Büttiker et al. (1993); Brouwer et al. (2005). For short RC times $\tau_{RC}^{\pm}\equiv R_{q}C_{\mu}^{\pm}$ such that $\omega\tau_{RC}^{\pm}\ll 1$, the mesoscopic capacitor is equivalent to an RC circuit with the impedance $Z_{0}^{\pm}(\omega)=R_{q}+i/(\omega C_{\mu}^{\pm})$. Here $R_{q}$ is the charge relaxation resistance, which at zero temperature and for a single channel capacitor is universal and given by half a resistance quantum, i.e. $R_{q}=h/(2e^{2})$. The electrochemical capacitance $C_{\mu}^{\pm}$ however depends on $C_{{\rm eff}}^{\pm}$ and on the density of states (DOS) of the capacitor and is thus sensitive to the state of the qubit. Explicitly one finds Büttiker et al. (1993) $\frac{1}{C_{\mu}^{\pm}}=\frac{1}{C_{\rm eff}^{\pm}}+\frac{1}{e^{2}\nu_{\pm}(E_{F})}\,.$ (11) Here $\nu_{\pm}(E_{F})$ is the DOS at the Fermi-energy of the QD with the shifted spectrum $\\{\varepsilon_{i}^{\pm}\\}$. The electrochemical capacitance thus acts like the pointer of a measurement device. At the degeneracy point $\epsilon=0$, the shift of the levels vanishes, while the correction to the capacitance is maximal. If to the contrary $\epsilon\gg\Delta$ then the correction to the capacitance vanishes while the dot spectrum is maximally shifted by the amount $\pm\kappa/2$. Let us now discuss a way of probing the electrochemical capacitance in the high frequency regime. Using a dispersive read-out scheme similar to Schoelkopf et al. (1998); Johansson et al. (2006), we embed the effective capacitor into an LC tank-circuit and via a standard homodyne detection scheme Gardiner and Zoller (2000), probe the phase shift of waves reflected from the tank-circuit (see Fig. 1 (b)). It is important to note that in contrast to Schoelkopf et al. (1998), we here do not want to measure the resistance of our effective capacitor. Indeed, owing to the universality of the charge relaxation resistance in the single channel limit, this quantity is actually insensitive to the state of the qubit. Instead, we propose to detect the phase shift of a reflected signal, which is determined by the non-dissipative part of the response of the mesoscopic capacitor. To second order in $C_{\mu}^{\pm}/C$, the shifted resonance frequency of the tank-circuit is given by $\omega_{osc}^{\pm}\approx\omega_{0}\left(1-\frac{1}{2}\frac{C_{\mu}^{\pm}}{C}\right)-i\omega_{0}^{2}\frac{C_{\mu}^{\pm}}{2C}\tau_{RC}^{\pm}\,,$ (12) where $\omega_{0}=1/\sqrt{LC}$ is the bare oscillator resonance frequency. Notice that because of the finite resistance $R_{q}$, the oscillation of the LC circuit is damped. This is reflected in the non-vanishing imaginary part of $\omega_{osc}^{\pm}$ in Eq. (12). In oder words, photons coming down the transmission line toward the LC-tank circuit, will be dissipated with some finite probability. The reflected photons however will experience a phase shift, which depends on the state of the qubit. It is this phase shift which we propose to measure. The impedance of the tank-circuit which terminates the transmission line is $Z_{\pm}(\omega)=iL({\omega_{osc}^{\pm}}^{2}-\omega^{2})/\omega$. From this, we can calculate the complex reflection coefficient $\mathcal{R}^{\pm}$ of the transmission line with characteristic impedance $Z_{0}$, relating incoming and outgoing modes via $a_{out}^{\pm}(\omega)=\mathcal{R}^{\pm}(\omega)a_{in}(\omega)$. We find $\mathcal{R}^{\pm}(\omega)=\frac{Z_{0}-Z_{\pm}(\omega)}{Z_{0}+Z_{\pm}(\omega)}=\frac{\omega^{2}-(\omega_{osc}^{\pm})^{2}-i\eta_{0}\omega}{\omega^{2}-(\omega_{osc}^{\pm})^{2}+i\eta_{0}\omega}\,,$ (13) with $\eta_{0}=Z_{0}/L$. Because $(\omega_{osc}^{\pm})^{2}$ has a non- vanishing imaginary part, $\mathcal{R}^{\pm}$ is not unitary. At the bare resonance frequency, we obtain, $\mathcal{R}_{\pm}(\omega_{0})=\gamma_{\pm}e^{i\phi_{\pm}}$, with $\gamma_{\pm}=1-2\frac{R_{q}}{Z_{0}}\left(\frac{C_{\mu}^{\pm}}{C}\right)^{2}+O\left(\left(C_{\mu}^{\pm}/C\right)^{3}\right)\,,$ (14) and $\phi_{\pm}=Q_{0}\frac{C_{\mu}^{\pm}}{C}\left(2-\frac{1}{2}\left(\frac{C_{\mu}^{\pm}}{C}\right)\right)+O\left(\left(C_{\mu}^{\pm}/C\right)^{3}\right)\,.$ (15) Here we have introduced the quality factor $Q_{0}=\sqrt{L/C}/Z_{0}$ of the resonator plus transmission line circuit. To leading order, the probability of a photon to be dissipated is thus given by $1-\gamma_{\pm}^{2}=4(R_{q}/Z_{0})(C_{\mu}^{\pm}/C)^{2}$. Also, we remark that the leading order correction to the reflection phase due to a finite $R_{q}$ is of order $(C_{\mu}^{\pm}/C)^{3}$. Finally note that the leading order correction to $\gamma_{\pm}$ is independent of $L$. This is ultimately the reason why we can achieve a large Heisenberg efficiency; increasing $L$ increases the signal without increasing the dissipation. We next derive expressions for the measurement and dephasing rates Gardiner and Zoller (2000); Johansson et al. (2006); Clerk et al. (2009). The measured quantity is the number of photons reflected from the load in time $T$. By mixing this signal with a strong signal from a local oscillator driven at the same frequency $\omega_{0}$ as the drive and afterwards taking the average, the measured number of photons $n_{\pm}(T)$ becomes sensitive to the reflection phase shift, which in turn depends on the state of the qubit. The two eigenstates are said to be resolved, when the difference $\Delta n(T)=n_{+}(T)-n_{-}(T)$ becomes larger than the noise. The time when this happens defines the measurement time $T_{m}$. Let us consider a monochromatic coherent state input with amplitude $\beta_{0}$. $\mathinner{\|{\psi}\rangle}_{in}=\exp\left[T(\beta_{0}a^{\dagger}_{L}(\omega_{0})-\beta_{0}^{}a_{L}(\omega_{0}))\right]\mathinner{\|{0}\rangle}\,,$ (16) where $a_{L}^{\dagger}(\omega)$ creates an incoming photon at frequency $\omega$. Using the same definition for the signal to noise ratio as in Clerk et al. (2009), we find a measurement rate given by $\Gamma_{m}\equiv{T_{m}}^{-1}=\|\beta_{0}\|^{2}\frac{(\gamma_{+}+\gamma_{-})^{2}}{\gamma_{+}^{2}+\gamma_{-}^{2}}\sin^{2}(\Delta\phi/2)\,,$ (17) where $\Delta\phi=\phi_{+}-\phi_{-}$. We note that $\Gamma_{m}$ is bounded from above by $2\|\beta_{0}\|^{2}$, or twice the photon injection rate. Incidentally, this is the maximally achievable measurement rate in the absence of dissipation, where $\gamma_{\pm}=1$. To derive the dephasing rate, we essentially follow the quantum information theoretic argument of Clerk et al. (2009) and adapt it to a dissipative system. The resistor is replaced by a semi-infinite transmission line with characteristic impedance $R_{q}$ (see Fig. 1 (b)), which is then quantized Yurke and Denker (1984). Hence we determine the transmission coefficient $\mathcal{T}^{\pm}$ for photons to be dissipated. The measurement can be represented as the entangling process $(\alpha\mathinner{\|{+}\rangle}+\beta\mathinner{\|{-}\rangle})\mathinner{\|{\beta_{0}}\rangle}\rightarrow\alpha^{\prime}\mathinner{\|{+}\rangle}\mathinner{\|{\beta_{+}}\rangle}+\beta^{\prime}\mathinner{\|{-}\rangle}\mathinner{\|{\beta_{-}}\rangle}\,,$ (18) where detector states after the scattering are given by a product of phase shifted and damped coherent states as $\mathinner{\|{\beta_{\pm}}\rangle}=\mathinner{\|{\beta_{0}\gamma_{\pm}e^{i\phi_{\pm}}}\rangle}\otimes\mathinner{\|{\beta_{0}\sqrt{1-\gamma_{\pm}^{2}}e^{i\theta_{\pm}}}\rangle}\,.$ (19) Here $\theta_{\pm}=\arg(\mathcal{T}^{\pm})$ is the phase shift of the dissipated photons. The off-diagonal elements of the reduced density matrix of the qubit are proportional to the overlap of the detector states, i.e. $\|\rho_{12}\|\sim\|\mathinner{\langle{\beta_{+}\|\beta_{-}}\rangle}\|$. For long times, these elements decay exponentially defining the dephasing rate by $\|\rho_{12}\|\sim\exp[-\Gamma_{\phi}T]$. We find explicitly $\Gamma_{\phi}=\|\beta_{0}\|^{2}\left[1-D_{1}\cos(\Delta\phi)-D_{2}\cos(\Delta\theta)\right]\,,$ (20) with $D_{1}=\gamma_{+}\gamma_{-}$ and $D_{2}=\sqrt{(1-\gamma_{+}^{2})(1-\gamma_{-}^{2})}$. From Eqs. (17) and (20) we finally obtain the Heisenberg efficiency of our detector $\eta\equiv\frac{\Gamma_{m}}{\Gamma_{\phi}}=\frac{L/C}{L/C+R_{q}Z_{0}}+O\left((C_{\mu}^{\pm}/C)^{2}\right)\,.$ (21) Figure 2: (Color online) Efficiency $\eta$ as a function of inductance $L$. The inset shows $\eta(\omega)$ for $L=10\,{\mu\rm H}$ and $C=10^{-13}\,{\rm}F$. Thus we find that to leading order the Heisenberg efficiency does not depend on $C_{\mu}^{\pm}$. This result holds as long as $\omega_{0}\tau_{RC}^{\pm}\ll 1\ll C/C_{\mu}^{\pm}$. To reach acceptable efficiency, we need to have $L/C\gg R_{q}Z_{0}$. Fig. 2 shows the Heisenberg efficiency as a function of $L$ for realistic parameters. Decreasing $C$, which increases $\Delta\phi$, increases the efficiency and at the same time increases the measurement frequency $\omega_{0}=1/\sqrt{LC}$. For example for $L=10\,\mu\rm H$, and $C=100\,{\rm fF}$, we have $\omega_{0}=1\,{\rm GHz}$ and $\eta=99.4\%$ (see full thick (red) curves on Fig. 2). In conclusion, we have shown that the mesoscopic capacitor can in principle be operated as an efficient detector in the GHz regime. We find that to leading order its efficiency is universal, i.e. independent of the microscopic details of the detector and qubit. This universality can be directly traced back to the experimentally demonstrated Gabelli et al. (2006) universality Büttiker et al. (1993) of the charge relaxation resistance of a mesoscopic capacitor. This work is supported by the Swiss NSF, MaNEP and the STREP project SUBTLE. ## References von Neumann (1932) J. von Neumann, _Mathematische Grundlagen der Quantenmechanik._ (Springer, Berlin, 1932). * Katz et al. (2008) N. Katz et al., Phys. Rev. Lett. 101, 200401 (2008). * Korotkov and Jordan (2006) A. N. Korotkov and A. N. Jordan, Phys. Rev. Lett. 97, 166805 (2006). * Pilgram and Büttiker (2002) S. Pilgram and M. Büttiker, Phys. Rev. Lett. 89, 200401 (2002). * Clerk et al. (2003) A. A. Clerk, S. M. Girvin, and A. D. Stone, Phys. Rev. B. 67, 165324 (2003). * Korotkov and Averin (2001) A. N. Korotkov and D. V. Averin, Phys. Rev. B 64, 165310 (2001). * Makhlin et al. (2001) Y. Makhlin, G. Schön, and A. Shnirman, RMP 73, 357 (2001). * Gurvitz (1997) S. A. Gurvitz, Phys. Rev. B. 56, 15215 (1997). * Aleiner et al. (1997) I. L. Aleiner, N. S. Wingreen, and Y. Meir, Phys. Rev. Lett. 79, 3740 (1997). * Clerk and Stone (2004) A. A. Clerk and A. D. Stone, Phys. Rev. B. 69, 245303 (2004). * Averin and Sukhorukov (2005) D. V. Averin and E. V. Sukhorukov, Phys. Rev. Lett. 95, 126803 (2005). * Schoelkopf et al. (1998) R. J. Schoelkopf, P. Wahlgren, A. A. Kozhevnikov, P. Delsing, and D. E. Prober, Science 280, 1238 (1998). * Büttiker et al. (1993) M. Büttiker, H. Thomas, and A. Prêtre, Phys. Lett. A 180, 364 (1993). * Gabelli et al. (2006) J. Gabelli, J. M. Berroir, G. Fève, B. Plaçais, Y. Jin, B. Etienne, and D. C. Glattli, Science 313, 499 (2006). * Nigg et al. (2006) S. E. Nigg, R. López, and M. Büttiker, Phys. Rev. Lett. 97, 206804 (2006). * Johansson et al. (2006) G. Johansson, L. Tornberg, and C. M. Wilson, Phys. Rev. B. 74, 100504(R) (2006). * Duty et al. (2005) T. Duty, G. Johansson, K. Bladh, D. Gunnarsson, C. Wilson, and P. Delsing, Phys. Rev. Lett. 95, 206807 (2005). * Sillanpää et al. (2005) M. A. Sillanpää, T. Lehtinen, A. Paila, Y. Makhlin, L. Roschier, and P. J. Hakonen, Phys. Rev. Lett. 95, 206806 (2005). * Brouwer et al. (2005) P. W. Brouwer, A. Lamacraft, and K. Flensberg, Phys. Rev. B. 72, 075316 (2005). * Gardiner and Zoller (2000) C. W. Gardiner and P. Zoller, _Quantum Noise, 2nd ed._ (Springer-Verlag, Berlin Heidelberg New York, 2000). * Clerk et al. (2009) A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf (2009), preprint:arXiv:0810.4729v1 [cond-mat.mes-hall]. * Yurke and Denker (1984) B. Yurke and J. S. Denker, Phys. Rev. A. 29, 1419 (1984).	arxiv-papers	2009-02-04T19:52:48	2024-09-04T02:49:00.444838	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Simon E. Nigg and Markus Buttiker", "submitter": "Simon Nigg", "url": "https://arxiv.org/abs/0902.0686" }
0902.1091	# Controlling the gap of fullerene microcrystals by applying pressure: the role of many-body effects Murilo L. Tiago Oak Ridge National Laboratory, Oak Ridge, TN, 37831 Fernando A. Reboredo Oak Ridge National Laboratory, Oak Ridge, TN, 37831 ###### Abstract We studied theoretically the optical properties of C60 fullerene microcrystals as a function of hydrostatic pressure with first-principles many-body theories. Calculations of the electronic properties were done in the GW approximation. We computed electronic excited states in the crystal by diagonalizing the Bethe-Salpeter equation (BSE). Our results confirmed the existence of bound excitons in the crystal. Both the electronic gap and optical gap decrease continuously and non-linearly as pressure of up to 6 GPa is applied. As a result, the absorption spectrum shows strong redshift. We also obtained that “negative” pressure shows the opposite behavior: the gaps increase and the optical spectrum shifts toward the blue end of the spectrum. Negative pressure can be realized by adding cubane (C8H8) or other molecules with similar size to the interstitials of the microcrystal. For the moderate lattice distortions studied here, we found that the optical properties of fullerene microcrystals with intercalated cubane are similar to the ones of an expanded undoped microcrystal. Based on these findings, we propose doped C60 as active element in piezo-optical devices. ###### pacs: 1.48.-c,2.50.-p,71.35.Cc ## I Introduction Since its discovery, C60 KrotoHOCS85 has been characterized as the most stable member in the series of fullerenes, which are pure-carbon molecules with the shape of spherical shells DresselhausDE96 . C60 can be produced economically and in abundance KratschmerLFH90 . Its chemical bonds have strong $sp^{2}$ character, making the shell very stiff and at the same time free of dangling bonds. Together with clathrates and other carbon-based materials, fullerenes are being actively investigated as building blocks for novel materials with unusual mechanical properties San-Miguel06 ; BlaseGSM04 . Fullerenes has remarkable properties, for example: C60 doped with alkali and alkali-earth atoms is superconductor KortanKGGRFTH92 ; HebardRHMGPRK91 ; it has been claimed that C60 can be heavily hydrogenated with up to 36 hydrogen atoms per molecule, making it a promising material for hydrogen storage MeletovK05 . In pure form, C60 crystallizes in a molecular solid (fullerite), bound by weak forces between molecules. Its phase diagram is very rich, with a low- temperature face-centered cubic (FCC) phase, amorphous phases at intermediate temperature, and a diamond-like phase at high temperature. The FCC phase has rotational disorder and it is stable under pressure in excess of 15 GPa at room temperature Sundqvist99 . Its energy gap is in the visible range, around 2 eV DresselhausDE96 ; MeletovD98 . The softness and stability of fullerite could make it a good candidate for piezo-optical devices, where external pressure is applied reversibly and it modifies the optical response of the material. In addition, when fullerite is heavily doped with molecules of appropriate size, it behaves as if it is under “negative pressure” PekkerKOBKBJJBKBKTF05 . To that end, it is important to characterize the pressure dependence of the optical properties of this material. Extensive experimental work has been done on this direction PoloniFFPTMCPS08 ; Duclos91 ; NunezMHBP95 ; SnokeSM93 ; MeletovD98 ; MosharyCSBDVB92 ; KozlovY95 . A review of the literature can be found in Reference Sundqvist99, . Theoretical analyses are not so extensive, mostly concentrated on characterization at zero pressure ShirleyL93 ; ShirleyBL96 ; HartmannZMSBLFZ95 . In order to fill this vacuum, we present a systematic analysis of the optical properties of fullerite at pressures ranging from zero to 6 GPa. We also investigate the optical response of the crystal at “negative” pressure, which can be realized in laboratory, for instance, by doping fullerite with weakly interacting molecules such as cubane (C8H8). Numerical accuracy is essential, which is why we use state-of-the-art theories, namely many-body Green’s function theories based on the GW-BSE approach. This article is organized as follows: we outline the theoretical framework in section II. That section is followed by a discussion of results at zero pressure in section III, results at finite hydrostatic pressure in section IV and finally a description of the cubane-fullerene compound, which resembles fullerite with “negative pressure”, in section V. Finally, we conclude with some perspectives of future applications and a summary. ## II Theory The underlying electronic structure of fullerene is determined using density- functional theory (DFT) Martin . We use a plane-wave basis to solve the Kohn- Sham equations with cut-off in the kinetic energy of 50 Ry. Interactions involving valence electrons and core electrons are taken into account using norm-conserving pseudopotentials of the Troullier-Martins type Martin . We use the Perdew-Burke-Ernzerhof functional PerdewBE96 (PBE) for exchange and correlation, based on the generalized-gradient approximation (GGA). It is well-known that DFT in the local-density approximation (LDA) or the PBE severely underestimates electronic gaps in general, making it unsuitable for detailed studies of optical properties of electronic systems. Accurate bandwidths and electronic energy gaps are calculated in a many-body framework within the GW approximation HybertsenL86 . In that approximation, the electron self-energy is computed by summing up Feynman diagrams to lowest order in the screened Coulomb interaction. At lowest order, the self-energy becomes a product between the one-electron Green’s function $G$ and the screened Coulomb interaction $W$, hence the name. We ignore vertex diagrams and we assume that Kohn-Sham eigenvalues and eigenvectors give a good approximation to the Green’s function. Formally, the self-energy in space-energy representation is written as $\Sigma({\bf r},{\bf r}^{\prime};E)={i\over 2\pi}\int{\rm d}E^{\prime}e^{-i0^{+}E^{\prime}}G_{0}({\bf r},{\bf r}^{\prime};E-E^{\prime})W_{0}({\bf r},{\bf r}^{\prime};E^{\prime})\;\;,$ (1) where $G_{0}$ denotes the DFT-PBE Green’s function and $W_{0}$ is the screened Coulomb interaction, related to the random phase approximation (RPA) dielectric function by: $W_{0}({\bf r},{\bf r}^{\prime};E)=\int{\rm d}{\bf r}^{\prime\prime}\epsilon^{-1}({\bf r},{\bf r}^{\prime\prime};E){q_{e}^{2}\over\|{\bf r}^{\prime\prime}-{\bf r}^{\prime}\|}\;\;.$ (2) In the present formulation, the dielectric function is expanded in a basis of plane waves with cut-off 9.5 Ry. Its energy dependence is described by a generalized plasmon pole model HybertsenL86 . After the self-energy is computed, we diagonalize the quasi-particle Hamiltonian, $H=H_{PBE}+\Sigma- V_{xc}$ HybertsenL86 ; AulburJW00 . Eigenvalues of that Hamiltonian provide the electronic band structure of the real material. This formulation is one of the simplest ab initio formulations of the GW approximation. Extensive applications of this formulation to a wide class of carbon-based materials have shown it to predict electronic band gaps with an accuracy of 0.1 to 0.2 eV AulburJW00 ; OnidaRR02 . In the specific case of fullerite, the first calculation of electronic gap within the GW approximation was consistent with direct/inverse photoemission spectra ShirleyL93 . Owing to the fact that hydrostatic pressure on fullerite microcrystals does not affect their electronic properties besides an increase in intermolecular interactions, we expect our theoretical methodology to be equally reliable in describing the pressure dependence of the electronic gap. Technical details about the theory can be found in review articles AulburJW00 ; OnidaRR02 . The electronic band structure often does not give access to optical spectra because, after electron-hole pairs are excited, they interact and produce bound excitons, with energy lower than the electronic gap OnidaRR02 ; RohlfingL00 . We describe the dynamics of excitons by diagonalizing the Bethe- Salpeter equation (BSE) for electrons and holes. The BSE is an equation for the two-particle Green’s function. Written as an eigenvalue equation, its solution gives the energy of optical excitations in the material: $(E_{c}-E_{v})A^{s}_{vc}+\sum_{v^{\prime}c^{\prime}}K^{vc}_{v^{\prime}c^{\prime}}A^{s}_{v^{\prime}c^{\prime}}=\Omega^{s}A^{s}_{vc}\;\;.$ (3) where $\Omega^{s}$ is the excitation energy of optical modes, indexed by $s$, and $A^{s}_{vc}$ are the corresponding eigenvectors. $K^{vc}_{v^{\prime}c^{\prime}}$ is the electron-hole interaction kernel, written in the basis of pair transitions. In the absence of electron-hole interactions ($K=0$), each excitation energy is simply the difference between quasi-particle energies of electrons ($E_{c}$) and holes ($E_{v}$). The kernel $K$ adds two types of interactions: an electrostatic interaction mediated again by $W_{0}$; and a repulsive exchange interaction between electron and hole, which is related to the fact that they can annihilate each otherFetterW . We follow the standard procedure to build and solve the BSE. We ignore the energy dependence of the interaction kernel and we assume the Tamm-Dancoff approximation FetterW when computing $K$. Both approximations have been used extensively and they were shown to simplify considerably the numerical complexity, with little impact on numerical accuracy. This methodology has been presented in great detail elsewhere OnidaRR02 ; RohlfingL00 ; TiagoC06 . Although numerically expensive, the GW-BSE theory has been remarkably successful in predicting electronic and optical properties of real materials without the need for phenomenological parametersAulburJW00 ; OnidaRR02 . All the approximations involved, such as the plasmon pole model and the non-self- consistent assumption, are unambiguously defined. In addition, sp-bonded systems such as carbon-based nanostructures seem to be the ideal materials for this theory, owing to the fact that they have very weak correlation effects OnidaRR02 ; SpataruICL05 ; TiagoKHR08 . ## III Fullerite at zero pressure The phase diagram of C60 is extremely complex. At zero pressure and temperature, it crystallizes in a structure where the orientation of molecules is random but the molecules form a face-centered cubic (FCC) structure with lattice parameter around 14.2 ÅSundqvist99 . At room temperature and under mechanical pressure of 8 GPa, the crystals were found to polymerize in several phases, with substantial distortion of the cage NunezMHBP95 ; MosharyCSBDVB92 . Since we are primarily concerned with hydrostatic pressure, we do not consider anisotropic pressure in this article. With increasing hydrostatic pressure, the lattice parameter decreases continuously according to Vinet equation of state Duclos91 . The energy threshold of optical transmission also decreases SnokeSM93 , following the reduction in lattice parameter. Other absorption edges are also known to redshift with applied pressure MeletovD98 . In our calculations, we apply pressure indirectly by fixing the lattice parameter and using Vinet equation to map lattice parameter into hydrostatic pressure Duclos91 : $p(a)=3\kappa_{0}{1-x\over x^{2}}\exp{\left[{3\over 2}(\kappa_{0}^{\prime}-1)(1-x)\right]}\;\;,$ (4) using a bulk modulus $\kappa_{0}=18.1\pm 1.8$ GPa and its pressure derivative $\kappa^{\prime}_{0}=5.7\pm 0.6$ Duclos91 . The parameter $x$ is the ratio between lattice parameters, $x=a/a_{0}$. The lattice is built in the $Pa{\overline{3}}$ (= $cP12$) structure, with one molecule per periodic cell. Figure 1 shows the calculated electronic and optical gaps for several choices of lattice parameter. At zero pressure, we obtain an electronic gap of 2.1 eV, in full agreement with previous work and compatible with photoemission and inverse photoemission data ShirleyL93 . Our DFT-PBE gap is 1.2 eV. The minimum gap is direct, around the crystallographic $X$ point. The C60 molecule has icosahedral symmetry, belonging to the Ih point group Cotton . Owing to its high symmetry, most molecular orbitals cluster in degenerate multiplets. The three highest occupied multiplets belong to symmetry representations denoted as Hu, Gg, and Hg (ordered from highest energy to lowest energy), with degeneracies 5, 4 and 5 respectively. The lowest unoccupied multiplet in molecular C60 has symmetry T1u, followed by a T1g multiplet, both with degeneracy 3. In fullerite, each molecular multiplet originates a set of quasi-degenerate bands. The wavefunctions retain most of the shape of the molecular orbitals, so that they can still be labeled by symmetry representations of the molecular orbitals. The bandwidth of the $H_{u}$ quintuplet, at the top of the valence bands, is 0.5 eV. The next multiplet is a $T_{1u}$ triplet, with approximately the same bandwidth. Within the GW theory, these bandwidths are slightly larger than the ones calculated with DFT-PBE. There are two major differences between band structures predicted with GW and DFT-PBE: (1) widening of the electronic gap, and (2) small stretch of bands according to the expressions: $\displaystyle E_{GW}^{val.}=E_{PBE}^{val.}\times 1.2+0.6{\rm\;\;eV}\;\;$ $\displaystyle E_{GW}^{cond.}=E_{PBE}^{cond.}\times 1.2+1.25{\rm\;\;eV}\;\;,$ (5) respectively for valence and conduction bands. In the equation above, the energies $E_{GW}$ and $E_{PBE}$ are given with respect to the DFT-PBE valence band maximum. We determine the optical gap as the minimum excitation energy obtained after diagonalizing the BSE. This gap at equilibrium lattice constant is calculated to be 1.7 eV. The oscillator strength associated to this excitation is very weak, owing to a molecular selection rule that prevents optical absorption from the $H_{u}$ to the $T_{1u}$ multiplets. Significant absorption is found around 2.2 eV, corresponding to $H_{u}$-$T_{1g}$ transitions, as shown on Figure 3. The measured transmission edge is 1.9 eV SnokeSM93 . This is compatible with our calculated results, considering the difficulties in determining the onset of absorption experimentally and the orientational disorder in the lattice, which is not included in our calculations. One of the earliest measurements of absorption spectra of crystalline C60 identified peaks at 2.0 eV, 2.7 eV and 3.5 eV. Our calculations show peaks at 2.2 eV and 3.6 eV. The inset of Figure 1 shows the maximum exciton binding energy, defined as the difference between electronic gap and optical gap. The binding energy is high: around 0.4 eV. As discussed above, it arises primarily from the Coulomb attraction between electrons and holes, which is large compared with other solids because of a weak dielectric screening. The first bound exciton has well-defined Frenkel character, which is compatible with the fact that its binding energy is close to the electron and hole bandwidths. Figure 2 depicts the probability distribution of the electron given that the hole is fixed on the surface of the central molecule. There are sharp maximums of probability on the central molecule, with more diffuse features in the neighbor molecules. In order to quantify the exciton radius, we have computed the integrated electron-hole probability and listed it on the third column of Table 1. For the first bound exciton, the probability of locating electron and hole on the same molecule is 62%. The probability of locating the electron on any of the nearest neighbors molecule relative to the hole site in substantially smaller (30%), decreasing then to 2% if the electron is on any second nearest neighbor. Excitons with lower binding energy (and higher excitation energy) have more pronounced charge-transfer character, with the probability at nearest neighbor higher than the probability at the hole site. In order to address the validity of our calculations, based on the $Pa{\overline{3}}$ lattice, with respect to the real, glassy crystal, we repeated the zero-pressure calculations with five different orientations of the molecule. As a result, the electronic gap fluctuated from 2.0 to 2.2 eV. That establishes an uncertainty in the determination of energy gaps arising from orientational disorder of the molecules. We find that orientational disorder affects similarly the electronic and optical gaps. The exciton binding energy fluctuates by less than 0.1 eV upon rotation of the molecular unit. Fine features in the absorption spectrum and in the density of states are smoothed out by molecular disorder while the broader features (energy position and width of major peaks) are very robust. ## IV Fullerite at hydrostatic pressure Figure 1 shows that the electronic and optical gaps decrease continuously as an hydrostatic pressure of up to 6 GPa is applied on crystalline C60. The overall decrease in electronic gap is 0.9 eV with pressure ranging from zero to 6 GPa. To our knowledge, the electronic gap at high hydrostatic pressure has not been measured yet. The optical gap (i.e., the excitation energy of the first bound exciton) decreases by 0.7 eV in the same pressure range. Since that exciton is optically inactive, the best comparison of optical activity as a function of pressure should be done following the position of the first peak in the absorption spectrum, on Figure 3. The peak moves from 2.2 eV to 1.75 eV in the pressure range from zero to 6 GPa. This is compatible with the first determinations of transmission edge as a function of pressure SnokeSM93 : the transmission edge decreases from 1.9 eV (zero pressure) to 1.5 eV (5 GPa). Figure 1 also shows that the profile of energy gap versus lattice parameter is not linear. A suitable model for the dependence of the gap with respect to pressure should take into account the behavior of dielectric screening for different amounts of intermolecular spacing and hence different amounts of overlap between molecular orbitals at different molecules. Snoke and collaborators SnokeSM93 have proposed a phenomenological model for the gap. Meletov and Dolganov MeletovD98 have also found a decrease in the optical gap as a function of pressure. In their experiment, microcrystals of fullerite were placed inside a diamond anvil cell, with pressure of up to 2.5 GPa. Several phenomena were observed in that experiment: 1. 1. At zero pressure, a low-energy line and two well pronounced lines in the absorption spectrum were found, labeled A (at 2.0 eV), B (2.7 eV) and C (3.5 eV) respectively. Line A is very weak and it could originate from transitions $H_{u}\to T_{1u}$, which gain finite oscillator strength from mixing with higher transitions. That interpretation is supported by our calculations, which indicate an onset of the line at 1.7 eV and very small but non-vanishing oscillator strength. 2. 2. Lines B and C have similar strength. It was found experimentally that optical activity migrates from C to B as the microcrystals are compressed. That effect is found in Figure 3, where we see enhancement of the peak at 2.2 eV and reduction of the peak at 3.5 eV, while both peaks redshift from zero to 3.4 GPa. Since we also see mixing between transitions $H_{u}\to T_{1g}$ and $H_{g}\to T_{1u}$, the major components of peaks B and C respectively, our calculations confirm the assumption that migration of optical activity is caused by mixing between different optical transitions MeletovD98 . 3. 3. The energy dependence of the measured absorption spectrum was reported to be weakly dependent on pressure in the pressure range from zero to 2.5 GPa MeletovD98 . Figure 3 confirms that observation. At the next pressure value (6.1 GPa), the two peaks merge into an asymmetric wide peak. That indicates that bands derived from different molecular multiplets start to overlap, as shown in Figure 4. Hydrostatic pressure also modifies the character of bound excitons, making them more delocalized. Comparing the distribution of probabilities at zero pressure (lattice parameter $14.2~{}\AA$) and 3.4 GPa (lattice parameter $13.6~{}\AA$), Table 1 shows that the first bound exciton becomes substantially more delocalized, with a correlation radius between the first and third nearest-neighbor distances. We believe that two mechanisms contribute to delocalization: applied pressure increases the overlap of molecular orbitals on different molecules, thus increasing the probability of one electron moving from one molecule to its neighbor; and pressure also increases the mixing between bands, particularly between transition $H_{u}\to T_{1u}$ and higher transitions. As in the zero-pressure regime, we use the $Pa{\overline{3}}$ lattice to perform calculations at hydrostatic pressure, with no orientational disorder. Our estimates of the impact of disorder on the energy gap, mentioned at the end of the previous section, also apply to the regime of finite pressure. Since our calculations do not contain accurate van der Waals forces, we have not attempted to investigate the emergence of different glassy phases as a function of pressure. Including accurate van der Waals forces would remove the inaccuracy of the calculated gaps with respect to orientational disorder in fullerite. ## V Fullerite with intercalated molecules Fullerene C60 is very stable, which favors the engineering of microcrystals with intercalated molecules. At equilibrium, the FCC crystal has two large types of voids: an octahedral site with radius 3.5 Å and a tetrahedral site with radius 1.15 Å. Isolated atoms and small molecules can be easily placed in one of those voids. Doped C60 has very interesting properties, for instance K3C60 is superconductor at 18 K HebardRHMGPRK91 . Ca3C60 is superconductor at 8.4 K KortanKGGRFTH92 . Those compounds also show significant electron transfer from dopant atom to cage. Doping fullerite with wide-gap molecules produce different phenomena. Depending on the concentration and symmetry of the dopant, it can lower the symmetry of the host crystal and enhance the oscillator strength of otherwise dark optical transitions of fullerite. Highly-symmetric dopants are expected to produce less distortions in the host. In particular, cubane (C8H8) has been proposed as an ideal intercalator PekkerKOBKBJJBKBKTF05 . It has perfect cubic symmetry. If placed in an octahedral void, it will preserve the cubic symmetry of the lattice. With doping, the crystal is forced to expand isotropically in order to accommodate the extra molecules but no additional structural distortion is necessary. In addition, solid cubane is bound by weak van der Waals forces YildirimGNEE92 , which means that cubane is not likely to segregate into clusters. The ionization potential of molecular cubane is 8.6 eV LifshitzE83 . Its electron affinity is negative, indicating that it has an energy gap in the ultraviolet range. Since the band edges of C60 are inside the ones of cubane, cubane can be used to mimic negative hydrostatic pressure in fullerite without altering the optical properties of the host PekkerKOBKBJJBKBKTF05 . We built a lattice of cubane-fullerene with maximum doping by filling all octahedral voids in the FCC lattice with cubane. The lattice parameter is taken as 14.8 Å, following experimental determination of the rotor-stator phase PekkerKOBKBJJBKBKTF05 . The band structure of this compound around its energy gap is very similar to undoped fullerite with the same lattice constant. Cubane derived bands are found no less than 4 eV away from the gap. The electronic gap of fullerite with intercalated cubane (C8H8-C60), obtained from our GW calculations is 2.6 eV, similar to the electronic gap obtained for fullerite with the same lattice parameter (2.7 eV). The difference of 0.1 eV is close to the numerical precision of our calculations. C8H8-C60 and pristine fullerite also have similar optical gaps: 2.0 eV and 1.85 eV. Figure 5 shows that the absorption spectra of C8H8-C60 and pristine fullerite differ from each other only above 3.5 eV. These findings confirm that the effect of adding cubane is, by and large expand the lattice of fullerene molecules. All other phenomena in its electronic structure are direct consequences of lattice expansion. Other intercalants can also produce lattice expansion. Atoms of noble gases are good candidates, owing to their low reactivity and high energy gap. One shortcoming is that they are smaller than cubane. While these molecules will not expand the lattice significantly, a small change in gap can be measured. Therefore, fullerite could be used as a sensor of inert molecules. Significant expansion could be obtained by overdoping fullerite with several atoms per interstitial site. Other candidates are small molecules such as methane (CH4), hydrogen (H2) or nitrogen (N2). ## VI Summary and perspectives The results presented above show that several properties of fullerite, particularly the threshold of its optical absorption, can be tuned by applying pressure. By applying hydrostatic pressure of up to 6 GPa, easily obtained in diamond anvil cell devices, the first peak of optical absorption redshifts from 2.2 eV (yellow-green) to 1.8 eV (red), thus making microcrystals less transparent. Similar reduction in the gap as a function of pressure has been reported in alkali-doped fullerite PoloniFFPTMCPS08 . Applications of this phenomenon are plenty. One of them is in piezo-optical sensors: one can put clean microcrystals of fullerite in an environment under unknown pressure and infer the pressure by measuring their transmittance or absorbance. This is particularly useful if the microcrystals are part of a microdevice, subject to pressure gradients and where usual pressure gauges cannot be used. One can expand the range of colors where fullerite gauges operate by doping microcrystals with weakly interacting molecules. It has been shown experimentally that saturating microcrystals with cubane increases its lattice parameter KortanKGGRFTH92 . Our results show that the lattice expansion produces a blueshift of the first absorption peak from 2.2 eV to around 2.6 eV. Other dopants can produce larger lattice expansion and hence larger blueshifts, depending on their size and concentration. In summary, we have done first-principles calculations of electronic and optical properties of fullerite in order to characterize their pressure dependence. Comparison between available experimental data and our calculations at equilibrium lattice parameter show that our methodology predicts gaps with an accuracy of 0.1 to 0.2 eV. The absorption edge shifts toward the red end of the spectrum as we apply hydrostatic pressure of up to 6 GPa. There is little distortion in the electronic structure of the material in the pressure range investigated. We have also confirmed earlier hypotheses that cubane-intercalated fullerite has optical properties very similar to fullerite with an artificial lattice expansion. These findings show that pure fullerite or fullerite with inert dopants can be used as active element in piezo-optical sensors. We would like to thank discussions with E. Schwegler, T. Oguitsu and H. Whitley for discussions. Research sponsored by the Division of Materials Sciences and Engineering BES, U.S. DOE under contract with UT-Battelle, LLC. Computational support was provided by the National Energy Research Scientific Computing Center. ## References * (1) H. Kroto, J. Heath, S. O’Brien, R. Curl, and R. Smalley, Nature 318, 162 (1985). * (2) M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund, _Science of fullerenes and carbon nanotubes: their properties and applications_ (Academic Press, San Diego, 1996). * (3) W. Kratschmer, L. Lamb, K. Fostiropoulos, and D. Huffman, Nature 347, 354 (1990). * (4) X. Blase, P. Gillet, A. San Miguel, and P. Mélinon, Phys. Rev. Lett. 92, 215505 (2004). * (5) A. San Miguel, Chem. Soc. Rev. 35, 876 (2006). * (6) A. Kortan, N. Kopylov, S. Glarum, E. Gyorgy, A. Ramirez, R. Fleming, F. Thiel, and R. Haddon, Nature 355, 529 (1992). * (7) A. Hebard, M. Rosseinsky, R. Haddon, D. Murphy, S. Glarum, T. Palstra, A. Ramirez, and A. Kortan, Nature 350, 600 (1991). * (8) K. Meletov and G. Kourouklis, J. Exp. Theor. Phys. 100, 760 (2005). * (9) B. Sundqvist, Adv. Phys. 48, 1 (1999). * (10) K. Meletov and V. Dolganov, J. Exp. Theor. Phys. 86, 177 (1998). * (11) S. Pekker, E. Kovats, G. Oszlanyi, G. Benyei, G. Klupp, G. Bortel, I. Jalsovszky, E. Jakab, F. Borondics, K. Kamaras, et al., Nature Mat. 4, 764 (2005). * (12) R. Poloni, M. V. Fernandez-Serra, S. Le Floch, S. De Panfilis, P. Toulemonde, D. Machon, W. Crichton, S. Pascarelli, and A. San-Miguel, Phys. Rev. B 77, 035429 (2008). * (13) S. Duclos, K. Brister, R. Haddon, A. Kortan, and F. Thiel, Nature 351, 380 (1991). * (14) M. Núñez Regueiro, L. Marques, J. L. Hodeau, O. Béthoux, and M. Perroux, Phys. Rev. Lett. 74, 278 (1995). * (15) D. W. Snoke, K. Syassen, and A. Mittelbach, Phys. Rev. B 47, 4146 (1993). * (16) F. Moshary, N. H. Chen, I. F. Silvera, C. A. Brown, H. C. Dorn, M. S. de Vries, and D. S. Bethune, Phys. Rev. Lett. 69, 466 (1992). * (17) M. Kozlov and K. Yakushi, J. Physics - Condens. Matter 7, L209 (1995). * (18) E. L. Shirley and S. G. Louie, Phys. Rev. Lett. 71, 133 (1993). * (19) E. L. Shirley, L. X. Benedict, and S. G. Louie, Phys. Rev. B 54, 10970 (1996). * (20) C. Hartmann, M. Zigone, G. Martinez, E. L. Shirley, L. X. Benedict, S. G. Louie, M. S. Fuhrer, and A. Zettl, Phys. Rev. B 52, R5550 (1995). * (21) R. W. Martin, _Electronic structure: basic theory and practical methods_ (Cambridge University Press, Cambridge, UK, 2004). * (22) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett 77, 3865 (1996). * (23) M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986). * (24) W. Aulbur, L. Jönsson, and J. Wilkins, _Solid State Physics_ (Academic Press, New York, 2000), vol. 54, p. 1. * (25) G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002). * (26) M. Rohlfing and S. G. Louie, Phys. Rev. B 62, 4927 (2000). * (27) A. L. Fetter and J. D. Walecka, _Quantum Theory of Many-Particle Systems_ (Mc-Graw Hill, New York, 1971). * (28) M. L. Tiago and J. R. Chelikowsky, Phys. Rev. B 73, 205334 (2006). * (29) C. D. Spataru, S. Ismail-Beigi, R. B. Capaz, and S. G. Louie, Phys. Rev. Lett. 95, 247402 (2005). * (30) M. L. Tiago, P. R. C. Kent, R. Q. Hood, and F. A. Reboredo, J. Chem. Phys. 129, 084311 (2008). * (31) F. A. Cotton, _Chemical Applications of Group Theory_ (J. Wiley and Sons, New York, 1990). * (32) T. Yildirim, P. M. Gehring, D. A. Neumann, P. E. Eaton, and T. Emrick, Phys. Rev. Lett. 78, 4938 (1997). * (33) C. Lifshitz and P. Eaton, Int. J. Mass Spectrom. Ion Phys. 49, 337 (1983). Figure 1: (Color online) Electronic gap (squares) and optical gap (crosses) calculated for fullerite as functions of lattice parameter. The equivalent hydrostatic pressure was obtained using Vinet equation, Equation 4. The electronic gap was calculated within the GW approximation. The optical gap shown is the energy of the first excitation energy obtained from the BSE. It is a lower bound to the experimental optical gap since the first excitation has very low oscillator strength (see text). The inset shows the maximum exciton binding energy, i.e. difference between the electronic and optical gaps. Figure 2: (Color online) Isocontour plot of the electron probability distribution function of the first bound exciton provided that the hole is at a position where the highest occupied molecular orbital has maximum amplitude. The plot corresponds to fullerite at zero pressure. Only molecules up to second neighbor from the hole site are depicted in the figure. The isocontour shown corresponds to a value 10% lower than the maximum value of the probability distribution. This exciton is composed primarily by transitions in the $H_{u}\to T_{1u}$ multiplet. Figure 3: (Color online) Imaginary part of the dielectric function for several choices of lattice parameter. Vertical bars on each panel indicate the calculated optical and electronic gaps. An artificial Gaussian broadening of 0.02 eV was added to all absorption spectra. Sharp features in the spectrum are expected to fade away with inclusion of rotational disorder. Line A in the measured spectrum MeletovD98 (see text) is very weak to be visible. Line B is the first absorption line, at around 2.2 eV at zero pressure. Line C is the second absorption line, at around 3.6 eV at zero pressure. Figure 4: (Color online) Density of states in fullerite for several choices of lattice parameter, obtained within the GW theory. Energies are defined with respect to the valence band maximum. A Gaussian broadening of 0.1 eV was added to all density distributions. The five major features in the density of states (well separated at zero and “negative” pressure) correspond to different molecular multiplets, from lower to higher energy: $H_{g}$+$G_{g}$ (superimposed) , $H_{u}$, $T_{1u}$, $T_{1g}$. Figure 5: Imaginary part of the dielectric function of C8H8-C60 (a) and pure C60 (b). An artificial Gaussian broadening of 0.02 eV was added to all absorption spectra. Location of Electron \| Probability ---\|--- \| $a=13.6\AA$ \| $a=14.2\AA$ \| $a=14.8\AA$ Hole’s molecule \| 15 % \| 62 % \| 84 % 1st Nearest Neighbor \| 25 % \| 30 % \| 13 % 2nd Nearest Neighbor \| 10 % \| 2 % \| $<$ 1% 3rd Nearest Neighbor \| 25 % \| 3 % \| $<$ 1 % Table 1: Electron probability distribution of the first bound exciton calculated as a function of the distance to the hole. The probability was integrated over each shell of molecules around the molecule that contains the hole. The integration was performed over a Wigner-Seitz cell centered on each molecule. Three different lattice parameters are shown: 13.6 $\AA$ (3.4 GPa), 14.2 $\AA$ (zero pressure) and 14.8 $\AA$ (negative pressure -1.6 GPa).	arxiv-papers	2009-02-06T13:56:38	2024-09-04T02:49:00.460638	{ "license": "Public Domain", "authors": "Murilo L. Tiago, Fernando A. Reboredo", "submitter": "Fernando Reboredo", "url": "https://arxiv.org/abs/0902.1091" }
0902.1361	# Spectral and optical properties in the antiphase stripe phase of the cuprate superconductors Hong-Min Jiang National Laboratory of Solid State of Microstructure and Department of Physics, Nanjing University, Nanjing 210093, China Cui-Ping Chen National Laboratory of Solid State of Microstructure and Department of Physics, Nanjing University, Nanjing 210093, China Jian-Xin Li National Laboratory of Solid State of Microstructure and Department of Physics, Nanjing University, Nanjing 210093, China ###### Abstract We investigate the superconducting order parameter, the spectral and optical properties in a stripe model with spin (charge) domain-derived scattering potential $V_{s}$ ($V_{c}$). We show that the charge domain-derived scattering is less effective than the spin scattering on the suppression of superconductivity. For $V_{s}\gg V_{c}$, the spectral weight concentrates on the ($\pi,0$) antinodal region, and a finite energy peak appears in the optical conductivity with the disappearance of the Drude peak. But for $V_{s}\approx V_{c}$, the spectral weight concentrates on the ($\pi/2,\pi/2$) nodal region, and a residual Drude peak exists in the optical conductivity without the finite energy peak. These results consistently account for the divergent observations in the ARPES and optical conductivity experiments in several high-$T_{c}$ cuprates, and suggest that the ”insulating” and ”metallic” properties are intrinsic to the stripe state, depending on the relative strength of the spin and charge domain-derived scattering potentials. ###### pacs: 74.20.Mn, 74.25.Ha, 74.25.Jb, 74.72.Bk ## I introduction The nature of spin and/or charge inhomogeneities, especially in the form of stripes, in some cuprates and their involvement to high-temperature superconductivity are currently debate issues. kive1 The stripe state is characterized by the self-organization of the charges and spins in the CuO2 planes in a peculiar manner, where the doped holes are arranged in one- dimensional (1D) lines and form the so-called ”charge stripe” separating the antiferromagnetic domains. The stripe-ordered state minimizes the energy of the hole-doped antiferromagnetic system, thus leading to an inhomogeneous state of matter. Static one-dimensional charge and spin stripe order have been observed experimentally in a few special cuprate compounds, specifically in La1.6-xNd0.4SrxCuO4 tran2 ; zhou1 and La2-xBaxCuO4 with $x=1/8$. abba1 ; tran3 Similar signatures identified in La2-xSrxCuO4 (LSCO) cheo1 ; maso1 ; bian1 ; yama1 and other high temperature superconductors well1 ; lee4 ; mook2 point to the possible existence of stripes, albeit of a dynamical or fluctuating nature. A pivotal issue about this new electronic state of matter concerns whether it is compatible with superconductivity, and possibly even essential for the high transition temperatures, or it competes with the pairing correlations. A prerequisite for addressing these issues is to understand the electronic structures of various stripe states in different cuprates, and to answer the question whether the stripe phase is intrinsically ”metallic” or ”insulating”, given its spin- and charge-ordered nature. Angle-resolved photoemission spectroscopy (ARPES) study by Zhou et al. in (La1.28Nd0.6Sr0.12)CuO4 with static stripes have found the depletion of the low-energy excitation near the ($\pi/2,\pi/2$) nodal region. zhou1 In another compound La1.875Ba0.125CuO4, a system where the superconductivity is heavily suppressed due to the development of the static spin and charge orders, Valla et al. have detected the high spectral intensity of the low-energy excitation in the vicinity of the ($\pi/2,\pi/2$) nodal region [while antinodal low-energy quasiparticle near $(0,\pi)$ are gapped]. vall1 The compound (La1.4-xNd0.6Srx)CuO4 ($x=0.10$ and $0.15$) with static one-dimensional stripe, seems to be an in- between system, in where the existence of spectral weight around the nodal region, though weak, has been identified. zhou2 Meanwhile, optical conductivity measurements on the systems with a stripe phase also display the divergent results. In La1.275Nd0.6Sr0.125CuO4 dumm1 and La1.875Ba0.125-xSrxCuO4, orto1 a finite frequency absorption peak with almost disappearance of the Drude mode in the low-frequency conductivity in several experiments has been interpreted as collective excitations of charge stripes or as charge localization from the disorder created by Nd or Ba substitutions. These observations may support the suggestion that such stripe- ordered state should be ”insulating” in nature. cast1 On the other hand, optical experiment on La1.875Ba0.125CuO4 home1 has observed a residual Drude peak with a loss of the low-energy spectral weight below the temperature corresponding to the onset of charge stripe order, which indicates that stripes are compatible with the so-called nodal-metal state. ando1 ; zhou3 ; dumm2 ; suth1 ; lee3 ; home1 Although, there have been some theoretical studies on the spectral and optical properties in the stripe phase in the past years , tohy1 ; mark1 ; mart1 ; lore1 the contradictory observations in recent experiments as mentioned above have yet not been explained consistently in theoretical frame by adopting a realistic stripe model. In this paper, by using a stripe model in which the experimentally observed spin and charge structures at 1/8 doping are well reflected, we show that the spin domain-derived scattering will depress the zero-energy spectral weight around the nodal regions, while the charge domain- derived scattering will suppress mostly those around the antinodal regions and the hot spots. Compared to the ARPES data, this suggests that the different spectral weight distribution may result from the different relative strength of the spin and charge domain-derived scattering potentials inherently existing in these compounds. Meanwhile, a finite frequency peak in the optical conductivity appears with the disappearance of the Drude peak in the case of the dominant spin domain-derived scattering. While, when the charge domain- derived scattering is comparable to the spin one, a residual Drude peak exists with the disappearance of the finite energy peak. This suggests that both the ”insulating” and ”metallic” properties are intrinsic to the stripe state without introducing another distinct metallic phase. The rest of this paper is organized as follows. In Sec. II, we introduce the model Hamiltonian and carry out the analytical calculations. In Sec. III, we present the numerical calculations and discuss the results. In Sec. IV, we present the conclusion. ## II THEORY AND METHOD As the above discussed compounds have a doping density at or near 1/8, we will in this paper consider the 1/8 doping antiphase vertical stripe state. A schematic illustration of its charge and spin pattern is presented in Fig. 1. The charge stripes, with a unit cell of 8 lattice sites (Note for 1/8 doping, there is one hole for every two sites along the length of a charge stripe), act as antiphase domain walls for the magnetic order, so that the magnetic unit cell is twice as long as that for charge order. Due to the periodical modulation of the stripe order, the electrons moving in the state will be scattered by the modulation potentials. After Fourier transformation, the potential $V_{n}$ can be written as the scattering term between the state $k$ and those at $k\pm nQ$ with $Q=(3\pi/4,\pi)$. Following Ref. mill1, , we expect that the terms $V_{1}$ and $V_{2}$ will be the dominant spin and charge domain-derived scattering term, and will be relabeled as $V_{s}$ and $V_{c}$ in the following, respectively. The weaker higher harmonic terms will be neglected here. In the coexistence with the superconducting (SC) order, the model Hamiltonian can be written as a $16\times 16$ matrix for $k$ in the reduced Brillouin zone, $\displaystyle\hat{H}=\sum_{k}{{}^{\prime}}\hat{C}^{{\dagger}}(k)\left(\begin{array}[]{cc}\hat{H}_{k}&\hat{\Delta}_{k}\\\ \hat{\Delta}_{k}&-\hat{H}_{k}\\\ \end{array}\right)\hat{C}(k),$ (3) where, the prime denotes the summation over the reduced Brillouin zone. $\hat{C}_{k}$ is a column vector with its elements $C_{i}(k)=C_{k+(i-1)Q,\uparrow}$ for $i=1,2,\cdots,8$, and $C^{{\dagger}}_{-k-(i-9)Q,\downarrow}$ for $i=9,10,\cdots,16$. Both $\hat{H}_{k}$ and $\hat{\Delta}_{k}$ are $8\times 8$ matrix with $\displaystyle\hat{H}_{k}=\left(\begin{array}[]{cccccccc}\varepsilon_{k}&V_{s}&V_{c}&0&0&0&V_{c}&V_{s}\\\ V_{s}&\varepsilon_{k+Q}&V_{s}&V_{c}&0&0&0&V_{c}\\\ V_{c}&V_{s}&\varepsilon_{k+2Q}&V_{s}&V_{c}&0&0&0\\\ 0&V_{c}&V_{s}&\varepsilon_{k+3Q}&V_{s}&V_{c}&0&0\\\ 0&0&V_{c}&V_{s}&\varepsilon_{k+4Q}&V_{s}&V_{c}&0\\\ 0&0&0&V_{c}&V_{s}&\varepsilon_{k+5Q}&V_{s}&V_{c}\\\ V_{c}&0&0&0&V_{c}&V_{s}&\varepsilon_{k+6Q}&V_{s}\\\ V_{s}&V_{c}&0&0&0&V_{c}&V_{s}&\varepsilon_{k+7Q}\end{array}\right),$ (12) and $\displaystyle\hat{\Delta}_{k}=\left(\begin{array}[]{cccccccc}\Delta_{k}&0&0&0&0&0&0&0\\\ 0&\Delta_{k+Q}&0&0&0&0&0&0\\\ 0&0&\Delta_{k+2Q}&0&0&0&0&0\\\ 0&0&0&\Delta_{k+3Q}&0&0&0&0\\\ 0&0&0&0&\Delta_{k+4Q}&0&0&0\\\ 0&0&0&0&0&\Delta_{k+5Q}&0&0\\\ 0&0&0&0&0&0&\Delta_{k+6Q}&0\\\ 0&0&0&0&0&0&0&\Delta_{k+7Q}\end{array}\right).$ (21) As for the tight-binding energy band, we will choose the following form, lee5 ; li1 $\displaystyle\varepsilon_{k}=$ $\displaystyle-2(\delta t+J^{{}^{\prime}}\chi_{0})(\cos k_{x}+\cos k_{y})$ (22) $\displaystyle-4\delta t^{{}^{\prime}}\cos k_{x}\cos k_{y}-\mu.$ where, $\delta$ is the doping density, and a $d$-wave SC order parameter $\Delta_{k}=2J^{{}^{\prime}}\Delta_{0}(\cos k_{x}-\cos k_{y})$ is assumed. Generally, the charge modulation will induce the modulation of the SC order leading to the finite momentum pairs. However, in the present study, one of our aim is to examine the effect of the spin (charge) domain-derived scattering on the SC order. In this regard, the average value of the SC order parameter is relevant and the modulation of the SC order will be ignored. We have checked the effect of this modulation and found no qualitative change in the results presented in Fig. 2. In the following, $J=100\textmd{meV}$ is taken as the energy unit, $t=2J$, $t^{{}^{\prime}}=-0.45t$, $J^{{}^{\prime}}=\frac{3}{8}J$. This dispersion can be derived from the slave- boson mean-field calculation of the $t-t^{{}^{\prime}}-J$ model lee5 ; li1 , and in this way the parameters $\Delta_{0}$, $\chi_{0}$ and $\mu$ are determined self-consistently. Here we take it as a phenomenological form. In a self-consistent calculation, the Hamiltonian is first diagonalized by a unitary matrix $\hat{U}(k)$ with a set of trial values of $\Delta_{0}$, $\chi_{0}$ and $\mu$ for given potentials $V_{s}$ and $V_{c}$. Then $\Delta_{0}$, $\chi_{0}$ and $\mu$ are self-consistently calculated by using the relations: $\pm\Delta_{0}=\langle c_{i\uparrow}c_{i+\tau\downarrow}-c_{i\downarrow}c_{i+\tau\uparrow}\rangle$ (To get the $d$-wave pairing, the sign before $\Delta_{0}$ takes $+$ for $\tau=\pm\hat{x}$ and $-$ for $\tau=\pm\hat{y}$, where $\hat{x}$ and $\hat{y}$ denote the unit vectors along $x$ and $y$ directions, respectively.), $\chi_{0}=\sum_{\sigma}\langle c^{{\dagger}}_{i\sigma}c_{j\sigma}\rangle$, and $n=\sum_{\sigma}\langle c^{{\dagger}}_{i\sigma}c_{i\sigma}\rangle$, respectively. Reformularization of the expressions of $\Delta_{0}$, $\chi_{0}$ and $\mu$ in terms of eigenfunctions and eigenvalues of the Hamiltonian, one obtains the self-consistency relations $\displaystyle\Delta_{0}$ $\displaystyle=$ $\displaystyle-\frac{1}{N}\sum_{k}(\cos k_{x}-\cos k_{y})\sum^{16}_{m=1}U_{1m}(k)U^{{\dagger}}_{m9}(k)f[E_{m}(k)]$ $\displaystyle\chi_{0}$ $\displaystyle=$ $\displaystyle\frac{1}{N}\sum_{k}(\cos k_{x}+\cos k_{y})\sum^{16}_{m=1}U_{1m}(k)U^{{\dagger}}_{m1}(k)f[E_{m}(k)]$ $\displaystyle n$ $\displaystyle=$ $\displaystyle\frac{2}{N}\sum_{k}\sum^{16}_{m=1}U_{1m}(k)U^{{\dagger}}_{m1}(k)f[E_{m}(k)],$ (23) where, $E_{m}(k)$ is the eigenvalue of the Hamiltonian, $U_{mn}(k)$ the elements of the matrix $\hat{U}(k)$, and $f[E_{m}(k)]$ is the Fermi-Dirac distribution function. Then, the single particle Green functions $G_{ij}(k,i\omega_{n})=-\int^{\beta}_{0}d\tau\exp^{i\omega_{n}\tau}\langle T_{\tau}C_{i}(k,i\tau)C^{{\dagger}}_{j}(k,0)\rangle$ can be expressed as $\displaystyle G_{ij}(k,i\omega_{n})=\sum^{16}_{m=1}\frac{U_{im}(k)U^{{\dagger}}_{mj}(k)}{i\omega_{n}-E_{m}(k)},$ (24) and the spectral functions is $\displaystyle A_{ij}(k,\omega)=-\frac{1}{\pi}\textmd{Im}G_{ij}(k,\omega+i0^{+}).$ (25) ## III results and discussion ### III.1 Self-consistent calculation of the SC order parameter We first present in Fig. 2 the self-consistent results of the SC order parameter as a function of $V_{s}$ and $V_{c}$. While the scattering from both spin and charge domain-derived scattering potentials in the stripe state leads to the suppression of the SC order parameter, the charge domains are more compatible with superconductivity than spin domains, as can be seen from Fig. 2(a). This may support the statement that the SC pairing in the stripe state occurs most strongly within the charge stripes. berg1 On the other hand, an interesting feature is that the SC order parameter will be zero at the spin domain-derived scattering potential $V_{sc}\approx 0.14$ in the absence of the charge domain-derived scattering, however, it will develop a noticeable value after turning on the charge domain-derived scattering potential, as shown in Fig. 2(b). This shows that the charge domain-derived scattering will lead to the emergency of the SC order which is otherwise destroyed by the spin only scattering. ### III.2 Distribution of spectral weight In Fig. 3, we present the distribution of the low-energy spectral weight in the original Brillouin zone (integrated over an energy window $\Delta\epsilon=0.1J$ about $\epsilon_{F}$) in the 1/8 antiphase stripe state for different spin (charge) domain-derived scattering potential $V_{s}$ ($V_{c}$). Let us first look at the limit where only the spin domain-derived scattering is included, i.e., $V_{c}=0$ with $V_{s}=0.15$, one will find that the spectral weight around the nodal region is suppressed heavily[See Fig. 3(a)]. At another limit where only the charge domain-derived scattering is included ($V_{c}=0.17$ with $V_{s}=0$), the spectral weight around the nodal region is recovered and those around the hot spot (the cross of the Fermi surface with the line $k_{x}\pm k_{y}=\pm\pi$) and near the antinodal region are suppressed[See Fig. 3(b)]. Starting from the limit of $V_{c}=0$ and fixing $V_{s}=0.15$, the spectral weight will redistribute gradually from the antinodal region to the nodal region with the increase of the charge domain- derived scattering potential $V_{c}$, as shown in Figs. 3(c) and (d). When two scattering potentials are comparable, the strongest spectral weight situates around the nodal region, and at the meantime noticeable spectral weights along the whole Fermi surface is presented. Therefore, the divergent features observed in ARPES measurements by Zhou et al. in (La1.28Nd0.6Sr0.15)CuO4 zhou2 in which the low-energy excitations near the nodal region are depleted, and by Valla et al. in La1.875Ba0.125CuO4 vall1 in which the high spectral intensity of the low-energy excitation in the vicinity of the nodal region is detected are consistently reproduced here by a change of the relative strength between the charge and spin domain-derived scatterings. This consistent accounting enables us to propose that the spin domain-derived scattering dominates over the charge one in the former system while the scattering strengthes of them are comparable in the latter system. In the presence of the spin (charge) domain-derived potential, quasiparticles near the Fermi surface will be scattered from k to $\textbf{k}\pm n\textbf{Q}$ (n=1 for the spin domain-derived potential, n=2 for the charge one), for the 1/8 antiphase vertical stripe configuration shown as Fig. 1. This gives rise to two scattering channels from the spin domain with potential $V_{s}$, k $\displaystyle\rightarrow$ $\displaystyle\textbf{k}+Q=\textbf{k}+(3\pi/4,\pi),$ k $\displaystyle\rightarrow$ $\displaystyle\textbf{k}-Q=\textbf{k}+(5\pi/4,\pi),$ (26) and two scattering channels from the charge domain with potential $V_{c}$, k $\displaystyle\rightarrow$ $\displaystyle\textbf{k}+2Q=\textbf{k}+(3\pi/2,0),$ k $\displaystyle\rightarrow$ $\displaystyle\textbf{k}-2Q=\textbf{k}+(\pi/2,0).$ (27) Strong potential scattering will destruct those parts of the Fermi surface connected by the above mentioned scattering wave vectors. Because the scattering wave vectors $Q$ and $-Q$ are close to the transferred momenta from the node to node scattering, so it will lead to a depletion of the spectral weight near the nodal region as shown in Fig. 3(a). On the other hand, the scattering wave vectors $2Q$ and $-2Q$, which is near the connecting wave vectors between the two approximately parallel segments of the Fermi surface near the antinodal and hot spot region, the scatterings with these wave vectors will suppress the spectral weights around the antinodal and hot spot regions [Fig. 3(b)]. ### III.3 In-plane optical conductivity Now, we turn to the discussion of the in-plane optical properties in the 1/8 antiphase stripe state, and to see how they are influenced by the scattering from the spin and charge domains. We will fix the temperature at $T=0.05$ in all calculations, in order to avoid the influence from the temperature induced change in the scattering rate. We consider an electric field applied in the $x$ direction, which is perpendicular to the stripe. From the Kubo formula for the optical conductivity, the real part of the optical conductivity is $\sigma_{1}(\omega)=-\lim_{q\rightarrow 0}\textmd{Im}[\Pi(q,\omega)]/\omega$. The imaginary part of the current-current correlation function Im$[\Pi(q\rightarrow 0,\omega)]$ is given by $\displaystyle\textmd{Im}[\Pi(q\rightarrow 0,\omega)]=$ $\displaystyle\frac{\pi}{N}\sum_{k}{{}^{\prime}}\sum^{16}_{j,l=1}v^{jj}(k)v^{ll}(k)$ (28) $\displaystyle\times\int d\omega^{{}^{\prime}}[f(\omega+\omega^{{}^{\prime}})-f(\omega^{{}^{\prime}})]$ $\displaystyle\times A_{jl}(k,\omega^{{}^{\prime}})A_{lj}(k,\omega+\omega^{{}^{\prime}}).$ Here, $v^{jj}(k)$ is the diagonal element of the quasiparticle group velocity in the matrix form $\displaystyle\hat{v}(k)=\left(\begin{array}[]{cc}\frac{\partial\hat{H}_{k}}{\partial k_{x}}&0\\\ 0&-\frac{\partial\hat{H}_{k}}{\partial k_{x}}\\\ \end{array}\right).$ (31) Figs. 4(a)-4(d) show the results for the optical conductivity calculated with the same scattering potentials as used to get Fig. 3(a)-3(d). With only spin domain-derived scattering[Fig. 4(a)], no Drude-like component appears at zero frequency in the optical conductivity, instead a finite frequency conductivity peak occurs around 0.3. This indicates that the system exhibits the ”insulating” property. note When only charge domain-derived scattering is considered[Fig. 4(b)], the Drude-like peak shows up and at the meantime the finite frequency peak remains. Optical conductivity involves the contribution from the quasiparticle excitations along the whole Fermi surface weighted by the quasiparticle group velocity. Due to the relative flat band structure near the antinodal region for the high-$T_{c}$ cuprates, the zero frequency optical conductivity mainly comes from the quasiparticle excitations around the nodal region. In the case of only spin domain-derived scattering, the nodal region of the Fermi surface is gapped and therefore the quasiparticle spectral weight is suppressed around the nodal region as shown in Fig. 3(a), so that the zero- frequency Drude-like peak is absent and a finite frequency peak with its position being equal to the gap ($\approx 2V_{s}=0.3$) occurs. For the charge domain-derived scattering, the gap opens around the hot spots and near the antinodal, but a large spectral weight situates around the nodal region, as can be seen from Fig. 3(b). Thus, the Drude-like peak emerges and the finite frequency peak remains (it is now situates at $\approx 2V_{c}=0.34$). As shown in Fig. 3(c), with the increase of the charge domain-derived scattering $V_{c}$, the gap near the nodal region which is resulted from the spin domain- derived scattering will be suppressed gradually and correspondingly the spectral weight will be enhanced. As a result, the finite frequency peak in the optical conductivity is shifted to lower frequency and the zero frequency component is lifted up gradually[Fig. 4(c)]. When the charge domain-derived scattering is comparable to the spin one, the quasiparticles have noticeable spectra weight along the entire Fermi surface with its largest weight around the nodal region[Fig. 3(d)], then the Drude-like mode occurs at the zero frequency, and the finite frequency peak fades away and merges into the Drude- like peak, as shown in Fig. 4(d). The calculated results for the optical conductivity presented in Figs. 4(c) and 4(d) are consistent well with the experimental observations in the stripe state of La1.275Nd0.6Sr0.125CuO4 dumm1 and La1.875Ba0.125CuO4 home1 , respectively. ### III.4 Discussion We now discuss the implication of our theoretical results. As noted in the introduction, in La1.275Nd0.6Sr0.125CuO4 system, ARPES experiment has found that there is little or no low-energy spectral weight near the nodal region, zhou1 and optical conductivity experiment has observed a finite frequency peak with almost the disappearance of the Drude mode, indicating an ”insulating” stripe state. dumm1 ; orto1 These spectroscopic features can be reproduced here with a strong spin domain-derived scattering potential $V_{s}=0.15$ and a weak charge domain-derived potential $V_{c}=0.08$ and $V_{c}=0$, as shown in Figs. 3(a), 3(c), 4(a) and 4(c). Interestingly, in this parameter regime for the spin and charge domain-derived scattering, the SC order is destroyed as can be seen from Fig. 2(b). This is in consistent with the experimental fact that La1.275Nd0.6Sr0.125CuO4 is nonsuperconducting. In another cuprate La1.875Ba0.125CuO4, ARPES spectra have identified the existence of high spectral intensity around the nodal region, vall1 and the optical conductivity measurement has observed a residual Drude peak without the finite frequency peak, home1 pointing to a so-called nodal metal state. ando1 ; zhou3 ; dumm2 ; suth1 ; lee3 ; home1 When comparable spin and charge domain-derived scattering potentials are assumed such as $V_{s}=0.15$ and $V_{c}=0.17$, we can reproduce these features consistently, as shown in Figs. 3(d) and 4(d). On the other hand, a weak superconductivity emerges in the otherwise nonsuperconducting regime (when only spin scattering potential $V_{sc}$ is considered) with the increase of the charge domain-derived scattering potential[see Fig. 2(b)]. This suggests that the weak superconductivity in La1.875Ba0.125CuO4 is likely beneficial from the metallic behaviors of the stripe state originated from a sufficient charge domain- derived scattering. The above mentioned consistent accounting for both divergent spectroscopic features observed in two families of high-$T_{c}$ cuprates indicates that the stripe state may be intrinsically ”insulating” or ”metallic”, depending on the relative strength of the spin and charge domain- derived scattering potentials. Specifically, a large spin domain-derived scattering potential favors the ”insulating” state, while a large charge domain-derived scattering potential the ”metallic” state. ## IV conclusion We have calculated the SC order parameter, the spectral function and the optical conductivity in a stripe model with spin and charge domain-derived scattering potentials ($V_{s}$ and $V_{c}$). The self-consistent calculation of the SC order parameter shows that the charge domain-derived scattering is less effective than the spin scattering on the suppression of superconductivity, and may even lead to the emergency of the SC order which is otherwise destroyed by the spin only scattering. For $V_{s}\gg V_{c}$, the zero-energy spectral weight disappears around the nodal points, and a finite energy peak appears in the optical conductivity with almost the disappearance of the Drude peak. But for $V_{s}\approx V_{c}$, the spectral weight concentrates on the nodal region, and a residual Drude peak exists in the optical conductivity without the finite energy peak. These results consistently account for the divergent spectroscopic properties observed experimentally in two families of high-$T_{c}$ cuprates, and demonstrate that both the ”insulating” and ”metallic” behavior may be the intrinsic properties of the stripe state, depending on the relative strength of the spin and charge domain-derived scattering potentials. ## V acknowledgement This project was supported by National Natural Science Foundation of China (Grant No. 10525415), the Ministry of Science and Technology of Science (Grants Nos. 2006CB601002, 2006CB921800), the China Postdoctoral Science Foundation (Grant No. 20080441039), and the Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 0801008C). Figure 1: (Color online) Schematic illustration of the charge and spin patterns in the 1/8 doped antiphase stripe state. Circles represent the charge domain wall (An empty circle indicates a hole density of one per site), and arrows the copper spins. Figure 2: (Color online) (a) Superconducting order parameter as a function of $V_{s}$ and $V_{c}$, respectively. (b) A two- dimensional map of the superconducting order parameter in the parameter space of $V_{s}$ and $V_{c}$. Figure 3: (Color online) Spectral weight distribution for different spin (charge) domain-derived scattering potentials in the normal state with (a) $V_{s}=0.15$ and $V_{c}=0$, (b) $V_{s}=0$ and $V_{c}=0.17$, (c) $V_{s}=0.15$ and $V_{c}=0.08$, and (d) $V_{s}=0.15$ and $V_{c}=0.17$, respectively. Figure 4: In-plane optical conductivity as a function of frequency for different spin (charge) domain-derived scattering potentials in the 1/8 antiphase stripe with the SC order parameter $\Delta=0$. (a) $V_{s}=0.15$ and $V_{c}=0$, (b) $V_{s}=0$ and $V_{c}=0.17$, (c) $V_{s}=0.15$ and $V_{c}=0.08$, and (d) $V_{s}=0.15$ and $V_{c}=0.17$. ## References * (1) S. A. Kivelson and I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, A. Kapitulnik, and C. Howald, Rev. Mod. Phys. 75, 1201 (2003). * (2) J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Nature 375, 561 (1995). * (3) X. J. Zhou, P. Bogdanov, S. A. Kellar, T. Noda, H. Eisaki, S. Uchida, Z. Hussain, and Z.-X. Shen, Science 286, 268 (1999). * (4) J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, Nature 429 534 (2004). * (5) P. Abbamonte, A. Rusydi, S. Smadici, G. D. Gu, G. A. Sawatzky and D. L. Feng, Nat. Phys. 1 155 (2005). * (6) S-W. Cheong, G. Aeppli, T. E. Mason, H. Mook, S. M. Hayden, P. C. Canfield, Z. Fisk, K. N. Clausen, and J. L. Martinez, Phys. Rev. Lett. 67, 1791 (1991). * (7) T. E. Mason, G. Aeppli, and H. A. Mook, Phys. Rev. Lett. 68, 1414 (1992). * (8) A. Bianconi, N. L. Saini, A. Lanzara, M. Missori, T. Rossetti, H. Oyanagi, H. Yamaguchi, K. Oka, and T. Ito, Phys. Rev. Lett. 76, 3412 (1996). * (9) K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, S. Wakimoto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shirane, R. J. Birgeneau, M. Greven, M. A. Kastner, and Y. J. Kim, Phys. Rev. B 57, 6165 (1998). * (10) B. O. Wells, Y. S. Lee, M. A. Kastner, R. J. Christianson, R. J. Birgeneau, K. Yamada, Y. Endoh, and G. Shirane, Science 277, 1067 (1997). * (11) Y. S. Lee, R. J. Birgeneau, M. A. Kastner, Y. Endoh, S. Wakimoto, K. Yamada, R. W. Erwin, S.-H. Lee, and G. Shirane, Phys. Rev. B 60, 3643 (1999). * (12) H. A. Mook, P. Dai, S. M. Hayden, G. Aeppli, T. G. Perring, and F. Doǧan, Nature 395, 580 (1998). * (13) T. Valla, A. V. Fedorov, J. Lee, J. C. Davis, and G. D. Gu, Science 314, 1914 (2006). * (14) X. J. Zhou, T. Yoshida, S. A. Kellar, P. V. Bogdanov, E. D. Lu, A. Lanzara, M. Nakamura, T. Noda, T. Kakeshita, H. Eisaki, S. Uchida, A. Fujimori, Z. Hussain, and Z.-X. Shen, Phys. Rev. Lett. 86, 5578 (2001). * (15) M. Dumm, and D. N. Basov, Seiki Komiya, Yasushi Abe, and Yoichi Ando, Phys. Rev. Lett. 88, 147003 (2002). * (16) M. Ortolani, P. Calvani, S. Lupi, U. Schade, A. Perla, M. Fujita, and K. Yamada, Phys. Rev. B 73, 184508 (2006). * (17) C. Castellani, C. Di Castro, M. Grilli, A. Perali, Physica C 341-348, 1739 (2000). * (18) C. C. Homes, S. V. Dordevic, G. D. Gu, Q. Li, T. Valla, and J. M. Tranquada, Phys. Rev. Lett. 96, 257002 (2006). * (19) Y. Ando, A. N. Lavrov, S. Komiya, K. Segawa, and X. F. Sun, Phys. Rev. Lett. 87, 017001 (2001). * (20) X. J. Zhou, T. Yoshida, A. Lanzara, P. V. Bogdanov, S. A. Kellar, K. M. Shen, W. L. Yang, F. Ronning, T. Sasagawa, T. Kakeshita, T. Noda, H. Eisaki, S. Uchida, C. T. Lin, F. Zhou, J. W. Xiong, W. X. Ti, Z. X. Zhao, A. Fujimori, Z. Hussain, and Z.-X. Shen, Nature (London) 423, 398 (2003). * (21) M. Dumm, S. Komiya, Y. Ando, and D. N. Basov, Phys. Rev. Lett. 91, 077004 (2003). * (22) M. Sutherland, D. G. Hawthorn, R. W. Hill, F. Ronning, S. Wakimoto, H. Zhang, C. Proust, E. Boaknin, C. Lupien, L. Taillefer, R. Liang, D. A. Bonn, W. N. Hardy, R. Gagnon, N. E. Hussey, T. Kimura, M. Nohara, and H. Takagi, Phys. Rev. B 67, 174520 (2003). * (23) Y. S. Lee, K. Segawa, Y. Ando, and D. N. Basov, Phys. Rev. B 70, 014518 (2004). * (24) T. Tohyama, S. Nagai, Y. Shibata, and S. Maekawa, Phys. Rev. Rev. 82, 4910 (1999). * (25) R. S. Markiewicz, Phys. Rev. B 62, 1252 (2000). * (26) I. Martin, G. Ortiz, A. V. Balatsky, and A. R. Bishop, Europhys. Lett. 56, 849 (2001). * (27) J. Lorenzana, and G. Seibold, Phys. Rev. Rev. 90, 066404 (2003). * (28) A. J. Millis, and M. R. Norman, Phys. Rev. B 76, 220503(R) (2007). * (29) P. A. Lee, N. Nagaosa, and X. G. Wen, Rev. Mod. Phys. 78, 17 (2006). * (30) J. X. Li, C. Y. Mou, and T. K. Lee, Phys. Rev. B 62, 640 (2000). * (31) E. Berg, E. Fradkin, E.-A. Kim, S. A. Kivelson, V. Oganesyan, J. M. Tranquada, and S. C. Zhang, Phys. Rev. Lett. 99, 127003 (2007). * (32) The term ”insulating” state used here follows Refs. zhou2, ; orto1, ; home1, to indicate a strong suppression of Drude peak in the optical conductivity. In fact, the spectral weight at the Fermi level will not be fully gapped out, so it is not a true insulating state. We use the term here is to facilitate our comparison with the experiments[Refs. zhou2, ; orto1, ; home1, ].	arxiv-papers	2009-02-09T03:14:54	2024-09-04T02:49:00.471111	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hong-Min Jiang, Cui-Ping Chen, and Jian-Xin Li", "submitter": "Hong-Min Jiang", "url": "https://arxiv.org/abs/0902.1361" }
0902.1506	# Needles in the Haystack: Identifying Individuals Present in Pooled Genomic Data Rosemary Braun, William Rowe, Carl Schaefer Jinghui Zhang, and Kenneth Buetow National Cancer Institute, NIH, Bethesda, MD. ###### Abstract Recent publications have described and applied a novel metric that quantifies the genetic distance of an individual with respect to two population samples, and have suggested that the metric makes it possible to infer the presence of an individual of known genotype in a sample for which only the marginal allele frequencies are known. However, the assumptions, limitations, and utility of this metric remained incompletely characterized. Here we present an exploration of the strengths and limitations of that method. In addition to analytical investigations of the underlying assumptions, we use both real and simulated genotypes to test empirically the method’s accuracy. The results reveal that, when used as a means by which to identify individuals as members of a population sample, the specificity is low in several circumstances. We find that the misclassifications stem from violations of assumptions that are crucial to the technique yet hard to control in practice, and we explore the feasibility of several methods to improve the sensitivity. Additionally, we find that the specificity may still be lower than expected even in ideal circumstances. However, despite the metric’s inadequacies for identifying the presence of an individual in a sample, our results suggest potential avenues for future research on tuning this method to problems of ancestry inference or disease prediction. By revealing both the strengths and limitations of the proposed method, we hope to elucidate situations in which this distance metric may be used in an appropriate manner. We also discuss the implications of our findings in forensics applications and in the protection of GWAS participant privacy. ## 1 Introduction In the recently published article “Resolving Individuals Contributing Trace Amounts of DNA to Highly Complex Mixtures Using High-Density SNP Genotyping Microarrays” [1], the authors describe a method by which the presence of a individual with a known genotype may be inferred as being part of a mixture of genetic material for which marginal minor allele frequencies (MAFs), but not sample genotypes, are known. The method [1] is motivated by the idea that the presence of a specific individual’s genetic material will bias the MAFs of a sample of which they are part in a subtle but systematic manner, such that when considering multiple loci, the bias introduced by a specific individual can be detected even when his DNA comprises only a small fraction of the mixture. More generally, it is well known that samples of a population will exhibit slightly different MAFs due to sampling variance following a binomial distribution; the genotype of the individual in question contributes to this variation, and so may be “closer” to a sample containing him than to a sample which does not. Based on this intuition, the article [1] defines a genetic distance statistic to measure the distance of an individual relative to two samples, summarized as follows: Consider an underlying population $P$ from which two samples $F$ (of size $n_{F}$) and $G$ (of size $n_{G}$) are drawn independently and identically distributed (i.i.d.) [in [1], these are referred to as “reference” and “mixture” respectively]. Consider now an additional sample $Y$; we wish to detect whether $Y$ was drawn from $G$, versus the null hypothesis that $Y$ was drawn from $P$ independent of $G$ and $F$. Given the MAFs $f_{i}$ and $g_{i}$ at locus $i$ for $F$ and $G$, respectively, and given the MAFs $y_{i}$ for sample $Y$ with $y_{i}\in\\{0,0.5,1\\}$ (corresponding to homozygous major, heterozygous, and homozygous minor alleles) at each locus $i$, [1] defines the relative distance of sample $Y$ from $F$ and $G$ at $i$ as: $D_{i}(Y)=\left\|{y_{i}-f_{i}}\right\|-\left\|{y_{i}-g_{i}}\right\|\,.$ (1) By assuming only independent loci are chosen and invoking the central limit theorem for the large number of loci genotyped in modern studies, the article [1] asserts that the $Z$-score of $D_{i}$ across all loci will be normally distributed, $\displaystyle T(Y)$ $\displaystyle=\frac{\langle{D_{i}}\rangle-\mu_{0}}{\sqrt{\mathsf{Var}({D_{i}})/s}}=\frac{\langle{D_{i}}\rangle}{\sqrt{\mathsf{Var}({D_{i}})/s}}\sim N(0,1)$ (2) where $\langle{\cdot}\rangle$ denotes the average over all SNPs $i$, $s$ is the number of SNPs, and Eq. 2 exploits the assumption [1] that an individual who is in neither $F$ nor $G$ will be on average equidistant to both under the null hypothesis, i.e., $\mu_{0}=0$. The article [1] proposes using this approach in a forensics context, in which $G$ is a mixture of genetic material of unknown composition (e.g., from a crime scene), and $Y$ is suspect’s genotype; by choosing an appropriate reference sample for group $F$, it is hypothesized that large, positive $T$ will be obtained for individuals whose genotypes are included in $G$, and hence bias $g_{i}$, while individuals whose genotypes are not in $G$ should have insignificant $T$ since they should intuitively be no more similar to the mixture sample $G$ than they are to the reference sample $F$. In [1], the authors applied this test to a multitude of individuals $Y$, each of which are present in the samples constructed by them for $F$ or $G$, and report near-zero false negative rates. The article concludes that it is possible to identify the presence of DNA of specific individuals within a series of highly complex genomic mixtures, and that these “findings show a clear path for identifying whether specific individuals are within a study based on summary-level statistics.” In response, many GWAS data sources have retracted the publicly available frequency data pending further study of this method due to the concern that the privacy of study participants can be compromised. However, because no samples absent from both $F$ and $G$ were used, false positive rates—significant $T$ for individuals neither in $G$ nor $F$—are not assessed in practice; rather, they are simply assumed to follow the nominal false-positive rate $\alpha$ given by quantiles of the putative null distribution in Eq. 2. In this manuscript, we expand on [1] by investigating the method’s robustness to several inherent assumptions: 1. 1. that $F$, $G$, and $Y$ are all i.i.d. samples of the same population $P$ and hence the difference of MAFs $f_{i}$ and $g_{i}$ in the two samples is small; 2. 2. that the loci $i$ are independent, such that the central limit theorem may be invoked in Eq 2; and 3. 3. that an individual $Y_{-}$ in neither $G$ nor $F$ does not have sufficient genotype identity (e.g., via inheritance) to true positive individual $Y_{+}$ that $D_{i}(Y_{-})\approx D_{i}(Y_{+})$ for enough $i$ to bias $T(Y_{-})$. To investigate the effect of these assumptions, we begin with a statement of the problem that [1] attempts to address, analytically derive the effect of deviations from the assumptions, and empirically explore the accuracy of the method in practice using real and simulated genotype data. We conclude with a discussion of the implications of our findings, both in forensics as well as regarding identification of individuals contributing DNA in GWAS. The results presented here reveal that membership classification via Eq. 2 is sensitive to the choice of $F$ and $G$; that even a small amount of LD will alter the distribution of $T$ for null samples; and that individuals who are related to members of $F$ or $G$ are frequently assigned significant $T$ values. Our findings suggest that Eq. 2 will in practice yield a high false- positive rate if used to discern the membership of an individual in a specific sample, and when used for this purpose is likely be accurate only if the above assumptions are exceedingly well-met and the individual $Y$ is believed a priori to be present in exactly one of $F$ or $G$. However, although these findings suggest that Eq. 2 may have limited utility to reliably detect the identity of an individual in $F$ or $G$ without prior knowledge, it may be valuable for verifying that an individual is not in either sample, and we find some suggestion that the metric (Eq. 1) proposed in [1] could perhaps be extended to other genetic-similarity problems (e.g., in ancestry inference). ## 2 Materials and Methods We explore the performance of the method described in [1] both analytically and empirically. For the empirical studies, we attempt to classify real and simulated samples into pools derived from publicly available data sources in order to assess the chances that an individual is mistakenly classified into a group which does not contain his specific genotype. The data used in these tests is described below: ### 2.1 Experimental genotypes and MAFs Real-world genotypes from publicly available data sets were retrieved as follows: 2287 samples with known genotypes were obtained from the Cancer Genomic Markers of Susceptibility (CGEMS) breast cancer study. The samples were sourced as described in [2]. Briefly, the samples comprised 1145 breast cancer cases (sample group C+) and a comparable number (1142) of matched controls (group C–) from the participants of the Nurses Health Study. All the participants were American women of European descent. The samples were genotyped against the Illumnina 550K arrays, which assays over 550,000 SNPs across the genome. To assess the genetic identity shared between samples, we computed the fraction of SNPs with identical alleles for all possible pairs of individuals; none exceeded $0.62$. Additionally, 90 genotypes of individuals of European descent (CEPH) and 90 genotypes of individuals of Yoruban descent (YRI) were obtained from the HapMap Project [3]. In both cases, the 90 individuals were members of 30 family trios comprising two unrelated parents and their offspring. SNPs in common with those assayed by the CGEMS study and located on chromosomes 1–22 were kept in the analysis (sex chromosomes were excluded since the CGEMS participants were uniformly female); a total of 481,482 SNPs met these criteria. Minor allele frequencies for case and control groups were computed from the CGEMS genotypes. Publicly-available minor allele frequencies from the 60 unrelated CEPH individuals were retrieved directly from the HapMap Project [3]. The distribution of MAF differences for each group may be seen in Fig. 1. ### 2.2 Simulated Genotypes I To explore the potential for a sample whose genotype is drawn on $f_{i}$ or $g_{i}$ (without being a member of $F$ or $G$) to be misclassified, five sets of 320 simulated genotypes were created by drawing a genotype for each SNP independently as a pair of Bernoulli trials from given allele frequencies: * S.1: For each locus in each sample, genotypes were drawn on the CGEMS control allele frequencies for that locus. * S.2: For each locus in each sample, genotypes were drawn on the CGEMS case allele frequencies for that locus. * S.3: For each locus in each sample, genotypes were drawn on the HapMap CEPH [3] allele frequencies for that locus. * S.4: For each sample, 50% of the loci were selected at random to have genotypes drawn on CGEMS case frequencies, and the other 50% had genotypes drawn on CGEMS control frequencies. * S.5: For each sample, 50% of the loci were selected at random to have genotypes drawn on HapMap CEPH frequencies, 25% of the the of the loci were selected at random to have genotypes drawn on CGEMS case frequencies, and the other 25% had genotypes drawn on CGEMS control frequencies. ### 2.3 Simulated Genotypes II To further explore the influence of genetic similarity, two other simulation sets were created. Beginning with the MAFs from CGEMS controls, here denoted by $p_{i}$, we create the first set as follows: 1. 1. Draw $f_{i}$ from $\mathsf{Bin}(2000,p_{i})/2000$ to simulate the MAFs of a sample of 1000 individuals; 2. 2. Draw 1000 genotypes on $\mathsf{Bin}(2,p_{i})/2$ to simulate genotypes of 1000 individuals who will comprise $G$; 3. 3. Construct 200 genotypes ($Y$s) for which $q$ percent of SNPs are chosen at random to be identical to a specific $G$ individual (selected at random for each of the 200 samples), and the other $1-q$ fraction of SNPs are drawn on $\mathsf{Bin}(2,p_{i})/2$; 4. 4. Perform step 3 for values of $q$ in 0.01 increments from 0 to 1, thus generating 100 pools of 200 samples each who bear $q$ identity to a true- positive individual, and apply Eqs. 1,2 to classify them against the $F$ and $G$ generated in steps 1 and 2. A second set is created as follows, also using the MAFs from CGEMS controls as $p_{i}$: 1. 1. Draw $f_{i}$, $g_{i}$ independently from $\mathsf{Bin}(2000,p_{i})/2000$ to simulate the MAFs of two samples of 1000 individuals each; 2. 2. Draw 200 genotypes ($Y$s) on $\mathsf{Bin}(2,(1-q)p_{i}+(q)g_{i})/2$ to simulate 200 individuals from a population with MAFs biased toward $G$ by $q$ percent; 3. 3. Perform step 2 for values of $q$ in 0.01 increments from 0 to 1, thus generating 100 pools of 200 samples each to be classified against the $F$ and $G$ generated in step 1. By creating these sets, we ensure that we have samples for which all SNPs are independent in $F$ and $G$, and that $F$ and $G$ are samples of the same underlying population; the classification can then be observed as a function of the similarity parameter $q$ in both cases. ### 2.4 Classification of real and simulated genotypes The method as described in [1] and summarized in the Introduction was implemented using R [4]. Subsets of the real data (Sect. 2.1) and simulated data (Sect. 2.2) described above were classified in a total of 17 tests, starting with a total of 481,382 SNPs and excluding those which did not achieve a minor allele frequency $>$0.05 in both $F$ and $G$ for a given test. A summary of the tests is provided in Table 1. Additionally, a series of 200 tests using $Y$, $f_{i}$, and $g_{i}$ as described in Sect. 2.3 were performed. ## 3 Results We begin with an analytical exploration of the assumptions underlying Eq. 1,2, followed by the results of the tests as described in Methods. ### 3.1 $D_{i}$ and $T$ under the null hypothesis To address the need for a fully rigorous examination of the problem which [1] tries to address, we here attempt to set up an idealized situation to which the theory and methods in [1] apply, and consider the properties of $D_{i}$ and $T$ (Eqs. 1, 2) in that setting versus deviations from that setting. Let us assume an underlying population $P$ with MAFs $p_{i}$ from which samples $F$ (of size $n_{F}$) and $G$ (of size $n_{G}$) are drawn i.i.d. Consider now an additional sample $Y$. The null hypothesis is that $Y$ was drawn from $P$, independent of $F$ and $G$; the alternative of interest is that $Y$ is drawn from $G$ (or, symmetrically, $F$). Under these idealized circumstances, we observe that: $\displaystyle f_{i}$ $\displaystyle\sim$ $\displaystyle\mathsf{Bin}(2n_{F},p_{i})/2n_{F}\,,$ (3) $\displaystyle g_{i}$ $\displaystyle\sim$ $\displaystyle\mathsf{Bin}(2n_{G},p_{i})/2n_{G}\,,$ (4) $\displaystyle y_{i}$ $\displaystyle\sim$ $\displaystyle\mathsf{Bin}(2,p_{i})/2\,,$ (5) where the factors of two are a consequence of each sample possessing two independent alleles per locus. In [1], it is proposed that $T$ (the $Z$-score of $D_{i}$ across all SNPs) follows a standard normal distribution (Eqs. 1,2). This proposition rests upon two assumptions: namely, that the mean $\langle{D_{i}}\rangle$ across all SNPs under the null hypothesis is zero, i.e., $\mu_{0}=0$ in Eq. 2; and that the SNPs $i$ are completely independent such that we can write the variance of the mean as the mean variance, ie, ${\mathsf{Var}({\langle{D_{i}}\rangle})=\mathsf{Var}({D_{i}})/s}$ in the denominator of Eq. 2. Below, we consider sources of deviation from $T\sim N(0,1)$ under the null hypothesis. #### 3.1.1 Deviations from $\mu_{0}=0$ In the large-sample limit, under the null hypothesis, $\lim_{n_{F}\rightarrow\infty}f_{i}=p_{i}\;;\;\;\lim_{n_{G}\rightarrow\infty}g_{i}=p_{i}\,,$ (6) and hence $\lim_{n_{F},n_{G}\rightarrow\infty}D_{i}=\lim_{n_{F},n_{G}\rightarrow\infty}\bigl{(}\left\|{y_{i}-f_{i}}\right\|-\left\|{y_{i}-g_{i}}\right\|\bigr{)}=0\;.$ (7) Intuition might further suggest that since $f_{i}$ and $g_{i}$ are both drawn from binomial distributions which are symmetric about $p_{i}$, any sampling deviations resulting from finite $n_{F},n_{G}$ will fall symmetrically, and hence $\mu_{0}=0$. As we will show below, however, this conclusion is sensitive to two assumptions: 1. 1. that the MAF differences between samples $F$ and $G$, $f_{i}-g_{i}$ are small; 2. 2. that the sample sizes $n_{F}$ and $n_{G}$ are not only large, but comparable. Because the number of SNPs $s$ is quite large, slight deviations away from $\mu_{0}=0$ have the power to shift the location of the null distribution of $T$ considerably, rendering $T$ incomparable to a standard normal unless the true $\mu_{0}$ is known. Consider that the difference in $T$ with and without the $\mu_{0}=0$ assumption is $T-T_{\mu_{0}=0}=\frac{\mu_{0}}{\sqrt{\mathsf{Var}({D_{i}})/s}}\,$ (8) and that because $D_{i}$ ranges on $(-1,1)$, $\max(\mathsf{Var}({D_{i}}))=2$. This means that $\min(T-T_{\mu_{0}=0})=\frac{\sqrt{s}}{\sqrt{2}}\mu_{0}$ (9) which can be quite large for even small values of $\mu_{0}$ since the number of SNPs $s$ is on the order of $10^{5}$. It is thus essential that $\mu_{0}$ be known or controllable. Dependence of $\mu_{0}$ on slight differences in MAFs $f_{i}-g_{i}$. Let us begin by writing the difference between MAFs $f_{i}$ and $g_{i}$ at locus $i$ as $\tau_{i}$, $f_{i}=g_{i}+\tau_{i}\,.$ (10) We can then write $D_{i}=\left\|{y_{i}-g_{i}-\tau_{i}}\right\|-\left\|{y_{i}-g_{i}}\right\|\,,$ (11) and thus $\displaystyle\mu_{0}$ $\displaystyle=\langle{\left\|{y_{i}-g_{i}-\tau_{i}}\right\|-\left\|{y_{i}-g_{i}}\right\|}\rangle\,,$ (12) where $\mu_{0}$ is $\langle{D_{i}}\rangle$ under the null hypothesis. We next make a simplifying assumption: since $p_{i}$ are the minor allele frequencies and thus $0\leq p_{i}\leq 0.5$, and since $f_{i}$ and $g_{i}$ are estimates of $p_{i}$, with few exceptions we will have $0\leq f_{i}\leq 0.5$ and $0\leq g_{i}\leq 0.5$ (eliminating this assumption does not significantly alter the results). Under this assumption we can write $\left\|{y_{i}-g_{i}-\tau_{i}}\right\|-\left\|{y_{i}-g_{i}}\right\|=\begin{cases}\tau_{i}&\text{for $y_{i}=0$;}\\\ -\tau_{i}&\text{for $y_{i}=0.5$;}\\\ -\tau_{i}&\text{for $y_{i}=1$.}\end{cases}$ (13) and hence Eq. 12 may be written $\displaystyle\mu_{0}$ $\displaystyle=\sum_{i}\Bigl{[}\tau_{i}\cdot\mathbb{P}({y_{i}=0\|p_{i}})-\tau_{i}\cdot\mathbb{P}({y_{i}=0.5\|p_{i}})-\tau_{i}\cdot\mathbb{P}({y_{i}=1\|p_{i}})\Bigr{]}\;\mathbb{P}({p_{i}})\;\mathbb{P}({\tau_{i}})\,,$ (14) where $\mathbb{P}({\cdot})$ denotes probability and where we have exploited the fact that because $F$, $G$ are independent samples of $P$, $\tau_{i}$ is independent of $p_{i}$, i.e., $\mathbb{P}({\tau_{i}\|p_{i}})=\mathbb{P}({\tau_{i}})$. Observing that $\displaystyle\mathbb{P}({y_{i}=0\|p_{i}})=(1-p_{i})^{2}\,;$ (15) $\displaystyle\mathbb{P}({y_{i}=0.5\|p_{i}})=2p_{i}(1-p_{i})\,;$ $\displaystyle\mathbb{P}({y_{i}=1\|p_{i}})=p_{i}^{2}\,,$ Eq. 14 becomes $\displaystyle\mu_{0}$ $\displaystyle=\sum_{i}\bigl{(}1-4p_{i}+2p_{i}^{2}\bigr{)}\,\tau_{i}\;\mathbb{P}({p_{i}})\;\mathbb{P}({\tau_{i}})$ (16) $\displaystyle=\langle{(1-4p_{i}+2p_{i}^{2})\,\tau_{i}}\rangle\,,$ (17) which is readily verified by simulation. Eq. 17 implies that when $\tau_{i}$ deviates from zero, either due to systematic differences in $F$ and $G$ (i.e., violation of the assumption that both are drawn on the same population $P$) or due to sampling variation, the location of the null distribution of the test statistic given by Eq. 2 will be shifted by an amount equal to ${\langle{(1-4p_{i}+2p_{i}^{2})\tau_{i}}\rangle\cdot\sqrt{s/\mathsf{Var}({D_{i}})}}$ relative to that under the assumption that $\mu_{0}=0$. It is important to note that the shift is a weighted average of $\tau_{i}$; ie, it depends not only on the differences in MAFs $\tau_{i}$ but also on $p_{i}$, and hence it is not sufficient that $\langle{\tau_{i}}\rangle=0$, since small $\tau_{i}$ will be amplified when $p_{i}$ is small and reduced when $p_{i}$ is large. As a result, predicting the deviation away from $\mu_{0}=0$ to properly calibrate $T$ requires knowing not only $\tau_{i}=f_{i}-g_{i}$, but $p_{i}$ as well. In practice, $\tau_{i}$ is easily calculated (examples of the distribution of $\tau_{i}$ for the CGEMS and HapMap CEPH groups are given in Fig. 1). On the other hand, knowing $p_{i}$ requires making assumptions about the population from which $Y$ is drawn. In the case where $Y$ is, in fact, drawn from a different underlying population than are $F$ and $G$, the $p_{i}$ are difficult to obtain from the given data and the shift in $T$ resulting from Eq. 17 is not readily calculated. (This effect is revealed in the empirical tests shown in Fig. 4, discussed in the empirical results section 3.2.1 below, wherein the HapMap samples are shifted by differing amounts.) Dependence of $\mu_{0}$ on sample sizes $n_{F}$ and $n_{G}$. The effect of deviations from the second assumption above is intuitively obvious: if $n_{G}>n_{F}$, $G$ will better approximate the underlying population $P$ and so will be closer on average to a future sample $Y$. The dependence is derived explicitly in the Appendix. We can demonstrate this effect by simulation, as shown in Fig. 2. Here, we begin by creating $10^{5}$ SNP MAFs $p_{i}$ uniformly distributed on the interval $(0,0.5)$. From these $p_{i}$, we simulate the $g_{i}$ with sample size $n_{G}=1000$ as given by Eq. 4 (i.e., a binomial sample) as well as 200 independent samples $Y$ with $y_{i}$ as given by Eq. 5. By simulating $f_{i}$ per Eq. 3 as $n_{F}$ is varied and computing $\langle{D_{i}}\rangle$ for each sample $Y$ per Eq. 1, we can observe the dependence of $\langle{D_{i}}\rangle$ under the null hypothesis (i.e. $\mu_{0}$) on the sample size of $n_{F}$. A plot of the result is provided in Fig. 2. As seen in the plot and derived explicitly in the Appendix, the dependence in this case varies indirectly with ${n_{F}}$; as expected based on the intuition above, smaller $n_{F}$ leads to larger values of $\langle{D_{i}}\rangle$, indicating that $Y$ is closer to $G$ (the larger, more representative sample of $P$) than it is to $F$. Although the difference is small, $\langle{D_{i}}\rangle/\sqrt{\mathsf{Var}({D_{i}})/s}$ – given in Fig. 2(B) – is quite large, which would lead to a high false-positive rate in practice if the $\mu_{0}=0$ assumption were used and $T$ values compared to the presumed null distribution $N(0,1)$. Thus, we see that as $n_{F}$ decreases, the distribution of $T$ under the null hypothesis diverges from the standard normal distribution, resulting in a higher false positive rate than that predicted by the nominal $\alpha$ from the standard normal. #### 3.1.2 Deviations from $\mathsf{Var}({\langle{D_{i}}\rangle})=\mathsf{Var}({D_{i}})/s$ Invocation of the central limit theorem to compare $T$ to a standard normal distribution (as given in Eq. 2) requires that the variance of the mean of $D_{i}$ be estimable by the mean of the variance, ie, ${\mathsf{Var}({\langle{D_{i}}\rangle})=\mathsf{Var}({D_{i}})/s}$. This, in turn, requires that the $D_{i}$ are uncorrelated. However, if the various $D_{i}$ are correlated—most notably due to linkage disequilibrium—this is no longer true. Specifically, the variance of the mean for $s$ variables $D_{i}$ with variance $\mathsf{Var}({D_{i}})$ and average correlation $\rho$ amongst the distinct $D_{i}$ is given by $\mathsf{Var}({\langle{D_{i}}\rangle})=\left(\frac{1}{s}+\frac{s-1}{s}\rho\right)\mathsf{Var}({D_{i}})\,.$ (18) In the case where the average correlation amongst the $D_{i}$’s is zero, Eq. 18 yields the result which is found in the denominator of Eq. 2; on the other hand, $\rho\neq 0$ generates a $\bigl{(}1+(s-1)\rho\bigr{)}$ multiplicative increase over the correlationless variance. The large number of SNPs $s$ results in little room for any correlation between them: consider that Eq. 18 dictates that for a modest number of SNPs $s=5\cdot 10^{4}$ even a very slight average correlation between all pairs of SNPs $\rho=0.002$ would result in a tenfold increase in $\mathsf{Var}({T})$; for 500K SNPs ($s=5\cdot 10^{5}$), $\rho=0.0002$ causes a a two order of magnitude increase in $\mathsf{Var}({T})$. However, it is impossible to ascertain $\rho$ simply from $y_{i}$, $f_{i}$, and $g_{i}$. Instead, this issue may be addressed by choosing fewer SNPs and assuming that $\rho$ is sufficiently small. ### 3.2 Results of Empirical Tests To demonstrate the results derived in Sect. 3.1 above, as well as to explore the performance of the method in realistic situations, we carried out the computations described by Eqs. 1,2 for various $F$, $G$, and $Y$ as described in Table 1. Distributions of $T$ for each of the 17 tests described in Table 1 are shown in the corresponding figures listed in the table. Bearing in mind the fact that $\left\|{T}\right\|>1.64$ yields a nominal $\alpha$ ($p$-value) of 0.05 and $\left\|{T}\right\|>4.75$ yields a nominal $\alpha=10^{-6}$ when compared to a standard normal distribution, the vast majority of samples we tested which were in neither $F$ nor $G$ were misclassified as being members of one or the other group when using the $\alpha=0.05$ threshold for rejection of the null hypothesis; the misclassification rate was also higher than expected when using a nominal $\alpha=10^{-6}$ threshold. The high false- positive rate in practice is attributable to sensitivity to the assumptions which underlie the method, as described above in Sect. 3.1. We present the results under the assumptions from [1] and then discuss the possibility of improving them based on our analytical and empirical findings. #### 3.2.1 Deviation from putative null distribution Choice of $F$ and $G$. In Sect. 3.1.1, we saw that $T$ will depend on the characteristics of the samples $F$ and $G$. The effect is demonstrated in the results shown in Fig. 3. In these plots, $T$ statistics (Eq. 1, 2) are given for all the CGEMS and S.1–S.5 samples for three choices of $F$ and $G$: * • $F$ = HapMap CEPH, $G$ = CGEMS case; * • $F$ = HapMap CEPH, $G$ = CGEMS control; * • $F$ = CGEMS control, $G$ = CGEMS case. The distribution of minor allele frequencies for each of these three groups (CGEMS cases, controls, and HapMap CEPHs) and the distribution of MAF differences for all three pairs of these groups may be seen in Fig 1. Notably, even though it may reasonably be expected that the HapMap CEPH sample closely resembles the Caucasian subjects in CGEMS, the distributions of the allele frequencies is much more similar in CGEMS cases and CGEMS controls than in either group and HapMap CEPHs. (The most striking difference in the HapMap and CGEMS distributions occurs around $0.5$, where it can be seen that the minor (MAF$<0.5$) allele in the CGEMS samples sometimes has a frequency $>0.5$ in HapMap CEPHs.) Importantly, the width of the the distribution of MAF differences $\tau_{i}=f_{i}-g_{i}$ is much greater when HapMap CEPHs are one of $F/G$: although the mean difference in allele frequencies is quite small (0.0003–0.001) in all cases, $\mathsf{Var}({\tau_{i}})$ is an order of magnitude larger when HapMap CEPH is used as one of the the groups, leading to non-zero $\mu_{0}$ via Eq. 17. Additionally, the sample size of the HapMap group is much smaller than that of CGEMS, thus biasing classification of an unknown sample toward the larger (and hence more representative) CGEMS sample when HapMap is used for one of the groups (cf. Sect. 3.1.1 and Appendix for associated derivations). As expected, using the HapMap CEPHs for $F$ fails to separate the CGEMS case and control distributions, such that CGEMS controls and cases all yield high $T$ (and hence would all be classified as cases) when $G$ = CGEMS cases; the situation is analogous for $G$ = CGEMS controls (Fig. 3, top and center left). Only in the situation where $F$ and $G$ have similar large sample sizes and similar MAFs (when $G$ = CGEMS cases and $F$ = CGEMS controls) is good separation achieved, with the $T$ statistics generally falling on the appropriate side of 0 (Fig. 3, bottom left); even so, 15 of the controls were misclassified as cases. This final case, which achieves 99.4% accuracy using $\left\|{T}\right\|>1.64$ (nominal $\alpha=0.05$), is analogous to the data presented in [1], for which all samples are in either $F$ or $G$. As anticipated, the accuracy of the classification of cases and controls is dependent on the choice of $F$ and $G$. The classification of the 1600 samples described in Sect. 2.2 with the same choices of $F$ and $G$ (right column of Fig. 3) is also instructive. In all three cases, all samples achieve high $T$ statistics despite the fact that they are in neither $F$ nor $G$, frequently with $\left\|{T}\right\|\gg 4.75$, i.e., a nominal $p$-value less than $10^{-6}$. (No simulated sample genotype was identical to any true positive genotype at greater than 62% of loci, comparable to the degree of genetic identity observed in the real samples.) That is to say, the method classifies as positive individuals who possess a genotype $y_{i}$ that is drawn on $f_{i}$ or $g_{i}$, but who are not necessarily in $F$ or $G$. This is unsurprising, since Eqs. 1,2 quantify the degree to which $Y$ is not equidistant from $F$ and $G$. Furthermore, this suggests that relatives of true positives may be misclassified (we consider this below in Sect. 3.2.4). Classification of null samples when $F$ and $G$ are well-chosen. Having observed the sensitivity of the classifier to the appropriate choice of $F$ and $G$, we now explore the classification of samples which are in neither $F$ nor $G$ in the case where $F$ and $G$ are well-chosen. Here, we randomly select 100 cases and 100 controls from CGEMS to form an out-of-pool test sample set comprising 200 individuals, and recompute the MAFs for the remaining 1045 CGEMS cases ($G$) and 1042 CGEMS controls ($F$). (Several such random subsets were created; the results were consistent and hence we present a single representative one.) SNPs were kept subject to the same constraint (MAF$>0.05$ in both $F$ and $G$) as above, and $T$ statistics (Eq. 1, 2) were computed for all the test samples using $f_{i}$ and $g_{i}$ as described. For the positives samples (those in $F$ or $G$), the classifier performs fairly well, correctly classifying 2083 samples (and calling 4 as in neither $F$ nor $G$). However, of the 200 test samples which were in neither $F$ nor $G$, only 62 have $\left\|{T}\right\|<1.64$, and the bulk are misclassified into the reduced group of CGEMS cases. The rate of false positives is thus 69% if $T$ is used as an indicator of group membership under the assumptions in [1] at the nominal $\alpha=0.05$ (see Table 2). A plot of the $T$ values for all samples is given in Fig. 4(A). A similar test, in which HapMap individuals unrelated to the CGEMS participants (90 each from CEPH and YRI groups) were classified against the same subsets of 1045 CGEMS cases ($G$) and 1042 CGEMS controls ($F$), yields similar results: all the YRI individuals and 85/90 of the CEPH individuals were misclassified into the group of CGEMS cases at $\alpha=0.05$; a plot of the $T$ value distributions are given in Fig. 4(B). Selecting a more stringent $\alpha=10^{-6}$ (the minimum reported in [1]) results in a 29.5% false-positive rate amongst the 200 out-of-pool CGEMS samples, 72% false-positive rate amongst HapMap CEPHs, and 100% false-positive rate amongst HapMap YRIs. A summary of the specificity and sensitivities obtained in this test is given in Table 2. The reason for the high false-positive rates in practice despite the stringent nominal false positive rate is clear from the plots Fig. 4(A,B): namely, it can be seen that the putative null distribution (light grey line, $N(0,1)$, cf Eq. 2) does not correspond to the observed distribution for samples for which the null hypothesis is correct, with differences in both the location and width. The overall shift to the right is a product of the small differences in $f_{i}-g_{i}$ which accumulate as given by Eq. 17. Because in this test we happen to know the MAFs $p_{i}$ along with $f_{i}$ and $g_{i}$ for each of the CGEMS samples, we can compute $\mu_{0}$ given by Eq. 17 as $1.133\cdot 10^{-4}$ and verify that, when divided by the average $\sqrt{\mathsf{Var}({D_{i}})/s}\approx 5.6\cdot 10^{-5}$ amongst the samples, the center of the observed null distribution will be at $T\approx 2$. Indeed, visual inspection of Fig. 4(A) shows that shifting each $T$ distribution by -2 would result in $F$, $G$, and null-sample distributions which lie more symmetrically about $T=0$. Note, also, that the HapMap CEPHs and YRIs are shifted by different amounts than are the CGEMS samples, due to the fact that the $p_{i}$’s which underlie the HapMap samples differ from each other and from CGEMS. From this, we can see that samples $Y$ which are not drawn on the same population as $F$ and $G$ may in practice have a high false positive rate. The effect of LD, derived in Sect. 3.1.2, is also seen in these examples. In Fig. 4(B), we observe a narrower distribution of $T$ for the HapMap YRI samples versus the Caucasian CGEMS participants and HapMap CEPHs (the Yoruban individuals, who come from an older population, have lower average LD). The same effect is observed by comparing the distribution of $T$ for the simulated samples in Fig. 3 (for which each SNP was independently sampled and hence have artificially low LD) to those of real populations. #### 3.2.2 Correcting for deviations from $N(0,1)$ Although the empirical false-positive rates obtained the the tests described above are exceedingly high, the distributions of $T$ obtained in Fig. 4(A,B) are nonoverlapping. Hence, one might expect that if one could appropriately calibrate the thresholds of $T$ at which classification is made, the sensitivity and specificity of the test could be considerably improved. (Note that, in practice, one does not know where the true-positive $F$ and $G$ distributions of $T$ lie; this requires the genotypes of the $F$ and $G$ individuals.) Two approaches may be taken toward calibrating classification thresholds for $T$: an analytical approach, based on the results in Sect. 3.1 above; or an empirical approach, based on constructing a null distribution from available samples. As we will see, both these approaches pose substantial difficulties. Analytical approach. In order to correct for the deviations from $N(0,1)$ analytically, we need to know both the location and width of the distribution of $T$ in the non-ideal circumstances under which the test is being conducted. That is, we need to know deviations from $\mu_{0}=0$ resulting from MAF differences $f_{i}-g_{i}$ and sample size differences of $n_{F}$ and $n_{G}$ (cf. Sect. 3.1.1 and Appendix), as well as the average correlation amongst SNPs $\rho$ (cf. Sect. 3.1.2, Eq. 18). Let us first consider the result in Eq. 17, which shows that $\mu_{0}$ in practice will be a function of the MAF differences $\tau_{i}=f_{i}-g_{i}$ as well as the MAFs $p_{i}$ of the population $P$ of which $Y$ is a sample. If we are well-assured that $F$ and $G$ are large samples of the same population $P$ and that $Y$ is also a sample of that population, an average of $f_{i}$ and $g_{i}$ may be used to estimate $p_{i}$ (the $y_{i}$, while necessarily drawn on $p_{i}$, are too small a sample to be a good estimate) and thus obtain $\mu_{0}$. Results of this approach (for the tests shown in Fig. 4 and Table 2) are given in Table 3, in which $p_{i}$ was estimated as $(n_{G}\cdot g_{i}+n_{F}\cdot f_{i})/(n_{G}+n_{F})$ and $\mu_{0}$ was computed according to Eq. 17. A slight improvement in the performance of the method can be seen by comparing the first two columns of Table 2 to those of Table 3. However, the assumption used to compute $p_{i}$ (i.e., that $Y$, $F$, and $G$ are all i.i.d. samples of the same population $P$) is one on which the accuracy of the correction is strongly dependent; consider, for instance, that the $\mu_{0}\approx 1.133\cdot 10^{-4}$ obtained for the simulations in Fig. 4(A,B) and discussed above will produce the appropriate shift $T\approx T_{\mu_{0}=0}-2$ for the 200 CGEMS samples in Fig. 4(A) using this method, but will not centralize the HapMap $T$ distributions in Fig. 4(B) appropriately, because the $f_{i}$ and $g_{i}$ are not good estimates of the MAFs of the populations from which the HapMap samples are drawn. Applying this correction to the HapMap samples (equivalent to moving the HapMap $T$ distribution two units to the left in Fig. 4(B)) results in a misclassification rate of 86% (nominal $\alpha=0.05$) and 44% (nominal $\alpha=10^{-6}$) for the HapMap CEPHs and continued 100% misclassification of all HapMap YRIs. It is thus essential that if the $\mu_{0}$ given by Eq. 17 is to be used, sound estimates of $p_{i}$ need to be obtained. When $Y$ is not a sample of the same population as $F$ or $G$, estimates of $p_{i}$ are unobtainable from $f_{i}$, $g_{i}$ and $y_{i}$ alone, and hence this correction relies upon the assumption that $G$, $F$, and $Y$ are well-matched. The second influence on $\mu_{0}$, described in both Sect. 3.1.1 and the Appendix, is the effect of the sample sizes $n_{F}$ and $n_{G}$. Here, corrections are readily made, provided the sample sizes of $F$ and $G$ are known. In a forensics context, where $G$ is a sample of unknown composition, $n_{G}$ may not be known; on the other hand, in other contexts (such as when using case and control MAFs from a GWAS), sample sizes are known and readily adjusted for. (In this test, $n_{F}\approx n_{G}\approx 1000$, and the correction is negligible.) We also saw in Sect. 3.1.2 and Fig. 4(B) that the distribution of $T$ for null samples will depend on the degree of correlation between the SNPs. To accurately derive the width of the $T$ distribution for null samples, one would need to either select SNPs that yield vanishingly small $\rho$ or know the value of $\rho$ with high accuracy for the population of which $Y$ is a sample so that it can be discounted. The latter option requires knowledge beyond the MAFs of $F$ and $G$ and the genotype of individual $Y$; namely, it requires multiple genotypes from the population $P$ from which $Y$ was drawn such that the average correlation $\rho$ between SNPs can be computed; even with a collection of null genotypes, the computation of the average pairwise correlation for $10^{5}$ SNPs is a computationally unfeasible task. Rather, selecting fewer SNPs in order to reduce LD is a more workable solution; the results of this approach can be seen in Fig. 4(C,D) and in Table 2. Here, 50,000 SNPs were selected, uniformly distributed across of the $481,382$ SNPs used in Fig. 4(A,B). 50,000 SNPs was shown in [1] to be a reasonable lower bound to detect at nominal $\alpha\approx 10^{-5}$ one individual amongst 1000, which is the concentration of true positive individuals in this test. As is clear from Fig. 4, reducing the number of SNPs narrows the distributions considerably, yet at the same time brings them closer together such that the crisp separation previously obtained is reduced. Using this method, we see that the 200 CGEMS samples now have a distribution closer to that of the putative null $N(0,1)$ such that using a threshold of $\alpha=0.05$ yields an improved—yet still larger than nominal—21% false-positive rate while maintaining a high 96.3% true positive rate. However, the misclassification rate is still over 50% for both HapMap samples, and improving these values requires compromising the sensitivity, a direct result of the overlapping $T$ distributions for the $G$ and HapMap samples. Finally, we can consider applying both the SNP reduction and the $\mu_{0}$ correction applied above; the results here are given in the final two columns of Table 3. Because $F$ and $G$ are well-matched and the $\mu_{0}$ correction given by Eq. 17 is slight in the case of these 50,000 SNPs, the correction happens to offer little improvement over that achieved by subsetting the SNPs. Empirical approach. Another potential approach to obtaining a correct null distribution is purely empirical, namely, collecting a set of presumed-null genotypes (called $N$) which can be assumed to be drawn from the same population as $Y$, and determining the distribution of $T$ for the null samples $N$. However, once again the method’s sensitivity to the assumptions are a source of error. To see this, let us once more return to Fig. 4. In these figures, vertical bars represent the 0.05 and 0.95 quantiles of the 200 CGEMS (black), 90 HapMap CEPH (cyan) and 90 HapMap YRI (blue) $T$ distributions. Let us first consider a situation in which we have $f_{i}$ and $g_{i}$, along with an individual $Y$ who is one of the 200 CGEMS samples not in $F$ or $G$, but no other genotypes. We might reasonably turn to publicly available HapMap genotypes as our group $N$ from which we construct an empirical null distribution from which we set thresholds. The lines in Fig. 4(A,C) depict this case. Using thresholds obtained from the HapMap CEPH distribution (cyan lines) still incorrectly classifies half of the 200 CGEMS samples; the false positive rate is yet greater (and the true-positive rate smaller) when using the HapMap YRI distribution. These lines illustrate the importance of choosing for $N$ a sample which closely resembles $Y$—as with the choice of $F$ for a given $G$ in Sect. 3.2.1, HapMap CEPHs are insufficiently similar to CGEMS to provide accurate results, despite the fact that both samples are Americans of European descent. The converse is true as well: if we have $N$, $F$, and $G$ which are well matched—such as illustrated in Fig. 4(B,D), in which $N$, $F$, and $G$ all come from CGEMS data—yet $Y$ is not drawn from the same underlying population as $N/F/G$, the method will incorrectly classify $Y$; roughly a quarter of the HapMap CEPHs and the majority of HapMap YRIs lie outside the thresholds set by the 200 CGEMS samples in Fig. 4(B,D). Once again, this underscores the importance of the assumption that $F$, $G$, and $Y$ are all i.i.d. samples of the same population $P$, and—if a sample $N$ is being used to construct a null distribution empirically—it, too, must be an i.i.d. sample of $P$. Another empirical option is that of simulating genotypes from the $f_{i}$ and $g_{i}$ to simulate $T$ under the alternative hypothesis, with the assumption that the null and alternative hypothesis $T$ distributions do not strongly overlap. However, this method also requires that $F$ and $G$ are large and well-matched samples, since (as can be seen in the top- and middle-right graphs in Fig. 3) poorly-matched $F$ and $G$ will not produce crisply separated distributions. Furthermore, the thresholds derived by this approach will relate not to the false-positive rate but rather to the false-negative rate, i.e., these thresholds would control the power of the test, and the specificity in practice will remain unknown. We have thus seen that small deviations from the assumptions that $F$, $G$, and $Y$ are i.i.d. samples of the same population $P$ can produce false- positive rates which greatly exceed those predicted by the null hypothesis. Even when these sources of error were adjusted for, in our tests we still observed a false positive rate that was higher than expected, such that the false positive rate was never less than 20% in practice for a nominal false- positive rate of 5%, and never less than 13% at a nominal false-positive rate of 0.0001%. While the distributions of $T$ for the $F$, $G$, and various $Y$ samples were observed to be separate in Fig. 4, we find that calibrating the thresholds accurately in absence of genotype information for $F$ and $G$ is not obviously doable. More importantly, it is not clear that, once thresholds are chosen, the empirical specificity could be assessed without additional genotype information from subjects who are well-matched to $F$, $G$ and $Y$. #### 3.2.3 Positive predictive value of the method. The effect of the modest specificity—even in the best of cases described above—on the posterior probability that the individual $Y$ is in $F$ or $G$ is considerable, given that the prior probability is likely to be relatively small in most applications of this method. Let us consider the positive predictive value (PPV), which quantifies the post-test probability that an individual $Y$ with a positive result (i.e., significant $T$) is in $F$ or $G$. This probability depends on the prior probability that the individual is in $F$ or $G$, i.e., on the prevalence of being a member of $F$ or $G$. PPV follows directly from Bayes’ theorem, and is defined as $\mathsf{PPV}=\frac{\mathsf{Sens}\cdot\mathsf{Prev}}{\mathsf{Sens}\cdot\mathsf{Prev}+(1-\mathsf{Spec})(1-\mathsf{Prev})}\,,$ (19) where the PPV is the posterior probability that $Y$ is in $F/G$ given a prior probability of $\mathsf{Prev}$. We can write this equivalently in terms of the positive likelihood ratio $\mathsf{LR}_{+}$, $\displaystyle\mathsf{Posterior\,odds}$ $\displaystyle=\mathsf{LR}_{+}\cdot\mathsf{Prior\,odds}$ (20) $\displaystyle\mathsf{LR}_{+}$ $\displaystyle=\frac{\mathsf{Sens}}{(1-\mathsf{Spec})}\,.$ (21) A plot of PPV vs. prevalence is given in Fig. 5. Even with the best sensitivity (99.23%) and specificity (87%) obtained in our tests—that in which $F$, $G$, and $Y$ were drawn on the same underlying population $P$, $\mu_{0}$ was accurately computed, and a nominal $\alpha=10^{-6}$ was used as a threshold (cf. Table 3)—the prior probability (prevalence) of $Y$ being in $F/G$ needs to exceed 54% in order to achieve a 90% post-test probability that the subject is in $F/G$. For a PPV of 99%, the prior probability needs to exceed 72% for any specificity under 95%, assuming the observed sensitivity of 99%. We thus see the strong need for prior belief that $Y$ is in $F$ or $G$. The difficulty in assessing the (empirical) specificity of the test in absence of additional data makes the posterior probability difficult to ascertain since the false positive rate in practice is much greater than that given by the nominal false-positive rate $\alpha$. Eq. 21 underscores this fact; referring once more to the best result in Tables 2, 3, consider that $LR_{+}$ at 87% specificity and 99% sensitivity is 7.6, versus 990000 if the nominal false-positive rate $\alpha=10^{-6}$ were correct. For prior probability of 1/1000, the first case yields a posterior probability of 1.1/1000, while the second yields a posterior probability of 998/1000. These differences, which are difficult to measure without additional, well-matched null sample genotypes and which depend strongly on the degree to which the assumptions underlying the method are met (consider the differences between the CGEMS and HapMap CEPH specificities in Tables 2, 3), pose a severe limitation on the utility of using Eqs. 1,2 to resolve $Y$’s membership in samples $F$ or $G$. #### 3.2.4 Classification of relatives We now turn to the classification of individuals who are relatives of true positives. As discussed above in Sect. 3.2.1, the results from simulations S.1–S.5 in Fig. 3 suggest that individuals who are genetically similar, but not identical to, the subjects in pools $F$ and $G$, frequently exhibit high $\left\|{T}\right\|$. This effect can be investigated by using HapMap families, since we can reasonably expect that the children will bear a greater resemblance to their parents than their parents do to one another. Recalling that the HapMap pools consist of thirty individual mother-father-offspring pedigrees, we construct pools as follows: * • $F$ = Mothers from pedigrees 1–15 and fathers from pedigrees 1–15 * • $G$ = Children from pedigrees 1–15 and fathers from pedigrees 16–30 and then compute $T$ for mothers and children from pedigrees 16–30 using the same SNP criteria as before. The results of these tests for both the CEPH and YRI pedigrees, given in Fig. 6, are as expected, with the children having a significantly higher distribution of $T$ than the mothers; the $T$ values for all the children were so large that $p$-values $\ll 10^{-16}$ were obtained when comparing to $N(0,1)$. By contrast, 5/15 of the YRI mothers from pedigrees 16–30 and 10/15 of the CEPH mothers from pedigrees 16–30 yielded $\left\|{T}\right\|>1.64$ (with distributions roughly centered about $T=0$). The wider distribution amongst the CEPHS again reflects the effect of LD. In Fig. 6 we can see that the method has the power to resolve three groups: those in a group, those related to members of a group, and those who are neither (as the groups become bigger, and hence more homogeneous, we would expect the distributions to move closer together, as evidenced by the lower range of $T$ for the CGEMS-based tests in Fig. 3). Note, however, that without knowing the distribution of $T$ for true positives (which necessitates knowing the genotypes of true positives) setting a threshold to distinguish between true positives and their relatives is not possible by any of the methods described above. In order to explore the effect of genetic similarity in a controlled, ideal situation for which $F$ and $G$ are known to be samples of the same underlying population and for which all SNPs are known to be independent (i.e., in the ideal situation in which the putative null distribution $N(0,1)$ should hold), we carried out the simulations described in Sect. 2.3. In these simulations, the underlying population $P$ was taken to have MAFs $p_{i}$ as given by the CGEMS controls; $f_{i}$, $g_{i}$, and $y_{i}$ were derived as described in Sect. 2.3 as binomial samples of $p_{i}$. In the first of these simulations, the test samples were constrained to have a proportion $q$ of SNPs identical to a true positive individual, with the remaining SNPs drawn on $p_{i}$. A plot of the false positive rate, defined as the fraction of the 200 simulated samples that achieve significant $\left\|{T}\right\|>1.64$ ($\alpha<0.05$), as the similarity parameter $q$ is varied is shown in Fig. 7. Once simulated samples exceeded $65\%$ identity with a true positive individual, they universally achieve significant $T$, and significant values of $T$ are found over half the time for simulated samples exceeding $60\%$ identity. (It should be noted that of the real samples, no two had $>62\%$ fractional identity.) In the second set of these simulations, the test samples were drawn from a weighted mixture of MAFs: $\displaystyle y_{i}$ $\displaystyle\sim$ $\displaystyle\mathsf{Bin}(2,p^{\prime}_{i})/2\,,$ (22) $\displaystyle p^{\prime}_{i}$ $\displaystyle=$ $\displaystyle(1-q)p_{i}+(q)g_{i}\,,$ (23) i.e., the sample was drawn from MAFs $p^{\prime}_{i}$ which are $q$ percent like $G$ and $(1-q)$ like CGEMS controls (MAFs $p_{i}$). By simulating 200 samples for various $q$, computing $T$ for each sample using the simulated $F$ and $G$, and counting the number of samples that achieve significant $\left\|{T}\right\|>1.64$ at $\alpha=0.05$, we can see how the false positive rate varies with the percentage of $G$. Results are given in Fig 7. The misclassification rate exceeds 50% for $q=0.05$; at $q=0.1$, all samples yield significant $T$. The misclassification of relatives follows directly from the method’s premise. Eqs. 1,2 together answer whether individual’s genotype $y_{i}$ is closer to sample $G$’s MAFs $g_{i}$ than to sample $F$’s MAFs $f_{i}$ than would be expected by chance, and it is unsurprising that a relative of a true member of $G$ would appear closer to $G$ (via Eqs. 1,2) than to $F$. Put another way, $Y$ being a member of $G$ is sufficient but not necessary for $y_{i}$ to be closer (via Eq. 1) to $g_{i}$ than to $f_{i}$; it is possible for other sources of genetic variation to cause $y_{i}$ to be closer $g_{i}$ than to $f_{i}$. We can observe this by turning once again to Fig. 4(A,C), where the dashed red and green lines show that the not-in-$G$ CGEMS cases had a distribution of $T$ closer to the other CGEMS cases $G$, and the not-in-$F$ CGEMS controls had a distribution of $T$ closer to the other CGEMS controls $F$, indicating that small class-specific genetic differences can yield altered values of $T$. The erroneous inferential leap that significant $T$ results from $Y$’s presence in $F$ or $G$ is responsible for the misclassification of relatives as well as for misclassification of non- relatives in the previous examples. ## 4 Discussion and Conclusions In this work, we have further characterized and tested the genetic distance metric initially proposed in [1]. This metric, summarized here by Eqs. 1,2, quantifies the distance of an individual genotype $Y$ with respect to two samples $F$ and $G$ using the marginal minor allele frequencies $f_{i}$ and $g_{i}$ of the two samples and the genotype $y_{i}$. The article [1] proposes to use this metric to infer the presence of the individual in one of the two samples, and the authors demonstrate the utility of their classifier on known positive samples (i.e., samples which are in either $F$ or $G$) showing that in this situation their method yields classifications of high sensitivity. Our investigations reveal that while the sensitivity is quite high (correctly classifying true positives into groups $F$ and $G$) the specificity is considerably less than that predicted by the quantiles of the putative null distribution $N(0,1)$. As a result, Eqs. 1, 2 are severely limited in their utility for discerning $Y$’s presence in samples $F$ or $G$. In this work we have shown that high $T$ values, significant when compared against $N(0,1)$, may be obtained for samples that are in neither of the pools tested under several circumstances: * • when pools $F$ and $G$ are sufficiently dissimilar such that the differences in $f_{i}$ and $g_{i}$ dominate, as seen in Sects. 3.1.1 and 3.2.1 as well as the Appendix; * • when $Y$ is a sample of a different population than are $F$ and $G$, as seen in Sect. 3.2.1; * • when a small amount of average LD is present such that the putative null distribution in Eq. 2 does not hold (due to a violation of the CLT assumption of independence), as seen in Sects. 3.1.2 and 3.2.1; * • and when a sample is genetically similar, but not identical to, individuals comprising $F$ or $G$ (e.g., relatives of true positives), as seen in Sect. 3.2.4. The high false positive rates in the first two cases result from assumptions underlying the putative null distribution which are not met in practice, specifically, that the individual $Y$ along with samples $F$ and $G$ are all i.i.d. samples of the same underlying population $P$, and that the amount of correlation between all $s$ SNPs is vanishingly small. As we saw in Sect. 3.2.1 and 3.2.2, these assumptions are difficult to meet; for instance, HapMap CEPH and CGEMS samples are sufficiently dissimilar that they introduce error in violation of the first assumption, despite the fact that both samples are Americans of European descent. Adjusting for deviations from the putative null distribution also requires making strong assumptions or obtaining additional information, as seen in Sect. 3.2.2. Additionally, the conclusion that high $T$ values result from $Y$’s presence in $G$ relies upon the questionable assumption that individuals in neither $F$ nor $G$ will be equidistant from both, resulting in false positives even when the other assumptions are met. For instance, similarly genotyped individuals (both relatives and simulated samples) are often classified into the same group despite the fact that the other assumptions were met (Sect. LABEL:res4). Amongst non-relatives, even when the thresholds have been adjusted for violations of the above assumptions as in Sect. 3.2.2, Eqs. 1,2 produce misleading classifications at a rate that is considerably greater than expected (21% vs. nominal 5% and 13% vs. nominal 0.0001% in the best cases reported in Table 2). The unpredictable false positive rate in practice, resulting from the difficulty in accurately calibrating the significance of $T$, results in a likelihood ratio (and hence post-test probability) that is also unpredictable, with higher false positive rates yielding lower post-test probabilities. When the prior probability of $Y$’s presence in $F$ or $G$ is modest, strong evidence (i.e., high specificity) is needed to outweigh this prior, which was not achieved in our tests (Sect 3.2.4). On the other hand, when samples were known a priori to be in one of the groups $F$/$G$, Eqs. 1,2 correctly identify the sample of which the individual is part (Sect. 3.2.4). These findings have implications both in forensics (for which the method [1] was proposed) and GWAS privacy (which has become a topic of considerable interest in light of [1]). We briefly consider each: Forensics implications. The stated purpose of the method—namely, to positively identify the presence of a particular individual in a mixed pool of genetic data of unknown size and composition—is difficult to achieve. In this scenario, we have $g_{i}$ (from forensic evidence) and a suspect genotype $y_{i}$. To apply the method, we would need 1) to assume that $Y$ and $G$ are indeed i.i.d. samples of the same population $P$; 2) to obtain a sample $F$ which is also a sample of the underlying population $P$, well-matched in size and composition to $G$; 3) to obtain an estimate of the sample size of $G$ such that sample-size effects can be appropriately discounted; and 4) to assume that the $p$-values at the selected classification thresholds are accurate. We have seen in the Results section the sensitivity to the assumption that $Y$, $F$, and $G$ all come from the same population, the sensitivity to the sample size of $G$, and the difficulties in calibrating thresholds; the high false-positive rates which result from even small violations of these assumptions make it exceedingly likely that an innocent party will be wrongly identified as suspicious; its is even more likely for a relative of an individual whose DNA is present in $G$. GWAS privacy implications. Here the scenario of concern is that of a malefactor with the genotype of one (or many) individuals, and access to the case and control MAFs from published studies; could the malefactor use this method to discern whether one of the genotypes in his possession belongs to a GWAS subject? In this case, $F$ and $G$ are known to be samples of the same underlying population $P$ (due to the careful matching in GWAS), and their sample sizes are large and known. However, the malefactor still needs 1) to assure that $Y$ is a member of this population as well (as shown by the poor results when HapMap samples were classified using CGEMS MAFs) and 2) to assume that the $p$-values at the selected classification thresholds are accurate. Additionally, the prior probability that any of the genotypes in the malefactor’s possession comes from a GWAS subject is likely to be quite small, since GWAS samples are a tiny fraction of the population from which they are drawn. Even if the malefactor were able to narrow down the prior probability to one in three, a sensitivity of 99% and a specificity of 95% is needed to obtain a 90% posterior probability that the individual is truly a participant. On the other hand, if the malefactor does have prior knowledge that the individual $Y$ participated in a certain GWAS but does not know $Y$’s case status, Eqs. 1, 2 permit the malefactor to discover with high accuracy which group $Y$ was in. Additionally, in the case of a priori knowledge, the participant’s genotype is not strictly necessary, since a relative’s DNA will yield a large $T$ score that falls on the appropriate $F/G$ side of null. Despite these limitations, we have found that the distance metric (Eqs. 1, 2) may still have forensic and research utility. It is clear from both our studies and the original paper [1] that the sensitivity is quite high; in the (rare) case that a sample has an insignificant $\left\|{T}\right\|<1.64$, it is very likely that $Y$ is in neither $F$ nor $G$. We can also see that genetically distinct groups have $T$ distributions with little overlap (Fig. 4), and so it may be worth investigating the utility of Eqs. 1,2 for ancestry inference. On this note, let us once more consider the quantity which Eq. 1 measures, namely the distance of $y_{i}$ from $f_{i}$ relative to the distance of $y_{i}$ from $g_{i}$. Referring to Fig. 3 (right column) and Fig. 4(A,C), we can see that samples $Y$ which are more like those in sample $G$ have a distribution that lies to the right of samples which are more similar to $F$, as expected; for example, in Fig. 4(A,C), the distribution of null (not in $F,G$) CGEMS cases (dashed red line) is shifted to the right with respect to the distribution of null CGEMS controls, as might be expected from Eq. 1, i.e., the CGEMS case $Y$s are closer to CGEMS case $G$s than are the CGEMS control $Y$s. Although this difference is not statistically significant, one could imagine that it may be possible to select SNPs for which the shift is significant, i.e., a selection of SNPs for which unknown cases are statistically more likely to be closer (via Eq. 1) to the cases in $G$ and unknown controls are statistically more likely to be closer to the controls in $F$. In this case, a subset of SNPs known to be associated with disease may potentially be used with Eqs. 1, 2 to predict the case status of new individuals; conversely, finding a subset of SNPs which produce significant separations of the test samples may be indicative of a group of SNPs which play a role in disease. Because this type of application would use fewer SNPs and would involve the comparison of two distributions of $T$ (cases $\notin\\{F,G\\}$ vs. controls $\notin\\{F,G\\}$), it may be possible to circumvent some of the problems stemming from the unknown width and location of the null distribution described above; still, much work is needed to investigate this possible application. If successful, the metric proposed in [1], while failing to function as a tool to positively identify the presence of a specific individual’s DNA in a finite genetic sample, may if refined be a useful tool in the analysis of GWAS data. ## Appendix: Dependence of $\mu_{0}$ on the sample size of $F$ and $G$ Consider $\langle{D_{i}}\rangle$ (cf. Eq. 1) under the null hypothesis assumptions that $Y$, $F$, and $G$ are all drawn i.i.d. from the same underlying population $P$ with MAFs $p_{i}$. Writing the probability distribution of $p_{i}$ as $\mathbb{P}({p_{i}})$, $\langle{D_{i}}\rangle$ is given by $\displaystyle\langle{D_{i}}\rangle=$ $\displaystyle\langle{\left\|{y_{i}-f_{i}}\right\|-\left\|{y_{i}-g_{i}}\right\|}\rangle=\langle{\left\|{y_{i}-f_{i}}\right\|}\rangle-\langle{\left\|{y_{i}-g_{i}}\right\|}\rangle$ (A-1) $\displaystyle\begin{split}=&\iiint_{-\infty}^{\infty}\left\|{y_{i}-f_{i}}\right\|\;\mathbb{P}({y_{i}\|p_{i}})\,\mathbb{P}({f_{i}\|p_{i}})\,\mathbb{P}({p_{i}})\,dy_{i}\,df_{i}\,dp_{i}-\\\ &\qquad\qquad-\iiint_{-\infty}^{\infty}\left\|{y_{i}-g_{i}}\right\|\;\mathbb{P}({y_{i}\|p_{i}})\,\mathbb{P}({g_{i}\|p_{i}})\,\mathbb{P}({p_{i}})\,dy_{i}\,dg_{i}\,dp_{i}\;,\end{split}$ (A-2) where we exploit the fact that $Y$, $F$ and $G$ are independent of each other but depend on the underlying population MAFs. The dependence of the first (second) term in Eq. A-2 on $n_{F}$ ($n_{G}$) is derived as follows. First, we note that since each $y_{i}$ is two Bernoulli trials (two alleles) with probability $p_{i}$, we have the following values of $\left\|{y_{i}-f_{i}}\right\|$ with probability $\mathbb{P}({y_{i}\|p_{i}})$ for each allowable value of $y_{i}$: $\left\|{y_{i}-f_{i}}\right\|\cdot\mathbb{P}({y_{i}\|p_{i}})=\begin{cases}\bigr{(}1-f_{i}\bigl{)}\cdot\bigr{(}p_{i}^{2}\bigl{)}&\text{for $y_{i}=1$}\,;\\\ \bigr{(}\left\|{0.5-f_{i}}\right\|\bigl{)}\cdot\bigr{(}2p_{i}(1-p_{i})\bigl{)}&\text{for $y_{i}=0.5$}\,;\\\ \bigr{(}f_{i}\bigl{)}\cdot\bigr{(}(1-p_{i})^{2}\bigl{)}&\text{for $y_{i}=0$}\,.\end{cases}$ (A-3) Moreover, since each $f_{i}$ follows a binomial distribution of size $2n_{F}$ (two alleles per person), we invoke the normal approximation to the binomial for values of $n_{F}>10$ with mean $p_{i}$ and variance $p_{i}(1-p_{i})/(2n_{F})$. Hence: $\displaystyle\mathbb{P}({f_{i}\|p_{i}})$ $\displaystyle=$ $\displaystyle\sqrt{\frac{2n_{F}}{2\pi p_{i}(1-p_{i})}}\,\exp{\left[-\frac{2n_{F}(f_{i}-p_{i})^{2}}{2p_{i}(1-p_{i})}\right]}$ (A-4) $\displaystyle=$ $\displaystyle\frac{A_{F,i}}{\sqrt{\pi}}\exp{\bigl{[}-A_{F,i}^{2}(f_{i}-p_{i})^{2}\bigr{]}}\,,$ (A-5) where we introduce $\displaystyle A_{F,i}=\sqrt{n_{F}/(p_{i}(1-p_{i}))}$ (A-6) to simplify the notation. In consequence, the first term of Eq. A-2 can be written: $\iint_{-\infty}^{\infty}\biggl{[}(1-f_{i})(p_{i}^{2})+(\left\|{0.5-f_{i}}\right\|)(2p_{i}(1-p_{i}))+(f_{i})((1-p_{i})^{2})\biggr{]}\cdot\\\ \cdot\frac{A_{F,i}}{\sqrt{\pi}}\exp{\biggl{[}-A_{F,i}^{2}(f_{i}-p_{i})^{2}\biggr{]}}\mathbb{P}({p_{i}})\,df_{i}\,dp_{i}$ (A-7) and the second term may be written analogously for $G$. The absolute value in Eq. A-7 is dealt with by considering the $f_{i}\geq 0.5$ and $f_{i}\leq 0.5$ cases separately, i.e., treating Eq. A-7 as the sum of integrals $\int_{-\infty}^{\infty}\left[\int_{0.5}^{\infty}\biggl{(}(1-f_{i})(p_{i}^{2})+(f_{i}-0.5)(2p_{i}(1-p_{i}))+(f_{i})((1-p_{i})^{2})\biggr{)}\,\mathbb{P}({f_{i}\|p_{i}})\,df_{i}+\right.\\\ \left.\qquad+\int_{-\infty}^{0.5}\biggl{(}(1-f_{i})(p_{i}^{2})+(0.5-f_{i})(2p_{i}(1-p_{i}))+(f_{i})((1-p_{i})^{2})\biggr{)}\,\mathbb{P}({f_{i}\|p_{i}})\,df_{i}\right]\mathbb{P}({p_{i}})\,dp_{i}$ (A-8) Expanding the polynomials in Eq. A-8 and once more using Eq. A-6 to simplify notation, we rewrite the above as $\int_{-\infty}^{\infty}\frac{A_{F,i}}{\sqrt{\pi}}\left[\int_{0.5}^{\infty}\bigl{(}C_{1}f_{i}+C_{2}\bigr{)}e^{-A_{F,i}^{2}(f_{i}-p_{i})^{2}}df_{i}+\right.\\\ \left.+\int_{-\infty}^{0.5}\bigl{(}C_{3}f_{i}+C_{4}\bigr{)}e^{-A_{F,i}^{2}(f_{i}-p_{i})^{2}}df_{i}\right]\mathbb{P}({p_{i}})\;dp_{i}$ (A-9) where $C_{1},C_{2},C_{3},$ and $C_{4}$ are functions of $p_{i}$ but independent of $f_{i}$: $\displaystyle C_{1}$ $\displaystyle=1-2p_{i}^{2}\,,$ (A-10) $\displaystyle C_{2}$ $\displaystyle=2p_{i}^{2}-p_{i}\,,$ (A-11) $\displaystyle C_{3}$ $\displaystyle=1-4p_{i}+2p_{i}^{2}\,,$ (A-12) $\displaystyle C_{4}$ $\displaystyle=p_{i}\,.$ (A-13) Performing the interior integration in Eq. A-9 yields $\int_{-\infty}^{\infty}\frac{A_{F,i}}{\sqrt{\pi}}\;\Biggl{[}\bigl{(}C_{1}-C_{3})\Biggl{(}\frac{e^{-A_{F,i}^{2}(0.5-p_{i})^{2}}}{2A_{F,i}^{2}}\Biggr{)}+(C_{3}p_{i}+C_{4})\Biggl{(}\frac{\sqrt{\pi}}{A_{F,i}}\Biggr{)}+\Biggr{.}\\\ \Biggl{.}+\Bigl{(}(C_{1}-C_{3})p_{i}+(C_{2}-C_{4})\Bigr{)}\Biggl{(}\frac{\sqrt{\pi}\,\mathrm{erfc}\bigr{(}A_{F,i}(0.5-p_{i})\bigl{)}}{2A_{F,i}}\Biggr{)}\Biggr{]}\;\mathbb{P}({p_{i}})\;dp_{i}\;.$ (A-14) Expanding out the various $C$s as well as $A_{F,i}$, we now have for the first term of $\langle{D_{i}}\rangle$ $\int_{-\infty}^{\infty}\bigl{(}p_{i}(1-p_{i})\bigr{)}\;\Biggl{[}2\sqrt{\frac{p_{i}(1-p_{i})}{\pi\;n_{F}}}\exp{\Biggl{(}-\frac{n_{F}(0.5-p_{i})^{2}}{p_{i}(1-p_{i})}}\Biggr{)}+\\\ +2(1-p_{i})+(2p_{i}-1)\mathrm{erfc}\Biggl{(}\sqrt{\frac{n_{F}(0.5-p_{i})^{2}}{p_{i}(1-p_{i})}}\Biggr{)}\Biggr{]}\;\mathbb{P}({p_{i}})\;dp_{i}\;,$ (A-15) which has an indirect dependence on $n_{F}$. Performing the same integration for the second term in Eq. A-2 yields analogous indirect $n_{G}$ dependence. As a result, when $n_{F}<n_{G}$, the first term is greater than the second, yielding $\langle{D_{i}}\rangle>0$; in the limit $n_{F},n_{G}\rightarrow\infty$, this difference becomes smaller. The dependence is illustrated in Fig. 2A. Here, we assume a uniform distribution of $p_{i}$ on $(0,0.5)$ and construct $10^{5}$ $p_{i}$’s for the underlying population $P$ from which we draw, independently, a sample $G$ of size $n_{G}=1000$ and 200 samples $Y$ from which we estimate $\langle{D_{i}}\rangle$ under the null hypothesis. Sample $F$ is drawn i.i.d. from $P$ with sample sizes ranging from $n_{F}=10$ to $n_{F}=1000$, permitting us to plot $\langle{D_{i}}\rangle$ as $n_{F}$ is varied. The simulation results are shown as circles, overlayed with a plot of Eq. A-2 using the result in Eq. A-15 and assuming the uniform distribution of $p_{i}$. The values for $\langle{D_{i}}\rangle$ obtained from the simulation closely matches those derived from Eq. A-15. In Fig. 2B, the corresponding values of $T$ are presented. ## Acknowledgments This research was supported by the Intramural Research Program of the National Cancer Institute, National Institutes of Health, Bethesda, MD. RB was supported by the Cancer Prevention Fellowship Program, National Cancer Institute, National Institutes of Health, Bethesda, MD. ## References * [1] Nils Homer, Szabolcs Szelinger, Margot Redman, David Duggan, Waibhav Tembe, Jill Muehling, John V Pearson, Dietrich A Stephan, Stanley F Nelson, and David W Craig. Resolving individuals contributing trace amounts of DNA to highly complex mixtures using high-density SNP genotyping microarrays. PLoS Genetics, 4(8):e1000167, 2008. * [2] David J Hunter, Peter Kraft, Kevin B Jacobs, David G Cox, Meredith Yeager, Susan E Hankinson, Sholom Wacholder, Zhaoming Wang, Robert Welch, Amy Hutchinson, Junwen Wang, Kai Yu, Nilanjan Chatterjee, Nick Orr, Walter C Willett, Graham A Colditz, Regina G Ziegler, Christine D Berg, Saundra S Buys, Catherine A McCarty, Heather Spencer Feigelson, Eugenia E Calle, Michael J Thun, Richard B Hayes, Margaret Tucker, Daniela S Gerhard, Joseph F Fraumeni, Robert N Hoover, Gilles Thomas, and Stephen J Chanock. A genome-wide association study identifies alleles in FGFR2 associated with risk of sporadic postmenopausal breast cancer. Nature Genetics, 39(7):870–874. * [3] The International HapMap Consortium. The International HapMap Project. Nature, 426(6968):789–796. * [4] R Development Core Team. A language and environment for statistical computing. Vienna, Austria, 2004. $Y$ individuals \| $F$ population \| $G$ population \| $T$ distribution ---\|---\|---\|--- 1145 CGEMS cases \| 60 unrelated HapMap CEPH \| 1145 CGEMS cases \| Fig. 3 1142 CGEMS controls S.1 – S.5 1145 CGEMS cases \| 60 unrelated HapMap CEPH \| 1142 CGEMS controls \| Fig. 3 1142 CGEMS controls S.1 – S.5 1145 CGEMS cases \| 1142 CGEMS controls \| 1145 CGEMS cases \| Fig. 3 1142 CGEMS controls S.1 – S.5 100 CGEMS cases not in $G$ \| 1042 CGEMS controls \| 1045 CGEMS cases \| Fig. 4 100 CGEMS controls not in $F$ 90 HapMap CEPH 90 HapMap YRI HapMap YRI mothers 16–30 \| HapMap YRI mothers 1–15 and fathers 1–15 \| HapMap YRI children 1–15 and fathers 16–30 \| Fig. 6 HapMap YRI children 16–30 HapMap CEPH mothers 16–30 \| HapMap CEPH mothers 1–15 and fathers 1–15 \| HapMap CEPH children 1–15 and fathers 16–30 \| Fig. 6 HapMap CEPH children 16–30 Table 1: Summary of tests performed. In the last four rows, the numbers refer to the families in the HapMap YRI and CEPH populations, such that child 1 is the offspring of mother 1 and father 1, et cetera. \| 481,382 SNPs \| 50,000 SNPs ---\|---\|--- \| $\alpha=0.05$ \| $\alpha=10^{-6}$ \| $\alpha=0.05$ \| $\alpha=10^{-6}$ Sensitivity \| 99.8% \| 97.5% \| 96.3% \| 36.3% Specificity, 200 CGEMS \| 31.0% \| 70.5% \| 79.0% \| 99.5% Specificity, 90 HapMap CEPH \| 5.5% \| 27.7% \| 45.5% \| 100.0% Specificity, 90 HapMap YRI \| 0.0% \| 0.0% \| 4.4% \| 97.7% Table 2: Empirical sensitivity and specificity for the tests shown in Fig. 4 assuming $\mu_{0}=0$. Classification results are given for two different nominal false positive rates $\alpha=0.05$ and $\alpha=10^{-6}$. \| 481,382 SNPs \| 50,000 SNPs ---\|---\|--- \| $\alpha=0.05$ \| $\alpha=10^{-6}$ \| $\alpha=0.05$ \| $\alpha=10^{-6}$ Sensitivity \| 99.90% \| 99.23% \| 97.36% \| 31.09% Specificity, 200 CGEMS \| 40.0% \| 87.0% \| 78.0% \| 99.5% Specificity, 90 HapMap CEPH \| 14.4% \| 55.5% \| 54.4% \| 100.0% Specificity, 90 HapMap YRI \| 0.0% \| 0.0% \| 7.7% \| 100.0% Table 3: Empirical sensitivity and specificity for the tests shown in Fig. 4 using $\mu_{0}$ as given by Eq. 17 and assuming that $p_{i}=(n_{F}\cdot f_{i}+n_{G}\cdot g_{i})/(n_{F}+n_{G})$. Classification results are given for two different nominal false positive rates $\alpha=0.05$ and $\alpha=10^{-6}$. Figure 1: Distribution of minor allele frequencies (left) and differences (right) in CGEMS cases vs HapMap CEPHs (top), CGEMS controls vs HapMap CEPHs (center), and CGEMS cases vs CGEMS controls (bottom). Note that the distribution of MAF differences is much narrower when comparing CGEMS cases to controls (bottom) than when comparing either to HapMap CEPH. Only SNPs achieving frequencies of 0.05 or more were considered. Figure 2: Observed $\langle{D_{i}}\rangle$ and $T$ values for simulated data with varying sample sizes of $n_{F}$ under the $\mu_{0}=0$ assumption. In A, open circles represent the average $\langle{D_{i}}\rangle$ for each simulation; the solid line is the theoretical $\langle{D_{i}}\rangle$ based on numerical integration of Eq. A-15. In B, boxplots of the observed $T$s for each simulation are given assuming $\mu_{0}=0$; box boundaries correspond to the 0.25 and 0.75 quantiles, and whiskers indicate the 0.05 and 0.95 quantiles ($T$ values outside those limits are shown as square points). Horizontal lines at $T=0$ (green), $T=1.64$ (corresponding to $\alpha=0.05$, in amber), and $T=4.75$ (corresponding to $\alpha=10^{-6}$, in red) are shown for reference; note that for $n_{F}<600$, at least 25% of null samples yield significant $T$ at the nominal $\alpha=0.05$. Figure 3: Distribution of $T$ for real CGEMS samples (left column) and simulated samples S.1–S.5 (right column) using $F$/$G$ pairs as follows: top, $F=$ HapMap CEPHs, $G=$ CGEMS cases; center, $F=$ HapMap CEPHs, $G=$ CGEMS controls; bottom, $F=$ CGEMS controls, $G=$ CGEMS cases. Only SNPs achieving frequencies of 0.05 or more were considered. Note that $\left\|{T}\right\|>1.64$ is significant at the nominal $\alpha=0.05$ level and $\left\|{T}\right\|>4.75$ is significant at the nominal $\alpha=10^{-6}$ under the putative null distribution. Figure 4: Comparison of $T$ distributions for true positive and negative samples vs. putative null, starting with 481,382 SNPs in (A,B) and 50,000 SNPs in (C,D). In all plots, true positive $F$ (1042 CGEMS controls) is shown as a solid green curve, true positive $G$ (1045 CGEMS cases) is shown as a solid red curve, and the putative null $N(0,1)$ is given as a thin grey curve. The dark and light grey regions represent the areas for which the null hypothesis would be accepted at $\alpha=0.05$ and $\alpha=10^{-6}$, respectively. In plots (A,C), CGEMS test samples in neither $F$ nor $G$ (100 CGEMS cases and 100 CGEMS controls) are given by a heavy black curve. The CGEMS case and CGEMS control distributions within this group are shown as dashed red and green lines, respectively. In plots (B,D), $T$ distributions are given for HapMap CEPHs (cyan) and YRIs (blue). Vertical lines mark the 0.05 and 0.95 quantiles of the negative CGEMS samples (black), HapMap CEPHs (cyan), and HapMap YRIs (blue). Figure 5: Positive predictive value (PPV) as a function of prevalence and specificity given 99% sensitivity. In (A), PPV is shown on the $y$ axis and color corresponds to specificity. The black curve depicts the 87% sensitivity line—the best sensitivity obtained in the empirical tests in Tables 2, 3. In (B), PPV is shown by color, and the $y$ axis corresponds to specificity. Figure 6: Distributions of $T$ for out-of-group samples who are related (red line) and unrelated (blue line) to individuals in $G$ for HapMap YRI (A) and HapMap CEPH (B) populations. (C) and (D) show the same distributions as (A) and (B) respectively, with the addition (green line) of individuals who are in $G$ and unrelated to $F$ (i.e., true positives). Dashed black lines indicate the $T$ significance thresholds of $\pm 1.64$ at nominal $\alpha=0.05$. Figure 7: Misclassification rates for samples resembling true positives, as described in Sects. 2.3. In (A), samples were generated which had fractional genotype identity to a specific true positive; the false positive rate is given as a function of the pairwise similarity. In (B), samples drawn on a distribution that is a proportional mixture of $g_{i}$ and the reference population MAFs; the false positive rate is given as a function of the proportion of $g_{i}$.	arxiv-papers	2009-02-09T20:18:03	2024-09-04T02:49:00.478653	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Rosemary Braun, William Rowe, Carl Schaefer, Jinghui Zhang, and\n Kenneth Buetow", "submitter": "Rosemary Braun", "url": "https://arxiv.org/abs/0902.1506" }
0902.1575	# Dynamic sensitivity of photon-dressed atomic ensemble with quantum criticality Jin-Feng Huang Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Department of Physics, Hunan Normal University, Changsha 410081, China Yong Li Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China Jie-Qiao Liao Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100080, China Le-Man Kuang lmkuang@hunnu.edu.cn Key Laboratory of Low- Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Department of Physics, Hunan Normal University, Changsha 410081, China C. P. Sun suncp@itp.ac.cn http://www.itp.ac.cn/~suncp Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China ###### Abstract We study the dynamic sensitivity of an atomic ensemble dressed by a single- mode cavity field (called a photon-dressed atomic ensemble), which is described by the Dicke model near the quantum critical point. It is shown that when an extra atom in a pure initial state passes through the cavity, the photon-dressed atomic ensemble will experience a quantum phase transition, showing an explicit sudden change in its dynamics characterized by the Loschmidt echo of this quantum critical system. With such dynamic sensitivity, the Dicke model can resemble to the cloud chamber for detecting a flying particle by the enhanced trajectory due to the classical phase transition. ###### pacs: 42.50.Nn, 73.43.Nq, 03.65.Yz ## I Introduction The quantum phase transition (QPT) Book occurs at zero temperature when the external parameters of some interacting many-body systems change to reach the critical values. Generally, it is associated with the ground state with energy level crossing and symmetry breaking at the critical points. Recently, it was discovered that, near the quantum critical point the QPT system possesses the ultra-sensitivity in its dynamical evolution Quan2006 . This theoretical prediction has been demonstrated by an NMR experiment NMR . Similar sensitivity exists in some quantum systems Hepp ; Emary2003 ; Zanardi ; Fazio ; Zhang ; Wang possessing QPT. In this paper, we study the dynamic sensitivity of an atomic ensemble in a cavity with a single-mode electromagnetic field (called a photon-dressed atomic ensemble), which is described by the Dicke model Dicke . We assume the atoms in the Dicke model are resonant with the cavity field. When an extra two-level atom in large detuning goes through the cavity field, the frequency of cavity field will be shifted effectively according to the Stark effect so that the photon-dressed atomic ensemble near the QPT will be forced into its critical point. In this situation the dynamic evolution of the Dicke model becomes too sensitive in response to the passage of the extra atom. Here, this dynamic sensitivity is characterized by the Loschmidt echo (LE) LE , which is intrinsically defined by the structure of the photon-dressed atomic ensemble. For a short time approximation, we prove that the LE is just an exponential function of the photon number variance in the photon-dressed atomic ensemble. This finding means that the LE can be experimentally measured by detecting the photon correlation. Its sudden change may imply the passage of an extra atom through the cavity. With this reorganization we will demonstrate that such quantum sensitivity in the Dicke model is very similar to the classical sensitivity of the cloud chamber for detecting a flying particle, which is characterized by the macroscopically observable trajectories enhanced by the classical phase transitions. Figure 1: (Color online) Schematic of a cavity field coupled with an atomic gas consisting of $N$ two-level atoms. An extra detected two-level atom $S$ is injected into cavity field. This paper is organized as follows. In Sec. II, we describe the setup of the quantum critical model based on the Dicke model. The effective Hamiltonian is given in terms of the collective excitation of the atomic ensemble. Then the analytic calculation of LE (or the decoherence of the extra atom) is carried out in Sec. III for the normal and super-radiant phases, respectively, by short-time approximation. In the following Sec. IV we plot some figures to explicitly show the sensitive properties of the LE. In Sec. V, we address the similarity between the dynamic sensitivity of the photon-dressed atomic ensemble induced by an extra atom and the classical cloud chamber. Finally, we draw our conclusion in Sec. VI. The detailed coefficients for Bogoliubov transformation in Sec. IV are given in the Appendix. ## II Model and Hamiltonian As showed in Fig. 1, we consider an atomic ensemble confined in a gas cell coupled with a single-mode cavity field of frequency $\omega$, which is described by the annihilation (creation) operator $a$ ($a^{{\dagger}}$). We use the Pauli matrices $\sigma_{z}^{(j)}=\|e\rangle_{jj}\langle e\|-\|g\rangle_{jj}\langle g\|$, $\sigma_{+}^{(j)}=\|e\rangle_{jj}\langle g\|$, and $\sigma_{-}^{(j)}=\|g\rangle_{jj}\langle e\|$ to describe the atomic transition of the $j$th atom with energy level spacing $\omega_{0}$, where $\left\|e\right\rangle_{j}$ and $\left\|g\right\rangle_{j}$ are the excited and ground states of the $j$th atom, respectively. The system of the atomic ensemble coupled with the single-mode cavity field is described by the Dicke model (hereafter, we take $\hbar=1$), $H_{0}=\omega a^{{\dagger}}a+\sum_{j=1}^{N}\left[\frac{1}{2}\omega_{0}\sigma_{z}^{(j)}+g_{0}(a^{{\dagger}}+a)\left(\sigma_{-}^{(j)}+\sigma_{+}^{(j)}\right)\right].$ (1) Here, for small-dimension atomic gas Emary2003 , we have assumed that all the atoms locate near the origin point and interact with the cavity field with the identical coupling strength $g_{0}$. An extra two-level atom $S$ with transition operators $\sigma_{z}$, $\sigma_{+}$, and $\sigma_{-}$ couples to the original single-mode cavity field with Hamiltonian $H_{I}=\frac{1}{2}\omega_{s}\sigma_{z}+g_{s}(a^{{\dagger}}\sigma_{-}+a\sigma_{+}),$ (2) where we have made a rotating wave approximation. Similarly, $\omega_{s}$ is the transition frequency between the ground state $\left\|g\right\rangle$ and excited state $\left\|e\right\rangle$ of the atom $S$; $g_{s}$ is the corresponding coupling strength. It has been shown that the QPT will occur in the system described by Dicke Hamiltonian (1) Emary2003 , since it keeps Hermitian only for a small coupling strength $g_{0}$. But it is only a model to display QPT in quantum optical system. Actually it could not happen for the realistic atomic, molecular, and optical (AMO) system if the unreasonably ignored two-photon term $A^{2}$ is included Rzazewski1975 . To focus on our main idea in the work, we only regard the Dicke system as a simplified model. We would like to point out that many authors have recognized this problem, but there still exist many explorations by using this simplified model Dicke model . If the atom $S$ is far-off-resonant with the cavity field, that is, the detuning $\Delta_{s}$ ($\equiv\omega_{s}-\omega$) is much larger than the corresponding coupling strength $g_{s}$, i.e., $\|\Delta_{s}\|$ $\gg$ $g_{s}$, then one can use the so-called Fröhlich-Nakajima transformation Frohlich ; Nakajama (or other elimination methods) to obtain the effective total Hamiltonian $\displaystyle H_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle(\omega+\tilde{\delta}\sigma_{z})a^{{\dagger}}a+\frac{1}{2}(\omega_{s}+\tilde{\delta})\sigma_{z}+\frac{\omega_{0}}{2}\sum_{j=1}^{N}\sigma_{z}^{(j)}$ (3) $\displaystyle+\frac{g}{\sqrt{N}}\sum_{j=1}^{N}(a^{{\dagger}}+a)\left(\sigma_{-}^{\left(j\right)}+\sigma_{+}^{\left(j\right)}\right),$ where $\tilde{\delta}\equiv g_{s}^{2}/\Delta_{s}$ and $g\equiv g_{0}\sqrt{N}$. We note that the Fröhlich-Nakajima transformation is equivalent to the approach based on the adiabatical elimination. The Hilbert space of $N$ two-level atoms is spanned by $2^{N}$ basis states. In the current case all the atoms have the same free frequencies and coupling constants with the cavity field, we can consider these atoms being identical. Then the Hilbert space is reduced into a subspace of $(2N+1)$ dimension. In this subspace, Hamiltonian (3) is simplified by introducing the collective atomic operators $J_{\pm}=\sum_{j=1}^{N}\sigma_{\pm}^{\left(j\right)},\hskip 14.22636ptJ_{z}=\frac{1}{2}\sum_{j=1}^{N}\sigma_{z}^{\left(j\right)},$ (4) which obey the following angular momentum commutation relations, $[J_{z},J_{\pm}]=\pm J_{\pm},\hskip 14.22636pt[J_{+},J_{-}]=2J_{z}.$ (5) The collective atomic operator $J_{z}$ denotes the collective population of the atomic gas and $J_{\pm}$ represents the collective transitions. In terms of the above angular momentum operators, Hamiltonian (3) is written as $\displaystyle H_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle(\omega+\tilde{\delta}\sigma_{z})a^{{\dagger}}a+\frac{1}{2}(\omega_{s}+\tilde{\delta})\sigma_{z}$ (6) $\displaystyle+\omega_{0}J_{z}+\frac{g}{\sqrt{N}}(a^{{\dagger}}+a)\left(J_{+}+J_{-}\right),$ which is further reduced to $\displaystyle H_{\mathrm{eff}}$ $\displaystyle=$ $\displaystyle(\omega+\tilde{\delta}\sigma_{z})a^{{\dagger}}a+\omega_{0}b^{{\dagger}}b+\frac{1}{2}(\omega_{s}+\tilde{\delta})\sigma_{z}$ (7) $\displaystyle+g(a^{{\dagger}}+a)\left(b^{{\dagger}}\sqrt{1-b^{{\dagger}}b/N}+h.c.\right)$ (up to constant terms) through making use of the Holstein-Primakoff HP transformation, which represents the angular momentum operators in terms of a single bosonic mode as follows: $\displaystyle J_{+}$ $\displaystyle=$ $\displaystyle b^{{\dagger}}\sqrt{N-b^{{\dagger}}b},$ $\displaystyle J_{-}$ $\displaystyle=$ $\displaystyle\sqrt{N-b^{{\dagger}}b}b,$ $\displaystyle J_{z}$ $\displaystyle=$ $\displaystyle b^{{\dagger}}b-\frac{1}{2}N.$ (8) To see more explicitly the dynamic sensitivity of the photon-dressed atomic ensemble in response to the extra atom, corresponding to different state of the extra atom, the effective Hamiltonian in Eq. (7) reads $H_{\mathrm{eff}}=\left\|g\right\rangle\left\langle g\right\|\otimes H_{g}+\left\|e\right\rangle\left\langle e\right\|\otimes H_{e}$ (9) with $\displaystyle H_{g}$ $\displaystyle=$ $\displaystyle\omega_{g}a^{{\dagger}}a+\omega_{0}b^{{\dagger}}b+g(a^{{\dagger}}+a)$ (10) $\displaystyle\times\left(b^{{\dagger}}\sqrt{1-b^{{\dagger}}b/N}+h.c.\right),$ $\displaystyle H_{e}$ $\displaystyle=$ $\displaystyle\omega_{e}a^{{\dagger}}a+\omega_{0}b^{{\dagger}}b+g(a^{{\dagger}}+a)$ (11) $\displaystyle\times\left(b^{{\dagger}}\sqrt{1-b^{{\dagger}}b/N}+h.c.\right),$ where $\omega_{e}=\omega+\tilde{\delta}$ and $\omega_{g}=\omega-\tilde{\delta}$. Note that in the derivation of the above Hamiltonians (10) and (11), we have discarded some constant terms. ## III Quantum critical Effect Before the extra atom $S$ is sent into the cavity, the photon-dressed atomic ensemble (including the cavity field and the atomic gas) is described by the Dicke Hamiltonian $\displaystyle H_{G}$ $\displaystyle=$ $\displaystyle\omega a^{{\dagger}}a+\omega_{0}b^{{\dagger}}b+g(a^{{\dagger}}+a)$ (12) $\displaystyle\times\left(b^{{\dagger}}\sqrt{1-b^{{\dagger}}b/N}+h.c.\right).$ Comparing Eqs. (10) and (11) with Eq. (12), we find, as a result of the injection of the atom $S$, only the frequency of the optical field changes by a small shift $\tilde{\delta}$ in the dynamic evolution of the photon-dressed atomic ensemble. The photon-dressed atomic ensemble is initially prepared in the ground state $\left\|G\right\rangle$ of Hamiltonian (12) and the extra atom $S$ in a superposed state $\alpha\left\|g\right\rangle+\beta\left\|e\right\rangle$, where the normalization condition requires $\|\alpha\|^{2}+\|\beta\|^{2}=1$. When the extra atom $S$ interacts dispersively with the cavity field, the total system is governed by Hamiltonians (10) and (11) corresponding to the extra atom $S$ in states $\left\|g\right\rangle$ and $\left\|e\right\rangle$, respectively. Then at time $t$ the state of the total system becomes an entanglement one, $\displaystyle\|\Psi(t)\rangle$ $\displaystyle=$ $\displaystyle e^{-iH_{\mathrm{eff}}t}(\alpha\|g\rangle+\beta\|e\rangle)\otimes\|G\rangle$ (13) $\displaystyle=$ $\displaystyle\alpha\|g\rangle\otimes e^{-iH_{g}t}\|G\rangle+\beta\|e\rangle\otimes e^{-iH_{e}t}\|G\rangle$ $\displaystyle\equiv$ $\displaystyle\alpha\|g\rangle\otimes\left\|G_{g}(t)\right\rangle+\beta\|e\rangle\otimes\left\|G_{e}(t)\right\rangle,$ where we have defined $\displaystyle\left\|G_{g}(t)\right\rangle\equiv e^{-iH_{g}t}\|G\rangle,\hskip 14.22636pt\left\|G_{e}(t)\right\rangle\equiv e^{-iH_{e}t}\|G\rangle.$ (14) The generation of the above entanglement is due to the conditional dynamics of the total system. This is to say, corresponding to the detected atom prepared in states $\left\|g\right\rangle$ and $\left\|e\right\rangle$, the evolution of the photon-dressed atomic ensemble will be governed by the Hamiltonians $H_{g}$ and $H_{e}$, respectively. The central task of this paper is to show that the dynamic of the photon-dressed atomic ensemble is sensitive to the state of the extra atom. When the photon-dressed atomic ensemble stays in the vicinity of the QPT, the effect of QPT must impose on the state of the extra atom with some enhancement fashion, like the results in Ref. Quan2006 . This motivates us to study the quantum decoherence of the extra atom near the critical point of the photon-dressed atomic ensemble, which can also reflect the dynamic sensitivity of the photon-dressed atomic ensemble. By tracing over the degree of freedom of the photon-dressed atomic ensemble in evolution state (13), the reduced density matrix $\rho_{s}(t)=\mathtt{Tr}_{a,b}\\{\left\|\Psi\left(t\right)\right\rangle\left\langle\Psi\left(t\right)\right\|\\}$ of the detected atom is obtained as $\rho_{s}(t)=\|\alpha\|^{2}\left\|g\right\rangle\left\langle g\right\|+\|\beta\|^{2}\left\|e\right\rangle\left\langle e\right\|+(D\alpha^{\ast}\beta\left\|e\right\rangle\left\langle g\right\|+h.c.),$ (15) where we have introduced the decoherence factor $D(t)=\left\langle G\right\|\exp(iH_{g}t)\exp\left(-iH_{e}t\right)\left\|G\right\rangle.$ (16) Alternatively, we can investigate the decoherence of the extra atom by examining the so-called LE $L(t)\equiv\left\|D(t)\right\|^{2}$ (17) defined for the dynamic sensitivity of the photon-dressed atomic ensemble. For a short time $t$, the LE can be approximated as $L(t)\approx\left\|\left\langle G\right\|e^{-2i\tilde{\delta}ta^{{\dagger}}a}\left\|G\right\rangle\right\|^{2}.$ (18) The straightforward calculation can give $L(t)\approx\exp\left(-4\gamma\tilde{\delta}^{2}t^{2}\right).$ (19) Here, we have introduced the photon number variance $\gamma\equiv\left\langle\left(a^{{\dagger}}a\right)^{2}\right\rangle-\left\langle a^{{\dagger}}a\right\rangle^{2},$ (20) and the average $\langle\cdot\rangle$ is taken for the ground state $\left\|G\right\rangle$. We point out that, up to the second order of time $t$, the decay rate of the LE depends not only on $t^{2}$, but also on the photon number variance $\gamma$. It is well known that the photon-dressed atomic ensemble described by Dicke Hamiltonian (12) transits from the normal phase to the super-radiant one with the increase in the parameter $g$ from that less than the critical value $g_{c}=\sqrt{\omega\omega_{0}}/2$ to that larger than $g_{c}$. Going across the phase transition point, the ground state of the photon-dressed atomic ensemble experiences a complex change. We can predict that the photon number variance $\gamma$ of the ground state will exhibit some special features at the critical point. According to Eq. (13), we can imagine that the quantum criticality of the photon-dressed atomic ensemble can display which single state $\|g\rangle$ or $\|e\rangle$ that the extra atom stays. When $L(t)$ approaches zero, the photon-dressed atomic ensemble is forced into two orthogonal states $\|G_{g}(t)\rangle$ and $\|G_{e}(t)\rangle$, and thus it behaves as a measurement apparatus to detect the state of the extra atom. In this case, its measurement on the atom will induce the decoherence of the extra atom. In what follows, we will calculate the photon number variance $\gamma$ of the photon-dressed atomic ensemble in two different phases, that is, the normal phase and the super-radiant phase. ### III.1 Dynamic sensitivity in normal phase In this subsection, we explicitly calculate $\gamma$ to investigate the properties of the LE when the photon-dressed atomic ensemble is within the normal phase. In the case of low excitations at thermodynamic limit $N\rightarrow\infty$, Hamiltonian (12) becomes $H_{G}=\omega a^{{\dagger}}a+\omega_{0}b^{{\dagger}}b+g(a^{{\dagger}}+a)(b^{{\dagger}}+b)$ (21) for $\sqrt{1-b^{{\dagger}}b/N}\approx 1$, which is typical to describe two- coupled harmonic oscillators. It is well known that Hamiltonian (21) becomes non-Hermitian in the over-strong coupling region $g>g_{c}$, namely, the Hamiltonian possesses imaginary eigenvalues wagner . This means effective Hamiltonian (21) is ill-defined for $g>g_{c}$. Therefore, we now restrict the Hamiltonian within the so-called normal phase region $g<g_{c}$. Correspondingly, this limited Hamiltonian (21) describes the normal phase of the Dicke model. In the normal phase, Hamiltonian (21) can be diagonalized as $H_{G}=\omega_{A}A^{{\dagger}}A+\omega_{B}B^{{\dagger}}B$ (22) by introducing the polariton operators $A$ ($A^{{\dagger}}$) and $B$ ($B^{{\dagger}}$), which depict the mixed bosonic fields of photons and collective atomic excitations. The eigen-frequencies of the polaritons $A$ and $B$ are $\displaystyle\omega_{A}^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\omega_{0}^{2}+\omega^{2})-\frac{1}{2}\sqrt{(\omega_{0}^{2}-\omega^{2})^{2}+16g^{2}\omega_{0}\omega},$ (23) $\displaystyle\omega_{B}^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(\omega_{0}^{2}+\omega^{2})+\frac{1}{2}\sqrt{(\omega_{0}^{2}-\omega^{2})^{2}+16g^{2}\omega_{0}\omega}.$ (24) It is straightforward to see that $\omega_{A}^{2}<0$ when $g>g_{c}\equiv\sqrt{\omega\omega_{0}}/2$. That is, the eigen-frequency $\omega_{A}$ of mode $A$ becomes a complex number, which means Hamiltonian (22) will be non-Hermitian in the coupling region of $g>g_{c}$. The relations between the operators {$a$, $b$, $a^{{\dagger}}$, $b^{{\dagger}}$} and {$A$, $B$, $A^{{\dagger}},B^{{\dagger}}$} are given by $\displaystyle a^{{\dagger}}$ $\displaystyle=$ $\displaystyle f_{1}A^{{\dagger}}+f_{2}A+f_{3}B^{{\dagger}}+f_{4}B,$ $\displaystyle b^{{\dagger}}$ $\displaystyle=$ $\displaystyle h_{1}A^{{\dagger}}+h_{2}A+h_{3}B^{{\dagger}}+h_{4}B,$ (25) where the concrete forms of coefficients $f_{i}$ and $h_{i}$ ($i=1,2,3,4$) have been given by Ref. Emary2003 . Here we only give the detailed forms of $f_{i}$ in the Appendix. From Eq. (22), we can see that the ground state of the photon-dressed atomic ensemble in the polariton representation is $\|G\rangle=\|0\rangle_{A}\otimes\|0\rangle_{B}\equiv\|00\rangle_{AB}$. Making use of Eqs. (20) and (25), we can obtain the photon number variance $\gamma=2f_{1}^{2}f_{2}^{2}+2f_{3}^{2}f_{4}^{2}+\left(f_{1}f_{4}+f_{2}f_{3}\right)^{2}.$ (26) In the normal phase, all the coefficients $f_{i}$ ($i=1,2,3,4$) are real, then the photon variance is a positive number, which implies the coherence of the extra atom will vanish with time. We have mentioned that Hamiltonian (21) of two-coupled harmonic oscillators can not work well in the over-strong coupling region ($g>g_{c}$). This is because the approximation $\sqrt{1-b^{{\dagger}}b/N}\approx 1$ for the original one [Eq. (12)] can not make sense in this region. Thus, we need to consider a different approximation for Eq. (12) when $g>g_{c}$. ### III.2 Dynamic sensitivity in super-radiant phase Physically, when the atom-light coupling becomes stronger and stronger, the coupled system will acquire a macroscopic excitations of atomic ensemble. And then the system enters into a super-radiant phase when $g>g_{c}$. In this situation, the low-excitation approximation is no longer valid. We can use the coherent state $\left\|\beta\right\rangle$ of the collective atomic operator $b$ to depict these kinds of macroscopic excitations Hepp . To achieve the effective Hamiltonian over such background of macroscopic excitations, we need to do the displacement Hepp ; Emary2003 $b^{{\dagger}}\rightarrow b^{\prime{\dagger}}-\sqrt{\beta}$ (27) (or alternatively, $b^{{\dagger}}\rightarrow b^{\prime{\dagger}}+\sqrt{\beta}$). Correspondingly, we also displace the optical field by $a^{{\dagger}}\rightarrow a^{\prime{\dagger}}+\sqrt{\alpha}$ (28) (or alternatively, $a^{{\dagger}}\rightarrow a^{\prime{\dagger}}-\sqrt{\alpha}$). Here $a^{\prime{\dagger}}$ and $b^{\prime{\dagger}}$ describe quantum fluctuations about the semiclassical steady state Carmichael ; elsewhere, $\sqrt{\alpha}$ and $\sqrt{\beta}$ describe the macroscopic mean fields above $g_{c}$ in the order of $O(\sqrt{N})$ Emary2003 . Then Hamiltonian (12) becomes $\displaystyle H_{G}$ $\displaystyle=$ $\displaystyle\omega_{0}\left[b^{\prime{\dagger}}b^{\prime}-\sqrt{\beta}(b^{\prime{\dagger}}+b^{\prime})+\beta\right]$ (29) $\displaystyle+\omega\left[a^{\prime{\dagger}}a^{\prime}+\sqrt{\alpha}(a^{\prime{\dagger}}+a)+\alpha\right]$ $\displaystyle+g\sqrt{\frac{k}{N}}\left(a^{\prime{\dagger}}+a^{\prime}+2\sqrt{\alpha}\right)$ $\displaystyle\times\left(b^{\prime{\dagger}}\sqrt{\xi}+\sqrt{\xi}b^{\prime}-2\sqrt{\beta}\sqrt{\xi}\right),$ where $\sqrt{\xi}=\sqrt{1-[d^{{\dagger}}d-\sqrt{\beta}(d^{{\dagger}}+d)]/(N-\beta)}$ is introduced. In the thermodynamic limit $N\rightarrow\infty$, for Eq. (29), we follow Emary and Brandes Emary2003 : expand the square root $\sqrt{\xi}$ and keep terms up to the order of $N^{0}$ in the Hamiltonian. Then through choosing the appropriate displacements $\sqrt{\alpha}=\frac{g}{\omega}\sqrt{N(1-\mu^{2})},\hskip 14.22636pt\sqrt{\beta}=\sqrt{\frac{N}{2}(1-\mu)}$ with $\mu=\omega\omega_{0}/4g^{2}$, we can diagonalize Hamiltonian (29) as $H_{G}=\omega_{A}^{\prime}A^{\prime{\dagger}}A^{\prime}+\omega_{B}^{\prime}B^{\prime{\dagger}}B^{\prime}$ (30) by the Bogoliubov transformation $\displaystyle a^{\prime{\dagger}}$ $\displaystyle=$ $\displaystyle f_{1}^{\prime}A^{\prime{\dagger}}+f_{2}^{\prime}A^{\prime}+f_{3}^{\prime}B^{\prime{\dagger}}+f_{4}^{\prime}B^{\prime},$ $\displaystyle b^{\prime{\dagger}}$ $\displaystyle=$ $\displaystyle h_{1}^{\prime}A^{\prime{\dagger}}+h_{2}^{\prime}A^{\prime}+h_{3}^{\prime}B^{\prime{\dagger}}+h_{4}^{\prime}B^{\prime},$ (31) where the coefficients $f_{i}^{\prime}$ and $h_{i}^{\prime}$ ($i=1,2,3,4$) have been given in Ref. Emary2003 . Here we only give the detailed forms of $f_{i}^{\prime}$ in the Appendix. The eigen-frequencies $\omega_{A}^{\prime}$ and $\omega_{B}^{\prime}$ of the polaritons described by the operators $A^{\prime}$ and $B^{\prime}$ are given by $\displaystyle\omega_{A}^{{}^{\prime}2}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left[\frac{\omega_{0}^{2}}{\mu^{2}}+\omega^{2}-\sqrt{\left(\frac{\omega_{0}^{2}}{\mu^{2}}-\omega^{2}\right)^{2}+4\omega^{2}\omega_{0}^{2}}\right],$ (32) $\displaystyle\omega_{B}^{{}^{\prime}2}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left[\frac{\omega_{0}^{2}}{\mu^{2}}+\omega^{2}+\sqrt{\left(\frac{\omega_{0}^{2}}{\mu^{2}}-\omega^{2}\right)^{2}+4\omega^{2}\omega_{0}^{2}}\right].$ (33) It is known that if the coupling strength $g$ exceeds the critical value $g_{c}$, both the above eigen-frequencies are real, but not in the region of $g<g_{c}$. Namely, when $g>g_{c}$, Hamiltonian (30) is Hermitian. In the super-radiant phase, the ground state $\|G\rangle=\|00\rangle_{A^{\prime}B^{\prime}}$ satisfies $A^{\prime}\|G\rangle=B^{\prime}\|G\rangle=0$. Similar to the normal phase, we can calculate the photon number variance in the super-radiate phase as $\displaystyle\gamma$ $\displaystyle=$ $\displaystyle 2f_{1}^{\prime 2}f_{2}^{\prime 2}+2f_{3}^{\prime 2}f_{4}^{\prime 2}+(f_{1}^{\prime}f_{4}^{\prime}+f_{2}^{\prime}f_{3}^{\prime})^{2}$ (34) $\displaystyle+\alpha\left[(f_{1}^{\prime}+f_{2}^{\prime})^{2}+(f_{3}^{\prime}+f_{4}^{\prime})^{2}\right].$ Compared with the case of normal phase, the displacement $\alpha$ of the photon operator appears in the photon number variance. Figure 2: (Color online) 3D diagram of the LE plotted as a function of the time $t$ and the coupling strength $g$ both in the normal phase (the left panel) and in the super-radiant phase (the right panel). Here, in unit of $\omega$, $\omega_{0}=1.44\omega$, $\tilde{\delta}=g_{s}^{2}/\Delta_{s}=0.001\omega$ ($\Delta_{s}=0.1\omega$, $g_{s}=0.01\omega$), the critical point $g_{c}=\sqrt{\omega\omega_{0}}/2=0.6\omega$, the number of atoms $N=100$. ## IV Photon Number Variance for Loschmidt Echo We have separately calculated the LE of the photon-dressed atom ensemble perturbed by an extra atom in two quantum phases: normal phase and super- radiant phase. Our calculations are based on the short time approximation, but it can cover the main character of the QPT of the photon-dressed atomic ensemble induced by the extra atom. As follows, we illustrate the LE versus the coupling strength $g$ and time $t$ by plotting its three-dimensional (3D) contour. Figure 2 shows the LE as a function of the time $t$ and the coupling strength $g$ in the normal and super-radiant phases. It is obvious that the LE, which is calculated from Eqs. (19), (26), and (34), will have a sudden change near the critical point. Its decay is highly enhanced at the critical value $g_{c}$. In the normal phase, the LE decays rapidly to zero as the enlarged coupling strength $g$ of the photon-dressed atomic ensemble approaches the critical point $g_{c}$. In the super-radiant phase, similarly, the LE decays faster as the parameter $g$ decreases to the critical point $g_{c}$. Then the coherence of the extra atom is very sensitive to the dynamical perturbation of the photon-dressed atomic ensemble near the critical point. Meanwhile, in the vicinity of the critical point, the coherence of the extra atom decreases to zero sharply with time at fixed point of $g$. The more nearly the work point $g$ approaches the critical point $g_{c}$, the sharper the decay of the decoherence of the extra atom is. During this process, the detected atom evolves from a pure state to a mixed one. Therefore, we can measure the QPT of the photon-dressed atomic ensemble by exploring the coherence of the detected atom in the photon-dressed atomic ensemble. Figure 3 shows the LE at a fixed time ($\omega t=100$) for the photon-dressed atomic ensemble in both the normal and super-radiant phases. Contrary to the case of the transverse field Ising model, the LE in the present system will not approach $1$ when the coupling strength is much more than the critical point (seen from Fig. 3). The reason is that a large displacement $\sqrt{\alpha}\propto g\sqrt{N}$ appears in the super-radiant phase and will increase as the coupling strength increases. That means a small disparity ($\tilde{\delta}a^{{\dagger}}a$) in the initial Hamiltonian in the super- radiant phase may lead to a large difference (e.g., the decoherence factor will decay faster) after period of long-enough time. As pointed out in Ref. Emary2003 , the so-called quantum chaos always appears in the super-radiant phase. Figure 3: (Color online) The cross section of the 3D surface of the LE in Fig. 2 at $\omega t=100$. For other parameters see Fig. 2. It follows from Eqs. (19), (26), and (37) that, the LE is independent of $N$ in the normal phase. However, the LE depends on the number of the atoms $N$ in the super-radiant phase via $\sqrt{\alpha}\propto$ $\sqrt{N}$. In Fig. 4, the LE is plotted as a function of the coupling strength $g$ with $N=100$, $1000$, and $10000$ respectively. It can be observed from Fig. 4 that the LE line decays faster and faster in the super-radiant phase as the atom number $N$ increases. The reason is the same as that mentioned above. The photon number variance $\gamma$ proportional to the decay rate for the decoherence of the extra atom increases as $N$ increases via approximately $\displaystyle\gamma\propto\alpha\propto g^{2}N.$ (35) Accordingly, the LE decreases with the form $\displaystyle\ln{L}\propto-g^{2}N$ (36) in the super-radiant phase. Thus, as $N\rightarrow\infty$, the decay of the LE will be strongly enhanced at the critical point. Figure 4: (Color online) The LE of the systems for different $N$ at $\omega t=100$. In normal phase, the LE is independent of $N$. In super-radiant phase, $N=100$ (solid line), $1000$ (dashed line), and $10000$ (dotted line), respectively, from up to bottom. For other parameters see Fig. 2. ## V Analog to Cloud Chamber Now we can address the similarity of sensitive dynamics between the present system and the classical cloud chamber. In classical cloud chamber, when a charged particle (or a dust) flies into the cloud chamber, which is filled with supersaturated and supercooled water or alcohol, the water or alcohol vapor will condensate around the flying charged particle (or a dust) and form a liquid droplet, then a track is left. During this process, as a result of the sensitivity in response to the extra particle, the supersaturated vapor staying in the vicinity of the classical phase transition experiences a classical phase transition, transiting from vapor to liquid. In the present investigation, similarly, there exists very sensitive dynamics of the photon-dressed atomic ensemble when a far-off-resonant atom goes through the cavity. In view of the Stark effect, the far-off-resonant atom shifts the frequency of the cavity field. We assume that the photon-dressed atomic ensemble is initially prepared in a state near the quantum critical point of the QPT of the Dicke model. Then the frequency change induced by the far-off-resonant atom will lead the Dicke model to cross the quantum critical point, resulting in a sensitive dynamics of the LE. This quantum effect is similar to the classical phenomenon in the realistic cloud chamber that the vapor in the cloud chamber will condensate around the microscopic detected particle after experiencing the classical phase transition. Therefore, it is possible to realize the quantum version of the cloud chamber effect through observing the sensitive change in the LE of the photon-dressed atomic ensemble. Here, the enhancement of the decay of LE or its sudden change can be regarded as an indicator of the one-atom induced QPT to detect the passage of the atom. This fact properly resembles the cloud chamber effect. In this analogy, the photon-dressed atomic ensemble, which can be tuned to the vicinity of the QPT point, behaves as the supersaturated vapor in the classical cloud chamber, while the enhancement of the decay of LE just resembles the transition from vapor to liquid. Indeed, the LE in our paper is obtained from the decoherence factor for time evolution of the extra atom, but it actually represents the “mark” of this atom on the “cloud chamber” — the photon-dressed atomic ensemble. An obvious reason is that the LE only depends on the parameters of the “chamber” and, thus, is an intrinsic quantity of the chamber. Especially, the extra atom can only provide a small perturbation; thus, the LE is independent of the detected particle. In most of the references we cite, the LE can be defined without the detected particle by the chamber. It is only in our own paper Quan2006 where the detected particle is introduced and it is proved that the decoherence factor of the detected particle is just the LE of the chamber. Thus, the LE is obviously the mark of the detected particle left in the chamber. ## VI Conclusion with a remark In summary, based on the QPT of the Dicke model, we have proposed a quantum critical model to display the ultra-sensitivity of dynamic evolution of a QPT system of a photon-dressed atomic ensemble. We have also pointed out the analog of this one-atom induced QPT to the cloud chamber based on QPT. Frankly we have to point out that such a model can not be implemented easily with the generic AMO system, since the two-photon term could not be simply ignored in the over-strong coupling limit Rzazewski1975 . However, our present study is still heuristic and the toy model covers the principle ideas for QPT inducing the cloud chamber-like effect. Furthermore, with the great development of solid quantum device physics, the Dicke model may be realized in some solid- state systems such as the super-conducting quantum circuits and the nano- mechanical resonators integrated with some qubit array systems. Finally, we would like to mention a reference Carmichael , in which an effective Dicke model was derived in a multilevel atomic ensemble. In this reference, the two-photon term $A^{2}$ may be safely ignored originally; thus, the modified Dicke model based on such a practical setup may be used to display the QPT phenomena we found in this paper. ###### Acknowledgements. We would like thank Shuo Yang for helpful discussions. This work was supported by the National Natural Science Foundation of China with Grants No. 10935010 and No. 10775048, and the National Fundamental Research Program of China with Grants No. 2006CB921205 and No. 2007CB925204. ## Appendix A Coefficients of Bogoliubov transformation ### A.1 Normal phase The coefficients of Bogoliubov transformation in the normal phase are $\displaystyle f_{1,2}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{\cos\theta}{\sqrt{\omega\omega_{A}}}(\omega\pm\omega_{A}),$ $\displaystyle f_{3,4}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{\sin\theta}{\sqrt{\omega\omega_{B}}}(\omega\pm\omega_{B}),$ (37) where the rotating angle in the coordinate-momentum representation $\theta$ is given by $\tan 2\theta=\frac{4g\sqrt{\omega\omega_{0}}}{\omega_{0}^{2}-\omega^{2}}.$ (38) ### A.2 Super-radiant phase The coefficients of Bogoliubov transformation in the super-radiant phase are: $\displaystyle f_{1,2}^{\prime}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{\cos\theta^{\prime}}{\sqrt{\omega\omega_{A}^{\prime}}}(\omega\pm\omega_{A}^{\prime}),$ $\displaystyle f_{3,4}^{\prime}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{\sin\theta^{\prime}}{\sqrt{\omega\omega_{B}^{\prime}}}(\omega\pm\omega_{B}^{\prime}),$ (39) where the analogous rotating angle $\theta^{\prime}$ is $\tan 2\theta^{\prime}=\frac{2\omega\omega_{0}\mu^{2}}{\omega_{0}^{2}-\mu^{2}\omega^{2}}.$ (40) ## References * (1) S. Sachdev, Quantum Phase Transition (Cambridge University Press, Cambridge, 1999). * (2) H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. 96, 140604 (2006). * (3) J. Zhang, X. Peng, N. Rajendran, and D. Suter, Phys. Rev. Lett. 100, 100501 (2008). * (4) K. Hepp and E. H. Lieb, Ann. Phys. (N.Y.) 76, 360 (1973); Phys. Rev. A 8, 2517 (1973); Y. K. Wang and F. T. Hioe, Phys. Rev. A 7, 831 (1973). * (5) C. Emary and T. Brandes, Phys. Rev. Lett. 90, 044101 (2003); Phys. Rev. E 67, 066203 (2003). * (6) P. Zanardi and N. Paunković, Phys. Rev. E 74, 031123 (2006). * (7) D. Rossini, T. Calarco, V. Giovannetti, S. Montangero, and R. Fazio, Phys. Rev. A 75, 032333 (2007). * (8) J. Zhang, F. M. Cucchietti, C. M. Chandrashekar, M. Laforest, C. A. Ryan, M. Ditty, A. Hubbard, J. K. Gamble, and R. Laflamme, Phys. Rev. A 79, 012305 (2009). * (9) L. C. Wang, X. L. Huang, and X. X. Yi, Phys. Lett. A 368, 362 (2007). * (10) R. H. Dicke, Phys. Rev. 93, 99 (1954). * (11) Z. P. Karkuszewski, C. Jarzynski, and W. H. Zurek, Phys. Rev. Lett. 89, 170405 (2002); F. M. Cucchietti, D. A. R. Dalvit, J. P. Paz, and W. H. Zurek, ibid. 91, 210403 (2003); R. A. Jalabert and H. M. Pastawski, ibid. 86, 2490 (2001); T. Gorin, T. Prosen, T. H. Seligman, M. Žnidarič, Phys. Rep. 435, 33 (2006). * (12) K. Rzazewski, K. Wódkiewicz, and W. Zacowicz, Phys. Rev. Lett. 35, 432 (1975). * (13) G. Liberti and R. L. Zaffino, Phys. Rev. A 70 , 033808 (2004); Eur. Phys. J. B 44, 535 (2005); Y. Li, Z. D. Wang, and C. P. Sun, Phys. Rev. A 74, 023815 (2006); D. Tolkunov and D. Solenov, Phys. Rev. B 75, 024402 (2007); G. Chen, X. Wang, J. Q. Liang and Z. D. Wang, Phys. Rev. A 78, 023634 (2008); Y. Li and Z. D. Wang, arXiv:0904.4730. * (14) H. Fröhlich, Phys. Rev. 79, 845 (1950); Proc. R. Soc. London, Ser. A 215, 291 (1952); Adv. Phys. 3, 325 (1954). * (15) S. Nakajima, Adv. Phys. 4, 363 (1955). * (16) T. Holstein, H. Primakoff, Phys. Rev. 58, 1098 (1940). * (17) M. Wagner, Unitary Transformations in Solid State Physics (North-Holland, Amsterdam, 1986). * (18) F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, Phys. Rev. A 75, 013804 (2007).	arxiv-papers	2009-02-10T04:21:03	2024-09-04T02:49:00.487264	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jin-Feng Huang, Yong Li, Jie-Qiao Liao, Le-Man Kuang, C. P. Sun", "submitter": "Jin-Feng Huang", "url": "https://arxiv.org/abs/0902.1575" }
0902.1583	# Asymptotic flatness at spatial infinity in higher dimensions Kentaro Tanabe Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Norihiro Tanahashi Tetsuya Shiromizu Department of Physics, Kyoto University, Kyoto 606-8502, Japan ###### Abstract A definition of asymptotic flatness at spatial infinity in $d$ dimensions ($d\geq 4$) is given using the conformal completion approach. Then we discuss asymptotic symmetry and conserved quantities. As in four dimensions, in $d$ dimensions we should impose a condition at spatial infinity that the “magnetic” part of the $d$-dimensional Weyl tensor vanishes at faster rate than the “electric” part does, in order to realize the Poincare symmetry as asymptotic symmetry and construct the conserved angular momentum. However, we found that an additional condition should be imposed in $d>4$ dimensions. ###### pacs: 04.20.Ha ## I Introduction If one considers an “isolated” system in general relativity, one should impose some asymptotic boundary conditions on gravitational fields. As one of such conditions, there is the asymptotically flat condition, which states that the metric should approach to Minkowski metric at “far away” place from gravitational sources. In order to define the notion of this “far away” covariantly, one often uses the conformal completion method introduced by Penrose Penrose . In this method, physical space-time $M$ is conformally embedded to unphysical space-time $\hat{M}$ with boundary, and this boundary is constituted of spatial infinity and null infinity. Hence, one can define asymptotic flatness, imposing some proper boundary conditions at this spatial infinity or null infinity. In four dimensions, asymptotic flatness at spatial infinity was investigated using the conformal completion method by Ashtekar and Hansen AH . They revealed that asymptotic symmetry at spatial infinity can be reduced to the Poincare symmetry which is a symmetry associated with “background” flat metric, and constructed $4$-momentum and angular momentum. On the other hand, in higher dimensions, there is only a few works about asymptotic structure at spatial infinity Shiromizu:2004jt or null infinity Hollands:2003ie though recently the importance of higher dimensional black holes is increasing in string theory and TeV gravity scenario HDBH ; Review . While in four dimensions, uniqueness theorem was obtained Israel , we cannot prove the uniqueness for stationary black holes (counterexamples are Myers- Perry black hole MP and black ring BR with the same mass and angular momentum) in higher dimensions (although uniqueness was shown in GIS for static black holes). If one would like to classify these higher dimensional black holes using some parameters, the investigation on asymptotic structure at spatial infinity could play a key role. The purpose of this paper is to define asymptotic flatness and investigate asymptotic structure at spatial infinity in higher dimensions, following Ashtekar and Hansen AH . (The reference Shiromizu:2004jt investigates into asymptotic flatness in higher dimensions following Ashtekar and Romano Ashtekar:1991vb . This analysis is useful when one is interested only in spatial infinity. For full understanding of asymptotic structures, however, Ashtekar and Hansen’s work is appropriate.) The rest of this paper is organized as follows. In the section II, we define asymptotic flatness at spatial infinity following Ashtekar and Hansen AH . In the section III, we investigate asymptotic structure: asymptotic symmetry and conserved quantities. Finally, we give a summary and discussion in the section IV. In the appendix A we introduce some important concepts in this literature such as directional dependence, and in the appendix B we summarize basic features of conformal completion taking Minkowski space-time for an example. Some important equations in this literature are derived in the appendix C, and in the appendix D we prove the equivalence of our expressions for conserved quantities with the ADM formulae. ## II Definition We define asymptotic flatness at spatial infinity ($i^{0}$) in $d$ dimensions using the conformal completion method developed by Ashtekar and Hansen in four dimensions AH . In this paper, for simplicity we assume physical space-time $(M,g_{ab})$ satisfies the vacuum Einstein equation $R_{ab}=0$. It is easy to extend our current work to more general non-vacuum cases as long as one focuses on the asymptotically flat space-time. Definition: $d$-dimensional physical space-time $(M,g_{ab})$ will be said to be asymptotically flat at spatial infinity $i^{0}$ if there exists $(\hat{M},\hat{g}_{ab})$, where $\hat{g}_{ab}$ is $C^{>d-4}$ at $i^{0}$ (see Appendix A for the definition of $C^{>n}$), and embedding of $M$ into $\hat{M}$ satisfying the following conditions: 1. 1. $\bar{J}(i^{0})=\hat{M}-M$, where $\bar{J}(i^{0})$ is the closure of the union of chronological future and past of $i^{0}$. 2. 2. There exists a function $\Omega$ on $\hat{M}$ that is $C^{2}$ at $i^{0}$ such that $\hat{g}_{ab}=\Omega^{2}g_{ab}$ on $M$ and $\hat{\nabla}_{a}\hat{\nabla}_{b}\Omega\hat{=}2\hat{g}_{ab}$, $\Omega\hat{=}0$ and $\hat{\nabla}_{a}\Omega\hat{=}0$ at $i^{0}$ on $\hat{M}$. Here, and $\hat{\nabla}_{a}$ is the connection for $\hat{g}_{ab}$, and $\hat{=}$ implies the evaluation on $i^{0}$ (i.e. “$=\lim_{\rightarrow i^{0}}$” is equivalent to “$\hat{=}$”). The first condition requires that, in $\hat{M}$, $i^{0}$ is connected to the points on $M$ only via spacelike curves. The second condition says that $\Omega$ behaves $\sim 1/r^{2}$ near $i^{0}$. This is the same asymptotic behavior as in the Minkowski space-time (see Appendix B). Since we assume $\hat{g}_{ab}$ is $C^{>d-4}$ at $i^{0}$, $\hat{\partial}_{a_{1}}\cdots\hat{\partial}_{a_{(d-3)}}\hat{g}_{bc}$ has directional dependent limit at $i^{0}$ (where $\hat{\partial}$ is flat connection on $i^{0}$). This condition is equivalent to one such that $\Omega^{(5-d)/2}\hat{R}_{abcd}$ has directional dependent limit at $i^{0}$. When we discuss asymptotic structure, we often use the Weyl tensor $\hat{C}_{abcd}$ as asymptotic gravitational fields. Thus, it is convenient to use the latter condition on $\hat{R}_{abcd}$ for the discussions hereafter. ## III asymptotic structure In this section, we show how to derive the asymptotic structure from the asymptotic flatness definition. Firstly, we discuss asymptotic symmetry in the section III.1. We show that the asymptotic symmetry is constituted of the Lorentz group and supertranslation group (infinite group of angular-dependent translation) in higher dimensions. In the section III.2, we define asymptotic fields and study their transformation behavior under supertranslation. We find that supertranslation group reduces to the Poincare group if we impose an additional asymptotic condition $B_{a_{1}a_{2}\cdots a_{d-2}}\hat{=}0$ in the definition of asymptotic flatness. We define conserved quantities ($d$-momentum and angular momentum) associated to this Poincare symmetry in the section III.3. We confirm that the conserved quantities we define agree with the ADM formulae in this section and the appendix D. ### III.1 Asymptotic symmetry The asymptotic symmetry is a group of mappings which conserve asymptotic structure. Here, by asymptotic structure we mean $(\hat{g}_{ab},\Omega^{(4-d)/2}\hat{\partial}\hat{g}_{bc})$ at $i^{0}$, since we impose $C^{>d-4}$ condition on the behavior of $\hat{g}_{ab}$ at $i^{0}$. In order to investigate this asymptotic symmetry, we consider the generator $\hat{\xi}$ of the asymptotic symmetry on $\hat{M}$. This generator $\hat{\xi}$ should be an extension of $\xi$, which is a generator of diffeomorphism on $M$. This extension $\hat{\xi}$ of $\xi$ to $i^{0}$ should satisfy 1. 1. $\hat{\xi}\hat{=}0$ , 2. 2. $\hat{\nabla}_{(a}\hat{\xi}_{b)}\hat{=}0$ , 3. 3. $\hat{\nabla}_{(a}\hat{\xi}_{b)}$ is a $C^{>d-4}$ tensor at $i^{0}$. Roughly speaking, these conditions set the behavior of components of $\hat{\xi}$ near $i^{0}$ as $\hat{\xi}^{a}\,\sim\,\frac{1}{r}+\frac{1}{r^{d-2}}.$ (1) The first condition says that a generator $\hat{\xi}$ does not touch $i^{0}$. The second condition implies that $\hat{\xi}$ is asymptotically a Killing vector, i.e. $\hat{g}_{ab}$ at $i^{0}$ is not changed. Before explaining the meaning of the third condition, let us consider the gauge freedom of the conformal completion. First, let $\omega$ be a function on $\hat{M}$, $C^{>d-4}$ at $i^{0}$ and $\omega\hat{=}1$. Then, a conformal completion such that $\hat{g}^{\prime}_{ab}=(\omega\Omega)^{2}g_{ab}$ is equivalent to $\hat{g}_{ab}=\Omega^{2}g_{ab}$, because $\omega\Omega$ satisfies $\omega\Omega\hat{=}0\,,\,\hat{\nabla}_{a}(\omega\Omega)\hat{=}0\,,\,\hat{\nabla}_{a}\hat{\nabla}_{b}(\omega\Omega)\hat{=}2\hat{g}_{ab}.$ (2) Then, we cannot distinguish these two conformal completions under the asymptotic flatness definition in section II. This gauge freedom $\omega$ of the conformal completion reshuffles the value $\Omega^{(4-d)/2}\hat{\partial}\hat{g}_{bc}$ in the asymptotic structure as $\displaystyle\Omega^{(4-d)/2}\left(\hat{\nabla}_{a}^{\prime}-\hat{\nabla}_{a}\right)\hat{v}_{b}\quad$ $\displaystyle\;\hat{=}\;\frac{1}{\omega}\Bigl{[}\;\;\delta^{c}_{a}\Omega^{(4-d)/2}\hat{\nabla}_{b}\omega$ $\displaystyle+\delta^{c}_{b}\Omega^{(4-d)/2}\hat{\nabla}_{a}\omega$ $\displaystyle- g_{ab}\Omega^{(4-d)/2}\hat{\nabla}^{c}\omega\;\;\Bigr{]}\hat{v}_{c}\;,$ (3) where $\hat{\nabla}_{a}^{\prime}$ is the connection for $\hat{g}_{ab}^{\prime}$ and $\hat{v}_{a}$ is any vector. This equation can also be written as $\Omega^{(4-d)/2}\hat{\nabla}_{a}\hat{\nabla}_{(b}\hat{\xi}_{c)}\;\hat{=}\;2\Omega^{(4-d)/2}(\hat{\nabla}_{a}\omega)\hat{g}_{bc}.$ (4) Thus, asymptotic structure $\Omega^{(4-d)/2}\hat{\partial}\hat{g}_{bc}$ has an ambiguity coming from gauge freedom $\omega$, and this ambiguity is reshuffled by order $1/r^{d-2}$ part of $\hat{\xi}$. Hence, asymptotic symmetry is the group of transformations which does not change the asymptotic structure except for this gauge ambiguity. Then, we call this asymptotic symmetry transformation, which is induced by order $1/r^{d-2}$ component of $\hat{\xi}$, supertranslation group. As any two generators $\hat{\xi}^{1}$, $\hat{\xi}^{2}$ of supertranslation group commute: $\displaystyle\left[\hat{\xi}^{1},\hat{\xi}^{2}\right]^{a}$ $\displaystyle\sim\frac{1}{r^{d-2}}\frac{\partial}{\partial U}r^{2-d}\sim\mathcal{O}\left(\frac{1}{r^{2d-5}}\right)\;,$ (5) supertranslation group is abelian (where we use the fact that the contribution to $\Omega^{(4-d)/2}\hat{\partial}\hat{g}_{bc}$ from $\mathcal{O}(1/r^{2d-5})$ part of $\hat{\xi}$ is only $\mathcal{O}(1/r^{d-3})$, which is regarded as zero at $i^{0}$, and thus that part cannot transform the asymptotic structure). Because of angular dependence of $\omega$, however, supertranslation group has infinite translational directions. In this stage, asymptotic symmetry is not expected to be the Poincare symmetry. ### III.2 Asymptotic fields In order to construct conserved quantities associated with the asymptotic symmetry, we define asymptotic gravitational fields using the Weyl tensor $\hat{C}_{ambn}$ as111 In the definition of the magnetic part of the Weyl tensor (7), the power of $\Omega$ is determined by the following evaluation. Since $a_{1},\cdots,a_{d-3}$ are indices for angular coordinates and $m$ is for the radial coordinate in polar coordinates, one of $p$ and $q$ has to be for the time coordinate $t$ and the other one has to be for an angular coordinate $\varphi$. Each parts in the magnetic part behaves near $i^{0}$ as $\hat{\epsilon}_{a_{1}\cdots a_{d-3}mpq}=\mathcal{O}(\sqrt{-\hat{g}})=\mathcal{O}(r^{2-d})$, $\hat{C}^{p}{}_{qbn}=\mathcal{O}(r^{5-d})$, $g_{tt}=\mathcal{O}(1)$, $g_{\varphi\varphi}=\mathcal{O}(r^{-2})$ and $\hat{\eta}^{a}=\mathcal{O}(1)$. Thus, $\hat{\epsilon}_{a_{1}\cdots a_{d-3}mpq}\hat{C}^{pq}{}_{bn}\hat{\eta}^{m}\hat{\eta}^{n}=\mathcal{O}(r^{9-2d})\sim\Omega^{(2d-9)/2}$, and we have to multiply an inverse of this factor to define a regular quantity. $\hat{E}_{ab}$ is a symmetric traceless tensor since the Weyl tensor is traceless. $\hat{B}_{a_{1}\cdots a_{d-3}b}$ is also a traceless tensor; $\hat{B}_{a_{1}\cdots a_{d-3}b}\hat{g}^{a_{i}a_{j}}=0$ due to antisymmetry of $\hat{\epsilon}$ in Eq. (7); $\hat{B}_{a_{1}\cdots a_{d-3}b}\hat{g}^{a_{i}b}=0$ since it contains $\hat{C}^{[pqb]n}=0$. This $\hat{B}_{a_{1}\cdots a_{d-3}b}$ is antisymmetric on the first $d-3$ indices $a_{i}$ ($i=1,\cdots,d-3$). There are no symmetry between the last index $b$ and the other indices $a_{i}$ in general, though in the four-dimensional case the magnetic part $\hat{B}_{ab}$ is symmetric. $\displaystyle\hat{E}_{ab}\;\hat{=}\;\Omega^{(5-d)/2}\hat{C}_{ambn}\hat{\eta}^{m}\hat{\eta}^{n},$ (6) $\displaystyle\hat{B}_{a_{1}\cdots a_{d-3}b}\;\hat{=}\;\Omega^{(9-2d)/2}\hat{\epsilon}_{a_{1}\cdots a_{d-3}mpq}\hat{C}^{pq}{}_{bn}\hat{\eta}^{m}\hat{\eta}^{n},$ (7) where $\hat{\epsilon}_{a_{1}\cdots a_{d-3}mpq}\equiv\sqrt{-\hat{g}}E_{a_{1}\cdots a_{d-3}mpq}$ is a totally antisymmetric tensor in $\hat{M}$, and we take the convention that $E_{012\cdots d-1}=1$. $\hat{\eta}_{a}\hat{=}\hat{\nabla}_{a}\Omega^{1/2}$ is a normal vector to $\Omega=\text{constant}$ surface which becomes a unit vector at $i^{0}$. We call these asymptotic fields (6) and (7) electric and magnetic parts of the Weyl tensor respectively. As these fields do not have components parallel to $\hat{\eta}^{a}$, we can regard them as fields on a timelike hypersurface $\mathcal{S}$ normal to $\hat{\eta}^{a}$. Firstly, let us derive asymptotic field equations. Using the Bianchi identity in the physical vacuum space-time $\nabla_{[m}C_{ab]cd}=0$, we obtain the following equation in terms of the unphysical space-time quantities: $\hat{\nabla}_{[m}\hat{C}_{ab]cd}=\Omega^{-1}\left(\hat{g}_{c[m}\hat{C}_{ab]pd}\hat{\nabla}^{p}\Omega+\hat{g}_{d[m}\hat{C}_{ab]cp}\hat{\nabla}^{p}\Omega\right)\;.$ (8) It is better to rewrite the left-hand side as $\displaystyle\hat{\nabla}_{[m}\hat{C}_{ab]cd}=\Omega^{-1}\biggl{[}\Omega^{(d-3)/2}\hat{\nabla}_{[m}\\!\Bigl{(}\Omega^{(5-d)/2}\hat{C}_{ab]cd}\Bigr{)}\;\;$ $\displaystyle-\frac{5-d}{2}(\hat{\nabla}_{[m}\Omega)\hat{C}_{ab]cd}$ $\displaystyle\biggr{]},$ (9) since $\Omega^{(5-d)/2}\hat{C}_{abcd}$ have directional dependent limit at $i^{0}$. We project these equations into the timelike hypersurface $\mathcal{S}$, and contract with $\hat{\eta}^{a}$. Then, we get the equations for the electric part $\hat{D}_{a}\hat{E}_{bc}-\hat{D}_{b}\hat{E}_{ac}\;\hat{=}\;(4-d)\hat{h}_{a}^{~{}p}\hat{h}_{b}^{~{}q}\hat{h}_{c}^{~{}r}\Omega^{(5-d)/2}\hat{C}_{pqrm}\hat{\eta}^{m}$ (10) and for the magnetic part $\displaystyle\hat{D}_{b}\hat{B}_{a_{1}\cdots a_{d-3}c}-\hat{D}_{c}$ $\displaystyle\hat{B}_{a_{1}\cdots a_{d-3}b}$ (11) $\displaystyle\hat{=}-(d-3)$ $\displaystyle\Omega^{(9-2d)/2}\hat{\epsilon}_{a_{1}\cdots a_{d-3}}{}^{fpq}\,{}^{(d-1)}\hat{C}_{bcpq}\hat{\eta}_{f}\,,$ where $\hat{h}_{ab}$ is the induced metric on $\mathcal{S}$, and $\hat{D}_{a}\hat{v}_{b}\equiv\Omega^{1/2}\hat{h}_{a}^{~{}p}\hat{h}_{b}^{~{}q}\hat{\nabla}_{p}\hat{v}_{q}$ (12) is a regular differentiation with respect to $\hat{h}_{ab}$ on $\mathcal{S}$. ${}^{(d-1)}\hat{C}_{abcd}$ is the $(d-1)$-dimensional Weyl tensor with respect to $\hat{h}_{ab}$, and $\Omega^{(5-d)/2}\,{}^{(d-1)}\hat{C}^{a}{}_{bcd}$ have a directional dependent limit at $i^{0}$. (For detailed derivations of Eqs. (10) and (11), see Appendix C.1.) Next, in order to see how these fields transform under the supertranslation, we introduce potentials of the Weyl tensor. To do so, we will use the Bianchi identity in the unphysical space-time $\hat{\nabla}_{m}\hat{C}_{abc}{}^{m}+\frac{2(d-3)}{d-2}\hat{\nabla}_{[a}\hat{S}_{b]c}=0\;\;,$ (13) where $\hat{S}_{ab}\equiv\hat{R}_{ab}-\frac{\hat{R}}{2(d-1)}\hat{g}_{ab}\;\;.$ (14) Since we assume $\hat{g}_{ab}$ to be $C^{>d-4}$, $\Omega^{(5-d)/2}\hat{S}_{ab}$ admits directional dependent limit at $i^{0}$. Then, we define potentials as $\displaystyle\hat{E}\;$ $\displaystyle\hat{=}\;\Omega^{(5-d)/2}\hat{S}_{pq}\hat{\eta}^{p}\hat{\eta}^{q}\;\;\;,$ (15) $\displaystyle\hat{Q}_{a}\;$ $\displaystyle\hat{=}\;\Omega^{(5-d)/2}\hat{S}_{pq}\hat{h}_{a}^{~{}p}\hat{\eta}^{q}\;\;,$ (16) $\displaystyle\hat{U}_{ab}\;$ $\displaystyle\hat{=}\;\Omega^{(5-d)/2}\hat{S}_{pq}\hat{h}_{a}^{~{}p}\hat{h}_{b}^{~{}q}\;.$ (17) Using Eqs. (8) and (13), we can write down the electric and the magnetic part in terms of potentials as $\hat{E}_{ab}\,\hat{=}\,\frac{-1}{2(d-2)}\left[\frac{1}{d-3}\hat{D}_{a}\hat{D}_{b}\hat{E}+\hat{E}\hat{h}_{ab}+(4-d)\hat{U}_{ab}\right],$ (18) $\displaystyle\\!\hat{B}_{a_{1}\cdots a_{d-3}b}\,\hat{=}\,\frac{-1}{d-2}$ $\displaystyle\hat{\epsilon}_{a_{1}\cdots a_{d-3}mpq}\hat{\eta}^{m}\Omega^{(4-d)/2}$ $\displaystyle\qquad\times\hat{D}^{p}\left(\hat{U}^{q}_{~{}b}-\frac{1}{d-3}\hat{E}\hat{h}^{q}_{~{}b}\right)$ $\displaystyle\equiv\frac{-1}{d-2}$ $\displaystyle\hat{\epsilon}_{a_{1}\cdots a_{d-3}mpq}\Omega^{(4-d)/2}\hat{\eta}^{m}\hat{D}^{p}\hat{\mathcal{K}}^{q}_{~{}b}\,\,,$ (19) where we define a tensor $\hat{\mathcal{K}}_{ab}$ by Eq. (19). (Eqs. (18) and (19) are derived in Appendix C.2.) Now, we observe transformation behaviors of the asymptotic fields under the supertranslation. In a supertranslational transformation $\hat{g}_{ab}\rightarrow\hat{g}_{ab}^{\prime}=\omega^{2}\hat{g}_{ab}$, where $\omega$ is a $C^{>d-4}$ function ($\omega\hat{=}1$), $\hat{S}_{ab}$ transforms as $\displaystyle\hat{S}_{ab}^{\prime}=\hat{S}_{ab}$ $\displaystyle-(d-2)\omega^{-1}\hat{\nabla}_{a}\hat{\nabla}_{b}\omega$ $\displaystyle+2(d-2)\omega^{-2}(\hat{\nabla}_{a}\omega)(\hat{\nabla}_{b}\omega)$ $\displaystyle+\frac{2-d}{2}\omega^{-2}\hat{g}_{ab}(\hat{\nabla}_{m}\omega)(\hat{\nabla}^{m}\omega)\;.$ (20) Since $\omega$ is $C^{>d-4}$ and $\omega\hat{=}1$, it can be written as $\omega=1+\Omega^{(d-3)/2}\alpha\;,$ (21) where $\alpha$ is a function which has directional dependent limit at $i^{0}$. Then, the potentials $\hat{E}$ and $\hat{U}_{ab}$ transform under the supertranslational transformation as $\displaystyle\hat{E}^{\prime}\;\hat{=}\;\hat{E}-(d-2)(d-3)(d-4)\alpha\;,$ (22) $\displaystyle\hat{U}_{ab}^{\prime}\;\hat{=}\;\hat{U}_{ab}-(d-2)\left(\hat{D}_{a}\hat{D}_{b}\alpha+(d-3)\alpha\hat{h}_{ab}\right)\;.$ (23) To show these equations, we use a relation $\displaystyle\Omega^{(4-d)/2}\hat{\eta}^{a}\hat{\nabla}_{a}\omega$ $\displaystyle\;\hat{=}\;\Omega^{1/2}\hat{\eta}^{a}\hat{\nabla}_{a}\alpha+(d-3)\alpha$ $\displaystyle\;\hat{=}\;(d-3)\alpha\;.$ (24) The second equality in this relation holds since $\alpha$ has directional dependent limit at $i^{0}$ and $\hat{\eta}^{a}\hat{\nabla}_{a}\alpha\hat{=}0$. We note that only $\hat{\nabla}_{a}\hat{\nabla}_{b}\omega$ term of Eq. (20) contributes to the variation of $\hat{E}$ and $\hat{U}_{ab}$. It is easy to check that the electric part does not change in this transformation. On the other hand, the potential of the magnetic part $\hat{\mathcal{K}}_{ab}$ transforms as $\hat{\mathcal{K}}_{ab}^{\prime}\;\hat{=}\;\hat{\mathcal{K}}_{ab}-(d-2)(\hat{D}_{a}\hat{D}_{b}\alpha+\alpha\hat{h}_{ab})\;.$ (25) Hence, the magnetic part $\hat{B}_{a_{1}\cdots a_{d-3}b}$ does change under the supertranslational transformation. ### III.3 Conserved quantities and Poincare symmetry Let us construct conserved quantities and the asymptotic symmetry in this section. First, as in four dimensions, we impose an additional condition $\hat{B}_{a_{1}\cdots a_{d-3}b}\;\hat{=}\;0\;.$ (26) This condition implies that the Taub-NUT charge is zero. Although it is of course possible to consider asymptotically locally Minkowski space-time with $\hat{B}_{a_{1}\cdots a_{d-3}b}\neq 0$, we focus only on asymptotically globally Minkowski space-time in this paper. In order to impose the condition (26) consistently with Eq. (11), we must require a further additional condition $\Omega^{(5-d)/2}\,{}^{(d-1)}\hat{C}^{a}{}_{bcd}\;\hat{=}\;0$ (27) as one of the conditions in the definition of asymptotic flatness. Note that ${}^{(d-1)}C_{abcd}$ vanishes automatically in four dimensions. By the way, the condition (26) is not preserved under the supertranslation. To preserve the condition (26), we realise that one must impose $\hat{D}_{a}\hat{D}_{b}\alpha+\alpha\hat{h}_{ab}\;\hat{=}\;0\;.$ (28) As in four dimensions, we can write down the solution to Eq. (28) as $\alpha=\hat{\omega}_{a}\hat{\eta}^{a}$, where $\hat{\omega}_{a}$ is a fixed vector at $i^{0}$. The number of independent solutions is the number of dimensions. Thus, we can regard the transformation generated by $\alpha$ satisfying Eq. (28) as translation. Then, the asymptotic symmetry reduces to the Poincare group which is constituted of the Lorentz group and the translation group, and we can define conserved quantities associated with this Poincare symmetry. Now, it is ready to define conserved quantities. First, we define $d$-momentum $P_{a}$ for translation $\hat{\omega}^{a}$ as $P_{a}\omega^{a}\equiv\frac{-1}{8\pi G_{d}(d-3)}\int_{S^{d-2}}\\!\\!\\!\\!\\!\\!\\!\hat{E}_{ab}\hat{\omega}^{a}\hat{\epsilon}^{b}{}_{e_{1}\cdots e_{d-2}m}\hat{\eta}^{m}dS^{e_{1}\cdots e_{d-2}},$ (29) where $dS^{e_{1}\cdots e_{d-2}}$ is the volume element on $(d-2)$-dimensional unit sphere $S^{d-2}$ on $i^{0}$. From Eq. (10), we get $\hat{D}_{a}\hat{E}^{ab}\hat{=}0$ since $\hat{E}_{ab}$ is traceless. Then, the integral of Eq. (29) is independent of the choice of time slice at $i^{0}$, and thus $P_{a}\omega^{a}$ is conserved. After tedious calculations, we can show that Eq. (29) agrees with the ADM formula (see Appendix D.1 and D.2). Next, in order to define angular momentum using the magnetic part of the Weyl tensor, we consider the next-to-leading order part of $\hat{B}_{a_{1}\cdots a_{d-3}b}\,$: $\hat{\beta}_{a_{1}\cdots a_{d-3}b}\;\hat{=}\;\Omega^{4-d}\hat{\epsilon}_{a_{1}\cdots a_{d-3}mpq}\hat{C}^{pq}{}_{bn}\hat{\eta}^{m}\hat{\eta}^{n}.$ (30) Since $\hat{\beta}_{a_{1}\cdots a_{d-3}b}$ satisfies $\hat{D}_{b}\hat{\beta}^{ba_{2}\cdots a_{d-3}c}\hat{=}0$ due to Eq. (11) and the traceless property of $\hat{B}_{a_{1}\cdots a_{d-3}b}$, we can define conserved quantity $M_{ab}$ which is regarded as angular momentum: $\displaystyle M_{ab}F^{ab}\equiv\frac{-1}{8\pi G_{d}(d-2)!}\int_{S^{d-2}}\hat{\beta}_{a_{1}\cdots a_{d-3}b}$ $\displaystyle\xi^{a_{1}\cdots a_{d-3}}\;$ (31) $\displaystyle\times\hat{\epsilon}^{b}_{~{}e_{1}\cdots e_{d-2}m}$ $\displaystyle\hat{\eta}^{m}dS^{e_{1}\cdots e_{d-2}},$ where $\xi^{a_{1}\cdots a_{d-3}}\equiv\hat{\epsilon}^{a_{1}\cdots a_{d-3}mpq}\hat{\eta}_{m}F_{pq}$ (32) and $F_{ab}$ is any skew tensor in $\cal S$. The coefficients in (31) so that angular momentum $M_{ab}$ transforms properly under translation $\hat{\omega}_{a}$, including the coefficient: $M_{ab}\rightarrow M_{ab}^{\prime}=M_{ab}+2P_{[a}\hat{\omega}_{b]}\;.$ (33) See Appendix D.3 for details of the coefficient determination. ## IV Summary and discussion In this paper, we gave a definition of asymptotic flatness, and constructed conserved quantities, $d$-momentum and angular momentum in $d$ dimensions. As in four dimensions, by imposing an additional constraints on the behavior of the “magnetic” part of the Weyl tensor, we can remove the supertranslational ambiguity. Then, the asymptotic symmetry of the space-time reduces to the Poincare symmetry, which is a symmetry of “background” flat metric, and we can construct conserved quantities associated with this Poincare symmetry. It can be shown that the expressions of these conserved quantities agree with the ADM formulae. In four dimensions, the additional constraint is only $\hat{B}_{ab}=0$ to realize the Poincare symmetry as the asymptotic symmetry, and it is satisfied if there is a Killing vector in $M$, such as timelike Killing vector $(\partial/\partial t)$ or rotational Killing vector $(\partial/\partial\varphi)$ AM2 . On the other hand, in higher dimensions, due to the evolution equation (11) of $\hat{B}_{a_{1}\cdots a_{d-3}b}$, we need to impose a further condition $\Omega^{(5-d)/2}\,{}^{(d-1)}\hat{C}^{a}{}_{bcd}\hat{=}0$ to remove the supertranslational ambiguity and realize the Poincare symmetry. As in four dimensions, $\hat{B}_{a_{1}\cdots a_{d-3}b}=0$ would be satisfied in stationary or axisymmetric space-time in higher dimensions. However, it might be interesting to investigate asymptotic symmetry under more general conditions which $\Omega^{(5-d)/2}\,{}^{(d-1)}\hat{C}^{a}{}_{bcd}\hat{=}0$ does not hold. In this paper, we focused only on spatial infinity. However, it is interesting to explore the full asymptotic structure including null infinity. As our future work, we would investigate the relationship between the Bondi energy formula at null infinity and Weyl tensor formula in this paper at spatial infinity. We also would like to consider asymptotic structure and its symmetry at null infinity, and investigate its connection to the supertranslation at spatial infinity. Another future issue is the preparation for the uniqueness theorem in stationary black hole space-times. As mentioned in the introduction, at first glance, the uniqueness theorem does not hold in higher dimensions, although there are some partial achievements Hollands ; Morisawa ; Hollands2 ; Morisawa2 . However, we would guess that the reason why we fail to prove it is due to lack of asymptotic boundary conditions. If we can specify the boundary condition appropriately, we will be able to prove the uniqueness theorem. The mass, charge and angular momentum are not enough to specify the black hole space-time uniquely. The additional information for the uniqueness may be higher multipole moments. Therefore, the study on higher multipole moments in stationary space-time will be useful. ###### Acknowledgements. The work of TS was supported by Grant-in-Aid for Scientific Research from Ministry of Education, Science, Sports and Culture of Japan (Nos. 19GS0219 and 20540258). NT was supported by JSPS Grant-in-Aid for Scientific Research No. 20$\cdot$56381\. This work was supported by the Grant-in-Aid for the Global COE Program ”The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. ## Appendix A directional dependence In the conformal completion method, spatial infinity which has a non-zero size in the physical space-time $M$ contracts to a point $i^{0}$ in the unphysical space-time $\hat{M}$. Hence, the definition of differentiability and continuity of physical fields (e.g. electromagnetic fields or gravitational fields) on $i^{0}$ is more subtle. In this appendix, we give the notion of directional dependent limit and $C^{>n}$ class. First, the tensor $\hat{T}^{a\cdots b}_{c\cdots d}$ is said to have directional dependent limit at $i^{0}$ if $\hat{T}^{a\cdots b}_{c\cdots d}$ satisfies the following conditions: 1. 1. $\displaystyle\lim_{\rightarrow i^{0}}\hat{T}^{a\cdots b}_{c\cdots d}=\hat{T}^{a\cdots b}_{c\cdots d}(\hat{\eta})\;,$ where $\hat{\eta}$ is a vector on tangential space at $i^{0}$, which is tangent to the curve arriving at $i^{0}$. 2. 2. The derivative coefficients at $i^{0}$ defined by $\left(\Omega^{1/2}\hat{\nabla}_{e_{1}}\right)\cdots\left(\Omega^{1/2}\hat{\nabla}_{e_{n}}\right)\hat{T}^{a\cdots b}_{c\cdots d}$ are regular. The first condition says that, since $i^{0}$ has a non-zero size ($S^{d-2}$) in $M$, $\hat{T}^{a\cdots b}_{c\cdots d}$ may have an angular dependence even in the limit $r\rightarrow\infty$. The operator $\Omega^{1/2}\hat{\nabla}_{a}$ in the second condition gives regular derivative coefficients, since an application of a derivative operator $\hat{\nabla}_{a}$ in $\hat{M}$ corresponds to a multiplication of $r$ near $i^{0}$ (see Appendix B). The second condition says that these regular derivative coefficients should be finite and regular. Next, we define $C^{>n}$ class. A tensor $\hat{T}^{a\cdots b}_{c\cdots d}$ is $C^{>n}$ at $i^{0}$ if the $n+1$ derivatives of $\hat{T}^{a\cdots b}_{c\cdots d}$ have directional dependent limit at $i^{0}$. For example, when we set $\hat{g}_{ab}$ to be $C^{>n}$ at $i^{0}$, the behavior of $\hat{g}_{ab}$ near $i^{0}$ is $\hat{g}_{ab}\sim\text{const.}+\frac{f(\theta,\varphi,\cdots)}{r^{n+1}}\;,$ (34) where the dots stand for other angular coordinates. ## Appendix B conformal completion for Minkowski space-time In this appendix, we discuss conformal completion for Minkowski space-time. This analysis tells us how we can define asymptotically flat space-time in general. First, we introduce coordinates $(U,V)$ such that $\displaystyle ds^{2}=$ $\displaystyle-dt^{2}+dr^{2}+r^{2}d\Omega_{d-2}^{2}$ $\displaystyle=$ $\displaystyle-dudv+\frac{(u-v)^{2}}{4}d\Omega_{d-2}^{2}$ $\displaystyle=$ $\displaystyle-\frac{dUdV}{\cos^{2}U\cos^{2}V}+\frac{\sin^{2}(U-V)}{4\cos^{2}U\cos^{2}V}d\Omega_{d-2}^{2}\;,$ (35) where $u=t-r=\tan U~{},~{}v=t+r=\tan V\;,$ (36) and $d\Omega_{d-2}^{2}$ is a metric on unit $S^{d-2}$. Let us take $\Omega\equiv\cos U\cos V$ as a conformal factor. In this case, we can see that $\hat{\nabla}_{a}\hat{\nabla}_{b}\Omega\;\hat{=}\;2\hat{g}_{ab}$ (37) holds at $i^{0}$. The unit normal vector $\hat{\eta}_{a}$ to $\Omega=\text{constant}$ surface becomes $\hat{\eta}_{a}\hat{=}\hat{\nabla}_{a}\Omega^{1/2}$ (38) on $i^{0}$. It will be useful for discussions in the main text to look how the differential operators behave: $\hat{\nabla}_{U}\sim\frac{\partial}{\partial U}\sim r^{2}\frac{\partial}{\partial r}\;,$ (39) i.e. an application of $\hat{\nabla}_{a}$ corresponds to a multiplication of $r$. When we say that $\hat{g}_{ab}$ is $C^{>n}$ at $i^{0}$, by the way, we should take differentiation in the coordinates $(U,V)$, and so this condition implies that the metric in the unphysical space-time is given by $\hat{g}_{ab}=\hat{\eta}_{ab}\left(1+\frac{f(\theta,\varphi,\cdots)}{r^{n+1}}\right),$ (40) where $\hat{\eta}_{ab}dx^{a}dx^{b}\equiv- dUdV+\frac{\sin^{2}\left(U-V\right)}{4}d\Omega^{2}_{d-2}$ (41) is the unphysical space-time metric corresponding to the flat metric (35) in the physical space-time. ## Appendix C Derivations of Eqs. (10), (11), (18) and (19) In this appendix, we give detailed derivations of Eqs. (10), (11), (18) and (19). Since we compute quantities only at spatial infinity, we omit “hat” and $\lim_{\rightarrow i^{0}}$ throughout this appendix for convenience. ### C.1 Derivation of Eqs. (10) and (11) First, from Eqs. (8) and (9), we obtain $\displaystyle\Omega^{1/2}\nabla_{[m}\mathcal{X}_{ab]cd}$ $\displaystyle\;\;=\Omega^{-1/2}\Bigl{(}g_{c[m}\mathcal{X}_{ab]pd}\nabla^{p}\Omega+g_{d[m}\mathcal{X}_{ab]cp}\nabla^{p}\Omega$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+\frac{5-d}{2}(\nabla_{[m}\Omega)\mathcal{X}_{ab]cd}\Bigr{)},$ (42) where $\mathcal{X}_{abcd}\equiv\Omega^{(5-d)/2}C_{abcd}$ . Multiplying $\eta^{b}\eta^{d}h_{e}^{m}h_{f}^{a}h_{g}^{c}$ to the above, the left-hand side becomes $\displaystyle\Omega^{1/2}\eta^{b}\eta^{d}h_{e}^{m}h_{f}^{a}h_{g}^{c}\nabla_{[m}\mathcal{X}_{ab]cd}$ $\displaystyle\;\;\;\;=\frac{1}{3}(D_{e}E_{fg}-D_{f}E_{eg})-\frac{1}{3}(2W_{feg}+W_{gef}-W_{gfe}),$ (43) where we used the fact that $\Omega^{1/2}\nabla_{a}\eta_{b}=g_{ab}-\eta_{a}\eta_{b}=h_{ab}\;,$ (44) and the definition $W_{abc}\equiv h_{a}^{e}h_{b}^{f}h_{c}^{g}\mathcal{X}_{efgd}\eta^{d}.$ (45) In addition, we used the fact that $\mathcal{X}_{abcd}$ has directional dependent limit and thus $\eta^{e}\nabla_{e}\mathcal{X}_{abcd}$ vanishes. In the right-hand side of Eq. (42), the second and third terms become $\Omega^{-1/2}\eta^{b}\eta^{d}h_{e}^{m}h_{f}^{a}h_{g}^{c}g_{d[m}\mathcal{X}_{ab]cp}\nabla^{p}\Omega=\frac{2}{3}W_{efg}$ (46) and $\frac{5-d}{2}\Omega^{-1/2}\eta^{b}\eta^{d}h_{e}^{m}h_{f}^{a}h_{g}^{c}\nabla_{[m}\Omega\mathcal{X}_{ab]cd}=\frac{5-d}{3}W_{efg}.$ (47) The first term vanishes since $\nabla^{p}\Omega=2\Omega^{1/2}\eta^{p}$. Finally, we obtain Eq. (10) from Eq. (42), that is $\displaystyle D_{e}E_{fg}-D_{f}E_{eg}$ $\displaystyle=(d-5)W_{feg}+W_{gef}+W_{fge}$ $\displaystyle=(d-4)W_{feg}\;,$ (48) where we used $W_{[abc]}=0$ in the second line. Next, we multiply $\Omega^{(4-d)/2}\mathcal{E}_{a_{1}\cdots a_{d-3}}{}^{fcd}\eta_{f}\eta^{b}h_{g}^{m}h_{h}^{a}$ to Eq. (42), where $\mathcal{E}_{a_{1}\cdots a_{d-3}}{}^{fcd}\equiv h_{a_{1}}^{b_{1}}\cdots h_{a_{d-3}}^{b_{d-3}}\epsilon_{b_{1}\cdots b_{d-3}}{}^{fcd}$, and then obtain $\displaystyle\frac{1}{3}\left(D_{g}B_{a_{1}\cdots a_{d-3}h}-D_{h}B_{a_{1}\cdots a_{d-3}g}\right)$ $\displaystyle-\frac{1}{3}{\cal E}_{a_{1}\cdots a_{d-3}}{}^{fcd}\Omega^{(4-d)/2}$ $\displaystyle\;\;\;\;\times\left(h_{fg}h^{a}_{h}\eta^{b}+h_{g}^{b}h_{h}^{a}\eta_{f}-h_{hf}h_{g}^{a}\eta^{b}-h_{h}^{b}h_{g}^{a}\eta_{f}\right)\mathcal{X}_{abcd}$ $\displaystyle=\frac{1}{3}\Bigl{[}4\left(h_{dg}E_{hc}-h_{dh}E_{gc}\right)+(5-d)h_{g}^{a}h_{h}^{b}\mathcal{X}_{abcd}\Bigr{]}$ $\displaystyle~{}~{}~{}\times\Omega^{(4-d)/2}{\cal E}_{a_{1}\cdots a_{d-3}}{}^{fcd}\eta_{f}\;.$ (49) From this equation, we obtain Eq. (11): $\displaystyle\\!\\!D_{g}B_{a_{1}\cdots a_{d-3}h}-D_{h}B_{a_{1}\cdots a_{d-3}g}$ $\displaystyle=-(d-3)\Omega^{(9-2d)/2}h_{g}^{p}h_{h}^{q}\eta_{f}\mathcal{E}_{a_{1}\cdots a_{d-3}}^{~{}~{}~{}~{}~{}~{}~{}~{}~{}fcd}\,{}^{(d-1)}C_{pqcd}\;,$ (50) where ${}^{(d-1)}C_{abcd}$ is the $(d-1)$-dimensional Weyl tensor on $\Omega=\text{constant}$ surface at $i^{0}$. To transform Eq. (49) to Eq. (50), we used the following relations $\displaystyle{\cal E}_{a_{1}\cdots a_{d-3}}{}^{fcd}h_{hf}h_{g}^{a}\eta^{b}\mathcal{X}_{abcd}$ $\displaystyle~{}={\cal E}_{a_{1}\cdots a_{d-3}}{}^{fcd}h_{hf}h_{g}^{a}\eta^{b}(h_{c}^{i}+\eta_{c}\eta^{i})(h_{d}^{j}+\eta_{d}\eta^{j})\mathcal{X}_{abij}$ $\displaystyle~{}=2{\cal E}_{a_{1}\cdots a_{d-3}}{}^{fcd}h_{hf}E_{gc}\eta_{d}$ (51) and $\displaystyle\mathcal{X}_{abcd}h_{w}^{~{}a}h_{x}^{~{}b}h_{y}^{~{}c}h_{z}^{~{}d}=$ $\displaystyle\;\Omega^{(5-d)/2}{}^{(d-1)}C_{wxyz}$ $\displaystyle-\frac{2}{d-3}\left(E_{w[y}h_{z]x}-E_{x[y}h_{z]w}\right).$ (52) In Eqs. (51) and (52), we used the fact that the extrinsic curvature of $\Omega={\rm constant}$ surface at $i^{0}$ is $\displaystyle\pi_{ab}$ $\displaystyle\equiv(1/2)\mbox{\pounds}_{\eta}h_{ab}$ $\displaystyle=\frac{1}{2}(\eta^{c}\nabla_{c}h_{ab}+h_{ac}\nabla_{b}\eta^{c}+h_{bc}\nabla_{a}\eta^{c})$ $\displaystyle=\Omega^{-1/2}h_{ab}\;.$ (53) For the derivation of Eq. (52), see Eq. (A6) in SMS . (Note that the magnetic part defined there is different from ours.) ### C.2 Derivation of Eqs. (18) and (19) Hereafter in this appendix, we derive Eqs. (18) and (19). Firstly, to facilitate the derivation, we derive the following relation: $\mathcal{X}_{abcm}\eta^{m}=\frac{1}{d-2}\Bigl{[}\Omega^{1/2}\nabla_{[b}T_{a]c}+(d-5)\eta_{[b}T_{a]c}\Bigr{]},$ (54) where $T_{ab}\equiv\Omega^{(5-d)/2}S_{ab}$ is a tensor which have directional dependent limit at $i^{0}$, and $S_{ab}$ is defined in Eq. (14). The manipulation of $g^{md}\times$ Eq. (9) implies $\nabla_{m}C_{abc}{}^{m}=(d-3)\Omega^{-1}C_{abcp}\nabla^{p}\Omega\;.$ (55) Note that Eq. (8) was derived from the Bianchi identity in the physical vacuum spacetime ($\nabla_{[m}C_{ab]cd}=0$). On the other hand, from the Bianchi identity in the unphysical spacetime ($\hat{\nabla}_{[m}\hat{R}_{ab]cd}=0$), we can derive $\nabla_{m}C_{abc}{}^{m}+\frac{2(d-3)}{d-2}\nabla_{[a}S_{b]c}=0~{}.$ (56) From these two equations, we can see that $C_{abcm}\eta^{m}=-\frac{1}{d-2}\Omega^{1/2}\nabla_{[a}S_{b]c}\;.$ (57) It is easy to see that Eq. (54) holds from this equation. Now we are ready to derive Eqs. (18) and (19). Let us first take the manipulation of $h_{p}^{a}h_{q}^{c}\eta^{b}\times$ Eq. (54), which results in $\displaystyle E_{pq}$ $\displaystyle=\frac{1}{d-2}h_{p}^{a}h_{q}^{c}\eta^{b}\Bigl{[}\Omega^{1/2}\nabla_{[b}T_{a]c}+(d-5)\eta_{[b}T_{a]c}\Bigr{]}$ $\displaystyle=-\frac{1}{2(d-2)}\Bigl{[}D_{p}Q_{q}+h_{pq}E+(4-d)U_{pq}\Bigr{]}$ $\displaystyle=-\frac{1}{2(d-2)}\Bigl{[}\frac{1}{d-3}D_{p}D_{q}E+h_{pq}E$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\;+(4-d)U_{pq}\Bigr{]}.$ (58) This is Eq. (18). In the last line, we used the relation derived by the manipulation of $h^{a}_{e}\eta^{b}\eta^{c}\times$ Eq. (54): $D_{e}E=(d-3)Q_{e}\;.$ (59) Next, let us apply $\Omega^{(4-d)/2}\mathcal{E}_{a_{1}\cdots a_{d-3}}{}^{fab}\eta_{f}$ to Eq. (54). Then we obtain Eq. (19): $\displaystyle B_{a_{1}\cdots a_{d-3}c}$ $\displaystyle~{}~{}=\frac{1}{d-2}\mathcal{E}_{a_{1}\cdots a_{d-3}}{}^{fab}\eta_{f}\Omega^{(4-d)/2}(D_{b}U_{ac}+h_{bc}Q_{a})$ $\displaystyle~{}~{}=-\frac{1}{d-2}\mathcal{E}_{a_{1}\cdots a_{d-3}}{}^{fab}\eta_{f}\Omega^{(4-d)/2}D_{a}\Bigl{(}U_{bc}-\frac{h_{bc}}{d-3}E\Bigr{)}.$ (60) ## Appendix D $(d-1)+1$ decomposition In this appendix, we show that $d$-momentum defined in Eq. (29) agrees with the ADM formulae for energy and momentum: $\displaystyle E$ $\displaystyle=\frac{1}{16\pi G_{d}}\lim_{r_{0}\rightarrow\infty}\int_{S^{d-2}}\left(\partial^{a}h_{ab}-\partial_{b}h^{a}_{~{}a}\right)dS^{b}\Big{\|}_{r=r_{0}},$ (61) $\displaystyle Q_{N^{a}}$ $\displaystyle=\frac{-1}{8\pi G_{d}}\lim_{r_{0}\rightarrow\infty}\int_{S^{d-2}}\left(K_{ab}-K^{m}_{\,\,m}h_{ab}\right)N^{a}dS^{b}\Big{\|}_{r=r_{0}},$ (62) where $h_{ab}\equiv g_{ab}+t^{a}t^{b}$ and $K_{ab}\equiv h_{a}^{~{}c}h_{bd}\nabla_{c}t^{d}$ are the induced metric and the extrinsic curvature of a $t=\text{constant}$ surface whose unit normal is $t^{a}$, and $\partial_{a}$ is a coordinate derivative with respect to asymptotic Cartesian coordinates. $N^{a}$ is an asymptotic spacelike translational Killing vector such that $D_{a}N_{b}\rightarrow 0$ as $r\rightarrow\infty$, where $D_{a}$ is the connection for $h_{ab}$. We also show in this appendix that the angular momentum defined in Eq. (31) transforms in translational transformation as Eq. (33). This appendix may be regarded as an extension of the work by Ashtekar and Magnon in four dimensions AM . We will describe in much detail because it is very hard to check their result. ### D.1 Energy First, let us consider the energy. Let $\hat{\Sigma}$ be a spacelike hypersurface in $\hat{M}$ on $i^{0}$ which has unit timelike vector $\hat{t}^{a}$ as its normal. Then, the energy defined by Eq. (29) becomes $-P_{a}\hat{t}^{a}=-\frac{1}{8\pi G_{d}(d-3)}\int_{S^{d-2}}\hat{E}_{ab}\hat{t}^{a}\hat{t}^{b}dS\;,$ (63) where $dS$ is the volume element of a $(d-2)$-dimensional unit sphere $S^{d-2}$. In order to compare the above with the ADM formula, we must write it down in terms of quantities in physical space-time $M$. To do so, we introduce a spacelike hypersurface $\Sigma$ in $M$, unit timelike vector $t^{a}$ normal to $\Sigma$, and a unit radial vector $\eta^{a}=\partial^{a}r$. $t^{a}$ and $\eta^{a}$ are related to $\hat{t}^{a}$ and the unit radial vector in the unphysical space-time $\hat{\eta}^{a}$ as $\lim_{r_{0}\rightarrow\infty}\Omega^{-1}t^{a}=\hat{t}^{a}$ and $\lim_{r_{0}\rightarrow\infty}\Omega^{-1}\eta^{a}=\hat{\eta}^{a}$, respectively. Then, the above expression of energy (63) becomes $\displaystyle-P_{a}\hat{t}^{a}=$ $\displaystyle-\frac{1}{8\pi G_{d}(d-3)}\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}\\!\\!\\!\\!\\!\\!\\!\\!r^{d-1}C_{abcd}\eta^{b}\eta^{d}t^{a}t^{c}dS\,,$ (64) where we used the fact that $\Omega\simeq 1/r^{2}$ near $i^{0}$. Now, we define the usual electric part of the Weyl tensor $e_{ab}\equiv C_{ambn}t^{m}t^{n}$ in the physical space-time $M$. This electric part can be decomposed as $\displaystyle e_{ab}=$ $\,{}^{(d-1)}R_{ab}-K_{a}^{~{}m}K_{bm}+KK_{ab}$ $\displaystyle-\frac{1}{d-2}\Bigl{(}(d-3)h_{a}^{~{}m}h_{b}^{~{}n}+h_{ab}h^{mn}\Bigr{)}S_{mn}\;.$ (65) Taking into account of asymptotic behaviors $K_{ab}=\mathcal{O}(1/r^{d-2})$ and ${}^{(d-1)}R_{ab}=\mathcal{O}(1/r^{d-1})$ for $r\rightarrow\infty\,$, and the vacuum Einstein equation $R_{ab}=0$, we obtain $-P_{a}\hat{t}^{a}=-\frac{1}{8\pi G_{d}(d-3)}\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}\\!\\!\\!\\!\\!\\!r^{d-1}\,{}^{(d-1)}R_{ab}\eta^{a}\eta^{b}dS.$ (66) In order to integrate by parts in direction $r$, we rewrite the integral into the following form: $\displaystyle-P_{a}\hat{t}^{a}=$ $\displaystyle-\frac{1}{8\pi G_{d}(d-3)}\lim_{r_{0}\rightarrow\infty}$ (67) $\displaystyle\times\frac{1}{\Delta r}\int_{r=r_{0}}^{r=r_{0}+\Delta r}\int_{S^{d-2}}r^{d-1}\,{}^{(d-1)}R_{ab}\eta^{a}\eta^{b}drdS,$ where we used the fact that the integrand in Eq. (66) is independent of $r$ at large $r$. In this expression, the part which contribute to the integral is ${}^{(d-1)}R_{ab}\sim\frac{1}{2}\left(\partial^{c}\partial_{b}h_{ac}+\partial_{a}\partial^{c}h_{bc}-\partial^{c}\partial_{c}h_{ab}-\partial_{a}\partial_{b}h^{c}_{~{}c}\right).$ (68) Substituting (68) into (67) and integrating by parts, we can get the desired result. Since this calculation is a little difficult, we describe carefully. First, we integrate the first part in (68) by parts: $\displaystyle\frac{1}{\Delta r}\\!\\!\int_{S^{d-2}\times\Delta r}r\left(\partial^{c}\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}dV$ (69) $\displaystyle=$ $\displaystyle\frac{1}{\Delta r}\\!\\!\int_{S^{d-2}\times\Delta r}\\!\Bigl{[}\partial^{c}\bigl{(}r\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}\bigr{)}\\!-\\!\left(\partial_{b}h_{ac}\right)\partial^{c}(r\eta^{a}\eta^{b})\Bigr{]}dV,$ where $dV\equiv r^{d-2}drdS$. The first term in the right-hand side becomes $\displaystyle\frac{1}{\Delta r}$ $\displaystyle\int_{S^{d-2}\times\Delta r}\partial^{c}\bigl{(}r\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}\bigr{)}dV$ $\displaystyle=\;\;\;\,\frac{1}{\Delta r}\int_{S^{d-2}}r\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}dS^{c}\Big{\|}_{r=r_{0}+\Delta r}$ $\displaystyle\;\;\;\;-\frac{1}{\Delta r}\int_{S^{d-2}}r\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}dS^{c}\Big{\|}_{r=r_{0}}$ $\displaystyle=\int_{S^{d-2}}\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}dS^{c}\Big{\|}_{r=r_{0}}\quad,$ (70) where $dS^{c}\equiv\eta^{c}r^{d-2}dS$. In the first and the second equalities, we used the Gauss theorem, and the fact that $\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}r^{d-2}$ is independent of $r$ in the limit of $r_{0}\to\infty$. The second term in the right-hand side of Eq. (69) becomes $\displaystyle\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta r}\left(\partial_{b}h_{ac}\right)\partial^{c}(r\eta^{a}\eta^{b})dV$ $\displaystyle=\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta r}\\!\\!\\!\\!\\!\left(\partial_{b}h_{ac}\right)(\eta^{a}\eta^{b}\eta^{c}+q^{ac}\eta^{b}+q^{bc}\eta^{a})r^{d-2}drdS$ $\displaystyle=\int_{S^{d-2}}\\!\\!\left(\partial_{b}h_{ac}\right)(\eta^{a}\eta^{b}\eta^{c}+q^{ac}\eta^{b}+q^{bc}\eta^{a})r^{d-2}dS.$ (71) To transform the second into the third line, we used the fact that the integrand in the second line does not depend on $r$. Then, we obtain $\displaystyle\int_{S^{d-2}}r^{d-1}\left(\partial^{c}\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}dS$ $\displaystyle\;\;=-\int_{S^{d-2}}\left(\partial_{b}h_{ac}\right)(\eta^{a}q^{bc}+\eta^{b}q^{ac})r^{d-2}dS.$ (72) Here, we defined a metric $q_{ab}$ on $r=\text{constant}$ surface such that $\partial_{a}\eta_{b}=(h_{ab}-\eta_{a}\eta_{b})/r\equiv q_{ab}/r$. In the same way, the other terms in (68) are transformed as $\displaystyle\int_{S^{d-2}}r^{d-1}\left(\partial_{a}\partial^{c}h_{bc}\right)\eta^{a}\eta^{b}dS=-(d-2)\int_{S^{d-2}}\partial^{c}h_{ac}dS^{a},$ $\displaystyle\int_{S^{d-2}}r^{d-1}\left(\partial^{c}\partial_{c}h_{ab}\right)\eta^{a}\eta^{b}dS$ $\displaystyle\qquad\qquad\qquad=-\int_{S^{d-2}}\left(\partial_{c}h_{ab}\right)(q^{ac}\eta^{b}+q^{bc}\eta^{a})r^{d-2}dS\;,$ $\displaystyle\int_{S^{d-2}}r^{d-1}\left(\partial_{a}\partial_{b}h^{c}_{~{}c}\right)\eta^{a}\eta^{b}dS=-(d-2)\int_{S^{d-2}}\\!\partial_{a}h^{c}_{~{}c}dS^{a}.$ Finally, we obtain the desired result: $-P_{a}\hat{t}^{a}=\frac{1}{16\pi G_{d}}\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}\left(\partial^{a}h_{ab}-\partial_{b}h^{a}_{~{}a}\right)dS^{b}\;.$ (73) ### D.2 Momentum Next, let us consider momentum. The components of $(d-1)$-momentum along a spacelike vector $N^{a}$ at $i^{0}$ can be written as $P_{a}\hat{N}^{a}=\frac{1}{8\pi G_{d}(d-3)}\int_{S^{d-2}}\hat{E}_{ab}\hat{N}^{a}\hat{t}^{b}dS.$ (74) In terms of quantities of physical space-time, this equation becomes $P_{a}\hat{N}^{a}=\frac{1}{8\pi G_{d}(d-3)}\lim_{r\rightarrow\infty}\int_{r=r_{0}}r^{d-1}C_{abcd}\eta^{b}\eta^{d}N^{a}t^{c}dS\,,$ (75) where $N^{a}=\lim_{\rightarrow i^{0}}\Omega\hat{N}^{a}$. Using the Codacci equation and the vacuum Einstein equation, this expression becomes $\displaystyle P_{a}\hat{N}^{a}=\;$ $\displaystyle\frac{1}{8\pi G_{d}(d-3)}\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}r^{d-1}$ $\displaystyle\times\left(D_{d}K_{ab}-D_{a}K_{db}\right)\eta^{b}\eta^{d}N^{a}dS.$ (76) Using the fact that the leading part of $r^{d-1}D_{d}K_{ab}$ does not depend on $r$, the first term in the right-hand side is reexpressed as volume integral as $\displaystyle\int_{r=r_{0}}r^{d-1}\left(D_{d}K_{ab}\right)\eta^{b}\eta^{d}N^{a}dS$ $\displaystyle\;\;=\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta r}r\left(D_{d}K_{ab}\right)\eta^{b}\eta^{d}N^{a}dV$ $\displaystyle\;\;=\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta r}\Bigl{[}\;D_{d}(rK_{ab}\eta^{b}\eta^{d}N^{a})$ $\displaystyle\;\;\qquad\qquad\qquad\quad\;\;-K_{ab}D_{d}(r\eta^{b}\eta^{d}N^{a})\;\Bigr{]}dV.$ (77) Using the Gauss theorem to the first term in the last line, we see that $\displaystyle\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta r}D_{d}(rK_{ab}\eta^{b}\eta^{d}N^{a})dV$ $\displaystyle=\frac{1}{\Delta r}\biggl{[}\int_{S^{d-2}}\\!\\!\\!\\!\\!rK_{ab}N^{a}dS^{b}\Big{\|}_{r=r_{0}+\Delta r}\\!-\\!\int_{S^{d-2}}\\!\\!\\!\\!\\!rK_{ab}N^{a}dS^{b}\Big{\|}_{r=r_{0}}\biggr{]}$ $\displaystyle=\int_{S^{d-2}}K_{ab}N^{a}dS^{b}\Big{\|}_{r=r_{0}}\;\;,$ (78) where we used the fact that $K_{ab}N^{a}r^{d-2}$ does not depend on $r$. The second term in Eq. (77) can be rearranged as $\displaystyle-$ $\displaystyle\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta r}K_{ab}D_{d}(r\eta^{b}\eta^{d}N^{a})dV$ $\displaystyle=-\frac{d-1}{\Delta r}\int_{S^{d-2}\times\Delta r}K_{ab}\eta^{b}N^{a}dV$ $\displaystyle=-(d-1)\int_{S^{d-2}}K_{ab}N^{a}dS^{b}\Big{\|}_{r=r_{0}}\;\;.$ (79) In the same way, the second term of Eq. (76) is rearranged as $\displaystyle\int_{r=r_{0}}r^{d-1}\left(D_{a}K_{db}\right)\eta^{b}\eta^{d}N^{a}dS$ $\displaystyle=2\int_{S^{d-2}}K_{ab}\eta^{a}\eta^{b}N^{c}dS_{c}-2\int_{S^{d-2}}K_{ab}N^{a}dS^{b}$ (80) From this equation, we can obtain a relation $\displaystyle 2$ $\displaystyle\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}r^{d-2}K_{ab}N_{c}\eta^{a}\eta^{b}\eta^{c}dS$ $\displaystyle=$ $\displaystyle\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}\Big{(}K_{ab}-(d-3)Kh_{ab}\Big{)}N^{a}dS^{b}.$ (81) Derivation of this relation is a little non-trivial, so we describe it in detail. Note that the Gauss theorem makes the surface integral into the volume integral as $\displaystyle\int_{r=r_{0}}r^{d-2}K_{ab}N_{c}\eta^{a}\eta^{b}\eta^{c}dS$ $\displaystyle\\!=\\!\frac{1}{\Delta r}\\!\int_{S^{d-2}\times\Delta r}\partial^{a}\left(rK_{ab}N_{c}\eta^{b}\eta^{c}\right)dV$ $\displaystyle\\!=\\!\frac{1}{\Delta r}\\!\int_{S^{d-2}\times\Delta r}\Bigl{[}r\left(D_{b}K\right)\eta^{b}N^{c}\eta_{c}\\!+\\!K_{ab}D^{a}\left(r\eta^{b}N^{c}\eta_{c}\right)\Bigr{]}dV,$ (82) where we used the momentum constraint equation for the vacuum Einstein equation, $D_{a}K^{a}_{b}-D_{b}K=0$, in the last line. The first and the second terms are rearranged respectively as $\displaystyle\frac{1}{\Delta r}$ $\displaystyle\int_{S^{d-2}\times\Delta r}r\left(D_{b}K\right)\eta^{b}N^{c}\eta_{c}dV$ $\displaystyle=$ $\displaystyle\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta r}\Bigl{[}D_{b}\left(rK\eta^{b}N^{c}\eta_{c}\right)-KD_{b}\left(r\eta^{b}N^{c}\eta_{c}\right)\Bigr{]}dV$ $\displaystyle=$ $\displaystyle-(d-2)\int_{S^{d-2}}KN^{a}dS_{a}\Big{\|}_{r=r_{0}}\;\;,$ (83) and $\displaystyle\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta r}K_{ab}D^{a}\left(r\eta^{b}N^{c}\eta_{c}\right)dV$ $\displaystyle\\!=\\!\int_{S^{d-2}}\\!\\!\left(KN^{a}dS_{a}+K_{ab}N^{b}dS^{a}-K_{ab}\eta^{a}\eta^{b}N^{c}dS_{c}\right)\Big{\|}_{r=r_{0}}.$ (84) Then, we proceed as $\displaystyle\int_{r=r_{0}}r^{d-2}K_{ab}N_{c}\eta^{a}\eta^{b}\eta^{c}dS$ $\displaystyle~{}~{}=~{}\int_{S^{d-2}}\Bigl{(}K_{ab}N^{b}-(d-3)KN_{a}\Bigr{)}dS^{a}\Big{\|}_{r=r_{0}}$ $\displaystyle~{}~{}~{}~{}-\int_{S^{d-2}}K_{ab}\eta^{a}\eta^{b}N^{c}dS_{c}\Big{\|}_{r=r_{0}}\;\;.$ (85) The last term in the right-hand side is the same with the left-hand side except for the signature. Therefore, we have the relation of Eq. (81). Substituting Eq. (81) into Eq. (80), we can show $\displaystyle\int_{r=r_{0}}r^{d-1}(D_{d}K_{ab}-D_{a}K_{db})\eta^{b}\eta^{d}N^{a}dS$ $\displaystyle=-(d-3)\int_{S^{d-2}}(K_{ab}-Kh_{ab})N^{a}dS^{b}\Big{\|}_{r=r_{0}}\;\;,$ (86) and then, combining this equation with Eq. (76), we see that our formula (29) for $(d-1)$-momentum becomes the ADM formula, that is $P_{a}\hat{N}^{a}=-\frac{1}{8\pi G_{d}}\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}\left(K_{ab}-Kh_{ab}\right)N^{a}dS^{b}.$ (87) ### D.3 Angular momentum Finally, we consider translational transformation of angular momentum $M_{ab}$. We consider translation $\omega_{a}$ which is a fixed vector at $i^{0}$, and relate it with $\alpha$ as $\alpha=\omega_{a}\hat{\eta}^{a}$. This translation transforms $\hat{\eta^{a}}$ as $\hat{\eta}_{a}^{\prime}\;\hat{=}\;\hat{\eta}_{a}+\frac{1}{2}\Omega^{(d-3)/2}\left((d-2)\alpha\hat{\eta}_{a}+\Omega^{1/2}\hat{\nabla}_{a}\alpha\right)\;.$ (88) Then, the magnetic part of the Weyl tensor $\hat{\beta}_{a_{1}\cdots a_{d-3}b}$ transforms as $\displaystyle\hat{\beta}_{a_{1}\cdots a_{d-3}b}^{\prime}\,\hat{=}$ $\displaystyle\;\hat{\beta}_{a_{1}\cdots a_{d-3}b}+\frac{d-2}{d-3}\hat{\epsilon}_{a_{1}\cdots a_{d-3}mpq}\hat{\eta}^{m}\hat{E}^{q}_{~{}b}\hat{D}^{p}\alpha$ $\displaystyle+\frac{1}{d-3}\hat{\epsilon}_{a_{1}\cdots a_{d-3}mpb}\hat{\eta}^{m}\hat{E}^{pr}\hat{D}_{r}\alpha\;.$ (89) We used the projection formulae of the Weyl tensor (51) and (52) to derive this equation. Substituting (89) into (31), and noting that $F_{ab}\hat{\epsilon}^{b}_{~{}e_{1}\cdots e_{d-2}m}\hat{\eta}^{m}dS^{e_{1}\cdots e_{d-2}}$ vanishes since $d-1$ indices of $\hat{\epsilon}$ are projected onto $(d-2)$-dimensional surface, we find that usual translational transformation $M_{ab}^{\prime}=M_{ab}+2P_{\,[a}\hat{\omega}_{b]}\;,$ (90) where $\hat{D}_{a}\alpha=\hat{\omega}_{a}$, is correctly reproduced including the coefficient, if we define the angular momentum as (31). ## References * (1) R. Penrose, Phys. Rev. Lett. 10 , 66 (1963); Proc. Roy. Soc. A (London) 284 , 159 (1965) * (2) A. Ashtekar and R. O. Hansen, J. Math. Phys. 19, 1542 (1978). A. Ashtekar, General Relativity and Gravitation vol 2, ed A. Held (New York: Plenum); 1984 * (3) T. Shiromizu and S. Tomizawa, Phys. Rev. D 69, 104012 (2004) [arXiv:gr-qc/0401006]. * (4) S. Hollands and A. Ishibashi, J. Math. Phys. 46, 022503 (2005) [arXiv:gr-qc/0304054]. * (5) P. C. Argyres, S. Dimopoulos and J. March-Russell, Phys. Lett. B 441, 96 (1998) [arXiv:hep-th/9808138], R. Emparan, G. T. Horowitz and R. C. Myers, Phys. Rev. Lett. 85, 499 (2000) [arXiv:hep-th/0003118]. S. Dimopoulos and G. L. Landsberg, Phys. Rev. Lett. 87, 161602 (2001) [arXiv:hep-ph/0106295]. S. B. Giddings and S. D. Thomas, Phys. Rev. D 65, 056010 (2002) [arXiv:hep-ph/0106219]. * (6) R. Emparan and H. S. Reall, Living Rev. Rel. 11, 6 (2008) [arXiv:0801.3471 [hep-th]]. * (7) W. Israel, Phys. Rev. 164, 1776 (1967); B. Carter, Phys. Rev. Lett. 26, 331 (1971); S. W. Hawking, Commun. Math. Phys. 25, 152 (1972); D. C. Robinson, Phys. Rev. Lett. 34, 905 (1975); P. O. Mazur, J. Phys. A15, 3173 (1982); For review, M. Heusler, Black Hole Uniqueness Theorems, (Cambridge University Press, London, 1996); P. O. Mazur, hep-th/0101012; G. L. Bunting, PhD thesis, Univ. of New England, Armidale (1983). * (8) R. C. Myers and M. J. Perry, Ann. Phys. 172, 304 (1986). * (9) R. Emparan and H. S. Reall, Phys. Rev. Lett. 88, 101101 (2002) [arXiv:hep-th/0110260]. * (10) G. W. Gibbons, D. Ida and T. Shiromizu, Phys. Rev. Lett. 89, 041101 (2002); Phys. Rev. D66, 044010 (2002); Prog. Theor. Phys. Suppl. 148, 284 (2003); M. Rogatko, Class. Quantum Grav. 19, L151 (2002); Phys. Rev. D67, 084025 (2003); S. Hwang, Geometriae Dedicata 71, 5 (1998). * (11) A. Ashtekar and J. D. Romano, Class. Quant. Grav. 9, 1069 (1992). * (12) A. Ashtekar and A. Magnon-Ashtekar, J. Math. Phys. 20, 793 (1979). * (13) S. Hollands, A. Ishibashi and R. M. Wald, Commun. Math. Phys. 271, 699 (2007) [arXiv:gr-qc/0605106]; arXiv:0809.2659 [gr-qc]. * (14) Y. Morisawa and D. Ida, Phys. Rev. D 69, 124005 (2004) [arXiv:gr-qc/0401100]. * (15) S. Hollands and S. Yazadjiev, Commun. Math. Phys. 283, 749 (2008) [arXiv:0707.2775 [gr-qc]]. * (16) Y. Morisawa, S. Tomizawa and Y. Yasui, Phys. Rev. D 77, 064019 (2008) [arXiv:0710.4600 [hep-th]]. * (17) A. Ashtekar and A. Magnon, J. Math. Phys. 25, 2682 (1984). * (18) T. Shiromizu, K. i. Maeda and M. Sasaki, Phys. Rev. D 62, 024012 (2000) [arXiv:gr-qc/9910076].	arxiv-papers	2009-02-10T05:46:25	2024-09-04T02:49:00.492979	{ "license": "Public Domain", "authors": "Kentaro Tanabe, Norihiro Tanahashi, Tetsuya Shiromizu", "submitter": "Kentarou Tanabe", "url": "https://arxiv.org/abs/0902.1583" }
0902.1589	# New n-mode squeezing operator and squeezed states with standard squeezing ††thanks: Work supported by the National Natural Science Foundation of China under grants 10775097 and 10874174. Li-yun Hu1,2 and Hong-yi Fan1 1Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China 2College of Physics & Communication Electronics, Jiangxi Normal University, Nanchang 330022, China Corresponding author. E-mail addresses: hlyun2008@126.com or hlyun@sjtu.edu.cn. ###### Abstract We find that the exponential operator $V\equiv\exp\left[\mathtt{i}\lambda\left(Q_{1}P_{2}+Q_{2}P_{3}+\cdots+Q_{n-1}P_{n}+Q_{n}P_{1}\right)\right],$ $Q_{i},$ $P_{i}$ are respectively the coordinate and momentum operators, is an n-mode squeezing operator which engenders standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive $V$’s normally ordered expansion and obtain the n-mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations. PACS 03.65.-w—Quantum mechanics PACS 42.50.-p—Quantum optics ## 1 Introduction Quantum entanglement is a weird, remarkable feature of quantum mechanics though it implies intricacy. In recent years, various entangled states have brought considerable attention and interests of physicists because of their potential uses in quantum communication [1, 2]. Among them the two-mode squeezed state exhibits quantum entanglement between the idle-mode and the signal-mode in a frequency domain manifestly, and is a typical entangled state of continuous variable. Theoretically, the two-mode squeezed state is constructed by the two-mode squeezing operator $S=\exp[\lambda(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})]$ [3, 4, 5] acting on the two-mode vacuum state $\left\|00\right\rangle$, $S\left\|00\right\rangle=\text{sech}\lambda\exp\left[-a_{1}^{{}^{\dagger}}a_{2}^{{}^{\dagger}}\tanh\lambda\right]\left\|00\right\rangle,$ (1) where $\lambda$ is a squeezing parameter, the disentangling of $S$ can be obtained by using SU(1,1) Lie algebra, $[a_{1}a_{2},a_{1}^{\dagger}a_{2}^{\dagger}]=a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+1,$ or by using the entangled state representation $\left\|\eta=\eta_{1}+i\eta_{2}\right\rangle$ [6, 7] $\left\|\eta\right\rangle=\exp\left[-\frac{1}{2}\left\|\eta\right\|^{2}+\eta a_{1}^{\dagger}-\eta^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\right]\left\|00\right\rangle,$ (2) $\left\|\eta\right\rangle$ is the common eigenvector of two particles’ relative position $\left(Q_{1}-Q_{2}\right)$ and the tota momentum $\left(P_{1}+P_{2}\right)$, obeys the eigenvector equation, $\left(Q_{1}-Q_{2}\right)\left\|\eta\right\rangle=\sqrt{2}\eta_{1}\left\|\eta\right\rangle,$ $\left(P_{1}+P_{2}\right)=\left\|\eta\right\rangle=\sqrt{2}\eta_{2}\left\|\eta\right\rangle,$ and the orthonormal-complete relation $\int\frac{d^{2}\eta}{\pi}\left\|\eta\right\rangle\left\langle\eta\right\|=1,\text{ }\left\langle\eta^{\prime}\right\|\left.\eta\right\rangle=\pi\delta\left(\eta-\eta^{\prime}\right)\left(\eta^{\ast}-\eta^{\prime\ast}\right),$ (3) because the two-mode squeezing operator has its natural representation in $\left\langle\eta\right\|$ basis $S=\exp\left[\lambda\left(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger}\right)\right]=\int\frac{d^{2}\eta}{\pi\mu}\left\|\frac{\eta}{\mu}\right\rangle\left\langle\eta\right\|,\text{ }S\left\|\eta\right\rangle=\frac{1}{\mu}\left\|\frac{\eta}{\mu}\right\rangle,\text{ }\mu=e^{\lambda},$ (4) The proof of Eq.(4) is proceeded by virtue of the technique of integration within an ordered product (IWOP) of operators [8, 9, 10] $\displaystyle\int\frac{d^{2}\eta}{\pi\mu}\left\|\eta/\mu\right\rangle\left\langle\eta\right\|$ $\displaystyle=$ $\displaystyle\int\frac{d^{2}\eta}{\pi\mu}\colon\exp\left\\{-\frac{\mu^{2}+1}{2\mu^{2}}\|\eta\|^{2}+\eta\left(\frac{a_{1}^{{}^{\dagger}}}{\mu}-a_{2}\right)\right.$ (5) $\displaystyle\left.+\eta^{\ast}\left(a_{1}-\frac{a_{2}^{{}^{\dagger}}}{\mu}\right)+a_{1}^{\dagger}a_{2}^{{}^{\dagger}}+a_{1}a_{2}-a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2}\right\\}\colon$ $\displaystyle=$ $\displaystyle\frac{2\mu}{1+\mu^{2}}\colon\exp\left\\{\frac{\mu^{2}}{1+\mu^{2}}\left(\frac{a_{1}^{{}^{\dagger}}}{\mu}-a_{2}\right)\left(a_{1}-\frac{a_{2}^{{}^{\dagger}}}{\mu}\right)-\left(a_{1}-a_{2}^{{}^{\dagger}}\right)\left(a_{1}^{{}^{\dagger}}-a_{2}\right)\right\\}\colon$ $\displaystyle=$ $\displaystyle e^{-a_{1}^{{}^{\dagger}}a_{2}^{{}^{\dagger}}\tanh\lambda}e^{(a_{1}^{{}^{\dagger}}a_{1}+a_{2}^{{}^{\dagger}}a_{2}+1)\ln\text{sech}\lambda}e^{a_{1}a_{2}\tanh\lambda}\equiv S,$ Eq. (4) confirms that the two-mode squeezed state itself is an entangled state which entangles the idle mode and signal mode as an outcome of a parametric- down conversion process [11]. The $\left\|\eta\right\rangle$ state was constructed in Ref. [6, 7] according to the idea of Einstein, Podolsky and Rosen in their argument that quantum mechanics is incomplete [12]. Using the relation between bosonic operators and the coordinate $Q_{i},$ momentum $P_{i},$ $Q_{i}=(a_{i}+a_{i}^{\dagger})/\sqrt{2},\ P_{i}=(a_{i}-a_{i}^{\dagger})/(\sqrt{2}\mathtt{i}),$ and introducing the two- mode quadrature operators of light field as in Ref. [4], $x_{1}=(Q_{1}+Q_{2})/2,x_{2}=(P_{1}+P_{2})/2,$ the variances of $x_{1}$ and $x_{2}$ in the state $S\left\|00\right\rangle$ are in the standard form $\left\langle 00\right\|S^{\dagger}x_{2}^{2}S\left\|00\right\rangle=\frac{1}{4}e^{-2\lambda},\text{ \ }\left\langle 00\right\|S^{\dagger}x_{1}^{2}S\left\|00\right\rangle=\frac{1}{4}e^{2\lambda},$ (6) thus we get the standard squeezing for the two quadrature: $x_{1}\rightarrow\frac{1}{2}e^{\lambda}x_{1},$ $x_{2}\rightarrow\frac{1}{2}e^{-\lambda}x_{2}$. On the other hand, the two- mode squeezing operator can also be recast into the form $S=\exp\left[\mathtt{i}\lambda\left(Q_{1}P_{2}+Q_{2}P_{1}\right)\right].$ Then an interesting question naturally rises: what is the property of the $n$-mode operator $V\equiv\exp\left[\mathtt{i}\lambda\left(Q_{1}P_{2}+Q_{2}P_{3}+\cdots+Q_{n-1}P_{n}+Q_{n}P_{1}\right)\right],$ (7) and is it a squeezing operator which can engenders the standard squeezing for $n$-mode quadratures? What is the normally ordered expansion of $V$ and what is the state $V\left\|\mathbf{0}\right\rangle$ ($\left\|\mathbf{0}\right\rangle$ is the n-mode vacuum state)? In this work we shall study $V$ in detail. But how to disentangling the exponential of $V?$ Since all terms of the set $Q_{i}P_{i+1}\ $($i=1\cdots n$) do not make up a closed Lie algebra, the problem of what is $V^{\prime}$s the normally ordered form seems difficult. Thus we appeal to the IWOP technique to solve this problem. Our work is arranged in this way: firstly we use the IWOP technique to derive the normally ordered expansion of $V$ and obtain the explicit form of $\ V\left\|\mathbf{0}\right\rangle$; then we examine the variances of the $n$-mode quadrature operators in the state $V\left\|\mathbf{0}\right\rangle$, we find that $V$ just causes standard squeezing. Thus $V$ is a squeezing operator. The Wigner function of $V\left\|\mathbf{0}\right\rangle$ is calculated by using the Weyl ordering invariance under similar transformations. Some examples are discussed in the last section. ## 2 The normal product form of $V$ In order to disentangle operator $V$, let $A$ be $A=\left(\begin{array}[]{ccccc}0&1&0&\cdots&0\\\ 0&0&1&\cdots&0\\\ \vdots&\vdots&\ddots&\ddots&0\\\ 0&0&\cdots&\ddots&1\\\ 1&0&\cdots&\cdots&0\end{array}\right),$ (8) then $V$ in (7) is compactly expressed as $V=\exp\left[\mathtt{i}\lambda\underset{i,j=1}{\overset{n}{\sum}}Q_{i}A_{ij}P_{j}\right].$ (9) Using the Baker-Hausdorff formula, $e^{A}Be^{-A}=B+\left[A,B\right]+\frac{1}{2!}\left[A,\left[A,B\right]\right]+\frac{1}{3!}\left[A,\left[A,\left[A,B\right]\right]\right]+\cdots,$we have $($here and henceforth the repeated indices represent the Einstein summation notation) $\displaystyle V^{-1}Q_{k}V$ $\displaystyle=$ $\displaystyle Q_{k}-\lambda Q_{i}A_{ik}+\frac{1}{2!}\mathtt{i}\lambda^{2}\left[Q_{i}A_{ij}P_{j},Q_{l}A_{lk}\right]+...$ (10) $\displaystyle=$ $\displaystyle Q_{i}(e^{-\lambda A})_{ik}=(e^{-\lambda\tilde{A}})_{ki}Q_{i},$ $\displaystyle V^{-1}P_{k}V$ $\displaystyle=$ $\displaystyle P_{k}+\lambda A_{ki}P_{i}+\frac{1}{2!}\mathtt{i}\lambda^{2}\left[A_{ki}P_{j},Q_{l}A_{lm}P_{m}\right]+...$ (11) $\displaystyle=$ $\displaystyle(e^{\lambda A})_{ki}P_{i},$ From Eq.(10) we see that when $V$ acts on the n-mode coordinate eigenstate $\left\|\vec{q}\right\rangle,$ where $\widetilde{\vec{q}}=(q_{1},q_{2},\cdots,q_{n})$, it squeezes $\left\|\vec{q}\right\rangle$ in the way of $V\left\|\vec{q}\right\rangle=\left\|\Lambda\right\|^{1/2}\left\|\Lambda\vec{q}\right\rangle,\text{ }\Lambda=e^{-\lambda\tilde{A}},\text{ }\left\|\Lambda\right\|\equiv\det\Lambda.$ (12) Thus $V$ has the representation on the coordinate $\left\langle\vec{q}\right\|$ basis $V=\int d^{n}qV\left\|\vec{q}\right\rangle\left\langle\vec{q}\right\|=\left\|\Lambda\right\|^{1/2}\int d^{n}q\left\|\Lambda\vec{q}\right\rangle\left\langle\vec{q}\right\|,\text{ \ \ }V^{\dagger}=V^{-1},$ (13) since $\int d^{n}q\left\|\vec{q}\right\rangle\left\langle\vec{q}\right\|=1.$ Using the expression of eigenstate $\left\|\vec{q}\right\rangle$ in Fock space $\displaystyle\left\|\vec{q}\right\rangle=\pi^{-n/4}\colon\exp[-\frac{1}{2}\widetilde{\vec{q}}\vec{q}+\sqrt{2}\widetilde{\vec{q}}a^{{\dagger}}-\frac{1}{2}\tilde{a}^{{\dagger}}a^{{\dagger}}]\left\|\mathbf{0}\right\rangle,\text{ }$ (14) $\displaystyle\tilde{a}^{{\dagger}}=(a_{1}^{{\dagger}},a_{2}^{{\dagger}},\cdots,a_{n}^{{\dagger}})\text{,}$ and $\left\|\mathbf{0}\right\rangle\left\langle\mathbf{0}\right\|=\colon\exp[-\tilde{a}^{{\dagger}}a^{{\dagger}}]\colon,$ we can put $V$ into the normal ordering form , $\displaystyle V$ $\displaystyle=$ $\displaystyle\pi^{-n/2}\left\|\Lambda\right\|^{1/2}\int d^{n}q\colon\exp[-\frac{1}{2}\widetilde{\vec{q}}(1+\widetilde{\Lambda}\Lambda)\vec{q}+\sqrt{2}\widetilde{\vec{q}}(\widetilde{\Lambda}a^{{\dagger}}+a)$ (15) $\displaystyle-\frac{1}{2}(\widetilde{a}a+\tilde{a}^{{\dagger}}a^{{\dagger}})-\tilde{a}^{{\dagger}}a]\colon.$ To compute the integration in Eq.(15) by virtue of the IWOP technique, we use the mathematical formula $\int d^{n}x\exp[-\widetilde{x}Fx+\widetilde{x}v]=\pi^{n/2}(\det F)^{-1/2}\exp\left[\frac{1}{4}\widetilde{v}F^{-1}v\right],$ (16) then we derive $\displaystyle V$ $\displaystyle=$ $\displaystyle\left(\frac{\det\Lambda}{\det N}\right)^{1/2}\exp\left[\frac{1}{2}\tilde{a}^{{\dagger}}\left(\Lambda N^{-1}\widetilde{\Lambda}-I\right)a^{{\dagger}}\right]$ (17) $\displaystyle\times\colon\exp\left[\tilde{a}^{{\dagger}}\left(\Lambda N^{-1}-I\right)a\right]\colon\exp\left[\frac{1}{2}\widetilde{a}\left(N^{-1}-I\right)a\right],$ where $N=(1+\widetilde{\Lambda}\Lambda)/2$. Eq.(17) is just the normal product form of $V.$ ## 3 The squeezing property of $V\left\|\mathbf{0}\right\rangle$ Operating $V$ on the n-mode vacuum state $\left\|\mathbf{0}\right\rangle,$ we obtain the squeezed vacuum state $V\left\|\mathbf{0}\right\rangle=\left(\frac{\det\Lambda}{\det N}\right)^{1/2}\exp\left[\frac{1}{2}\tilde{a}^{{\dagger}}\left(\Lambda N^{-1}\widetilde{\Lambda}-I\right)a^{{\dagger}}\right]\left\|\mathbf{0}\right\rangle.$ (18) Now we evaluate the variances of the n-mode quadratures. The quadratures in the n-mode case are defined as $X_{1}=\frac{1}{\sqrt{2n}}\sum_{i=1}^{n}Q_{i},\text{ }X_{2}=\frac{1}{\sqrt{2n}}\sum_{i=1}^{n}P_{i},$ (19) obeying $[X_{1},X_{2}]=\frac{\mathtt{i}}{2}.$ Their variances are $\left(\Delta X_{i}\right)^{2}=\left\langle X_{i}^{2}\right\rangle-\left\langle X_{i}\right\rangle^{2}$, $i=1,2.$ Noting the expectation values of $X_{1}$ and $X_{2}$ in the state $V\left\|\mathbf{0}\right\rangle$, $\left\langle X_{1}\right\rangle=\left\langle X_{2}\right\rangle=0,$ and using Eqs. (10) and (11) we see that the variances are $\displaystyle\left(\triangle X_{1}\right)^{2}$ $\displaystyle=$ $\displaystyle\left\langle\mathbf{0}\right\|V^{-1}X_{1}^{2}V\left\|\mathbf{0}\right\rangle=\frac{1}{2n}\left\langle\mathbf{0}\right\|V^{-1}\sum_{i=1}^{n}Q_{i}\sum_{j=1}^{n}Q_{j}V\left\|\mathbf{0}\right\rangle$ (20) $\displaystyle=$ $\displaystyle\frac{1}{2n}\left\langle\mathbf{0}\right\|\sum_{i=1}^{n}Q_{k}(e^{-\lambda A})_{ki}\sum_{j=1}^{n}(e^{-\lambda\tilde{A}})_{jl}Q_{l}\left\|\mathbf{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{2n}\underset{i,j}{\sum^{n}}(e^{-\lambda A})_{ki}(e^{-\lambda\tilde{A}})_{jl}\left\langle\mathbf{0}\right\|Q_{k}Q_{l}\left\|\mathbf{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{4n}\underset{i,j}{\sum^{n}}(e^{-\lambda A})_{ki}(e^{-\lambda\tilde{A}})_{jl}\left\langle\mathbf{0}\right\|a_{k}a_{l}^{\dagger}\left\|\mathbf{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{4n}\underset{i,j}{\sum^{n}}(e^{-\lambda A})_{ki}(e^{-\lambda\tilde{A}})_{jl}\delta_{kl}=\frac{1}{4n}\underset{i,j}{\sum^{n}}(\widetilde{\Lambda}\Lambda)_{ij},$ similarly we have $\left(\triangle X_{2}\right)^{2}=\left\langle\mathbf{0}\right\|V^{-1}X_{2}^{2}V\left\|\mathbf{0}\right\rangle=\frac{1}{4n}\underset{i,j}{\sum^{n}}(\widetilde{\Lambda}\Lambda)_{ij}^{-1},$ (21) Eqs. (20) -(21) are the quadrature variance formula in the transformed vacuum state acted by the operator $\exp[\mathtt{i}\lambda\underset{i,j=1}{\overset{n}{\sum}}Q_{i}A_{ij}P_{j}].$ By observing that $A$ in (8) is a cyclic matrix, we see $\underset{i,j}{\sum^{n}}\left[(A+\tilde{A})^{l}\right]_{i\text{ }j}=2^{l}n,$ (22) then using $A\tilde{A}=\tilde{A}A,$ so $\widetilde{\Lambda}\Lambda=e^{-\lambda(A+\tilde{A})}$, a symmetric matrix, we have $\underset{i,j=1}{\sum^{n}}(\widetilde{\Lambda}\Lambda)_{i\text{ }j}=\sum_{l=0}^{\infty}\frac{(-\lambda)^{l}}{l!}\underset{i,j}{\sum^{n}}\left[(A+\tilde{A})^{l}\right]_{i\text{ }j}=n\sum_{l=0}^{\infty}\frac{(-\lambda)^{l}}{l!}2^{l}=ne^{-2\lambda},$ (23) and $\underset{i,j=1}{\sum^{n}}(\widetilde{\Lambda}\Lambda)_{i\text{ }j}^{-1}=ne^{2\lambda}.$ (24) it then follows $\displaystyle\left(\triangle X_{1}\right)^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{4n}\underset{i,j}{\sum^{n}}(\widetilde{\Lambda}\Lambda)_{ij}=\frac{e^{-2\lambda}}{4},$ (25) $\displaystyle\left(\triangle X_{2}\right)^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{4n}\underset{i,j}{\sum^{n}}(\widetilde{\Lambda}\Lambda)_{ij}^{-1}=\frac{e^{2\lambda}}{4}.$ (26) This leads to $\triangle X_{1}\cdot\triangle X_{2}=\frac{1}{4},$ which shows that $V$ is a correct n-mode squeezing operator for the n-mode quadratures in Eq.(19) and produces the standard squeezing similar to Eq. (6). ## 4 The Wigner function of $V\left\|\mathbf{0}\right\rangle$ Wigner distribution functions [13, 14, 15] of quantum states are widely studied in quantum statistics and quantum optics. Now we derive the expression of the Wigner function of $V\left\|\mathbf{0}\right\rangle.$ Here we take a new method to do it. Recalling that in Ref.[16, 17, 18] we have introduced the Weyl ordering form of single-mode Wigner operator $\Delta\left(q,p\right)$, $\Delta_{1}\left(q_{1},p_{1}\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{1}-Q_{1}\right)\delta\left(p_{1}-P_{1}\right)\genfrac{}{}{0.0pt}{}{:}{:},$ (27) its normal ordering form is $\Delta_{1}\left(q_{1},p_{1}\right)=\frac{1}{\pi}\colon\exp\left[-\left(q_{1}-Q_{1}\right)^{2}-\left(p_{1}-P_{1}\right)^{2}\right]\colon$ (28) where the symbols $\colon\colon$ and $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$ denote the normal ordering and the Weyl ordering, respectively. Note that the order of Bose operators $a_{1}$ and $a_{1}^{\dagger}$ within a normally ordered product and a Weyl ordered product can be permuted. That is to say, even though $[a_{1},a_{1}^{\dagger}]=1$, we can have $\colon a_{1}a_{1}^{\dagger}\colon=\colon a_{1}^{\dagger}a_{1}\colon$ and$\genfrac{}{}{0.0pt}{}{:}{:}a_{1}a_{1}^{\dagger}\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}a_{1}^{\dagger}a_{1}\genfrac{}{}{0.0pt}{}{:}{:}.$ The Weyl ordering has a remarkable property, i.e., the order-invariance of Weyl ordered operators under similar transformations [16, 17, 18], which means $U\genfrac{}{}{0.0pt}{}{:}{:}\left(\circ\circ\circ\right)\genfrac{}{}{0.0pt}{}{:}{:}U^{-1}=\genfrac{}{}{0.0pt}{}{:}{:}U\left(\circ\circ\circ\right)U^{-1}\genfrac{}{}{0.0pt}{}{:}{:},$ (29) as if the “fence” $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$did not exist. For n-mode case, the Weyl ordering form of the Wigner operator is $\Delta_{n}\left(\vec{q},\vec{p}\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\vec{q}-\vec{Q}\right)\delta\left(\vec{p}-\vec{P}\right)\genfrac{}{}{0.0pt}{}{:}{:},$ (30) where $\widetilde{\vec{Q}}=(Q_{1},Q_{2},\cdots,Q_{n})$ and $\widetilde{\vec{P}}=(P_{1},P_{2},\cdots,P_{n})$. Then according to the Weyl ordering invariance under similar transformations and Eqs.(10) and (11) we have $\displaystyle V^{-1}\Delta_{n}\left(\vec{q},\vec{p}\right)V$ $\displaystyle=$ $\displaystyle V^{-1}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\vec{q}-\vec{Q}\right)\delta\left(\vec{p}-\vec{P}\right)\genfrac{}{}{0.0pt}{}{:}{:}V$ (31) $\displaystyle=$ $\displaystyle\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{k}-(e^{-\lambda\tilde{A}})_{ki}Q_{i}\right)\delta\left(p_{k}-(e^{rA})_{ki}P_{i}\right)\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=$ $\displaystyle\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(e^{r\tilde{A}}\vec{q}-\vec{Q}\right)\delta\left(e^{-rA}\vec{p}-\vec{P}\right)\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=$ $\displaystyle\Delta\left(e^{r\tilde{A}}\vec{q},e^{-rA}\vec{p}\right),$ Thus using Eqs.(27) and (31) the Wigner function of $V\left\|\mathbf{0}\right\rangle$ is $\displaystyle\left\langle\mathbf{0}\right\|V^{-1}\Delta_{n}\left(\vec{q},\vec{p}\right)V\left\|\mathbf{0}\right\rangle$ (32) $\displaystyle=$ $\displaystyle\frac{1}{\pi^{n}}\left\langle\mathbf{0}\right\|\colon\exp[-(e^{r\tilde{A}}\vec{q}-\vec{Q})^{2}-(e^{-rA}\vec{p}-\vec{P})^{2}]\colon\left\|\mathbf{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\pi^{n}}\exp[-(e^{r\tilde{A}}\vec{q})^{2}-\left(e^{-rA}\vec{p}\right)^{2}]$ $\displaystyle=$ $\displaystyle\frac{1}{\pi^{n}}\exp\left[-\widetilde{\vec{q}}e^{rA}e^{r\tilde{A}}\vec{q}-\widetilde{\vec{p}}e^{-r\tilde{A}}e^{-rA}\vec{p}\right]$ $\displaystyle=$ $\displaystyle\frac{1}{\pi^{n}}\exp\left[-\widetilde{\vec{q}}\left(\Lambda\widetilde{\Lambda}\right)^{-1}\vec{q}-\widetilde{\vec{p}}\Lambda\widetilde{\Lambda}\vec{p}\right],$ From Eq.(32) we see that once the explicit expression of $\Lambda\tilde{\Lambda}=\exp[-\lambda(A+\tilde{A})]$ is deduced, the Wigner function of $V\left\|\mathbf{0}\right\rangle$ can be calculated. ## 5 Some examples of calculating the Wigner function Taking $n=2$ as an example, $V_{n=2}$ is the usual two-mode squeezing operator. The matrix $A=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right),$ it then follows that $\Lambda\tilde{\Lambda}=e^{-\lambda(\tilde{A}+A)}=\left(\begin{array}[]{cc}\cosh 2\lambda&-\sinh 2\lambda\\\ -\sinh 2\lambda&\cosh 2\lambda\end{array}\right),$ (33) and $\left(\Lambda\tilde{\Lambda}\right)^{-1}=\left(\begin{array}[]{cc}\cosh 2\lambda&\sinh 2\lambda\\\ \sinh 2\lambda&\cosh 2\lambda\end{array}\right).$ (34) Substituting Eqs.(33) and (34) into Eq.(32), we have $\left\langle 00\right\|V^{-1}\Delta_{2}\left(\vec{q},\vec{p}\right)V\left\|00\right\rangle=\frac{1}{\pi^{2}}\exp\left[-2\left(\alpha_{1}^{\ast}\alpha_{2}^{\ast}+\alpha_{1}\alpha_{2}\right)\sinh 2\lambda-2\left(\left\|\alpha_{1}\right\|^{2}+\left\|\alpha_{2}\right\|^{2}\right)\cosh 2\lambda\allowbreak\right],$ (35) where $\alpha_{i}=\frac{1}{\sqrt{2}}\left(q_{i}+\mathtt{i}p_{i}\right),(i=1,2.)$. Eq.(35) is just the Wigner function of the usual two-mode squeezing vacuum state. For $n=3,$ we have $\Lambda\tilde{\Lambda}=\allowbreak\left(\begin{array}[]{ccc}u&v&\allowbreak v\\\ \allowbreak v&u&\allowbreak v\\\ v&v&u\end{array}\right),\text{ }u=\frac{2}{3}e^{\lambda}+\frac{1}{3}e^{-2\lambda},\text{ }v=\frac{1}{3}\left(\allowbreak e^{-2\lambda}-e^{\lambda}\right),$ (36) and $\left(\Lambda\tilde{\Lambda}\right)^{-1}$ is obtained by replacing $\lambda$ with $-\lambda$ in $\Lambda\tilde{\Lambda}$. By using Eq.(32) the Wigner function is $\displaystyle\left\langle\mathbf{0}\right\|V^{-1}\Delta_{3}\left(\vec{q},\vec{p}\right)V\left\|\mathbf{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\pi^{3}}\exp\left[-\frac{2}{3}\left(\cosh 2\lambda+2\cosh\lambda\right)\sum_{i=1}^{3}\left\|\alpha_{i}\right\|^{2}\right]$ (37) $\displaystyle\times\exp\left\\{-\frac{1}{3}\allowbreak\left(\sinh 2\lambda-2\sinh\lambda\right)\sum_{i=1}^{3}\alpha_{i}^{2}\right.$ $\displaystyle-\left.\frac{2}{3}\sum_{j>i=1}^{3}\left[\left(\cosh 2\lambda-\cosh\lambda\right)\alpha_{i}\alpha_{j}^{\ast}+\left(\allowbreak\sinh\lambda+\sinh 2\lambda\right)\alpha_{i}\alpha_{j}\right]+c.c\right\\}.$ For $n=4$ case we have (see the Appendix) $\Lambda\tilde{\Lambda}=\allowbreak\left(\begin{array}[]{cccc}u^{\prime}&w^{\prime}&v^{\prime}&w^{\prime}\\\ w^{\prime}&u^{\prime}&w^{\prime}&v^{\prime}\\\ v^{\prime}&w^{\prime}&u^{\prime}&w^{\prime}\\\ w^{\prime}&v^{\prime}&w^{\prime}&u^{\prime}\end{array}\right),$ (38) where $u^{\prime}=\cosh^{2}\lambda,v^{\prime}=\sinh^{2}\lambda,w^{\prime}=-\sinh\lambda\cosh\lambda$. Then substituting Eq.(38) into Eq.(32) we obtain $\left\langle\mathbf{0}\right\|V^{-1}\Delta_{4}\left(\vec{q},\vec{p}\right)V\left\|\mathbf{0}\right\rangle=\frac{1}{\pi^{4}}\exp\left\\{-2\cosh^{2}\lambda\left[\sum_{i=1}^{4}\left\|\alpha_{i}\right\|^{2}+\left(M+M^{\ast}\right)\tanh^{2}\lambda+\left(R^{\ast}+R\right)\allowbreak\tanh\lambda\right]\right\\},$ (39) where $M=\alpha_{1}\alpha_{3}^{\ast}+\alpha_{2}\alpha_{4}^{\ast},$ $R=\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{4}+\alpha_{2}\alpha_{3}+\alpha_{3}\alpha_{4}.$ This form differs evidently from the Wigner function of the direct-product of usual two two-mode squeezed states’ Wigner functions (35). In addition, using Eq. (38) we can check Eqs.(25) and (26). Further, using Eq.(38) we have $N^{-1}=\frac{1}{2}\allowbreak\left(\begin{array}[]{cccc}2&\tanh\lambda&0&\tanh\lambda\\\ \tanh\lambda&2&\tanh\lambda&0\\\ 0&\tanh\lambda&2&\tanh\lambda\\\ \tanh\lambda&0&\tanh\lambda&2\end{array}\right),\text{ }\det N=\cosh^{2}\lambda.$ (40) Then substituting Eqs.(40) and (A.4) into Eq.(18) yields the four-mode squeezed state, $V\left\|0000\right\rangle=\text{sech}\lambda\exp\left[-\frac{1}{2}\left(a_{1}^{{\dagger}}+a_{3}^{{\dagger}}\right)\left(a_{2}^{{\dagger}}+a_{4}^{{\dagger}}\right)\tanh\lambda\right]\left\|0000\right\rangle,$ (41) from which one can see that the four-mode squeezed state is not the same as the direct product of two two-mode squeezed states in Eq.(1). In sum, by virtue of the IWOP technique, we have introduced a kind of an n-mode squeezing operator $V\equiv\exp\left[\mathtt{i}\lambda\left(Q_{1}P_{2}+Q_{2}P_{3}+\cdots+Q_{n-1}P_{n}+Q_{n}P_{1}\right)\right]$, which engenders standard squeezing for the n-mode quadratures. We have derived $V$’s normally ordered expansion and obtained the expression of n-mode squeezed vacuum states and evaluated its Wigner function with the aid of the Weyl ordering invariance under similar transformations. Appendix: Derivation of Eq.(38) For the completeness of this paper, here we derive analytically Eq.(38). Noticing, for the case of $n=4,$ $A^{4}=I,$ $I$ is the 4$\times$4 unit matrix, from the Cayley-Hamilton theorem we know that the expanding form of $\exp(-r\tilde{A})$ must be $\Lambda=\exp(-\lambda\tilde{A})=c_{0}(\lambda)I+c_{1}(\lambda)\tilde{A}+c_{2}(\lambda)\tilde{A}^{2}+c_{3}(\lambda)\tilde{A}^{3}.$ (A.1) To determine $c_{j}(\lambda)$ , we take $\tilde{A}$ being $e^{\mathtt{i}(j/2)\pi}$ $(j=0,1,2,3)$ respectively, then we have $\left\\{\begin{array}[]{l}\exp(-\lambda)=c_{0}(\lambda)+c_{1}(\lambda)+c_{2}(\lambda)+c_{3}(\lambda),\\\ \exp(-\lambda e^{\mathtt{i}(1/2)\pi})=c_{0}(\lambda)+c_{1}(\lambda)e^{\mathtt{i}(1/2)\pi}+c_{2}(\lambda)e^{\mathtt{i}\pi}+c_{3}(\lambda)e^{\mathtt{i}(3/2)\pi},\\\ \exp(-\lambda e^{\mathtt{i}\pi})=c_{0}(\lambda)+c_{1}(\lambda)e^{\mathtt{i}\pi}+c_{2}(\lambda)e^{\mathtt{i}2\pi}+c_{3}(\lambda)e^{\mathtt{i}3\pi},\\\ \exp(-\lambda e^{\mathtt{i}(3/2)\pi})=c_{0}(\lambda)+c_{1}(\lambda)e^{\mathtt{i}(3/2)\pi}+c_{2}(\lambda)e^{\mathtt{i}(6/2)\pi}+c_{3}(\lambda)e^{\mathtt{i}(9/2)\pi}.\end{array}\right.$ (A.2) Its solution is $\left\\{\begin{array}[]{l}c_{0}(\lambda)=\frac{1}{2}\left(\cosh\lambda+\cos\lambda\right)\\\ c_{1}(\lambda)=\frac{1}{2}\left(-\sinh\lambda-\sin\lambda\right)\\\ c_{2}(\lambda)=\frac{1}{2}\left(\cosh\lambda-\cos\lambda\right)\\\ c_{3}(\lambda)=\frac{1}{2}\left(-\sinh\lambda+\sin\lambda\right)\end{array}\right..$ (A.3) It follows that $\Lambda=\left(\begin{array}[]{cccc}c_{0}&c_{3}&c_{2}&c_{1}\\\ c_{1}&c_{0}&c_{3}&c_{2}\\\ c_{2}&c_{1}&c_{0}&c_{3}\\\ c_{3}&c_{2}&c_{1}&c_{0}\end{array}\right),\det\Lambda=1,$ (A.4) and $\displaystyle\widetilde{\Lambda}\Lambda$ $\displaystyle=\left[c_{0}(\lambda)I+c_{1}(\lambda)A+c_{2}(\lambda)A^{2}+c_{3}(\lambda)A^{3}\right]\cdot\left[c_{0}(\lambda)I+c_{1}(\lambda)\tilde{A}+c_{2}(\lambda)\tilde{A}^{2}+c_{3}(\lambda)\tilde{A}^{3}\right]$ $\displaystyle=\frac{1}{2}\left(\begin{array}[]{cccc}\allowbreak 2\cosh^{2}\lambda&-\sinh 2\lambda&\allowbreak 2\sinh^{2}\lambda&-\sinh 2\lambda\\\ -\sinh 2\lambda&\allowbreak 2\cosh^{2}\lambda&-\sinh 2\lambda&\allowbreak 2\sinh^{2}\lambda\\\ \allowbreak 2\sinh^{2}\lambda&-\sinh 2\lambda&\allowbreak 2\cosh^{2}\lambda&-\sinh 2\lambda\\\ -\sinh 2\lambda&\allowbreak 2\sinh^{2}\lambda&-\sinh 2\lambda&\allowbreak 2\cosh^{2}\lambda\end{array}\right),$ (A.5) this is just Eq.(38). ACKNOWLEDGEMENT Work supported by the National Natural Science Foundation of China under grants 10775097 and 10874174. ## References * [1] Bouwmeester D. et al., _The Physics of Quantum Information_ , (Springer, Berlin) 2000. * [2] Nielsen M. A. and Chuang I. L., Quantum Computation and Quantum Information (Cambridge University Press) 2000. * [3] Buzek V., _J. Mod. Opt._ , 37 (1990) 303. * [4] Loudon R., Knight P. L., _J. Mod. Opt._ , 34 (1987) 709\. * [5] Dodonov V. V., _J. Opt. B: Quantum Semiclass. Opt._ , 4 (2002) R1. * [6] Fan H.-y and Klauder J. R., _Phys. Rev. A_ 49 (1994) 704. * [7] Fan H.-y and Fan Y., _Phys. Rev. A_ , 54 (1996) 958. * [8] Fan H.-y, _Europhys. Lett._ , 23 (1993) 1. * [9] Fan H.-y, _Europhys. Lett._ , 17 (1992) 285; Fan H.-y, _Europhys. Lett._ , 19 (1992) 443. * [10] Fan H.-y, _J Opt B: Quantum Semiclass. Opt._ , 5 (2003) R147. * [11] Mandel L. and Wolf E., _Optical Coherence and Quantum Optics_ (Cambridge University Press 1995) and references therein * [12] Einstein A., Poldolsky B. and Rosen N., _Phys. Rev._ , 47 (1935) 777. * [13] Wigner E. P., _Phys. Rev._ , 40 (1932) 749. * [14] O’Connell R. F. and Wigner E. P., _Phys. Lett. A_ , 83 (1981) 145. * [15] Schleich W., _Quantum Optics_ (New York: Wiley) 2001. * [16] Fan H.-y, _J. Phys. A_ , 25 (1992) 3443; Fan H.-y, Fan Y., _Int. J. Mod. Phys. A_ , 17 (2002) 701. * [17] Fan H.-y, _Mod. Phys. Lett. A_ , 15 (2000) 2297. * [18] Fan H.-y, _Ann. Phys._ , 323 (2008) 500; 323 (2008) 1502.	arxiv-papers	2009-02-10T06:49:03	2024-09-04T02:49:00.499403	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Li-yun Hu and Hong-yi Fan", "submitter": "Liyun Hu", "url": "https://arxiv.org/abs/0902.1589" }
0902.1727	# Molecular states near a collision threshold Paul S. Julienne Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, 100 Bureau Drive Stop 8423, Gaithersburg, Maryland 20899-8423, USA ###### Abstract [To appear as Chapter 6 of Cold Molecules: Theory, Experiment, Applications, ed. by Roman Krems et al. (Taylor and Francis, 2009)] ## I Introduction Real atoms are typically complex, having ground and excited states with spin structure. The molecules formed from the atoms typically have a rich spectrum of near-threshold bound and quasi-bound molecular states when the molecular spin, rotational, and vibrational structure is taken into account. When an ultracold gas of atoms is produced, the atoms are prepared in specific quantum states, and collisions between the atoms occur with an extremely precisely defined energy close to the $E=0$ collision threshold of the interacting atoms, where $E$ denotes energy. The collision then makes the near-threshold spectrum of the molecular complex of the two atoms accessible to electromagnetic probing. An external magnetic or electromagnetic field can be precisely tuned to couple the colliding atoms to a specific molecular state, which can be viewed as a scattering resonance. This permits both extraordinary spectral accuracy in probing near-threshold level positions (order of $E/h=10$ kHz accuracy for 1 $\mu$K atoms) and precise resonant control of the collisions that determine both static and dynamical macroscopic properties of quantum gases. Consequently, understanding the near-threshold bound and scattering states is essential for understanding the collisions and interactions of ultracold atoms. This is also true for interactions of ultracold molecules. This Chapter concentrates on understanding molecules that can be made by combining two cold atoms using either magnetically tunable Feshbach resonance states Köhler et al. (2006) or optically tunable photoassociation resonance states Jones et al. (2006). Such resonances provide a mechanism for the formation of ultracold molecules from already cold atoms. In addition, magnetically tunable resonances have been used very successfully to control the properties of ultracold quantum gases. This Chapter treats both magnetically and optically tunable molecular resonances with the same scattering theory framework. The viewpoint from quantum defect theory is emphasized of conceptually separating the interaction of the atoms into short range and long range regions. These regions are characterized by very different energy and length scales. Much insight about near-threshold collisions and bound states, as well as practical tools for their study, can be gained by taking advantage of this separation Julienne and Mies (1989); Moerdijk et al. (1995); Vogels et al. (1998); J. P. Burke et al. (1998); Vogels et al. (2000); Mies and Raoult (2000); Gao (2000, 2001); Julienne and Gao (2006). While molecular physics is typically concerned with strong short range interactions associated with ”ordinary molecules,” ultracold physics is concerned with scattering states and very weakly bound molecular states in the threshold domain near $E=0$. The long range potential, which has a lead term that varies as $1/R^{n}$, plays an important role in connecting these two regimes. We briefly summarize here the theory of cold collisions, which is described in detail in Chapter XXX. The scattering wavefunction is expanded in states of relative angular momentum of the two atoms characterized by partial wave quantum number $\ell=0,1,2\ldots$. Generally, the atoms can be initially prepared in one of several quantum states, and the scattering ”channels” can be specified by a collective set of quantum numbers $\alpha$ representing the state of each atom and the partial wave. Upon solving the Schrödinger equation for the system, the effect of all short-range interactions during a collision with $E>0$ is summarized in the scattering wavefunction for $R\to\infty$ by a unitary $S$-matrix. Only the lowest few partial waves can contribute to cold collisions, and in the limit $E\to 0$, only $s$-wave channels with $\ell=0$ have non-negligible collision cross sections. Using the complex scattering length $a-ib$ to represent the $s$-wave $S$-matrix element $S_{\alpha\alpha}=\exp{[-2ik(a-ib)]}$ in the limit $E\to 0$, the contribution to the elastic scattering cross section from $s$-wave collisions in channel $\alpha$ is $\sigma_{\mathrm{el}}=\lim_{E\to 0}g\frac{\pi}{k^{2}}\left\|1-S_{\alpha\alpha}\right\|^{2}=4g\pi\left(a^{2}+b^{2}\right)\,,$ (1) where $\hbar k=\sqrt{2\mu E}$ is the relative collision momentum in the center of mass frame for the atom pair with reduced mass $\mu$. The rate coefficient $K_{\mathrm{loss}}=\sigma_{\mathrm{loss}}v$ for $E\to 0$ $s$-wave inelastic collisions that remove atoms from channel $\alpha$ is $K_{\mathrm{loss}}=\lim_{E\to 0}g\frac{\pi\hbar}{\mu k}\left(1-\|S_{\alpha\alpha}\|^{2}\right)=2g\frac{h}{\mu}b$ (2) where $v=\hbar k/\mu$ is the relative collision velocity. The symmetry factor $g=1$ when the atoms are bosons or fermions that are not in identical states, $g=2$ or $g=1$ respectively for two bosons in identical states in a normal thermal gas or a Bose-Einstein condensate, and $g=0$ for two fermions in identical states. If there are no exoergic inelastic channels present, then $b=0$ and only elastic collisions are possible. The Schroödinger equation also determines the bound states with discrete energies $E_{i}<0$. While the conventional picture of molecules counts the bound states by vibrational quantum number $v=0,1\ldots$ from the lowest energy ground state up, it is more helpful for the present discussion to count the near-threshold levels from the $E=0$ dissociation limit down by quantum numbers $i=-1,-2\ldots$. In the special case where $a\to+\infty$, the energy of the last bound $s$-wave state of the system with $i=-1$ depends only on $a$ and $\mu$ and takes on the following ”universal” form: $E_{-1}=-\frac{\hbar^{2}}{2\mu a^{2}}\,\,\mathrm{as}\,\,a\to+\infty\,.$ (3) Section II describes the bound and scattering properties of a single potential with a van der Waals long range form. Section III extends the treatment to multiple states and scattering resonances. Sections IV and V respectively discuss the properties of magnetically and optically tunable molecular resonance states. ## II Properties for a single potential In this section let us ignore any complex internal atomic structure and first consider two atoms $A$ and $B$ that interact by a single adiabatic Born- Oppenheimer interaction potential $V(R)$, illustrated schematically in Fig. 1. The wavefunction for the system is $\|\alpha\rangle\|\psi_{\ell}\rangle/R$, where $\|\alpha\rangle$ represents the electronic and rotational degrees of freedom, and the wavefunction for relative motion is found from the radial Schrödinger equation $-\frac{\hbar^{2}}{2\mu}\frac{d^{2}\psi_{\ell}}{dR^{2}}+\left(V(R)+\frac{\hbar^{2}\ell(\ell+1)}{2\mu R^{2}}\right)\psi_{\ell}=E\psi_{\ell}\,.$ (4) Solving Eq. 4 gives the spectrum of bound molecular states $\psi_{i\ell}$ with energy $E_{i\ell}=-\hbar^{2}k_{i\ell}^{2}/(2\mu)<0$ and the scattering states $\psi_{\ell}(E)$ with collision kinetic energy $E=\hbar^{2}k^{2}/(2\mu)>0$, where $k_{i\ell}$ and $k$ have units of $(\mathrm{length})^{-1}$. As $R\to\infty$, the bound states decay as $e^{-k_{i\ell}R}$ and the scattering states approach $\psi_{\ell}(E)\to c\sin(kR-\pi\ell/2+\eta_{\ell})/k^{1/2}\,.$ (5) Bound states are normalized to unity, $\|\langle\psi_{i\ell}\|\psi_{j\ell^{\prime}}\rangle\|^{2}=\delta_{ij}\delta_{\ell\ell^{\prime}}$. We choose the normalization constant $c=\sqrt{2\mu/\hbar^{2}\pi}$ so that scattering states are normalized per unit energy, $\langle\psi_{\ell}(E)\|\psi_{\ell^{\prime}}(E^{\prime})\rangle=\delta(E-E^{\prime})\delta_{\ell\ell^{\prime}}$. Thus, the energy density of states is included in the wavefunction when taking matrix elements involving scattering states. The long range potential between the two atoms varies as $-C_{n}/R^{n}$. We are especially interested in the case of $n=6$ for the van der Waals interaction between two neutral atoms. This is the lead term in the long-range expansion of the potential in inverse powers of $R$ that applies to many atoms that are used in ultracold experiments. This potential has a characteristic length scale of $R_{\mathrm{vdw}}=\sqrt[4]{2\mu C_{6}/\hbar^{2}}/2$ that depends only on the values of $\mu$ and $C_{6}$ Jones et al. (2006). Values of $C_{6}$ are tabulated by Derevianko Derevianko et al. (1999) for alkali-metal species and by Porsev and Derevianko Porsev and Derevianko (2006) for alkaline-earth species. We prefer to use a closely related van der Waals length introduced by Gribakin and Flambaum Gribakin and Flambaum (1993) $\bar{a}=4\pi/\Gamma(1/4)^{2}\,R_{\mathrm{vdw}}=0.955978\dots\,R_{\mathrm{vdw}}\,,$ (6) where $\Gamma(x)$ is the Gamma function. This length defines a corresponding energy scale $\bar{E}=\hbar^{2}/(2\mu\bar{a}^{2})$. The parameters $\bar{a}$ and $\bar{E}$ occur frequently in formulas based on the van der Waals potential. The wavefunction approaches its asymptotic form when $R\gg\bar{a}$ and is strongly influenced by the potential when $R\lesssim\bar{a}$. Table 1 gives the values of $\bar{a}$ and $\bar{E}$ for several species used in ultracold experiments. Table 1: Characteristic van der Waals scales $\bar{a}$ and $\bar{E}$ for several atomic species. (1 amu = 1/12 mass of a 12C atom, 1 au= 1 $E_{h}a_{0}^{6}$ where $E_{h}$ is a hartree and 1 $a_{0}$= 0.0529177 nm) Species \| mass \| C6 \| $\bar{a}$ \| $\bar{E}/h$ \| $\bar{E}/k_{B}$ ---\|---\|---\|---\|---\|--- \| (amu) \| (au) \| ($a_{0}$) \| (MHz) \| (mK) 6Li \| 6.015122 \| 1393 \| 29.88 \| 671.9 \| 32.25 23Na \| 22.989768 \| 1556 \| 42.95 \| 85.10 \| 4.084 40K \| 39.963999 \| 3897 \| 62.04 \| 23.46 \| 1.126 87Rb \| 86.909187 \| 4691 \| 78.92 \| 6.668 \| 0.3200 88Sr \| 87.905616 \| 3170 \| 71.76 \| 7.974 \| 0.3827 133Cs \| 132.905429 \| 6860 \| 96.51 \| 2.916 \| 0.1399 174Yb \| 173.938862 \| 1932 \| 75.20 \| 3.670 \| 0.1761 Figure 1: Schematic figure of the potential energy curve $V(R)$ as a function of the separation $R$ between two atoms $A$ and $B$. The horizontal lines labeled $AB$ indicate a spectrum of molecular bound states leading up to the molecular dissociation limit at $E=0$, indicated by the dashed line. The long range potential varies as $-C_{n}/R^{n}$. Samples of cold atoms can be prepared with kinetic temperatures on the order of nK to mK. The energy associated with temperature $T$ is $k_{B}T$ where $k_{B}$ is the Boltzmann constant. For example, at $T=1$ $\mu$K, $k_{B}T=0.86$ neV and $k_{B}T/h=21$ kHz. This ultracold energy scale is 9 to 10 orders of magnitude smaller than the energy scale of 1 to 10 eV associated with ground or excited state interaction energies when a molecule is formed at small interatomic separation $R_{\mathrm{bond}}$ on the order of a chemical bond length. In a cold collision, the initially separated atoms have very low collision energy $E=\hbar^{2}k^{2}/(2\mu)\approx 0$ and very long de Broglie wavelength $2\pi/k$. The atoms come together from large distance $R$ and are accelerated by the interatomic potential $V(R)$, so that when they reach distances on the order of $R_{\mathrm{bond}}$ they have very high kinetic energy on the order of $\|V(R_{\mathrm{bond}})\|$. The local de Broglie wavelength $2\pi/k(R,E)$ in the short range classical part of the potential, where $k(R,E)=\sqrt{2\mu(E-V(R))}/\hbar$, is orders of magnitude smaller than the separated atom de Broglie wavelength and is nearly independent of the value of $E$, which is close to $0$. Figure 2: Radial wavefunction $\psi_{0}(R)$ for $\ell=0$ at $E/k_{b}=1$ $\mu$K for the pairs 174Yb-174Yb (solid), 171Yb-171Yb (dashed), and 170Yb-173Yb (dotted), which have respective scattering lengths of 105 a0, -3 a0, and $-81$ a0 Kitagawa et al. (2008). The inset shows an expanded view of the wavefunction on a smaller length scale on the order of $\bar{a}$, the characteristic length of the van der Waals potential. The 174Yb-174Yb case shows the oscillations that develop when $R<\bar{a}$. This separation of scales is illustrated in Fig. 2, which shows examples of $s$-wavefunctions at a collision energy $E/k_{B}=$ 1 $\mu$K, where dividing $E$ by $k_{B}$ allows us to express energy in temperature units. This example uses three isotopic combinations of pairs of Yb atoms, which has a spinless 1S0 electronic configuration and a single ground state electronic Born- Oppenheimer potential $V(R)$. The species Yb makes a good example case to illustrate the principles in this section, since it has 7 stable isotopes and 28 different atom pairs of different isotopic composition for which the threshold properties have been worked out Kitagawa et al. (2008). All combinations have the same $V(R)$ but different reduced masses. This mass- scaling approximation, which ignores very small mass-dependent corrections to the potential, is normally quite good except for very light species such as Li. Fig. 2 shows that the three examples have similar phase-shifted sine waves with a common long de Broglie wavelength of $2\pi/k=6300$ a0. For small $R$ where $kR\ll 1$ the sine function vanishes as $c\sin k(R-a)/\sqrt{k}\to c\sqrt{k}(R-a)$. The actual wavefunction oscillates rapidly at small $R$ due to the influence of the potential. Since the asymptotic form for $kR\ll 1$ varies as $k^{1/2}$ as $k\to 0$, the short range oscillating part also has an amplitude proportional to $k^{1/2}$ in order to connect smoothly to the asymptotic form as $k\to 0$. This property ensures that the threshold matrix elements that characterize Feshbach resonances and $s$-wave inelastic scattering are proportional to $k^{1/2}$. Figure 3: wavefunctions for the last $s$-wave $i=-1$ bound state (solid line) with $E_{-1,0}/h=-10.6$ MHz and for the $s$-wave scattering state (dashed line) for $E/h=0.02$ MHz ($E/k_{B}=1$ $\mu$K) for two 174Yb atoms. Both wavefunctions are given a common JWKB normalization at small $R\ll\bar{a}$ and are nearly indistinguishable for $R<\bar{a}$. The potential supports $N=72$ bound states, and the wavefunction for this $i=-1$ and $v=71$ level has $N-1=71$ nodes. Figure 3 illustrates more clearly the nature of threshold short range scattering and bound state wavefunctions. When given an appropriate short range normalization, near-threshold scattering and bound state wavefunctions have a common amplitude and phase in the region of $R$ small compared to the range $\bar{a}$ of the long range potential. While this can be put on a rigorous quantitative ground within the framework of quantum defect theory Mies and Raoult (2000), it is easy to show using the familiar JWKB approximation Julienne and Mies (1989); Vogels et al. (2000); Jones et al. (2006). We can always write the wavefunction in phase-amplitude form $\psi_{\ell}(R,E)=\alpha_{\ell}(R,E)\sin{\beta_{\ell}(R,E)}$ and transform the Schrödinger equation (4) into a set of equations for $\alpha_{\ell}$ and $\beta_{\ell}$. The asymptotic $\psi_{\ell}(R,E)$ in Eq. (5) clearly corresponds to this form with $\alpha_{\ell}\to c/k^{1/2}$ as $R\to\infty$. Another familiar form is the JWKB semiclassical wavefunction $\psi_{\ell}^{JWKB}(R,E)$, for which $\displaystyle\alpha_{\ell}^{JWKB}(R,E)$ $\displaystyle=$ $\displaystyle c/k_{\ell}(R,E)^{1/2}$ (7) $\displaystyle\beta_{\ell}^{JWKB}(R,E)$ $\displaystyle=$ $\displaystyle\int_{R_{t}}^{R}k_{\ell}(R^{\prime},E)dR^{\prime}+\frac{\pi}{4}\,.$ (8) where $R_{t}$ is the inner classical turning point of the potential. When the collision energy $E$ is sufficiently large, so there are no threshold effects, the JWKB approximation is a excellent approximation at all $R$, and the form of $\alpha_{\ell}^{JWKB}(R,E)$ in Eq. (7) applies at all $R$, transforming into the correct quantum limit as $R\to\infty$. On the other hand, the JWKB approximation fails for $s$-waves with very low collision energy. This failure occurs in a region of $R$ near $\bar{a}$ and for collision energies $E$ on the order of $\bar{E}$ or less. The consequence is that the JWKB wavefunction, with the normalization in Eq. (7), is related to the actual wavefunction, with the asymptotic form in Eq. (5), by a multiplicative factor $C_{\ell}(E)$, so that as $E\to 0$ $\psi_{\ell}(R,E)=C_{\ell}(E)^{-1}\psi_{\ell}^{JWKB}(R,0)\,.$ (9) As $k\to 0$ for a van der Waals potential varying as $1/R^{6}$, the $s$-wave threshold form is $C_{0}(E)^{-2}=k\bar{a}[1+(r-1)^{2}]$, where $r=a/\bar{a}$ is the dimensionless scattering length in units of $\bar{a}$ Mies and Raoult (2000). Equation (9) gives an excellent approximation for the threshold $\psi_{0}(R,E)$ for $R<\bar{a}$ and $k<1/a$. At high energy, when $E\gg\bar{E}$, $C_{0}(E)^{-1}$ approaches unity and the JWKB approximation for $\psi_{0}(R,E)$ applies at all $R$. The unit normalized bound state wavefunction $\psi_{i\ell}(R)$ can be converted to an ”energy normalized” form by multiplying by $\|\partial i/\partial E_{i\ell}\|^{1/2}$, where $-\partial i/\partial E_{i\ell}>0$ is the energy density of states. Away from threshold, this is just the inverse of the mean spacing between levels, whereas for $s$-wave levels near threshold for a van der Waals potential , $\partial i/\partial E_{i0}\to r/(2\pi\bar{E})^{-1}$ as $k_{-1,0}=1/a\to 0$ Mies and Raoult (2000). The relation of $\psi_{i\ell}$ to the energy-normalized JWKB form in the classically allowed region of the potential is $\psi_{i\ell}(R,E_{i\ell})=\left\|\frac{\partial i}{\partial E_{i\ell}}\right\|_{E_{i\ell}}^{-1/2}\psi_{\ell}^{JWKB}(R,E_{i\ell})\,,$ (10) Figure 3 plots $C_{0}(E)\psi_{0}(R,E)\approx\psi_{0}^{JWKB}(R,0)$ for the scattering state and $\|\partial i/\partial E_{i0}\|^{1/2}\psi_{i0}(R,E_{i0})\approx\psi_{0}^{JWKB}(R,0)$ for the $i=-1$ bound state. Thus the near-threshold bound and scattering wavefunctions, when given a common short range normalization, are nearly identical and are well approximated by $\psi_{0}^{JWKB}(R,0)$ in the region $R<\bar{a}$. For $R>\bar{a}$ the wavefunctions begin to take on their asymptotic form as $R\to\infty$. The shape of the wavefunction at very small $R$ on the order of $R_{\mathrm{bond}}$ is usually independent of $E$ for ranges of $E/k_{B}$ on the order of many K. The short range shape is even independent of $\ell$ for small $\ell$, since the rotational energy is very small compared to typical values of $V(R_{\mathrm{bond}})$. However, the amplitudes of the wavefunctions depend strongly on the whole potential, which determines $a$, and are analytically related to the form of the long range potential. The separation of scales for $R>\bar{a}$ and $R<\bar{a}$ is a key feature of ultracold physics that enables much physical insight as well as practical approximations to be developed about molecular bound and quasibound states and collisions. Given that $C_{6}$, $\mu$, and the $s$-wave scattering length $a$ are known, the Schrödinger equation (4) can be integrated inward using the form of Eq. (5) as $k\to 0$ as a boundary condition, thus giving the wavefunction and nodal pattern for $R<\bar{a}$ as $E\to 0$. Assume that it is possible to pick some $R=R_{m}$ such that $R_{\mathrm{bond}}\ll R_{m}\ll\bar{a}$ and $V(R_{m})$ is well-represented by its van der Waals form. Then the log of the derivative of the wavefunction at $R_{m}$, which also can be calculated, provides an inner boundary condition, independent of $E$ over a wide range of $E$, for matching the wavefunction at $E$ propagated from large $R$. All that is needed to do this is to know $a$, $\mu$ and the long range potential. Thus, it is readily seen that all of the near threshold bound and scattering states, even those for $\ell>0$, can be calculated to a very good approximation for $R>R_{m}$ once $C_{6}$, $\mu$, and $a$ are known. Figure 4 shows the spectrum of bound states $E_{i\ell}$, in units of $\bar{E}$, for $\ell$ up to 5 for two cases of scattering length, based on the van der Waals quantum defect theory of Gao Gao (2000, 2001). Panel (a) shows the case of $a=\pm\infty$, where there is a bound state at $E=0$. The locations of the bound states for $a=\pm\infty$ define the boundaries of the ”bins” in which, for any $a$, there will be one and only one $s$-wave bound state, for example, $-36.1\bar{E}<E_{-1,0}<0$ and $-249\bar{E}<E_{-2,0}<-36.1\bar{E}$. The panel also shows the rotational progressions for each level as $\ell$ increases. The $a=\pm\infty$ van der Waals case also follows a ”rule of 4”, where partial waves $\ell=4,8,\ldots$ also have a bound state at $E=0$. Panel (b) shows how the spectrum changes when $a=\bar{a}$, for which the there is a $d$-wave level at $E=0$. Similar spectra can be calculated for any $a$. Gribakin and Flambaum Gribakin and Flambaum (1993) showed that the near- threshold $s$-wave bound state for a van der Waals potential in the limit $a\gg\bar{a}$ is modified from the universal form in Eq. (3) as $E_{-1}=-\frac{\hbar^{2}}{2\mu(a-\bar{a})^{2}}\,.$ (11) This approaches the universality limit when $a\gg\bar{a}$, in which case the $s$-wave wavefunction takes on the universal form $\psi_{0}(R,E)=\sqrt{2/a}e^{-R/a}$. Such an exotic bound state, known as a “halo molecule,” exists primarily in the nonclassical domain beyond the outer classical turning point of the long-range potential with an expectation value of $R$ of $a/2$, which grows without bound as $a\to+\infty$ Köhler et al. (2006). Figure 4: Dimensionless bound state energies $E_{i\ell}/\bar{E}$ for partial waves $\ell=0\ldots 5$ $(s,p,d,f,g,h)$. Panel (a) is for the case $a=\pm\infty$ and Panel (b) is for $a=\bar{a}$. Figure 5: The upper panel shows $s$-wave scattering length and the lower panel shows bound state binding energies $-E_{i0}(\lambda)$ for Yb2 molecular dimers versus the control parameter $\lambda=2\mu$. The vertical dashed lines show the points of singularity of $a(\lambda)$. The horizontal dashed lines show the boundaries of the bins in which the $i=-1$ and $i=-2$ levels must lie. Bound state and scattering properties are closely related. It is instructive to imagine that there is some control parameter $\lambda$ that can be varied to make the scattering length vary over its whole range from $+\infty$ and $-\infty$, changing the corresponding bound state spectrum. One way to do this would be to vary the reduced mass. Of course, this is not physically possible. However, there are elements with many isotopes, so that a wide range of discrete reduced masses are possible. An excellent physical system to illustrate this is the Ytterbium atom, used in the examples of Figs. 2 and 3. The stable isotopes with masses 168, 170, 172, 174, and 176 are all spinless bosons and the 171 and 173 isotopes are fermions with spin $1/2$ and $5/2$ respectively. Yb atoms can be cooled into the $\mu$K domain and all isotopes, including the fermionic ones in different spin states, have s-wave interactions.. The locations of several $\ell=0$ and 2 threshold bound states of different isotopic combinations of Yb atoms in Yb2 dimer molecules have been measured, and the long range potential parameters and scattering lengths determined Kitagawa et al. (2008). Figure 5 shows the $s$-wave scattering length and bound state binding energies versus the continuous control parameter $\lambda=2\mu$. Physically, there are 28 discrete values between $\lambda=$168 and 176. The scattering length has a singularity, and a new bound state occurs with increasing $\lambda$, at $\lambda=$ 167.3, 172.0, and 177.0. The range between 167.3 and 172 corresponds to exactly $N=71$ bound states in the model potential used. Near $\lambda=167.3$ the last $s$-wave bound state energy $E_{-1,0}\to 0$ as $-\hbar^{2}/(2\mu a^{2})$ as $a\to+\infty$. The binding energy $\|E_{-1,0}\|$ gets larger as $\lambda$ increases and $a$ decreases, so that for a van der Waals potential $E_{-1,0}$ approaches the lower edge of its ”bin” at $-36.1\bar{E}$ as $a\to-\infty$. As $\lambda$ increases beyond $172.0$, the $i=-1$ level becomes the $i=-2$ level as a new $i=-1$“last” bound state appears in the spectrum. The variation of scattering length with $2\mu$ is given by a remarkably simple formula. While semiclassical theory breaks down at threshold, Gribakin and Flambaum Gribakin and Flambaum (1993) showed that the correct quantum mechanical relation between $a$ and the potential is $a=\bar{a}\left[1-\tan{\left(\Phi-\frac{\pi}{8}\right)}\right]\,,$ (12) where $\Phi=\int_{R_{t}}^{\infty}\sqrt{-2\mu V(R)/\hbar^{2}}=\beta_{0}^{JWKB}(\infty,0)-\pi/4\,.$ (13) The number of bound states in the potential is $N=[\Phi/\pi-5/8]+1$, where $[\ldots]$ means the integer part of the expression. These expressions work remarkably well in practice. Although the results in Fig. 5 are obtained by solving the Schrödinger equation for a realistic potential, virtually identical results are obtained for $a$ from Eq. (12). In fact, $a$ and $E_{i0}$ are nearly the same on the scale of Fig. 5 if the simple hard-core van der Waals model of Gribakin and Flambaum (1993) is used for the potential, namely $V(R)=-C_{6}/R^{6}$ if $R\geq R_{0}$ and $V(R)=+\infty$ if $R<R_{0}$, where the cutoff $R_{0}$ is chosen to fit $a$ or $E_{-1,0}$ data from two different isotopes. With the mass scaling $\propto\sqrt{\mu}$ in Eq. (13), knowing $C_{6}$ and $E_{-1,0}$ for two isotopic pairs determines $a$ and $E_{-1,0}$ for all isotopic pairs. The approximation is fairly good even for levels with larger $\|i\|$ or $\ell>0$, although it will become worse as $\|i\|$ or $\ell$ increase. In summary, it is very useful to take advantage of the enormous difference in energy and length scales associated with the cold separated atoms and deeply bound molecular potentials. This allows us to introduce a generalized “quantum defect” approach for understanding threshold physics Julienne and Mies (1989); J. P. Burke et al. (1998); Gao (2001); Vogels et al. (2000); Mies and Raoult (2000). Threshold bound state and scattering properties are determined mainly by the long range potential, once the overall effect of the whole potential is known through the $s$-wave scattering length. A similar analysis can be developed for other long range potential forms, for example, $1/R^{4}$ ion- induced dipole or $1/R^{3}$ dipole-dipole interactions. ## III Interactions for multiple potentials Generally the cold atoms used in experiments have additional angular momenta (electron orbital and/or electron spin and/or nuclear spin), so that more than one scattering channel $\alpha$ can be involved in a collision. Each channel has a separated atom channel energy $E_{\alpha}$. Fig. 1 could be modified to illustrate such channels by adding additional potentials and their corresponding spectra dissociating to the $E_{\alpha}$ limits. If $E_{tot}$ is the total energy of the colliding system, the designation open or closed is used for channels with $E_{tot}>E_{\alpha}$ or $E_{tot}<E_{\alpha}$ respectively. Inelastic collisions from entrance channel $\alpha$ are possible to open exit channels $\beta$ when $E_{\alpha}>E_{\beta}$, whereas closed channels $\beta$ can support quasibound states as scattering resonances when $E_{\alpha}<E_{tot}<E_{\beta}$. The ability to tune resonance states to control scattering properties or to convert them into true molecular bound states is an important aspect of ultracold physics that has been exploited in a wide variety of experiments with bosonic or fermionic atoms Köhler et al. (2006). Let us first examine the basic magnitude of the $s$-wave inelastic collision rates that are possible when open channels are present. The rate constant is determined by the magnitude of $b$ in Eq. (2), for which a typical order of magnitude is $b\approx\bar{a}$ for an allowed transition, that is, one with a relatively large short-range interactions in the system Hamiltonian. The rate constant can be written $K_{\mathrm{loss}}=0.84\times 10^{-10}g\frac{b[\mathrm{au}]}{\mu[\mathrm{amu}]}\,\,\mathrm{cm}^{3}/\mathrm{s}\,,$ (14) where $b$ is expressed in atomic units (1 au $=$ 0.0529177 nm) and $\mu$ in atomic mass units ($\mu=12$ for 12C). Allowed processes will typically have the order of magnitude of 10-10 cm${}^{3}/$s for $K_{\mathrm{loss}}$. The $s$-wave $K_{\mathrm{loss}}$ can be even larger, with an upper bound of $b_{u}=1/(4k)$ being imposed by the unitarity property of the $S$-matrix, i. e., $0\leq 1-\|S_{\alpha\alpha}\|^{2}\leq 1$. Since the lifetime relative to collision loss is $\tau=1/(K_{\mathrm{loss}}n)$, where $n$ is the density of the collision partner, allowed processes result in fast loss with $\tau\lesssim 1$ ms at typical quantum degenerate gas densities. This applies to atom-molecule and molecule-molecule collisions as well as atom-atom collisions. Such losses need to be avoided by working with atomic or molecular states that do not experience fast loss collisions, such as the lowest energy ground state level, which does not have exoergic 2-body exit channels. Alternatively, placing the species in a lattice cell that confines a single atom or molecule can offer protection against collisional loss. An alternative formulation of the collision loss rate is possible by rewriting Eq. (2), not taking the $E\to 0$ limit but introducing a thermal average over a Maxwellian distribution of collision energies $E$, $K_{loss}=g\frac{1}{Q_{T}}\frac{k_{B}T}{h}\sum_{\alpha}\left\langle 1-\|S_{\alpha\alpha}\|^{2}\right\rangle_{T}\,,$ (15) where $Q_{T}$ is the translational partition function, $1/Q_{T}=(2\pi\mu k_{B}T/h^{2})^{3/2}=\Lambda_{T}^{3}$ where $\Lambda_{T}$ is the molecular thermal de Broglie wavelength. The $\langle\ldots\rangle_{T}$ expression implies a thermal average over the velocity distribution. The sum represents a dynamical factor $f_{D}$ that varies as $T^{1/2}$ as $T\to 0$ and has an upper bound of unity for $s$-waves and $\approx\ell_{\mathrm{max}}^{2}$ if $\ell_{\mathrm{max}}$ partial waves contribute at the unitarity limit. Although Eq. (15) reduces to Eq. (14) in the $T\to 0$ $s$-wave limit, it lets us see that the collision rate is given by an expression having the form $\tau^{-1}=K_{loss}n=g(n\Lambda_{T}^{3})\frac{k_{B}T}{h}f_{D}\,.$ (16) This form embodies some general principles for any collisions of atoms and molecules. The dimensionless $n\Lambda_{T}^{3}$ factor shows that the collision rate is proportional to phase space density of the collision partner (scale by mass ratios to convert to an atomic phase space density). The $k_{B}T/h$ factor sets an intrinsic rate scale (dimension of inverse time) associated with $T$. The dimensionless factor $f_{D}$ embodies all of the detailed collision dynamics. Even using fast time-dependent manipulations to control $f_{D}$ does not change the fundamental thermodynamic limits imposed by the phase space density and $k_{B}T/h$ factors. Given Eqs. (14) and (15) and plausible assumptions about $b$ or $f_{D}$, it is possible to estimate the time scales for a wide variety of atomic and molecular collision processes under various kinds of conditions. Now we will examine the important case of tunable resonant scattering when a closed channel is present. Assume that open entrance channel $\alpha$, with $E_{\alpha}$ chosen as $E_{\alpha}=0$, is coupled through terms in the system Hamiltonian to a closed channel $\beta$ with $0<E<E_{\beta}$. Then a molecular bound state in channel $\beta$ becomes a quasibound state that acts as a scattering resonance in channel $\alpha$. Using Fano’s form of resonant scattering theory Fano (1961), let us assume a ”bare” or uncoupled approximate bound state $\|C\rangle=\psi_{c}(R)\|c\rangle$ with energy $E_{c}$ in the closed channel $\beta=c$ and a ”’bare” or background scattering state $\|E\rangle=\psi_{bg}(R,E)\|bg\rangle$ at energy $E$ in the entrance channel $\alpha=bg$. The scattering phase shift $\eta(E)=\eta_{bg}(E)+\eta_{res}(E)$ of the coupled system picks up a resonant part due to the Hamiltonian coupling $W(R)$ between the ”bare” channels. Here $\eta_{bg}$ is the phase shift due to the uncoupled single background channel, as described in the last Section, and $\eta_{res}(E)=-\tan^{-1}\left(\frac{\frac{1}{2}\Gamma(E)}{E-E_{c}-\delta E(E)}\right)\,,$ (17) has the standard Breit-Wigner resonance scattering form. The two key features of the resonance are its width $\Gamma(E)=2\pi\|\langle C\|W(R)\|E\rangle\|^{2}\,,$ (18) and its shift $\delta E(E)={\cal{P}}\int_{-\infty}^{\infty}\frac{\|\langle C\|W(R)\|E^{\prime}\rangle\|^{2}}{E-E^{\prime}}dE^{\prime}\,.$ (19) The primary difference between an ”ordinary” resonance and a threshold one as $E\to 0$ is that for the former we normally make the assumption that $\Gamma(E)$ and $\delta E(E)$ are evaluated at $E=E_{c}$ and are independent of $E$ across the resonance. By contrast, the explicit energy dependence of $\Gamma(E)$ and $\delta E(E)$ are key features of threshold resonances Bohn and Julienne (1999); Julienne and Gao (2006); Marcelis et al. (2004). In the special case of the $E\to 0$ limit for $s$-waves, $\displaystyle\frac{1}{2}\Gamma(E)$ $\displaystyle\to$ $\displaystyle(ka_{bg})\Gamma_{0}$ (20) $\displaystyle E_{c}+\delta E(E)$ $\displaystyle\to$ $\displaystyle E_{0}\,,$ (21) where $\Gamma_{0}$ and $E_{0}$ are $E$-independent constants. Note that $\Gamma(E)$ is positive definite, so that $\Gamma_{0}$ has the same sign as $a_{bg}$. Assuming an entrance channel without inelastic loss, so that $\eta_{bg}(E)\to-ka_{bg}$, and for the sake of generality, adding a decay rate $\gamma_{c}/\hbar$ for the decay of the bound state $\|C\rangle$ by irreversible loss processes, gives in the limit of $E\to 0$, $\tilde{a}=a-ib=a_{bg}-\frac{a_{bg}\Gamma_{0}}{E_{0}-i(\gamma_{c}/2)}\,.$ (22) This formalism accounts for both kinds of tunable resonances that are used for making cold molecules from cold atoms, namely, magnetically or optically tuned resonances. We now give our attention to each of these in turn. ## IV Magnetically tunable resonances Cold alkali metal atoms have a variety of magnetically tunable resonances that have been exploited in a number of experiments to control the properties of ultracold quantum gases or to make cold molecules. For the most part, experiments have succeeded with species that either do not have inelastic loss channels, or if they do, the loss rates are very small. Thus, for practical purposes, we can set the resonance decay rate $\gamma_{c}=0$ in examining a wide class of magnetically tunable resonances. While general coupled channel methods can be set up to solve the multichannel Schrödinger equation Köhler et al. (2006), we will use simpler models to explain the basic features of tunable Feshbach resonance states. Many resonances occur for alkali metal species in their 2S electronic ground state because of their complex hyperfine and Zeeman substructure with energy splittings very large compared to $k_{B}T$. Thus, closed spin channels that have bound states near $E_{\alpha}$ of an entrance channel $\alpha$ can serve as tunable scattering resonances for threshold collisions in that channel. The key to magnetic tuning of a resonance is that the resonance state $\|C\rangle$ has a different magnetic moment $\mu_{c}$ than the moment $\mu_{\mathrm{atoms}}$ of the pair of separated atoms in the entrance channel. The bare bound state energy can be tuned by varying the magnetic field $B$ $E_{c}(B)=\delta\mu(B-B_{c})\,,$ (23) where $\delta\mu=\mu_{\mathrm{atoms}}-\mu_{c}$ is the magnetic moment difference and $B_{c}$ is the field where $E_{c}(B_{c})=0$ at threshold. The scattering length is real with $b=0$ and takes on the following resonant form $a(B)=a_{bg}-a_{bg}\frac{\Delta}{B-B_{0}}\,,$ (24) where $\Delta=\frac{\Gamma_{0}}{\delta\mu}\quad{\rm and}\quad B_{0}=B_{c}+\delta B\,.$ (25) Note that the interaction between the entrance and closed channels shifts the point of singularity of $a(B)$ from $B_{c}$ to $B_{0}$. Such magnetically tunable Feshbach resonances are characterized by four parameters, namely, the background scattering length $a_{bg}$. the magnetic moment difference $\delta\mu$, the resonance width $\Delta$, and position $B_{0}$. Figure 6: Molecular bound state energies (lower panel) and scattering length (upper panel) versus magnetic field $B$ in mT (1 mT $=$ 10 Gauss) for the lowest energy $\alpha=1$ $s$-wave spin channel of the 40K87Rb fermionic molecule. The bound state energies are shown relative to the channel energy $E_{1}$ of the two separated atoms taken to be zero. This $\alpha=1$ spin channel has respective 40K and 87Rb spin projection quantum numbers of $-9/2$ and $+1$, giving a total projection of $-7/2$. In this species there are 11 additional closed $s$-wave channels with $E_{\alpha}>E_{1}$ and with the same projection of $-7/2$. The bound state quantum numbers are $\alpha(i)$, where $i$ is the vibrational quantum number relative to the dissociation limit of closed channel $\alpha=2,\ldots,12$. Four bound states cross threshold in this range of $B$, giving rise to singularities in the scattering length. Figure 6 shows an example of the scattering length and bound state energies for the 40K87Rb molecule near the lowest energy spin channel of the separated atoms. The spin quantum numbers and hyperfine splitting in their respective electronic ground states are 1, 2, and 6.835 GHz for 87Rb and $9/2$, $7/2$ and $-1.286$ GHz (inverted) for 40K. There are 11 other closed spin channels in this system with $E_{\alpha}>E_{1}$ that have the same total projection quantum number as the lowest energy $\alpha=1$ spin channel. Because of their different magnetic moments the energy of a bound state of one of these closed channels can be tuned relative to the energy of the two separated atoms in the $\alpha=1$ $s$-wave channel, as shown in the Figure. Due to coupling terms in the Hamiltonian among the various channels, bound states that cross threshold couple to the entrance channel and give rise to resonance structure in its $a(B)$. The resonance with $B_{0}$ near 54.6 mT (546 G) has been used to associate a cold 40K atom and a cold 87Rb atom to make a 40K87Rb molecules in a near-threshold state with a small binding energy on the order of 1 MHz or less Ospelkaus et al. (2006). It is extremely useful to introduce the properties of the long range van der waals potential and take advantage of the separation of short and long range physics discussed in the previous Section. Assuming that the interaction $W(R)$ is confined to distances $R\ll\bar{a}$, the matrix element in Eq. (18) defining $\Gamma(E)$ can be factored as $\Gamma(E)=C_{bg,\ell}(E)^{-2}\bar{\Gamma}\,,$ (26) where $\bar{\Gamma}$ is a measure of resonance strength that depends only on the energy-independent short-range physics near $E=0$, and is completely independent of the asymptotic boundary conditions. It thus can be used in characterizing the properties of both scattering and bound states when $E\neq 0$. The extrapolation of resonance properties away from $E=0$ depends on two additional parameters associated with the long range potential, $\mu$ and $C_{6}$, which determine $\bar{a}$ and $\bar{E}$. Let us define a dimensionless resonance strength parameter $s_{res}=\frac{a_{bg}\delta\mu\Delta}{\bar{a}\bar{E}}=r_{bg}\frac{\Gamma_{0}}{\bar{E}}\,$ (27) where $r_{bg}=a_{bg}/\bar{a}$. Using the threshold van der Waals form of $C_{bg,0}(E)^{-1}$ given in the previous section, we can write $\frac{\bar{\Gamma}}{2}=(s_{res}\bar{E})\frac{1}{1+(1-r_{bg})^{2}}\,.$ (28) The above-threshold scattering properties are found from the scattering phase shift $\eta(E)=\eta_{bg}(E)+\eta_{res}(E)$, where $\eta_{res}(E)$ is found from Eq. (17) once $E_{c}$, $\Gamma(E)$ and $\delta E(E)$ are known. The first two are given by Eqs. (23) and (26), and $\delta E(E)=\frac{\bar{\Gamma}}{2}\tan{\lambda_{bg}(E)},\,$ (29) where $\tan{\lambda_{bg}(E)}$ is a function determined by the van der Waals potential, given $a_{bg}$. It has the limiting form $\tan{\lambda_{bg}(E)}=1-r_{bg}$ as $E\to 0$, and $\tan{\lambda_{bg}(E)=0}$ for $E\gg\bar{E}$ Julienne and Mies (1989); Mies and Raoult (2000). Thus the position of the scattering length singularity is shifted by $\delta B=B_{0}-B_{c}=\Delta\frac{r_{bg}(1-r_{bg})}{1+(1-r_{bg})^{2}}$ (30) from the crossing point $B_{c}$ of the “bare” bound state. Scattering phase shifts calculated from the van der Waals potential with the“quantum defect” forms in Eqs. (26) and (29) are generally in excellent agreement with complete coupled channels methods for energy ranges on the order of $\bar{E}$ and even larger Julienne and Gao (2006). The properties of bound molecular states near threshold can also be calculated from the general coupled-channels quantum defect method using the properties of the long range potential. When the energy $E_{b}(B)=-\hbar^{2}k_{b}(B)^{2}/(2\mu)$ of the threshold $s$-wave bound state is small, that is, $\|E_{b}(B)\|\ll\bar{E}$ or $k_{b}(B)\bar{a}\ll 1$, then the equation for $E_{b}(B)$ from the quantum defect method is $\left(E_{c}(B)-E_{b}(B)\right)\left(\frac{1}{r_{bg}-1}-k_{b}(B)\bar{a}\right)=\frac{\bar{\Gamma}}{2}\,.$ (31) If $\bar{\Gamma}=0$, we recover the uncoupled, or “bare,” bound states of the system, whereas when $\bar{\Gamma}>0$, this equation gives the coupled, or “dressed,” bound states. The threshold bound state “disappears” into the continuum at $B=B_{0}$, where $a(B)$ has a singularity. The shift in Eq. 30 follows immediately upon solving for $E_{c}(B_{0})$ where $E_{b}(B_{0})=0$. Threshold bound state properties are strongly affected by the magnitudes of $s_{res}$ and $r_{bg}$. When the coupled bound state wavefunction is expanded as a mixture of closed and background channel components, $\|c\rangle$ and $\|bg\rangle$ respectively, an important property is the norm $Z(B)$ of the closed channel component; the norm of the entrance channel component is $1-Z(B)$. The value of $Z$ can be calculated from a knowledge of $E_{b}(B)$, since $Z=\|\delta\mu^{-1}\partial E_{b}/\partial B\|$ Köhler et al. (2006). There are two basic classes of resonances. One, for which $s_{res}\gg 1$, are called entrance channel dominated resonances. These have $Z(B)\ll 1$ as $B-B_{0}$ varies over a range that is a significant fraction of $\|\Delta\|$. In addition, the bound state energy is given by Eq. (11) over a large part of this range. On the other hand, closed channel dominated resonances are those with $s_{res}\ll 1$. They have $Z(B)$ large, on the order of unity, as $\|B-B_{0}\|$ varies over a large fraction of $\|\Delta\|$, and only have a ”universal” bound state Eq. (3) over a quite small range $\ll\|\Delta\|$ near $B_{0}$. Entrance channel dominated resonances have $\Gamma(E,B)>E$ when $0<E<\bar{E}$, so that no sharp resonance feature persists above threshold, where $a(B)<0$ and the last bound state has disappeared. By contrast, closed channel dominated resonances with $\|r_{bg}\|$ not too large will have $\Gamma(E,B)<E$ when $0<E<\bar{E}$, so that a sharp resonant feature emerges just above threshold, continuing as a quasibound state with $E>0$ into the region where $a(B)<0$. Figure 7: The lower panel shows an expanded view of $E_{b}(B)$ near $B_{0}$ for the 40K87Rb resonance with $B_{0}=54.693$ mT (546.93 G) in Fig. 6. The solid line comes from a coupled channels calculation that includes all 12 channels with the same $-7/2$ projection quantum number. The dashed and dotted lines respectively show the universal energy of Eq. (3) and the van der Waals corrected energy of Eq. (11). The upper panel shows the closed channel norm $Z(B)$. The width $\Delta=0.310$ mT (3.10 G), $a_{bg}=-191$ a0, and $\delta\mu/h=33.6$ MHz/mT (3.36 MHz/G). With $\bar{a}=68.8$ a0 and $\bar{E}/h=13.9$ MHz, this is a marginal entrance channel dominated resonance with $s_{res}=2.08$. Figure 7 shows an expanded view of the $4(-2)$ resonance of 40K87Rb near 54.6 mT. The figure shows the character of the bound state as it merges into threshold at $B_{0}$. It tends to be a universal “halo” bound state over a range of $\|B-B_{0}\|$ that is less than about $1/3$ of $\Delta$. As $\|B-B_{0}\|$ increases, the bound state increasingly takes on the character of the closed channel $4(-2)$ level as $Z$ increases towards unity. Figure 8 shows an example of the very broad 6Li resonance in the lowest energy $\alpha=1$ $s$-wave channel, which requires two 6Li fermions in different spin states. This is a strongly entrance channel dominated resonance, where $Z\ll 1$ over a range of $\|B-B_{0}\|$ nearly as large as $\Delta$. The last bound state is a universal halo molecule over a range larger than 100 G. The corrected Eq. (11) is a good approximation over an even larger range. The scattering length graph shows that the size $\approx a(B)/2$ of the halo state is very large compared to $\bar{a}=30$ a0 (see Table 1) over this range. Figure 8: Molecular bound state energy (lower panel) and scattering length (upper panel) versus magnetic field $B$ for the lowest energy $\alpha=1$ $s$-wave spin channel of the 6Li2 molecule. This channel has one 6Li atom in the lowest $+1/2$ projection state and the other in the lowest $-1/2$ projection state for a total projection of $0$. There are 4 additional closed channels with projection $0$. In this range of $B$ there is only one bound state that crosses threshold at $B_{0}=83.4$ mT (834 G). The lower panel shows $E_{b}(B)$ from a coupled channels calculation (solid circles), the universal limit of Eq. (3) (dashed line) and the corrected limit of Eq. (11) (solid line). The width $\Delta=30.0$ mT (300 G), $a_{bg}=-1405$ a0, and $\delta\mu/h=28$ MHz/mT (2.8 MHz/G). This is a strongly entrance channel dominated resonance with $s_{res}=59$, and $Z<0.06$ over the range of $B$ shown. Magnetically tunable scattering resonances have proven very useful in associating two cold atoms to make a molecule in the weakly bound states near threshold. This work is reviewed in detail in Ref. Köhler et al. (2006). The magneto-association process works by first preparing a gas with a mixture of both atomic species at $B>B_{0}$ (assuming $\delta\mu>0$), where there is no threshold bound state. By ramping the $B$ field down in time so that $B<B_{0}$, colliding pairs of atoms with $E>0$ can be converted to diatomic molecules in a bound state with energy $E<0$. The conversion efficiency will depend on both the ramp rate and the phase space density of the initial gas. If the initial atom pair is held in a single cell of an optical lattice instead of a gas, the conversion efficiency can approach 100 per cent. A simple Landau-Zener picture has been found to be quite accurate for such lattice cells, where the conversion probability of the atom pair in the trap ground state $i=0$ is $1-e^{-A}$, where $A=\frac{2\pi}{\hbar}\frac{W_{ci}^{2}}{\dot{E_{c}}}\,.$ (32) Here $i\geq 0$ represents the above-threshold levels of the atom pair confined by the trap, continuing the below threshold series of dimer levels with $i\leq-1$. For a three dimensional harmonic trap with frequency $\omega_{x}=\omega_{y}=\omega_{z}=\omega$, the matrix element $W_{ci}=\langle C\|W(R)\|i\rangle$ is well-approximated as $W_{ci}=\sqrt{\Gamma(E_{i})/2\pi}\sqrt{\partial E_{i}/\partial i}$, where $\partial E_{i}/\partial i=2\hbar\omega$ and $\Gamma(E_{i})=2k_{i}a_{bg}\delta\mu\Delta$ for the $i=0$ trap ground state of relative motion with $k_{i}=\sqrt{3\mu\omega/\hbar}$ (see Eqs. 18, 20 and 25). The trick used here in getting a matrix element $W_{ci}$ between two bound states from the matrix element $\langle C\|W(R)\|E\rangle$ involving an energy- normalized scattering state is to introduce the density of states as in Eq. (10). In a similar manner, the matrix element can be obtained between the bare closed channel state and the bound states $i<0$ of the entrance channel. Such matrix elements characterize avoided crossings like the one in Fig. 6 for $E/h$ near $-0.4$ MHz and $B$ near 43 mT. Finally, it should be noted that a Landau-Zener model can also be used for molecular dissociation by a fast magnetic field ramp. An alternative phenomenological model has been developed to describe molecular association in cold gases, which are more complex than two atoms in a lattice cell Hodby et al. (2005). ## V Photoassociation Cold atoms can also be coupled to molecular bound states through photoassociation (PA), as discussed in Chapters XX, YY, and ZZ. Fig. 9 gives a schematic description of PA, a process by which the colliding atoms can be coupled to such bound state resonances through one- or two-photons. Reference Jones et al. (2006) reviews theoretical and experimental work on PA spectroscopy and molecule formation. Molecules made using the magnetically tunable resonances described in the last Section are necessarily very weakly bound, with binding energies limited by the small range of magnetic tuning. Photoassociation has the advantage that laser frequencies are widely tunable, so that a range of many bound states becomes accessible to optical methods, even the lowest $v=0$ vibrational level of the ground state. In addition, the light can be turned off and on or varied in intensity for time-dependent manipulations. Figure 9: Schematic representation of one- and two-color photoassociation (PA). The two colliding ground state atoms at energy $E$ can absorb a laser photon of frequency $\nu_{1}$ and be excited to an excited molecular bound state at energy $E_{v}^{}$. The bound state decays via spontaneous emission at rate $\gamma_{c}/\hbar$. If a second laser is present with frequency $\nu_{2}$, the excited level can also be coupled to a ground state vibrational level $v$ at energy $E_{v}$, if $h(\nu_{2}-\nu_{1})=E-E_{v}$. The PA process depends on the ground state wavefunction at the Condon point $R_{C}$ of the transition, where $h\nu_{1}$ equals the difference between the excited and ground state potentials. Photoassociation naturally lends itself to the resonant scattering treatment of a decaying resonance in Eq. (22), which applies to the one-color case with position $E_{c}=E_{v}^{}-h\nu_{1}$, strength $a_{bg}\Gamma_{0}(I)=\Gamma(E,I)/(2k)$, and shift $\delta E(I)$. The latter two are linear in laser intensity $I$ when $I$ is low enough. PA is usually detected by the inelastic collisional loss of cold atoms it causes, due to the spontaneous decay of the excited state to make hot atoms or deeply bound molecules. In the limit $E\to 0$ the complex scattering length is $\displaystyle a(\nu_{1},I)$ $\displaystyle=$ $\displaystyle a_{bg}-L_{opt}\frac{\gamma_{c}E_{0}}{E_{0}^{2}+(\gamma_{c}/2)^{2}}$ (33) $\displaystyle b(\nu_{1},I)$ $\displaystyle=$ $\displaystyle\frac{1}{2}L_{opt}\frac{\gamma_{c}^{2}}{E_{0}^{2}+(\gamma_{c}/2)^{2}}\,,$ (34) where $E_{0}=E_{v}^{}-h\nu_{1}+\delta E(I)$ is the detuning from resonance, including the intensity-dependent shift, and the optical length is defined by $L_{opt}=a_{bg}\Gamma_{0}(I)/\gamma_{c}$. Photoassociation spectra, line shapes, and shifts have been widely studied for a variety of like and mixed alkali-metal species. At the higher temperatures often encountered in magneto-optical traps, contributions to PA spectra from higher partial waves, e.g., $p$\- or $d$-waves, have been observed in a number of cases. The theory can be readily extended to higher partial waves. By introducing an energy-dependent complex scattering length the theory for $s$-waves can be extended to finite $E$ away from threshold and to account for effects due to reduced dimensional confinement in optical lattices Naidon and Julienne (2006). The optical length formulation of resonance strength is very useful for a decaying resonance. It also applies to decaying magnetically tunable resonances, if $\Gamma_{0}$ from Eq. (25) is used to define a resonance length $a_{bg}\delta\mu\Delta/\gamma_{c}$ equivalent to $L_{opt}$ Hutson (2007). The scattering length has its maximum variation of $a_{bg}\pm L_{opt}$ when the laser is tuned to $E_{0}=\pm\gamma_{c}/2$, and losses are maximum at $E_{0}=0$ where $b=L_{opt}$. When detuning is small, on the order of $\gamma_{c}$, significant changes to the scattering length on the order of $\bar{a}$ are thus normally accompanied by large loss rates (see Eq. 14). Losses can be avoided by going to large detuning, since when $(\gamma_{c}/E_{0})\ll 1$, $b=(L_{opt}/2)(\gamma_{c}/E_{0})^{2}$, whereas the change in $a$ only varies as $a-a_{bg}=-L_{opt}(\gamma_{c}/E_{0})$. To make the change $a-a_{bg}$ large enough while requiring $(\gamma_{c}/E_{0})\ll 1$ means that $L_{opt}$ has to be very large compared to $\bar{a}$. The magnitude of $L_{opt}$ depends on the matrix element $\langle C\|\hbar\Omega_{1}(R)\|E\rangle$ where $\hbar\Omega_{1}(R)$ represents the optical coupling between the ground and excited state. Using Eqs. (18) and (20) and the above definition of $L_{opt}$, and factoring out the relatively constant $\hbar\Omega_{1}$ value, $L_{opt}=\pi\frac{\|\hbar\Omega_{1}\|^{2}}{\gamma_{c}}\frac{F(E)}{k}\,.$ (35) The Franck-Condon overlap factor is $\displaystyle F(E)$ $\displaystyle=$ $\displaystyle\left\|\int_{0}^{\infty}\psi_{v}^{}(R)\psi_{0}(R,E)dR\right\|^{2}$ (36) $\displaystyle\approx$ $\displaystyle\frac{\partial E_{v}^{}}{\partial v}\frac{1}{D_{C}}\|\psi_{0}(R_{C},E)\|^{2}\,,$ (37) where $D_{C}$ is the derivative of the difference between the excited and ground state potentials evaluated at the Condon point $R_{C}$, and ${\partial E_{v}^{}}/{\partial v}$ is the excited state vibrational spacing. Equation (37) is known as the reflection approximation, generally an excellent approximation where $F(E)$ is proportional to the square of the ground state wavefunction at $R_{C}$, the Condon point where the molecular potential difference matches $h\nu_{1}$ (see Fig. 9). Thus, $F(E)$ can be evaluated using expressions like Eqs. (5) or (9) for $R_{C}\gg\bar{a}$ or $R_{C}\ll\bar{a}$ respectively. The reflection approximation is quite good over a wide range of $E$ and for higher partial waves than the $s$-wave. By selecting a range of excited levels $v$ by changing laser frequency $\nu_{1}$, thus changing $R_{C}$, the shape and nodal structure of the ground state wavefunction can be mapped out over a range of $R$. The optical length has several important properties evident from Eq. (35). First, since both $\Omega_{1}$ and $\gamma_{c}$ are proportional to the same squared transition dipole moment, $L_{opt}$ does not depend on whether the transition is strong or weak, but can be large for both kinds of transitions. Second, $L_{opt}\propto\|\Omega_{1}\|^{2}$ so it can be increased by increasing laser intensity. Third, since $F(E)\propto k$ as $E\to 0$ for entrance channel $s$-waves, $L_{opt}\propto F(E)/k$ is independent of $E$ or $k$ at low energy. However, it does depend strongly on the molecular structure through the Franck-Condon factor. In practice, using strong transitions with large decay rates such as those in alkali-metal species leads to the requirement to use excited molecular levels far from threshold with large binding energies. This is necessary to achieve large detuning from atomic and molecular resonance. This requirement means such levels have small $F(E)$ factors, due to the very large value of $D_{C}$ in Eq. (37). On the other hand, weak transitions with small decay rates, such as those associated with the 1S${}_{0}\to^{3}$P1 intercombination line transition for alkaline earth species such as Sr, can lead to quite large values of $L_{opt}$. This is because large detuning in $\gamma_{c}$ units can be achieved for levels that are still quite close to the excited state threshold. Such levels typically have large Franck-Condon factors. In fact, PA transitions near the weak intercombination line of Sr have been observed to have $L_{opt}$ several orders of magnitude larger than was observed for strongly allowed molecular transitions involving Rb Zelevinsky et al. (2006). Thus, there are good prospects for some degree of optical resonant control of collisions in ultracold gases of species like Ca, Sr, or Yb. Two-color PA is also possible when a second laser with frequency $\nu_{2}$ is added, as shown in Fig. 9. When the the frequency difference is chosen so that $h(\nu_{2}-\nu_{1})=E-E_{i\ell}$, the ground $i\ell$ molecular level is in resonance with collisions at energy $E$. By keeping $\nu_{1}$ fixed and tuning $\nu_{2}$, two-color PA spectroscopy can be used to probe the level energies. This is how the data on binding energies of the Yb2 molecule was obtained Kitagawa et al. (2008) so as to be able to construct Fig. 5. In this case, 12 different levels from different isotopic species were measured, among which were levels with $i=-1$ and $-2$ and $\ell=0$ and $2$. Two color spectroscopy has also been carried out for several alkali-metal homonuclear species. Two-color processes are also an excellent way to assemble two cold atoms into a translationally cold molecule. Early work along these lines was done using the spontaneous decay of the excited level to populate a wide range of levels in the ground state. The disadvantage of spontaneous decay is that it is not selective. However, by using a laser with a precise frequency, a specific level can be chosen as the target level. One early experiment did this to associate two 87Rb atoms in a Bose-Einstein condensate to make a molecular level at a specific energy of $h(-636)$ MHz Wynar et al. (2000). It is highly desirable to be able to make translationally cold molecules in their vibrational ground state $v=0$. This is especially true of polar molecules, which have large dipole moments in $v=0$. On the other hand, threshold levels have negligible dipole moments, since there is no charge transfer because of the large average atomic separation $\approx\bar{a}$. A promising technique is to use magnetoassociation using a tunable Feshbach resonance to associate the atoms into a threshold molecular level, then use a 2-color Raman process to move the population in that state to a much more deeply bound level. Although molecules in a gas are subject to fast destructive collisions with cold atoms or other molecules in the gas (see Eq. 14), the molecules can be protected against such collisions by forming them in individual optical lattice trapping cells. Then the 2-color Raman process could be used to produce much more deeply bound molecules that are stable against destructive collisions. This has been done successfully with 87Rb2 Winkler et al. (2007) molecules. In the future, such methods are likely to produce $v=0$ polar molecules, with which a range of interesting physics can be explored Lewenstein (2006); Büchler et al. (2007). ## References * Köhler et al. (2006) T. Köhler, K. Góral, and P. S. Julienne, Rev. Mod. Phys. 78, 1311 (2006). * Jones et al. (2006) K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne, Rev. Mod. Phys. 78, 483 (2006). * Julienne and Mies (1989) P. S. Julienne and F. H. Mies, J. Opt. Soc. Am. B 6, 2257 (1989). * Moerdijk et al. (1995) A. J. Moerdijk, B. J. Verhaar, and A. Axelsson, Phys. Rev. A 51, 4852 (1995). * Vogels et al. (1998) J. M. Vogels, B. J. Verhaar, and R. H. Blok, Phys. Rev. A 57, 4049 (1998). * J. P. Burke et al. (1998) J. J. P. Burke, C. H. Greene, and J. L. Bohn, Phys. Rev. Lett. 81, 3355 (1998). * Vogels et al. (2000) J. M. Vogels, R. S. Freeland, C. C. Tsai, B. J. Verhaar, and D. J. Heinzen, Phys. Rev. A 61, 043407 (2000). * Mies and Raoult (2000) F. H. Mies and M. Raoult, Phys. Rev. A 62, 012708 (2000). * Gao (2000) B. Gao, Phys. Rev. A 62, 050702 (2000). * Gao (2001) B. Gao, Phys. Rev. A 64, 010701 (2001). * Julienne and Gao (2006) P. S. Julienne and B. Gao, in _Atomic Physics 20_ , edited by C. Roos, H. Häffner, and R. Blatt (AIP, Melville, New York, 2006), pp. 261–268, physics/0609013. * Derevianko et al. (1999) A. Derevianko, W. R. Johnson, M. S. Safronova, and J. F. Babb, Phys. Rev. Lett. 82, 3589 (1999). * Porsev and Derevianko (2006) S. G. Porsev and A. Derevianko, JETP 102, 195 (2006), [Pis’ma Zh. Eksp. Teor. Fiz., 129, 227–238 (2006)]. * Gribakin and Flambaum (1993) G. F. Gribakin and V. V. Flambaum, Phys. Rev. A 48, 546 (1993). * Kitagawa et al. (2008) M. Kitagawa, K. Enomoto, K. Kasa, Y. Takahashi, R. Ciurylo, P. Naidon, and P. S. Julienne, Phys. Rev. A 77, 012719 (2008). * Fano (1961) U. Fano, Phys. Rev. A 124, 1866 (1961). * Bohn and Julienne (1999) J. L. Bohn and P. S. Julienne, Phys. Rev. A 60, 414 (1999). * Marcelis et al. (2004) B. Marcelis, E. G. M. van Kempen, B. J. Verhaar, and S. J. J. M. F. Kokkelmans, Phys. Rev. A 70, 012701 (2004). * Ospelkaus et al. (2006) C. Ospelkaus, S. Ospelkaus, L. Humbert, P. Ernst, K. Sengstock, and K. Bongs, Phys. Rev. Lett. 97, 120402 (2006). * Hodby et al. (2005) E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson, D. S. Jin, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 94, 120402 (2005). * Naidon and Julienne (2006) P. Naidon and P. S. Julienne, Phys. Rev. A 74, 022710 (2006). * Hutson (2007) J. M. Hutson, New J. Phys. 9, 152 (2007). * Zelevinsky et al. (2006) T. Zelevinsky, M. M. Boyd, A. D. Ludlow, T. Ido, J. Ye, R. Ciurylo, P. Naidon, and P. S. Julienne, Phys. Rev. Lett. 96, 203201 (2006). * Wynar et al. (2000) R. Wynar, R. S. Freeland, D. J. Han, C. Ryu, and D. J. Heinzen, Science 287, 1016 (2000). * Winkler et al. (2007) K. Winkler, F. Lang, G. Thalhammer, P. van der Straten, R. Grimm, and J. Hecker Denschlag, Phys. Rev. Lett. 98, 043201 (2007). * Lewenstein (2006) M. Lewenstein, Nature Physics 2, 309 (2006). * Büchler et al. (2007) H. P. Büchler, A. Micheli, and P. Zoller, Nature Physics 3, 726 (2007). ## References * (1)	arxiv-papers	2009-02-10T19:21:26	2024-09-04T02:49:00.506684	{ "license": "Public Domain", "authors": "Paul S. Julienne", "submitter": "Paul Julienne", "url": "https://arxiv.org/abs/0902.1727" }
0902.1800	# Optical transformation from chirplet to fractional Fourier transformation kernel Hong-yi Fan and Li-yun Hu Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, P.R. China Corresponding author. hlyun2008@126.com or hlyun@sjtu.edu.cn ###### Abstract We find a new integration transformation which can convert a chirplet function to fractional Fourier transformation kernel, this new transformation is invertible and obeys Parseval theorem. Under this transformation a new relationship between a phase space function and its Weyl-Wigner quantum correspondence operator is revealed. In the history of developing optics we have known that each optical setup corresponds to an optical transformation, for example, thick lens as a fractional Fourier transformer. In turn, once a new integration transform is found, its experimental implementation is expected, for example, the fractional Fourier transform (FrFT) of a function was originally introduced by Namias as a mathematical tool for solving theoretical physical problems 1 ; 2 , and later Mendlovic, Ozakatas et al explored its applications in optics by redefining it as the change of the field caused by propagation along a quadratic Graded-Index (GRIN) medium3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 . In this Letter we report a new integration transformation which can convert chirplet function to fractional Fourier transformation kernel, as this new transformation is invertible and obeys Parseval theorem, we expect it be realized by experimentalists. The new transform we propose here is $\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}h(p,q)\equiv f\left(x,y\right),$ (1) which differs from the usual two-fold Fourier transformation $\iint_{-\infty}^{\infty}\frac{dxdy}{4\pi^{2}}e^{ipx+iqy}f(x,y).$ In particular, when $h(p,q)=1,$ Eq. (1) reduces to $\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}=\int_{-\infty}^{\infty}dq\delta\left(q-y\right)e^{-2xi\left(q-y\right)}=1,$ (2) so $e^{2i\left(p-x\right)\left(q-y\right)}$ can be considered a basis funtion in $p-q$ phase space, or Eq. (1) can be looked as an expansion of $f\left(x,y\right)$ with the expansion coefficient being $h(p,q).$ We can prove that the reciprocal transformation of (1) is $\iint_{-\infty}^{\infty}\frac{dxdy}{\pi}e^{-2i(p-x)(q-y)}f(x,y)=h(p,q).$ (3) In fact, substituting (1) into the left-hand side of (3) yields $\displaystyle\iint_{-\infty}^{\infty}\frac{dp^{\prime}dq^{\prime}}{\pi}h(p^{\prime},q^{\prime})\iint\frac{dxdy}{\pi}e^{2i\left[\left(p^{\prime}-x\right)\left(q^{\prime}-y\right)-\left(p-x\right)\left(q-y\right)\right]}$ (4) $\displaystyle=$ $\displaystyle\iint_{-\infty}^{\infty}dp^{\prime}dq^{\prime}h(p^{\prime},q^{\prime})e^{2i\left(p^{\prime}q^{\prime}-pq\right)}\delta\left(p-p^{\prime}\right)\delta\left(q-q^{\prime}\right)=h(p,q).$ This transformation’s Parseval-like theorem is $\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}\|h(p,q)\|^{2}$ (5) $\displaystyle=$ $\displaystyle\iint\frac{dxdy}{\pi}\|f\left(x,y\right)\|^{2}\iint\frac{dx^{\prime}dy^{\prime}}{\pi}e^{2i\left(x^{\prime}y^{\prime}-xy\right)}$ $\displaystyle\times\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left[\left(-y^{\prime}p-x^{\prime}q\right)+\left(py+xq\right)\right]}$ $\displaystyle=$ $\displaystyle\iint\frac{dxdy}{\pi}\|f\left(x,y\right)\|^{2}\iint dx^{\prime}dy^{\prime}e^{2i\left(x^{\prime}y^{\prime}-xy\right)}$ $\displaystyle\times\delta\left(x-x^{\prime}\right)\delta\left(p-p^{\prime}\right)$ $\displaystyle=$ $\displaystyle\iint\frac{dxdy}{\pi}\|f\left(x,y\right)\|^{2}.$ Now we apply Eq. (1) to phase space transformation in quantum optics. Recall that a signal $\psi\left(q\right)$’s Wigner transform 12 ; 13 ; 14 ; 15 is $\psi\left(q\right)\rightarrow\int\frac{du}{2\pi}e^{ipu}\psi^{\ast}\left(q+\frac{u}{2}\right)\psi\left(q-\frac{u}{2}\right).$ (6) Using Dirac’s symbol 16 to write $\psi\left(q\right)=\left\langle q\right\|\left.\psi\right\rangle,$ $\left\|q\right\rangle$ is the eigenvector of coordinate $Q$, $Q\left\|q\right\rangle=q\left\|q\right\rangle,$ $\left[Q,P\right]=i\hbar,$ the Wigner operator emerges from (6), $\frac{1}{2\pi}\int_{-\infty}^{\infty}due^{-ipu}\left\|q-\frac{u}{2}\right\rangle\left\langle q+\frac{u}{2}\right\|=\Delta\left(p,q\right),\text{ }\hbar=1.$ (7) If $h\left(q,p\right)$ is quantized as the operator $\hat{H}\left(P,Q\right)$ through the Weyl-Wigner correspondence 17 $H\left(P,Q\right)=\iint_{-\infty}^{\infty}dpdq\Delta\left(p,q\right)h\left(q,p\right),$ (8) then $h\left(q,p\right)=\int_{-\infty}^{\infty}due^{-ipu}\left\langle q+\frac{u}{2}\right\|\hat{H}\left(Q,P\right)\left\|q-\frac{u}{2}\right\rangle,$ (9) this in the literature is named the Weyl transform, $h\left(q,p\right)$ is the Weyl classical correspondence of the operator $\hat{H}\left(Q,P\right)$. Substituting (9) into (1) we have $\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}h(p,q)$ (10) $\displaystyle=$ $\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}\int_{-\infty}^{\infty}due^{-ipu}$ $\displaystyle\times\left\langle q+\frac{u}{2}\right\|\hat{H}\left(Q,P\right)\left\|q-\frac{u}{2}\right\rangle$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}dq\int_{-\infty}^{\infty}du\left\langle q+\frac{u}{2}\right\|\hat{H}\left(Q,P\right)\left\|q-\frac{u}{2}\right\rangle$ $\displaystyle\times\delta\left(q-y-\frac{u}{2}\right)e^{-2ix\left(q-y\right)}$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}due^{-ixu}\left\langle y+u\right\|\hat{H}\left(Q,P\right)\left\|y\right\rangle.$ Using $\left\langle y+u\right\|=\left\langle u\right\|e^{iPy}$ and $(\sqrt{2\pi})^{-1}e^{-ixu}=\left\langle p_{=x}\right\|\left.u\right\rangle,$ where $\left\langle p\right\|$ is the momentum eigenvector, and $\displaystyle\int_{-\infty}^{\infty}due^{-ixu}\left\langle y+u\right\|$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}due^{-ixu}\left\langle u\right\|e^{iPy}$ (11) $\displaystyle=$ $\displaystyle\sqrt{2\pi}\int_{-\infty}^{\infty}du\left\langle p_{=x}\right\|\left.u\right\rangle\left\langle u\right\|e^{iPy}$ $\displaystyle=$ $\displaystyle\sqrt{2\pi}\left\langle p_{=x}\right\|e^{ixy},$ then Eq. (10) becomes $\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}h(p,q)$ (12) $\displaystyle=$ $\displaystyle\sqrt{2\pi}\left\langle p_{=x}\right\|\hat{H}\left(Q,P\right)\left\|y\right\rangle e^{ixy},$ thus through the new integration transformation a new relationship between a phase space function $h(p,q)$ and its Weyl-Wigner correspondence operator $\hat{H}\left(Q,P\right)$ is revealed. The inverse of (12), according to (3), is $\iint_{-\infty}^{\infty}\frac{dxdy}{\sqrt{\pi/2}}e^{-2i\left(p-x\right)\left(q-y\right)}\left\langle p_{=x}\right\|\hat{H}\left(Q,P\right)\left\|y\right\rangle e^{ixy}=h(p,q).$ (13) For example, when $\hat{H}\left(Q,P\right)=e^{f(P^{2}+Q^{2}-1)/2},$ its classical correspondence is $e^{f\left(P^{2}+Q^{2}-1\right)/2}\rightarrow h(p,q)=\frac{2}{e^{f}+1}\exp\left\\{2\frac{e^{f}-1}{e^{f}+1}\left(p^{2}+q^{2}\right)\right\\}.$ (14) Substituting (14) into (12) we have $\displaystyle\frac{2}{e^{f}+1}\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}\exp\left\\{2\frac{e^{f}-1}{e^{f}+1}\left(p^{2}+q^{2}\right)\right\\}$ (15) $\displaystyle=$ $\displaystyle\sqrt{2\pi}\left\langle p_{=x}\right\|e^{f\left(P^{2}+Q^{2}-1\right)/2}\left\|y\right\rangle e^{ixy}.$ Using the Gaussian integration formula $\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}e^{-\lambda\left(p^{2}+q^{2}\right)}$ (16) $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{\lambda^{2}+1}}\exp\left\\{\frac{-\lambda\left(x^{2}+y^{2}\right)}{\lambda^{2}+1}+\frac{2i\lambda^{2}}{\lambda^{2}+1}xy\right\\},$ in particular, when $\lambda=-i\tan\left(\frac{\pi}{4}-\frac{\alpha}{2}\right),$ (17) with $\frac{-\lambda}{\lambda^{2}+1}=\frac{i}{2\tan\alpha},\text{ }\frac{2\lambda^{2}}{\lambda^{2}+1}=1-\frac{1}{\sin\alpha},$ (18) Eq. (16) becomes $\displaystyle\frac{2}{ie^{-i\alpha}+1}\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}$ (19) $\displaystyle\times\exp\left\\{i\tan\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)\left(p^{2}+q^{2}\right)\right\\}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{i\sin\alpha e^{-i\alpha}}}\exp\left\\{\frac{i\left(x^{2}+y^{2}\right)}{2\tan\alpha}-\frac{ixy}{\sin\alpha}\right\\}e^{ixy},$ where $\exp\\{i\tan\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)\left(p^{2}+q^{2}\right)\\}$ represents an infinite long chirplet function. Comparing (19) with (15) we see $ie^{-i\alpha}=e^{f},$ $f=i\left(\frac{\pi}{2}-\alpha\right),$ it then follows $\displaystyle\left\langle p_{=x}\right\|e^{i(\frac{\pi}{2}-\alpha)\left(P^{2}+Q^{2}-1\right)/2}\left\|y\right\rangle$ (20) $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2\pi i\sin\alpha e^{-i\alpha}}}\exp\left\\{\frac{i\left(x^{2}+y^{2}\right)}{2\tan\alpha}-\frac{ixy}{\sin\alpha}\right\\},$ where the right-hand side of (20) is just the FrFT kernel. Therefore the new integration transformation (1) can convert spherical wave to FrFT kernel. We expect this transformation could be implemented by experimentalists. Moreover, when we notice $\displaystyle\frac{1}{\pi}\exp[2i\left(p-x\right)\left(q-y\right)]$ (21) $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}\frac{dv}{2\pi}\delta\left(q-y-\frac{v}{2}\right)\exp\left\\{i\left(p-x\right)v\right\\},$ so the transformation (1) is equivalent to $\displaystyle h(p,q)$ $\displaystyle\rightarrow$ $\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{\pi}e^{2i\left(p-x\right)\left(q-y\right)}h(p,q)$ (22) $\displaystyle=$ $\displaystyle\iint_{-\infty}^{\infty}dpdq\int_{-\infty}^{\infty}\frac{dv}{2\pi}\delta\left(q-y-\frac{v}{2}\right)e^{i\left(p-x\right)v}h(p,q)$ $\displaystyle=$ $\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{2\pi}h(p+x,y+\frac{q}{2})e^{ipq}.$ For example, using (7) and (22) we have $\displaystyle\Delta(p,q)$ $\displaystyle\rightarrow$ $\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{2\pi}\Delta(p+x,y+\frac{q}{2})e^{ipq}$ (23) $\displaystyle=$ $\displaystyle\iint_{-\infty}^{\infty}\frac{dpdq}{4\pi^{2}}\int_{-\infty}^{\infty}due^{-i\left(p+x\right)u}$ $\displaystyle\times\left\|y+\frac{q}{2}-\frac{u}{2}\right\rangle\left\langle y+\frac{q}{2}+\frac{u}{2}\right\|e^{ipq}$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}\frac{dq}{2\pi}\int_{-\infty}^{\infty}due^{-ixu}\delta\left(q-u\right)$ $\displaystyle\times\left\|y+\frac{q}{2}-\frac{u}{2}\right\rangle\left\langle y+\frac{q}{2}+\frac{u}{2}\right\|$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{\infty}\frac{du}{2\pi}e^{-ixu}\left\|y\right\rangle\left\langle y+u\right\|=\left\|y\right\rangle\left\langle y\right\|\int_{-\infty}^{\infty}\frac{du}{2\pi}e^{iu\left(P-u\right)}$ $\displaystyle=$ $\displaystyle\delta\left(y-Q\right)\delta\left(x-P\right),$ so $\frac{1}{\pi}\iint\mathtt{d}p^{\prime}\mathtt{d}q^{\prime}\Delta\left(q^{\prime},p^{\prime}\right)e^{2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}=\delta\left(q-Q\right)\delta\left(p-P\right),$ (24) thus this new transformation can convert the Wigner function of a density operator $\rho$, $W_{\psi}(p,q)\equiv\mathtt{Tr}\left[\rho\Delta(p,q)\right],$ to $\displaystyle\iint_{-\infty}^{\infty}\frac{dp^{\prime}dq^{\prime}}{\pi}\mathtt{Tr}\left[\rho\Delta(p^{\prime},q^{\prime})\right]e^{2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}$ (25) $\displaystyle=$ $\displaystyle\mathtt{Tr}\left[\rho\delta\left(q-Q\right)\delta\left(p-P\right)\right]$ $\displaystyle=$ $\displaystyle\int\frac{dudv}{4\pi^{2}}\mathtt{Tr}\left[\rho e^{i\left(q-Q\right)u}e^{i\left(p-P\right)v}\right],$ we may define $\mathtt{Tr}\left[\rho e^{i\left(q-Q\right)u}e^{i\left(p-P\right)v}\right]$ as the $Q-P$ characteristic function. Similarly, $\displaystyle\iint_{-\infty}^{\infty}\frac{dp^{\prime}dq^{\prime}}{\pi}\mathtt{Tr}\left[\rho\Delta(p^{\prime},q^{\prime})\right]e^{-2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}$ (26) $\displaystyle=$ $\displaystyle\mathtt{Tr}\left[\rho\delta\left(p-P\right)\delta\left(q-Q\right)\right]$ $\displaystyle=$ $\displaystyle\int\frac{dudv}{4\pi^{2}}\mathtt{Tr}\left[\rho e^{i\left(p-P\right)v}e^{i\left(q-Q\right)u}\right]$ we name $\mathtt{Tr}\left[\rho e^{i\left(p-P\right)v}e^{i\left(q-Q\right)u}\right]$ as the $P-Q$ characteristic function. In summary, we have found a new integration transformation which can convert chirplet function to FrFT kernel, this new transformation is worth paying attention because it is invertible and obeys Parseval theorem. Under this transformation the relationship between a phase space function and its Weyl- Wigner quantum correspondence operator is revealed. ACKNOWLEDGEMENT: Work supported by the National Natural Science Foundation of China under grants: 10775097 and 10874174. ## References * (1) V. Namias, ”The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980). * (2) V. Namias, “Fractionalization of Hankel Transforms,” 26, 187 (1980). * (3) D. Mendlovic and H. M. Ozaktas, ”Fractional fourier transforms and their optical implementation:I,” J. Opt. Soc. Am. A 10, 1875-1881 (1993). * (4) H. M. Ozakatas, D. Mendlovic. “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A, 10, 2522 (1993). * (5) H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional orders and their optical interpretation” Opt. Commun. 101, 163 (1993). * (6) Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769 (1993). * (7) R. g. Dorsch and A. W. Lohmann, “Fractional Fourier transform used for a lens-design problem,” Appl. Opt. 34, 4111 (1995). * (8) A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A,10, 2181 (1993). * (9) D. Mendlovic, H. M. Ozakatas, A. W. Lohmann. “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188 (1994) * (10) Hong-yi Fan, “Fractional Hankel transform studied by charge-amplitude state representations and complex fractional Fourier transformation,” Opt. Lett. 28, 2177 (2003). * (11) Hong-yi Fan and Hai-liang Lu, “Eigenmodes of fractional Hankel transform derived by the entangled-state method,” Opt. Lett. 28, 680 (2003). * (12) E. Wigner, “On the Quantum Correction For Thermodynamic Equilibrium,” Phys. Rev., 40, 749 (1932). * (13) G. S. Agarwal and E. Wolf, “Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. I. Mapping Theorems and Ordering of Functions of Noncommuting Operators,” Phys. Rev. D 2, (1970) 2161. * (14) W. Schleich, Quantum Optics in Phase Space, (Wiley-VCH, Berlin 2001) * (15) H. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147 (1995). * (16) P. A. M. Dirac, The Principles of Quantum Mechanics, (Oxford: Clarendon Press, 1930) * (17) H. Weyl, “Quantenmechanik und Gruppentheorie,” Z. Phys. 46, 1 (1927).	arxiv-papers	2009-02-11T14:07:12	2024-09-04T02:49:00.514416	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hong-yi Fan and Li-yun Hu", "submitter": "Liyun Hu", "url": "https://arxiv.org/abs/0902.1800" }
0902.1880	# Mutually unbiased bases and generalized Bell states Andrei B. Klimov Departamento de Física, Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico Denis Sych Max-Planck-Institut für die Physik des Lichts, Günther-Scharowsky-Straße 1, Bau 24, 91058 Erlangen, Germany Universität Erlangen-Nürnberg, Staudtstraße 7/B2, 91058 Erlangen, Germany Luis L. Sánchez-Soto Max-Planck-Institut für die Physik des Lichts, Günther- Scharowsky-Straße 1, Bau 24, 91058 Erlangen, Germany Gerd Leuchs Max-Planck- Institut für die Physik des Lichts, Günther-Scharowsky-Straße 1, Bau 24, 91058 Erlangen, Germany Universität Erlangen-Nürnberg, Staudtstraße 7/B2, 91058 Erlangen, Germany ###### Abstract We employ a straightforward relation between mutually unbiased and Bell bases to extend the latter in terms of a direct construction for the former. We analyze in detail the properties of these new generalized Bell states, showing that they constitute an appropriate tool for testing entanglement in bipartite multiqudit systems. ###### pacs: 03.65.Ca, 03.65.Ta, 03.65.Ud, 42.50.Dv ## I Introduction Entanglement is probably the most intriguing feature of the quantum world, the hallmark of correlations that delimits the boundary between classical and quantum behavior. Although some amazing aspects of this phenomenon were already noticed by Schrödinger in the early stages of quantum theory Schrodinger:1935 , it was not until quite recently that it attracted a considerable attention as a crucial resource for quantum information processing Nielsen:2000 . The simplest instance of entanglement is most clearly illustrated by the maximally entangled states between a pair of qubits (known as Bell states), whose properties can be found in many textbooks Peres:1993 . Despite their simplicity, they are of utmost importance for the analysis of many experiments Wei:2007 . In consequence, as any sound concept, Bell states deserve an appropriate generalization. However, this is a touchy business, since thoughtful notions for a pair of qubits, may become fuzzy for more complex systems. There are two sensible ways to proceed: the first, is to investigate multipartite entanglement of qubits. While the standard Bell basis defines (for pure states) a natural unit of entanglement, it has recently become clear that for qubits shared by more parties there is a rich phenomenology of entangled states Dur:2000 ; Durr:2000b ; Acin:2000 ; Briegel:2001 ; Verstraete:2002 ; Rigolin:2006 ; Facchi:2008 . The second possibility involves examining bipartite entanglement between two multidimensional systems Bechmann:2000 ; Bourennane:2001 ; Cerf:2002 ; Sych:2004 ; Sych:2009 . Again there is no unique way of looking at the problem, and different definitions focus on different aspects and capture different features of this quantum phenomenon. We wish to approach this subject from a new perspective: our starting point is the notion of mutually unbiased bases (MUBs), which emerged in the seminal work of Schwinger Schwinger:1960 and it has turned into a cornerstone of quantum information, mainly due to the elegant work of Wootters and coworkers Wootters:1987 ; Wootters:1989 ; Wootters:2004 ; Gibbons:2004b ; Wootters:2006 . Since MUBs contain complete single-system information and Bells bases about bipartite entanglement, one is led to look for a relation between them. In this paper we confirm such a relation for qudits Planat:2005 and take advantage of the well-established MUB machinery (in prime power dimensions) to propose a straightforward generalization of Bell states for any dimension. The resulting bases are analyzed in detail, paying special attention to their symmetry properties. In view of the results, we conclude that these states constitute an ideal instrument to analyze bipartite multiqudit systems. ## II Bipartite qudit systems ### II.1 Mutually unbiased bases for qudits We start by considering a qudit, which lives in a Hilbert space $\mathcal{H}_{d}$, whose dimension $d$ is assumed for now to be a prime number. The different outcomes of a maximal test constitute an orthogonal basis of $\mathcal{H}_{d}$. One can also look for other orthogonal bases that, in addition, are “as different as possible”. To formalize this idea, we suppose we have a number of orthonormal bases described by vectors $\|\psi_{\ell}^{n}\rangle$, where $\ell$ ($\ell=0,1,\ldots,d-1$) labels the vectors in the $n$th basis. These are MUBs if each state of one basis gives rise to the same probabilities when measured with respect to other basis: $\|\langle\psi_{\ell^{\prime}}^{n^{\prime}}\|\psi_{\ell}^{n}\rangle\|^{2}=\frac{1}{d}\,,\qquad n\neq n^{\prime}\,.$ (1) Equivalently, this can be concisely reformulated as $\|\langle\psi_{\ell^{\prime}}^{n^{\prime}}\|\psi_{\ell}^{n}\rangle\|^{2}=\delta_{\ell\ell^{\prime}}\delta_{nn^{\prime}}+\frac{1}{d}(1-\delta_{nn^{\prime}})\,.$ (2) Note in passing that the Hermitian product of two MUBs is then a generalized Hadamard matrix, i.e., a unitary matrix whose entries all have the same absolute value Bengtsson:2007 . If one wants to determine the state of a system, given only a limited supply of copies, the optimal strategy is to perform measurements with respect to MUBs. They have also been used in cryptographic protocols Asplund:2001 , due to the complete uncertainty about the outcome of a measurement in some basis after the preparation of the system in another, if the bases are mutually unbiased. MUBs are also important for quantum error correction codes Gottesman:1996 ; Calderbank:1997 and in quantum game theory Englert:2001 ; Aravind:2003 ; Paz:2005 ; Kimura:2006 . The maximum number of MUBs can be at most $d+1$ Ivanovic:1981 . Actually, it is known that if $d$ is prime or power of prime (which is precisely our case), the maximal number of MUBs can be achieved. Unbiasedness also applies to measurements: two nondegenerate tests are mutually unbiased if the bases formed by their eigenstates are MUBs. For example, the measurements of the components of a qubit along $x$, $y$, and $z$ axes are all unbiased. It is also obvious that for these finite quantum systems unbiasedness is tantamount of complementarity Kraus:1987 ; Lawrence:2002 . The construction of MUBs is closely related to the possibility of finding of $d+1$ disjoint classes, each one having $d-1$ commuting operators, so that the corresponding eigenstates form sets of MUBs Bandyopadhyay:2002 . Different explicit methods in prime power dimensions have been suggested in a number of recent papers Klappenecker:2004 ; Lawrence:2004 ; Pittenger:2005 ; Wocjan:2005 ; Durt:2005 ; Klimov:2007 , but we follow here the one introduced in Ref. Klimov:2005 , since it is especially germane for our purposes. First, we choose a computational basis $\|\ell\rangle$ in $\mathcal{H}_{d}$ and introduce the basic operators $X\|\ell\rangle=\|\ell+1\rangle\,,\qquad\qquad Z\|\ell\rangle=\omega(\ell)\|\ell\rangle\,,$ (3) where addition and multiplication must be understood modulo $d$ and, for simplicity, we employ the notation $\omega(\ell)=\omega^{\ell}=\exp(i2\pi\ell/d)\,,$ (4) $\omega=\exp(i2\pi/d)$ being a $d$th root of the unity. These operators $X$ and $Z$, which are generalizations of the Pauli matrices, were studied long ago by Weil Weil:1964 . They generate a group under multiplication known as the generalized Pauli group and obey $ZX=\omega\,XZ$, which is the finite- dimensional version of the Weyl form of the commutation relations Putnam:1987 . We consider the following sets of operators: $\tilde{\Lambda}(m)=X^{m}\,,\qquad\Lambda(m,n)=Z^{m}X^{nm}\,,$ (5) with $m=1,\ldots,d-1$ and $n=0,\ldots,d-1$. They fulfill the pairwise orthogonality relations $\displaystyle\mathop{\mathrm{Tr}}\nolimits[\tilde{\Lambda}(m)\,\tilde{\Lambda}^{\dagger}(m^{\prime})]=d\,\delta_{mm^{\prime}}\,.$ (6) $\displaystyle\mathop{\mathrm{Tr}}\nolimits[\Lambda(m,n)\,\Lambda^{\dagger}(m^{\prime},n^{\prime})]=d\,\delta_{mm^{\prime}}\,\delta_{nn^{\prime}}\,,$ which indicate that, for every value of $n$, we generate a maximal set of $d-1$ commuting operators and that all these classes are disjoint. In addition, the common eigenstates of each class $n$ form different sets of MUBs. If one recalls that the finite Fourier transform $F$ is Vourdas:2004 $F=\frac{1}{\sqrt{d}}\sum_{\ell,\ell^{\prime}=0}^{d-1}\omega(\ell\,\ell^{\prime})\,\|\ell\rangle\langle\ell^{\prime}\|\,,$ (7) then one easily verifies that $Z=F\,X\,F^{\dagger}\,,$ (8) much in the spirit of the standard way of looking at complementary variables in the infinite-dimensional Hilbert space: the position and momentum eigenstates are Fourier transform one of the other. The operators $\Lambda(m,n)$ can be written as $\Lambda(m,n)=e^{i\phi(m,n)}\,V^{n}\,Z^{m}\,V^{\dagger n}\,,$ (9) where $V$ turns out to be ($d>2$) $V=\sum_{\ell=0}^{d-1}\omega(-2^{-1}\ell^{2})\,\|\widetilde{\ell}\rangle\langle\widetilde{\ell}\|\,,$ (10) and the phase $\phi(m,n)$ is Klimov:2006 ; Bjork:2008 $\phi(m,n)=\omega(2^{-1}nm^{2})\,.$ (11) Here $2^{-1}$ denotes the multiplicative inverse of 2 modulo $d$ [that is, $2^{-1}=(d+1)/2$] and $\|\widetilde{\ell}\rangle$ is the conjugate basis, which is defined by the action of the Fourier transform on the computational basis, namely $\|\widetilde{\ell}\rangle=F\,\|\ell\rangle$. The case of qubits ($d=2$) requires minor modifications: $V$ is now $V=\frac{1}{2}\left(\begin{array}[]{cc}1+i&1-i\\\ 1-i&1+i\end{array}\right)\,,$ (12) while its action reads as $V\,Z\,V^{\dagger}=-iZX$. The operator $V$ has quite an important property: its powers generate MUBs when acting on the computational basis: indeed, if $\|\psi_{\ell}^{n}\rangle=V^{n}\|\ell\rangle\,,$ (13) one can check by a direct calculation that the states $\|\psi_{\ell}^{n}\rangle$ fulfill (2), which confirms the unbiasedness. If we denote $\Lambda_{\ell\ell^{\prime}}(m,n)=\langle\ell\|\Lambda(m,n)\|\ell^{\prime}\rangle$, according to Eq. (9), we have $\Lambda_{\ell\ell^{\prime}}(m,n)=e^{i\phi(m,n)}\,\langle\psi_{c}^{n}\|Z^{m}\|\psi_{d}^{n}\rangle\,.$ (14) Therefore, up to an unessential phase factor, $\Lambda_{\ell\ell^{\prime}}(m,n)$ are the matrix elements of the powers of the diagonal operator $Z$ in the corresponding MUB. This provides an elegant interpretation of these objects, which will play an essential role in what follows. ### II.2 Qudit Bell states For the case of two qudits, a sensible generalization of Bell states was devised in Ref. Bennett:1993 , namely $\|\Psi_{mn}\rangle=\frac{1}{\sqrt{d}}\sum_{\ell=0}^{d-1}\omega(m\ell)\,\|\ell\rangle_{A}\|\ell+n\rangle_{B}\,,$ (15) where, to simplify as much as possible the notation, we drop the subscript $AB$ from $\|\Psi_{mn}\rangle$, since we deal only with bipartite states. For further use, we also define $\|\tilde{\Psi}_{m}\rangle=\frac{1}{\sqrt{d}}\sum_{\ell=0}^{d-1}\|\ell\rangle_{A}\|\ell+m\rangle_{B}\,.$ (16) In the same vein, some generalized gates have been proposed to create these $d^{2}$ states Alber:2001 ; Durt:2003 . This set of states is orthonormal $\displaystyle\langle\Psi_{mn}\|\Psi_{m^{\prime}n^{\prime}}\rangle=\delta_{mm^{\prime}}\,\delta_{nn^{\prime}},\qquad\langle\tilde{\Psi}_{m}\|\tilde{\Psi}_{m^{\prime}}\rangle=\delta_{mm^{\prime}}\,,$ (17) $\displaystyle\langle\Psi_{mn}\|\tilde{\Psi}_{m^{\prime}}\rangle=\delta_{m0}\,\delta_{m^{\prime}0}\,,$ and allows for a resolution of the identity $\sum_{m=1}^{d-1}\sum_{n=0}^{d-1}\|\Psi_{mn}\rangle\langle\Psi_{mn}\|+\sum_{m=1}^{d-1}\|\tilde{\Psi}_{m}\rangle\langle\tilde{\Psi}_{m}\|=\openone\,,$ (18) so they constitute a bona fide basis for any bipartite qudit system. As anticipated in the Introduction, there must be then a connection with MUBs. And this is indeed the case: it suffices to observe that the states (15) and (16) can be recast as $\displaystyle\displaystyle\|\Psi_{mn}\rangle=\frac{1}{\sqrt{d}}\sum_{\ell,\ell^{\prime}=0}^{d-1}\Lambda_{\ell\ell^{\prime}}(m,n)\,\|\ell\rangle_{A}\|\ell^{\prime}\rangle_{B}\,,$ (19) $\displaystyle\displaystyle\|\tilde{\Psi}_{m}\rangle=\frac{1}{\sqrt{d}}\sum_{\ell,\ell^{\prime}=0}^{d-1}\tilde{\Lambda}_{\ell\ell^{\prime}}(m)\,\|\ell\rangle_{A}\|\ell^{\prime}\rangle_{B}\,,$ which can be checked by a direct calculation and $\Lambda_{\ell\ell^{\prime}}(m,n)$ and $\tilde{\Lambda}_{\ell\ell^{\prime}}(m)$ are the matrix elements of the operators (5). The matrices $\Lambda$ possess quite an interesting symmetry property $\Lambda_{\ell\ell^{\prime}}(m,n)=\omega(m^{2}n)\,\Lambda_{\ell^{\prime}\ell}(m,n)\,,\quad\tilde{\Lambda}_{\ell\ell^{\prime}}(m)=\tilde{\Lambda}_{\ell^{\prime}\ell}(m)\,.$ (20) In consequence, $\tilde{\Lambda}(m)$ are always totally symmetric under the permutation of subsystems $A$ and $B$ and so are the corresponding Bell states. Whenever $\omega(m^{2}n)=\pm 1$, $\Lambda(m,n)$ are either symmetric or antisymmetric. This happens for $mn=0$ $\pmod{d}$, and this is only possible for qubits: the symmetric matrices are $\tilde{\Lambda}(0)$, $\tilde{\Lambda}(1)$, and $\Lambda(1,0)$, while the antisymmetric is $\Lambda(1,1)$. The corresponding symmetric states are $\|\tilde{\Psi}_{0}\rangle=\|\Phi_{+}\rangle$, $\|\tilde{\Psi}_{1}\rangle=\|\Psi_{+}\rangle$, and $\|\Psi_{1,0}\rangle=\|\Phi_{-}\rangle$, and $\|\Psi_{1,1}\rangle=\|\Psi_{-}\rangle$ is the antisymmetric one. Finally, we can sum up the projectors of the bipartite states (15) over $m$, obtaining the following interesting novel property: $\displaystyle\displaystyle\sum_{m=0}^{d-1}\|\Psi_{mn}\rangle\langle\Psi_{mn}\|=\frac{1}{d}\sum_{\ell=0}^{d-1}(X^{n\ell}Z^{-\ell})_{A}\otimes(X^{n\ell}Z^{\ell})_{B}\,,$ (21) $\displaystyle\displaystyle\sum_{m=0}^{d-1}\|\tilde{\Psi}_{m}\rangle\langle\tilde{\Psi}_{m}\|=\frac{1}{d}\sum_{\ell=0}^{d-1}(X^{\ell})_{A}\otimes(X^{\ell})_{B}\,.$ In words, this means that the sum of projectors over the index $m$ is the sum of direct product of commuting operators for each particle. The proof of this statement involves a tedious yet direct calculation. For the case of two qubits, this implies that $\displaystyle\displaystyle\sum_{m=0,1}\|\Psi_{m1}\rangle\langle\Psi_{m1}\|=\frac{1}{2}[\openone+(XZ)_{A}\otimes(XZ)_{B}]\,,$ (22) $\displaystyle\displaystyle\sum_{m=0,1}\|\tilde{\Psi}_{m}\rangle\langle\tilde{\Psi}_{m}\|=\frac{1}{2}[\openone+(X)_{A}\otimes(X)_{B}]\,.$ ## III Bipartite multiqudit systems ### III.1 Mutually unbiased bases for $n$ qudits The previous ideas can be extended for a system of $n$ qudits. Instead of natural numbers, it is then convenient to use elements of the finite field $\mathbb{F}_{d^{n}}$ to label states, since then we can almost directly translate all the properties studied before for a single qudit. In the Appendix we briefly summarize the basic notions of finite fields needed to proceed. We denote as $\|\lambda\rangle$ (from here on, Greek letters will represent elements in the field $\mathbb{F}_{d^{n}}$) an orthonormal basis in the Hilbert space of the quantum system. Operationally, the elements of the basis can be labelled by powers of the primitive element, which can be found as roots of a minimal irreducible polynomial of degree $n$ over $\mathbb{Z}_{d}$. The generators of the generalized Pauli group are now $X_{\mu}\|\lambda\rangle=\|\lambda+\mu\rangle\,,\qquad Z_{\mu}\|\lambda\rangle=\chi(\lambda\mu)\|\lambda\rangle\,,$ (23) where $\chi(\lambda)$ is an additive character (defined in the Appendix). The Weyl form of the commutation relations reads as $Z_{\mu}X_{\nu}=\chi(\mu\nu)X_{\nu}Z_{\mu}$. In agreement with (5), we introduce the set of monomials $\tilde{\Lambda}(\mu)=X_{\mu}\,,\qquad\Lambda(\mu,\nu)=Z_{\mu}X_{\nu\mu}\,,$ (24) and their corresponding eigenstates also form a complete set of $d^{n}+1$ MUBs. The finite Fourier transform now is Vourdas:2005 $F=\frac{1}{\sqrt{d^{n}}}\sum_{\lambda,\lambda^{\prime}\in}\chi(\lambda\,\lambda^{\prime})\|\lambda\rangle\langle\lambda^{\prime}\|\,,$ (25) and thus $Z_{\mu}=F\,X_{\mu}\,F^{\dagger}\,.$ (26) The rotation operator $V_{\nu}$ transforms the diagonal $Z_{\mu}$ into an arbitrary monomial according to $\Lambda(\mu,\nu)=e^{i\varphi(\mu,\nu)}\,V_{\nu}\,Z_{\alpha}\,V_{\nu}^{\dagger}\,,$ (27) and is diagonal in the conjugate basis (defined, as before, via the Fourier transform $\|\widetilde{\lambda}\rangle=F\,\|\lambda\rangle$) $V_{\nu}=\sum_{\lambda}c_{\lambda\nu}\,\|\widetilde{\lambda}\rangle\langle\widetilde{\lambda}\|\,,$ (28) where the coefficients $c_{\lambda\nu}$ satisfy the following relation $c_{0\nu}=1\,,\qquad c_{\lambda+\lambda^{\prime}\,\nu}\,c_{\lambda\nu}^{\ast}=c_{\lambda^{\prime}\nu}\chi(-\nu\lambda^{\prime}\lambda),$ (29) When $d\neq 2$, a particular solution of Eq. (29) is $c_{\lambda\nu}=\chi(-2^{-1}\lambda^{2}\nu).$ (30) Again, if we define the states $\|\psi_{\lambda}^{\mu}\rangle=V_{\mu}\|\lambda\rangle\,,$ (31) they are unbiased and $\Lambda_{\lambda\lambda^{\prime}}(\mu,\nu)$ are the matrix elements of the diagonal operator $Z_{\mu}$ on the corresponding MUB $\Lambda_{\lambda\lambda^{\prime}}(\mu,\nu)=e^{i\varphi(\mu,\nu)}\langle\psi_{\lambda}^{\nu}\|Z_{\mu}\|\psi_{\lambda^{\prime}}^{\mu}\rangle\,.$ (32) ### III.2 Multiqudit Bell states For a bipartite system of $n$ qudits, it seems natural to extend the previous construction (II.2) by introducing the $d^{2n}$ states $\displaystyle\|\Psi_{\mu\nu}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{d^{n}}}\sum_{\lambda,\lambda^{\prime}}\Lambda_{\lambda\lambda^{\prime}}(\mu,\nu)\,\|\lambda\rangle_{A}\|\lambda^{\prime}\rangle_{B}\,,$ $\displaystyle\|\tilde{\Psi}_{\mu}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{d^{n}}}\sum_{\lambda,\lambda^{\prime}}\tilde{\Lambda}_{\lambda\lambda^{\prime}}(\mu)\,\|\lambda\rangle_{A}\|\lambda^{\prime}\rangle_{B}\,.$ Accordingly, the associated Bell states are (apart from an unessential global phase) $\displaystyle\|\Psi_{\mu\nu}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{d^{n}}}\sum_{\lambda}\chi(\mu\lambda)\,\|\lambda\rangle_{A}\|\lambda+\nu\rangle_{B}\,,$ $\displaystyle\|\tilde{\Psi}_{\mu}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{d^{n}}}\sum_{\lambda}\|\lambda\rangle_{A}\|\lambda+\nu\rangle_{B}\,,$ which look as quite a reasonable generalization. One can prove the orthogonality $\displaystyle\langle\Psi_{\mu\nu}\|\Psi_{\mu^{\prime}\nu^{\prime}}\rangle=\delta_{\mu\mu^{\prime}}\,\delta_{\nu\nu^{\prime}},\qquad\langle\tilde{\Psi}_{\mu}\|\tilde{\Psi}_{\mu^{\prime}}\rangle=\delta_{\mu\mu^{\prime}}\,,$ (35) $\displaystyle\langle\Psi_{\mu\nu}\|\tilde{\Psi}_{\mu^{\prime}}\rangle=\delta_{\mu 0}\,\delta_{\mu^{\prime}0}\,,$ and the completeness relation $\sum_{\mu\neq 0,\nu}\|\Psi_{\mu\nu}\rangle\langle\Psi_{\mu\nu}\|+\sum_{\mu}\|\tilde{\Psi}_{\mu}\rangle\langle\tilde{\Psi}_{\mu}\|=\openone\,,$ (36) which confirms that they constitute a basis. Moreover, the reduced density matrices for both subsystems are completely random $\mathop{\mathrm{Tr}}\nolimits_{A}(\|\Psi_{\mu\nu}\rangle\langle\Psi_{\mu\nu})=\frac{1}{d^{n}}\sum_{\lambda}\|\lambda\rangle_{B}\,\,{}_{B}\langle\lambda\|\,,$ (37) (and other equivalent equation with $A$ and $B$ interchanged) showing that they are maximally entangled states. The concept of symmetric and antisymmetric states can be worked out for systems of $n$ qubits, which constitutes a nontrivial generalization of our previous discussion Sych:2009 ; Jex:2003 . The symmetric states [i.e., $\Lambda_{\lambda\lambda^{\prime}}(\mu,\nu)=\Lambda_{\lambda^{\prime}\lambda}(\mu,\nu)]$, correspond to those pairs $(\mu,\nu)$ such that $\mathop{\mathrm{tr}}\nolimits(\nu\mu^{2})=0\,,$ (38) where $\mathop{\mathrm{tr}}\nolimits$, in small case, denotes the trace map in the field. Clearly, all the states $\|\Psi_{\mu 0}\rangle$ and $\|\tilde{\Psi}_{\mu}\rangle$ are symmetric. The antisymmetric states [i.e., $\Lambda_{\lambda\lambda^{\prime}}(\mu,\nu)=-\Lambda_{\lambda^{\prime}\lambda}(\mu,\nu)]$ are defined by the pairs $(\mu,\nu)$ such that $\mathop{\mathrm{tr}}\nolimits(\nu\mu^{2})=1\,.$ (39) Finally, a property similar to (II.2) is fulfilled: summing up the projectors over $\mu$ one obtains $\displaystyle\displaystyle\sum_{\mu}\|\Psi_{\mu\nu}\rangle\langle\Psi_{\mu\nu}\|=\sum_{\lambda}(X_{\lambda\nu}Z_{-\lambda})_{A}\otimes(X_{\lambda\nu}Z_{\lambda})_{B}\,,$ (40) $\displaystyle\displaystyle\sum_{\mu}\|\tilde{\Psi}_{\mu}\rangle\langle\tilde{\Psi}_{\mu}\|=\sum_{\lambda}(X_{\lambda})_{A}\otimes(X_{\lambda})_{B}\,,$ whose interpretation is otherwise the same as for qudits. ### III.3 Examples Since we are dealing with $n$-qudit systems, we can map the abstract Hilbert space $\mathcal{H}_{d^{n}}$ into $n$ single-qudit Hilbert spaces. This is achieved by expanding any field element in a convenient basis $\\{\theta_{j}\\}$ (with $j=1,\ldots,n$), so that $\lambda=\sum_{j}\ell_{j}\,\theta_{j}\,,$ (41) where $\ell_{j}\in\mathbb{Z}_{d}$. Then, we can represent the states as $\|\lambda\rangle=\|\ell_{1},\ldots,\ell_{n}\rangle$ and the coefficients $\ell_{j}$ play the role of quantum numbers for each qudit. For example, for two qubits, the abstract state $(\|0\rangle+\|\sigma^{3}\rangle)/\sqrt{2}$, where $\sigma$ is a primitive elements, can be mapped onto the physical state $\|00\rangle+\|10\rangle)/\sqrt{2}$ in the polynomial basis $\\{1,\sigma\\}$, whereas in the selfdual basis $\\{\sigma,\sigma^{2}\\}$ it is associated with $(\|00\rangle+\|11\rangle)/\sqrt{2}$. Observe that, while the first state is factorizable, the other one is entangled. The use of the selfdual basis (or the almost selfdual, if the latter does not exist) is especially advantageous, since only then the Fourier transform and the basic operators factorize in terms of single-qudit analogues: $X_{\lambda}=X^{\ell_{1}}\otimes\ldots\otimes X^{\ell_{n}}\,,\qquad Z_{\lambda}=Z^{\ell_{1}}\otimes\ldots\otimes Z^{\ell_{n}}\,.$ (42) For a bipartite $4\times 4$ system the states are represented as $\|\lambda\rangle=\|\ell_{1},\ell_{2}\rangle$ with $\ell_{j}\in\mathbb{Z}_{2}$. The Bell basis can be expressed as $\displaystyle\|m_{1},n_{1};m_{2},n_{2}\rangle$ $\displaystyle=$ $\displaystyle\frac{(-1)^{m_{1}n_{2}+m_{2}n_{1}}}{2}\sum_{\ell_{1},\ell_{2}}(-1)^{m_{1}\ell_{1}+m_{2}\ell_{2}}\,\|\ell_{1}+m_{1}n_{2}+m_{2}n_{1},\ell_{2}+m_{1}n_{1}+m_{2}n_{2}\rangle_{A}\|\ell_{1},\ell_{2}\rangle_{B}\,,$ $\displaystyle\|\widetilde{m_{1},m_{2}}\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{\ell_{1},\ell_{2}}\|\ell_{1}+n_{1},\ell_{2}+m_{2}\rangle_{A}\|\ell_{1};\ell_{2}\rangle_{B}\,.$ The conditions $m_{1}n_{2}+m_{2}n_{1}=\left\\{\begin{array}[]{l}0\,,\\\ 1\,,\end{array}\right.$ (44) determine the symmetric and antisymmetric states, respectively. The solutions of this equation show that there are 10 symmetric states and 6 antisymmetric ones, whose explicit form can be computed from previous formulas. Before ending, we wish to stress that so far we have been dealing with systems made of $n$ qudits. However, sometimes they can be treated instead as a single ‘particle’ with $d^{n}$ levels. For example, a four-dimensional system can be taken as two qubits or as a ququart. If, for some physical reason, we choose for the quqart, we can still use Eq. (15), as in Ref. Bennett:1993 , even if now the dimension is not a prime number. However, if we proceed in this way the resulting basis contains 6 symmetric and 2 antisymmetric states, while the other 8 do not have any explicit symmetry, contrary to our results. ## IV Concluding remarks In summary, we have provided a complete MUB-based construction of Bell states that fulfills all the requirements needed for a good description of maximally entangled states of bipartite multiqudit systems. Mutually unbiasedness is a very deep concept arising from the exact formulation of complementarity. The deep connection shown in this paper with Bell bases is more than a mere academic curiosity, for it is immediately applicable to a variety of experiments involving qudit systems. ## Appendix A Finite fields In this appendix we briefly recall the minimum background needed in this paper. The reader interested in more mathematical details is referred, e.g., to the excellent monograph by Lidl and Niederreiter Lidl:1986 . A commutative ring is a nonempty set $R$ furnished with two binary operations, called addition and multiplication, such that it is an Abelian group with respect the addition, and the multiplication is associative. Perhaps, the motivating example is the ring of integers $\mathbb{Z}$, with the standard sum and multiplication. On the other hand, the simplest example of a finite ring is the set $\mathbb{Z}_{n}$ of integers modulo $n$, which has exactly $n$ elements. A field $F$ is a commutative ring with division, that is, such that 0 does not equal 1 and all elements of $F$ except 0 have a multiplicative inverse (note that 0 and 1 here stand for the identity elements for the addition and multiplication, respectively, which may differ from the familiar real numbers 0 and 1). Elements of a field form Abelian groups with respect to addition and multiplication (in this latter case, the zero element is excluded). The characteristic of a finite field is the smallest integer $d$ such that $d\,1=\underbrace{1+1+\ldots+1}_{\mbox{\scriptsize$d$ times}}=0$ (45) and it is always a prime number. Any finite field contains a prime subfield $\mathbb{Z}_{d}$ and has $d^{n}$ elements, where $n$ is a natural number. Moreover, the finite field containing $d^{n}$ elements is unique and is called the Galois field $\mathbb{F}_{d^{n}}$. Let us denote as $\mathbb{Z}_{d}[x]$ the ring of polynomials with coefficients in $\mathbb{Z}_{d}$. Let $P(x)$ be an irreducible polynomial of degree $n$ (i.e., one that cannot be factorized over $\mathbb{Z}_{d}$). Then, the quotient space $\mathbb{Z}_{d}[X]/P(x)$ provides an adequate representation of $\mathbb{F}_{d^{n}}$. Its elements can be written as polynomials that are defined modulo the irreducible polynomial $P(x)$. The multiplicative group of $\mathbb{F}_{d^{n}}$ is cyclic and its generator is called a primitive element of the field. As a simple example of a nonprime field, we consider the polynomial $x^{2}+x+1=0$, which is irreducible in $\mathbb{Z}_{2}$. If $\sigma$ is a root of this polynomial, the elements $\\{0,1,\sigma,\sigma^{2}=\sigma+1=\sigma^{-1}\\}$ form the finite field $\mathbb{F}_{2^{2}}$ and $\sigma$ is a primitive element. A basic map is the trace $\mathop{\mathrm{tr}}\nolimits(\lambda)=\lambda+\lambda^{2}+\ldots+\lambda^{d^{n-1}}\,.$ (46) It is always in the prime field $\mathbb{Z}_{d}$ and satisfies $\mathop{\mathrm{tr}}\nolimits(\lambda+\lambda^{\prime})=\mathop{\mathrm{tr}}\nolimits(\lambda)+\mathop{\mathrm{tr}}\nolimits(\lambda^{\prime})\,.$ (47) In terms of it we define the additive characters as $\chi(\lambda)=\exp\left[\frac{2\pi i}{p}\mathop{\mathrm{tr}}\nolimits(\lambda)\right]\,,$ (48) which posses two important properties: $\chi(\lambda+\lambda^{\prime})=\chi(\lambda)\chi(\lambda^{\prime}),\qquad\sum_{\lambda^{\prime}\in\mathbb{F}_{d^{n}}}\chi(\lambda\lambda^{\prime})=d^{n}\delta_{0,\lambda}\,.$ (49) Any finite field $\mathbb{F}_{d^{n}}$ can be also considered as an $n$-dimensional linear vector space. Given a basis $\\{\theta_{j}\\}$, ($j=1,\ldots,n$) in this vector space, any field element can be represented as $\lambda=\sum_{j=1}^{n}\ell_{j}\,\theta_{j},$ (50) with $\ell_{j}\in\mathbb{Z}_{d}$. In this way, we map each element of $\mathbb{F}_{d^{n}}$ onto an ordered set of natural numbers $\lambda\Leftrightarrow(\ell_{1},\ldots,\ell_{n})$. Two bases $\\{\theta_{1},\ldots,\theta_{n}\\}$ and $\\{\theta_{1}^{\prime},\ldots,\theta_{n}^{\prime}\\}$ are dual when $\mathop{\mathrm{tr}}\nolimits(\theta_{k}\theta_{l}^{\prime})=\delta_{k,l}.$ (51) A basis that is dual to itself is called selfdual. There are several natural bases in $\mathbb{F}_{d^{n}}$. One is the polynomial basis, defined as $\\{1,\sigma,\sigma^{2},\ldots,\sigma^{n-1}\\},$ (52) where $\sigma$ is a primitive element. An alternative is the normal basis, constituted of $\\{\sigma,\sigma^{d},\ldots,\sigma^{d^{n-1}}\\}.$ (53) The choice of the appropriate basis depends on the specific problem at hand. For example, in $\mathbb{F}_{2^{2}}$ the elements $\\{\sigma,\sigma^{2}\\}$ are both roots of the irreducible polynomial. The polynomial basis is $\\{1,\sigma\\}$ and its dual is $\\{\sigma^{2},1\\}$, while the normal basis $\\{\sigma,\sigma^{2}\\}$ is selfdual. The selfdual basis exists if and only if either $d$ is even or both $n$ and $d$ are odd. However for every prime power $d^{n}$, there exists an almost selfdual basis of $\mathbb{F}_{d^{n}}$, which satisfies the properties: $\mathop{\mathrm{tr}}\nolimits(\theta_{i}\theta_{j})=0$ when $i\neq j$ and $\mathop{\mathrm{tr}}\nolimits(\theta_{i}^{2})=1$, with one possible exception. For instance, in the case of two qutrits $\mathbb{F}_{3^{2}}$, a selfdual basis does not exist and two elements $\\{\sigma^{2},\sigma^{4}\\}$, $\sigma$ being a root of the irreducible polynomial $x^{2}+x+2=0$, form a self dual basis $\mathop{\mathrm{tr}}\nolimits(\sigma^{2}\sigma^{2})=1\,,\quad\mathop{\mathrm{tr}}\nolimits(\sigma^{4}\sigma^{4})=2\,,\quad\mathop{\mathrm{tr}}\nolimits(\sigma^{2}\sigma^{4})=0\,.$ (54) ## References * (1) E. Schrödinger, Math. Proc. Cambridge Philos. Soc. 31, 555 (1935). * (2) M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000). * (3) A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Boston, 1993). * (4) T. C. Wei, J. T. Barreiro, and P. G. Kwiat, Phys. Rev. A 75, 060305(R) (2007). * (5) W. Dür and J. I. Cirac, Phys. Rev. A 61, 042314 (2000). * (6) W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000). * (7) A. Acín, A. Andrianov, L. Costa, E. Jané, J. I. Latorre, and R. Tarrach, Phys. Rev. Lett. 85, 1560 (2000). * (8) H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86, 910 (2001). * (9) F. Verstraete, J. Dehaene, B. DeMoor, and H. Verschelde, Phys. Rev. A 65, 052112 (2002). * (10) G. Rigolin, T. R. de Oliveira, and M. C. de Oliveira, Phys. Rev. A 74, 022314 (2006). * (11) P. Facchi, G. Florio, G. Parisi, and S. Pascazio, Phys. Rev. A 77, 060304 (2008). * (12) H. Bechmann-Pasquinucci and W. Tittel, Phys. Rev. A 61, 062308 (2000). * (13) M. Bourennane, A. Karlsson, and G. Björk, Phys. Rev. A 64, 012306 (2001). * (14) N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, Phys. Rev. Lett. 88, 127902 (2002). * (15) D. Sych, B. Grishanin, and V. Zadkov, Phys. Rev. A 70, 052331 (2004). * (16) D. Sych and G. Leuchs, New J. Phys. 11, 013006 (2009). * (17) J. Schwinger, Proc. Natl. Acad. Sci. USA 46, 570 (1960). * (18) W. K. Wootters, Ann. Phys. (NY) 176, 1 (1987). * (19) W. K. Wootters and B. D. Fields, Ann. Phys. (NY) 191, 363 (1989). * (20) W. K. Wootters, IBM J. Res. Dev. 48, 99 (2004). * (21) K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A 70, 062101 (2004). * (22) W. K. Wootters, Found. Phys. 36, 112 (2006). * (23) M. Planat and H. C. Rosu, Eur. Phys. J. D 36, 133 (2005). * (24) I. Bengtsson, W. Bruzda, Å. Ericsson, J. Å. Larsson, W. Tadej, and K. Życzkowski, J. Math. Phys. 48, 052106 (2007). * (25) R. Asplund, G. Björk, and M. Bourennane, J. Opt. B 3, 163 (2001). * (26) D. Gottesman, Phys. Rev. A 54, 1862 (1996). * (27) A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Phys. Rev. Lett. 78, 405 (1997). * (28) B.-G. Englert and Y. Aharonov, Phys. Lett. A 284, 1 (2001). * (29) P. K. Aravind, Z. Naturforsch. A: Phys. Sci. 58, 2212 (2003). * (30) J. P. Paz, A. J. Roncaglia, and M. Saraceno, Phys. Rev. A 72, 012309 (2005). * (31) G. Kimura, H. Tanaka, and M. Ozawa Phys. Rev. A 73 050301(R) (2006). * (32) I. D. Ivanović, J. Phys. A 14, 3241 (1981). * (33) K. Kraus, Phys. Rev. D 35, 3070 (1987). * (34) J. Lawrence, Č. Brukner, A. Zeilinger, Phys. Rev. A 65 032320 (2002). * (35) S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and V. Vatan, Algorithmica 34, 512 (2002). * (36) A. Klappenecker and M. Rötteler, Lecture Notes in Comput. Sci. 2948, 137 (2004). * (37) J. Lawrence, Phys. Rev. A 70, 012302 (2004). * (38) A. O. Pittenger and M. H. Rubin, J. Phys. A 38, 6005 (2005). * (39) P. Wocjan and T. Beth, Quantum Inform. Compu. 5, 93 (2005). * (40) T. Durt, J. Phys. A 38, 5267 (2005). * (41) A. B. Klimov, J. L. Romero, G. Björk, and L. L. Sánchez-Soto, J. Phys. A 40, 3987 (2007). * (42) A. B. Klimov, L. L. Sánchez-Soto, and H. de Guise, 2005, J. Phys. A 38, 2747 (2005). * (43) A. Weil, Acta Math. 111, 143 (1964). * (44) C. P. Putnam, Commutation Properties of Hilbert Space Operators (Springer, Heidelberg, 1987). * (45) A. Vourdas, Rep. Prog. Phys. 67, 267 (2004). * (46) A. B. Klimov, C. Muñoz, and J. L. Romero, J. Phys. A 39, 14471 (2006). * (47) G. Björk,, A. B. Klimov, and L. L. Sánchez-Soto, Prog. Opt. 51, 469 (2008). * (48) Ch. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). * (49) G. Alber, A. Delgado, N. Gisin, and I. Jex, J. Phys. A 34, 8821 (2001). * (50) T. Durt and B. Nagler, Phys. Rev. A 68, 042323 (2003). * (51) A. Vourdas, J. Phys. A 40, R285 (2005). * (52) I. Jex, G. Alber, S. M. Barnett, and A. Delgado, Fortschr. Phys. 51, 172 (2003). * (53) R. Lidl and H. Niederreiter Introduction to Finite Fields and their Applications (Cambridge University Press, Cambridge, 1986).	arxiv-papers	2009-02-11T12:56:37	2024-09-04T02:49:00.521626	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "A. B. Klimov, D. Sych, L. L. Sanchez-Soto and G. Leuchs", "submitter": "Luis L. Sanchez. Soto", "url": "https://arxiv.org/abs/0902.1880" }
0902.1895	# Coherent State Quantum Key Distribution with Multi Letter Phase-Shift Keying Denis Sych and Gerd Leuchs Max Planck Institut für die Physik des Lichts, Günther–Scharowsky–Strasse 1 / Bau24, D-91058 Erlangen, Germany Institut für Optik, Information und Photonik, Universität Erlangen–Nürnberg, Staudtstrasse 7 / B2, 91058 Erlangen, Germany ###### Abstract We present a protocol for quantum key distribution using discrete modulation of coherent states of light. Information is encoded in the variable phase of coherent states which can be chosen from a regular discrete set ranging from binary to continuous modulation, similar to phase–shift–keying in classical communication. Information is decoded by simultaneous homodyne measurement of both quadratures and requires no active choice of basis. The protocol utilizes either direct or reverse reconciliation, both with and without postselection. We analyze the security of the protocol and show how to enhance it by the optimal choice of all variable parameters of the quantum signal. ###### pacs: 03.67.Dd, 42.50.Ex, 89.70.Cf ## 1 Introduction Quantum key distribution (QKD) is a procedure of information exchange between two parties, the sender Alice and the receiver Bob, which allows to distribute absolutely secure data between them [Gisin:02RMP, Dusek:06, Scarani:09]. The distinctive part of QKD with respect to classical communication is the use of a quantum information channel, where the signal is protected from unauthorized duplication [Wooters:82, Dieks:82, Bennett:84, Ekert:91]. Mathematically, the information signal can be described by discrete variables (DV) or by continuous variables (CV) [Cerf:07book], although physically the signal can be of various kinds: single photons [Bennett:84], weak coherent pulses [Bennett:92], squeezed states [Ralph:99, Hillery:00] and other systems where a signal possesses essentially quantum properties. A universal characteristic that can be compared for different QKD protocols (apart from their experimental realizations) is the secret key generation rate (shortly, key rate) — i.e. the average amount of secure information per elementary transmission (e.g. per light pulse). In order to transmit higher total amount of secure data one can increase the pulse repetition rate or increase the key rate. The first way is limited mainly by experimental techniques, while the second one is defined by mathematical properties of the given QKD protocol. In the case of DV QKD, it has been shown that extensions of the standard four–letter BB84 protocol [Bennett:84] to higher number of letters in the alphabet can improve performance in terms of higher critical error rate or longer communication lines [Brus:98, Bech:99, Sych:04, Sych:05]. In the case of CV QKD, the first protocols were based on Gaussian alphabets consisting of either coherent or squeezed states. The common weak point of these protocols is high sensitivity to losses in the quantum communication channel, which initially was believed to lead to the so–called “3 dB loss limit”: the key rate is equal to zero when channel losses are higher than $50\%$. After the invention of postselection [Silb:02] and reverse reconciliation [Gross:03], this limit was overcome, and the key rate was substantially improved. Another interesting option for improving the key rate is to consider discrete alphabets [Namiki:03, Heid:06] instead of Gaussian ones. Protocols with Gaussian alphabets have an advantage of simpler security analysis, whilst the protocols based on discrete modulation are easier to realize in practice. In this work we address the question how one can further increase the key rate of CV QKD with discrete modulation by varying the geometry of the quantum alphabet. Having in mind the idea of improving the properties of DV QKD by use of more symmetric alphabets with higher number of letters [Sych:05], we present a new CV QKD protocol which generalizes previous ones with discrete modulation [Namiki:03, Lorenz:04, Heid:06]. Namely, the protocol employs an alphabet with $N$ coherent states $\mathop{\left\|\alpha_{k}\right\rangle}\nolimits=\mathop{\left\|ae^{i\frac{2\pi}{N}k}\right\rangle}\nolimits$ which have relative phases $\frac{2\pi}{N}k$ and a fixed amplitude $a$. In classical communication, this type of encoding is known as phase–shift–keying (PSK). We perform a security analysis of the proposed protocol for lossy but noiseless quantum channels, providing full optimization of all parameters of the protocol, and show how the number of letters affects the key rate. ## 2 Description of the protocol An elementary information transmission in the multi letter PSK protocol is as follows: * • The sender (Alice) chooses a random equiprobable number $k=1\ldots N$ and sends the respective coherent state $\displaystyle\mathop{\left\|\alpha_{k}\right\rangle}\nolimits=\mathop{\left\|ae^{i\frac{2\pi}{N}k}\right\rangle}\nolimits$. * • The receiver (Bob) measures the state by splitting the signal at a 50/50 beam splitter and measuring two conjugate quadratures $\hat{x}$ and $\hat{p}$ at the output ports by homodyning each signal — such a scheme is called heterodyne measurement [Lorenz:04, Weed:04], where the two conjugate measurements can be separated in time or space. The results of the measurements are $\beta_{x}$ and $\beta_{p}$, which we write as a pure coherent state $\mathop{\left\|\beta\right\rangle}\nolimits=\mathop{\left\|\beta_{x}+i\beta_{p}\right\rangle}\nolimits$. * • Bob assigns a classical number $l$ to the measured state $\mathop{\left\|\beta\right\rangle}\nolimits$ by finding a state $\mathop{\left\|\alpha_{l}\right\rangle}\nolimits$ which is the closest alphabet’s state to the state $\mathop{\left\|\beta\right\rangle}\nolimits$: $\|\mathop{\left\langle\alpha_{l}\right\|\left.\beta\right\rangle}\nolimits\|^{2}=\max\limits_{n}\|\mathop{\left\langle\alpha_{n}\right\|\left.\beta\right\rangle}\nolimits\|^{2}$. Generally speaking, this classical value $l$ decoded by Bob can be different from the initial value $k$ sent by Alice because of intrinsic quantum uncertainty even in the absence of eavesdropping or any channel noise. The elementary information transmission from Alice to Bob can be schematically shown as a “$k\rightarrow l$” channel: $k\stackrel{{\scriptstyle encoding}}{{\longrightarrow}}\mathop{\left\|\alpha_{k}\right\rangle}\nolimits\stackrel{{\scriptstyle measurement}}{{\longrightarrow}}\mathop{\left\|\beta\right\rangle}\nolimits\stackrel{{\scriptstyle decoding}}{{\longrightarrow}}l.$ (1) As long as the considered quantum alphabet has a regular $2\pi/N$ phase–shift symmetry, and all states have an equal probability to be sent, all channel inputs and outputs (1) are equiprobable. We note, that there is no active choice of measurement basis in our protocol, therefore there is no basis reconciliation needed. All transmissions contribute to the total secure key, and nothing is discarded unlike in previous protocols with discrete modulation and homodyne detection [Namiki:06]. After the measurement, Bob can (but not necessarily has to) use the postselection idea [Silb:02], when he decides whether to keep the transmission depending on the value $\beta$. The elementary transmission (1), possibly followed by postselection, is repeated until Alice and Bob collect enough data to perform classical error correction and privacy amplification procedures [Gisin:02RMP]. The resulting data is a secret key. The amplitude $a$ and the number of letters $N$ can be flexibly adjusted for a given information channel. The exact optimization for a given channel transmittance will be discussed later. These parameters are supposed to be publicly known, particularly by the potential eavesdropper. We note that the number of states can be arbitrary large. In the limit of infinity $N\rightarrow\infty$ we have a continuous phase modulation. Thus our protocol can be viewed as a smooth transition between discrete [Namiki:03, Lorenz:04, Heid:06] and continuous [Silb:02, Gross:02, Weed:04] modulation of CV. ## 3 Security analysis We investigate the security of our protocol assuming there is no excess noise in the quantum channel. In this case the best possible attack is the beam splitting attack [Heid:06]. We allow Eve to have unlimited access to all the losses, as she could replace the real lossy information channel with an ideal lossless one. The beam splitting transformation is $\mathop{\left\|\alpha_{k}\right\rangle}\nolimits_{A}\rightarrow\mathop{\left\|\beta_{k}\right\rangle}\nolimits_{B}\otimes\mathop{\left\|\epsilon_{k}\right\rangle}\nolimits_{E}$, where Alice’s initial state $\mathop{\left\|\alpha_{k}\right\rangle}\nolimits$ is split to Bob’s state $\mathop{\left\|\beta_{k}\right\rangle}\nolimits$ and Eve’s state $\mathop{\left\|\epsilon_{k}\right\rangle}\nolimits$: $\mathop{\left\|\beta_{k}\right\rangle}\nolimits=\mathop{\left\|\sqrt{\eta}\alpha_{k}\right\rangle}\nolimits,\quad\mathop{\left\|\epsilon_{k}\right\rangle}\nolimits=\mathop{\left\|\sqrt{1-\eta}\alpha_{k}\right\rangle}\nolimits,$ (2) and $\eta$ is the channel transmittance. In this beam splitting scenario, Eve does not introduce any excess noise on Bob’s side, whereas in any other better eavesdropping strategy Eve necessarily does. For example, if Eve would make an intercept–resend attack, then she adds at least one unit of shot noise. Afterwards, she can attenuate the signal, and the excess noise will be proportionally reduced. As a remark on the side, we see in this way, that the maximum tolerable excess noise cannot exceed that of the intercept–resend strategy, i.e. cannot be higher then channel transmittance $\eta$. In real communication lines, such as optical fibres or free space, excess noise (typically, about $1\%$ of shot noise [Lorenz:04, Lorenz:06, Elser:08]) is introduced mainly by imperfections of the experimental setup, and there is almost no measurable excess noise due to the channel itself. If the absence of channel excess noise is experimentally verified, then the eavesdropping strategy based on beam splitting is the best possible attack, at least for the values of the channel transmittance $\eta\gg 0.01$. ### 3.1 Information between Alice and Bob The amount of classical mutual Shannon information $I_{AB}$ transmitted from Alice to Bob via the channel (1) is equal to the difference of a priori (before measurement) and a posteriori (after measurement) entropies [Shan:48]. Before any measurement, all channel outcomes are equiprobable for Bob, so his a priori entropy $H_{Bob}^{prior}$ is the unconditional “pure guess” entropy equal to $\log_{2}N$ bit per transmission. The conditional probability density to measure the state $\mathop{\left\|\beta\right\rangle}\nolimits$ when a state $\mathop{\left\|\alpha_{k}\right\rangle}\nolimits$ has been sent is $p(\beta\|k)\sim\|\mathop{\left\langle\beta_{k}\right\|\left.\beta\right\rangle}\nolimits\|^{2}\sim e^{-\|\beta-\beta_{k}\|^{2}}.$ The total unconditional probability density to measure a state $\mathop{\left\|\beta\right\rangle}\nolimits$ is $p(\beta)=\frac{1}{N}\sum\limits_{k=1}^{N}p(\beta\|k)$. Its normalization $\int p(\beta)d\beta=1$ also yields the normalization of $p(\beta\|k)=\frac{1}{\pi}e^{-\|\beta-\beta_{k}\|^{2}}.$ After the measurement of $\mathop{\left\|\beta\right\rangle}\nolimits$, the probability $p_{l}(\beta)$ that the state $\mathop{\left\|\alpha_{l}\right\rangle}\nolimits$ was initially sent is $p_{l}(\beta)=\frac{p(\beta\|l)}{Np(\beta)}=\frac{1}{\pi Np(\beta)}e^{-\|\beta-\beta_{l}\|^{2}}.$ (3) As we discussed above, the measured state $\mathop{\left\|\beta\right\rangle}\nolimits$ is decoded by Bob to a classical value $l$ such that the state $\mathop{\left\|\alpha_{l}\right\rangle}\nolimits$ is the closest alphabet’s state to the measured state $\mathop{\left\|\beta\right\rangle}\nolimits$. Corresponding regions in the phase space are shown by different shades of grey in Fig. 1. In the case when $l=k$ the value (3) is the probability of decoding the correct result, otherwise (3) is the error probability of decoding a wrong result $l\neq k$. Bob’s a posteriori entropy $H_{Bob}^{post}$ is the Shannon entropy of the total probability distribution $P(\beta)=\\{p_{1}(\beta),p_{2}(\beta),\ldots,p_{N}(\beta)\\}$ conditioned on the measured state $\mathop{\left\|\beta\right\rangle}\nolimits$: $H_{Bob}^{post}[P(\beta)]=-\sum\limits_{k=1}^{N}p_{k}(\beta)\log_{2}p_{k}(\beta).$ (4) Finally, the amount of classical information transmitted from Alice to Bob via the channel (1): $I_{AB}=\int p(\beta)I_{AB}(\beta)d\beta,$ (5) where $I_{AB}(\beta)=\log_{2}N+\sum\limits_{k=1}^{N}p_{k}(\beta)\log_{2}p_{k}(\beta)$. ### 3.2 Eve’s information To calculate Eve’s potential information we consider two strategies of classical communication between Alice and Bob during the post processing step: direct reconciliation and reverse reconciliation. In the first strategy Alice sends correcting information to Bob, and in the second one Bob sends it to Alice. We also assume, that after the beam splitting Eve is not restricted to any practical way of information extraction from this state, thus her potential knowledge is bounded by the Holevo information [Holevo:73]. In the general case, the Holevo information $\chi$ sets the upper bound on the information which can be transmitted by a state randomly chosen from a set of $N$ states $\hat{\rho}_{k}$ with a respective probability $p_{k}$: $\chi=S\left(\sum\limits_{k=1}^{N}p_{k}\hat{\rho}_{k}\right)-\sum\limits_{k=1}^{N}p_{k}S(\hat{\rho}_{k}),$ (6) where $S(\hat{\rho})$ is the von Neumann entropy $S(\hat{\rho})=-{\rm Tr}\hat{\rho}\log_{2}\hat{\rho}$. #### 3.2.1 Direct reconciliation In the direct reconciliation case, Eve has a state (2) conditioned only on Alice’s sent state $\mathop{\left\|\alpha_{k}\right\rangle}\nolimits$. Eve’s conditional state is pure, thus her information is equal to the von Neumann entropy $I_{AE}=S(\hat{\rho}_{E})$ of her unconditional state $\hat{\rho}_{E}=\frac{1}{N}\sum\limits_{k=1}^{N}\hat{\rho}_{k}$. To calculate $S(\hat{\rho}_{E})$ we need to find the eigenvalues of $\hat{\rho}_{E}$. The rotational symmetry of the phase–shift alphabet allows us to write Eve’s conditional states in an orthogonal basis $\\{\mathop{\left\|m\right\rangle}\nolimits\\}$ as [Chefles:98]: $\mathop{\left\|\epsilon_{k}\right\rangle}\nolimits=\sum\limits_{m=1}^{N}c_{m}e^{i\frac{2\pi}{N}km}\mathop{\left\|m\right\rangle}\nolimits.$ (7) In the basis $\\{\mathop{\left\|m\right\rangle}\nolimits\\}$ Eve’s unconditional state takes the diagonal form: $\hat{\rho}_{E}=\frac{1}{N}\sum\limits_{k=1}^{N}\mathop{\left\|\epsilon_{k}\right\rangle}\nolimits\mathop{\left\langle\epsilon_{k}\right\|}\nolimits=\sum\limits_{m=1}^{N}\|c_{m}\|^{2}\mathop{\left\|m\right\rangle}\nolimits\mathop{\left\langle m\right\|}\nolimits,$ (8) so Eve’s information is equal to $I_{AE}=S(\hat{\rho}_{E})=H[C]$, where $H[C]$ is the Shannon entropy $(\ref{ShanEnt})$ of the probability distribution $C=\\{\|c_{1}\|^{2},\|c_{2}\|^{2},\ldots,\|c_{N}\|^{2}\\}$. The coefficients $\|c_{m}\|^{2}$ are derived from a system of $N$ linear equations enumerated by an index $k=1\ldots N$: $\sum\limits_{m=1}^{N}e^{i\frac{2\pi}{N}km}\|c_{m}\|^{2}=\mathop{\left\langle\epsilon_{N}\right\|\left.\epsilon_{k}\right\rangle}\nolimits.$ (9) It has a formal analytical solution $\|c_{m}\|^{2}=\frac{1}{N}\sum\limits_{n=1}^{N}e^{-i\frac{2\pi}{N}mn-a^{2}(1-\eta)\left(1-e^{i\frac{2\pi}{N}n}\right),}$ (10) where the coefficients $c_{m}$ depend on the signal amplitude $a$ and the channel transmittance $\eta$. #### 3.2.2 Reverse reconciliation In the case of reverse reconciliation Eve has a state (2) conditioned on Bob’s measured state $\mathop{\left\|\beta\right\rangle}\nolimits$. After classical communication Eve can possibly find out the amount of information (5) between Alice and Bob in each transmission, so we assume this value is publicly open. Additionally we assume that Bob announces the amplitude of the measured state, so Eve knows the measured state $\mathop{\left\|\beta\right\rangle}\nolimits$ up to a cyclic phase shift $2\pi/N$. We denote these possible states as $\mathop{\left\|\beta^{(l)}\right\rangle}\nolimits$. On her side, Eve has to distinguish between the states $\hat{\rho}_{E}^{(l)}=\sum\limits_{k}p_{k}(\beta^{(l)})\mathop{\left\|\epsilon_{k}\right\rangle}\nolimits\mathop{\left\langle\epsilon_{k}\right\|}\nolimits$. After averaging, Eve’s state is $\frac{1}{N}\sum\limits_{l}\hat{\rho}_{E}^{(l)}=\hat{\rho}_{E}$, so the left entropy term in (6) is the same as we calculated before for the case of direct reconciliation. Due to the $2\pi/N$ phase–shift symmetry of the alphabet, the averaging in the right entropy term in (6) is just equal to the entropy of any of the states $\hat{\rho}_{E}^{(l)}$, let it be the first one $\hat{\rho}_{E}^{(1)}$. Again, we can rewrite Eve’s states $\mathop{\left\|\epsilon_{k}\right\rangle}\nolimits$ in the orthogonal basis (7). Unfortunately, the state $\hat{\rho}_{E}^{(1)}$ in this basis takes a non–diagonal form $\hat{\rho}_{E}^{(1)}=\sum\limits_{k,m,n}p_{k}(\beta^{(1)})c_{m}c^{}_{n}e^{i\frac{2\pi}{N}k(m-n)}\mathop{\left\|m\right\rangle}\nolimits\mathop{\left\langle n\right\|}\nolimits$. We analytically calculate eigenvalues of this state for a given $N$, but the result is too large to be presented here. Finally, Eve’s information is $I_{BE}(\beta)=S[\hat{\rho}_{E}]-S[\hat{\rho}_{E}^{(1)}]$. ### 3.3 Postselection The key rate $G$, i.e. the amount of secret information per transmission (1), is equal to the difference between Bob’s and Eve’s informations [Devetak:05, Renner:07]: $G=\int p(\beta)G(\beta)d\beta,\quad G(\beta)=I_{AB}-I_{AE,BE}.$ (11) where $I_{AE}$ and $I_{BE}$ refer to direct and reverse reconciliation respectively. Figure 1: Reconciliation and postselection areas for 5 letter protocol. Different shades of grey correspond to the regions in the phase space where measurement results $\mathop{\left\|\beta\right\rangle}\nolimits$ are associated with a given letter. Letters are shown as black circles. Dashed lines show the borders of the postselection areas for a case when amplitude is $a=1.4$, and transmittance varies from $0.95$ (the smallest area) to $0.4$ (the biggest area) with a step $0.05$. In the case of direct reconciliation, Eve has to guess what was sent by Alice. If the channel transmittance is lower than $50\%$, Eve can potentially have a better signal than Bob, thus Eve’s information can be higher than Bob’s information and no secure communication is possible. To overcome this “3 dB limit” we use the postselection idea [Silb:02], so that Bob has an information advantage over Eve ($I_{AB}>I_{AE}$), i.e. we select only that part of transmissions which give us positive terms $G(\beta)$. The postselection procedure in the direct reconciliation scenario can be qualitatively described as follows: Eve’s information $I_{AE}$ does not depend on Bob’s measured state $\beta$, so the key rate can be increased if Bob accepts only those transmissions where $\beta$ is such that he has higher information than Eve ($I_{AB}(\beta)>I_{AE}$). Instead of integration over the whole phase space (11) we have integration over the postselected area (PSA): $G_{PS}=\int\limits_{PSA}p(\beta)\left(I_{AB}(\beta)-I_{AE}\right)d\beta.$ (12) To find this PSA we numerically solve an equation $I_{AB}(\beta)>I_{AE}$. As an example in Fig. 1 we show the borders of the PSA for 5 letter protocol as the dashed lines. Different dashed lines correspond to different values of transmittance $\eta=0.4,0.45,0.5,\ldots,0.95$, and the amplitude is fixed $a=1.4$. The PSA is the phase space except the central region bounded by a dashed line. If the measured state $\mathop{\left\|\beta\right\rangle}\nolimits$ lies inside the region the transmission should be omitted, otherwise it is accepted. The higher the transmittance, the smaller the omitted region, the bigger the PSA, and the higher the key rate (12). To find the key rate $G_{PS}$ as a function of transmittance $\eta$ we optimize the amplitude $a$ such as to maximize the key rate $G_{PS}(\eta)=\max\limits_{a}G_{PS}(\eta,a)$. In the case of reverse reconciliation, Eve has to guess Bob’s measurement result, thus her information cannot be higher than Bob’s information: $I_{AB}\geq I_{BE}$. Therefore, $G(\beta)$ is always nonnegative, and the postselection procedure does not have to be applied. ## 4 Results The calculated secret key rate $G_{PS}(\eta)$ and the optimal amplitude $a_{0}(\eta)$ for several alphabets are shown in Fig. 2111Discontinuity of the curve $a_{0}(\eta)$ for the 5–letter alphabet is not a mistake. Due to the fact that the function $G(a)$ at a fixed value $\eta$ can have two slightly different global maxima, the exact optimization for variable $\eta$ may cause a “jump” from one maximum to another.. We can see, that for the channel transmittance $\eta<0.9$ all the curves of the key rate are almost the same when the number of letters is more than 4. Figure 2: The secret key rate $G$ in logarithmic scale (upper plot) and optimal signal amplitude (bottom plot). Solid and dashed lines correspond to direct and reverse reconciliation, respectively. In the case of reverse reconciliation we have an interesting result: the higher the number of letters, the higher the key rate. We don’t have an analytical expression for the $\infty$–letter alphabet, but its approximation by a high number of letter confirms that this is the best choice for all values of $\eta$. The key rate is higher than for the 2–letter alphabet of almost an order of magnitude. In the case of direct reconciliation, we can see that lines $G(\eta)$ for various numbers of letters are intersecting in different points, which are presented in Table 1. This means that for different values of transmittance $\eta$ there are different optimal numbers of letters. The higher the transmittance, the higher the optimal number of states. Table 1: Values of transmittance $\eta$, where a curve $G_{PS}(N,\eta)$ intersects with a curve $G_{PS}(N+1,\eta)$. N \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 ---\|---\|---\|---\|---\|---\|---\|--- $\eta$ \| 0.493 \| 0.705 \| 0.797 \| 0.797 \| 0.753 \| 0.725 \| 0.696 Again, we don’t have have an analytical expression for the curve $G(N=\infty,\eta)$, but we found that left side of the curves quickly saturates (there is no essential difference between $G(N=5,\eta)$ and $G(N=64,\eta)$ for $\eta<0.9$). So we can conjecture, that the optimal alphabets can consist of 2, 3, 4, or $\infty$ letters. A curve $G(N=\infty,\eta)$ intersects with $G(N=4,\eta)$ at the value $\eta\simeq 0.795$, so the most significant advantage of the multi letter protocol over the two letter protocol one can get in the case of high transmittance. The intuitive explanation is as follows: In the case of high losses, Eve has a stronger signal than Bob. Thus the amplitude of the signal $a_{0}$ must be small, and Bob relies basically on the postselection. In postselection it is harder for Bob to distinguish between many letters than between two. Thus with an increasing number of letters his information essentially decreases. In this case the alphabet with two letters outperforms the multi letter alphabet. In the opposite case of low losses Eve has a weaker signal, and Alice can increase the amplitude of the signal and number of letters. With higher amplitude of the signal Bob can better distinguish between many letters and increase his information. In the limit $\eta\rightarrow 1$ Alice can use signals with high amplitude and Bob can get almost $\log_{2}N$ bit per transmission. Therefore, the more letters in the alphabet are, the higher Bob’s information is. In principle, one can use an arbitrary high number of letters, and in the limit $N\rightarrow\infty$ (continuous phase modulation) Bob’s information seems to be infinite. However, there are limiting factors from both experimental and theoretical viewpoints. First, as one can see in Fig. 2 the curves $G(\eta)$ and $a_{0}(\eta)$ start to essentially increase from the values $\eta>0.99$. In any real experimental setup there are imperfections (inaccuracy, losses, etc.), so the case $\eta>0.99$ can hardly be achieved. Second, any real signal has certain energy limit, which sets maximum amplitude. Also taking into account excess noise in the channel might somewhat change the situation. ## 5 Conclusions To summarize, we have presented a new CV QKD protocol with coherent states. The protocol employs multi letter phase–shift–keying and heterodyne measurement. Security analysis of the proposed protocol is performed for the case of lossy but noiseless quantum channels. We have shown that for each given channel transmittance one can find a certain optimal number of letters (2, 3, 4, or $\infty$), optimal amplitude of the signal (typically, 1 to 4 photons per pulse), and optimal postselection threshold, which increase the secret key rate about one order of magnitude comparing to the protocol with binary modulation. ## Acknowledgments The authors thank Norbert Lütkenhaus for helpful discussions, Dominique Elser and Christoffer Wittmann for valuable comments on the manuscript. D.S. acknowledges the Alexander von Humboldt Foundation for a fellowship. ## Bibliography ## References [1] Bechmann-Pasquinucci Gisin1999Bech:99 Bechmann-Pasquinucci H Gisin N 1999 Phys. Rev. A 59, 4238 – 4248. * [2] Bennett1992Bennett:92 Bennett C H 1992 Phys. Rev. Lett. 68, 3121 – 3124. * [3] Bennett Brassard1984Bennett:84 Bennett C H Brassard G 1984 in ‘Proceedings IEEE Int. Conf. on Computers, Systems ad Signal Processing’ IEEE New York p. 175. * [4] Bruß1998Brus:98 Bruß D 1998 Phys. Rev. Lett. 81, 3018 – 3021. * [5] Cerf et al.2007Cerf:07book Cerf N J, Leuchs G Polzik E 2007 Quantum Information with Continuous Variables of Atoms and Light Imperial College Press. * [6] Chefles Barnett1998Chefles:98 Chefles A Barnett S M 1998 Phys. Lett. A 250, 223–229. * [7] Devetak Winter2005Devetak:05 Devetak I Winter A 2005 Proc. R. Soc. London Ser. A 461, 207. * [8] Dieks1982Dieks:82 Dieks D 1982 Phys. Lett. A 92(6), 271. * [9] Dušek et al.2006Dusek:06 Dušek M, Lütkenhaus N Hendrych M 2006 Progress in Optics 49, 381–454. * [10] Ekert1991Ekert:91 Ekert A K 1991 Phys. Rev. Lett. 67, 661 – 663. * [11] Elser et al.2008Elser:08 Elser D, Bartley T, Heim B, Wittmann C, Sych D Leuchs G 2008 arXiv:0811:4756 . * [12] Gisin et al.2002Gisin:02RMP Gisin N, Ribordy G, Tittel W Zbinden H 2002 Rev. Mod. Phys. 74(1), 145. * [13] Grosshans et al.2003Gross:03 Grosshans F, Assche G V, Wenger J, Brouri R, Cerf N J Grangier P 2003 Nature 421, 238. * [14] Grosshans Grangier2002Gross:02 Grosshans F Grangier P 2002 Phys. Rev. Lett. 88, 057902. * [15] Heid Lütkenhaus2006Heid:06 Heid M Lütkenhaus N 2006 Phys. Rev. A 73, 052316. * [16] Hillery2000Hillery:00 Hillery M 2000 Phys. Rev. A 61, 022309. * [17] Holevo1973Holevo:73 Holevo A 1973 Probl. Inf. Trans. 9, 177. * [18] Lorenz et al.2004Lorenz:04 Lorenz S, Korolkova N Leuchs G 2004 Appl. Phys. B: Lasers Opt. 79, 273. * [19] Lorenz et al.2006Lorenz:06 Lorenz S, Rigas J, Heid M, Andersen U L, Lütkenhaus N Leuchs G 2006 Phys. Rev. A 74, 042326. * [20] Namiki Hirano2003Namiki:03 Namiki R Hirano T 2003 Phys. Rev. A 67, 022308. * [21] Namiki Hirano2006Namiki:06 Namiki R Hirano T 2006 Phys. Rev. A 74, 032302. * [22] Ralph1999Ralph:99 Ralph T C 1999 Phys. Rev. A 61, 010303. * [23] Renner2007Renner:07 Renner R 2007 Nature Physics 3, 645 – 649. * [24] Scarani et al.2009Scarani:09 Scarani V, Bechmann-Pasquinucci H, Cerf N J, Dušek M, Lütkenhaus N Peev M 2009 Rev. Mod. Phys. 81(4), 1301. * [25] Shannon1948Shan:48 Shannon C 1948 Bell Syst. Tech. J. 27, 379. * [26] Silberhorn et al.2002Silb:02 Silberhorn C, Ralph T C, Lütkenhaus N Leuchs G 2002 Phys. Rev. Lett. 89(167901). * [27] Sych et al.2004Sych:04 Sych D V, Grishanin B A Zadkov V N 2004 Phys. Rev. A 70, 052331. * [28] Sych et al.2005Sych:05 Sych D V, Grishanin B A Zadkov V N 2005 Quant. Electron. 35, 80. * [29] Weedbrook et al.2004Weed:04 Weedbrook C, Lance A M, Bowen W P, Symul T, Ralph T C Lam P K 2004 Phys. Rev. Lett. 93, 170504. * [30] Wooters Zurek1982Wooters:82 Wooters W K Zurek W H 1982 Nature 299, 802. * [31]	arxiv-papers	2009-02-11T14:26:30	2024-09-04T02:49:00.527239	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Denis Sych and Gerd Leuchs", "submitter": "Denis Sych", "url": "https://arxiv.org/abs/0902.1895" }
0902.1989	# O VI Absorption in the Milky Way Disk, and Future Prospects for Studying Absorption at the Galaxy-IGM Interface D. V. Bowen E. B. Jenkins T. M. Tripp D. G. York ###### Abstract We present a brief summary of results from our FUSE program designed to study O VI absorption in the disk of the Milky Way. As a full analysis of our data has now been published, we focus on the improvements that FUSE afforded us compared to Copernicus data published thirty years ago. We discuss FUSE’s limitations in studying O VI absorption from nearby galaxies using background QSOs, but present FUSE spectra of two probes which indicate the absence of O VI (but the presence of Ly$\beta$) absorption 8 and 63 kpc from a foreground galaxy. Finally, we discuss the need for a more sensitive UV spectrograph to map out the physical conditions of baryons around galaxies. ###### Keywords: Galaxy:disk — ultraviolet:ISM — quasars: absorption lines ###### : 95.85.Mt, 98.35.Hj, 98.38.Kx, 98.58.-w, 98.62.Ra ## 1 The FUSE Survey of O VI Absorption in the Milky Way Disk The FUSE survey of O VI absorption lines in the disk of the Milky Way was a program that used spectra of 153 early-type stars at latitudes $<10^{\rm{o}}$ and distances of more than 1 kpc to characterize O VI absorption in the plane of the Galaxy. The results from the survey have now been published in full (Bowen et al., 2008), so in this contribution, we highlight just a few of the results from that paper. We begin, however, in celebrating the accomplishments of FUSE, by comparing some of the data obtained with the satellite to the data available prior to its launch. ### 1.1 A Copernicus/FUSE comparison The results of our FUSE survey were published thirty years and three months after the seminal survey of Jenkins (1978) (which built upon the initial work of Jenkins and Meloy (1974) and York (1974)) using the Copernicus satellite. Comparing data from the two telescopes might simply be considered amusing if the goal was to merely demonstrate the obvious superiority of current instrumentation and detectors over those available three decades ago. A more serious intent, however, for such a comparison is to verify the integrity of the older data. So, for example, the physical parameters of O VI absorbing clouds along any particular sightline should be the same whether measured with Copernicus or with FUSE. Unfortunately, few lines of sight were actually observed by both satellites; Copernicus regularly recorded spectra of stars brighter than $\sim 7$ mag, but these objects were too bright to be observed with FUSE (Sahnow et al., 2000). Figure 1: Comparison of Copernicus and FUSE spectra of two stars, HD 186994 (top panels) and HD 41161 (bottom panels). The left-hand panels show the coadded Copernicus scans, spanning only a small wavelength range; the right hand panels show the FUSE spectra, with and without the removal of the HD 6$-$0 R(0) line from the O VI $\lambda 1032$ absorption, as well as the adopted fit to the continuum and, for HD 186994, the theoretical Voigt profile fits (red line) to the data. The O VI column densities derived by Jenkins (1978) for Copernicus, and Bowen et al. (2008) for FUSE, are shown bottom right of each panel. For HD 186994, errors in $N$(OVI) are given first from continuum fitting, then from counting statistics (see Bowen et al. (2008) for more details). The flux scales are arbitrary, and are scaled to allow comparison of the spectra. The strong absorption lines flanking the O VI line are the H2 6$-$0 P(3) and 6$-$0 R(4) transitions at 1031.2 and 1032.4 Å. Nevertheless, in our survey, two stars — HD 186994 and HD 41161 — were observed by both Copernicus and FUSE, and the data from each satellite are shown in Fig. 1. This comparison is somewhat cruel, since with magnitudes of $B=7.4$ and 6.7 for the two stars, respectively, these were some of the faintest objects Copernicus could observe. In this respect, the apparently poor signal-to-noise (S/N) of the two spectra are unrepresentative of the data used by Jenkins (1978). Still, one clear difference between the two is the much smaller wavelength coverage of the Copernicus spectra; the satellite used a scanning spectrophotometer to record stellar spectra while FUSE was fitted with the multi-spectral-element detector arrays that observers now take for granted. The coadded Copernicus scans of HD 186994 (taken here from the Multimission Archive at STScI) represent an exposure time of 106 min; the 8 min FUSE data taken with the LWRS aperture provide a larger wavelength range, and so superior a S/N, than the Copernicus spectrum, that fitting the stellar continuum is much more straightforward. The resulting measurement of the physical parameters of the O VI absorption [column density $N$(O VI), Doppler parameter and absorption velocity] are more precise than those that were derived with Copernicus, and with the FUSE spectrum we could estimate the errors in the physical parameters arising from both counting statistics and errors in the continuum fit. For HD 186994, $\log N$(O VI) derived from the Copernicus data was 14.0 or $<13.85$, as measured from O VI $\lambda 1032$ line or from the O VI $\lambda 1037$ line, respectively. These numbers, taken together, are broadly consistent with the value derived from FUSE, $\log N$(O VI)$=13.88$, which is reassuring given the low quality of this particular Copernicus spectrum. The FUSE data also allow an accurate subtraction of the HD 6$-$0 R(0) line which contaminates the O VI profile. This contamination was well understood by Jenkins (1978), and accounted for in the Copernicus spectrum of HD 41161, where some residual O VI absorption was detected after subtraction of the HD line. Interestingly, in our FUSE spectrum (taken, unusually, using the MDRS aperture) we determined that all of the observed absorption was due to the HD line. Nevertheless, our upper limit to $N$(O VI) is still consistent with the value given by Jenkins (1978). Verifying the integrity of the earlier Copernicus data is far from academic: since early-type stars at distances of $d<1$ kpc were too bright to be observed with FUSE, we needed the Copernicus data to probe these distances when examining how O VI absorption varies with path length in the Galactic disk. ### 1.2 Summary of Program Results To explore the characteristics of O VI absorption in the Galactic disk, we combined our FUSE observations of stars at $d\sim 1-4$ kpc with several other datasets. We included stars at $d<1$ kpc observed by Copernicus, as well as halo stars Zsargó et al. (2003), extragalactic sightlines (Wakker et al., 2003; Savage et al., 2003), and nearby white dwarfs Savage and Lehner (2006), all targeted by FUSE. As noted above, these results are now published (Bowen et al., 2008), and our measurements are available at http://www.astro.princeton.edu/~dvb/o6home.html. Our data confirmed that O VI absorbing clouds are ubiquitous throughout the Alpha and Beta quadrants of the Galaxy. The O VI volume density $n$ falls off exponentially with height above the Galactic plane, as had been shown from previous studies Widmann et al. (1998); Savage et al. (2003). With the FUSE data, however, we were able to measure the mid-plane density to be precisely $1.3\times 10^{-8}$ cm-3, with scale heights of 4.6 and 3.2 kpc for sightlines in the southern and northern Galactic hemispheres, respectively. However, even though the O VI density falls off with height above the plane, the O VI absorbing material is not smooth, but clumpy, with a range of cloud sizes. We were also able to settle a long standing question as to how much O VI absorption towards a target star actually comes from hot circumstellar material around the star itself — only a small amount of $N$(O VI) arises in such regions. We found that $N$(O VI) correlates with $d$, demonstrating that O VI absorbing clouds are truly interstellar, and composed of many individual, overlapping, components. The dispersion of $N$(O VI) with $d$ is large though, and very different from what would be expected from absorption by an ensemble of identical clouds. The velocity extent of O VI lines follow those of lower ionization lines observed along the same sightlines, showing that hot and cold gas are coupled. There are different ways to interpret our results, and in the future, our data should provide the observations necessary to test theoretical predictions of how hot gas is produced in the Galaxy. We note that concurrent with our investigations, detailed hydrodynamical simulations of hot gas in the local Galactic disk were being engineered by de Avillez and Breitschwerdt (2005). In these models, the ISM contains a hot, turbulent multi-phase medium churned by shock heated gas from multiple supernovae (SNae) explosions. Hot gas arises in bubbles around SNae, which is then sheered through turbulent diffusion, destroying the bubbles and stretching the hot absorbing gas into filaments that dissipate with time. Although these simulations are unlikely to be the last word in modelling the hot Milky Way ISM, they do provide a contemporary context in which to interpret our data. For example, they successfully predict the mid-plane O VI density that we measure in our survey. ## 2 O VI Absorption in other galaxies Outside of the Galactic disk, FUSE demonstrated the existence of copious amounts of O VI absorption in the Milky Way halo (Wakker et al., 2003; Savage et al., 2003) and in the star-forming regions of the Magellanic Clouds (Howk et al., 2002; Hoopes et al., 2002). O VI was also detected in Galactic High Velocity Clouds (HVCs) (Sembach et al., 2003), which posed the interesting question: how far out from a galaxy can O VI be detected? The distances to the HVCs are not well constrained, and O VI absorbing HVCs may arise in material infalling into or outflowing from the Galactic disk (from areas of active star formation, for example), or further away, from accretion of gas from the intergalactic medium (IGM) into the extended Milky Way halo, or even the Local Group. In addition, the relationship between all these local O VI absorbers, and the population of weak O VI absorption systems detected towards QSOs at redshifts of a few tenths (Tripp et al., 2000; Danforth and Shull, 2005; Tripp et al., 2008) is far from clear. The latter systems may contain a large fraction of baryons, as much as that currently found in stars, cool gas in galaxies, and X-ray emitting gas in galaxy clusters. Where these O VI lines actually come from, however, is unclear. Although individual galaxies have been detected at similar redshifts to O VI absorption systems (within impact parameters of $\sim 0.2-1.5$ Mpc (Tripp and Savage, 2000; Savage et al., 2002; Sembach et al., 2004; Tripp et al., 2006; Lehner et al., 2008)), redshift information for objects in these fields is incomplete, and the environment of the absorbing gas is hard to establish at redshifts of $z>0.2$. Figure 2: Spectra of two QSOs that lie close to nearby galaxies, taken as part of GI program G020: the top panel shows a 20.0 ksec exposure of ESO 185$-$G013 (including data from supplementary program Z909), whose sightline passes 63 kpc from IC 4889; the bottom panel shows a 99.2 ksec exposure of PG 0838+770, whose line of sight passes 8 kpc from a low luminosity Im galaxy UGC 4527. Both spectra are from the LiF1A channel, taken using the LWRS aperture, and have been reduced to the rest-frame wavelength of the foreground galaxies. The positions of Ly$\beta$ and the O VI doublet are marked, as are the wavelengths of strong airglow lines (by $\oplus$ symbols). For ESO 185$-$G013, we have drawn a representative theoretical Voigt profile for the Ly$\beta$ absorption, assuming a Doppler parameter of $b=20$ km s-1, and $\log N$(H I)$=19.6$. $N$(HI) could be nearly a dex lower than this for higher values of $b$ (see text). The depression at 1035 Å is likely to be broad Ly$\alpha$ absorption at the emission redshift of ESO 185$-$G013. One way to address these questions is to search for O VI absorption in the disks and halos of low-$z$ galaxies. Working at low redshifts has several advantages: there is no ambiguity in the origin of any detected lines, the properties of the galaxies can be more readily quantified than at high-$z$, and the physical conditions of the absorbing gas can be directly linked to those observable properties. Moreover, the environment of nearby galaxies – whether they are isolated, are interacting with companions, reside in loose groups, or in clusters — can be more easily determined than at higher-$z$. The problem in performing these types of experiments has always been the difficulty in finding QSOs which are close to galaxies in projection, and that are bright enough to be observed in the UV with the available instrumentation. For FUSE, the challenge was almost insurmountable. With generous allocations of observing time, of order $100$ ksec, FUSE could obtain “adequate” S/N (at least in the LiF1a channel) of QSOs with fluxes of $\sim 0.5\times 10^{-14}$ ergs cm-2 s-1 Å-1. Most of the interesting QSOs close to low-$z$ foreground galaxies have fluxes less than this, and could not be targeted by FUSE. Fortunately, there were a few exceptions. Fig. 2 shows the results from a program (G020) we designed to search for Ly$\beta$ and O VI absorption from the outer regions of two low-$z$ galaxies. Again, these pairs were selected in part because the impact parameters between QSO and galaxy were small, but largely because the QSOs were predicted to have high UV fluxes. That is, we could not select pairs based on particular galaxy properties that we might be interested in. ESO 185$-$013 is an AGN at $z_{\rm{em}}=0.019$ which lies behind the bright E5 galaxy IC 4889. The galaxy has a redshift of 2570 km s-1, and the sightline to the AGN passes 63 kpc from its center. Strong Ly$\beta$ absorption is detected; unfortunately, the O VI $\lambda 1031$ line falls at the position of an O I∗ airglow feature, making it hard to determine whether the line is present. Nevertheless, O VI $\lambda 1037$ is not detected to a limit of $\approx 0.15$ Å. The H I column density is difficult to constrain since the Ly$\beta$ line is strongly saturated, and the S/N of the data are not sufficient to show the onset of damping wings in the line profile. For Doppler parameters of $b\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$<$}}}20$ km s-1, $\log N$(H I)$\simeq 19.8$, but if $b$ is large, e.g., $\sim 30$ kms, $N$(H I) could be one dex less. However, the H I absorption extends in size the structure seen in 21 cm emission around IC 4889 (Oosterloo et al., 2007) by a factor of two. The radio data measures $\log N$(H I) to a limit of $\sim 19$; our data suggests that $N$(H I) remains relatively high at a radius twice that seen at 21 cm. The second QSO-galaxy pair studied in our program was PG 0838+770/UGC 4527. The QSO sightline passes only 8 kpc from the UGC 4527, which is a low surface brightness Im galaxy at a redshift of 720 km s-1. We again detect Ly$\beta$ at the redshift of the galaxy, although it is likely that the line profile is contaminated with O I∗∗ airglow. However, the non-detection of O VI is clear, to a limit of $\approx 0.1$ Å. Little is known about UGC 4527; deciding whether the lack of O VI so close to an irregular dwarf galaxy is surprising will depend on a better understanding of the galaxy itself, and ultimately, obtaining data of better quality. ## 3 Future Studies of Absorption in galactic disks and halos Of course, the study of O VI provides insights into only one phase of the gas in and around galaxies. To fully characterize the physical conditions of gas in galaxy disks and halos, absorption lines from many different species (each probing gas at different temperatures, densities, etc.) must be observed. The need for a sophisticated analysis of what is likely to be a multiphase medium at the boundary between a galaxy and the IGM is now more compelling than ever, because our view of galaxies and their relationship to the IGM has changed dramatically over the last decade. The exponential growth in available computing power has allowed detailed modelling of the large scale structure (LSS) of the Dark Matter (DM) in the universe, along with its evolution over a significant period of time. More importantly, these models incorporate the gas hydrodynamics required to predict the formation and development of galaxies, and incorporate the likely symbiosis between the physical process at work in the evolution of a galaxy, and the IGM itself. So, for example, galaxies must interact and enrich the IGM at all epochs via various feedback mechanisms: gas may be expelled either from intense bursts of star formation via strong Galactic winds, or from outflows from a central AGN. Conversely, the IGM must influence galaxy evolution by the action of channeling baryons along DM filaments into galaxy groups. These infusions of gas will most likely change the metallicity of a galaxy. Arguably, our ability to test the simulations with observations lags behind the development of these models. The use of QSO absorption lines enables us to directly probe the galaxy-IGM interface, but the data are sparse. There are two obvious goals for future observations. First, we need to study gas on galaxy scale-lengths around a large number of galaxies. For example, we still have little (unambiguous) information on how the density of gas and its ionization state declines with radius from a galaxy — a seemingly fundamental piece of information for models of galaxy evolution. Second, we need to probe individual galaxies along multiple sightlines, to examine how the properties we measure for an ensemble of galaxies might actually vary in a single system. Indeed, mapping the gaseous structures around single galaxies with multiple lines of sight may be the best way to determine how gas accretes onto a galaxy and/or how it escapes. Studying galaxies with multiple probes can only be accomplished at low-$z$, where the angular extent of a galaxy is large, and the background surface density of QSOs is high. On the other hand, probing the inner regions of nearby galaxies is more difficult, because QSOs which shine through the hearts of bright low$-z$ galaxies are not readily detected. Instead, a different approach to exploring the inner regions of galaxies is to work at a somewhat higher redshift. Over the last few years we have been identifying galaxies cataloged by the Sloan Digital Sky Survey (SDSS) that lie close to QSOs on the sky, with a emphasis on finding QSO-galaxy pairs with very small separations. One technique has been to identify multiple emission lines ([O II], [O III], H$\alpha$, etc.) from low-$z$ galaxies in the spectra of background QSOs. The fibers used by SDSS to obtain spectra of selected objects are 3′′ in diameter, and can collect light from both a QSO and any galaxy along the line of sight. This “spectroscopic” technique has enabled us to find galaxies at $z\sim 0.2$ probed only a few kpc from their centers by a QSO (York et al. in prep). One of these Galaxies on top of QSOs (GOTOQs) is shown in Fig. 3. Figure 3: Identification of a galaxy in front of a background QSO using the QSO spectrum. Right: The flux from the $z=2.67$ QSO J104257.58+074850.5 dominates the spectrum taken with the SDSS spectrograph — broad emission lines of Ly$\alpha$ C IV and CIII] are clearly visible. Superimposed on the spectrum, however, are narrow emission lines from a galaxy at $z=0.032$. Left: In this case, the intervening galaxy can be seen in SDSS imaging data. Galaxies discovered using this technique are inevitably probed at very small impact parameters, and are thus candidates for future studies of the inner disks and halos of a wide variety of galaxies. Studying galaxies close to QSOs at $z\sim 0.2$ is not so easy, compared to studying QSOs behind $z\sim 0$ galaxies. Nevertheless, the GOTOQs offer a special opportunity to probe gas in the inner regions of galaxies, which can complement the studies on larger scales discussed above. The problem in achieving these goals is the same one mentioned in the previous section — finding QSOs that are bright enough to be observed with available satellites. The Cosmic Origins Spectrograph (COS) which will be installed in HST in 2009 will certainly make significant advances in probing the galaxy-IGM interface, but difficulties remain. For example, if the goal is to map low-$z$ galaxies with multiple QSO sightlines, it is quite possible to find a sufficient number of QSOs beyond several hundred kpc, but at smaller distances (where much of the galaxy-IGM interaction is probably taking place) too few QSOs are bright enough for observation with COS. Further, for studying the inner regions of galaxies, whether we select nearby galaxies or those at redshifts of a few tenths, the number of suitable pairs will still be relatively small. Yet in order to characterize the gas around galaxies selected by their properties — their luminosity, morphology, star-formation rates, environment, etc. — we will, in the end, need to probe several hundred systems to fully characterize the galaxy-IGM interface. Achieving these goals will require a new facility. The requirements for the ideal UV spectrograph are obvious, and have been stated by many previous authors: it would be able to reach sensitivities of a few $\mu$Jy, about a factor of ten times more sensitive than COS; it would have a resolution of less than 10 km s-1, to enable accurate measurements of column densities and Doppler parameters, and permit mapping of the velocity distribution of multicomponent complexes; and it would cover the entire UV wavelength range, from 912 Å through to the atmospheric limit of $\sim 3200$ Å. With such an instrument, we could map out the physical conditions of gas on scales ranging from galactic disks to IGM large scale structures. Eventually, we would chart the variations in these conditions over a significant fraction of galactic history, as we extended our techniques to higher redshifts. Comparing our results to simulations which will continue to grow ever more sophisticated would enable us to understand comprehensively the life-cycle of baryons in the universe. The work described in this contribution was funded by subcontract 2440$-$60014 from the Johns Hopkins University under NASA prime subcontract NAS5$-$32985, and by LTSA NASA grant NNG05GE26G. ## References * Bowen et al. (2008) D. V. Bowen, et al, _ApJS_ 176, 59 (2008). * Jenkins (1978) E. B. Jenkins, _ApJ_ 219, 845 (1978). * Jenkins and Meloy (1974) E. B. Jenkins, and D. A. Meloy, _ApJ_ 193, L121 (1974). * York (1974) D. G. York, _ApJ_ 193, L127 (1974). * Sahnow et al. (2000) D. J. Sahnow, et al, _ApJ_ 538, L7 (2000). * Zsargó et al. (2003) J. Zsargó, K. R. Sembach, J. C. Howk, and B. D. Savage, _ApJ_ 586, 1019 (2003). * Wakker et al. (2003) B. P. Wakker, et al, _ApJS_ 146, 1 (2003). * Savage et al. (2003) B. D. Savage, et al, _ApJS_ 146, 125 (2003). * Savage and Lehner (2006) B. D. Savage, and N. Lehner, _ApJ_ 162, 134 (2006). * Widmann et al. (1998) H. Widmann, et al, _A &A_ 338, L1 (1998). * de Avillez and Breitschwerdt (2005) M. A. de Avillez, and D. Breitschwerdt, _ApJ_ 634, L65 (2005). * Howk et al. (2002) J. C. Howk, K. R. Sembach, B. D. Savage, D. Massa, S. D. Friedman, and A. W. Fullerton, _ApJ_ 569, 214 (2002). * Hoopes et al. (2002) C. G. Hoopes, K. R. Sembach, J. C. Howk, B. D. Savage, and A. W. Fullerton, _ApJ_ 569, 233 (2002). * Sembach et al. (2003) K. R. Sembach, et al, _ApJS_ 146, 165 (2003). * Tripp et al. (2000) T. M. Tripp, B. D. Savage, and E. B. Jenkins, _ApJ_ 534, L1 (2000). * Danforth and Shull (2005) C. W. Danforth, and J. M. Shull, _ApJ_ 624, 555 (2005). * Tripp et al. (2008) T. M. Tripp, K. R. Sembach, D. V. Bowen, B. D. Savage, E. B. Jenkins, N. Lehner, and P. Richter, _ApJS_ 177, 39 (2008). * Tripp and Savage (2000) T. M. Tripp, and B. D. Savage, _ApJ_ 542, 42 (2000). * Savage et al. (2002) B. D. Savage, K. R. Sembach, T. M. Tripp, and P. Richter, _ApJ_ 564, 631 (2002). * Sembach et al. (2004) K. R. Sembach, T. M. Tripp, B. D. Savage, and P. Richter, _ApJS_ 155, 351 (2004). * Tripp et al. (2006) T. M. Tripp, B. Aracil, D. V. Bowen, and E. B. Jenkins, _ApJ_ 643, L77 (2006). * Lehner et al. (2008) N. Lehner, J. X. Prochaska, H. A. Kobulnicky, K. L. Cooksey, J. C. Howk, G. M. Williger, and S. L. Cales, _ArXiv:_ 0821.4231 (2008). * Oosterloo et al. (2007) T. A. Oosterloo, R. Morganti, E. M. Sadler, T. van der Hulst, and P. Serra, _A &A_ 465, 787 (2007).	arxiv-papers	2009-02-11T21:03:16	2024-09-04T02:49:00.533310	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "D.V. Bowen, E.B. Jenkins, T.M. Tripp and D.G. York", "submitter": "David V. Bowen", "url": "https://arxiv.org/abs/0902.1989" }
0902.2307	# Quantum Transport in Bridge Systems Santanu K. Maiti E-mail: santanu.maiti@saha.ac.in 1Theoretical Condensed Matter Physics Division Saha Institute of Nuclear Physics 1/AF, Bidhannagar, Kolkata-700 064, India 2Department of Physics Narasinha Dutt College 129, Belilious Road, Howrah-711 101, India ###### Contents 1. 1 Introduction 2. 2 Theoretical Description 3. 3 Quantum Transport in Molecular Wires 1. 3.1 Model 2. 3.2 Results and Discussion 4. 4 Quantum Transport in a Thin Film 1. 4.1 Model 2. 4.2 Results and Discussion 5. 5 Concluding Remarks Abstract We study electron transport properties of some molecular wires and a unconventional disordered thin film within the tight-binding framework using Green’s function technique. We show that electron transport is significantly affected by quantum interference of electronic wave functions, molecule-to- electrode coupling strengths, length of the molecular wire and disorder strength. Our model calculations provide a physical insight to the behavior of electron conduction across a bridge system. Keywords: Molecular wires; Thin film; Conductance and $I$-$V$ characteristic. ## 1 Introduction Recent advances in nanoscience and technology have made feasible to growth nanometer sized systems like, quantum wires [1, 2, 3], quantum dots [4, 5, 6, 7], molecular wires [8], etc. Quantum transport in such systems provides several novel features due to their reduced dimensionality and lateral quantum confinement. The geometrical sensitivity of low-dimensional systems makes them truly unique in offering the possibility of studying quantum transport in a very tunable environment. In the present age, designing of electronic circuits using a single molecule or a cluster of molecules becomes much more widespread since the molecules are the fundamental building blocks for future generation of electronic devices where electron transmits coherently [9, 10]. Based on the pioneering work of Aviram and Ratner [11] where an innovative idea of a molecular electronic device was predicted for the first time, the development of a theoretical description of molecular devices has been pursued. Later, many experiments [12, 13, 14, 15, 16] have been carried out in different molecular bridge systems to justify the basic mechanisms underlying such transport. Though there exists a vast literature of theoretical as well as experimental study on electron transport in bridge systems, but yet the complete knowledge of conduction mechanism in such systems is not very well established even today. Many significant factors are there which can control the electron transport across a bridge system, and all these effects have to be taken into account properly to characterize such transport. For our illustrative purposes, here we mention very briefly some of them as follows. (I) The molecular coupling with side attached electrodes and the electron- electron correlation [17] provide important signatures in the electron transport. The understanding of the molecular coupling to the electrodes under non-equilibrium condition is a major challenge in this particular study. (II) The molecular geometry itself has a typical role. To emphasize it, Ernzerhof et al. [18] have predicted several model calculations and provided some new interesting results. (III) The quantum interference effect [19, 20, 21, 22, 23, 24, 25, 26, 27] of electron waves passing through a bridge system probably the most important aspect for controlling the electron transport, and a clear idea about it is needed to reveal the transport mechanism. (IV) The dynamical fluctuation in the small-scale devices is another important factor which plays an active role and can be manifested through the measurement of shot noise, a direct consequence of the quantization of charge. It can be used to obtain information on a system which is not available directly through the conductance measurements, and is generally more sensitive to the effects of electron-electron correlations than the average conductance [28, 29]. Beside these, several other factors are there which may control the electron transport in a bridge system. There exist several ab initio methods for the calculation of conductance [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40] through a molecular bridge system. At the same time, tight-binding model has been extensively studied in the literature, and it has also been extended to DFT transport calculations [41, 42]. The study of static density functional theory (DFT) [43, 44] within the local-density approximation (LDA) to investigate the electron transport through nanoscale conductors, like atomic-scale point contacts, has met with great success. But when this similar theory applies to molecular junctions, theoretical conductances achieve much larger values compared to the experimental predictions, and these quantitative discrepancies need extensive and proper study in this particular field. In a recent work, Sai et al. [45] have predicted a correction to the conductance using the time-dependent current-density functional theory since the dynamical effects give significant contribution in the electron transport, and illustrated some important results with specific examples. Similar dynamical effects have also been reported in some other recent papers [46, 47], where authors have abandoned the infinite reservoirs, as originally introduced by Landauer, and considered two large but finite oppositely charged electrodes connected by a nanojunction. In this dissertation, we reproduce an analytic approach based on the tight-binding model to characterize the electron transport properties through some bridge systems, and utilize a simple parametric approach [48, 49, 50, 51, 52, 53, 54, 55] for these calculations. The model calculations are motivated by the fact that the ab initio theories are computationally much more expensive, while the model calculations by using the tight-binding formulation are computationally very cheap, and also provide a physical insight to the behavior of electron conduction through such bridge systems. This dissertation can be organized in this way. Following the introductory part (Section $1$), in Section $2$ we illustrate very briefly the methodology for the calculation of transmission probability, conductance and current through a finite size conductor attached to two metallic electrodes by using Green’s function formalism. Section $3$ describes electron transport in some molecular wires. In Section $4$, we focus our study on electron transport through a unconventional disordered thin film in which disorder strength varies smoothly from layer to layer with the distance from its surface. Finally, we conclude our results in Section $5$. ## 2 Theoretical Description This section follows the methodology for the calculation of the transmission probability ($T$), conductance ($g$) and current ($I$) through a finite size conductor attached to two one-dimensional semi-infinite metallic electrodes by using Green’s function technique. Let us refer to Fig. 1, where a finite size conductor is attached to two metallic electrodes, viz, source and drain through the lattice sites $S$ and $S$. At sufficient low temperature and bias voltage, we use the Landauer conductance formula [56, 57] to calculate the conductance $g$ of the conductor which can be expressed as, $g=\frac{2e^{2}}{h}T$ (1) where $T$ becomes the transmission probability of an electron through the conductor. It can be expressed Figure 1: Schematic view of a finite size conductor attached to two metallic electrodes, viz, source and drain through the lattice sites $S$ and $S$. in terms of the Green’s function of the conductor and its coupling to the two electrodes by the relation [56, 57], $T=Tr\left[\Gamma_{S}G_{c}^{r}\Gamma_{D}G_{c}^{a}\right]$ (2) where $G_{c}^{r}$ and $G_{c}^{a}$ are respectively the retarded and advanced Green’s functions of the conductor including the effects of the electrodes. The parameters $\Gamma_{S}$ and $\Gamma_{D}$ describe the coupling of the conductor to the source and drain respectively, and they can be defined in terms of their self-energies. For the complete system i.e., the conductor with the two electrodes the Green’s function is defined as, $G=\left(\epsilon-H\right)^{-1}$ (3) where $\epsilon=E+i\eta$. $E$ is the injecting energy of the source electron and $\eta$ gives an infinitesimal imaginary part to $\epsilon$. Evaluation of this Green’s function requires the inversion of an infinite matrix as the system consists of the finite conductor and the two semi-infinite electrodes. However, the entire system can be partitioned into sub-matrices corresponding to the individual sub-systems and the Green’s function for the conductor can be effectively written as, $G_{c}=\left(\epsilon-H_{c}-\Sigma_{S}-\Sigma_{D}\right)^{-1}$ (4) where $H_{c}$ is the Hamiltonian of the conductor which can be written in the tight-binding model within the non-interacting picture like, $H_{c}=\sum_{i}\epsilon_{i}c_{i}^{\dagger}c_{i}+\sum_{<ij>}t\left(c_{i}^{\dagger}c_{j}+c_{j}^{\dagger}c_{i}\right)$ (5) where $\epsilon_{i}$’s are the site energies and $t$ is the hopping strength between two nearest-neighbor atomic sites in the conductor. Similar kind of tight-binding Hamiltonian is also used to describe the two semi-infinite one- dimensional perfect electrodes where the Hamiltonian is parametrized by constant on-site potential $\epsilon_{0}$ and nearest neighbor hopping integral $t_{0}$. In Eq. (4), $\Sigma_{S}=h_{Sc}^{\dagger}g_{S}h_{Sc}$ and $\Sigma_{D}=h_{Dc}g_{D}h_{Dc}^{\dagger}$ are the self-energy operators due to the two electrodes, where $g_{S}$ and $g_{D}$ correspond to the Green’s functions of the source and drain respectively. $h_{SC}$ and $h_{DC}$ are the coupling matrices and they will be non-zero only for the adjacent points of the conductor, $S$ and $S$ as shown in Fig. 1, and the electrodes, respectively. The matrices $\Gamma_{S}$ and $\Gamma_{D}$ can be calculated through the expression, $\Gamma_{S(D)}=i\left[\Sigma_{S(D)}^{r}-\Sigma_{S(D)}^{a}\right]$ (6) where $\Sigma_{S(D)}^{r}$ and $\Sigma_{S(D)}^{a}$ are the retarded and advanced self-energies respectively, and they are conjugate with each other. Datta et. al. [58] have shown that the self-energies can be expressed like as, $\Sigma_{S(D)}^{r}=\Lambda_{S(D)}-i\Delta_{S(D)}$ (7) where $\Lambda_{S(D)}$ are the real parts of the self-energies which correspond to the shift of the energy eigenvalues of the conductor and the imaginary parts $\Delta_{S(D)}$ of the self-energies represent the broadening of these energy levels. This broadening is much larger than the thermal broadening and this is why we restrict our all calculations only at absolute zero temperature. All the informations about the conductor-to-electrode coupling are included into these two self-energies as stated above and are described through the use of Newns-Anderson chemisorption theory [48, 49]. The detailed description of this theory is available in these two references. By utilizing the Newns-Anderson type model, we can express the conductance in terms of the effective conductor properties multiplied by the effective state densities involving the coupling. This allows us to study directly the conductance as a function of the properties of the electronic structure of the conductor within the electrodes. The current passing across the conductor is depicted as a single-electron scattering process between the two reservoirs of charge carriers. The current $I$ can be computed as a function of the applied bias voltage $V$ through the relation [56], $I(V)=\frac{e}{\pi\hbar}\int_{E_{F}-eV/2}^{E_{F}+eV/2}T(E,V)dE$ (8) where $E_{F}$ is the equilibrium Fermi energy. For the sake of simplicity, we assume that the entire voltage is dropped across the conductor-electrode interfaces and this assumption doesn’t greatly affect the qualitative aspects of the $I$-$V$ characteristics. Such an assumption is based on the fact that, the electric field inside the conductor especially for short conductors seems to have a minimal effect on the conductance-voltage characteristics. On the other hand, for quite larger conductors and high bias voltages the electric field inside the conductor may play a more significant role depending on the internal structure and size of the conductor [58], yet the effect is quite small. ## 3 Quantum Transport in Molecular Wires In this section, we narrate electron transport properties of some molecular wires consisting with polycyclic hydrocarbon molecules. These molecules are named as benzene, napthalene, anthracene and tetracene respectively. The transport properties in the molecular wires are significantly affected by the (i) quantum interference effects, (ii) molecule-to-electrode coupling strength, and (iii) length of the molecular wire, and here we discuss our results in these aspects. ### 3.1 Model In Fig. 2, we show the model of the four different polycyclic hydrocarbon molecules. To reveal the quantum interference Figure 2: Molecular model for the four different polycyclic hydrocarbon molecules. The molecules are benzene (one ring), napthalene (two rings), anthracene (three rings) and tetracene (four rings) respectively. These molecules are attached to the electrodes, at the $\alpha$-$\alpha$ positions called the cis configuration, and at the $\beta$-$\beta$ positions called the trans configuration via thiol (SH) groups. effects, we consider two different arrangements of the molecular wires. In one case, the molecules are attached to the electrodes at the $\alpha$-$\alpha$ sites (see the first column of Fig. 2). This is so-called the cis configuration. In the other case, the electrodes are attached to these molecules at the $\beta$-$\beta$ sites, as presented in the second column of Fig. 2. This particular arrangement is so-called the trans configuration. In actual experimental set-up, the electrodes made from gold (Au) are used and the molecule coupled to the electrodes through thiol (SH) groups in the chemisorption technique where hydrogen (H) atoms remove and sulfur (S) atoms reside. To describe the polycyclic hydrocarbon molecules here we use the similar kind of non-interacting tight-binding Hamiltonian as illustrated in Eq. (5). ### 3.2 Results and Discussion Here we describe all the essential features of the electron transport for the two distinct regimes. One is so-called the weak coupling regime, defined by the condition $\tau_{\\{S,D\\}}<<t$. The other one is so-called the strong- coupling regime, denoted by the condition $\tau_{\\{S,D\\}}\sim t$, where $\tau_{S}$ and $\tau_{D}$ correspond to the hopping strengths of the molecule to the source and drain respectively. For these two limiting cases we take the values of the different parameters as follows: $\tau_{S}=\tau_{D}=0.5$, $t=2.5$ (weak-coupling) and $\tau_{S}=\tau_{D}=2$, $t=2.5$ (strong-coupling). Here we set the on-site energy $\epsilon_{0}=0$ (we can take any constant value of it instead of zero, since it gives only the reference energy level) for the electrodes, and the hopping strength $t_{0}=4$ in the two semi- infinite metallic electrodes. For the sake of simplicity, we set the Fermi energy $E_{F}=0$. Let us begin our discussion with the variation of the conductance $g$ as a function of the injecting electron energy $E$. As representative examples, in Fig. 3, we plot the $g$-$E$ characteristics for the molecular wires in which the molecules are attached to the electrodes in the trans configuration. Figures 3(a), (b), (c) and (d) correspond to the results for the wires with benzene, napthalene, anthracene and tetracene molecules respectively. The solid and dotted curves represent the results in the weak and strong molecular coupling limits respectively. It is observed that, in the limit of weak molecular coupling, the conductance shows very sharp resonance peaks for some particular energy values, while almost for all other energies it ($g$) drops to zero. At these resonances, the conductance approaches the value $2$, and therefore, the transmission probability $T$ goes to unity since we have the relation $g=2T$ from the Landauer conductance formula (see Eq.(1) with $e=h=1$ in the present description). These resonance peaks are associated with the energy eigenvalues of the single hydrocarbon molecules, and therefore we can say that the conductance spectrum manifests itself the electronic structure of the molecules. Now in the strong molecule-to-electrode coupling limit, all the resonances get substantial widths, which emphasize that the electron conduction takes place almost for all energy values. Such an enhancement of the resonance widths is due to the broadening of the molecular energy levels in the limit of strong molecular coupling, where the contribution comes from the imaginary parts of the self-energies $\Sigma_{S}$ and $\Sigma_{D}$ [56] as mentioned earlier in the previous section. To illustrate the quantum interference effects on electron transport, in Fig. 4, we plot the conductance-energy ($g$-$E$) characteristics for the molecular wires where the molecules are attached to the electrodes Figure 3: $g$-$E$ characteristics of the molecular wires in the trans configuration, where (a), (b), (c) and (d) correspond to the wires with benzene, napthalene, anthracene and tetracene molecules respectively. The solid and dotted curves represent the results in the weak and strong molecule- to-electrode coupling limits respectively. in the cis configuration. Figures 4(a), (b), (c) and (d) correspond to the results of the wires with benzene, napthalene, anthracene and tetracene molecules respectively. The solid and dotted lines indicate the same meaning as in Fig. 3. These results predict that, some of the conductance peaks do not reach to unity anymore, and get much reduced value. This behavior can be understood in this way. During the motion of the electrons from the source to the drain through the molecules, the electron waves propagating along the different possible pathways can get a phase shift among themselves according to the result of quantum interference. Therefore, the probability amplitude of getting the electron across the molecules either becomes strengthened or weakened. This causes the transmittance cancellations and provides anti- resonances in the conductance spectrum. Thus it can be emphasized that the electron transmission is strongly affected by the quantum interference effects and hence the molecule to electrodes interface structures. The scenario of the electron transfer through the molecular junction becomes much more clearly visible by investigating the current-voltage ($I$-$V$) characteristics. The current through the molecular systems Figure 4: $g$-$E$ characteristics of the molecular wires in the cis configuration, where (a), (b), (c) and (d) correspond to the wires with benzene, napthalene, anthracene and tetracene molecules respectively. The solid and dotted curves represent the results in the weak and strong molecule- to-electrodes coupling limits respectively. can be computed by the integration procedure of the transmission function $T$ (see Eq.(8)), where the function $T$ varies exactly similar to the conductance spectra, differ only in magnitude by the factor $2$, since the relation $g=2T$ holds from the Landauer conductance formula (Eq.(1)). To reveal this fact, in Fig. 5 we plot the current-voltage characteristics for the molecular wires in which the molecules attached to the electrodes in the trans configuration. Figures 5(a) and (b) correspond to results for the weak- and strong-coupling limits respectively. The solid, dotted, dashed and dot-dashed curves represent the variations of the currents with the bias voltage $V$ for the molecular wires consisting with benzene, napthalene, anthracene and tetracene molecules respectively. In the weak molecular coupling, the current exhibits staircase- like structure with fine steps as a function of the applied bias voltage. This is due to the existence of the sharp resonance peaks in the conductance spectra in this limit of coupling, since the current is computed by the integration method of the transmission function $T$. With the increase of the applied bias voltage, the electrochemical potentials on the electrodes are shifted gradually, and finally cross Figure 5: $I$-$V$ characteristics of the molecular wires in the trans configuration, where the solid, dotted, dashed and dot-dashed curves correspond to the results for the wires with benzene, napthalene, anthracene and tetracene molecules respectively. (a) weak-coupling limit and (b) strong- coupling limit. one of the quantized energy levels of the molecule. Therefore, a current channel is opened up and the current-voltage characteristic curve provides a jump. The other important feature is that the threshold bias voltage of the electron conduction across the wire significantly depends on the length of the wire in this weak-coupling limit. On the other hand, for the strong molecular coupling, the current varies almost continuously with the applied bias voltage and achieves much large amplitude than the weak-coupling case. This is because the resonance peaks get broadened due to the broadening of the energy levels in the strong-coupling limit which provide much larger current amplitude as we integrate the transmission function $T$ to get the current. Thus by tuning the molecule-to-electrode coupling, one can achieve very high current from the very low one. For this strong-coupling limit, the electron starts to conduct as long as the bias voltage is applied, in contrary to that of the weak- coupling case, for all these molecular wires. Thus we can say that, for this strong molecular coupling limit, the threshold bias voltage of the electron conduction is almost independent of the length of the molecular wire. The effects of the quantum interference on electron transport can be much more clearly understood from the current-voltage characteristics Figure 6: $I$-$V$ characteristics of the molecular wires in the cis configuration, where the solid, dotted, dashed and dot-dashed curves correspond to the results for the wires with benzene, napthalene, anthracene and tetracene molecules respectively. (a) weak-coupling limit and (b) strong- coupling limit. plotted in Fig. 6. In this case, the molecular wires are attached to the electrodes in the cis configuration, where Figs. 6(a) and (b) correspond to the results for the weak- and strong-coupling limits respectively. The solid, dotted, dashed and dot-dashed curves represent the same meaning as in Fig. 5. Our results show that, for these wires the current amplitudes get reduced enormously compared to the results obtained for the wires when the molecules are attached with the electrodes in the trans configuration. This is solely due to the quantum interference effects among all the possible pathways that the electron can take. Therefore, we can predict that designing a molecular device is significantly influenced by the quantum interference effects i.e., the molecule to electrodes interface structures. In conclusion of this section, we have introduced a parametric approach based on the tight-binding model to investigate the electron transport properties in some polycyclic hydrocarbon molecules attached to two semi-infinite one- dimensional metallic electrodes. This technique may be utilized to study the electronic transport in any complicated molecular bridge system. The conduction of electron through the hydrocarbon molecules is strongly influenced by the molecule-to-electrode coupling strength, length of the molecule, and the quantum interference effects. This study reveals that designing a whole system that includes not only the molecule but also the molecule-to-electrode coupling and the interface structures are highly important in fabricating molecular electronic devices. ## 4 Quantum Transport in a Thin Film Here we explore a novel feature of electron transport in a unconventional disordered thin film where disorder strength varies smoothly from its surface. In the present age of nanoscience and technology, it becomes quite easy to fabricate a nano-scale device where charge carriers are scattered mainly from its surface boundaries [59, 60, 61, 62, 63, 64, 65, 66, 67, 68], and not from the inner core region. It is completely opposite to that of a traditional doped system where the dopant atoms are distributed uniformly along the system. For example, in shell-doped nanowires the dopant atoms are spatially confined within a few atomic layers in the shell region of a nanowire. In such a shell-doped nanowire, Zhong and Stocks [60] have shown that the electron dynamics undergoes a localization to quasi-delocalization transition beyond some critical doping. In other very recent work [62], Yang et al. have also observed such a transition in edge disordered graphene nanoribbons upon varying the strength of edge disorder. From extensive studies of electron transport in such unconventional systems, it has been suggested that the surface states [69], surface scattering [70] and the surface reconstructions [71] may be responsible to exhibit several diverse transport properties. Motivated with these systems, here we focus our study of electron transport in a special type of thin film, in which disorder strength varies smoothly from layer to layer with the distance from its surface. This system shows a peculiar behavior of electron transport where the current amplitude increases with the increase of the disorder strength in the limit of strong disorder, while it decreases in the weak disorder limit. On the other hand, for the traditional disordered thin film i.e., the film subjected to uniform disorder, the current amplitude always decreases with the increase of the disorder strength. ### 4.1 Model Let us refer to Fig. 7, where a thin film is attached to two metallic electrodes, viz, source and drain. In this film, disorder strength varies smoothly from the top most disordered layer (solid line) to-wards the bottom layer, keeping the lowest bottom layer (dashed line) as disorder free. The electrodes are symmetrically attached at the two extreme corners of the bottom layer. Figure 7: Schematic view of a smoothly varying disordered thin film attached to two metallic electrodes (source and drain). The top most front layer (solid line) is the highest disordered layer and the disorder strength decreases smoothly to-wards the bottom layer keeping the lowest bottom layer (dashed line) as disorder free. Two electrodes are attached at the two extreme corners of the bottom layer. Both this film and the two side attached electrodes are described by the similar kind of tight-binding Hamiltonian as prescribed in Eq. (5). Now to achieve our required unconventional thin film, we choose the site energies ($\epsilon_{i}$’s in Eq. (5)) randomly from a “Box” distribution function such that the top most front layer becomes the highest disordered layer with strength $W$, and the strength of disorder decreases smoothly to-wards the bottom layer as a function of $W/(N_{l}-m)$, where $N_{l}$ gives the total number of layers and $m$ represents the total number of ordered layers from the bottom side of the film. On the other hand, in the conventional disordered thin film, all the layers are subjected to the same disorder strength $W$. Here, we concentrate our study on the determination of the typical current amplitude which is obtained from the relation, $I_{typ}=\sqrt{<I^{2}>_{W,V}}$ (9) where $W$ and $V$ correspond to the impurity strength and the applied bias voltage respectively. ### 4.2 Results and Discussion All the numerical calculations we present here are performed for some particular values of the different parameters, and all the basic features remain also invariant for some the other parametric values. The values of the required parameters are as follows. The coupling strengths of the film to the electrodes are taken as $\tau_{S}=\tau_{D}=1.5$, the nearest-neighbor hopping integral in the film is fixed to $t=1$. The on-site potential and the hopping integral in the electrodes are set as $\epsilon_{0}=0$ and $t_{0}=2$ respectively. In addition to these, here we also introduce another three parameters $N_{x}$, $N_{y}$ and $N_{z}$ to specify the system size of the thin film, where they correspond to the total number of lattice sites along the $x$, $y$ and $z$ directions of the film respectively. In our numerical calculations, the typical current amplitude ($I_{typ}$) is determined by taking the average over the disordered configurations and bias voltages (see Eq.(9)). Since in this particular model the site energies are chosen randomly, we compute $I_{typ}$ by taking the average over a large number ($60$) of disordered configurations in each case to get much accurate result. On the other hand, for the averaging over the bias voltage $V$, we set the range of it from $-10$ to $10$. In this presentation, we focus only on the systems with small sizes since all the qualitative behaviors remain also invariant even for the large systems. Figure 8 represents the variation of the typical current amplitude ($I_{typ}$) as a function of disorder ($W$) for some typical thin films with $N_{x}=10$, $N_{y}=8$ and $N_{z}=5$. Here we set $m=1$, i.e., only the lowest bottom layer of the unconventional disordered thin film is free from any disorder. Figure 8: $I_{typ}$ vs. $W$ for the two different types of thin films with $N_{x}=10$, $N_{y}=8$ and $N_{z}=5$. Here we set $m=1$. The solid and dotted curves correspond to the smoothly varying and complete disordered films respectively. The solid and dotted curves correspond to the results of the smoothly varying and complete disordered thin films respectively. A remarkably different behavior is observed for the smoothly varying disordered film compared to the film with complete disorder. In the later system, it is observed that $I_{typ}$ decreases rapidly with $W$ and eventually it drops to zero for the higher value of $W$. This reduction of the current is due to the fact that the eigenstates become more localized [72] with the increase of disorder, and it is well established from the theory of Anderson localization [73]. The appreciable change in the variation of the typical current amplitude takes place only for the unconventional disordered film. In this case, the current amplitude decreases initially with $W$ and after reaching to a minimum at $W=W_{c}$ (say), it again increases. Thus the anomalous behavior is observed beyond the critical disorder strength $W_{c}$, and we are interested particularly in this regime where $W>W_{c}$. In order to illustrate this peculiar behavior, we consider the smoothly varying disordered film as a coupled system combining two sub-systems. The coupling exists between the lowest bottom ordered layer and the other disordered layers. Thus the system can be treated, in other way, as a coupled order-disorder separated thin film. For this coupled system we can write the Schrödinger equations as: $(H_{0}-H_{1})\psi_{0}=E\psi_{0}$ and $(H_{d}-H_{2})\psi_{d}=E\psi_{d}$. Here $H_{0}$ and $H_{d}$ represent the sub-Hamiltonians of the ordered and disordered regions of the film respectively, and $\psi_{0}$ and $\psi_{d}$ are the corresponding eigenfunctions. The terms $H_{1}$ and $H_{2}$ in the above two expressions are the most significant and they can be expressed as: $H_{1}=H_{od}(H_{d}-E)^{-1}H_{do}$ and $H_{2}=H_{do}(H_{o}-E)^{-1}H_{od}$. $H_{od}$ and $H_{do}$ correspond to the coupling between the ordered region and the disordered region [60, 61]. From these mathematical expressions, the anomalous behavior of the electron transport in the film Figure 9: $I_{typ}$ vs. $W$ for the two different types of thin films with $N_{x}=12$, $N_{y}=10$ and $N_{z}=6$. Here we set $m=2$. The solid and dotted curves correspond to the identical meaning as in Fig. 8. can be described clearly. In the absence of any interaction between the ordered and disordered regions, we can assume the full system as a simple combination of two independent sub-systems. Therefore, we get all the extended states in the ordered region, while the localized states are obtained in the disordered region. In this situation, the motion of an electron in any one region is not affected by the other. But for the coupled system, the motion of the electron is no more independent, and we have to take the combined effects coming from both the two regions. With the increase of disorder, the scattering effect becomes dominated more, and thus the reduction of the current is expected. This scattering is due to the existence of the localized eigenstates in the disordered regions. Therefore, in the case of strong coupling between the two sub-systems, the motion of the electron in the ordered region is significantly influenced by the disordered regions. Now the degree of this coupling between the two sub-systems solely depends on the two parameters $H_{1}$ and $H_{2}$, those are expressed earlier. In the limit of weak disorder, the scattering effect from both the two regions is quite significant since then the terms $H_{1}$ and $H_{2}$ have reasonably high values. With the increase of disorder, $H_{1}$ decreases gradually and for a very large value of $W$ it becomes very small. Hence the term $(H_{0}-H_{1})$ effectively goes to $H_{0}$ in the limit $W\rightarrow 0$, which indicates that the ordered region becomes decoupled from the disordered one. Therefore, in the higher disorder regime the scattering effect becomes less significant from the ordered region, and it decreases with $W$. For the low regime of $W$, the eigenstates of both the two effective Hamiltonians, $(H_{0}-H_{1})$ and $(H_{d}-H_{2})$, are localized. With the increase of $W$, $H_{1}$ gradually decreases, resulting in much weaker localization in the states of $(H_{0}-H_{1})$, while the states of $(H_{d}-H_{2})$ become more localized. At a critical value of $W=W_{c}$ (say) ($\simeq$ band width of $H_{0}$), we get a separation between the much weaker localized states and the strongly localized states. Beyond this value, the weaker localized states become more extended and the strongly localized states become more localized with the increase of $W$. In this situation, the current is obtained mainly from these nearly extended states which provide the larger current with $W$ in the higher disorder regime. To illustrate the size dependence of the film on the electron transport, in Fig. 9 we plot the variation of the typical current amplitude for some typical thin films with $N_{x}=12$, $N_{y}=10$ and $N_{z}=6$. For these films we take $m=2$, i.e., two layers from the bottom side of the smoothly varying disordered film are free from any disorder. The solid and dotted curves correspond to the identical meaning as in Fig. 8. For both the unconventional and traditional disordered films, we get almost the similar behavior of the current as described in Fig. 8. This study shows that the typical current amplitude strongly depends on the finite size of the thin film. In summary of this section, we have provided a numerical study to exhibit the anomalous behavior of electron transport in a unconventional disordered thin film, where the disorder strength varies smoothly from its surface. Our numerical results have predicted that, in the smoothly varying disordered film, the typical current amplitude decreases with $W$ in the weak disorder regime ($W<W_{c}$), while it increases in the strong disorder regime ($W>W_{c}$). On the other hand for the conventional disordered film, the current amplitude always decreases with disorder. In this present investigations, we have also studied the finite size effects which reveal that the typical current amplitude strongly depends on the size of the film. Similar type of anomalous quantum transport can also be observed in lower dimensional systems like, edge disordered graphene sheets of single-atom- thick, surface disordered finite width rings, nanowires, etc. ## 5 Concluding Remarks In this dissertation, we have demonstrated the quantum transport properties in different types of bridge systems like, molecular wires and thin films. The physics of electron transport through these nanoscale systems is surprisingly rich. Many fundamental experimentally observed phenomena in such systems can be understood by using simple arguments. In particular, the formal relation between conductance and transmission coefficients (the Landauer formula) has enhanced the understanding of electronic transport in the bridge system. We have investigated the electron transport properties of some molecular bridge systems and unconventional disordered thin films within the tight-binding framework using Green’s function technique and tried to explain how electron transport is affected by the quantum interference of the electronic wave functions, molecule-to-electrode coupling strengths, length of the molecular wire and disorder strength. Our model calculations provide a physical insight to the behavior of electron conduction in these bridge systems. First, we have studied the electron transport in some molecular wires consisting with some polycyclic hydrocarbon molecules. Most interestingly, it has been observed that the transport properties are significantly influenced by the molecular coupling strength to the side attached electrodes, quantum interference effects and length of the molecule. Our study has emphasized that the molecule to electrodes interface structures are highly important in fabricating molecular electronic devices. Secondly, we have investigated the electron transport in a unconventional disordered thin film. Most remarkably, we have noticed that the typical current amplitude increases with the disorder strength in the strong disorder regime, while it decreases with the strength of disorder in the weak disorder regime. This particular study has suggested that the carrier transport in an order-disorder separated mesoscopic device may be tailored to desired properties through doping for different applications. ## References * [1] P. A. Orellana, M. L. Ladron de Guevara, M. Pacheco and A. Latge, Phys. Rev. B 68, 195321 (2003). * [2] P. A. Orellana, F. Dominguez-Adame, I. Gomez and M. L. Ladron de Guevara, Phys. Rev. B 67, 085321 (2003). * [3] A. T. Tilke, F. C. Simmel, H. Lorenz, R. H. Blick and J. P. Kotthaus, Phys. Rev. B 68, 075311 (2003). * [4] S. M. Cronenwett, T. H. Oosterkamp and L. P. Kouwenhoven, Science 281, 5 (1998). * [5] A. W. Holleitner, R. H. Blick, A. K. Huttel, K. Eber and J. P. Kotthaus, Science 297, 70 (2002). * [6] A. W. Holleitner, C. R. Decker, H. Qin, K. Eberl and R. H. Blick, Phys. Rev. Lett. 87, 256802 (2001). * [7] W. Z. Shangguan, T. C. Au Yeung, Y. B. Yu and C. H. Kam, Phys. Rev. B 63, 235323 (2001). * [8] A. I. Yanson, G. Rubio-Bollinger, H. E. van den Brom, N. Agrait and J. M. van Ruitenbeek, Nature (London) 395, 780 (1998). * [9] A. Nitzan, Annu. Rev. Phys. Chem. 52, 681 (2001). * [10] A. Nitzan and M. A. Ratner, Science 300, 1384 (2003). * [11] A. Aviram and M. Ratner, Chem. Phys. Lett. 29, 277 (1974). * [12] T. Dadosh, Y. Gordin, R. Krahne, I. Khivrich, D. Mahalu, V. Frydman, J. Sperling, A. Yacoby and I. Bar-Joseph, Nature 436, 677 (2005). * [13] R. M. Metzger et al., J. Am. Chem. Soc. 119, 10455 (1997). * [14] C. M. Fischer, M. Burghard, S. Roth and K. V. Klitzing, Appl. Phys. Lett. 66, 3331 (1995). * [15] J. Chen, M. A. Reed, A. M. Rawlett and J. M. Tour, Science 286, 1550 (1999). * [16] M. A. Reed, C. Zhou, C. J. Muller, T. P. Burgin and J. M. Tour, Science 278, 252 (1997). * [17] T. Kostyrko and B. R. BuÅa, Phys. Rev. B 67, 205331 (2003). * [18] M. Ernzerhof, M. Zhuang and P. Rocheleau, J. Chem. Phys. 123, 134704 (2005). * [19] M. Magoga and C. Joachim, Phys. Rev. B 59, 16011 (1999). * [20] J.-P. Launay and C. D. Coudret, in: A. Aviram and M. A. Ratner (Eds.), Molecular Electronics, New York Academy of Sciences, New York, (1998). * [21] R. Baer and D. Neuhauser, Chem. Phys. 281, 353 (2002). * [22] R. Baer and D. Neuhauser, J. Am. Chem. Soc. 124, 4200 (2002). * [23] D. Walter, D. Neuhauser and R. Baer, Chem. Phys. 299, 139 (2004). * [24] K. Tagami, L. Wang and M. Tsukada, Nano Lett. 4, 209 (2004). * [25] K. Walczak, Cent. Eur. J. Chem. 2, 524 (2004). * [26] R. H. Goldsmith, M. R. Wasielewski and M. A. Ratner, J. Phys. Chem. B 110, 20258 (2006). * [27] M. Ernzerhof, H. Bahmann, F. Goyer, M. Zhuang and P. Rocheleau, J. Chem. Theory Comput. 2, 1291 (2006). * [28] Y. M. Blanter and M. Buttiker, Phys. Rep. 336, 1 (2000). * [29] K. Walczak, Phys. Stat. Sol. (b) 241, 2555 (2004). * [30] S. N. Yaliraki, A. E. Roitberg, C. Gonzalez, V. Mujica and M. A. Ratner, J. Chem. Phys. 111, 6997 (1999). * [31] M. Di Ventra, S. T. Pantelides and N. D. Lang, Phys. Rev. Lett. 84, 979 (2000). * [32] Y. Xue, S. Datta and M. A. Ratner, J. Chem. Phys. 115, 4292 (2001). * [33] J. Taylor, H. Guo and J. Wang, Phys. Rev. B 63, 245407 (2001). * [34] P. A. Derosa and J. M. Seminario, J. Phys. Chem. B 105, 471 (2001). * [35] P. S. Damle, A. W. Ghosh and S. Datta, Phys. Rev. B 64, R201403 (2001). * [36] W. W. Cheng, H. Chen, R. Note, H. Mizuseki and Y. Kawazoe, Physica E 25, 643 (2005). * [37] W. W. Cheng, Y. X. Liao, H. Chen, R. Note, H. Mizuseki and Y. Kawazoe, Phys. Lett. A 326, 412 (2004). * [38] M. Ernzerhof and M. Zhuang, Int. J. Quantum Chem. 101, 557 (2005). * [39] M. Zhuang, P. Rocheleau and M. Ernzerhof, J. Chem. Phys. 122, 154705 (2005). * [40] M. Zhuang and M. Ernzerhof, Phys. Rev. B 72, 073104 (2005). * [41] M. Elstner et al., Phys. Rev. B 58, 7260 (1998). * [42] T. Frauenheim et al., J. Phys.: Condens. Matter 14, 3015 (2002). * [43] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). * [44] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). * [45] N. Sai, M. Zwolak, G. Vignale and M. D. Ventra, Phys. Rev. Lett. 94, 186810 (2005). * [46] N. Bushong, N. Sai and M. D. Ventra, Nano Lett. 5, 2569 (2005). * [47] M. D. Ventra and T. N. Todorov, J. Phys.: Condens. Matter 16, 8025 (2004). * [48] V. Mujica, M. Kemp and M. A. Ratner, J. Chem. Phys. 101, 6849 (1994). * [49] V. Mujica, M. Kemp A. E. Roitberg and M. A. Ratner, J. Chem. Phys. 104, 7296 (1996). * [50] M. P. Samanta, W. Tian, S. Datta, J. I. Henderson and C. P. Kubiak, Phys. Rev. B 53, R7626 (1996). * [51] M. Hjort and S. Staftröm, Phys. Rev. B 62, 5245 (2000). * [52] S. K. Maiti, Org. Electron. 8, 575 (2007); [Corrigendum: Org. Electron. 9, 418 (2008)]. * [53] S. K. Maiti, Physica B 394, 33 (2007). * [54] S. K. Maiti, Physica E 36, 199 (2007). * [55] S. K. Maiti, Physica E 40, 2730 (2008). * [56] S. Datta, Electronic transport in mesoscopic systems, Cambridge University Press, Cambridge (1997). * [57] M. B. Nardelli, Phys. Rev. B 60, 7828 (1999). * [58] W. Tian, S. Datta, S. Hong, R. Reifenberger, J. I. Henderson and C. I. Kubiak, J. Chem. Phys. 109, 2874 (1998). * [59] L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R. M. Westervelt and N. S. Wingreen, in Mesoscopic Electron Transport: Proc. NATO Advanced Study Institutes (NATO Advanced Study Institute, Series E: Applied Sciences) 345, (1997). * [60] J. X. Zhong and G. M. Stocks, Nano. Lett. 6, 128 (2006). * [61] J. X. Zhong and G. M. Stocks, Phys. Rev. B 75, 033410 (2007). * [62] C. Y. Yang, J. W. Ding and N. Xu, Physica B 394, 69 (2007). * [63] H. B. Chen and J. W. Ding, Physica B 403, 2015 (2008). * [64] S. K. Maiti, Int. J. Nanosci. 7, 51 (2008). * [65] S. K. Maiti, J. Nanosci. Nanotechnol. 8, 4096 (2008). * [66] S. K. Maiti, Int. J. Nanosci. 7, 171 (2008). * [67] S. K. Maiti, J. Comput. Theor. Nanosci. 5, 1398 (2008). * [68] S. K. Maiti, Chem. Phys. Lett. 446, 365 (2007); [Addendum: Chem. Phys. Lett. 454, 419 (2008)]. * [69] J. Y. Yu, S. W. Chung and J. R. Heath, J. Phys. Chem. B 104, 11864 (2000). * [70] Y. Cui, X. F. Duan, J. T. Hu and C. M. Lieber, J. Phys. Chem. B 104, 5213 (2000). * [71] R. Rurali and N. Lorente, Phys. Rev. Lett. 94, 026805 (2005). * [72] P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985). * [73] P. W. Anderson, Phys. Rev. 109, 1492 (1958).	arxiv-papers	2009-02-13T13:23:31	2024-09-04T02:49:00.541627	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Santanu K. Maiti", "submitter": "Santanu Maiti Kumar", "url": "https://arxiv.org/abs/0902.2307" }
0902.2316	###### Abstract Let $C_{1}$ and $C_{2}$ be codes with code distance $d$. Codes $C_{1}$ and $C_{2}$ are called weakly isometric, if there exists a mapping $J:C_{1}\rightarrow C_{2}$, such that for any $x,y$ from $C_{1}$ the equality $d(x,y)=d$ holds if and only if $d(J(x),J(y))=d$. Obviously two codes are weakly isometric if and only if the minimal distance graphs of these codes are isomorphic. In this paper we prove that Preparata codes of length $n\geq 2^{12}$ are weakly isometric if and only if these codes are equivalent. The analogous result is obtained for punctured Preparata codes of length not less than $2^{10}-1$. On weak isometries of Preparata codes Ivan Yu. Mogilnykh Sobolev Institute of Mathematics, Novosibirsk, Russia e-mail: ivmog84@gmail.com Submitted to Problems of Information Transmission on 11th of January 2009. ## 1 Introduction Let $E^{n}$ denote all binary vectors of length n. The Hamming distance between two vectors from $E^{n}$ is the number of places where they differ. The weight of vector $x\in E^{n}$ is the distance between this vector and the all-zero vector $0^{n}$, and the support of $x$ is the set $supp(x)=\\{i\in\\{1,\ldots,n\\}:x_{i}=1\\}$. A set $C,$ $C\subset E^{n}$, is called a code with parameters $(n,M,d)$, if $\|C\|=M$ and the minimal distance between two codewords from $C$ equals $d$. We say that a code $C$ is reduced if it contains all-zero vector. A collection of $k$-subsets (referred to as blocks) of a $n$-set such that any $t$-subset occurs in $\lambda$ blocks precisely is called a $(\lambda,n,k,t)$-design. The minimal distance graph of a code $C$ is defined as the graph with all codewords of $C$ as vertices, with two vertices being connected if and only if the Hamming distance between corresponding codewords equals to the code distance of the code $C$. Two codeword of $C$ are called $d$-adjacent if the Hamming distance equals code distance $d$ of the code $C$. Two codes $C_{1}$ and $C_{2}$ of length $n$ are called equivalent, if an automorphism $F$ of $E^{n}$ exists such that $F(C_{1})=C_{2}$. A mapping $I:C_{1}\rightarrow C_{2}$ of two codes $C_{1}$ and $C_{2}$ is called an isometry between codes $C_{1}$ and $C_{2}$, if the equality $d(x,y)=d(I(x),I(y))$ holds for all $x$ and $y$ from $C_{1}$. Then codes $C_{1}$ and $C_{2}$ are called isometric. A mapping $J:C_{1}\rightarrow C_{2}$ is called a weak isometry of codes $C_{1}$ and $C_{2}$ (and codes $C_{1}$ and $C_{2}$ weakly isometrical), if for any $x,y$ from $C_{1}$ the equality $d(x,y)=d$ holds if and only if $d(J(x),J(y))=d$ where $d$ is the code distance of code $C_{1}$. Obviously two codes are weakly isometric if and only if the minimal distance graphs of these codes are isomorphic. In [2] Avgustinovich established that any two weakly isometric 1-perfect codes are equivalent. In [5] it was proved that this result also holds for extended 1-perfect codes. In this paper any weak isometry of two Preparata codes (punctured Preparata codes) is proved to be an isometry of these codes. Moreover, weakly isometric Preparata codes (punctured Preparata codes) of length $n\geq 2^{12}$ (of length $n\geq 2^{10}-1$ respectively) are proved to be equivalent. This topic is closely related with problem of metrical rigidity of codes. A code $C$ is called metrically rigid if any isometry $I:C\rightarrow E^{n}$ can be extended to an isometry (automorphism) of the whole space $E^{n}$. Obviously any two metrically rigid isometric codes are equivalent. In [4] it was established that any reduced binary code of length $n$ containing 2-$(n,k,\lambda)$-design is metrically rigid for any $n\geq k^{4}$. A maximal binary code of length $n=2^{m}$ for even $m$, $m\geq 4$ with code distance 6 is called a Preparata code $\overline{P^{n}}$. Punctured Preparata code is a code obtained from Preparata code by deleting one coordinate. By $P^{n}$ we denote a punctured Preparata code of length $n$. Preparata codes and punctured Preparata codes have some useful properties. All of them are distance invariant [1], strongly distance invariant [3]. Also a punctured Preparata code is contained in the unique 1-perfect code [6]. An arbitrary punctured Preparata code is uniformly packed [1]. As a consequence of this property, codewords of minimal weight of a Preparata code (punctured Preparata code) form a design. The last property is crucial in proving the main result of this paper. ## 2 Weak isomery of punctured Preparata codes In this section we prove that any two punctured Preparata codes of length $n$ with isomorphic minimum distance graphs are isometric. Moreover, these codes are equivalent for $n\geq 2^{10}-1$. First we give some preliminary statements. ###### Lemma 1. [1]. Let $P^{n}$ be an arbitrary reduced punctured Preparata code. Then codewords of weight 5 of the code $P^{n}$ form 2-(n,5,(n-3)/3) design. Taking into account a structure of the design from this lemma we obtain ###### Corollary 1. Let $P^{n}$ be an arbitrary reduced punctured Preparata code and $r,s$ be arbitrary elements of the set $\\{1,\ldots,n\\}$. Then there exists exactly one coordinate $t$ such that all codewords of minimal weight of the code $P^{n}$ with ones in coordinates $r$ and $s$ has zero in the coordinate $t$. Let $C$ be a code with code distance $d$ and $x$ be an arbitrary codeword of $C$ of weight $i$. Denote by $D_{i,j}(x)$ the set of all codewords of $C$ of weight $j$ which are $d$-adjacent with vector $x$. In case when $C$ is a punctured Preparata code we give some properties of the set $D_{i,j}(x)$ that make the structure of minimal distance graph of this code more clear. ###### Lemma 2. Let $x$ be an arbitrary codeword of a punctured Preparata code $P^{n}$. Then any vector from $D_{i,i-1}(x)$ ($D_{i,i-3}(x)$ $D_{i,i-5}(x)$ respectively) has exactly 3 (4 and 5 resp.) zero coordinates from $supp(x)$ and exactly 2 (1 and 0 resp.) nonzero coordinates from $\\{1,\ldots,n\\}\backslash supp(x)$. Proof. Suppose a vector $y\in D_{i,i-k}(x)$ has $m_{k}$ zero coordinates from $supp(x)$. Then it has exactly $m_{k}-k$ nonzero coordinates from the set $\\{1,\ldots,n\\}\backslash supp(x)$. Since $d(x,y)=5$ we have $m_{k}=(5+k)/2$, which implies the required property for $k=1,3,5$ . $\blacktriangle$ Let $x$ be a codeword of weight $i$ from a $P^{n}$ ; $m,l$ be arbitrary coordinates from $supp(x)$. We denote by $A_{m,l}(x)$ $(B_{m,l}(x)$ and $C_{m,l}(x)$) the sets $D_{i,i-1}(x)$ ($D_{i,i-3}(x)$ and $D_{i,i-5}(x)$ respectively) with coordinates $m$ and $l$ equal to zero. ###### Lemma 3. Let $x\in P^{n}$, $m,l\in supp(x)$ and $u,v$ be arbitrary codewords of $P^{n}$ with zeros in coordinates $m$ and $l$ that are at distance $5$ from $x$. Then $u,v$ do not share zero coordinates in $supp(x)\setminus\\{m,l\\}$ and do not share coordinates equal to one in the set $\\{1,\ldots,n\\}\backslash supp(x)$. Proof. Let us suppose the opposite. Then the vectors $x+u$ and $x+v$ of weight five share at least three coordinates with ones in them and therefore $d(u,v)=d(x+u,x+v)\leq 4$ holds. Since code distance of the code $P^{n}$ equals 5 we get a contradiction. $\blacktriangle$ ###### Lemma 4. Let $x$ be an arbitrary codeword of weight $i$ from a punctured Preparata code. Then the following inequalities hold: $(i-3)C_{i}^{2}\leq 3\|D_{i,i-1}(x)\|+12\|D_{i,i-3}(x)\|+30\|D_{i,i-5}(x)\|\leq(i-2)C_{i}^{2}.$ (1) Proof. Fix two coordinates $m$ and $l$ from $supp(x)$. By Lemma 2 an arbitrary vector from $A_{m,l}(x)$ ($B_{m,l}$ and $C_{m,l}$) has exactly one zero coordinate (two and three respectively) from $supp(x)\backslash\\{m,l\\}$. Then taking into account Lemma 3 the number of coordinates from $supp(x)\backslash\\{m,l\\}$ which are zero for vectors from $A_{m,l}$, $B_{m,l}$ and $C_{m,l}$ equals $\|A_{m,l}(x)\|$, $2\|B_{m,l}\|$ and $3\|C_{m,l}\|$ respectively. Therefore the number of coordinates from the $supp(x)\backslash\\{m,l\\}$ which are zero for vectors from $A_{m,l}\cup B_{m,l}\cup C_{m,l}$ equals $\|A_{m,l}(x)\|+2\|B_{m,l}(x)\|+3\|C_{m,l}(x)\|.$ Since $x$ is a vector of weight $i$ and $m,l\in supp(x)$, this number does not exceed $i-2$. From the other hand by Corollary 1 there exists at most one coordinate from $supp(x)\backslash\\{m,l\\}$ such that all vectors from $A_{m,l}\cup B_{m,l}\cup C_{m,l}$ have one in it. Thus we have: $i-3\leq\|A_{m,l}(x)\|+2\|B_{m,l}(x)\|+3\|C_{m,l}(x)\|\leq i-2.$ Summing these inequalities for all $m,l\in supp(x)$ we obtain $(i-3)C_{i}^{2}\leq\sum_{m,l\in supp(x)}\|A_{m,l}(x)\|+2\sum_{m,l\in supp(x)}\|B_{m,l}(x)\|+3\sum_{m,l\in supp(x)}\|C_{m,l}(x)\|\leq(i-2)C_{i}^{2}$ (2) As an arbitrary vector from $D_{i,i-1}(x)$ has exactly 3 zero coordinates from $supp(x)$, any such vector is counted $C_{3}^{2}$ times in the sum $\sum_{m,l\in supp(x)}\|A_{m,l}(x)\|$. Then $\sum_{m,l\in supp(x)}\|A_{m,l}(x)\|=C_{3}^{2}\|D_{i,i-1}(x)\|.$ Analogously we get: $\sum_{m,l\in supp(x)}\|B_{m,l}(x)\|=C_{4}^{2}\|D_{i,i-3}(x)\|,$ $\sum_{m,l\in supp(x)}\|C_{m,l}(x)\|=C_{5}^{2}\|D_{i,i-5}(x)\|.$ So from (2) we get (1). $\blacktriangle$ Now we prove the main result using Lemmas 2 and 4. ###### Theorem 1. The minimal distance graphs of two punctured Preparata codes are isomorphic if and only if these codes are isometric. Proof. It is obvious that if two punctured Preparata codes are isometric then they are weakly isometric. Let $J:P^{n}_{1}\rightarrow P^{n}_{2}$ be a weak isometry of two punctured Preparata codes $P_{1}^{n}$ and $P_{2}^{n}$ of length $n$. Without loss of generality suppose that $0^{n}\in P^{n}_{1}$, $J(0^{n})=0^{n}$. We now show that mapping $J$ is an isometry. For proving this it is sufficient to show that $wt(J(x))=wt(x)$ for all $x\in P^{n}_{1}$. Suppose $z$ is a codeword of the code $P^{n}_{1}$, such that $wt(J(z))\neq wt(z)=i$ holds and the mapping $J$ preserves weight of all codewords of weight smaller that $i$. The vector $z$ satisfying these conditions we call critical Since $J(0^{n})=0^{n}$ and the mapping $J$ preserves the distance between all codewords at distance 5, we have $i\geq 6$. We prove that there is no critical codewords in $P^{n}_{1}$. From $0^{n}\in P^{n}_{1}$ holds that the weak isometry $J$ preserves a parity of weight of a vector and therefore $wt(J(z))$ equals either $i+2$ or $i+4$. Suppose $wt(J(z))=i+2$. Since $J$ is a weak isometry and $z$ is a critical vector we have the following: $\|D_{i+2,i-1}(J(z))\|=\|D_{i,i-1}(z)\|$, $\|D_{i+2,i-3}(J(z))\|=\|D_{i,i-3}(z)\|$, $\|D_{i,i-5}(z)\|=0$. Taking into account these equalities, from the inequalities of Lemma 4 for vectors $z$ and $J(z)$ we get $(i-3)C_{i}^{2}\leq 3\|D_{i,i-1}(z)\|+12\|D_{i,i-3}(z)\|,$ (3) $3\|D_{i+2,i+1}(J(z))\|+12\|D_{i,i-1}(z)\|+30\|D_{i,i-3}(z)\|\leq iC_{i+2}^{2}.$ (4) Multiplying both sides of inequality (3) by $-4$ we get $-12\|D_{i,i-1}(z)\|-48\|D_{i,i-3}(z)\|\leq-4(i-3)C_{i}^{2}.$ Summing this inequality with (4) we get $3\|D_{i+2,i+1(J(z))}\|-18\|D_{i,i-3}(z)\|\leq iC_{i+2}^{2}-4(i-3)C_{i}^{2},$ and therefore $\|D_{i,i-3}(z)\|\geq\frac{4(i-3)C_{i}^{2}-iC_{i+2}^{2}}{18}.$ (5) In particular, from the inequality (5) we have $\|D_{i,i-3}(z)\|\geq 1$ for $i=6$ and $i=7$. But there is no codewords of weight 3 and 4 in the $P_{1}$ since $P_{1}$ is reduced code with code distance 5. Therefore $i\geq 8$. From Lemma 4 we have the following $\|D_{i,i-3}(z)\|\leq\frac{(i-2)C_{i}^{2}}{12}.$ (6) But for $i\geq 10$ the inequality $3(i-2)C_{i}^{2}<2(4(i-3)C_{i}^{2}-iC_{i+2}^{2})$ holds. This contradicts with (5) and (6). So it is only remains to prove that there are no codewords of weight 8 and 9, such that their images under the mapping $J$ have weights 10 and 11 respectively. Obviously the Hamming distance between any two vectors from $D_{i+2,i-3}(J(z))$ is not less than 6\. By Lemma 2 all ones coordinates of each vector from $D_{i+2,i-3}(J(z))$ are in set $supp(J(z))$. So $\|D_{i+2,i-3}(J(z))\|$ does not exceed the cardinality of maximal constant weight code of length $i+2$, with all code words of weight being equal $i-3$ and being at distance not less than 6 pairwise. For $i=8$ and $i=9$ the cardinalities of such codes equal to 6 and 11 respectively, but from (5) we have $\|D_{10,5}(J(z))\|=\|D_{8,5}(z)\|\geq 12,\|D_{11,6}(J(z))\|=\|D_{9,6}(z)\|\geq 21,$ a contradiction. Therefore there is no critical vectors $z$ in $P_{1}^{n}$, $wt(z)=i$, such that $wt(J(z))=i+2$. Suppose $wt(J(z))=i+4$. In this case we have $\|D_{i,i-3}(z)\|=\|D_{i,i-5}(z)\|=0$, $\|D_{i+4,i-1}(J(z))\|=\|D_{i,i-1}(z)\|$. Using these equalities we have from the inequalities of Lemma 4 for the vectors $z$ and $J(z)$ the following: $(i-3)C_{i}^{2}\leq 3\|D_{i,i-1}(z)\|,$ $30\|D_{i,i-1}(z)\|\leq(i+2)C_{i+4}^{2}.$ From these last two inequalities we obtain $\frac{(i-3)C_{i}^{2}}{3}\leq\frac{(i+2)C_{i+4}^{2}}{30},$ and therefore $10i(i-1)(i-3)\leq(i+4)(i+3)(i+2)$ that implies $10i(i-1)(i-3)\leq 2i(i+3)(i+2).$ Since last inequality does not hold for $i\geq 6$ there is no critical vectors in $P_{1}^{n}$ and therefore the mapping $J$ is an isometry. $\blacktriangle$ In [4] the following theorem was proved ###### Theorem 2. Any reduced code of length $n$, that contains a $2-(n,k,\lambda)$-design is metrically rigid for $n\geq k^{4}$. Taking into account that by Lemma 1 any punctured reduced Preparata code contains 2-$(n,5,(n-3)/4)$-design applying Theorems 1 and 2 we get ###### Corollary 2. Let $n\geq 2^{10}-1$. Two punctured Preparata codes of length $n$ are equivalent if and only if the minimal distance graphs of these codes are isomorphic. ## 3 Weak isometry of Preparata codes Using the analogous considerations, Theorems 1,2 and Corollary 2 can easily be extended for extended Preparata codes. We now give the analogues of Lemmas 1-4 omitting their proofs. ###### Lemma 5. ([1]) Let $\overline{P^{n}}$ be an arbitrary reduced Preparata code. Then codewords of weight 6 of code $\overline{P^{n}}$ form 3-(n,6,(n-4)/3)-design. ###### Lemma 6. Let $x$ be an arbitrary codeword of a Preparata code $\overline{P^{n}}$, $wt(x)=i$. Then any vector from $D_{i,i-2}(x)$ ($D_{i,i-4}(x)$ $D_{i,i-6}(x)$ respectively) has exactly 4 (5 and 6 respectively) zero coordinates from $supp(x)$ and exactly 2 (1 and 0 respectively) nonzero coordinates from $\\{1,\ldots,n\\}\backslash supp(x)$. ###### Lemma 7. Let $x\in\overline{P^{n}}$, $m,l,k\in supp(x)$, and $u,v$ be arbitrary codewords of $\overline{P^{n}}$ at distance $6$ from the vector $x$ with zero coordinates in positions $m$, $l$, $k$. Then there is no coordinate from $supp(x)\setminus\\{m,l,k\\}$ such that $u,v$ have zeros in it and there is no coordinate from $\\{1,\ldots,n\\}\backslash supp(x)$ such that $u,v$ have ones in it. ###### Lemma 8. Let $x$ be an arbitrary codeword of weight $i$ from a Preparata code. Then the following inequalities hold: $C_{i}^{3}(i-4)\leq 4\|D_{i,i-2}(x)\|+20\|D_{i,i-4}(x)\|+60\|D_{i,i-6}(x)\|\leq C_{i}^{3}(i-3).$ (7) Using Lemmas 5-8 and the same arguments as in the proof of Theorem 1 the following theorem it is not difficult to prove ###### Theorem 3. The minimal distance graphs of two Preparata codes are isomorphic if and only if the codes are isometric. From this theorem, Lemma 5 and Theorem 2 we get ###### Corollary 3. Let $n\geq 2^{12}$. Two Preparata codes of length $n$ are equivalent if and only if the minimal distance graphs of these codes are isomorphic. The Author is deepfuly grateful to Faina Ivanovna Soloveva for introducing into the topic, problem statement and all around support of this work. ## References * [1] Semakov N.V., Zinoviev V.A., Zaitsev G.V. Uniformly packed codes // Probl. Inf. Trans. 1971. V. 7. 1. P. 30–39. * [2] Avgustinovich S.V. Perfect binary (n,3) codes: the structure of graphs of minimum distances // Discrete Appl. Math. 2001. V. 114\. P. 9–11. * [3] Vasil’eva, A.Yu. Strong distance invariance of perfect binary codes// Diskr. Anal. Issled. Oper., 2002. Iss. 1. V. 9. 4\. P. 33–40. * [4] Avgustinovich S.V., Soloveva F.I. To the Metrical Rigidity of Binary Codes // Problems of Inform. Transm. 2003. V. 39. 2. P. 23–28. * [5] I. Y. Mogilnykh, P. R. J. Östergård, O. Pottonen and F. I. Solov eva, accepted to IEEE Inform. Theory, _Reconstructing Extended Perfect Binary One-Error-Correcting Codes from Their Minimum Distance Graphs_ , Arxiv preprint arXiv:0810.5633, 2008. * [6] Semakov N.V., Zinoviev V.A., Zaitsev G.V. Interrelation of Preparata and Hamming codes and extension of Hamming codes to new double-error correcting codes // Proc.2nd Intern. Sympos. Information Theory. Tsakhadsor, Armenia, 1971. Budapest: Akad.Kiado, 1973. P. 257-263.	arxiv-papers	2009-02-13T14:16:43	2024-09-04T02:49:00.548262	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ivan Yu. Mogilnykh", "submitter": "Mogilnykh Ivan Yurievich", "url": "https://arxiv.org/abs/0902.2316" }
0902.2426	# On the BL Lacertae objects/radio quasars and the FRI/II dichotomy Ya-Di Xu1, Xinwu Cao2, Qingwen Wu3 1Physics Department, Shanghai Jiao Tong University,800 Dongchuan Road, Min Hang, Shanghai, 200240, China 2Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China 3International Center for Astrophysics, Korean Astronomy and Space Science Institute, Daejeon 305348, Republic of Korean Email: ydxu@sjtu.edu.cn, cxw@shao.ac.cn,qwwu@shao.ac.cn. ###### Abstract In the frame of unification schemes for radio-loud active galactic nuclei (AGNs), FR I radio galaxies are believed to be BL Lacertae (BL Lac) objects with the relativistic jet misaligned to our line of sight, and FR II radio galaxies correspond to misaligned radio quasars. The Ledlow-Owen dividing line for FR I/FR II dichotomy in the optical absolute magnitude of host galaxy–radio luminosity ($M_{R}$–$L_{\rm Rad}$) plane can be translated to the line in the black hole mass–jet power ($M_{\rm bh}$–$Q_{\rm jet}$) plane by using two empirical relations: $Q_{\rm jet}$–$L_{\rm Rad}$ and $M_{\rm bh}$–$M_{R}$. We use a sample of radio quasars and BL Lac objects with measured black hole masses to explore the relation of the jet power with black hole mass, in which the jet power is estimated from the extended radio emission. It is found that the BL Lac objects are clearly separated from radio quasars by the Ledlow & Owen FR I/II dividing line in the $M_{\rm bh}$–$Q_{\rm jet}$ plane. This strongly supports the unification schemes for FR I/BL Lac object and FR II/radio quasar. We find that the Eddington ratios $L_{\rm bol}/L_{\rm Edd}$ of BL Lac objects are systematically lower than those of radio quasars in the sample with a rough division at $L_{\rm bol}/L_{\rm Edd}\sim 0.01$, and the distribution of Eddington ratios of BL Lac objects/quasars exhibits a bimodal nature, which imply that the accretion mode of BL Lac objects may be different from that of radio quasars. ###### Subject headings: black hole physics —galaxies: active—galaxies: nuclei—quasars: emission lines—BL Lacertae objects: general ††slugcomment: Received 2008 December 17; accepted 2009 February 12 ## 1\. Introduction FR I radio galaxies (defined by edge-darkened radio structure) have lower radio power than FR II galaxies (defined by edge-brightened radio structure due to compact jet-terminating hot spots) (Fanaroff & Riley, 1974). Relativistic jets are observed in many radio-loud active galactic nuclei (AGNs). In the frame of unification schemes of radio-loud AGNs, FR I radio galaxies are believed to be misaligned BL Lacertae (BL Lac) objects, and FR II radio galaxies correspond to misaligned radio quasars (see Urry & Padovani, 1995, for a review). Most BL Lac objects have featureless optical and ultraviolet continuum spectra, and only a small fraction of BL Lac objects show very weak broad emission lines, while quasars usually have strong broad- line emission. The broad emission lines of quasars are produced by distant gas clouds in broad-line regions (BLR), which are photo-ionized by the optical/UV continua radiated from the accretion disks surrounding massive black holes. The difference of the broad-line emission between radio-loud quasars and BL Lac objects may be attributed to their different central engines (e.g., Cavaliere & D’Elia, 2002; Cao, 2002, 2003). The unified scheme of BL Lac objects and FR I radio galaxies have been extensively explored by many previous authors with different approaches, such as the comparisons of the spectral energy distributions (SEDs) in different wavebands (e.g., Owen et al., 1996; Capetti et al., 2000; Bai & Lee, 2001), the radio morphology, the radio luminosity functions (LFs) (e.g., Padovani & Urry, 1991; Kollgaard et al., 1992; Laurent-Muehleisen et al., 1993) and the optical line emission (e.g., Marchã et al., 2005). Padovani & Urry (1992) derived the radio LFs of flat spectrum radio quasars (FSRQs) and FR II galaxies from a sample of radio-loud AGNs. They considered a two-component model, in which the total luminosity is the sum of an unbeamed part and a beamed jet luminosity. The beamed LFs of FR II radio galaxies are consistent with the observed LFs of FSRQs and steep spectrum radio quasars (SSRQs), which strengthens the unification of FR II galaxies and radio quasars (see Padovani & Urry, 1992, for the details). Similar analyses were carried out on the relation between FR I galaxies and BL Lac objects (Padovani & Urry, 1991; Urry & Padovani, 1995), which is also consistent with the unification of FR Is and BL Lac objects. Even though the main observational features of different types of radio-loud AGNs can be successfully explained in the frame of the unification schemes, some authors have found observations indicating that the unification may be more complex than usually portrayed in these schemes (e.g., Marchã et al., 2005; Landt & Bignall, 2008). Landt & Bignall (2008) found that a considerable number of BL Lac objects can be identified with the relativistically beamed counterparts of FR II radio galaxies in a sample of BL Lac objects selected from the Deep X-ray Radio Blazar Survey (DXRBS). Ledlow & Owen (1996) found that FR I and FR II radio galaxies can be clearly divided in the host galaxy optical luminosity–radio luminosity ($M_{R}$–$L_{\rm Rad}$) plane, by a dividing line showing that radio power is proportional to the optical luminosity of the host galaxy. What causes the FR I/FR II division is still unclear, and there are two categories of models to explain it: (1) the morphological differences being caused by the interaction of the jets with the ambient medium of different physical properties (e.g., Gopal-Krishna & Wiita, 2000); and/or (2) different intrinsic nuclear properties of accretion and jet formation processes (e.g., Baum et al., 1995; Bicknell, 1995; Reynolds et al., 1996; Ghisellini & Celotti, 2001; Marchesini et al., 2004; Hardcastle et al., 2007). Ghisellini & Celotti (2001) used the optical luminosity of the host galaxy to estimate the central black hole masses of FR I/FR II radio galaxies, and the bolometric luminosity is estimated from the radio power of jets in FR I/FR II galaxies. They suggested that most FR I radio galaxies are accreting at lower rates compared with FR IIs, which could correspond to different accretion modes in FR I and FR II radio galaxies. If the black hole is spinning rapidly, the rotational energy of the black hole is expected to be transferred to the jets by the magnetic fields threading the holes, namely, the Blandford-Znajek (BZ) mechanism (Blandford & Znajek, 1977). The jet can also be accelerated by the large-scale fields threading the rotating accretion disk (i.e., the Blandford-Payne (BP) mechanism, Blandford & Payne, 1982). Cao & Rawlings (2004) found that the BZ mechanism for rapidly spinning black holes surrounded by advection dominated accretion flows (ADAFs) (Narayan & Yi, 1995) provides insufficient power to explain the jets in some 3CR FR I radio galaxies. Wu & Cao (2008) calculated the maximal jet power available from ADAFs around Kerr black holes as a function of black hole mass with an hybrid jet formation model (i.e., BP+BZ mechanism). They found that it can roughly reproduce the dividing line of the Ledlow-Owen relation for FR I/FR II dichotomy in the black hole mass–jet power ($M_{\rm bh}$–$Q_{\rm jet}$) plane with the mass accretion rate $\dot{M}\sim 0.01\dot{M}_{\rm Edd}$, if the black hole spin parameter $a\sim 0.9-0.99$ is adopted. This accretion rate indicates that FR I and FR II galaxies have different accretion modes, supporting the results of Ghisellini & Celotti (2001) and suggesting that FR I sources are in the ADAF mode. Wu & Cao (2008)’s results imply that the black hole spin may play an important role in the jet formation at least for FR I radio galaxies (also see Sikora et al., 2007, for the discussion of the impact of black hole spin on the jet formation in AGNs). In this work, we use a sample of BL Lac objects and radio quasars with measured radio power, black hole masses, and Eddington ratios, to explore the relationship between BL Lac objects and radio quasars, and to compare it with the FR I/FR II division. The sample and the estimates of black hole mass/jet power are described in §2 and §3. We show the results in §4, and §5 contains the discussion. The cosmological parameters $H_{0}=70~{}\rm km~{}s^{-1}Mpc^{-1}$, $\Omega_{M}=0.3$ and $\Omega_{\Lambda}=0.7$ have been adopted in this work. ## 2\. The sample The host galaxies of 132 BL Lac objects have been observed with the Hubble Space Telescope WFPC2 by Urry et al. (2000), among which there are 48 sources with measured redshifts and extended radio emission. We add additional 18 BL Lac objects compiled in the work of Wu et al. (2008) to the Urry et al. (2000)’s sample, which leads to 66 BL Lac objects (including 28 low-energy- peaked BL Lac objects (LBLs) and 38 high-energy-peaked BL Lac objects (HBLs)) with measured redshifts and extended radio emission data for our present investigation. We search the literature for the emission line data of these sources, and find 44 sources including 23 LBLs and 21 HBLs. We use the luminosity of narrow line [O ii] at 3727 $\rm\AA$ to estimate the bolometric luminosity. For the sources in which the emission line data of [O ii] being unavailable, we estimate the [O ii] luminosity using other narrow emission lines. In order to compare the difference between BL Lac objects and radio quasars, we need a sample of radio quasars. In this work, we adopt the sample of radio quasars compiled by Liu et al. (2006), which is selected from the 1 Jy, S4, and S5 radio source catalogs. Their sample consists of 146 radio quasars including 79 FSRQs (with $\alpha_{2-8\rm GHz}<0.5$) and 67 steep-spectrum radio quasars (SSRQs) (with $\alpha_{2-8\rm GHz}>0.5$). All quasars in their sample have estimated black hole masses and jet power (see Liu et al., 2006, for the details of the quasar sample). ## 3\. The black hole mass and jet power The relation between black hole mass $M_{\rm bh}$ and host galaxy luminosity $L_{K}$ at $K$-band (Eq. 1 in McLure & Dunlop, 2004) is derived from $M_{\rm bh}$–$M_{R}$ by using an average color correction of $R-K=2.7$ for the same cosmology adopted in this paper. We convert this relation back to $M_{\rm bh}$–$M_{R}$ as $\log_{10}(M_{\rm bh}/M_{\odot})=-0.50(\pm 0.02)M_{R}-2.75(\pm 0.53),$ (1) to estimate the central black hole masses of BL Lac objects in this sample. For a few BL Lac objects, their black hole masses can also be estimated from their stellar dispersion velocity $\sigma$ with the empirical $M_{\rm bh}$–$\sigma$ relation. It is found that the black hole masses of three BL Lac objects estimated with $M_{\rm bh}$–$\sigma$ relation are roughly consistent with those estimated with Eq. (1) (see Cao, 2004, for the details, and references therein). The jet power can be estimated with the relation between jet power and radio luminosity proposed by Willott et al. (1999), $Q_{\rm jet}\simeq 3\times 10^{38}f^{3/2}L^{6/7}_{\rm ext,151}~{}~{}(\rm W),$ (2) where $L_{\rm ext,151}$ is the extended radio luminosity at 151 MHz in units of 1028 W Hz-1 sr-1. Willott et al. (1999) have argued that the normalization is uncertain and introduced the factor $f$ ($1\leq f\leq 20$) to account for these uncertainties. This relation was proposed for FR II radio galaxies and quasars. Cao & Rawlings (2004) compared the power of the jet in M87 (a typical FR I radio galaxy) derived with different approaches, and found that Eq. (2) may probably be suitable even for FR Is (see Cao & Rawlings, 2004, for the details, and references therein). Following Cao (2003), we adopt this relation to estimate the power of jets in BL Lac objects, which is believed to be a good approximation if BL Lacs can be unified with FR Is. For most BL Lac objects, their radio/optical continuum emission is strongly beamed to us due to their relativistic jets and small viewing angles of the jets with respect to the line of sight (e.g., Fan & Zhang, 2003; Gu et al., 2006). The low-frequency radio emission (e.g. 151 MHz) may still be Doppler beamed. We therefore use the extended radio emission detected by VLA to estimate the jet power, as adopted in Cao (2003). The observed extended radio emission is $K$-corrected to 151 MHz in the rest frame of the source assuming $\alpha_{\rm e}=0.8$ ($f_{\nu}\propto\nu^{-\alpha_{\rm e}}$) (Cassaro et al., 1999). We take the black hole masses of radio quasars from Liu et al. (2006), which are estimated from the broad line widths of ${\rm H\,{\beta}}$, Mg ii, or C iv, as well as the line luminosities of these lines (see Liu et al., 2006, for the details). In Liu et al. (2006)’s work, the jet power is estimated from the extended radio emission at 151 MHz with the formula derived by Punsly (2005), which is slightly different from Eq. (2) proposed by Willott et al. (1999). To be self-consistent, we estimate the jet power of quasars in Liu et al. (2006)’s sample from their extended radio luminosities with Eq. (2), which is the same as the estimates of jet power for BL Lac objects in this work. For BL Lac objects, the observed optical continuum emission may be dominated by the beamed synchrotron emission from the relativistic jets (e.g., Gu et al., 2006). The narrow-line regions are believed to be photo-ionized by the radiation from the accretion disk, and the narrow-line emission can be used to estimate the bolometric luminosity for BL Lac objects. We convert the luminosity of the narrow-line [O ii] to bolometric luminosity using the relation proposed by Willott et al. (1999), $L_{\rm bol}=5\times 10^{3}L_{\rm[O\,{II}]}~{}{\rm W},$ (3) for the BL Lac objects in this sample. For the objects which lack [O ii] line emission data, we convert the luminosities of other narrow lines ([O iii] or Hα+[N ii]) to the luminosity of [O ii] using the ratios suggested by Zirbel & Baum (1995) for FR I galaxies. The narrow-line emission data for the BL Lac objects are taken from the literature (Sbarufatti et al., 2006; Carangelo et al., 2003; Rector et al., 2000; Rector & Stocke, 2001; Stickel et al., 1993; Marchã et al., 1996; Morganti et al., 1992). We note that Eq. (3) is derived for FR IIs/quasars, while ADAFs may be present in these BL Lac objects. The SED of an ADAF is significantly different from that of a standard thin disk (e.g., Narayan et al., 1995). Nagao et al. (2002) calculated the emission of narrow-line regions photo-ionized by two different SED templates respectively, i.e., a standard thin disk SED template with a bump in UV/soft X-ray bands and a hot ADAF SED template described by a power-law continuum in hard X-ray bands with an exponential cutoff. They found that the narrow-line regions are more efficiently photo-ionized by the ADAF SED template than the standard thin disk case (see the bottom panel of Fig. 5 in Nagao et al., 2002), which implies that the present estimates on the bolometric luminosity with Eq. (3) may be over-estimated to some extent (a factor of $\sim 2-3$ for the narrow-line regions with hydrogen column density $\lesssim 10^{20}~{}{\rm cm^{-2}}$). For radio quasars, we estimate their bolometric luminosities from the total broad-line luminosities $L_{\rm BLR}$ calculated by Liu et al. (2006), as the optical continua for most radio-loud quasars may probably be contaminated by the beamed emission from relativistic jets. Liu et al. (2006) derived a tight correlation: $\lambda L_{\lambda}(5100{\rm\AA)}=84.3L_{{\rm H}\beta}^{0.998}$, for the sample of radio-quiet AGNs in Kaspi et al. (2000). Given that the luminosity of the broad-line ${\rm H}_{\beta}$ corresponds to $\sim$4 per cent of $L_{\rm BLR}$ (see Liu et al., 2006, and references therein) and using the relation $L_{\rm bol}\simeq 9\lambda L_{\lambda}(5100{\rm\AA)}$ (Kaspi et al., 2000), the bolometric luminosity can be estimated as $L_{\rm bol}\simeq 30L_{\rm BLR}$. ## 4\. The results The division between FR I and FR II radio galaxies is clearly shown by a line in the plane of total radio luminosity and optical luminosity of the host galaxy (Ledlow & Owen, 1996). The optical luminosity of the host galaxy can be converted to black hole mass $M_{\rm bh}$ by using the empirical relation (1), while the jet power $Q_{\rm jet}$ can be estimated from the radio luminosity with relation (2). Thus, the dividing line between FR I and II radio galaxies is translated to $\log Q_{\rm jet}({\rm ergs~{}s^{-1}})=1.13\log M_{\rm bh}(M_{\odot})+33.18+1.50\log f,$ (4) in $M_{\rm bh}$–$Q_{\rm jet}$ plane (see Wu & Cao, 2008, for the details), which is modified for the cosmology adopted in this paper. In Figure 4, we plot the relation between the black hole masses $M_{\rm bh}$ and jet power $Q_{\rm jet}$ for radio quasars and BL Lac objects. It is found that BL Lac objects can be roughly separated from quasars by the FR I/II dividing line. The distributions of Eddington ratios for BL Lac objects and quasars are plotted in Fig. 4, where only the BL Lac objects with measured line emission have been included, because the bolometric luminosity is derived from the emission lines for these sources. We estimate the statistical significance of a possible bimodal distribution of Eddington ratios for BL Lac objects and quasars using the KMM algorithm (Ashman et al., 1994). The distribution for the entire sample is strongly inconsistent with being unimodal ($P$-value$<0.001$), and the KMM algorithm separates the entire sample into two groups. No significant difference is found in the distributions of black hole masses for BL Lac objects and radio quasars. The relation between black hole mass $M_{\rm bh}$ and jet power $Q_{\rm jet}$ for the BL Lac objects and quasars. The open squares and filled squares represent FSRQs and SSRQs respectively, while the circles and triangles represent BL Lac objects. The filled circles/triangles represent the LBLs/HBLs with measured line emission, while the open circles/triangles the LBLs/HBLs without measured line emission. The dashed line represents the Ledlow-Owen dividing line between FR I and FR II radio galaxies given by Eq. (4). The distributions of Eddington ratios ($L_{\rm bol}/L_{\rm Edd}$) for BL Lac objects (dashed line) and quasars (solid line), respectively. ## 5\. Discussion Figure 4 shows that the FR I/FR II dividing line given by Ledlow & Owen (1996) roughly separates the radio-loud quasars from BL Lac objects in the $M_{\rm bh}$–$Q_{\rm jet}$ plane, which strongly supports the FR I/BL Lac objects and FR II/radio quasars unification schemes. This conclusion is independent of the value of the uncertainty factor $f$ in Eq. (2). We find that only a small fraction of LBLs/quasars are above/below the dividing line, which is similar to the FR I/II division (see for instance the Fig. 1 in Ledlow & Owen, 1996). The HBLs have relatively lower jet power than LBLs, and only one HBL appears above the dividing line. This means that the BL Lacs/quasars and the FR I/II divisions may be true only in a statistical sense. The exceptional sources in the $M_{\rm bh}$–$Q_{\rm jet}$ plane may provide useful clues to investigations on the central engines in radio-loud AGNs (e.g., Cao & Rawlings, 2004; Landt & Bignall, 2008). In Fig. 4, we show that the distributions of Eddington ratios for BL Lacs and quasars in our sample exhibit a bimodal nature. The BL Lac objects are roughly seperated from the quasars at $L_{\rm bol}/L_{\rm Edd}\sim 0.01$, with most BL Lac objects having $L_{\rm bol}/L_{\rm Edd}\lesssim 0.01$ and almost all the quasars having $L_{\rm bol}/L_{\rm Edd}\gtrsim 0.01$. We suggest that this bimodal behavior of the distribution may imply different accretion modes in BL Lac objects and quasars, and furthermore the transition between the accretion states happens at $L_{\rm bol}/L_{\rm Edd}\sim 0.01$ according to Fig. 4. Since this is roughly the critical luminosity above which ADAFs are not possible (e.g., Narayan & Yi, 1995), this suggests that ADAFs are present in BL Lac objects and standard thin disks are in quasars. We note that a similar explanation is invoked to explain the FR I/II division, in which ADAFs would be present in FR I galaxies while standard thin disks are in FR II galaxies (e.g., Ghisellini & Celotti, 2001; Wu & Cao, 2008). Interestingly enough, Marchesini et al. (2004) found a similar bimodal distribution of Eddington ratios for a sample of FR I and FR II radio galaxies. As discussed in §3, the bolometric luminosities of BL Lac objects may be over- estimated, if ADAFs are present in these sources. This would strengthen the bimodality in the distribution of Eddington ratios of BL Lac objects and quasars. The similarity between the division of BL Lac objects/quasars and FR I/II found in this Letter strongly supports the unification schemes for FR I/BL Lac object and FR II/radio quasar. We thank the anonymous referee for the helpful comments/suggestions. This work is supported by the NSFC (grants 10778621, 10703003, 10773020, 10821302 and 10833002), the CAS (grant KJCX2-YW-T03), and the National Basic Research Program of China (grant 2009CB824800). ## References * Ashman et al. (1994) Ashman, K. M., Bird, C. M., & Zepf, S. E. 1994, AJ, 108, 2348 * Bai & Lee (2001) Bai, J. M., & Lee, M. G. 2001, ApJ, 548, 244 * Baum et al. (1995) Baum, S. A., Zirbel, E. L., & O’Dea, C. P. 1995, ApJ, 451, 88 * Bicknell (1995) Bicknell, G. V. 1995, ApJS, 101, 29 * Blandford & Payne (1982) Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883 * Blandford & Znajek (1977) Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433 * Cao (2002) Cao, X. 2002, ApJ, 570, L13 * Cao (2003) Cao, X. 2003, ApJ, 599, 147 * Cao (2004) Cao, X. 2004, ApJ, 609, 80 * Cao & Rawlings (2004) Cao, X., & Rawlings, S. 2004, MNRAS, 349, 1419 * Capetti et al. (2000) Capetti, A., Trussoni, E., Celotti, A., Feretti, L., & Chiaberge, M. 2000, MNRAS, 318, 493 * Cassaro et al. (1999) Cassaro, P., Stanghellini, C., Bondi, M., Dallacasa, D., della Ceca, R., & Zappalà, R. A. 1999, A&AS, 139, 601 * Carangelo et al. (2003) Carangelo, N., Falomo, R., Kotilainen, J., Treves, A., & Ulrich, M.-H. 2003, A&A, 412, 651 * Cavaliere & D’Elia (2002) Cavaliere, A., & D’Elia, V. 2002, ApJ, 571, 226 * Fan & Zhang (2003) Fan, J. H., & Zhang, J. S. 2003, A&A, 407, 899 * Fanaroff & Riley (1974) Fanaroff, B. L., & Riley, J. M. 1974, MNRAS, 167, 31 * Ghisellini & Celotti (2001) Ghisellini, G., & Celotti, A. 2001, A&A, 379, L1 * Gopal-Krishna & Wiita (2000) Gopal-Krishna, & Wiita, P. J. 2000, A&A, 363, 507 * Gu et al. (2006) Gu, M. F., Lee, C.-U., Pak, S., Yim, H. S., & Fletcher, A. B. 2006, A&A, 450, 39 * Hardcastle et al. (2007) Hardcastle, M. J., Evans, D. A., & Croston, J. H. 2007, MNRAS, 376, 1849 * Kaspi et al. (2000) Kaspi, S., Smith, P. S., Netzer, H., Maoz, D., Jannuzi, B. T., & Giveon, U. 2000, ApJ, 533, 631 * Kollgaard et al. (1992) Kollgaard, R. I., Wardle, J. F. C., Roberts, D. H., & Gabuzda, D. C. 1992, AJ, 104, 1687 * Landt & Bignall (2008) Landt, H., & Bignall, H. E. 2008, MNRAS, 391, 967 * Laurent-Muehleisen et al. (1993) Laurent-Muehleisen, S. A., Kollgaard, R. I., Moellenbrock, G. A., & Feigelson, E. D. 1993, AJ, 106, 875 * Ledlow & Owen (1996) Ledlow, M. J., & Owen, F. N. 1996, AJ, 112, 9 * Liu et al. (2006) Liu, Y., Jiang, D. R., & Gu, M. F. 2006, ApJ, 637, 669 * Marchã et al. (1996) Marchã, M. J. M., Browne, I. W. A., Impey, C. D., & Smith, P. S. 1996, MNRAS, 281, 425 * Marchã et al. (2005) Marchã, M. J. M., Browne, I. W. A., Jethava, N., & Antón, S. 2005, MNRAS, 361, 469 * Marchesini et al. (2004) Marchesini, D., Celotti, A., & Ferrarese, L. 2004, MNRAS, 351, 733 * McLure & Dunlop (2004) McLure, R. J., & Dunlop, J. S. 2004, MNRAS, 352, 1390 * Morganti et al. (1992) Morganti, R., Ulrich, M.-H., & Tadhunter, C. N. 1992, MNRAS, 254, 546 * Nagao et al. (2002) Nagao, T., Murayama, T., Shioya, Y., & Taniguchi, Y. 2002, ApJ, 567, 73 * Narayan & Yi (1995) Narayan, R., & Yi, I. 1995, ApJ, 452, 710 * Narayan et al. (1995) Narayan, R., Yi, I., & Mahadevan, R. 1995, Nature, 374, 623 * Owen et al. (1996) Owen, F. N., Ledlow, M. J., & Keel, W. C. 1996, AJ, 111, 53 * Padovani & Urry (1991) Padovani, P., & Urry, C. M. 1991, ApJ, 368, 373 * Padovani & Urry (1992) Padovani, P., & Urry, C. M. 1992, ApJ, 387, 449 * Punsly (2005) Punsly, B. 2005, ApJ, 623, L9 * Rector et al. (2000) Rector, T. A., Stocke, J. T., Perlman, E. S., Morris, S. L., & Gioia, I. M. 2000, AJ, 120, 1626 * Rector & Stocke (2001) Rector, T. A., & Stocke, J. T. 2001, AJ, 122, 565 * Reynolds et al. (1996) Reynolds, C. S., di Matteo, T., Fabian, A. C., Hwang, U., & Canizares, C. R. 1996, MNRAS, 283, L111 * Sbarufatti et al. (2006) Sbarufatti, B., Falomo, R., Treves, A., & Kotilainen, J. 2006, A&A, 457, 35 * Sikora et al. (2007) Sikora, M., Stawarz, Ł., & Lasota, J.-P. 2007, ApJ, 658, 815 * Stickel et al. (1993) Stickel, M., Fried, J. W., & Kuehr, H. 1993, A&AS, 98, 393 * Urry & Padovani (1995) Urry, C. M., & Padovani, P. 1995, PASP, 107, 803 * Urry et al. (2000) Urry, C. M., Scarpa, R., O’Dowd, M., Falomo, R., Pesce, J. E., & Treves, A. 2000, ApJ, 532, 816 * Willott et al. (1999) Willott, C. J., Rawlings, S., Blundell, K. M., & Lacy, M. 1999, MNRAS, 309, 1017 * Wu & Cao (2008) Wu, Q., & Cao, X. 2008, ApJ, 687, 156 * Wu et al. (2008) Wu, Z., Gu, M., & Jiang, D. R. 2008, ChJAA, accepted (astro-ph/08041180). * Zirbel & Baum (1995) Zirbel, E. L., & Baum, S. A. 1995, ApJ, 448, 521	arxiv-papers	2009-02-14T02:08:47	2024-09-04T02:49:00.553980	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ya-Di Xu (1), Xinwu Cao (2), Qingwen Wu (3) ((1) Shanghai Jiao Tong\n University; (2) Shanghai Astron. Obs.; (3) Korean Astronomy and Space Science\n Institute)", "submitter": "Ya-Di Xu", "url": "https://arxiv.org/abs/0902.2426" }
0902.2435	# Hadron production by quark combination in central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV Chang-en Shao Department of Physics, Qufu Normal University, Shandong 273165, People’s Republic of China Jun Song Department of Physics, Shandong University, Shandong 250100, People’s Republic of China Feng-lan Shao Department of Physics, Qufu Normal University, Shandong 273165, People’s Republic of China Qu-bing Xie Department of Physics, Shandong University, Shandong 250100, People’s Republic of China ###### Abstract The quark combination mechanism of QGP hadronization is applied to nucleus- nucleus collisions at top SPS energy. The yields, rapidity and transverse momentum distributions of identified hadrons in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV are systematically studied. The calculated results are in agreement with the experimental data from NA49 Collaboration. The longitudinal and transverse collective flows and strangeness of the hot and dense quark matter produced in nucleus-nucleus collisions at top SPS energy are investigated. It is found that the collective flow of strange quarks is stronger than light quarks, which is compatible with that at RHIC energies, and the strangeness is almost the same as those at $\sqrt{s_{NN}}=$ 62.4, 130, 200 GeV. ###### pacs: 25.75.Dw, 25.75.Ld, 25.75.Nq, 25.75.-q ## I Introduction Lattice QCD predicts that at extremely high temperature and density, the confined hadronic matter will undergo a phase transition to a new state of matter called quark gluon plasma (QGP) Susskind1979 ; qpg2004 . The relativistic heavy ion collisions can provide the condition to create this deconfined partonic matter Gyulassy200530 . In general, two approaches are used to study the properties of the deconfined hot and dense quark matter produced in AA collisions. One is studying the high $p_{T}$ hadrons from initial hard jets, in which one can recur to the perturbative QCD to a certain degree WangXN1998 . The other is investigating the properties of thermal hadrons frozen out from the hot and dense quark matter. For the latter, the hadronization of the hot and dense quark matter (a typical non-perturbative process) is of great significance. Only through a reliable hadronization mechanism, can we reversely obtain various information of QGP properties from the final state hadrons measured experimentally. The abundant experimental data abelev:152301 ; abelev:2007prl and phenomenological studies Fries22003prl ; Greco2003prl ; Fries:2003prc ; Greco2003prc ; Hwa:2004prc ; Hwa:2008prc at RHIC energies suggest that quark combination mechanism is one of the most hopeful candidates. The two most noticeable results are the successful explanation of the high baryon/meson ratios and the constituent quark number scaling of the hadronic elliptic flow in the intermediate transverse momentum range Greco2003prl ; Fries:2003prc , which can not be understood at all in the partonic fragmentation picture. Recently, the NA49 Collaboration have measured the elliptic flow of identified hadrons at top SPS energy Alt2007prc , and found that the quark number scaling of elliptic flow was shown to hold also. It immediately gives us an inspiration of the applicability for the quark combination at top SPS energy. On the other hand, the NA49 collaboration have found three interesting phenomena around 30A GeV Alt08onset , i.e. the steepening of the energy dependence for pion multiplicity, a maximum in the energy dependence of strangeness to pion ratio and a characteristic plateau of the effective temperature for kaon production. These phenomena are indicative of the onset of the deconfinement at low SPS energies. One can estimate via Bjorken method that the primordial spatial energy density in Pb+Pb collisions at top SPS energy is about 3.0 GeV/$fm^{3}$ Stock2008 , exceeding the critical energy density (about 1 GeV/$fm^{3}$) predicted by Lattice QCD. Therefore, the deconfined hot and dense quark matter has been probably created, and we can extend the quark combination mechanism to SPS energies. As is well known, hadron yield is one of the most basic and important observables which can help us to test the understanding of the hadronization mechanism for the hot and dense quark matter created in the relativistic heavy ion collisions. In most of recombination/coalescence models, the hadron wave function is necessary to get the hadron yield. As the wave functions for almost all hadrons are unknown at present, it is difficult for these models to study this issue quantitatively Fries22003prl ; Greco2003prl ; Molnar03 . In addition, these models do not satisfy the unitarity which is important to the issue as well yang04 . Different from those models, the quark combination model Xieqb:1988 ; Shao2005prc uses the near-correlation in phase space and SUf(3) symmetry, instead of hadron wave function, to determine the hadron multiplicity. In addition, the model satisfies unitarity as well and has reproduced many experimental data at RHIC shao2007prc ; Yao:2006fk ; WangYF08 ; Yao08prc . Therefore, we apply it in this paper to systematically study the yields, rapidity and transverse momentum distributions of various hadrons in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. On one hand, one tests the applicability of the quark combination mechanism at this collision energy. We note that the first attempt of the mechanism at SPS energies, for hadron yields alone, was the ALCOR model ALCOR95 ; Levai2000 . Now, the rich experimental data of hadron multiplicities and momentum spectra provide an opportunity to make a further, systematical and even decisive test of the mechanism at SPS energies. On the other hand, the parton momentum distributions at hadronization, which carry the information on the evolution of the hot and dense quark mater, are extracted from the final hadrons at top SPS energy and compared with those at RHIC energies. We concentrate the comparison on two properties related to strange hadron production. One is the difference in collective flow between light and strange quarks, which occurs at RHIC energies ChenJH2008 ; WangYF08 . The other is the strangeness enhancement, a significant property of QGP Rafelski1982 . The paper is arranged as follows. In the next section, we make a brief introduction to the quark combination model. In section III, we calculate the yields and rapidity distributions of identified hadrons in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. In section IV, the results of transverse momentum distributions of various hadrons are shown. In section V, we firstly make a detailed analysis of the longitudinal and transverse collective flow of the hot and dense quark matter at top SPS energy. Secondly, the energy dependence of the strangeness in the hot and dense quark matter is extracted and analyzed. Section VI presents summary. ## II An introduction of the Quark Combination Model The starting point of the model is a color singlet system which consists of constituent quarks and antiquarks. All kinds of hadronization models demand that they satisfy rapidity or momentum correlation for quarks in the neighborhood of phase space. The essence of this correlation is in agreement with the fundamental requirement of QCD Xie:ap . According to QCD, a $q\overline{q}$ may be in a color octet or a singlet. The color factors $\langle(q\bar{q})_{8}\|\frac{-\lambda^{a}\cdot\lambda^{a}}{4}\|(q\bar{q})_{8}\rangle=\frac{1}{6}$ and $\langle(q\bar{q})_{1}\|\frac{-\lambda^{a}\cdot\lambda^{a}}{4}\|(q\bar{q})_{1}\rangle=-\frac{4}{3}$, which means a repulsive or an attractive interaction between them. Here $\lambda^{a}$ are the Gell-Mann matrices. If they are close with each other in phase space, they can interact with sufficiently time to be in the color singlet and form a meson. Similarly, a $qq$ can be in a sextet or an anti- triplet, and the color factors $\langle(qq)_{6}\|\frac{\lambda^{a}\cdot\lambda^{a}}{4}\|(qq)_{6}\rangle=\frac{1}{3}$ and $\langle(qq)_{\bar{3}}\|\frac{\lambda^{a}\cdot\lambda^{a}}{4}\|(qq)_{\bar{3}}\rangle=-\frac{2}{3}$. If its nearest neighbor in phase space is a $q$, they form a baryon. If the neighbor is a $\overline{q}$, because the attraction strength of the singlet is two times that of the anti-triplet, $q\overline{q}$ will win the competition to form a meson and leave a $q$ alone to combine with other quarks or antiquarks. Based on the above QCD and near-correlation in phase space requirements, we had proposed a quark combination rule(QCR) Xieqb:1988 ; Xie:ap which combines all these quarks and antiquarks into initial hadrons. When the transverse momentum of quarks are negligible, all $q$ and $\overline{q}$ can always line up stochastically in rapidity. The QCR reads as follows: 1. 1. Starting from the first parton ($q$ or $\overline{q}$) in the line; 2. 2. If the baryon number of the second in the line is of the different type from the first, i.e. the first two partons are either $q\overline{q}$ or $\overline{q}q$, they combine into a meson and are removed from the line, go to point 1; Otherwise they are either $qq$ or $\overline{q}\,\overline{q}$, go to the next point; 3. 3. Look at the third, if it is of the different type from the first, the first and third partons form a meson and are removed from the line, go to point 1; Otherwise the first three partons combine into a baryon or an anti-baryon and are removed from the line, go to point 1. The following example shows how the above QCR works $\displaystyle q_{1}\overline{q}_{2}\overline{q}_{3}\overline{q}_{4}\overline{q}_{5}q_{6}\overline{q}_{7}q_{8}q_{9}q_{10}\overline{q}_{11}q_{12}q_{13}q_{14}\overline{q}_{15}q_{16}q_{17}\overline{q}_{18}\overline{q}_{19}\overline{q}_{20}$ $\displaystyle\rightarrow M(q_{1}\overline{q}_{2})\;\overline{B}(\overline{q}_{3}\overline{q}_{4}\overline{q}_{5})\;M(q_{6}\overline{q}_{7})\;B(q_{8}q_{9}q_{10})\;M(\overline{q}_{11}q_{12})\;$ $\displaystyle M(q_{13}\overline{q}_{15})\;B(q_{14}q_{16}q_{17})\;\overline{B}(\overline{q}_{18}\overline{q}_{19}\overline{q}_{20})$ If the quarks and antiquarks are stochastically arranged in rapidity, the probability distribution for $N$ pairs of quarks and antiquarks to combine into $M$ mesons, $B$ baryons and $B$ anti-baryons is $X_{MB}(N)=\frac{2N(N!)^{2}(M+2B-1)!}{(2N)!M!(B!)^{2}}3^{M-1}\delta_{N,M+3B}.$ (1) Hadronization is the soft process of the strong interaction and is independent of flavor, so the net flavor number remains constant during the process. In the quark combination scheme, this means that the quark number for each certain flavor prior to hadronization equals to that of all initially produced hadrons after it. Obviously the quark number conservation is automatically satisfied in the model. It is different from the non-linear algebraic method in ALCOR model ALCOR95 where normalization factor for each quark flavor is introduced with the constraint of the quark number conservation. The average number of initially produced mesons $M(N)$ and baryons $B(N)$ are given by $\displaystyle\langle M(N)\rangle$ $\displaystyle=$ $\displaystyle\sum_{M}\sum_{B}MX_{MB}(N)\;,$ (2) $\displaystyle\langle B(N)\rangle$ $\displaystyle=$ $\displaystyle\sum_{M}\sum_{B}BX_{MB}(N)\;.$ (3) Then the multiplicity of various initial hadrons is obtained according to their production weights $\langle M^{initial}_{j}\rangle=C_{M_{j}}\langle M(N)\rangle,\hskip 22.76228pt\langle B^{initial}_{j}\rangle=C_{B_{j}}\langle B(N)\rangle,$ (4) where $C_{M_{j}}$ and $C_{B_{j}}$ are normalized production weights for the meson $M_{j}$ and baryon $B_{j}$, respectively. If three quark flavors are considered only, we can obtain the production weights using the SUf(3) symmetry with a strangeness suppression factor $\lambda_{s}$ Xieqb:1988 ; Shao2005prc , which are listed in Table 1. The extension of the symmetry to excited states, exotic states and more quark flavors is also straightforward excit95 ; Shao2005prc ; Yao08prc . Considering the decay contributions from the resonances, we can obtain the yields of final state hadrons $\langle{h_{i}^{final}}\rangle=\langle{h_{i}^{initial}}\rangle+\sum\limits_{j}B_{r}(j\rightarrow i)\langle{h_{j}}\rangle,$ (5) where the $B_{r}(j\rightarrow i)$ is the weighted decay branching ratio for $h_{j}$ to $h_{i}$ pdg08p355 . In principle, the hadron production probability should be calculated from the matrix element $\langle{q}\overline{q}\|M\rangle$ for meson or $\langle{qqq}\|B\rangle$ for baryon. However, the wave functions for almost all hadrons which are governed by the non-perturbative QCD are unknown at present. It is difficult to study the production of hadrons quantitatively through their wave functions. In view of this, the hadron production probability in our model is determined by the SUf(3) symmetry with a strangeness suppression. This symmetry has been supported by many experiments, particularly by the coincidence of the observed $\lambda_{s}$ obtained from various mesons and baryons Hofmann:1988gy . Therefore, the model can quantitatively describe many global properties for the bulk system by virtue of the Monte Carlo method Shao2005prc ; shao2007prc ; Yao:2006fk ; WangYF08 ; Yao08prc ; excit95 . Table 1: The normalized production weight for baryons and mesons in the $\texttt{SU}_{f}(3)$ ground state. ${r_{i}}$ is the number of strange quarks in hadron. The ratio of the vector ($J^{P}=1^{-}$) to pseudoscalar ($J^{P}=0^{-}$) meson follows the spin counting, while that of the decouplet ($J^{P}=\frac{3}{2}^{+}$) to octet ($J^{P}=\frac{1}{2}^{+}$) baryon suffers a spin suppression effect; see Ref. excit95 ; Shao2005prc for details. $C_{M}$ \| $C_{M_{i}}=\frac{2J_{i}+1}{4(2+\lambda_{s})^{2}}\lambda_{s}^{r_{i}}$, except ---\|--- $C_{\eta}=\frac{2J_{\eta}+1}{4(2+\lambda_{s})^{2}}\frac{1+2\lambda_{s}^{2}}{3}$ $C_{\eta^{\prime}}=\frac{2J_{\eta^{\prime}}+1}{4(2+\lambda_{s})^{2}}\frac{2+\lambda_{s}^{2}}{3}$ \| $C_{B_{i}}=\frac{4}{(2+\lambda_{s})^{3}(2J_{i}+1)}\lambda_{s}^{r_{i}}$, except $C_{B}$ \| $C_{\Lambda}=C_{\Sigma^{0}}=C_{\Sigma^{\ast 0}}=C_{\Lambda(1520)}=\frac{3}{2\,(2+\lambda_{s})^{3}}\lambda_{s}$ When applying the model to describe the hadronizaton of the hot and dense quark matter produced in heavy ion collisions, the net-baryon quantum number of the system perplexes the analysis formula of Eq. 1 but it can be easily evaluated in Monte Carlo program. On the other hand, the the transverse momentum of quarks is not negligible due to the strong collective flow of quark matter. In principle, we should define the QCR in three-dimensional phase space, but it is quite complicated to have it because one does not have an order or one has to define an order in a sophisticated way so that all quarks can combine into hadrons in a particular sequence. In practice, the combination is still put in rapidity and meanwhile the maximum transverse momentum difference $\Delta_{p}$ between (anti)quarks are constrained as they combine into hadrons. The transverse spectra of hadrons have a relationship with the quark spectra as follows (e.g. for meson) $\displaystyle\frac{dN_{M}}{d^{2}\mathrm{\textbf{p}_{T}}}\varpropto$ $\displaystyle\int d^{2}\mathrm{\textbf{p}_{1,T}}d^{2}\mathrm{\textbf{p}_{2,T}}f_{q}(\mathrm{\textbf{p}_{1,T}})f_{\overline{q}}(\mathrm{\textbf{p}_{2,T}})\delta^{2}(\mathrm{\textbf{p}_{T}}-\mathrm{\textbf{p}_{1,T}}-\mathrm{\textbf{p}_{2,T}})$ (6) $\displaystyle\times\,\Theta(\Delta_{p}-\|\mathrm{\textbf{p}^{\ast}_{1,T}}-\mathrm{\textbf{p}^{\ast}_{2,T}}\|),$ where $f_{q/\overline{q}}(\mathrm{\textbf{p}_{T}})$ is the transverse momentum distribution of the quark/antiquark, assumed to be rapidity-independent in present work. The superscript asterisk denotes the quark momentum in the center-of-mass frame of formed hadron. The limitation $\Delta_{p}$ is treated as parameter in our study, and fixed to be $\Delta_{p}=0.3$ GeV for mesons and $\Delta_{p}=0.6$ GeV for baryons both at RHIC and SPS energies. Note that the spectrum normalization is determined by the multiplicity in Eq. 4, i.e. the constraint of the parameter $\Delta_{p}$ on the hadron yield is neglected. One issue that is often questioned is the energy and entropy conservation in quark combination process. As the non-perturbative QCD is unsolved, there is no rigorous theory which can incorporate the partonic phase as well as hadronic phase, thus it is difficult to justify or condemn this issue in essence at the moment. As we know, a lot of the experimental phenomena in intermediate transverse momentum range at RHIC can be explained beautifully only in the quark combination scenario. It suggests that maybe this ’puzzling’ issue does not exist. As far as the quark combination itself is concerned, there is no difference for the combination occurred in the different (intermediate or low) transverse momentum range. Therefore, whether the properties of low $p_{T}$ hadrons can be reproduced or not is also a significant test of the quark combination mechanism, as the vast majority of hadrons observed experimentally are just these with low transverse momentum. ## III Hadron yields and rapidity distributions In high energy nucleus-nucleus collisions, the energy deposited in the collision region excites large numbers of newborn quarks and antiquarks from the vacuum. Subsequently, the hot and dense quark matter mainly composed of these newborn quarks will expend hydrodynamically until hadronization. The net-quarks from the colliding nuclei still carry a fraction of beam energy, thus their evolution is different from the newborn quarks. One part of net- quarks are stopped in the hot and dense quark matter, and hadronize together with it. The other part of net-quarks penetrate the hot quark matter, and run up to the forward rapidity region. The latter, together with small amount newborn quarks, form the leading fireball. Their hadronization should be earlier than that of the hot and dense quark matter with a prolonged expansion stage, and the hadronization outcomes consist of nucleons and small mount of mesons. Figure 1: (Color online) Rapidity spectra of newborn quarks and net-quarks at hadronization in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. The current version of quark combination model simulates only the hadronization of the hot and dense quark matter and subsequently decays of resonances. One indispensable input is the momentum distributions of thermal quarks and antiquarks at hadronization, which are the results of the hydrodynamic evolution in partonic phase. In order to focus attentions on the test of quark combination mechanism in this and next sections, we reversely extract the quark distributions by fitting the experimental data in the model. A detailed analysis of quark distributions at hadronization will be in section V. The solid and dashed lines in Fig. 1 show the rapidity distributions of newborn light and strange quarks at hadronization respectively, obtained from the $\pi^{-}$ and $K^{+}$ data Afanasiev2002prc .The dotted-dashed line is the rapidity distribution of net-quarks in the hot and dense quark matter, which is extracted from net-proton data Appelshauser1999prl . Firstly, we calculate the yields and rapidity densities at midrapidity of various hadrons in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. The results are shown in Table 2. From the energy dependence of the rapidity density for net-baryon bearden04stop , one can see that the nucleus-nucleus collisions at SPS energies exhibit a strong stopping power. Therefore, the leading particles contribute little to yields and rapidity distributions of various hadrons. The calculated yields and rapidity densities of vector meson $\phi$ are shown to be about twice as high as the experimental data. The results of other hadrons are basically in agreement with the experimental data, but slight deviations exist also. The overpredictions of $\phi$ meson may be associated with the exotic particle $f_{0}(980)$, which has a possible tetraquark structure containing a strange quark and a strange antiquark Hirar07prc . As a bond state containing strange components, it has a slightly lower mass than $\phi$ meson but is not included in the SUf(3) ground states. In the present work, we consider only the production of $36-plets$ of meson and $56-plets$ of baryon in the SUf(3) ground states, and the excited states and exotic states are not taken into account. The $f_{0}(980)$ multiplicity is found to be nearly the same as $\phi$ meson in the $e^{+}e^{-}$ annihilations pdg08p355 . The $m_{T}$ distribution of $f_{0}(980)$ measured by STAR Collaboration in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV is also shown to be comparable to that of $\phi$ Fachini2003f980 ; Adams2005phi . Therefore, the overprediction of $\phi$ meson can be removed by incorporating the $f_{0}(980)$ production. Table 2: The yields (left) and rapidity densities at midrapidity (right) of identified hadrons in central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. The experimental data are taken from Ref. Afanasiev2002prc ; Antinorik2005 ; Alt2008phi ; Alt2006prc ; Alt08Xi ; Mischke03Lam \| yield \| $\frac{dN}{dy}\|_{y=0}$ ---\|---\|--- \| data \| model \| data \| model $\pi^{+}$ \| $619\pm 17\pm 31$ \| $566$ \| $170.1\pm 0.7\pm 9$ \| 168.2 $\pi^{-}$ \| $639\pm 17\pm 31$ \| $630$ \| $175.4\pm 0.7\pm 9$ \| 183.5 $K^{+}$ \| $103\pm 5\pm 5$ \| $92.5$ \| $29.5\pm 0.3\pm 1.5$ \| 27.3 $K^{-}$ \| $51.9\pm 1.6\pm 3$ \| $45.3$ \| $16.8\pm 0.2\pm 0.8$ \| 15.7 $K^{0}_{s}$ \| $75\pm 4$ \| 66.7 \| $26.0\pm 1.7\pm 2.6$ \| 20.7 $\phi$ \| $8.46\pm 0.38\pm 0.33$ \| $15.2$ \| $2.44\pm 0.1\pm 0.08$ \| 5.26 $p$ \| \| 120 \| $29.6\pm 0.9\pm 2.96$ \| 25.9 $\overline{p}$ \| \| 3.2 \| $1.66\pm 0.17\pm 0.17$ \| 1.53 $\Lambda$ \| $44.9\pm 0.6\pm 8$ \| 52.9 \| $9.5\pm 0.1\pm 1.0$ \| 13.3 $\overline{\Lambda}$ \| $3.07\pm 0.06\pm 0.31$ \| 2.88 \| $1.24\pm 0.03\pm 0.13$ \| 1.35 $\mathrm{\Xi^{-}}$ \| $4.04\pm 0.16\pm 0.57$ \| 4.9 \| $1.44\pm 0.1\pm 0.15$ \| 1.43 $\mathrm{\overline{\Xi}^{\,{}_{+}}}$ \| $0.66\pm 0.04\pm 0.08$ \| 0.58 \| $0.31\pm 0.03\pm 0.03$ \| 0.26 Subsequently, we will calculate the longitudinal rapidity distributions of various hadrons. Due to the deviations in hadron yields, it is difficult to directly compare the calculated hadron spectra with the experimental data. In order to focus attentions on the property of hadron momentum spectra, we will scale the calculated rapidity densities to the center value of the experimental data when we show the hadronic rapidity and $p_{T}$ spectra in Fig. 2 and 4 respectively, thereby removing these deviations in hadron yields. Figure 2: (Color online) The scaled rapidity distributions of identified hadrons in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. The contributions from leading particles are not included. The open circles of $\phi$ data in the second panel are the latest results measured by NA49 Collaboration Alt2008phi , and filled circles are the previous ones Afanasiev2000plb . Other experimental data are taken from Ref. Afanasiev2002prc ; Mischke03Lam ; Alt08Xi . The open symbols of $K_{s}^{0}$, $\Lambda$ and $\Xi$ show data points reflected around midrapidity. Pion is the lightest and most abundant hadron produced in AA collisions, and its momentum distribution can best reflect the global evolution property of the hot and dense quark matter. In various models of high energy heavy ion collisions, the reproduction of pion meson is always taken as a paramount test of models. In Landau hydrodynamic model Landau , the rapidity distribution of pion can be well described and the sound velocity (which is an important physical quantity standing for the property of the hot and dense quark matter) can be extracted from the pion distribution. For other hadrons, such as kaons, protons, $\Lambda$, $\Xi$ and so on, the Landau model can not describe their rapidity distributions with the same sound velocity or freeze-out temperature Mohanty2003prc ; Satarov2007prc ; Sarkisyan:2006 . For a systematic description of the rapidity distributions of various hadrons, the detailed longitudinal dynamics, e.g. the evolution of net-baryon density which will result in the yield and spectrum asymmetry between hadron and antihadron, should be included. In addition, the hadronization mechanism is especially important to describe the differences in the yield and momentum distribution of various hadron species. Using the extracted quark distributions in Fig. 1, we calculate the rapidity distributions of pions, kaons, $\Lambda(\overline{\Lambda})$, $\Xi^{-}(\overline{\Xi}^{{}_{+}})$ and $\phi$ in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. The results are shown in Fig. 2 and are compared with the experimental data. The calculated rapidity spectrum of $\phi$ meson is narrow than the latest data of NA49 Collaboration (open circles in the second panel), but is in good agreement with previous data (filled circles). The rapidity spectra of other hadrons are well reproduced. One can see that the quark combination mechanism is applicable for describing the longitudinal distributions of various hadrons at top SPS energy. ## IV Hadron transverse momentum distributions In this section, we calculate the transverse momentum distributions of various hadrons in the midrapidity range. In this paper, we only consider the hadronization of the hot and dense quark matter. The transverse momentum invariant distribution of constituent quarks at hadronization is taken to be an exponential form $\exp(-m_{T}/T)$, where $T$ is the slope parameter which is also called effective production temperature. Fig. 3 shows the midrapidity $p_{T}$ spectra of constituent quarks at hadronization in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. The spectra of newborn light and strange quarks are extracted from the data of $\pi^{+}$ and $K^{+}$ respectively Alt2008hpt . The net-quark distribution is fixed by the data of $K^{-}/K^{+}$ ratio as a function of $p_{T}$ Alt2008hpt . A detailed analysis of the quark $p_{T}$ spectra will be shown in section V. Figure 3: (Color online) The transverse momentum distributions of constituent quarks in the midrapidity region at hadronization in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. Fig. 4 shows the calculated $p_{T}$ spectra of pions, kaons, protons, $\Lambda(\overline{\Lambda})$, $\Xi^{-}(\overline{\Xi}^{{}_{+}})$ and $\phi$ in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. For kaons, protons, $\Lambda$ and $\Omega$, the spectral slopes of antihadrons measured experimentally are all steeper than those of hadrons Alt2008hpt ; Anticic2004L ; Anticic2005O . However, the spectrum of $\Xi^{-}$ is abnormally steeper than that of $\overline{\Xi}^{{}_{+}}$ Alt08Xi . Our predictions are in good agreement with all the data except $\Xi^{-}$. Figure 4: (Color online) The scaled transverse momentum distributions of identified hadrons at midrapidity in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. Only the combination of thermal quarks is taken into account. Solid lines are the calculated results of hadrons and dashed lines for antihadrons. The experimental data are from Ref. Alt2008hpt ; Alt08Xi ; Alt2008phi The exponential function $\exp(-m_{T}/T)$ is often used experimentally to fit the transverse momentum distributions of identified hadrons in the low $p_{T}$ range, and to extract the effective production temperature $T$ of various hadrons. It is found at top SPS energy that all final-state hadrons except pion meson have much higher $T$ than the critical temperature Alessandro2003 , which indicates a strong collective flow at this collision energy. It is regarded in Ref. XuNu1998 that this observed flow mainly develops in the late hadronic rescattering stage. But results in Fig. 2 and Fig. 4 all show that both longitudinal and transverse spectra of various hadrons can be coherently explained by the same quark distributions, respectively. It suggests that the observed flow should mainly come from the expansive evolution stage of the hot and dense quark matter before hadronization, but not from the post- hadronization stage. In addition, the same constituent quark spectra contained in light, single- and multi- strange hadrons also imply that the hot quark matter hadronize into these initial hadrons almost at the same time, i.e. the hadronization is a rapid process. Figure 5: (Color online) The transverse momentum distributions of identified hadrons at midrapidity in most central Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV. Only the combination of thermal quarks is taken into account. Solid lines are the calculated results of hadrons and dashed lines for antihadrons. The experimental data are from Ref. abelev:152301 ; Adams07hyperon ; Abelev07phiv2 Figure 6: (Color online) The ratios of $\overline{p}/\pi^{-}$, $\Lambda/K_{s}^{0}$ and $\Omega/\phi$ at midrapidity in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV and Au+Au collisions at 200 GeV. Only the combination of thermal quarks is taken into account. Solid lines are the calculated results in Au+Au collisions and dashed lines for Pb+Pb collisions. The experimental data are from Ref. abelev:152301 ; Abelev06ks0 ; Abelev07phiv2 ; Alt2008hpt ; Andr06ksL In Fig. 5, we calculate the $p_{T}$ spectra of pions, kaons, protons, $\Lambda(\overline{\Lambda})$, $\Xi^{-}(\overline{\Xi}^{{}_{+}})$, $\phi$ and $\Omega(\overline{\Omega})$ in most central Au+Au collisions at top RHIC energy. The momentum distributions of constituent quarks at hadronization are taken to be $\exp(-m_{T}/0.375)$ for strange quarks and $\exp(-m_{T}/0.34)$ for light quarks. The numbers and rapidity spectra of the light and strange quarks and antiquarks have been obtained in the study of the longitudinal hadron production songjun09 . Here, the result of $\phi$ meson is multiplied by a factor 0.5. One can see that the $p_{T}$ spectra of various hadrons are well reproduced. The baryon/meson ratio as a function of $p_{T}$ is sensitive to the hadronization mechanism. As we know, the observed high baryon/meson ratios in the intermediate $p_{T}$ range at RHIC energies abelev:152301 can not be understood at all in the scheme of parton fragmentation, but can be easily explained in the quark combination mechanism. Fig. 6 shows the model predictions of $\overline{p}/\pi^{-}$, $\Lambda/K_{s}^{0}$ and $\Omega/\phi$ at midrapidity in both central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV and central Au+Au collisions at$\sqrt{s_{NN}}=200$ GeV. In the intermediate $p_{T}$ range where the hadron production is dominated by the combination of thermal quarks, the baryon/meson ratios increase with the increasing $p_{T}$. One can see that the experimental data in this region are well reproduced. The falling tendency of measured baryon/meson ratios after peak position is owing to the abundant participation of jet quarks, which is beyond the concern of the present paper. Besides the hadronization mechanism, the baryon/meson ratio in the intermediate $p_{T}$ region is also influenced by other two factors. One is the nuclear stopping power in collisions. Comparing with the strong collision transparency at top RHIC energy bearden04stop , the strong nuclear stopping at top SPS energy causes the detention of abundant net-quarks in the midrapidity region. These net-quarks significantly suppress the production of anti-baryons and enhance that of the baryons. Therefore, the $\overline{p}/\pi^{-}$ ratio at top SPS energy is much lower than that at top RHIC energy while the $\Lambda/K_{s}^{0}$ ratio at top SPS energy is higher than that at top RHIC energy. The other is the momentum distribution of constituent quarks at hadronization. This can be illustrated by $\Omega/\phi$ ratio because the production of these two hadron species is less influenced by the net-quarks. The calculated $\Omega/\phi$ ratio shows a weak dependence on the collision energy in the intermediate $p_{T}$ range. The well description of various baryon/meson ratios in such a wide energy range is an indication of the universality for the quark combination mechanism. ## V Analysis of parton distributions at hadronization The constituent quark distributions at hadronization carry the information on the evolution of the hot and dense quark matter in partonic phase. In this section, we focus attentions on the longitudinal and transverse collective flows and strangeness enhancement of the hot and dense quark matter produced at top SPS energy. ### V.1 The longitudinal and transverse collective flow Due to the thermal pressure, the hot and dense quark matter created in high energy heavy ion collisions will expand during the evolution before hadronization. The longitudinal and transverse collective flow of final hadrons measured experimentally is the exhibition of this early thermal expansion in the partonic phase. Utilizing the relativistic hydrodynamic evolution of the hot and dense quark matter, one can obtain the collective flow in quark level from the extracted quark momentum distributions, and compare it with that at RHIC energies. There are two well known hydrodynamic models for the description of the space time evolution of the hot and dense quark matter produced in heavy ion collisions. One is Bjorken model Bjorken84 which supposes that the collision is transparent, and it is appropriate to extremely high energy collisions, such as LHC. The other is Landau model Landau with an assumption of the full stopping for nucleus-nucleus collisions. The longitudinal evolution result is equivalent to the superposition of a set of thermal sources in rapidity axis, with a (Bjorken) uniform or (Landau) Gaussian weight. In general, when applying the model to describe the hadron rapidity distributions, different parameter values are required to make a good fit of different hadron species Mohanty2003prc . In this paper, we apply the hydrodynamic description to the evolution in quark level, thus the collective flow of various hadrons can be coherently explained. One can see from the energy dependence of the net-baryon rapidity distribution bearden04stop that the collisions at top SPS energy are neither full transparent nor full stopping. The suppositions of nuclear stopping power in the two models are inappropriate to the nucleus-nucleus collisions at top SPS energy. For the description of the rapidity distribution for constituent quarks, one can limit the boost invariance into a finite rapidity range in the framework of Bjorken model. This modification is often used to analyze the longitudinal collectivity in hadronic level Heinz:1993 ; Mohanty2003prc . The rapidity distribution in a isotropic, thermalized fluid element moving with a rapidity $\eta$ is $\displaystyle\frac{dN_{th}}{dy}(y-\eta)$ $\displaystyle=A\,T_{f}^{3}\exp\ \bigg{(}-\frac{m}{T_{f}}cosh\,(y-\eta)\bigg{)}\times$ (7) $\displaystyle\bigg{(}\frac{m^{2}}{T_{f}^{2}}+\frac{m}{T_{f}}\frac{2}{cosh\,(y-\eta)}+\frac{2}{cosh^{2}(y-\eta)}\bigg{)}.$ The rapidity distribution of constituent quarks in the hot and dense quark matter is the longitudinal boost-invariant superposition of multiple isotropic, thermalized fluid elements $\frac{dN}{dy}=\int_{-\eta_{max}}^{\eta_{max}}\frac{dN_{th}}{dy}(y-\eta)\,d\eta,$ (8) $\eta_{max}$ is the maximal boot rapidity of fluid elements. The average longitudinal collective velocity is taken to be $<\beta_{L}>=\tanh(\eta_{max}/2)$. Here, $T_{f}$ is the temperature of the locally-thermalized hot and dense quark matter at hadronization. It is taken to be $T_{f}=170$ MeV. $m$ is the constituent mass of quarks when they evolve to the transition point. It is taken to be $340$ MeV for light quarks and $500$ MeV for strange quarks. We have mentioned in above section that the net-quarks, still carrying a fraction of initial collision energy, have a more complex evolution than hydrodynamic expansion in longitudinal axis Wolschin04RDM . Therefore, we extract the longitudinal collective flow from the rapidity distribution of newborn quarks. Since most of the data are measured in the rapidity range about [-1.5, 1.5], the rapidity spectra of constituent quarks extracted from experimental data are valid only in this region. Using above equations to fit the rapidity distribution of newborn constituent quarks in Fig. 1, we obtain $<\beta_{L}>=0.58$ for light quarks and $<\beta_{L}>=0.65$ for strange quarks. It is interesting to find that the average longitudinal collective velocity of strange quarks is obviously greater than that of light quarks. For the transverse expansion of the hot and dense quark matter, we adopt a blast-wave model proposed by Heniz Heinz:1993 within the boost-invariant scenario. The quarks and antiquarks transversely boost with a flow velocity $\beta_{r}(r)$ as a function of transverse radial position $r$. $\beta_{r}(r)$ is parameterized by the surface velocity $\beta_{s}$: $\beta_{r}(r)=\beta_{s}\,\xi^{\,n}$, where $\xi=r/R_{max}$, and $R_{max}$ is the thermal source maximum radius ($0<\xi<1$). The transverse momentum distribution of constituent quarks in the hot and dense quark matter can be equivalently described by a superposition of a set of thermalized fluid elements, each boosted with transverse rapidity $\rho=tanh^{-1}\beta_{r}$ $\dfrac{dN}{{2\pi\hskip 2.84526pt\mathrm{{p}_{T}}d{p}_{T}}}=A\int_{0}^{1}\xi\,d\xi\,m_{T}\,\times{}I_{0}\bigg{(}\dfrac{{p}_{T}\,sinh\,\rho}{T_{f}}\bigg{)}K_{1}\bigg{(}\dfrac{m_{T}\,cosh\,\rho}{T_{f}}\bigg{)}.$ (9) Here, $I_{0}$ and $K_{1}$ are the modified Bessel functions. $m_{T}=\surd{\overline{{\mathrm{{p}_{T}}}^{2}+m^{2}}}$ is the transverse mass of the constituent quark. The average transverse velocity can be written as $\langle\beta_{r}\rangle=\dfrac{\int\beta_{s}\,\xi^{\,n}\xi\,d\xi}{\int\xi\,d\xi}=\dfrac{2}{n+2}\beta_{s}.$ (10) With fixed parameter $n=0.3$, the average transverse velocity $\langle\beta_{r}\rangle$ is able to characterize the transverse collective flow of the hot and dense quark matter. Using the above equations to fit the transverse momentum distributions of the newborn quarks in Fig. 3, we obtain $\langle\beta_{r}\rangle=0.41$ for strange quarks and $\langle\beta_{r}\rangle=0.36$ for light quarks. One can see that the $\langle\beta_{r}\rangle$ of strange quarks is obviously greater than that of light quarks. Both longitudinal and transverse results at top SPS energy show that the strange constituent quarks get a stronger collective flow than the light quarks in the hydrodynamic evolution of partonic matter. By analyzing the data of multi-strange hadrons ChenJH2008 and primary charged hadrons WangYF08 , the same property is found also at top RHIC energy. It suggests that the hot and dense quark matter produced at top SPS energy undergoes a similar hydrodynamic evolution to that at RHIC energies. It is generally believed that the decoupled quark and gluon plasma (QGP) has been created at RHIC energies Gyulassy05qcdMater . This similarity of collective flow property in quark level may be regarded as a signal of QGP creation at top SPS energy. ### V.2 The enhanced strangeness An interesting phenomenon in high energy heavy ion collisions is the enhanced production of strange hadrons, which is absent in elementary particle collisions. In relativistic heavy ion collisions, enormous amounts of energy are deposited in the collision region to create a deconfined hot and dense quark matter. The multiple scatterings between partons in the hot and dense quark matter will cause the large production rate of strangeness by $gg\rightarrow s\bar{s}$ Rafelski1982 . The high strangeness of the hot and dense quark matter, after hadronization, finally leads to the abundant production of the strange hadrons. This phenomenon is regarded as a signal of QGP creation. As we know, the enhancement of strangeness production at top RHIC energy is quite obvious Abelev08enhan , and it is generally believed that the QGP has been created at RHIC energies. When the collision energy drops to the SPS and AGS energies, it is found that the strangeness production peaks at about 30A GeV and turns into a plateau at higher collision energies Alt08onset . It is an indication of the onset of deconfinement. Table 3: The strange suppression factor $\lambda_{s}$ and the calculated hadron $dN/dy$ at midrapidity in central AA collisions at different energies. The experimental data are taken from Ref. Afanasiev2002prc ; Alt2006prc ; Alt08Xi ; Arsene08Kpi ; Abelev62GeV ; Taka05sqm ; Speltz04sqm ; Adcox130GeV ; Adcox02Lam ; Adams04Mults ; Adler04Light ; Adams07hyperon . \| Pb Pb 17.3 GeV \| Au Au 62.4 GeV \| Au Au 130 GeV \| Au Au 200 GeV ---\|---\|---\|---\|--- \| data \| model \| data \| model \| data \| model \| data \| model $\pi^{+}$ \| $170.1\pm 0.7\pm 9$ \| 168.3 \| $212\pm 5.8\pm 14$ \| $211$ \| $276\pm 3\pm 35.9$ \| $268.7$ \| $286.4\pm 24.2$ \| 287.3 $\pi^{-}$ \| $175.4\pm 0.7\pm 9$ \| 183.5 \| $204\pm 7.4\pm 14$ \| $217$ \| $270\pm 3.5\pm 35.1$ \| $272.4$ \| $281.8\pm 22.8$ \| 288.3 $K^{+}$ \| $29.6\pm 0.3\pm 1.5$ \| 27.3 \| $33.35\pm 2.15$ \| $36.3$ \| $46.7\pm 1.5\pm 7$ \| $46.6$ \| $48.9\pm 6.3$ \| 48.35 $K^{-}$ \| $16.8\pm 0.2\pm 0.8$ \| 15.7 \| $28.16\pm 1.76$ \| $29.9$ \| $40.5\pm 2.3\pm 6$ \| $43.1$ \| $45.7\pm 5.2$ \| 46.48 $p$ \| $29.6\pm 0.9\pm 2.9$ \| 25.9 \| $27\pm 1.8\pm 4.6$ \| $26.17$ \| $19.3\pm 0.6\pm 3.3$ \| $16.45$ \| $18.4\pm 2.6$ \| 17.41 $\overline{p}$ \| $1.66\pm 0.17\pm 0.16$ \| 1.53 \| $11.5\pm 1.5\pm 2.9$ \| $11.15$ \| $13.7\pm 0.7\pm 2.3$ \| $11.63$ \| $13.5\pm 1.8$ \| 13.48 $\Lambda$ \| $9.5\pm 0.1\pm 1$ \| 13.3 \| $14.9\pm 0.2\pm 1.49$ \| $13.42$ \| $17.3\pm 1.8\pm 2.7$ \| $14.99$ \| $16.7\pm 0.2\pm 1.1$ \| 15.76 $\overline{\Lambda}$ \| $1.24\pm 0.03\pm 0.13$ \| 1.35 \| $8.02\pm 0.11\pm 0.8$ \| $6.77$ \| $12.7\pm 1.8\pm 2$ \| $11.4$ \| $12.7\pm 0.2\pm 0.9$ \| 12.6 $\mathrm{\Xi^{-}}$ \| $1.44\pm 0.1\pm 0.15$ \| 1.43 \| $1.64\pm 0.03\pm 0.014$ \| $1.63$ \| $2.04\pm 0.14\pm 0.2$ \| $1.99$ \| $2.17\pm 0.06\pm 0.19$ \| 2.12 $\mathrm{\overline{\Xi}^{\,{}_{+}}}$ \| $0.31\pm 0.03\pm 0.03$ \| 0.26 \| $0.989\pm 0.057\pm 0.057$ \| $0.96$ \| $1.74\pm 0.12\pm 0.17$ \| $1.67$ \| $1.83\pm 0.05\pm 0.2$ \| 1.72 $\mathrm{\Omega+{\overline{\Omega}}}$ \| \| 0.17 \| $0.356\pm 0.046\pm 0.014$ \| $0.369$ \| $0.56\pm 0.11\pm 0.05$ \| 0.551 \| $0.53\pm 0.04\pm 0.04$ \| 0.539 $\chi^{2}/ndf$ \| 10.7/7 \| 6.2/8 \| $1.6/8$ \| $0.88/8$ $\lambda_{s}$ \| $0.48\pm 0.09$ \| $0.44\pm 0.02$ \| $0.44\pm 0.04$ \| $0.42\pm 0.025$ In our model, the strangeness of the hot and dense quark matter is characterized by the suppression factor $\lambda_{s}=N_{\bar{s}}:N_{\bar{u}}=N_{\bar{s}}:N_{\bar{d}}$, i.e. the ratio of $s$ quark number to newborn $u$ (or $d$) quark number. By fitting the experimental data of identified hadrons, we use the model to extract the $\lambda_{s}$ of hot and dense quark matter at midrapidity in central AA collisions at four energies $\sqrt{s_{NN}}=17.3$, 62.4, 130 and 200 GeV, and the results are shown in Table 3. The data of midrapidity $dN/dy$ and the calculated results with minimum deviations at different collision energies are shown also. The statistical uncertainty of $\lambda_{s}$ is fixed by the twice minimum deviation. The model reproduces the hadron yield in reasonably good way, and the chi-square fit seems to indicate that with increasing collision energy the agreement with data significantly improves. One can see that $\lambda_{s}$ in such a broad energy range is nearly unchanged within statistical uncertainties, exhibiting an obvious saturation phenomenon. The results of $\lambda_{s}$ are consistent with the grand canonical limit ($\approx 0.45$) of strangeness Stock2008 . Using the Bjorken model, one can estimate that the primordial spatial energy density of the hot and dense quark matter produced in collisions at top RHIC energy is about $6.0\,GeV/fm^{3}$ Stock2008 , double of that in Pb+Pb collisions at top SPS energy. The difference in primordial energy density is large while the final strangeness is nearly equal. The hot and dense quark matter created in heavy ion collisions is shown to be very close to a perfect fluid visHD08 . It means that the local relaxation time toward to thermal equilibrium is much shorter than the macroscopic evolution time of the hot and dense quark matter. When the hot and dense quark matter evolves to the point of hadronization, the strangeness abundance should be mainly determined by the current temperature, irrelevant to the initial energy density and temperature. The same strangeness is an indication of the universal hadronization temperature for the hot and dense quark matter with low baryon chemical potential. ## VI Summary In this paper, we have systematically studied the longitudinal and transverse production of various hadrons at top SPS energy in the scheme of quark combination. Using the quark combination model, we firstly calculate the yields and rapidity distributions of various hadrons in most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV. The calculated results are in agreement with the experimental data. This indicates that the quark combination mechanism is applicable in describing the longitudinal hadron production at this collision energy. Secondly the $p_{T}$ distributions of various hadrons at top SPS energy are calculated and compared with the data. It is found that the light, single and multi-strange hadrons are well reproduced by the same quark distributions. It indicates that the hadronization of the hot and dense quark matter is a rapid process. The well reproduced baryon/meson ratios in the intermediate $p_{T}$ range at different collision energies are indicative of the universality for the quark combination mechanism. By fitting the extracted constituent quark distributions at hadronization with the hydrodynamic scenario, we further obtain the longitudinal and transverse collective flow of the hot and dense quark matter produced at top SPS energy. It is found that the strange quarks get a stronger collective flow than light quarks, which is consistent with that at RHIC energies. The strangeness in the hot and dense quark matter produced at $\sqrt{s_{NN}}=17.3$, 62.4, 130, 200 GeV are extracted. The almost unchanged strangeness may be associated with a universal hadronization temperature for the hot and dense quark matter with low baryon chemical potential. ### ACKNOWLEDGMENTS The authors thank Q. Wang, Z. T. Liang and R. Q. Wang for helpful discussions. The work is supported in part by the National Natural Science Foundation of China under the grant 10775089 and the science fund of Qufu Normal University. ## References * (1) L. Susskind, Phys. Rev. D 20 2610 (1979). * (2) F. Karsch and E. Laerman, in Quark-Gluon Plasma 3, edited by R.C. Hwa and X. N. Wang ( World Scientific 2004), p.1. * (3) M. Gyulassy and L. McLerran, Nucl. Phys. A 750, 30 (2005). * (4) X. N. Wang, Phys. Rev. C 58, 2321 (1998). * (5) B. I. Abelev, et al. (STAR Collaboration), Phys. Rev. Lett. 97, 152301 (2006). * (6) A. Adare, et al. (PHENIX Collaboration), Phys. Rev. Lett. 98, 162301 (2007). * (7) R. J. Fries, B. Müller, C. Nonaka, and S. A. Bass, Phys. Rev. Lett. 90, 202303 (2003). * (8) V. Greco, C. M. Ko, and P. Lévai, Phys. Rev. Lett. 90, 202302 (2003). * (9) R. J. Fries, B. Müller, C. Nonaka, and S. A. Bass, Phys. Rev. C 68, 044902 (2003). * (10) V. Greco, C. M. Ko, and P. Lévai, Phys. Rev. C 68, 034904 (2003). * (11) Rudolph C. Hwa and C. B. Yang, Phys. Rev. C 70, 024905 (2004). * (12) Rudolph C. Hwa and Li-Lin Zhu, Phys. Rev. C 78, 024907 (2008). * (13) C. Alt et al. (NA49 Collaboration), Phys. Rev. C 75, 044901 (2007). * (14) C. Alt et al. (NA49 Collaboration), Phys. Rev. C 77, 024903 (2008). * (15) Reinhard Stock, arXiv: nucl-ex/0807.1610v1 (2008). * (16) D. Molnar and S. A. Voloshin, Phys. Rev. Lett. 91, 092301 (2003) * (17) C. B. Yang, J.Phys. G32 L11 (2006). * (18) Q. B. Xie and X. M. Liu , Phys. Rev. D 38, 2169 (1988). * (19) F. L. Shao, Q. B. Xie and Q. Wang, Phys. Rev. C 71, 044903 (2005). * (20) F. L. Shao, T. Yao, and Q. B. Xie, Phys. Rev. C 75, 034904 (2007). * (21) T. Yao, Q. B. Xie and F. L. Shao, Chinese Physics C 32, 356 (2008). * (22) T. Yao, W. Zhou and Q. B. Xie, Phys. Rev. C 78, 064911 (2008); E. S. Chen, Sci. China, 33, 955 (1990). * (23) Y. F. Wang et al., Chinese Physcis C 32 976 (2008). * (24) T. S. Biró, Lévai and J. Zimányi, Phys. Lett. B 347 6 (1995). * (25) J. Zimányi, T. S. Biró, T. Csörgö and Lévai and , Phys. Lett. B 472 243 (2000). * (26) J. H. Chen et al., Phys. Rev. C 78, 034907 (2008). * (27) J. Rafelski and B. Müller, Phys. Rev. Lett. 48 1066 (1982). * (28) Q. B. Xie, Proceedings of the 19th International Symposium on Multiparticle Dynamics, Arles, France,1988, edited by D.Schiff and J.Tran Thanh Vann (world scitific,1988)p369. * (29) Q. Wang and Q. B. Xie, J. Phys. G 21, 897 (1995). * (30) Review of Particle Physics, Particle Data Group, Phys. Lett. B 667 1 (2008). * (31) W. Hofmann, Ann. Rev. Nucl. Part. Sci. 38, 279 (1988). * (32) S. V. Afanasiev et al. (NA49 Collaboration), Phys. Rev. C 66, 054902 (2002). * (33) H. Appelshäuser et al. (NA49 Collaboration), Phys. Rev. Lett. 82, 2471 (1999). * (34) F. Antinorik et al. (NA57 Collaboration), J. Phys. G 31 1345 (2005). * (35) C. Alt et al. (NA49 Collaboration), Phys. Rev. C 78, 044907 (2008). * (36) C. Alt et al. (NA49 Collaboration), Phys. Rev. C 73, 044910 (2006). * (37) C. Alt et al. (NA49 Collaboration), Phys. Rev. C 78, 034918 (2008). * (38) A. Mischke for the NA49 Collaboration, Nucl. Phys. A 715 453c (2003). * (39) I. G. Bearden $et~{}al$. (BRAHMS Collaboration), Phys. Rev. Lett. 93, 102301 (2004). * (40) M. Hirai, S. Kumano, M. Oka, and K. Sudoh, Phys. Rev. D 77, 017504 (2008). * (41) P. Fachini for the STAR Collaboration, Nucl. Phys. A 715 462c (2003). * (42) J. Adams et al. (STAR Collaboration), Phys. Lett. B 612 181 (2005). * (43) L. D. Landau, Izv. Akad. Nauk Ser. Fiz. 17, 51 (1953); in Collected Papers of L. D. Landau (Pergamon, Oxford, 1965), p. 665. * (44) B. Mohanty and Jan-e Alam, Phys. Rev. C 68, 064903 (2003). * (45) L. M. Satarov, I. N. Mishustin, A. V. Merdeev, and H. Stöcker, Phys. Rev. C 75, 024903 (2007). * (46) E. K. G. Sarkisyan and A. S. Sakharov, AIP Conf. Proceedings 828, 35-41 (2006);hep-ph/0410324, CERN-PH-TH/2004-213. * (47) S. V. Afanasiev et al. (NA49 Collaboration), Phys. Lett. B 491 59 (2000). * (48) C. Alt et al. (NA49 Collaboration), Phys. Rev. C 77, 034906 (2008). * (49) T. Anticic et al. (NA49 Collaboration), Phys. Rev. Lett. 93 022302 (2004). * (50) C. Alt et al. (NA49 Collaboration), Phys. Rev. Lett. 94 192301 (2005). * (51) B. Alessandro et al. (NA50 Collaboration), Phys. Lett. B 555 147 (2003). * (52) H. van Hecke, H. Sorge, and N. Xu, Phys. Rev. Lett 81 5764 (1998). * (53) J. Adams, et al. (STAR Collaboration), Phys. Rev. Lett. 98, 062301 (2007). * (54) B. I. Abelev, et al. (STAR Collaboration), Phys. Rev. Lett. 99, 112301 (2007). * (55) J. Song, F. L. Shao and X. B. Xie, Int. J. Mod. Phys. A 24 1161 (2009). * (56) B. I. Abelev, et al. (STAR Collaboration), arXiv: nucl-ex/0601042 (2006). * (57) András László and Tim Schuster for the NA49 Collaboration, Nucl. Phys. A 774 473 (2006). * (58) J. D. Bjorken, Phys. ReV. D 27 140 (1984). * (59) E. Schnedermann, J. Sollfrank, and U. Heinz, Phys. Rev. C 48, 2462 (1993). * (60) G. Wolschin, Phys. Rev. C 69, 024906 (2004). * (61) M. Gyulassy and L. McLerran, Nucl. Phys. A 750 30 (2005). * (62) B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C 77 044908 (2008). * (63) I. C. Arsene (for BRAHMS Collaboration) J. Phys. G 35 104056 (2008). * (64) B.I. Abelev et al. (STAR Collaboration), Phys. Lett. B 655 104 (2007). * (65) J Takahashi (for the STAR Collaboration), J. Phys. G 31 s1061 (2005). * (66) Jeff Speltz (for the STAR Collaboration),J. Phys. G 31 s1025 (2005). * (67) K. Adcox et al. (PHENIX Collaboration), Phys. Rev. C 69 024904 (2004). * (68) K. Adcox et al. (PHENIX Collaboration), Phys. Rev. Lett. 89 092302 (2002). * (69) J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 92 182301 (2004). * (70) S. S. Adler et al. (PHENIX Collaboration), Phys. Rev. C 69 034909 (2004). * (71) Huichao Song, and U. Heinz, Phys. Rev. C 77 064901 (2008).	arxiv-papers	2009-02-14T06:46:50	2024-09-04T02:49:00.559437	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Chang-en Shao, Jun Song, Feng-lan Shao, Qu-bing Xie", "submitter": "FengLan Shao", "url": "https://arxiv.org/abs/0902.2435" }
0902.2456	Permutability of Backlund Transformation for $N=1$ Supersymmetric Sinh-Gordon 111 key words:Backlund Transformation, Supersymmetric sinh-Gordon, non linear superposition formula Pacs 02.30.Ik J.F. Gomes222corresponding author jfg@ift.unesp.br, L.H. Ymai and A.H. Zimerman Instituto de Física Teórica-UNESP Rua Pamplona 145 fax (55)11 31779080 01405-900 São Paulo, Brazil ###### Abstract The permutability of two Backlund transformations is employed to construct a non linear superposition formula to generate a class of solutions for the $N=1$ super sinh-Gordon model. Backlund transformations (BT) relating two different soliton solutions are known to be characteristic of certain class of nonlinear equations. A remarkable consequence is that from a particular soliton solution, a second solution can be generated by Backlund transformation. This second solution, in turn, generates a third one and such structure allows to construct conditions for the permutability of two sequences of BT. The study of superposition principle for soliton solutions of the sine-Gordon equation was employed to show that the order of two BT is, in fact, irrelevant [1]. Such condition became known as the permutability theorem as was applied to the KdV and for the $N=1$ super KdV equations in [2] and in [3] respectively. Soliton solutions for the $N=1$ super sine-Gordon were obtained in [4] from the super Hirota’s formalism and in [5] using dressing transformation and vertex operators. Backlund solutions for super mKdV were considered in [6]. In this paper we use the superfield approach for the BT as proposed in [7] to derive a closed algebraic superposition formula for soliton solutions of the $N=1$ super sinh-Gordon model assuming that two sucessive BT commute. The model is described by the following equation of motion written within the superfield formalism [7] $\displaystyle D_{x}D_{t}\Phi=2i\sinh\Phi,$ (1) where the bosonic superfield $\Phi$ is given in components by $\displaystyle\Phi=\phi+\theta_{1}\bar{\psi}+i\theta_{2}\psi-\theta_{1}\theta_{2}2i\sinh\phi,$ (2) where $\theta_{1}$ and $\theta_{2}$ are Grassmann variables (i.e. $\theta_{1}^{2}=\theta_{2}^{2}=0$ and $\theta_{1}\theta_{2}+\theta_{2}\theta_{1}=0$ ) The superderivatives $\displaystyle D_{x}=\partial_{\theta_{1}}+\theta_{1}\partial_{x},\qquad D_{t}=\partial_{\theta_{2}}+\theta_{2}\partial_{t},$ (3) satisfy $\displaystyle D_{x}^{2}=\partial_{x},\qquad D_{t}^{2}=\partial_{t},\qquad D_{x}D_{t}=-D_{t}D_{x}.$ (4) The Backlund transformation for eqn. (1) is given by [7] $\displaystyle D_{x}(\Phi_{0}-\Phi_{1})$ $\displaystyle=$ $\displaystyle-\frac{4i}{\beta_{1}}f_{0,1}\cosh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right),$ (5) $\displaystyle D_{t}(\Phi_{0}+\Phi_{1})$ $\displaystyle=$ $\displaystyle 2\beta_{1}f_{0,1}\cosh\left(\frac{\Phi_{0}-\Phi_{1}}{2}\right),$ (6) where $\beta_{1}$ is an arbitrary parameter (spectral parameter) and the auxiliary fermionic superfield $f_{0,1}$ (i.e. $f_{0,1}^{2}=0$) $\displaystyle f_{0,1}=f_{1}^{(0,1)}+\theta_{1}b_{1}^{(0,1)}+\theta_{2}b_{2}^{(0,1)}+\theta_{1}\theta_{2}f_{2}^{(0,1)},$ (7) satisfy $\displaystyle D_{x}f_{0,1}=\frac{2i}{\beta_{1}}\sinh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right),\qquad D_{t}f_{0,1}=\beta_{1}\sinh\left(\frac{\Phi_{0}-\Phi_{1}}{2}\right).$ (8) Consider now two successive Backlund transformations. The first one involving superfields $\Phi_{0}$ and $\Phi_{1}$ and the parameter $\beta_{1}$ whilst the second involves $\Phi_{1}$ and $\Phi_{3}$ with $\beta_{2}$. The Permutability theorem states that the order in which such Backlund transformations are employed is irrelevant, i.e. we might as well consider the first involving $\Phi_{0}$ and $\Phi_{2}$ with $\beta_{1}$ followed by a second, involving $\Phi_{2}$ and $\Phi_{3}$ with $\beta_{2}$. Similar to (5) we have, $\displaystyle D_{x}(\Phi_{0}-\Phi_{1})$ $\displaystyle=$ $\displaystyle-\frac{4i}{\beta_{1}}f_{0,1}\cosh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right),$ (9) $\displaystyle D_{x}(\Phi_{1}-\Phi_{3})$ $\displaystyle=$ $\displaystyle-\frac{4i}{\beta_{2}}f_{1,3}\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right),$ (10) $\displaystyle D_{x}(\Phi_{0}-\Phi_{2})$ $\displaystyle=$ $\displaystyle-\frac{4i}{\beta_{2}}f_{0,2}\cosh\left(\frac{\Phi_{0}+\Phi_{2}}{2}\right),$ (11) $\displaystyle D_{x}(\Phi_{2}-\Phi_{3})$ $\displaystyle=$ $\displaystyle-\frac{4i}{\beta_{1}}f_{2,3}\cosh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right).$ (12) and from (8), $\displaystyle D_{x}f_{0,1}$ $\displaystyle=$ $\displaystyle\frac{2i}{\beta_{1}}\sinh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right),$ (13) $\displaystyle D_{x}f_{1,3}$ $\displaystyle=$ $\displaystyle\frac{2i}{\beta_{2}}\sinh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right),$ (14) $\displaystyle D_{x}f_{0,2}$ $\displaystyle=$ $\displaystyle\frac{2i}{\beta_{2}}\sinh\left(\frac{\Phi_{0}+\Phi_{2}}{2}\right),$ (15) $\displaystyle D_{x}f_{2,3}$ $\displaystyle=$ $\displaystyle\frac{2i}{\beta_{1}}\sinh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right).$ (16) The equality of the sum of equations (9) and (10) with the sum of (11) and (12) yields the following relation $\displaystyle\frac{1}{\beta_{1}}f_{0,1}\cosh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right)+\frac{1}{\beta_{2}}f_{1,3}\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right)$ $\displaystyle=\frac{1}{\beta_{2}}f_{0,2}\cosh\left(\frac{\Phi_{0}+\Phi_{2}}{2}\right)+\frac{1}{\beta_{1}}f_{2,3}\cosh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right).$ (17) Analogously from (6) we find $\displaystyle D_{t}(\Phi_{0}+\Phi_{1})$ $\displaystyle=$ $\displaystyle 2\beta_{1}f_{0,1}\cosh\left(\frac{\Phi_{0}-\Phi_{1}}{2}\right),$ (18) $\displaystyle D_{t}(\Phi_{1}+\Phi_{3})$ $\displaystyle=$ $\displaystyle 2\beta_{2}f_{1,3}\cosh\left(\frac{\Phi_{1}-\Phi_{3}}{2}\right),$ (19) $\displaystyle D_{t}(\Phi_{0}+\Phi_{2})$ $\displaystyle=$ $\displaystyle 2\beta_{2}f_{0,2}\cosh\left(\frac{\Phi_{0}-\Phi_{2}}{2}\right),$ (20) $\displaystyle D_{t}(\Phi_{2}+\Phi_{3})$ $\displaystyle=$ $\displaystyle 2\beta_{1}f_{2,3}\cosh\left(\frac{\Phi_{2}-\Phi_{3}}{2}\right).$ (21) Equating now the difference of the first two, (18) and (19) and the last two equations, (20) and (21) we get $\displaystyle\beta_{1}f_{0,1}\cosh\left(\frac{\Phi_{0}-\Phi_{1}}{2}\right)-\beta_{2}f_{1,3}\cosh\left(\frac{\Phi_{1}-\Phi_{3}}{2}\right)$ $\displaystyle=\beta_{2}f_{0,2}\cosh\left(\frac{\Phi_{0}-\Phi_{2}}{2}\right)-\beta_{1}f_{2,3}\cosh\left(\frac{\Phi_{2}-\Phi_{3}}{2}\right).$ (22) Solving (17) and (22), for $f_{1,3}$ and $f_{2,3}$, we get $\displaystyle f_{1,3}$ $\displaystyle=$ $\displaystyle\Lambda_{1,3}^{(1)}f_{0,1}+\Lambda_{1,3}^{(2)}f_{0,2},$ (23) $\displaystyle f_{2,3}$ $\displaystyle=$ $\displaystyle\Lambda_{2,3}^{(1)}f_{0,1}+\Lambda_{2,3}^{(2)}f_{0,2},$ (24) where the coefficients $\Lambda$ are given as $\displaystyle\Lambda_{1,3}^{(1)}=-\beta_{1}\beta_{2}\frac{\left[\cosh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right)\cosh\left(\frac{\Phi_{2}-\Phi_{3}}{2}\right)+\cosh\left(\frac{\Phi_{0}-\Phi_{1}}{2}\right)\cosh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right)\right]}{\left[\cosh\left(\frac{\Phi_{2}-\Phi_{3}}{2}\right)\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right)\beta_{1}^{2}-\cosh\left(\frac{\Phi_{1}-\Phi_{3}}{2}\right)\cosh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right)\beta_{2}^{2}\right]},$ $\displaystyle\Lambda_{1,3}^{(2)}=\frac{\left[\cosh\left(\frac{\Phi_{0}+\Phi_{2}}{2}\right)\cosh\left(\frac{\Phi_{2}-\Phi_{3}}{2}\right)\beta_{1}^{2}+\cosh\left(\frac{\Phi_{0}-\Phi_{2}}{2}\right)\cosh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right)\beta_{2}^{2}\right]}{\left[\cosh\left(\frac{\Phi_{2}-\Phi_{3}}{2}\right)\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right)\beta_{1}^{2}-\cosh\left(\frac{\Phi_{1}-\Phi_{3}}{2}\right)\cosh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right)\beta_{2}^{2}\right]},$ $\displaystyle\Lambda_{2,3}^{(1)}=-\frac{\left[\cosh\left(\frac{\Phi_{0}-\Phi_{1}}{2}\right)\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right)\beta_{1}^{2}+\cosh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right)\cosh\left(\frac{\Phi_{1}-\Phi_{3}}{2}\right)\beta_{2}^{2}\right]}{\left[\cosh\left(\frac{\Phi_{2}-\Phi_{3}}{2}\right)\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right)\beta_{1}^{2}-\cosh\left(\frac{\Phi_{1}-\Phi_{3}}{2}\right)\cosh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right)\beta_{2}^{2}\right]},$ $\displaystyle\Lambda_{2,3}^{(2)}=\beta_{1}\beta_{2}\frac{\left[\cosh\left(\frac{\Phi_{0}+\Phi_{2}}{2}\right)\cosh\left(\frac{\Phi_{1}-\Phi_{3}}{2}\right)+\cosh\left(\frac{\Phi_{0}-\Phi_{2}}{2}\right)\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right)\right]}{\left[\cosh\left(\frac{\Phi_{2}-\Phi_{3}}{2}\right)\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right)\beta_{1}^{2}-\cosh\left(\frac{\Phi_{1}-\Phi_{3}}{2}\right)\cosh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right)\beta_{2}^{2}\right]},$ (25) Acting with $D_{x}$ in eqn. (9)-(12) and using (13)- (16), we find $\displaystyle\partial_{x}(\Phi_{0}-\Phi_{1})$ $\displaystyle=$ $\displaystyle\frac{4}{\beta_{1}^{2}}\sinh(\Phi_{0}+\Phi_{1})+\frac{4i}{\beta_{1}}f_{0,1}D_{x}\left[\cosh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right)\right].$ (26) $\displaystyle\partial_{x}(\Phi_{1}-\Phi_{3})$ $\displaystyle=$ $\displaystyle\frac{4}{\beta_{2}^{2}}\sinh(\Phi_{1}+\Phi_{3})+\frac{4i}{\beta_{2}}f_{1,3}D_{x}\left[\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right)\right].$ (27) $\displaystyle\partial_{x}(\Phi_{0}-\Phi_{2})$ $\displaystyle=$ $\displaystyle\frac{4}{\beta_{2}^{2}}\sinh(\Phi_{0}+\Phi_{2})+\frac{4i}{\beta_{2}}f_{0,2}D_{x}\left[\cosh\left(\frac{\Phi_{0}+\Phi_{2}}{2}\right)\right].$ (28) $\displaystyle\partial_{x}(\Phi_{2}-\Phi_{3})$ $\displaystyle=$ $\displaystyle\frac{4}{\beta_{1}^{2}}\sinh(\Phi_{2}+\Phi_{3})+\frac{4i}{\beta_{1}}f_{2,3}D_{x}\left[\cosh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right)\right].$ (29) Notice that equations (26)-(29) correspond formally to the pure bosonic case when the terms proportional to the fermionic superfields $f_{i,j}$ are neglected. Equating the R.H.S. of the sum of eqns. (26) with (27) and (28) with (29) we find $\displaystyle\frac{1}{\beta_{1}^{2}}\left(\sinh(\Phi_{0}+\Phi_{1})-\sinh(\Phi_{2}+\Phi_{3})\right)+\frac{1}{\beta_{2}^{2}}\left(\sinh(\Phi_{1}+\Phi_{3})-\sinh(\Phi_{0}+\Phi_{2})\right)$ (30) $\displaystyle=$ $\displaystyle-\frac{i}{\beta_{1}}f_{0,1}D_{x}\left[\cosh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right)\right]-\frac{i}{\beta_{2}}f_{1,3}D_{x}\left[\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right)\right]$ $\displaystyle+$ $\displaystyle\frac{i}{\beta_{2}}f_{0,2}D_{x}\left[\cosh\left(\frac{\Phi_{0}+\Phi_{2}}{2}\right)\right]+\frac{i}{\beta_{1}}f_{2,3}D_{x}\left[\cosh\left(\frac{\Phi_{2}+\Phi_{3}}{2}\right)\right].$ Factorizing the L.H.S. of (30) $\displaystyle\frac{1}{\beta_{1}^{2}}\left(\sinh(\Phi_{0}+\Phi_{1})-\sinh(\Phi_{2}+\Phi_{3})\right)+\frac{1}{\beta_{2}^{2}}\left(\sinh(\Phi_{1}+\Phi_{3})-\sinh(\Phi_{0}+\Phi_{2})\right)$ $\displaystyle=$ $\displaystyle 2\cosh({{\Phi_{0}+\Phi_{1}+\Phi_{2}+\Phi_{3}}\over{2}})\left({{1}\over{\beta_{1}^{2}}}\sinh({{\Phi_{0}+\Phi_{1}-\Phi_{2}-\Phi_{3}}\over{2}})+{{1}\over{\beta_{2}^{2}}}\sinh({{-\Phi_{0}+\Phi_{1}-\Phi_{2}+\Phi_{3}}\over{2}})\right)$ $\displaystyle=2\cosh({{\Phi_{0}+\Phi_{1}+\Phi_{2}+\Phi_{3}}\over{2}})\left(\sinh({{\Phi_{0}-\Phi_{3}}\over{2}})\cosh({{\Phi_{1}-\Phi_{2}}\over{2}}){{(\beta_{1}^{2}-\beta_{2}^{2})}\over{\beta_{1}^{2}\beta_{2}^{2}}}\right.$ $\displaystyle\left.+\sinh({{\Phi_{1}-\Phi_{2}}\over{2}})\cosh({{\Phi_{0}-\Phi_{3}}\over{2}})({{{\beta_{1}^{2}+\beta_{2}^{2}}\over{\beta_{1}^{2}\beta_{2}^{2}}}})\right).$ (31) The vanishing of this expression leads to $\displaystyle\Phi_{3}-\Phi_{0}=2arctanh\left(\delta\;\tanh({{\Phi_{1}-\Phi_{2}}\over{2}})\right)\equiv\Gamma(\Phi_{1}-\Phi_{2}),\qquad\delta=-\left({{{\beta_{1}^{2}+\beta_{2}^{2}}\over{\beta_{1}^{2}-\beta_{2}^{2}}}}\right).$ (32) For the more general case taking into account the fermionic superfields $f_{i,j}$ we propose the following ansatz, $\displaystyle\Phi_{3}=\Phi_{0}+\Gamma(\Phi_{1}-\Phi_{2})+\Delta,$ (33) where $\Delta$ is a bosonic superfield proportional to the product $f_{0,1}f_{0,2}$, i.e., $\displaystyle\Delta=\lambda f_{0,1}f_{0,2},\qquad\lambda=\lambda(\Phi_{1}-\Phi_{2}).$ (34) Due to the fact that $\Delta^{2}=0$, eqns. (25) take the general form $\displaystyle\Lambda_{1,3}^{(1)}$ $\displaystyle=$ $\displaystyle-a+c_{1}f_{0,1}f_{0,2},$ $\displaystyle\Lambda_{1,3}^{(2)}$ $\displaystyle=$ $\displaystyle-b+c_{2}f_{0,1}f_{0,2},$ $\displaystyle\Lambda_{2,3}^{(1)}$ $\displaystyle=$ $\displaystyle b+c_{3}f_{0,1}f_{0,2},$ $\displaystyle\Lambda_{2,3}^{(2)}$ $\displaystyle=$ $\displaystyle a+c_{4}f_{0,1}f_{0,2},$ (35) where $c_{i}=c_{i}(\Phi_{0},\Phi_{1},\Phi_{2}),i=1,\cdots 4$ do not contribute to eqns (23) and (24). Substituting (33) and (34) in (25) we obtain $\displaystyle a=\frac{\delta_{1}}{\sqrt{1-\delta^{2}\tanh^{2}\left(\frac{\Phi_{1}-\Phi_{2}}{2}\right)}},\qquad b=\frac{\delta\,\mathrm{sech}\left(\frac{\Phi_{1}-\Phi_{2}}{2}\right)}{\sqrt{1-\delta^{2}\tanh^{2}\left(\frac{\Phi_{1}-\Phi_{2}}{2}\right)}}$ (36) where $\delta_{1}=\frac{2\beta_{1}\beta_{2}}{(\beta_{1}^{2}-\beta_{2}^{2})}$. Inserting (35) in (23) and (24) we find, since $f_{01}^{2}=f_{02}^{2}=0$, $\displaystyle f_{1,3}$ $\displaystyle=$ $\displaystyle-af_{0,1}-bf_{0,2},$ $\displaystyle f_{2,3}$ $\displaystyle=$ $\displaystyle bf_{0,1}+af_{0,2}.$ (37) From the fact that $\delta^{2}-\delta_{1}^{2}=1$, it follows that $\displaystyle f_{1,3}f_{2,3}=f_{0,1}f_{0,2}.$ Adding (9) and (10) we find $\displaystyle D_{x}(\Phi_{3}-\Phi_{0})=\frac{4i}{\beta_{1}}f_{0,1}\cosh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right)+\frac{4i}{\beta_{2}}f_{1,3}\cosh\left(\frac{\Phi_{1}+\Phi_{3}}{2}\right).$ (38) Substituting (33) and (37) in (38) we find $\displaystyle f_{0,1}\Sigma_{1}+f_{0,2}\Sigma_{2}+(D_{x}\lambda)f_{0,1}f_{0,2}=0,$ (39) where $\displaystyle\Sigma_{1}$ $\displaystyle=$ $\displaystyle\partial_{\xi}\Gamma_{\|_{\xi=(\Phi_{1}-\Phi_{2})}}\frac{4i}{\beta_{1}}\cosh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right)-\lambda\frac{2i}{\beta_{2}}\sinh\left(\frac{\Phi_{0}+\Phi_{2}}{2}\right)$ $\displaystyle-\frac{4i}{\beta_{1}}\cosh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right)-\Lambda_{1,3}^{(1)}\frac{4i}{\beta_{1}}\cosh\left(\frac{\Phi_{0}+\Phi_{1}+\Gamma}{2}\right),$ $\displaystyle\Sigma_{2}$ $\displaystyle=$ $\displaystyle-\partial_{\xi}\Gamma_{\|_{\xi=(\Phi_{1}-\Phi_{2})}}\frac{4i}{\beta_{2}}\cosh\left(\frac{\Phi_{0}+\Phi_{2}}{2}\right)+\lambda\frac{2i}{\beta_{1}}\sinh\left(\frac{\Phi_{0}+\Phi_{1}}{2}\right)$ $\displaystyle-\Lambda_{1,3}^{(2)}\frac{4i}{\beta_{1}}\cosh\left(\frac{\Phi_{0}+\Phi_{1}+\Gamma}{2}\right).$ The last term in (39) vanishes since $\displaystyle D_{x}\lambda=\partial_{\xi}\lambda_{\|_{\xi=(\Phi_{1}-\Phi_{2})}}D_{x}(\Phi_{1}-\Phi_{2}),$ and $D_{x}(\Phi_{1}-\Phi_{2})$, from (9) and (11), is proportional to $f_{0,1}$ and $f_{0,2}$. Since $f_{0,1}$ and $f_{0,2}$ are independent, (39) yields a pair of algebraic equations for $\lambda$, i.e. $\Sigma_{1}=\Sigma_{2}=0$ which are satisfied by $\displaystyle\lambda=-\frac{4\sinh\left(\frac{\Phi_{1}-\Phi_{2}}{2}\right)\beta_{1}\beta_{2}(\beta_{1}^{2}+\beta_{2}^{2})}{\beta_{1}^{4}+\beta_{2}^{4}-2\cosh(\Phi_{1}-\Phi_{2})\beta_{1}^{2}\beta_{2}^{2}}.$ (40) and therefore $\displaystyle\Phi_{3}=\Phi_{0}+2\,\mathrm{Arctanh}\left[\left(\frac{\beta_{2}^{2}+\beta_{1}^{2}}{\beta_{2}^{2}-\beta_{1}^{2}}\right)\tanh\left(\frac{\Phi_{1}-\Phi_{2}}{2}\right)e^{\Omega f_{0,1}f_{0,2}}\right],$ (41) where $\displaystyle\Omega=\delta_{1}\mathrm{sech}\left(\frac{\Phi_{1}-\Phi_{2}}{2}\right).$ In order to write eqn. (41) in components, we need to specify the superfields $f_{0,1},f_{0,2}$ in terms of the components of $\Phi_{0},\Phi_{1},\Phi_{2}$. These are given by eqns. (5.91), (5.94), (5.96) and (5.98) in the the appendix of ref. [8]. Introducing $\sigma_{k}=-\frac{2}{\beta_{k}^{2}}$ $(k=1,2)$, the solution (41) in components according to (2) becomes $\displaystyle\phi_{3}$ $\displaystyle=$ $\displaystyle\phi_{0}+2\,\mathrm{Arctanh}\left[\delta\tanh\left(\frac{\phi_{1}-\phi_{2}}{2}\right)\right]$ (42) $\displaystyle-\frac{\Delta_{2}}{8\sqrt{\sigma_{1}\sigma_{2}}}\left[\frac{\bar{\psi}_{0}(\bar{\psi}_{1}-\bar{\psi}_{2})+\bar{\psi}_{1}\bar{\psi}_{2}}{\cosh\left(\frac{\phi_{0}+\phi_{1}}{2}\right)\cosh\left(\frac{\phi_{0}+\phi_{2}}{2}\right)}\right],$ $\displaystyle\bar{\psi}_{3}$ $\displaystyle=$ $\displaystyle\bar{\psi}_{0}+\Delta_{1}(\bar{\psi}_{1}-\bar{\psi}_{2})$ (43) $\displaystyle-\frac{\Delta_{2}}{2}\left[\sqrt{\frac{\sigma_{2}}{\sigma_{1}}}\frac{\sinh\left(\frac{\phi_{0}+\phi_{2}}{2}\right)}{\cosh\left(\frac{\phi_{0}+\phi_{1}}{2}\right)}(\bar{\psi}_{0}-\bar{\psi}_{1})-\sqrt{\frac{\sigma_{1}}{\sigma_{2}}}\frac{\sinh\left(\frac{\phi_{0}+\phi_{1}}{2}\right)}{\cosh\left(\frac{\phi_{0}+\phi_{2}}{2}\right)}(\bar{\psi}_{0}-\bar{\psi}_{2})\right],$ $\displaystyle\psi_{3}$ $\displaystyle=$ $\displaystyle\psi_{0}+\Delta_{1}(\psi_{1}-\psi_{2})$ (44) $\displaystyle-\frac{\Delta_{2}}{2}\left[\sqrt{\frac{\sigma_{2}}{\sigma_{1}}}\frac{\sinh\left(\frac{\phi_{0}-\phi_{1}}{2}\right)}{\cosh\left(\frac{\phi_{0}-\phi_{2}}{2}\right)}(\psi_{0}+\psi_{2})-\sqrt{\frac{\sigma_{1}}{\sigma_{2}}}\frac{\sinh\left(\frac{\phi_{0}-\phi_{2}}{2}\right)}{\cosh\left(\frac{\phi_{0}-\phi_{1}}{2}\right)}(\psi_{0}+\psi_{1})\right],$ where $\displaystyle\Delta_{1}$ $\displaystyle=$ $\displaystyle\frac{2}{\sinh\left(\phi_{1}-\phi_{2}\right)}\left[\frac{\delta\tanh\left(\frac{\phi_{1}-\phi_{2}}{2}\right)}{1-\delta^{2}\tanh^{2}\left(\frac{\phi_{1}-\phi_{2}}{2}\right)}\right],$ $\displaystyle\Delta_{2}$ $\displaystyle=$ $\displaystyle\frac{A\sinh\left(\frac{\phi_{1}-\phi_{2}}{2}\right)}{B-\sinh^{2}\left(\frac{\phi_{1}-\phi_{2}}{2}\right)},$ $\displaystyle\delta$ $\displaystyle=$ $\displaystyle\frac{\sigma_{1}+\sigma_{2}}{\sigma_{1}-\sigma_{2}},\qquad A=\frac{\sigma_{1}+\sigma_{2}}{\sqrt{\sigma_{1}\sigma_{2}}},\qquad B=\frac{(\sigma_{1}-\sigma_{2})^{2}}{4\,\sigma_{1}\sigma_{2}}.$ (45) One soliton Solution The Backlund equations in components for $\Phi_{0}=0$ take the form )from (7): $\displaystyle\partial_{x}\phi_{1}$ $\displaystyle=$ $\displaystyle 2\,\sigma_{1}\sinh\phi_{1},\qquad\partial_{t}\phi_{1}=\frac{2}{\sigma_{1}}\sinh\phi_{1},$ (46) $\displaystyle\bar{\psi}_{1}$ $\displaystyle=$ $\displaystyle 2\sqrt{2\sigma_{1}}\cosh\left(\phi_{1}/2\right)f_{1}^{(0,1)},\qquad\psi_{1}=2\sqrt{{{2}\over{\sigma_{1}}}}\cosh\left(\phi_{1}/2\right)f_{1}^{(0,1)}$ (47) $\displaystyle\partial_{x}f_{1}^{(0,1)}$ $\displaystyle=$ $\displaystyle\sqrt{{{\sigma_{1}}\over{2}}}\cosh\left(\phi_{1}/2\right)\bar{\psi}_{1},\qquad\partial_{t}f_{1}^{(0,1)}={{1}\over{\sqrt{2\sigma_{1}}}}\cosh\left(\phi_{1}/2\right)\psi_{1}$ (48) By direct integration we obtain $\displaystyle\phi_{1}$ $\displaystyle=$ $\displaystyle\ln\left(\frac{1+E_{1}}{1-E_{1}}\right),\qquad E_{1}=b_{1}\exp\left(2\sigma_{1}x+2\sigma_{1}^{-1}t\right),$ (49) $\displaystyle\bar{\psi}_{1}$ $\displaystyle=$ $\displaystyle\epsilon_{1}\frac{a_{1}}{b_{1}}E_{1}\left(\frac{1}{1+E_{1}}+\frac{1}{1-E_{1}}\right),\qquad\psi_{1}=\frac{\bar{\psi}_{1}}{\sigma_{1}},$ (50) where $a_{1}$ and $b_{1}$ are arbitrary constants and $\epsilon_{1}$ is a fermionic parameter. Two soliton Solution Choosing $\Phi_{0}=0$ and $\Phi_{1},\Phi_{2}$ as one soliton solutions with components $\displaystyle\phi_{k}$ $\displaystyle=$ $\displaystyle\ln\left(\frac{1+E_{k}}{1-E_{k}}\right),\qquad E_{k}=b_{k}\exp\left(2\sigma_{k}x+2\sigma_{k}^{-1}t\right),$ $\displaystyle\bar{\psi}_{k}$ $\displaystyle=$ $\displaystyle\epsilon_{k}\frac{a_{k}}{b_{k}}E_{k}\left(\frac{1}{1+E_{k}}+\frac{1}{1-E_{k}}\right),$ $\displaystyle\psi_{k}$ $\displaystyle=$ $\displaystyle\frac{\bar{\psi}_{k}}{\sigma_{k}},\qquad k=1,2$ where $a_{k}$, $b_{k}$ are arbitrary constants and $\epsilon_{k}$ fermiônic parameters we find from (41), $\displaystyle\phi_{3}$ $\displaystyle=$ $\displaystyle 2\,\mathrm{Arctanh}\left[\delta\tanh\left(\frac{\phi_{1}-\phi_{2}}{2}\right)\right]-\frac{\Delta_{2}}{8\sqrt{\sigma_{1}\sigma_{2}}}\left[\frac{\bar{\psi}_{1}\bar{\psi}_{2}}{\cosh\left(\frac{\phi_{1}}{2}\right)\cosh\left(\frac{\phi_{2}}{2}\right)}\right],$ (51) $\displaystyle\bar{\psi}_{3}$ $\displaystyle=$ $\displaystyle\left[\Delta_{1}+\frac{\Delta_{2}}{2}\sqrt{\frac{\sigma_{2}}{\sigma_{1}}}\frac{\sinh\left(\frac{\phi_{2}}{2}\right)}{\cosh\left(\frac{\phi_{1}}{2}\right)}\right]\bar{\psi}_{1}-\left[\Delta_{1}+\frac{\Delta_{2}}{2}\sqrt{\frac{\sigma_{1}}{\sigma_{2}}}\frac{\sinh\left(\frac{\phi_{1}}{2}\right)}{\cosh\left(\frac{\phi_{2}}{2}\right)}\right]\bar{\psi}_{2},$ (52) $\displaystyle\psi_{3}$ $\displaystyle=$ $\displaystyle\left[\Delta_{1}-\frac{\Delta_{2}}{2}\sqrt{\frac{\sigma_{1}}{\sigma_{2}}}\frac{\sinh\left(\frac{\phi_{2}}{2}\right)}{\cosh\left(\frac{\phi_{1}}{2}\right)}\right]\psi_{1}-\left[\Delta_{1}-\frac{\Delta_{2}}{2}\sqrt{\frac{\sigma_{2}}{\sigma_{1}}}\frac{\sinh\left(\frac{\phi_{1}}{2}\right)}{\cosh\left(\frac{\phi_{2}}{2}\right)}\right]\psi_{2},$ (53) By rescaling of parameters $\displaystyle\sigma_{k}\to\gamma_{k},\qquad\epsilon_{k}\to c_{k}\qquad k=1,2$ $\displaystyle b_{1}\to\frac{b_{1}}{2}\left(\frac{\gamma_{1}-\gamma_{2}}{\gamma_{1}+\gamma_{2}}\right),\qquad b_{2}\to-\frac{b_{2}}{2}\left(\frac{\gamma_{1}-\gamma_{2}}{\gamma_{1}+\gamma_{2}}\right),$ $\displaystyle a_{1}\to-\gamma_{1}\left(\frac{\gamma_{1}-\gamma_{2}}{\gamma_{1}+\gamma_{2}}\right),\qquad a_{2}\to\gamma_{2}\left(\frac{\gamma_{1}-\gamma_{2}}{\gamma_{1}+\gamma_{2}}\right).$ we verify that solution (51) and (52) coincide precisely with solution (3.28)-(3.29) of ref. [5] (after choosing $\gamma_{3}=-\gamma_{1}$ and $\gamma_{4}=-\gamma_{2}$). Acknowledgements LHY acknowledges support from Fapesp, JFG and AHZ thank CNPq for partial support. ## References * [1] C. Rogers, in “Soliton Theory:a survey of results”, Ed. A. Fordy, Manchester Univ. Press (1990) * [2] H. Wahlquist and F. Estabrook, Phys. Rev. Lett. 31 (1973) 1386 * [3] Q.P. Liu and Y.F. Xie Phys. Lett. 325A (2004) 139; Q.P. Liu and Xing-Biao Hu J. Physics A38 (2005) 6371 * [4] B. Grammaticos, A. Ramani and A.S. Carstea, J. Physics A34 (2001) 4881 * [5] J.F. Gomes, L. H. Ymai and A.H. Zimerman, Phys. Lett. 359A (2006) 630, hep-th/0607107 * [6] Q.P. Liu and Meng-Xia Zhang, Nonlinearity 18 (2005) 1597 * [7] M. Chaichian and P. Kulish, Phys. Lett. 78B (1978) 413 * [8] J.F. Gomes, L. H. Ymai and A.H. Zimerman, J. Physics A39 (2006) 7471, hep-th/0601014	arxiv-papers	2009-02-14T11:34:43	2024-09-04T02:49:00.565541	{ "license": "Public Domain", "authors": "J.F. Gomes, L.H. Ymai and A.H. Zimerman", "submitter": "Jose Francisco Gomes", "url": "https://arxiv.org/abs/0902.2456" }
0902.2474	# A mixing-like property and inexistence of invariant foliations for minimal diffeomorphisms of the 2-torus Alejandro Kocsard Universidade Federal Fluminense, Instituto de Matemática, Rua Mário Santos Braga S/N, 24020-140 Niteroi, RJ, Brasil alejo@impa.br and Andrés Koropecki Universidade Federal Fluminense, Instituto de Matemática, Rua Mário Santos Braga S/N, 24020-140 Niteroi, RJ, Brasil koro@mat.uff.br ###### Abstract. We consider diffeomorphisms in $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$, the $C^{\infty}$-closure of the conjugancy class of translations of $\mathbb{T}^{2}$. By a theorem of Fathi and Herman, a generic diffeomorphism in that space is minimal and uniquely ergodic. We define a new mixing-type property, which takes into account the “directions” of mixing, and we prove that generic elements of $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ satisfy this property. As a consequence, we obtain a residual set of strictly ergodic diffeomorphisms without invariant foliations of any kind. We also obtain an analytic version of these results. The authors were supported by CNPq-Brasil. ## 1\. Introduction In [FH77], Fathi and Herman combined generic arguments with the so-called _fast approximation by conjugations_ method of Anosov and Katok [AK70], to study a particular class of diffeomorphisms of a compact manifold: the $C^{\infty}$-closure of the set of diffeomorphisms which are $C^{\infty}$ conjugate to elements of a locally free $\mathbb{T}^{1}$-action on the manifold. They proved that a generic element of that space is minimal and uniquely ergodic (i.e. there is a residual subset of such diffeomorphisms), in particular proving that every compact manifold admitting a locally free $\mathbb{T}^{1}$-action supports a minimal and uniquely ergodic diffeomorphism. Surprisingly, the space studied by Fathi and Herman contains many elements with unexpected dynamical properties; for example, a generic diffeomorphism in that space is weak mixing [Her92, FS05], and if the underlying space is $\mathbb{T}^{n}$, the action of its derivative on the unit tangent bundle is minimal [Kor07]. For a very complete survey on the technique of Anosov-Katok and its variations, see [FK04]. In [FS05], Fayad and Saprikyna use a real analytic version of this method to construct minimal weak mixing diffeomorphisms. In this short article, we restrict our attention to diffeomorphisms of $\mathbb{T}^{2}$. In this setting, the closure of maps $C^{\infty}$ conjugated to elements of any locally free $\mathbb{T}^{1}$-action coincides with the $C^{\infty}$-closure of the conjugancy class of the rigid translations $R_{(\lambda_{1},\lambda_{2})}\colon(x,y)\mapsto(x+\lambda_{1},y+\lambda_{2}),$ that is, the set $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$, where $\mathcal{O}(\mathbb{T}^{2})=\left\\{hR_{\alpha}h^{-1}:h\in\operatorname{Diff}^{\infty}(\mathbb{T}^{2}),\,\alpha\in\mathbb{T}^{2}\right\\}.$ As we mentioned above, a generic element of $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ is topologically weak mixing; however, no topologically mixing elements are known. It is also unknown if a minimal diffeomorphism of $\mathbb{T}^{2}$ can be topologically mixing. In fact, no examples of minimal $C^{\infty}$ diffeomorphisms of $\mathbb{T}^{2}$ in the homotopy class of the identity are known other than the ones in $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$. Recall that a homeomorphism $f\colon\mathbb{T}^{2}\to\mathbb{T}^{2}$ is topologically weak mixing if $f\times f$ is transitive. An equivalent definition is the following: for each open $U\subset\mathbb{T}^{2}$ and $\epsilon>0$, there is $n>0$ such that $f^{n}(U)$ is $\epsilon$-dense in $\mathbb{T}^{2}$. We will be interested in a similar property, which implies weak-mixing but is stronger in that it requires open sets to be mixed in every homological direction. ###### Definition 1.1. A homeomorphism $f\colon\mathbb{T}^{2}\to\mathbb{T}^{2}$ is _weak spreading_ if for a lift $\hat{f}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ of $f$ the following holds: for each open set $U\in\mathbb{R}^{2}$, $\epsilon>0$ and $N>0$, there is $n>0$ such that $\hat{f}^{n}(U)$ is $\epsilon$-dense in a ball of radius $N$. Let $\overline{\mathcal{O}}^{\infty}_{\mu}(\mathbb{T}^{2})$ denote the area- preserving elements of $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$. Now we can state our main theorem. ###### Theorem 1.2. Weak spreading diffeomorphisms are generic in $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ and $\overline{\mathcal{O}}^{\infty}_{\mu}(\mathbb{T}^{2})$. As a consequence, we prove a result about invariant foliations announced by Herman in [FH77] without proof. By a topological foliation we mean a codimension-$1$ foliation of class $C^{0}$; that is, a partition $\mathcal{F}$ of $\mathbb{T}^{2}$ into one-dimensional topological sub-manifolds which is locally homeomorphic to the partition of the unit square by horizontal segments. We say that the foliation is invariant by $f$ if $f(F)\in\mathcal{F}$ for every $F\in\mathcal{F}$. We then have: ###### Corollary 1.3. The set of diffeomorphisms in $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ (resp. $\overline{\mathcal{O}}^{\infty}_{\mu}(\mathbb{T}^{2})$) without any invariant topological foliation is residual in $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ (resp. $\overline{\mathcal{O}}^{\infty}_{\mu}(\mathbb{T}^{2})$). Since the set of minimal and uniquely ergodic diffeomorphisms in $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ (or $\overline{\mathcal{O}}^{\infty}_{\mu}(\mathbb{T}^{2})$) is also residual, this provides a residual set of minimal, uniquely ergodic diffeomorphisms with no invariant foliations. Using the ideas of [FS05], it is possible to construct real analytic examples, working with diffeomorphisms which have an analytic extension to a band of fixed width in $\mathbb{C}^{2}$ (see the precise definitions in §5). ###### Theorem 1.4. The set of real analytic diffeomorphisms of $\mathbb{T}^{2}$ which are weak spreading is residual in $\overline{\mathcal{O}}^{\omega}_{\rho}(\mathbb{T}^{2})$. ###### Remark 1.5. We use the word “weak” in the definition of weak spreading because there is an analogy with the topological weak mixing property. We could also define _strong_ spreading (or just _spreading_) as the property that for any open set $U\subset\mathbb{R}^{2}$, $\epsilon>0$ and $N>0$ there is $n_{0}$ such that that $\hat{f}^{n}(U)$ is $\epsilon$-dense in a ball of radius $N$ whenever $n>n_{0}$. This would be in analogy with the definition of topological mixing, but it is clearly a stronger property. In fact, the typical examples of topologically mixing systems in $\mathbb{T}^{2}$ mix only in one direction (e.g. Anosov systems and time-one maps of some minimal flows [Fay02]). It is not obvious that strong spreading diffeomorphisms exist; however, as P. Boyland kindly explained to us, an example of a strong spreading diffeomorphism can be constructed using Markov partitions and the techniques of [Boy08]. ### 1.1. Acknowledgments We are grateful to E. Pujals and P. Boyland for useful discussions, and the anonymous referee for bringing the results of [FS05] to our attention and suggesting various improvements, in particular the content of $\S\ref{sec:analytic}$. ## 2\. The method of Fathi-Herman As usual, we identify $\mathbb{T}^{2}\simeq\mathbb{R}^{2}/\mathbb{Z}^{2}$ with quotient projection $\pi\colon\mathbb{R}^{2}\to\mathbb{T}^{2}$, and denote by $\operatorname{Diff}^{\infty}(\mathbb{T}^{2})$ the space of $C^{\infty}$ diffeomorphisms of $\mathbb{T}^{2}$. A lift of one such diffeomorphism $f$ to $\mathbb{R}^{2}$ is a map $\hat{f}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ such that $f\pi=\pi\hat{f}$. If $f$ is homotopic to the identity, this is equivalent to saying that $\hat{f}$ commutes with integer translations, i.e. $\hat{f}(z+v)=\hat{f}(z)+v$ for $v\in\mathbb{Z}^{2}$. Two different lifts of a diffeomorphism of $\mathbb{T}^{2}$ always differ by a constant $v\in\mathbb{Z}^{2}$. We will denote by $\hat{R}_{\alpha}$ the translation $z\mapsto z+\alpha$ of $\mathbb{R}^{2}$ and by $R_{\alpha}$ the rotation of $\mathbb{T}^{2}$ lifted by $\hat{R}_{\alpha}$. The method used in [FH77], adapted to our case, can be resumed as follows: ###### Lemma 2.1. Let $P\subset\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ (or $\overline{\mathcal{O}}^{\infty}_{\mu}(\mathbb{T}^{2})$) be such that 1. (1) $P=\bigcap_{n\geq 0}P_{n}$, where the $P_{n}$ are open; 2. (2) For each $g\in\operatorname{Diff}^{\infty}(\mathbb{T}^{2})$ (resp. $\operatorname{Diff}^{\infty}_{\mu}(\mathbb{T}^{2})$) and $m\in\mathbb{N}$, there is $N>0$ such that $\\{gfg^{-1}:f\in P_{n}\\}\subset P_{m}$ whenever $n>N$; 3. (3) For each $n\in\mathbb{N}$, $p/q\in\mathbb{Q}$, there exists $h\in\operatorname{Diff}^{\infty}(\mathbb{T}^{2})$ (resp. $\operatorname{Diff}^{\infty}_{\mu}(\mathbb{T}^{2})$) such that * • $hR_{(1/q,0)}=R_{(1/q,0)}h$; * • $hR_{\alpha_{k}}h^{-1}\in P_{n}$ for some sequence $\alpha_{k}\to(p/q,0)$. Then, $P$ is residual in $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ (resp. $\overline{\mathcal{O}}^{\infty}_{\mu}(\mathbb{T}^{2})$). ###### Proof. Given $m\in\mathbb{N}$, $p/q\in\mathbb{Q}$, and $g\in\operatorname{Diff}^{\infty}(\mathbb{T}^{2})$, let $n$ be as in (2), and then $h$ and $\\{\alpha_{k}\\}$ as in (3). Then $P_{n}\ni hR_{\alpha_{k}}h^{-1}\xrightarrow[k\to\infty]{C^{\infty}}hR_{(p/q,0)}h^{-1}=R_{(p/q,0)},$ so that $P_{m}\ni g(hR_{\alpha_{k}}h^{-1})g^{-1}\xrightarrow[k\to\infty]{C^{\infty}}gR_{(p/q,0)}g^{-1}.$ This proves that $gR_{(p/q,0)}g^{-1}\in\overline{P}_{m}^{\infty}$. Since this holds for all $g$ and $p/q$, it follows that $P_{m}$ is dense in $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ because so is the set $\left\\{hR_{(p/q,0)}h^{-1}:h\in\operatorname{Diff}^{\infty}(\mathbb{T}^{2}),\,p/q\in\mathbb{Q}\right\\}.$ Since this holds for all $m$ and each $P_{m}$ is open, this proves that $P$ is residual in $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$. The proof in the area-preserving case is the same. ∎ The property of having no invariant topological foliations is hard to deal with in the $C^{\infty}$ topology in order to apply the above lemma. However, the weak spreading property can be adequately described as an intersection of countably many properties that fit well into the lemma; thus we first prove Theorem 1.2 using the above method, and then we prove that weak spreading is not compatible with the existence of invariant foliations of any kind, which implies Corollary 1.3. ## 3\. Proof of Theorem 1.2 Let $P_{n}$ denote the set of all $f\in\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ such that if $\hat{f}$ is a lift of $f$, for any ball $B$ of radius $1/n$ in $\mathbb{R}^{2}$, there is $k>0$ such that $\hat{f}^{k}(B)$ is $1/n$-dense in a ball of radius $n$. Note that if this property holds for some lift, it holds for any lift of $f$. It is clear that $P=\cap P_{n}$ is the set of weak spreading elements of $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$. Denote by $B(z,\epsilon)$ the ball of radius $\epsilon$ centered at $z$. Given a lift $\hat{f}$ of some $f\in P_{n}$, and $z\in\mathbb{R}^{2}$, let $k_{z}$ be the smallest positive integer such that $\hat{f}^{k_{z}}(B(z,1/n))$ is $1/n$-dense in a ball of radius $n$. By continuity of $\hat{f}$, the map $z\mapsto k_{z}$ is upper semi-continuous, and therefore it attains a maximum $K$ when $z\in[0,1]^{2}$. But since $\hat{f}$ lifts a map homotopic to the identity, $k_{z}=k_{z+v}$ when $v\in\mathbb{Z}^{2}$, so that $k_{z}\leq K$ for all $z\in\mathbb{R}^{2}$. Hence, if $g$ is close enough to $f$ in the $C^{0}$ topology and $\hat{g}$ is the lift of $g$ closest to $\hat{f}$, it also holds that $\hat{g}^{k_{z}}(B(z,1/n))$ is dense in a ball of radius $n$ for any $z\in\mathbb{R}^{2}$. Hence $P_{n}$ is open in the $C^{0}$ topology (and, in particular, in the $C^{\infty}$ topology). To see that condition (2) of Lemma 2.1 holds, note that any lift $\hat{g}$ of a diffeomorphism $g$ of $\mathbb{T}^{2}$ is bi-Lipschitz. Fix $m\in\mathbb{N}$, let $C$ be a Lipschitz constant for $\hat{g}$ and $\hat{g}^{-1}$, and let $n>0$ be such that $C<n/m$. If $\hat{f}$ is a lift of $f\in P_{n}$, and if $U$ is an open set, then there is $k$ such that $\hat{f}^{k}(\hat{g}^{-1}(U))$ is $1/n$-dense in a ball of radius $n$. Thus, $\hat{g}\hat{f}^{k}\hat{g}^{-1}(U)$ is $C/n$-dense in a ball of radius $n/C$, which implies that $gfg^{-1}\in P_{m}$ as required. To finish the proof, it remains to see that condition (3) of Lemma 2.1 holds. To do this, it suffices to construct, for each $q,n\in\mathbb{N}$, a diffeomorphism $h\in\operatorname{Diff}^{\infty}(\mathbb{T}^{2})$ which commutes with $R_{(1/q,0)}$ and such that $hR_{(\alpha,0)}h^{-1}\in P_{n}$ whenever $\alpha$ is irrational. Note that it is enough to prove this for some multiple of $q$ instead of $q$. We will assume that $q$ is a multiple of $n$ and $q\geq 2n$, since otherwise we may use $2qn$ instead of $q$. We define $h$ by constructing a lift $\hat{h}=\hat{v}\circ\hat{u}$, where $\hat{v},\hat{u}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ are the maps $\hat{u}(x,y)=\left(x,\,y+m\cos(2\pi qx)\right),\quad\hat{v}(x,y)=\left(x+n\cos(2\pi qy),y\right)$ and $m$ is a sufficiently large integer that we will choose later. It is clear that $\hat{u}$ and $\hat{v}$ are lifts of $C^{\infty}$ torus diffeomorphisms in the homotopy class of the identity, because they commute with integer translations. They also commute with $\hat{R}_{(1/q,0)}$ and $\hat{R}_{(0,1/q)}$ as well. The same properties hold for $\hat{h}=\hat{v}\circ\hat{u}$. Moreover, since both $\hat{u}$ and $\hat{v}$ are area-preserving, so is $\hat{h}$ and the rest of the proof also works in the area-preserving setting. Figure 1. Image of $J_{\delta}$ by $\hat{h}$ Let $\delta=2n(\pi qm)^{-1}$, $I_{\delta}=[-\delta,\delta]\times\\{0\\}$, and $J_{\delta}=[(4q)^{-1}-\delta/2,(4q)^{-1}+\delta/2]\times\\{0\\}$. ###### Claim 1. If $m$ is large enough, then $\hat{h}(I_{\delta})$ is contained in the ball of radius $1/(2n)$ centered at $(n,m)$, and $\hat{h}(J_{\delta})$ is $1/n$-dense in $[-n,n]\times[-n,n]$. ###### Proof. First observe that from the inequality $1-\cos(x)\leq x^{2}/2\quad\forall x\in\mathbb{R}$ it follows that (denoting by $(x_{1},x_{2})_{i}$ the coordinate $x_{i}$) $\left\lvert{(\hat{u}(x,0)-\hat{u}(0,0))_{2}}\right\rvert\leq 2m(\pi qx)^{2}<2m(\pi q\delta)^{2}=8n^{2}/m$ if $\left\lvert{x}\right\rvert<\delta$. Since $\hat{u}(0,0)=(0,m)$, this means that $\hat{u}(I_{\delta})$ is contained in the rectangle $[-\delta,\delta]\times[m-b,m+b]$ where $b=8n^{2}/m$. By the definition of $\hat{v}$ and a similar argument (since $m$ is an integer), we can conclude that $\hat{v}(\hat{u}(I_{\delta}))\subset[n-a,n+a]\times[m-b,m+b]$, where $a=\delta+2n(\pi qb)^{2}=2n(\pi qm)^{-1}+128\pi^{2}q^{2}n^{5}m^{-2}.$ Since both $a$ and $b$ can be made arbitrarily small if $m$ is large enough, $\hat{h}(I_{\delta})$ is contained in a ball around $(n,m)$ of radius $1/(2n)$ if $m$ is large enough. For the second part of the claim, note that $\cos(x+\pi/2)=\sin(x)\geq x/2\quad\text{ if }0\leq x\leq\pi/2$ so that $\left(\hat{u}((4q)^{-1}+\delta/2,0)-\hat{u}((4q)^{-1},0)\right)_{2}=m\cos(\pi q\delta+\pi/2)\geq m\pi q\delta/2=n,$ and similarly $\left(\hat{u}((4q)^{-1}-\delta/2,0)-\hat{u}((4q)^{-1},0)\right)_{2}=-n.$ Thus $\hat{u}(J_{\delta})$ is an arc that transverses vertically the rectangle $[-\delta/2,\delta/2]\times[-n,n]$. Let $L=\\{0\\}\times[-n,n]$. Note that $\hat{v}(L)$ is $1/q$-dense in $[-n,n]\times[-n,n]$, since every rectangle of the form $[-n,n]\times[-n+k/q,-n+(k+1)/q]$, $0\leq k\leq 2qn-1$ is horizontally transversed by $\hat{v}(L)$. By the previous paragraph, $\hat{u}(J_{\delta})$ contains a point of the form $(s,y)$ with $\left\lvert{s}\right\rvert<\delta/2$ for each $(0,y)\in L$. Since $\hat{v}(s,y)=\hat{v}(0,y)+(s,0)$, it follows from the previous facts that, if $m$ is so large that $\delta/2<1/q$, $\hat{h}(J_{\delta})=\hat{v}(\hat{u}(J_{\delta}))$ is $2/q$-dense in $[-n,n]\times[-n,n]$ (see Figure 1). Since we assumed earlier that $q\geq 2n$, we conclude that $h(J_{\delta})$ is $1/n$-dense in $[-n,n]\times[-n,n]$ as claimed. This proves the claim. ∎ Let $B\subset\mathbb{R}^{2}$ be a ball of radius $1/n$. Then $B$ contains a ball $B^{\prime}$ of radius $1/(2n)$ around some point of coordinates $(i/q,j/q)$, with $i,j$ integers (because $q\geq 2n$). Since $\hat{h}$ commutes with $R_{(1/q,0)}$, and using Claim 1, we see that $\hat{h}(I_{\delta}+(i/q,j/q)-(n,m))=\hat{h}(I_{\delta})-(n,m)+(i/q,j/q)\subset B^{\prime}$ In particular, $I_{\delta}+(i/q,j/q)-(n,m)\subset\hat{h}^{-1}(B)$. Since $J_{\delta}$ lies on the same horizontal line as $I_{\delta}$ and is shorter than $I_{\delta}$, if $\alpha$ is an irrational number we can find $k\in\mathbb{N}$ and $r\in\mathbb{Z}$ such that $J_{\delta}+(r,0)\subset\hat{R}^{k}_{(\alpha,0)}(I_{\delta})$, and we have $J_{\delta}+(i/q,j/q)-(n,m)+(r,0)\subset\hat{R}^{k}_{(\alpha,0)}(I_{\delta}+(i/q,j/q)-(n,m)).$ Thus, if $\hat{f}=\hat{h}\hat{R}_{(\alpha,0)}\hat{h}^{-1}$, $\hat{f}^{k}(B)=\hat{h}\hat{R}^{k}_{(\alpha,0)}\hat{h}^{-1}(B)\supset\hat{h}\hat{R}^{k}_{(\alpha,0)}(I_{\delta}+(i/q,j/q)-(n,m))\\\ \supset\hat{h}(J_{\delta}+(i/q,j/q)-(n+r,m))=\hat{h}(J_{\delta})+(i/q,j/q)-(n+r,m)$ which is just a translation of $\hat{h}(J_{\delta})$, and thus by Claim 1 it is $1/n$-dense in some ball of radius $n$. That is, $\hat{f}^{k}(B)$ is $1/n$-dense in some ball of radius $n$, which means that $hR_{(\alpha,0)}h^{-1}\in P_{n}$. Since $\alpha$ was an arbitrary irrational number, this completes the proof. ∎ ## 4\. Invariant foliations Corollary 1.3 is a direct consequence of Theorem 1.2 and the next two propositions. ###### Proposition 4.1. If $\mathcal{F}$ is a foliation of $\mathbb{T}^{2}$ and $\hat{\mathcal{F}}$ is the lift of $\mathcal{F}$ to $\mathbb{R}^{2}$, then there is a leaf $F\in\hat{\mathcal{F}}$ which is contained in a strip bounded by two parallel straight lines $L$ and $L^{\prime}$, such that both lines belong to different connected components of $\mathbb{R}^{2}-F$. ###### Proof. If $\mathcal{F}$ has a compact leaf, there is $z\in\mathbb{R}^{2}$ and a leaf $F$ of $\hat{\mathcal{F}}$ such that $F+(p,q)=F$, for some pair of integers $(p,q)\neq(0,0)$. Thus, assuming $p\neq 0$, if $L_{0}$ is a line of slope $q/p$, it holds that $s=\sup\\{d(z,L_{0}):z\in F\\}<\infty$, and the proposition follows by choosing $L$ and $L^{\prime}$ a distance greater than $s$ apart from $L_{0}$, one on each side. If $p=0$, then $q\neq 0$ an analogous argument holds. Now suppose $\mathcal{F}$ has no compact leaves. By [HH83, Theorem 4.3.3], $\mathcal{F}$ is equivalent to a foliation $\mathcal{F}^{\prime}$ obtained by suspension of the trivial foliation $\mathbb{R}\times\mathbb{T}^{1}$ over an orientation preserving circle homeomorphism $f\colon\mathbb{T}^{1}\to\mathbb{T}^{1}$ with irrational rotation number. Such a foliation has a lift $\hat{\mathcal{F}}^{\prime}$ to $\mathbb{R}^{2}$ such that the intersection of the leaf through $(0,y)$ with the line $\\{n\\}\times\mathbb{R}$ is at $(n,\hat{f}^{n}(y))$, where $\hat{f}\colon\mathbb{R}\to\mathbb{R}$ is a lift of $f$. If $\phi(y)$ denotes the length of the arc of leaf joining $(0,y)$ to $(1,\hat{f}(y))$, then $\phi\colon\mathbb{R}\to\mathbb{R}$ is a continuous function and it is $\mathbb{Z}$-periodic, because $\hat{\mathcal{F}}^{\prime}$ is a lift of a foliation of $\mathbb{T}^{2}$. Thus there is a constant $C$ such that $\phi(x)<C$ for all $x\in\mathbb{R}$. Note that the length of the arc joining $(n,y)$ to $(n+1,\hat{f}(y))$ is also bounded by $C$. If $\rho$ is the rotation number of $\hat{f}$, by classic results for circle homeomorphisms (see, for example, [dMvS93]) we have $\|\hat{f}^{n}(y)-y-n\rho\|\leq 1$ for all $n\in\mathbb{Z}$ and $y\in\mathbb{R}$. Let $F^{\prime}$ be a leaf of $\hat{\mathcal{F}}^{\prime}$ containing the point $(0,y)$. Then $F^{\prime}=\cup_{n\in\mathbb{Z}}F_{n}^{\prime}$ where $F_{n}$ is the arc joining $(n,\hat{f}^{n}(y))$ to $(n+1,\hat{f}^{n+1}(y))$. Note that the distance from $(n,\hat{f}^{n}(y))$ to the line $L_{0}$ of slope $\rho$ through $(0,y)$ is at most $1$, and the length of $F_{n}^{\prime}$ is at most $C$. Thus the distance from any point of $F^{\prime}$ to $L_{0}$ is at most $C+1$. We know that $\mathcal{F}$ is equivalent to $\mathcal{F}^{\prime}$, which means there is a homeomorphism $h\colon\mathbb{T}^{2}\to\mathbb{T}^{2}$ mapping leaves of $\mathcal{F}^{\prime}$ to leaves of $\mathcal{F}$. If $\hat{h}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ is a lift of $h$, then we can write $\hat{h}(z)=A(z)+\psi(z)$ where $A\in\mathrm{GL}(2,\mathbb{Z})$ and $\psi$ is a $\mathbb{Z}^{2}$-periodic function, bounded by some constant $K$. If $L_{1}=AL_{0}$, $z=\hat{h}(z^{\prime})$ is a point in $F=h(F^{\prime})$, and $w=A(w^{\prime})$ is a point in $L_{1}$ then $\left\lvert{z-w}\right\rvert=\left\lvert{A(z^{\prime}-w^{\prime})+\psi(z)}\right\rvert\leq\left\\|{A}\right\\|\cdot\left\lvert{z^{\prime}-w^{\prime}}\right\rvert+K\leq\left\\|{A}\right\\|(C+1)+K,$ the last inequality following from the fact that $z^{\prime}\in F^{\prime}$ and $w^{\prime}\in L_{0}$. It follows that $s=\sup_{z\in F}d(z,L_{1})<\infty$, and as before we complete the proof choosing $L$ and $L^{\prime}$ parallel to $L_{1}$ and a distance at least $s$ apart from $L_{1}$, one on each side. ∎ ###### Proposition 4.2. If $f$ is weak spreading and homotopic to the identity, then $f$ has no invariant topological foliations. ###### Proof. By Proposition 4.1, if $\mathcal{F}$ is a foliation invariant by $f$ and $\hat{\mathcal{F}}$ is the lift of this foliation to $\mathbb{R}^{2}$ (hence invariant by $\hat{f}$), there is a leaf $\hat{F}_{0}\in\hat{\mathcal{F}}$ which is contained in a strip bounded by two parallel lines $L$ and $L^{\prime}$, and which contains each of those lines in a different component of its complement. Let $u$ be a unit vector orthogonal to $L$. We will assume without loss of generality that $u$ has a nonzero second coordinate. If $S$ is the strip bounded by $\hat{F}_{0}$ and $\hat{F}_{0}+(0,1)$, denoting by $\phi_{u}\colon\mathbb{R}^{2}\to\mathbb{R}$ the orthogonal projection onto the direction of $u$, it is clear that $0pt_{u}(S)\doteq\operatorname{diam}(\phi_{u}(S))<\infty.$ Moreover, $\cup_{n\in\mathbb{Z}}S+(0,n)=\mathbb{R}^{2}$. For each $n\in\mathbb{Z}$, since $\hat{f}^{n}(\hat{F}_{0})$ cannot cross $\hat{F}_{0}+(0,k)$ for any $k\in\mathbb{Z}$, we see that $\hat{f}^{n}(\hat{F}_{0})\subset S+(0,m)$ for some $m\in\mathbb{Z}$. This implies that $0pt_{u}({f}^{n}(\hat{F}_{0}))\leq M=0pt_{u}(S).$ But then $\hat{f}^{n}(\hat{F}_{0}+(0,1))\subset S+(0,m+1),$ so that $\hat{f}^{n}(S)$ is contained in the strip bounded by $\hat{F}_{0}+(0,m)$ and $\hat{F}_{0}+(0,m+2)$. This means that $0pt_{u}(\hat{f}^{n}(S))\leq 2M$. However, if $f$ is weak spreading, then there is $n>0$ such that $\hat{f}^{n}(S)$ is $1/3$-dense in some ball of radius $3M$, so that $0pt_{u}(\hat{f}^{n}(S))>2M$, contradicting the previous claim. This completes the proof. ∎ ## 5\. The real analytic case In this section we briefly explain how to obtain minimal weak spreading analytic diffeomorphisms of $\mathbb{T}^{2}$. We kindly thank the anonymous referee for bringing this to our attention. First we introduce some notation, following [FS05]. Fix $\rho>0$, and let $g:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be any real analytic $\mathbb{Z}^{2}$-periodic function which can be holomorphically extended to $A_{\rho}=\\{(z,w)\in\mathbb{C}^{2}:\left\lvert{\operatorname{Im}{z}}\right\rvert<\rho,\,\left\lvert{\operatorname{Im}{w}}\right\rvert<\rho\\}$. We define $\left\\|{g}\right\\|_{\rho}=\sup_{A_{\rho}}\left\lvert{g(z,w)}\right\rvert$, and we denote by $C^{\omega}_{\rho}(\mathbb{T}^{2})$ the space of all functions of this kind which satisfy $\left\\|{g}\right\\|_{\rho}<\infty$. Let $\operatorname{Diff}_{\rho}^{\omega}(\mathbb{T}^{2})$ be the space of all diffeomorphisms $f$ of $\mathbb{T}^{2}$ which are homotopic to the identity, and which have a lift whose periodic part is in $C_{\rho}^{\omega}(\mathbb{T}^{2})$. There is a metric in $\operatorname{Diff}_{\rho}^{\omega}(\mathbb{T}^{2})$ defined by $d_{\rho}(h,k)=\inf_{(p,q)\in\mathbb{Z}^{2}}\left\\|{\hat{h}-\hat{k}+(p,q)}\right\\|_{\rho},$ where $\hat{h}$ and $\hat{k}$ are lifts of $h$ and $k$, respectively. Since $C_{\rho}^{\omega}(\mathbb{T}^{2})$ is a Banach space, it is easy to see that the metric $d_{\rho}$ turns $\operatorname{Diff}^{\omega}_{\rho}(\mathbb{T}^{2})$ into a complete metric space. To apply the arguments of the previous sections we work in the space $\overline{\mathcal{O}}_{\rho}^{\omega}(\mathbb{T}^{2})$ defined as the closure in the $d_{\rho}$ metric of the set of diffeomorphisms of the form $hR_{\alpha}h^{-1}$ where $\alpha\in\mathbb{T}^{1}$, and $h\in\operatorname{Diff}_{\rho}^{\omega}(\mathbb{T}^{2})$ is any diffeomorphism whose lifts to $\mathbb{R}^{2}$ have a bi-holomorphic extension to $\mathbb{C}^{2}$. We observe that the proof of Lemma 2.1 applies to this setting if we use $\overline{\mathcal{O}}_{\rho}^{\omega}(\mathbb{T}^{2})$ instead of $\overline{\mathcal{O}}^{\infty}(\mathbb{T}^{2})$ (and the topology induced by $d_{\rho}$ instead of the $C^{\infty}$ topology). To complete the proof of Theorem 1.4, we note that everything in §3 works without any modifications, because the function $h$ constructed to obtain property (3) of Lemma 2.1 has an analytic extension to all of $\mathbb{C}^{2}$ which is a bi-holomorphism. ## References * [AK70] D. Anosov and A. Katok, _New examples in smooth ergodic theory. Ergodic diffeomorphisms._ , Transactions of the Moscow Mathematical Society 23 (1970), 1–35. * [Boy08] P. Boyland, _Transitivity of surface dynamics lifted to abelian covers_ , Preprint, 2008. * [dMvS93] W. de Melo and S. van Strien, _One-dimensional dynamics_ , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. * [Fay02] B. Fayad, _Weak mixing for reparameterized linear flows on the torus_ , Ergodic Theory & Dynamical Systems (2002), no. 22, 187–201. * [FH77] A. Fathi and M. Herman, _Existence de difféomorphismes minimaux_ , Asterisque 49 (1977), 37–59. * [FK04] B. Fayad and A. Katok, _Constructions in elliptic dynamics_ , Ergodic Theory & Dynamical Systems 24 (2004), 1477–1520. * [FS05] B. Fayad and M. Saprykina, _Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary_ , Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 3, 339–364. * [Her92] M. R. Herman, _On the dynamics of Lagrangian tori invariant by symplectic diffeomorphisms_ , Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations (L’Aquila, 1990), Longman Science and Technology, Harlow, 1992, (Pitman Research Notes Mathematical Series, 243), pp. 92–112. * [HH83] G. Hector and U. Hirsch, _Introduction to the geometry of foliations, part a: Fundamentals_ , Friedr Vieweg & Sohn, 1983. * [Kor07] A. Koropecki, _On the dynamics of torus homeomorphisms_ , Ph.D. thesis, IMPA, 2007.	arxiv-papers	2009-02-14T15:57:21	2024-09-04T02:49:00.570175	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Alejandro Kocsard and Andres Koropecki", "submitter": "Andres Koropecki", "url": "https://arxiv.org/abs/0902.2474" }
0902.2504	Hyperset Approach to Semi-structured Databases and the Experimental Implementation of the Query Language Delta Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Richard Molyneux Thesis Supervisors: Dr. Vladimir Sazonov Dr. Alexei Lisitsa External examiner: Dr. Ulrich Berger Internal examiner: Dr. Grant Malcolm Department of Computer Science The University of Liverpool January, 2009 ### Abstract This thesis presents practical suggestions towards the implementation of the hyperset approach to semi-structured databases and the associated query language $\Delta$ (Delta). This work can be characterised as part of a top- down approach to semi-structured databases, from theory to practice. Over the last decade the rise of the World-Wide Web has lead to the suggestion for a shift from structured relational databases to semi-structured databases, which can query distributed and heterogeneous data having unfixed/non-rigid structure in contrast to ordinary relational databases. In principle, the World-Wide Web can be considered as a large distributed semi-structured database where arbitrary hyperlinking between Web pages can be interpreted as graph edges (inspiring the synonym ‘Web-like’ for ‘semi-structured’ databases also called here WDB). In fact, most approaches to semi-structured databases are based on graphs, whereas the hyperset approach presented here represents such graphs as systems of set equations. This is more than just a style of notation, but rather a style of thought and the corresponding mathematical background leads to considerable differences with other approaches to semi- structured databases. The hyperset approach to such databases and to querying them has clear semantics based on the well established tradition of set theory and logic, and, in particular, on non-well-founded set theory because semi- structured data allow arbitrary graphs and hence cycles. The main original part of this work consisted in implementation of the hyperset $\Delta$-query language to semi-structured databases, including worked example queries. In fact, the goal was to demonstrate the practical details of this approach and language. The required development of an extended, practical version of the language based on the existing theoretical version, and the corresponding operational semantics. Here we present detailed description of the most essential steps of the implementation. Another crucial problem for this approach was to demonstrate how to deal in reality with the concept of the equality relation between (hyper)sets, which is computationally realised by the bisimulation relation. In fact, this expensive procedure, especially in the case of distributed semi-structured data, required some additional theoretical considerations and practical suggestions for efficient implementation. To this end the “local/global” strategy for computing the bisimulation relation over distributed semi-structured data was developed and its efficiency was experimentally confirmed. Finally, the XML-WDB format for representing any distributed WDB as system of set equations was developed so that arbitrary XML elements can participate and, hence, queried by the $\Delta$-language. The query system with the syntax of the language and several example queries from this thesis is available online at http://www.csc.liv.ac.uk/~molyneux/t/ Keywords: Semi-structured, Web-like, distributed databases, hypersets, bisimulation, query language $\Delta$ (Delta) ### Dedication This thesis is dedicated to my loving grandparents. ### Acknowledgement The research presented in this thesis was undertaken at the Department of Computer Science under the supervision of Dr. Vladimir Sazonov and Dr. Alexei Lisitsa. This work was inspired by the research of my primary supervisor Dr. Vladimir Sazonov, his encouragement and dedication was invaluable in developing those ideas presented here. Additionally, I am grateful to the help and support given by Dr. Alexei Lisitsa and Prof. Michael Fisher. This work was made possible by the scholarship awarded to me by the Department of Computer Science. I wish to thank my parents whose love and support has been the foundation of all my achievements. Also, to my brothers and sister for their encouragement and support. ###### Contents 1. 1 Introduction 2. I Hyperset approach to querying Web-like databases 1. 2 Semi-structured or Web-like databases 1. 2.1 Set theoretic view of structured and semi-structured data 1. 2.1.1 Structured relational data 2. 2.1.2 Relaxation of structural restrictions on relational data 3. 2.1.3 Semi-structured data 4. 2.1.4 Syntactical and conceptual set nesting 2. 2.2 Hyperset theoretic view of semi-structured data 3. 2.3 Graph or Web-like view 1. 2.3.1 Graph representation of systems of set equations 2. 2.3.2 Graphs or systems of set equations as Web-like databases 3. 2.3.3 Distributed WDB 4. 2.4 Hyperset data considered abstractly 1. 2.4.1 Bisimulation – preliminary considerations 2. 2.4.2 Redundancies in WDB 3. 2.4.3 Bisimulation invariance 4. 2.4.4 Anti-Foundation Axiom 2. 3 Query language $\Delta$ 1. 3.1 The syntax 2. 3.2 Intuitive denotational semantics 1. 3.2.1 Boolean valued expressions — $\Delta$-formulas 2. 3.2.2 Set valued expressions — $\Delta$-terms 3. 3.3 Operational semantics 1. 3.3.1 Examples of reduction 4. 3.4 Implemented $\Delta$-query language 1. 3.4.1 Queries with declarations 2. 3.4.2 Library 5. 3.5 Example $\Delta$-queries 1. 3.5.1 Example of a non-well-typed query 2. 3.5.2 Example of valid and executable query 3. 3.5.3 Restructuring query 4. 3.5.4 Horizontal transitive closure 5. 3.5.5 Dealing with proper hypersets 6. 3.5.6 Query optimisation by removing redundancies 6. 3.6 Imitating path expressions 7. 3.7 Linear ordering query 3. 4 Bisimulation 1. 4.1 Hyperset equality and the problem of efficiency 1. 4.1.1 Bisimulation relation 2. 4.2 Computing bisimulation over WDB 1. 4.2.1 Implemented algorithm for computing bisimulation over distributed WDB 3. II Local/global approach to optimise bisimulation and querying 1. 5 The Oracle 1. 5.1 Computing bisimulation with the help of the Oracle 2. 5.2 Imitating the Oracle for testing purposes 3. 5.3 Empirical testing of the trivial Oracle 2. 6 Local/global bisimulation 1. 6.1 Defining the ordinary bisimulation relation $\approx$ 2. 6.2 Defining the local upper approximation $\approx^{L}_{+}$ of $\approx$ 3. 6.3 Defining the local lower approximation $\approx^{L}_{-}$ of $\approx$ 4. 6.4 Using local approximations to aid computation of the global bisimulation 1. 6.4.1 Granularity of sites 2. 6.4.2 Local approximations giving rise to global bisimulation facts 3. 6.4.3 Practical algorithm for computation of local approximations 3. 7 The Oracle based on the idea of local/global bisimulation 1. 7.1 Description of the bisimulation engine (implementation of a more realistic Oracle) 1. 7.1.1 Strategies 2. 7.1.2 Exploiting local approximations to aid in the computation of bisimulation 2. 7.2 Empirical testing of the bisimulation engine 1. 7.2.1 Determining the benefit of background work by the bisimulation engine on query performance 2. 7.2.2 Determining the benefit of exploiting local approximations by the bisimulation engine on query performance 3. 7.2.3 Determining the benefits of background work by the bisimulation engine exploiting local approximations 3. 7.3 Overall conclusion 1. 7.3.1 Claims and limitations 4. III Implementation issues 1. 8 $\Delta$ Query Execution 1. 8.1 Implementation of $\Delta$-query execution by reduction process 1. 8.1.1 Separation construct 2. 8.1.2 Quantification 3. 8.1.3 Recursive separation 4. 8.1.4 Decoration 5. 8.1.5 Transitive closure 2. 8.2 Representation of query output 2. 9 $\Delta$ Query Syntax 1. 9.1 Parsing (well-formed queries) 1. 9.1.1 Implemented $\Delta$-language grammar 2. 9.1.2 BNF forking 3. 9.1.3 Query parsing 4. 9.1.4 Parsing ambiguities 5. 9.1.5 Grammar classification 2. 9.2 Contextual analysis (well-typed queries) 1. 9.2.1 Aim of contextual analysis 2. 9.2.2 Some useful definitions 3. 9.2.3 Bottom-up contextual analysis in detail 4. 9.2.4 Extension of contextual analysis to support libraries 3. 10 XML Representation of Web-like Databases (XML-WDB Format) 1. 10.1 Represention of WDB by graph or set equations 2. 10.2 Practical representation of WDB as XML 1. 10.2.1 XML-WDB document format 2. 10.2.2 Distributed WDB 3. 10.2.3 Transformation rules from XML to systems of set equations 4. 10.2.4 XML schema for XML-WDB format 5. IV Evaluation 1. 11 Comparative analysis 1. 11.1 Preliminary comparison 2. 11.2 SETL 3. 11.3 UnQL 4. 11.4 Lore 5. 11.5 Strudel 6. 11.6 G-Log 7. 11.7 Tree (XML) model approaches 2. 12 Conclusion and future outlook 1. 12.1 Hyperset approach to semi-structured databases 2. 12.2 Novel contributions 1. 12.2.1 Implementation of the hyperset approach to semi-structured databases 2. 12.2.2 Local/global approach towards efficient implementation of bisimulation 3. 12.2.3 Further optimisation 3. 12.3 Comparisons with other approaches 4. 12.4 Further work 3. A Appendix 1. A.1 Implemented BNF grammar of $\Delta$-query language 2. A.2 Example XML-WDB files 3. A.3 Predefined library queries ### Chapter 1 Introduction Before the emergence of the database culture in the late 1960’s data processing involved the ad hoc manipulation of data on tape or disk. The complexity of developing and managing such systems inspired new research into the principles of data organisation. Three models were suggested during the late 1960’s and early 1970’s: i) the hierarchical model [72], ii) the network model [70] proposed by the Data Base Task Group, and iii) Codd’s relational model [16]. The hierarchical and network models are closely related to the notion of _object-orientation_ as is argued in [73] and are, in fact, based on the idea of object identity, i.e. an object whose meaning is determined not only by records of values of its fields (or attributes) but also by a pointer or address of this object within files or memory. Note that, two objects are identical if they have the same address or pointer, whereas two objects are equivalent if they share the same fields. Links $T_{1}\rightarrow T_{2}$ denoting many-to-one relationships between record types constitute a graph in the case of the network model, and a forest (consisting of trees) in the case of the hierarchical model. Physically, each such graph or tree edge is represented by real relationships between OIDs of records of types $T_{1}$ and $T_{2}$. On the other hand, the great success of Codd’s relational model, which can be considered as a value-oriented approach, was based on taking the most fundamental concepts of logic and set theory as its foundation. Thus, any relation is a set of tuples, with each tuple also being represented111 under our interpretation by a set of a special kind (a set of attribute labelled values). In fact, this approach assumes an abstract view on data values where the concept of object identity is not needed. (Note that the concept of object identity may play a role in implementation but not in the abstract model itself.) The relational model was further extended by object-orientation during the early 1990’s [32], thus again absorbing the idea of object identity and additionally allowing complex data values with possibly nested structure and the idea of abstract data type with encapsulated methods. However, object-relational databases are still restricted by an imposed relational schema, that is they have a rigid structure. Note that complex, nested structures considered in this approach are somewhat related with the idea of semi-structured databases discussed in this thesis, but the latter approach does not assume in general a rigid structure. Moreover, the hyperset approach to semi-structured databases presented in this thesis is crucially based on the value-oriented rather than the object-oriented view #### From relations to semi-structured or Web-like data From the second half of the 1990s a new idea of semi-structured databases emerged (see [1] as a general reference). In the age of the Internet and the World-Wide Web (WWW), allowing accessibility of remote and heterogeneous databases, the relational paradigm has become too narrow and restrictive. Indeed, the structure of the data over the WWW is typically non-fixed or non- uniform. The idea of graph representation of data was introduced with the interpretation of graph edges like hyperlinks on the Web. Due to this analogy such graph-like semi-structured databases can also be reasonably called Web- like databases (WDB) [41]. An important example of the graph approach (in its pure form) is the system Lore [46] and the corresponding query language Lorel [2], which considers graph vertices as object identities (OIDs) with equality between vertices understood as essentially literal coincidence of OIDs irrespectively of their information content (presented by outgoing edges according to our hyperset approach). In fact, this is typical for most semi-structured database approaches [2, 8, 13, 14, 15, 18, 19, 22, 26, 27, 31, 33, 46, 51], except in the case of the query language UnQL [11] (as discussed briefly below). On the other hand, because of this idea of browsing by “picturing” the informational content (data value) of a graph vertex, considering such graphs merely as a binary (or ternary, if taking labels on edges into account) relation is not fully adequate in this context. Thus, we view the notion of semi-structured data as more than just a relation, that is more than just a graph where vertices are (uniquely presented by) object identities. In our hyperset theoretic approach, which is value-oriented, it becomes more appropriate to consider those target vertices of outgoing edges from any given vertex $v$ as children or even as _elements_ of $v$ with $v$ understood as a _set_ of its elements. It is the latter view on graph vertices which makes it value oriented. In fact, similar terminology is used in Extensible Markup Language (XML), which is a widely adopted approach to semi-structured data. However, this is only a superficial similarity with the set theoretic approach. XML only allows to syntactically represent semi-structured data whereas treating such data as sets requires an additional level of abstraction (supported by an appropriate technique such as some set theoretic query language) which is more than just using the rudiments of set theoretic terminology. XML documents, in fact, represent ordered tree structured data rather than arbitrary graph structured data, however, using the attributes `id` and `ref` allows one to imitate in XML arbitrary graphs as well. Considering the ordering of data in XML documents as an essential feature is related mainly with numerous software implementations which are deliberately sensitive to the order of such data. But, XML documents can also be treated as unordered, as we do in this thesis. Note that XML plays only an auxiliary role in our approach as a particular way of representing semi-structured data (XML-WDB format). Our main terminology and abstract data model is based on (hyper)set theory. #### The graph model and set theoretic model The interpretation of graph vertices as sets of their “children” leads us again to a set theoretic idea of representation of data, semi-structured data, a far going generalisation of the relational (value-oriented) approach. It is also worth noting that in the foundations of mathematics the previous century was marked by the triumph of the set theoretic approach for representing mathematical data as well as the style of mathematical language and reasoning. Mathematical logicians also developed generalised computability theory over abstract sets (of sets of sets, etc.) in the form of admissible set theory [6]. In computer science, the set theoretic programming language SETL [62, 63] was created, quite naturally, for the case of finite sets only. Also some theoretical considerations on computability and query languages over hereditarily finite sets were done in [20, 21, 43, 56, 57, 59, 61] with the perspective of a generalised set-theoretically presented databases – in fact semi-structured – even before the term “semi-structured databases” had arisen. Moreover, the set theoretic approach is closely related with a special version of the graph approach when graphs are considered up to bisimulation (see below). The first mathematical result relating both the set and graph approaches was Mostowski’s Collapsing Lemma, allowing the interpretation of graph vertices as sets of sets corresponding to children of these vertices. This, however, worked properly only for well-founded graphs and sets (which in the finite case, especially interesting for database applications, means the absence of cycles). But arbitrary graphs with cycles can also be “collapsed” into sets (interrelated by the membership relation) in the more general non-well-founded set theory also called hyperset theory [3, 5]. Here, for example the set $\Omega=\\{\Omega\\}$ consisting of itself is quite natural and meaningful, and corresponds to the simplest graph cycle $\circlearrowleft$. These two trends, from abstract set theory to more concrete graph model of semi-structured data (which is closer to implementation), and vice versa were called in [61] top-down and bottom-up approaches. They meet most closely in the work on UnQL query language [11] which is devoted to a specific graph model approach to semi-structured data considered up to bisimulation. The latter concept is also the key one in the works [41, 43, 56, 57, 61] (serving as the theoretical background for this thesis) for interpreting graph vertices as a system of (hyper)sets belonging one to another according to the graph edges. Nevertheless, [11] is still rather a graph approach than hyperset one according to the special, however related to, but not a genuine set theoretical way in which [11] treats graphs (see Section 11.3 and [61]). The main difference is that graphs considered in [11] have multiple “input” and “output” vertices, whereas graphs as considered in our hyperset approach have only one “input” corresponding to the set itself (and possibly one “output” corresponding to the empty set if it is contained in the transitive closure of this set). In fact, working with these “inputs” and “outputs” (used for appending one graph to another, etc.) is conceptually rather graph-theoretical than set-theoretical. #### Hyperset approach to semi-structured or Web-like databases As discussed above, the hyperset approach to semi-structured databases interprets graph structured data as abstract hypersets. Moreover, for the purposes of implementation, such graphs are represented as systems of set equations e.g. $\Omega=\\{\Omega\\}$ for the graph $\circlearrowleft$. In fact, arbitrary finite graphs can be rewritten into systems of set equations and vice versa, where graph vertices (or object identities) represent set names. Moreover, elements of sets in these set equations should be labelled according to labelling of graph edges, and, in fact, these labels are the carriers of atomic information in the hyperset approach to semi-structured databases. Furthermore, graph structure or, respectively, set-element nesting organises such atomic data, just like relational tables in the relational or nested relational approaches. The notion of equality between sets can be represented in graph terms by the bisimulation relation on vertices or set names whose idea consists, roughly speaking, in (recursively) ignoring the order and repetition. Thus, any two graph vertices or set names denote the same set if they are bisimilar, that is contain the same (recursively, up to bisimulation) elements. In fact, the bisimulation relation is very important in our approach being a fundamental concept underlying hyperset theory. ##### Hyperset query language $\Delta$ The associated $\Delta$-query language is based on set theory and predicate logic, being an extension of the basic or rudimentary operations [30, 39] – the _core_ fragment of $\Delta$. The set theoretic operators of the $\Delta$-language, like in the relational calculus, have clear and well- understood semantics. In fact, the expressive power of $\Delta$ (the core fragment plus transitive closure, decoration and recursion set theoretic operations) was shown in [57] and [43, 58] to capture all polynomial time computable operations over hereditarily finite sets and, respectively, hypersets. Also, another version of the language was shown in [40, 42] to capture exactly all LogSpace computable operations over hereditarily finite sets (without cycles). Therefore, in principle, the $\Delta$-query language can be reasonably considered as computationally viable and worthy of implementation. Some earlier preliminary work on the implementation of the $\Delta$-query language to WDB was done earlier by Yuri Serdyuk in [66], as well as in some practical attempt towards a new implementation based on multiple distributed agents working cooperately over the Internet [35] (taking into account the earlier theoretical work [60]). More recently the implementation work leading to this thesis was done in [49]. However, the latter implementation was insufficiently perfect. This antecedent work subsequently inspired the proposal for further research and the development of a sufficiently detailed implementation, that is, the point of the work done here. Note that some details of the implementation described here were published in [50]. ##### Implementation of the hyperset approach The goal of this work was to demonstrate how the hyperset approach to semi- structured or Web-like databases could be implemented, with the aim of presenting this approach in a practical rather than theoretical context and making it accessible to a more practically oriented audience. In particular, the practical characteristic of this work assumes representation of hyperset data as files distributed over the World-Wide Web and the implementation of the hyperset query language $\Delta$ allowing queries over such distributed data. Importantly, the implemented language should preserve the original high level, declarative character222 Recall that, for example, Prolog initially intended to be a logical, declarative programming language, eventually has both declarative and imperative features. This mixture of ideologies was the result of making this language more efficient. and retain its set theoretic style. Further, this approach should demonstrate the power of the set theoretic style of thought towards semi-structured databases. Note that the query system (which is implemented in Java) and the example queries described in this thesis can be found at http://www.csc.liv.ac.uk/~molyneux/t/ ##### Efficiency issues Another goal consisted in the subsequent investigation of theoretical considerations arising from this experimental implementation, specifically the problem of efficient implementation of the equality or the bisimulation relation – which crucially underlies this hyperset theoretic approach. Moreover, our proposed solution was restricted to making the bisimulation relation efficient only in context of distributed WDB which may require numerous and particularly expensive downloads of files from the World-Wide Web. However, this work does not consider the problem of efficiency in the non-distributed case, especially taking into account the previous works on efficient bisimulation algorithms that, on the other hand, do not consider distribution [24, 25]. Note that, many other aspects of efficiency of the implementation (such as indexing, hashing and other physical data organisation techniques [73]) as well as various other questions which should be resolved for creating a sufficiently realistic database management system were inevitably postponed here. In fact, the primary aim of this work was the correct and meaningful implementation of a non-trivial and user friendly version of the $\Delta$-language. #### Organisation of the thesis Details of the implementation are rather technical, thus it makes sense to firstly explain the intuitive (or high level) meaning of the hyperset approach and demonstrate example queries of the implemented $\Delta$-query language. Secondly, technical details of the implementation appear towards the end of the thesis detailing the lower level aspects of our approach. Note that, the material presented in this thesis follows an intuitive perception of this approach towards semi-structured databases rather than a strict logical dependency. The thesis is organised into four parts: Part I, “Hyperset approach to querying Web-like databases”, gives an overview of the implemented hyperset approach to semi-structured or Web-like databases and the associated query language $\Delta$, including worked example queries. The point of this part is to introduce this approach on an intuitive level before discussing the technical details of implementation. Part II, “Local/global approach to optimise bisimulation and querying”, is concerned with the problem of efficient implementation of the equality or bisimulation relation. Here two joint strategies were suggested for resolving this problem: i) implementation of an Internet service for resolving bisimulation questions, and ii) the computation of bisimulation approximations on fragments of distributed Web-like databases to aid the computation of global bisimulation. The viability of these suggestions as solutions is supported by empirical testing. Part III, “Implementation issues”, presents the technical details of the implementation of the hyperset approach towards semi-structured or Web-like databases. We start by detailing query execution (which we feel is potentially more important for readers) followed with query parsing and contextual analysis, although query execution is, in fact, formally dependent on the latter syntactical considerations. Finally, XML representation of WDB systems of set equation has a quite isolated role in our approach and is presented at the end of this technical material, but this discussion is actually quite self-contained and can be read independently of the rest of this thesis. Part IV, “Evaluation”, concludes with comparative analysis with other known approaches towards semi-structured databases, and finishes with some future prospects and closing remarks. ## Part I Hyperset approach to querying Web-like databases ### Chapter 2 Semi-structured or Web-like databases The term _semi-structured data_ denotes data which has a characteristically unfixed or non-rigid structure, thus semi-structured data is considered as “schemaless” or “self-describing”111 The consideration of semi-structured data as “self-describing” is somewhat misleading as it might be wrongly thought to suggest clear semantic description of such data. In particular, when considering the graph representation of semi-structured data, labels have only an informal meaning dependant on subjective interpretation of language, e.g. the imprecise term “location” could have many interpretations – address, map coordinates, URI, anatomical, etc. having no complete structural description or schema [1]. However, typically semi-structured data is similar to structured data e.g. relational data (as described below) but without strictly imposed structure. More specifically our approach to semi-structured databases is based on (hyper)set theory [3, 5]. #### 2.1 Set theoretic view of structured and semi-structured data ##### 2.1.1 Structured relational data Structured data has a fixed and rigid structure such as relational data [17] described by relational schema $R(A_{1},A_{2},...,A_{n})$, where $R$ is _relation_ name and $A_{i}$ are _attributes_ (constrained by the domain $D_{i}$). In the relational model, relations are naturally represented as tables with attributes as named columns of a table. For example, the `Stud` relation shown in Figure 2.1 has the attributes `forename`, `surname`, `DOB` (date of birth) and `department`. Figure 2.1: Relational table of students. The relational approach is essentially based on set theory, as well as on logic. For example, the `Stud` relation (above) can be represented as set of student _tuples_ (rows or records), Stud = { st1, st2, ... } or, better, as Stud = { student:st1, student:st2, ... } where each student tuple is represented as a set of labelled atomic values, with labels being _attribute names_ , and _attribute values_ as atomic values (strings of symbols between quotation marks to distinguish them from set names and attribute names), st1 = { forename:"Jack", surname:"Jones", DOB:"30/6/1986", department:"DeptChemistry" } st2 = { forename:"Sarah", surname:"Smith", DOB:"27/11/1988", department:"DeptBiology" }. Let us consider the relational database `Univ` as the following set of (labelled) relations, Univ = { departments:Dept, students:Stud, lecturers:Lect, modules:Mod, courses:Course, ... }. The relations `Dept`, `Lect`, `Mod` and `Course` will not be further described, they are plausible example relations, like `Stud`, that could belong to a University database. Here the labels (or attributes) `departments`, `students`, `lecturers`, etc., give an informal description of what the sets `Dept`, `Stud`, `Lect`, etc., are about. These sets could be denoted differently, say as `D`, `S`, `L`, etc. Thus, strictly speaking the denotation of sets does not necessarily carry informational content. Hence the important role of labels (attributes e.g. `forename`) and atomic values (e.g. `"Jack"`), which are the proper carriers of basic information. ##### 2.1.2 Relaxation of structural restrictions on relational data Relational data with the given schema $R(A_{1},A_{2},...,A_{n})$ has a rigid structure with mandatory attributes $A_{i}$ for associated tuple components. It is also known of the more general approaches to _nested_ relational databases [52, 54, 71] where attribute values could be relations. Say, in the above example we could reconsider `DeptChemistry` as a set (instead of an atomic value) by omitting the quotation marks around `DeptChemistry` and adding the corresponding set equation further detailing the chemistry department: DeptChemistry = { name:"Department_of_Chemistry", lecturers:ChemLect, modules:ChemMod, ... }. Moreover, we could relax the requirement on students tuples to have a value for each attribute `forename`, `surname`, `age` and `department`. For example, the DOB of a student could be absent by some reason, but some other information could be present, such as email:"jones@liv.ac.uk" or, sex:"male". Thus, relaxation of traditional structural restrictions on relational databases leads naturally to semi-structured databases, in fact, to the set theoretic approach where such data are considered as _arbitrary_ set of (labelled) sets of sets, etc., to any depth, represented by set equations like above. ##### 2.1.3 Semi-structured data For simplicity, we consider semi-structured data as systems of _flat_ set equations where a set equation consists of set name $s_{i}$ equated to a bracket expression $B_{i}(\bar{s})$ like those considered in the above example. In vector form this can be summarised as $\displaystyle\bar{s}=\bar{B}(\bar{s}).$ Flat bracket expression $\\{l_{1}:s_{i_{1}},\ldots,l_{n}:s_{i_{n}}\\}$ is thought of as a set of labelled elements. In the flat (non-nested form) only set names $s_{i}$ from the list of all set names $\bar{s}=s_{1},s_{2},...,s_{n}$, may participate as elements. Labels $l_{j}$ can be considered as analogous to attributes in the relational approach, however, element labelling is optional with the default label being the empty label $\Box$ (or null) which can be considered as invisible, such as the absence of labelling in the `Stud` set above. Formally our general approach does not consider atomic values such as `"Jack"`, `"Jones"`, etc., from the example above. However, any atomic value can be simulated as a set consisting of one labelled empty set [41, 57, 61], such as "Jack" = {’Jack’:{}}. Strictly speaking, we should use single quotation marks for labels (often omitted for simplicity) and double quotation marks for atomic values. Of course, we can still use the denotation for atomic data like `"Jack"`, but it should be understood as above. ##### 2.1.4 Syntactical and conceptual set nesting In the case where nesting is allowed (like the participation of `{}` in the above definition of atomic values, and also in more complicated cases) any set name $s_{i}$ can be substituted with the corresponding nested bracket expression $B_{i}$, and vice versa. For example, the `Stud` set equation could be rewritten with the nested right-hand side (and adding the `student` attribute) as follows, Stud = { student:{ forename:"Jack", surname:"Jones", DOB:"30/6/1986", department:"DeptChemistry" }, student:{ forename:"Sarah", surname:"Smith", DOB:"27/11/1988", department:"DeptBiology" } }. Here the nesting of data inside the `Stud` set equation proves useful in avoiding the introduction of new set names, and thus eliminating `st1` and `st2`. Moveover, this demonstrates that set names in set equations play an auxiliary role, and can even be readily renamed in an analogous way to renaming variables in any ordinary algebraic equations. Thus the real information of such semi-structured data is carried by labels and set/element nesting. More generally, we could allow (and, in fact, will consider later) arbitrary nesting in the right-hand sides of set equations $\bar{s}=\bar{B}(\bar{s})$. This can be evidently “unnested” or “flattened” by introducing new (fresh) set names and appropriate set equations. So, our restriction for non-nested systems of set equations (i.e. with non-nested right-hand sides) is not essential, but can simplify some considerations. In fact, the notion of non-nested or flat system of set equations is only syntactical and, conceptually, flat systems of set equations allow arbitrary nesting with the participation of set names (corresponding to set equation) as elements #### 2.2 Hyperset theoretic view of semi-structured data In the above approach to semi-structured data via systems of set equations $\bar{s}=\bar{B}(\bar{s})$ there was, in fact, no restriction on the form of these equations. Thus allowing not only arbitrarily nested, but also cycling data like in the simplest example of a set consisting of itself $\displaystyle\Omega=\\{\Omega\\}.$ Mathematically, such kind of sets are considered as non-traditional, although they have already been deeply investigated in _hyperset theory_ , as represented in the books [3, 5]. From the point of view of semi-structured data there is nothing strange in such sets. Imagine that we have a relational table where some cells can represent other relational tables, etc. Such nesting can be implemented so that “clicking” on such a cell leads to the corresponding nested relational table shown instead of the original table. There is no technical or conceptual problem to have such a situation that after several such “clicks” we will arrive back to the original table we started “clicking” with – like in the World-Wide Web by successive “clicking” we can possibly return to the Web page we started with. Moreover, from the informational or database point of view this can be quite meaningful. For example, let us consider the University database where formally the student set `st1` has the chemistry department set `DeptChemistry` as the member, and (possibly many) students are members of the `ChemStud` set of enrolled chemistry students, as described by mutually recursive set definitions, st1 = { forename:"Jack", surname:"Jones", DOB:"30/6/1986", department:DeptChemistry } DeptChemistry = { ..., enrolled:ChemStud, ... } ChemStud = { student:st1, ... } with `ChemStud` a subset of the set `Stud` of all university students. Any set (name) $s_{i}$ can be defined by referring to other set (names) as elements, etc., so that eventually we could possibly come to the original set $s_{i}$ – thus, arbitrary cycling is allowed. There is more to say about the hyperset approach to semi-structured data on the conceptual level, in particular, on the concept of equality between sets (possibly denoted by different set names) but we will postpone this discussion to Section 2.4.1. On the current very preliminary level of consideration sets are thought simply as syntactical bracket expressions, or as represented by formal systems of set equations. In fact, we need an abstract concept of hypersets amongst which we could find a (unique) solution to any given system of set equations. #### 2.3 Graph or Web-like view ##### 2.3.1 Graph representation of systems of set equations Representation of semi-structured databases by systems of set equations presents a clear and mathematically well-understood222 taking into account Section 2.4 conceptual view of semi-structured data as (hyper)sets. But it also makes sense to consider visualisation of systems of set equations by the equivalent representation as (finite) labelled directed graphs. In fact, it is important for all considerations of this work that any given system of set equations can be considered as a labelled directed graph. Figure 2.2: Semi-structured database Univ represented as directed graph. In fact, most approaches to semi-structured databases typically consider them as labelled directed graphs, that is, semi-structured data is modelled as (finite) directed graph $G=\langle N,E\rangle$ with $L$-labelled edges, where $L$ is an infinite set of possible labels ($l_{1},l_{2},\ldots$, etc., and the empty label $\Box$), $N$ is a finite set of nodes ($s_{1},s_{2},\ldots$, etc.), and $E$ is a finite set of edges with each edge $s_{i}\stackrel{{\scriptstyle l_{k}}}{{\rightarrow}}s_{j}$ being formally an ordered triple of the form $\langle s_{i},s_{j},l_{k}\rangle$. For example, the University database considered in Section 2.1 has the corresponding representation by directed graph shown in Figure 2.2. The membership of labelled element $label\\!:\\!s_{2}$ to the set $s_{1}$ ($label\\!:\\!s_{2}\in s_{1}$) corresponds to the labelled edge $s_{1}\stackrel{{\scriptstyle label}}{{\longrightarrow}}s_{2}$ (and vice versa), where set names $s_{i}$ serve as (the unique names of) graph nodes. In general, each set equation $s_{i}=\\{l_{1}\\!:\\!s_{i_{1}},\ldots,l_{n}\\!:\\!s_{i_{n}}\\}$ from the system generates a fork of labelled edges $s_{i}\stackrel{{\scriptstyle l_{1}}}{{\longrightarrow}}s_{i_{1}},\ldots,s_{i}\stackrel{{\scriptstyle l_{n}}}{{\longrightarrow}}s_{i_{n}}$ outgoing from $s_{i}$, as depicted in Figure 2.3. All those forks generated from every set equation give the corresponding representation as graph. Vice versa, any graph with labelled edges is evidently visualising a system of set equations, with one equation for each node so that each node is thought as a (hyper)set. Thus, graphs and (formal) systems of set equations are essentially equivalent concepts. Figure 2.3: Forking of labelled edges generated by the set equation $s_{i}=\\{l_{1}\\!:\\!s_{i_{1}},\ldots,l_{n}\\!:\\!s_{i_{n}}\\}$. ##### 2.3.2 Graphs or systems of set equations as Web-like databases The World-Wide Web (WWW) can, in principle, be considered as a large semi- structured database, consisting of an arbitrarily organised collection of hyperlinked HTML documents. Each HTML document has a corresponding URL (WWW address), and contains textual data with markup tags denoting visualisation and hyperlink information. The following fragment of HTML code is an example of a hyperlink, <a href="http://www.liv.ac.uk/">University of Liverpool</a> what in our symbolism of labelled elements can be represented as $\texttt{University of Liverpool}\,:\,\mbox{\url{http://www.liv.ac.uk/}}$ and visually (in Web browser) this hyperlink would appear as “clickable” fragment of text $\underline{\texttt{University of Liverpool}}$ with the URL hidden. Hiding of URLs corresponds to the idea mentioned above that set names (names of graph nodes) actually do not matter from the point of view of the proper information. Only labels on edges or the “clickable” links (and other text and visual content) on Web pages carry information, plus, of course, the graphical structure. That is, URLs play a different role than proper information in the WWW. In Figure 2.4 we consider browsing between hyperlinked HTML documents by “clicking” on such links. It is evident from this example that hyperlinked HTML documents can express arbitrary relationships, for example the cycle when browsing by “clicking” on the links, `Departments`, `Medicine`, `University of Liverpool`, and so on. Thus, any hyperlink can be denoted by the labelled edge $url_{i}\stackrel{{\scriptstyle label}}{{\longrightarrow}}url_{j}$, suggesting the intuitive understanding of hyperlinking as arbitrary labelled directed graph. Therefore, systems of set equations or equivalently labelled direct graphs, can be more generally named by the analogy _Web-like Databases_ (WDB) [19, 41, 60, 61]. Furthermore, our approach also considers WDB as Web-like with distribution over the Internet (in a similar manner to hyperlinks), however, it is intended to be smaller, simpler and better organised than the WWW. Such WDB graphs can, in principle, be quite arbitrary but in real applications it is assumed to be governed by some organisation or company, and possibly not allowed to be arbitrarily extended by anybody in the world (like typical databases). Additionally, WDB (or semi-structured data) can also have a schema restricting the shape of the WDB, but not necessarily so rigid like in the case of relational databases, see for example [9, 41, 57]. However, we will not go further into these details. Figure 2.4: Browsing of hyperlinked HTML documents on the University of Liverpool website. ##### 2.3.3 Distributed WDB Any WDB represented as a system of set equations $\bar{s}=\bar{B}(\bar{s})$ can be quite big, and naturally divided into subsystems of set equations. Each subsystem corresponds to a XML-WDB file (see Chapter 10 for details of the XML-WDB representation) containing only some of the equations (desirably closely interrelated by a subject matter). Moreover, these files could be distributed between various servers over the world, like HTML files on the World-Wide Web. It may happen that set equations defined in some WDB file may involve set names defined by equations in other (non-local) WDB files. Furthermore, when considering the real application of WDB distribution proves useful in the creation and management of (potentally large) databases, such as the plausible distribution of the University WDB. Let us consider that in the case of the University WDB, set equations might be distributed between many WDB files, let us say by department. Therefore, the WDB file http://www.liv.ac.uk/ChemistryDepartment.xml could contain the following subsystem of set equations333This is still not very realistic situation to assume that the file ChemistryDepartment.xml contains all set equations related with this department (on students, lecturers, etc.). These set equations should be further divided into natural fragments (WDB files). : DeptChemistry = { ..., enrolled:ChemStud, ... } ChemStud = { student:st1, ... } Likewise, the WDB file http://www.liv.ac.uk/BiologyDepartment.xml could contain the subsystem of set equations: DeptBiology = { ..., enrolled:BiolStud, ... } BiolStud = { student:st2, ... } Moreover, there could also be the WDB file `Students.xml` containing the set equations `st1 = {...}` and `st2 = {...}`. Thus, the set names `st1`, `st2`, etc. participating, respectively, in `ChemistryDepartment.xml` and `BiologyDepartment.xml` would now be described as sets in another file. In this case, we should consider the full versions of the simple set names, `st1`, `st2`, etc., described in http://www.liv.ac.uk/Students.xml, as discussed below. ###### 2.3.3.1 Full versus simple set names Taking into account the above example, any given set name should be considered as a _full set name_ , consisting of WDB file URL and _simple set name_ (with the simple set name described within the WDB file). For example, in the distributed University WDB considered above, the full set name of the biology student `st2` would be http://www.liv.ac.uk/Students.xml#st2 with the WDB file URL and simple set name delimited by `#` symbol. However, in practice it suffices to use simple set names in the left-hand side of set equations, and also for those occurrences of set names appearing in the right- hand side of set equation definitions if they are defined in the same WDB file. In particular, the author of a WDB file can freely use any simple set name (as such or as part of full set names) without the danger of clashes with simple names participating in the other WDB files. However, there is one subtle point: if a simple set name `set_name` occurs twice in some WDB file, once as a simple set name and again as part of a full set name `url#set_name` (with `url` referring to some different WDB file). Then in the latter case it refers to another file where the corresponding equation is defined, even if the current file already contains the equation `set_name = {...}`. Thus, these two occurrences are actually different set names because their corresponding full set names are indeed different. Of course, each set name must be defined either in the same or some other WDB file. Otherwise it is considered as syntactical error. Thus, it is necessary to download some WDB files whose URLs appear in full set names of the given file to confirm the existence of defining equations of the referenced set names. #### 2.4 Hyperset data considered abstractly The notion of WDB as a system of set equations presents a low level, syntactical understanding of semi-structured data. However, conceptually (and semantically) WDB is understood as consisting of _abstract_ hypersets (like relational database consists of abstract relations). The hyperset approach considers WDB as an arbitrary finite system of set equations, each set equation consisting of set name equated to corresponding bracket expression. But the intended meaning of such a syntactical expression is a set of labelled elements, _not_ an ordered sequence. Therefore according to this (hyper)set theoretic approach ordering and repetition of elements in a bracket expression should be completely ignored. That is, ignoring ordering and repetitions has some both _operational_ and _conceptual_ consequences. This can possibly lead to equality between different set names $s_{i}$ and $s_{j}$ denoted as $s_{i}=s_{j}$ and meaning that $s_{i}$ and $s_{j}$ denote the same abstract hyperset, or strictly denoted as $s_{i}\approx s_{j}$ (to avoid possible misunderstanding of $s_{i}=s_{j}$ as the assertion that these set names are identical, and to stress on the particularly important role of this concept of equality). In fact, $\approx$ is the well known concept in the context of graphs called _bisimulation relation_ between graph nodes or, in our case, between set names [3, 5, 61]. As the role of this relation is crucial for the hyperset approach to semi-structured databases, this approach is therefore more than pure graph theoretic, as considered in the approaches to semi-structured databases as graphs e.g. in [1, 2, 11, 18, 19, 36, 46] or as XML tree-like data e.g. in [23, 33]. Note that, however, [11] is also heavily based on the bisimulation relation, it is rather a graph than a hyperset approach as was argued in [61]. ##### 2.4.1 Bisimulation – preliminary considerations In general, the bisimulation relation between set names (graph nodes) of a WDB, i.e. a system of set equations, and the corresponding recursive algorithm is based on the idea that any two sets are equal if for each (labelled) element of the first set there exists an equal (bisimilar) element in the second set (and vice versa). Bisimilar set names are said to denote the same abstract (hyper)set. The bisimulation relation will be further described in Chapter 4, with formal theoretical definition, and practical considerations for its implementation. We consider that this hyperset approach to WDB is worth implementing as it suggests a clear and mathematically well-understood view on querying such semi-structured data. A WDB is called _strongly extensional_ [3] or non-redundant, if different set names (nodes) are non-bisimilar i.e. denote different hypersets. In the case of strongly extensional WDB, equality between set names (nodes) trivially becomes the syntactical identity relationship. Otherwise, even the simplest queries like $x=y$ or $x\in y$ can be quite expensive to evaluate, especially in the case of distributed WDB. Therefore, we devote Part II to some approach of dealing with this problem practically. ###### 2.4.1.1 Example Consider the set equations below, where trivially $x\approx x^{\prime}$ holds because our (hyper)set approach ignores the ordering and repetition of elements: $\displaystyle x=\\{y,z\\}$ $\displaystyle x^{\prime}=\\{z,y,z\\}.$ However, set names (or graph nodes) may be equal (bisimilar) for some “deeper” reason than for $x$ and $x^{\prime}$ above. Let us consider the above example extended with the (recursive) definitions of the sets $z$, $y$ and $y^{\prime}$: $\displaystyle z=\\{\\}$ $\displaystyle y=\\{x\\}$ $\displaystyle y^{\prime}=\\{x^{\prime}\\}.$ The sets $y$ and $y^{\prime}$ both contain one element of syntactically differing set names ($x$ and $x^{\prime}$ respectively), thus suggesting that $y$ and $y^{\prime}$ might not be equal. However, the bisimulation relation defines two sets as equal if for each element of the first set there exists an equal (or bisimilar) element in the second set, and vice versa. In the case above we already know that $x\approx x^{\prime}$ holds, and according to this informal definition of bisimulation all of the elements of $y$ are bisimilar to the elements of $y^{\prime}$, and vice versa. Therefore we can deduce that, in fact, $y\approx y^{\prime}$ holds. Let us now consider the strongly extensional version of this system of set equations obtained by eliminating the redundant set names $x^{\prime}$ and $y^{\prime}$, and omitting repetitions. Thus, after “collapsing” the bisimilar nodes $x^{\prime}$ to $x$ and $y^{\prime}$ to $y$, and omitting element repetitions, the resulting system of set equations is $\displaystyle x=\\{y,z\\}$ $\displaystyle y=\\{x\\}$ $\displaystyle z=\\{\\}.$ Thus, the elimination of redundancies (in the above system of set equations) is visualised by Figure 2.5. (a) Redundant version, with red dashed edges relating bisimilar nodes (or sets) (b) Non-redundant (strongly extensional) version Figure 2.5: Graphical representation of a trivial WDB (cf. corresponding set equations above). ##### 2.4.2 Redundancies in WDB The above example, although artificial, demonstrates that bisimilarity between set names introduces redundancies into WDB. However, the crucial question in implementing the hyperset approach to WDB is whether the bisimulation relation ($\approx$) can be computed in any reasonable and practical way. Some possible approaches and views are outlined below. In principle, the occurrence of bisimilar nodes in a realistic WDB (i.e. redundancies) should be infrequent. Therefore, such rare redundancies can be eliminated by supporting WDB in a _strongly extensional_ state, with redundancies detected or even eliminated instantly as soon as they might potentially appear. Trivially, after eliminating redundancies equality between sets (i.e. bisimulation relation between set names or graph nodes) becomes the identity relation. However, eliminating redundancies is more expensive than only detecting them i.e. just computing bisimulation relation on the WDB. Thus, supporting WDB in strongly extensional form may be reasonable option when WDB is not large. WDB should not be assumed to be just another version of WWW, freely extensible by anybody in the world. That is, an appropriate discipline of working with WDB could make the problem of bisimulation practically resolvable. Let us now consider several ways by which redundancies can appear. ###### 2.4.2.1 Redundancies arising during query execution Execution of queries leads to the temporary extension $\textrm{WDB}^{\prime}$ of the original WDB (as detailed later in Section 3.3), with the addition of new set names and set equations locally. Such extensions $\textrm{WDB}^{\prime}$ may potentially give rise to new redundancies, so that equality subqueries applied to these newly generated sets becomes non-trivial. Note that the set names in original WDB do not refer to new ones in $\textrm{WDB}^{\prime}$, thus WDB remains self-contained. Therefore, the new bisimulation relation ($\approx^{\prime}$) on $\textrm{WDB}^{\prime}$ restricted to those set names in WDB coincides with the identity relation on WDB. Moreover, the algorithm of query execution could be amended in such a way that as soon as new (auxiliary) set names are generated (like $res$ in Section 3.3) any possible redundancies will be eliminated immediately. It should also be taken into account that the extensions $\textrm{WDB}^{\prime}$ arising during query execution have several specific types, and are sufficiently simple and small, thus making the process of detecting/eliminating redundancies easier, see also [40, 42], but we will not go into the details here. ###### 2.4.2.2 Redundancies which can appear during a local update Local updates of WDB files are more problematic because previously non- bisimilar nodes outside this file may become bisimilar due to possible links (or paths) to the local nodes with changed/added meaning. The appropriate (more efficient than the standard) strategy of detecting/removing all such redundancies is not so straightforward and needs to be developed yet. However, taking into account the locality of changes, this task does not seem to be unrealistic. ###### 2.4.2.3 Deliberate redundancies Deliberate redundancies in WDB can also appear with the same aim as mirroring in WWW. But, if there is a requirement to officially registered such mirroring in the WDB, then such deliberate redundancies should most plausibly be dealt with in a quite feasible way. ###### 2.4.2.4 Local versus global bisimulation Unlike the other considerations above, we will consider the “local/global” approach and its implementation for supporting bisimulation relation on WDB (in background time) in more detail (see Part II). Now we present only some general introductory comments on this idea. Assume that all WDB nodes are divided into classes $L_{i}$ according to their sites (WDB servers) or even files. There is a quite natural definition of local (i.e. computed locally) lower and upper approximations ($\approx_{-}^{L},\approx_{+}^{L}$) to the global bisimulation relation ($\approx$) on the whole WDB: $\displaystyle n_{1}\approx_{-}^{L}n_{2}\Rightarrow n_{1}\approx n_{2}\Rightarrow n_{1}\approx_{+}^{L}n_{2}$ These approximations can help to compute and to permanently support global bisimulation in a distributed way in background time. Moreover, we could require _local independence_ (${\approx^{L}_{-}}={\approx^{L}_{+}}$, and hence ${}={\approx\upharpoonright L}$) and additionally _local non-redundancy_ (${\approx^{L}_{-}}={\approx^{L}_{+}}={=^{L}}$). ##### 2.4.3 Bisimulation invariance The hyperset approach assumes considering WDB (graphs or systems of set equations) up to bisimulation. Therefore, it is an important requirement for set theoretic operations and relations to be _bisimulation invariant_ , that is to preserve the bisimulation relation. Although not fully proven here, it can be shown [58] that all definable queries $q$ of the hyperset $\Delta$-query language444 The operational meaning of $\Delta$-queries are defined graph theoretically or in terms of set equations. (see Chapter 3) are bisimulation invariant: $\displaystyle\bar{x}\approx\bar{y}\Longrightarrow q(\bar{x})\approx q(\bar{y})\quad\textrm{(for set valued queries)}$ $\displaystyle\bar{x}\approx\bar{y}\Longrightarrow q(\bar{x})\Leftrightarrow q(\bar{y})\quad\textrm{(for boolean queries).}$ For example, in the case of the set theoretic operation union we have: $\displaystyle x_{1}\approx y_{1}\;\&\;x_{2}\approx y_{2}\Rightarrow(x_{1}\cup x_{2})\approx(y_{1}\cup y_{2}).$ This actually means that we work with (abstract) hypersets rather than just with graph nodes or set names, however the operational semantics of the language $\Delta$ is based on the syntactical manipulations of set equations [61]. The point is that the semantics of the language $\Delta$ respects bisimulation and completely agrees with the hyperset theory [3, 5]. In particular, $x_{1}\cup x_{2}$ is defined as a new set name, say $u$, with corresponding new set equation $u=\\{\ldots,\ldots\\}$, where the first “$\ldots$” is the content of the right-hand side of the equation $x_{1}=\\{\ldots\\}$ from the given WDB, and similarly for the second “$\ldots$” and the equation $x_{2}=\\{\ldots\\}$. The union $y_{1}\cup y_{2}$ is computed in the same way from set equations for $y_{1}$ and $y_{2}$ giving rise to new set name, $u^{\prime}$, and the corresponding set equation $u^{\prime}=\\{\ldots,\ldots\\}$. Then the conclusion of the above bisimulation invariance condition for $\cup$ actually means $u\approx u^{\prime}$, and can evidentially be shown. Note that the membership relation $x\in y$ for two sets (considering the unlabelled case for simplicity) is defined to be true if the set equation for $y$ involves some set name $x^{\prime}$, where $y=\\{\ldots,x^{\prime},\ldots\\}$ and, moreover, $x\approx x^{\prime}$. Additionally, it can be shown that the membership relation is also bisimulation invariant: $\displaystyle x_{1}\approx y_{1}\;\&\;x_{2}\approx y_{2}\implies x_{1}\in x_{2}\iff y_{1}\in y_{2}$ For all other constructs of the $\Delta$-language the operational semantics maybe more complicated, however, it follows that they also agree with this intuitive (abstract) set theoretical meaning. The syntax and semantics of the $\Delta$-query language will be further detailed in Sections 3.1 and 3.2, with some further indications of the operational semantics in terms of set equations detailed in Section 3.3. ##### 2.4.4 Anti-Foundation Axiom Finally, we do not go into full mathematical details on hypersets, however, we could assert the following form of Anti-Foundation Axiom (AFA) [3, 5], which holds in the universe of abstract (in our case finite) hypersets: > _Any system of set equations $\bar{s}=\bar{B}(\bar{s})$ has a unique > abstract hyperset solution for set names $\bar{s}$ making these equations > true._ Therefore, set names of any WDB (as system of set equations) denote quite concrete, uniquely defined abstract hypersets. In this sense each set name (in a $\Delta$-query) serves as a set constant (relative to the given WDB) denoting a unique hyperset. Note that, the $\Delta$-language also has set variables which can be quantified unlike constants. Strictly speaking all of this makes precise mathematical sense only in context of Chapter 4, which further details the bisimulation relation (with some additional mathematical considerations) beyond the general informal description of bisimulation relation so far. ### Chapter 3 Query language $\Delta$ #### 3.1 The syntax There has already been much theoretical considerations on (some versions of) the $\Delta$ (Delta) query language to hyperset/WDB databases [40, 41, 43, 57, 61]. The two main syntactical categories of $\Delta$ are: * • $\Delta$-_terms_ representing set valued operations over hypersets (_set queries_), and * • $\Delta$-_formulas_ representing truth valued operations (_boolean queries_). Note that the denotation $\Delta$ bears partly from the well-known class $\Delta_{0}$ of bounded formulas introduced by Levy, although $\Delta$, as defined here, denotes a wider language. It is based on the _basic_ or _rudimentary_ set theoretic languages of Gandy [30] and Jensen [39]. Moreover, inclusion of set theoretic operators: transitive closure (TC), recursion (Rec) and, for the case of hypersets, decoration (Dec) (the latter due to Forti and Honsell [29] and Aczel [3]), allows to define in $\Delta$ exactly all polynomial time computable operations over hypersets represented as WDB, thus demonstrating and characterising theoretically its rich expressive power (assuming that a linear order on labels is given) [43, 56, 57, 58]. The operators of $\Delta$ are defined as follows: $\displaystyle\langle\mbox{$\Delta$-term}\rangle::=\;$ $\displaystyle\langle\mbox{set variable or constant}\rangle\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}\emptyset\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}\\{l_{1}:a_{1},\ldots,l_{n},a_{n}\\}\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}\bigcup a\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}\mbox{\sf TC}(a)\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}$ $\displaystyle\\{l:t(x,l)\mid l:x\in a\mathrel{\&}\varphi(x,l)\\}\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}\mbox{\sf Rec}\;p.\\{l:x\in a\mid\varphi(x,l,p)\\}\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}\mbox{\sf Dec}(a,b)$ $\displaystyle\langle\mbox{$\Delta$-formula}\rangle::=\;$ $\displaystyle a=b\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}l_{1}=l_{2}\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}l_{1}<l_{2}\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}l_{1}\mathrel{R}l_{2}\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}l:a\in b\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}\varphi\mathrel{\&}\psi\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}\varphi\vee\psi\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}\neg\varphi\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}$ $\displaystyle\forall l:x\mathrel{\in}a.\varphi(x,l)\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}\exists l:x\mathrel{\in}a.\varphi(x,l)$ The intuitive set theoretic semantics of the majority of the above constructs should be well-understood by anyone with the minimal mathematical background in set theory and logic. In the above constructs we denote: $a,b,\ldots$ as (set valued) $\Delta$-terms; $x,y,z,\ldots$ as set variables; $l,l_{i}$ as label values or variables (depending on the context); $l:t(x,l)$ is any $l$-labelled $\Delta$-term $t$ possibly involving the label variable $l$ and the set variable $x$; and $\varphi,\psi$ as (boolean valued) $\Delta$-formulas. Note that labels $l_{i}$ participating in the $\Delta$-term $\\{l_{1}\\!:\\!a_{1},\ldots,l_{n}\\!:\\!a_{n}\\}$ need not be unique, that is, multiple occurrences of labels are allowed. This means that we consider arbitrary sets of labelled elements rather than records or tuples of a relational table where $l_{i}$ serve as names of fields (columns). The binding label and set variables $l,x,p$ of quantifiers, collect, and recursion constructs should not appear free in the bounding term $a$ (denoting a finite set). Otherwise, these operators may become unbounded and thus, in general, non-computable. For example, let us consider the universal quantifier $\forall l\\!:\\!x\mathrel{\in}\\{\ldots,l\\!:\\!x,\ldots\\}.\varphi(x,l)$ which becomes unbounded due to the quantified variables $l\\!:\\!x$ participating in the bounding term $\\{\ldots,l\\!:\\!x,\ldots\\}$. In fact, as $l\\!:\\!x\in\\{\ldots,l\\!:\\!x,\ldots\\}$ is always true the above quantified formula proves to be equivalent to unbounded one: $\forall l\\!:\\!x.\varphi(x,l)$. #### 3.2 Intuitive denotational semantics Any $\Delta$-query without free variables has either: i) (hyper)set value in the case of $\Delta$-terms, or ii) boolean value in the case of $\Delta$-formulas. Those participating set variables or set constants represent abstract hypersets (and thus correspond to set names in WDB), whereas participating label variables or label constants represent label values (corresponding to strings of symbols). The intuitive meaning of $\Delta$-queries is described by the _denotational semantics_ , that is what any expression denotes111 There is a deep mathematical theory of denotational semantics of programming languages based on Domain Theory [65, 68] (also see the contemporary reference [28]) to represent denotational values of a programming language expressions. The language $\Delta$, where all computations evaluating queries are finite, does not require this theory which is based on the idea of potentially infinite computations (embodied in the so called “undefined” element $\perp$). Anyway, it makes sense to use the term denotational semantics, although we will describe this semantics on a very intuitive level by reference to the “domain” of sets and hypersets. . For the purposes of implementation $\Delta$-queries are also described by means of their _operational_ or computational semantics (see Section 3.3) which must be coherent with our intuitive denotational semantics. Here we will also rely on intuition, without presenting any precise argument. In fact, the required coherence will be pretty much evident. So, we can concentrate on examples of queries and implementation aspects. ##### 3.2.1 Boolean valued expressions — $\Delta$-formulas _Equality_ ($=$) and the _alphabetic ordering_ ($<$) between labels is understood standardly. In the theoretical $\Delta$-language the relation $\mathrel{R}$ over labels is any easily computable relation over labels, however, in the implemented $\Delta$-language described in this thesis we consider $R$ as any of the following _substring_ relations $\displaystylel_{1}=l_{2}\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}l_{1}=l_{2}\mathrel{\rule[-2.5pt]{1.00006pt}{10.00002pt}}l_{1}=l_{2}$ where the wildcard $$ represents any string of symbols. In principle we could include into the language more relations over labels, but in the implementation there are only $<$ and substring relations, and the user currently has no way to define more primitive relations over labels. It should be noted that equality between $\Delta$-terms, $a=b$ or, for technical reasons, $a\approx b$, is understood as the equality of abstract hypersets denoted by these terms and, as such, is computed by the bisimulation algorithm discussed in Chapter 4. That is, when we discuss hypersets abstractly, we use $=$. But when considering bisimulation algorithm to determine whether two set names or graph nodes denote the same abstract hyperset, we use $\approx$. In the implemented version of the language we have only $=$ which, of course, involves calling the bisimulation algorithm, but this is hidden from the user who, therefore can think on hypersets abstractly. Moreover, bisimulation is implicitly involved in the (computational) meaning of the _membership_ relation according to the equivalence $\displaystyle l\\!:\\!a\in b\iff\exists m\\!:\\!x\in b.(m\\!=\\!l\mathrel{\&}x\\!\approx\\!a)$ informally having the meaning: find an outgoing $l$-labelled edge from $b$ which leads to some node $x$ bisimilar to $a$. But, thinking abstractly, $l\\!:\\!a\in b$ says simply that $a$ is an $l$-labelled element of $b$. The _logical operators_ ($\mathrel{\&},\vee,\neg$) have the usual meaning from propositional logic and can be used to form logical sentences from $\Delta$-formulas. _Universal quantification_ can be understood in terms of conjunction: $\displaystyle\forall l\\!:\\!x\in a.\varphi(x,l)$ $\displaystyle\iff\bigwedge_{l_{i}:x_{i}\in a}\varphi(x_{i},l_{i})$ and _existential quantification_ in terms of disjunction: $\displaystyle\exists l\\!:\\!x\in a.\varphi(x,l)$ $\displaystyle\iff\bigvee_{l_{i}:x_{i}\in a}\varphi(x_{i},l_{i})$ assuming that $a=\\{l_{1}:x_{1},\ldots,l_{n}:x_{n}\\}$. It is evident from this definition that quantification occurs over those elements of the set denoted by $a$ which satisfy the formula $\varphi$. That is, quantification is bounded by (elements of) the set $a$, with the $\Delta$ formula $\varphi$ being called the scope of the quantifier. Note that when a quantified formula participates as a subformula of a bigger formula or of a term the technical problem arises where exactly this (sub)formula is finished, that is what is the scope of the quantifier. In the implemented $\Delta$-language (Appendix A.1) there is a discipline of using parentheses to find unambiguously the scope of quantifiers, both intuitively and by the implemented parser (and contextual analysis algorithm). Say, in $\displaystyle\forall l:x\in a\,.\,(\varphi\;\&\;\psi\;\&\;\chi)$ the scope of the quantifier is the whole expression in the parentheses. But the general informal rule is: the scope of any quantifier is as small as possible. For example, in $\displaystyle(\forall l:x\in a\,.\,\varphi\;\&\;\psi\;\&\;\chi)$ the multiple conjunctions requires some compulsory external parentheses (exactly as shown), and then the scope of the quantifier is either $\varphi$ (excluding $\psi$ and $\chi$) or some initial part of $\varphi$, if syntactically meaningful at all. We will not give the formal definition which is usually widely known and intuitively evident. For the precise definition of the scope of quantifiers, declarations, etc. the reader should, first, inspect the relevant part of the $\Delta$-language syntax in Appendix A.1 and, most importantly, read the Section 9.2 on contextual analysis which, in fact, served as a rigorous conceptual guidance for us to implement the language correctly. ##### 3.2.2 Set valued expressions — $\Delta$-terms The set constant _empty set_ ($\emptyset$) denotes the set $\\{\\}$ having no elements. In general, set values are represented symbolically by either: set constants, set variables or $\Delta$-terms. Furthermore, “literal” set values can be introduced with the _enumeration_ expression $\\{l_{1}\\!:\\!a_{1},...,l_{n}\\!:\\!a_{n}\\}$ which can create new sets, possibly with nesting if some $a_{i}$ are also enumeration expressions, however, $a_{i}$ may also be arbitrary $\Delta$-terms. The _collection_ operation $\\{l\\!:\\!t(x,l)\mid l\\!:\\!x\in a\mathrel{\&}\varphi(x,l)\\}$ denotes the set of labelled elements $l\\!:\\!t(x,l)$ with $t(x,l)$ a $\Delta$-term depending on the set and label variables $l$ and $x$, where $l\\!:\\!x$ ranges over the set $a$, for which the $\Delta$-formula $\varphi(x,l)$ holds. We can also consider the more special case of collection called the _separation_ operation $\\{l\\!:\\!x\in a\mid\varphi(x,l)\\}$ which denotes the set of labelled elements $l\\!:\\!x$ in $a$ for which $\varphi(x,l)$ holds. The (unary) _union_ operation $\bigcup a$ is understood as the (multiple) ordinary union over the elements of $a$. Let us assume $a=\\{l_{1}\\!:\\!a_{1},\ldots,l_{n}\\!:\\!a_{n}\\}$ then $\displaystyle\bigcup a=a_{1}\cup\ldots\cup a_{n}$ with the ordinary union used in the right-hand side of equality. In particular, this also shows that the ordinary union is definable by means of the unary union and enumeration operators. This is only the simplest example of expressibility in $\Delta$. As we mentioned, this language has, in fact, very high expressive power exactly corresponding to polynomial time computability over hereditarily-finite hypersets222 Any hyperset set is hereditarily-finite if and only if it contains a finite number of elements, and these elements are also hereditarily-finite hypersets, etc. Moreover, it is required that the transitive closure of this hyperset is also finite. . The _transitive closure_ $\mbox{\sf TC}(a)$ denotes the set of (labelled) elements of elements, $\ldots$ , of elements of $a$ including $a$ itself. This can also be written (not fully formally, say, due to $\ldots$ present) as: $\displaystyle l\\!:\\!x\in\mbox{\sf TC}(a)\iff$ $\displaystyle l\\!:\\!x\in x_{0}\in\ldots\in x_{n}=a\;\vee$ $\displaystyle(l=\Box\mathrel{\&}x=a)$ with $x_{i}$ some intermediate elements in the membership chain, each belonging to the next $x_{i+1}$ with some label $l_{i}$ whose value is not important. In particular, we let $\Box:a\in\mbox{\sf TC}(a)$. The above core constructs of the $\Delta$-language extended with the two additional constructs recursion and decoration (introduced below) define all polynomial time computable operations and relations over hypersets (represented as WDB); see the precise formulations in [41, 43, 57]. ###### 3.2.2.1 Recursion operation The _recursion_ operator $\mbox{\sf Rec}\;p.\\{l\\!:\\!x\in a\mid\varphi(x,l,p)\\}$ defines a subset $\pi$ of the set denoted by (the $\Delta$-term) $a$, obtained as the result of stabilising (due to finiteness of $a$) the inflating sequence of subsets of $a$ defined iteratively as: $\displaystyle p_{0}$ $\displaystyle=\emptyset$ $\displaystyle p_{1}$ $\displaystyle=p_{0}\cup\\{l\\!:\\!x\in a\mid\varphi(x,l,p_{0})\\}$ $\displaystyle p_{2}$ $\displaystyle=p_{1}\cup\\{l\\!:\\!x\in a\mid\varphi(x,l,p_{1})\\}$ $\displaystyle\ldots$ $\displaystyle p_{k+1}$ $\displaystyle=p_{k}\cup\\{l\\!:\\!x\in a\mid\varphi(x,l,p_{k})\\}.$ Evidently, all $\emptyset=p_{0}\subseteq p_{1}\subseteq\ldots$ are subsets of $a$. As $a$ is finite, $p_{k}=p_{k+1}=p_{k+2},\ldots$ for some $k$, and this stabilised value, denoted above as $\pi$, is taken as the value of the recursion operator. ###### 3.2.2.2 Decoration operation Recall that in Chapter 2 graph nodes were shown to denote (hyper)sets, and vice versa, arbitrary hereditarily-finite hyperset can be represented in this way. Now, we shall consider finite graphs in set theoretic terms. Traditionally, this is done by defining a graph as a set of ordered pairs where ordered pairs represent graph edges, for example $\langle a,b\rangle$ denoting the edge $a\rightarrow b$. Here (the arbitrary sets) $a$ and $b$, play the role of the source and target vertices of the edge $a\rightarrow b$. Thus, any set $g$ of ordered pairs can be treated as a graph. Formally such ordered pairs are represented as the sets containing two elements labelled by $fst$ and $snd$ respectively, such as $\\{fst\\!:\\!a,snd\\!:\\!b\\}$. That is, we define $\langle a,b\rangle=\\{fst\\!:\\!a,snd\\!:\\!b\\}$. Any labelled ordered pair $l:\\{fst\\!:\\!a,snd\\!:\\!b\\}$ represents a labelled edge $a\stackrel{{\scriptstyle l}}{{\rightarrow}}b$. In general, we can consider absolutely arbitrary hyperset $g$ as representing a graph. Indeed, we can take into account only those elements of $g$ which happen to be ordered pairs, and ignore the other non-pair elements. This will make the operation of decoration defined below applicable to the arbitrary hyperset $g$ what is convenient. Otherwise the formulation of the language $\Delta$ would be more complicated. Also, the arbitrary set $v$ may either participate as an element of the ordered pairs of $g$, i.e. serving as a $g$-vertex, or, otherwise, it is considered as an isolated vertex of the graph $g$. In this sense each set $v$ serves as a $g$-vertex. ###### Definition 1. The abstract set theoretic _decoration operator_ $\mbox{\sf Dec}(g,v)=d$ takes two arbitrary input sets $g$ and $v$ where the former represents a graph as a set of ordered pairs, and the latter represents some vertex $v$ of this graph. It outputs a new (hyper)set $d$ corresponding to the $v$-rooted graph $g$ according to the first paragraph of this section. Note that decoration is the only operator in $\Delta$ which allows for the construction of cyclic hypersets, like $\Omega=\\{\Omega\\}$, from the ordinary “uncycled” sets (of sets of sets,…) of finite depth. For example, consider the _trivial cyclic_ graph $g$ defined by the following system of set equations, $\displaystyle g$ $\displaystyle=\\{\;\\{fst\\!:\\!a,snd\\!:\\!a\\}\;\\}$ $\displaystyle a$ $\displaystyle=\\{\\}$ The result of applying decoration to the graph $g$ and the participating vertex $a$ would be, $\displaystyle\Omega$ $\displaystyle=\\{\Omega\\}$ where $\Omega$ denotes the result $\mbox{\sf Dec}(g,a)$. Indeed this leads to the construction of the cyclic membership represented by the unique $g$-edge $a\rightarrow a$. In fact, here the Anti-Foundation Axiom from Section 2.4.4 guarantees that $\Omega$ is a unique hyperset denoted by $\mbox{\sf Dec}(g,a)$ (and the same for arbitrary $g$ and $a$). This operator can also be reasonably called the _plan performance operator_ [61] because its input(s) can be considered as a graphical plan for the construction of a hyperset with the output being the resulting abstract hyperset. Imagine that we have a plan of a Web site (i.e. of a system of hyperlinked Web pages) and that Dec is a tool (or query) which automatically creates all the required Web pages. See also Section 3.5.3 for a more involved example of using the decoration operation for defining a restructuring query. #### 3.3 Operational semantics Consider any set or boolean query $q$ which involves no free variables and whose participating set names (constants) are taken from the given WDB system of set equations. Resolving $q$ consists in the following two macro steps: • Extending this system by new equation $res=q$ with $res$ a fresh (i.e. unused in WDB) set or boolean name, and * • Simplifying the extended system: $\displaystyle\textrm{WDB}_{0}=\textrm{WDB}+(res=q)$ until it will contain only flat bracket expressions as the right-hand sides of the equations or the truth values _true_ or _false_ (if the left-hand side is boolean name). After simplification is complete, these set equations will contain no complex set or boolean queries (like $q$ above). In fact, the resulting version $\textrm{WDB}_{res}$ of WDB will consist (alongside the old equations of the original WDB) of new set equations (new set names equated to flat bracket expressions) and boolean equations (boolean names equated to boolean values, _true_ or _false_). This process of computation by _extension_ and _simplification_ was described in [61] as reduction steps $\displaystyle WDB_{0}\rhd WDB_{1}\rhd\ldots\rhd WDB_{res}$ where $WDB_{0}$ is the initial state of $WDB$ extended by the equation $res=q$, and $WDB_{res}$ is the final step of reduction consisting of only flat set equations including the flattened version of set equation $res=q$ (or boolean equation, if $q$ is a $\Delta$-formula). Each reduction step represents simplification by applying rewrite rules which transform set equations involving complicated $\Delta$ expressions into simpler, semantically equivalent, equations. Note that the rewrite rules described here are based on those in [61] but extended to the labelled case as considered in this thesis. In general, rewrite steps are denoted by the $\rhd$ symbol which means “transforms to”. Firstly, let us assume participation of the set names $s,p,r$ in the rewrite rules below, which correspond to the set equations $\displaystyle s$ $\displaystyle=\\{l_{1}\\!:\\!s_{1},...,l_{a}\\!:\\!s_{a}\\},$ $\displaystyle p$ $\displaystyle=\\{m_{1}\\!:\\!p_{1},...,m_{b}\\!:\\!p_{b}\\},$ $\displaystyle\ldots$ $\displaystyle r$ $\displaystyle=\\{n_{1}\\!:\\!r_{1},...,n_{c}\\!:\\!r_{c}\\}$ existing either in the initial $WDB$ or in the current reduction $WDB_{i}$. The operational semantics for the $\Delta$ operators (except for recursion, decoration, transitive closure, bisimulation and label relation operators) are described as the reduction rules $\displaystyle res$ $\displaystyle=t(t_{1},\ldots,t_{a})\rhd\begin{cases}res&=t(res_{1},\ldots,res_{a}),\\\ res_{1}&=t_{1},\\\ &\ldots\\\ res_{a}&=t_{a}.\end{cases}$ $\displaystyle res$ $\displaystyle=\\{l\\!:\\!s,m\\!:\\!p,\ldots,n\\!:\\!r\\}\mbox{ -- no further reduction required once $s,p\ldots,r,$ are set names},$ $\displaystyle res$ $\displaystyle=s\cup p\cup\ldots\cup r\rhd res=\\{l_{1}\\!:\\!s_{1},...,l_{a}\\!:\\!s_{a},\;m_{1}\\!:\\!p_{1},...,m_{b}\\!:\\!p_{b},\;\ldots,n_{1}\\!:\\!r_{1},...,n_{c}\\!:\\!r_{c}\\},$ $\displaystyle res$ $\displaystyle=\bigcup s\rhd res=s_{1}\cup\ldots\cup s_{a},$ $\displaystyle res$ $\displaystyle=\mbox{\sf TC}(p)\mbox{ -- operational semantics described in Section~{}\ref{sec:impl_tc}},$ $\displaystyle res$ $\displaystyle=\\{l:x\in p\mid\varphi(l,x)\\}\rhd res=\\{m_{i_{1}}\\!:\\!p_{i_{1}},\ldots,m_{i_{b^{\prime}}}\\!:\\!p_{i_{b^{\prime}}}\\}$ $\displaystyle\mbox{where }m_{i_{j}}\\!:\\!p_{i_{j}}\mbox{ are all those }m_{i}\\!:\\!p_{i}\in p\mbox{ for which }res_{i}=\varphi(m_{i},p_{i})\rhd res_{i}=\mbox{\bf true},$ $\displaystyle res$ $\displaystyle=\\{t(l,x)\mid l\\!:\\!x\in p\;\&\;\varphi(l,x)\\}\rhd res=\\{t(m_{i_{1}}\\!:\\!p_{i_{1}}),\ldots,t(m_{i_{b^{\prime}}}\\!:\\!p_{i_{b^{\prime}}})\\}$ $\displaystyle\mbox{where }m_{i_{j}}\\!:\\!p_{i_{j}}\mbox{ are all those }m_{i}\\!:\\!p_{i}\in p\mbox{ for which }res_{i}=\varphi(m_{i},p_{i})\rhd res_{i}=\mbox{\bf true},$ $\displaystyle res$ $\displaystyle=\mbox{\sf Rec}\;p.\\{l:x\in a\mid\varphi(l,x,p)\\}\mbox{ -- operational semantics described in Section~{}\ref{sec:impl_rec_sep}},$ $\displaystyle res$ $\displaystyle=\mbox{\sf Dec}(a,b)\mbox{ -- operational semantics described in Section~{}\ref{sec:impl_dec}},$ $\displaystyle res$ $\displaystyle=\forall l\\!:\\!x\in p\;.\;\varphi(l,x)\rhd res=\varphi(m_{1},p_{1})\;\&\;...\;\&\;\varphi(m_{n},p_{n}),$ $\displaystyle res$ $\displaystyle=\exists l\\!:\\!x\in p\;.\;\varphi(l,x)\rhd res=\varphi(m_{1},p_{1})\vee...\vee\varphi(m_{n},p_{n}),$ $\displaystyle res$ $\displaystyle=\mbox{\bf true}\;\&\;\mbox{\bf true}\rhd res=\mbox{\bf true},$ $\displaystyle res$ $\displaystyle=\mbox{\bf false}\;\&\;\varphi\rhd res=\mbox{\bf false},$ $\displaystyle res$ $\displaystyle=\varphi\;\&\;\mbox{\bf false}\rhd res=\mbox{\bf false},$ $\displaystyle res$ $\displaystyle=\varphi\vee\psi\rhd res=\neg(\neg\varphi\;\&\;\neg\psi),$ $\displaystyle res$ $\displaystyle=\neg\mbox{\bf false}\rhd res=\mbox{\bf true},$ $\displaystyle res$ $\displaystyle=\neg\mbox{\bf true}\rhd res=\mbox{\bf false},$ $\displaystyle res$ $\displaystyle=l\\!:\\!s\in p\rhd res=\exists m\\!:\\!x\in p\;.\;(s=x\;\&\;l=m),$ $\displaystyle res$ $\displaystyle=x=y\rhd x\approx y\mbox{ -- operational semantics described in Section~{}\ref{sec:impl_bisim_algo}},$ $\displaystyle res$ $\displaystyle=l\mathrel{R}m\mbox{ -- operational semantics described in Section~{}\ref{sec:denotational_semantics_formulas}}.$ The implementation of $\Delta$-query execution is based on this process of reduction except for the $\Delta$-terms: recursion, decoration, transitive closure described in Section 8.1.3, Section 8.1.4 and Section 8.1.5 respectively; and the $\Delta$-formulas: set equality (bisimulation) and label relation operators described in Section 4.2.1 and Section 3.2.1 respectively. ##### 3.3.1 Examples of reduction The above process of computation by _reduction_ is quite natural as shown in the following examples. ###### 3.3.1.1 Example elimination of complicated subterms Let us consider the reduction of the query $q=\bigcup q_{1}$ containing the complex subquery $q_{1}$. In general, any complicated term $t(t_{1},\ldots,t_{n})$ can be simplified by invoking the splitting rule which transforms the equation $res=t(t_{1},\ldots,t_{n})$ to the resultant equations $\displaystyle res$ $\displaystyle=t(res_{1},\ldots,res_{n})$ $\displaystyle res_{1}$ $\displaystyle=t_{1}$ $\displaystyle\ldots$ $\displaystyle res_{n}$ $\displaystyle=t_{n}$ Therefore, the complicated query $res=\bigcup q_{1}$ can be split into two subqueries, $res=\bigcup res_{1}$ and $res_{1}=q_{1}$ where $res_{1}$ is a new set name. ###### 3.3.1.2 Example reduction of union In the case of our union query having the particular form $q=\bigcup\\{l\\!:\\!s,m\\!:\\!p,n\\!:\\!r\\}$ where $s,p,r$ represent set names, it follows that the equation $res=q$ is reduced by the following steps: 1. 1. Split the complicated equation $res=\bigcup\\{l\\!:\\!s,m\\!:\\!p,n\\!:\\!r\\}$ resulting in the equations: $\displaystyle res$ $\displaystyle=\bigcup res_{1}$ $\displaystyle res_{1}$ $\displaystyle=\\{l\\!:\\!s,m\\!:\\!p,n\\!:\\!r\\}$ where $s,p,r$ are set names, and hence do not require further splitting. 2. 2. Reduce unary union $res=\bigcup res_{1}$ to multiple union resulting in the equation: $\displaystyle res=s\cup p\cup r$ with the unary union reduced to multiple unions over the elements of the set $res_{1}$ (the set names $s,p,r$). 3. 3. Reduce multiple union $res=s\,\cup\,p\,\cup\,r$ to the bracket expression resulting in the equation: $\displaystyle res=\\{l_{1}\\!:\\!s_{1},...,l_{i}\\!:\\!s_{i},\;m_{1}\\!:\\!p_{1},...,m_{j}\\!:\\!p_{j},\;n_{1}\\!:\\!r_{1},...,n_{k}\\!:\\!r_{k}\\}$ assuming that the current extension of the original WDB already contains the simplified equations $s=\\{l_{1}\\!:\\!s_{1},...,l_{i}\\!:\\!s_{i}\\}$, $p=\\{m_{1}\\!:\\!p_{1},...,m_{j}\\!:\\!p_{j}\\}$ and $q=\\{n_{1}\\!:\\!q_{1},...,n_{k}\\!:\\!r_{k}\\}$. Here multiple union over the sets $s,p,r$ is reduced to the bracket expression containing the elements of these sets. In general, most of the $\Delta$ operators can be resolved using the above reduction rules except for recursion, decoration, transitive closure, bisimulation and label relation operators. In fact, there is no common framework for describing the operational semantics for all the $\Delta$ operators, with the latter exceptions described as lower-level algorithms in Chapters 4 and 8. The main conclusion is that after reduction we will have the equation $res=\\{\ldots\\}$ of the required form whose right-hand side should involve no complicated terms or formulas, only set names either from the original WDB or new set names introduced during reduction (like $res_{1}$ above) together with the corresponding equations of the required form. Thus, execution of a query extends the original WDB to $\textrm{WDB}_{res}$ (simplification of $\textrm{WDB}_{0}$ above). This extension with the set name $res$ as an “entrance point” to the result of the query can be considered as a temporary one until we need this result. In principle, we could also consider _update queries_ which would change the original WDB (not only extend it as above), but this is beyond the scope of this work. #### 3.4 Implemented $\Delta$-query language The implemented $\Delta$-query language can express all operations definable in the original (as described above). For the purpose of writing queries the grammar of this language is expressed as BNF (see Appendix A.1) which the reader should take into consideration whilst reading the current section. (See Chapter 8 for technical details of the implementation of the $\Delta$-query language.) Note that, not every computable set theoretic operation is definable within the $\Delta$-language but everything which is polynomial time computable (and generic; cf. [41]) is already definable in the original language. Additional features (not present in the theoretical version of the language) have also been included in the implemented language making the language more practically convenient, but not increasing its theoretical expressive power. These additions, however important practically, are just “syntactic sugaring” of the above theoretical version of $\Delta$. ##### 3.4.1 Queries with declarations Like in many programming languages allowing procedure declarations and calls we also introduce in the language $\Delta$ query declarations and calls. Thus, a query once declared can be invoked as many times as we want by using its name with various parameters. Besides queries, we allow also constant declarations. Each declaration has its own scope especially delimited (unlike quantifiers) by the keywords `in` and `endlet` where the declared queries or constants can be used (called). For example, let us show how full set names333 Recall that full set name consists of XML-WDB file URL extended by simple set name (delimited by # symbol). (which can be quite long and unmanageable) can be declared and then used as set constants. The following query declares the set constant BibDB as an abbreviation of the corresponding full set name: set query let set constant BibDB be http://www.csc.liv.ac.uk/~molyneux/t/BibDB.xml#BibDB in QUERY( BibDB ) endlet; Here QUERY denotes any subquery (according to the syntax in Appendix A.1) which may involve (possibly many times) the set constant BibDB declared once in the `let` declaration at the beginning of the whole query. However in general `let` declarations of constants and queries can appear at any depth of a query. Let us now consider the more useful case of the query declaration `getBooks`, which in the following example gives the set of all books in the bibliography database illustrated by the graph in Figure 3.1 in Section 3.5 below. We first declare the query getBooks with one set variable argument input and then call it with the argument value BibDB: set query let set constant BibDB be http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#BibDB, set query getBooks (set input) be separate { pub-type:pub in input where pub-type=’book’ } in call getBooks(BibDB) endlet; Here the keyword `call` means that we invoke the set query `getBooks` defined above. In general, any query can be declared once and invoked many times, e.g. getBooks(BibDB1), getBooks(BibDB2), etc., each time with various <parameters> which may be either any <delta-term> or <label> according to the BNF. Those relevant parts of the BNF for this set query are as follows, <delta-term with declarations> ::= "let" <declarations> "in" <delta-term> "endlet" <set constant declaration> ::= "set constant" <set constant> ("be"\|"=") <delta-term> <set query declaration> ::=ΨΨ "set query" <set query name> "(" <variables> ")" ("be"\|"=") <delta-term> <set query call> ::= "call" <set query name> "(" <parameters> ")" In general, there are also <label constant declaration> and <boolean query declaration> syntactical categories. Note that in the syntactic category <delta-term with declarations> the keyword `in` evidently does not play the role of the membership relation such as in the case of the other contexts of the $\Delta$-language. Recursive calls are not allowed in query declarations, that is the declared query name or constant should not occur in the scope of the declaration. For <recursion> (see the syntax in Appendix A.1) we have the special construct recursive separation already discussed above and illustrated below in Section 3.5.4. ##### 3.4.2 Library The library allows to create query or constant declarations independent of a query. Library commands allow creation and modification of user defined queries and constants. Predefined and also user defined queries and constants can then be used, i.e. called, (globally) in any query. For example, the following library command adds the set constant some-book for the appropriate full set name to the library: library add set constant some_book = http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b1; where the identifier `some-book` may now participate in any subsequent queries in the current query session444 Query session is the period of time between opening the query system (for running queries and library commands) and closing it. When query system is restarted, only build in query and constant declarations (see the current list in the Appendix A.3) can be used. . Queries and constants can be modified or redeclared by rerunning the library `add` command. For example, the set constant `some_book` (above) could be redeclared as follows: library add set constant some_book = http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2; Predefined and user defined555 added in the current query session library queries/constants can be listed, in brief without the full declarations, with the command, library list; with result of this command (including predefined queries/constants) being, Library command is well-formed and well-typed, but not executable Warning, library command successful but no query executed. Warning, in the case of duplicate declaration names those declarations at the bottom of the list have precedence. List of library declaration(s): set query Pair (set x,set y), boolean query isPair (set p), set query First (set p), set query Second (set p), set query CartProduct (set x,set y), set query Square (set z), set query LabelledPairs (set v), set query Nodes (set g), set query Children (set x,set g), set query Regroup (set g), set query CanGraph (set x), set query Can (set x), set query TCPure (set x), set query HorizontalTC (set g), set query TC_along_label (label l,set z), set query SuccessorPairs (set L), boolean query Precedes5 (set R,label l,set x,label m,set y), set query StrictLinOrder_on_TC (set z), set constant some_book, set constant some_book The order of query/constant declarations depends on the order in which the corresponding library add commands were executed. Note that, the duplicate declarations named `some_book` is the result of running above the library add commands, and those declarations appearing at the bottom of the list have precedence over those at the top of the list. Thus, the set constant `some_book` appearing globally in any query would, in fact, have the redeclared set name http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2. However, there is one subtle point: if a query $q$ is declared in the library which calls another library query $q_{1}$ (or constant), then $q$ will invoke the latest declaration of $q_{1}$ _preceding_ this declaration of $q$ even if $q_{1}$ is redeclared again after $q$. Note that the modification or deletion of user defined declarations is not yet implemented, but it can be done easily. Also, the full declarations of user defined and predefined queries/constants can be listed with the command, library list verbose; with the result being, Library command is well-formed and well-typed, but not executable Warning, library command successful but no query executed. List of library declaration(s): set query Pair (set x,set y) be { ’fst’:x, ’snd’:y }, boolean query isPair (set p) be ( exists l: x in p . ( l=’fst’ and forall m:z in p . ( m=’fst’ => z=x ) ) and exists l:y in p . ( l=’snd’ and forall m:z in p .( m=’snd’ => z=y ) ) ), ... set constant some_book be http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b1 set constant some_book be http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2 Here the list of queries/constants follows as above, but including the full declaration for all other default library declarations (omitted here for brevity; see the full listing of predefined library declarations in Appendix A.3). Those relevant parts of the BNF for the library commands are as follows: <library commands> ::= "add" <declarations> \| "list" [ "verbose" ] Note that, only the predefined library declarations will remain in the library after finishing the query session. In principle the ability to work with several libraries (as well as user defined libraries) should also be implemented. The queries Pair, isPair, `First`, `Second` will be formally explained below; CartProduct, Square and HorizontalTC in Section 3.5.4; LabelledPairs, CanGraph and Can in Section 3.5.6; TC_along_label in Section 3.6; SuccessorPairs, Precedes5, TCPure, StrictLinOrder_on_TC in Section 3.7 and Appendix A.3; whereas Nodes, Children and Regroup in Section 8.1.4.1. ###### 3.4.2.1 The queries Pair, isPair, First and Second Thus, let us now define several auxiliary queries dealing with ordered pairs. According to the syntax in Appendix A.1 query declarations have the general form: $\displaystyle\mbox{\tt set query }\;q(\bar{x})=t(\bar{x}),$ $\displaystyle\mbox{\tt boolean query }\;q(\bar{x})=\varphi(\bar{x}).$ Here $q$ is either set or boolean query name, respectively, with query parameters defined by the list $\bar{x}$ of participating set or label variables. ###### 3.4.2.1.1 Pair: Our first query defines the operation creating an ordered pair: set query Pair(set x,set y) = {’fst’:x,’snd’:y} where `’fst’` and `’snd’` are label values helping to distinguish the first element `x` from the second element `y` of the ordered pair, with `x`,`y` as set variables denoting any (hyper)sets. Recall that the order of elements in a set is ignored, playing no role. But, labels of elements such as `fst` and `snd` add the required structure. ###### 3.4.2.1.2 isPair: Now we consider the boolean valued query `isPair(p)` which given a set `p` says whether it is an ordered pair `p={’fst’:x,’snd’:y}` for some sets `x` and `y`: boolean query isPair(set p) = (exists l:x in p . ( l=’fst’ and forall m:z in p . (m=’fst’ implies z = x) ) and exists l:y in p . ( l=’snd’ and forall m:z in p . (m=’snd’ implies z = y) ) ) Note that the equalities `z=x` and `z=y` in this query are actually based on the bisimulation relation. It follows that `isPair(p)` can hold even if the set equation `p={...}` contains syntactically more than two elements between braces. It is required that there exists only one element in `p` labelled by `’fst’` and one labelled by `’snd’` only up to bisimulation. ###### 3.4.2.1.3 First and Second: Let us also define the set valued operations `First(p)` and `Second(p)` giving the first and the second elements of any pair $p$: set query First(set p) = union separate {l:x in p where l=’fst’ } set query Second(set p) = union separate {l:x in p where l=’snd’ } Note that the union operation is necessary here. Indeed, assuming that the input is an ordered pair `p = {’fst’:u,’snd’:v}`, then we would get without union just singleton sets `{’fst’ : u}` and `{’snd’ : v}`, respectively, generated by the separation operator whereas we need their elements `u` and `v`, respectively. Therefore, we need to use the general set theoretic identity $\bigcup\\{l:u\\}=u$ where $u$ is any set. Of course, in the case of arbitrary set input `p` separation will not necessary generate a singleton set. Anyway, `First(p)` and `Second(p)` will give some set values so that these operations are always defined. ###### 3.4.2.2 Implementation of the library Although general implementation issues will be postponed till Part III, we can easily comment here how implementation of the library can be reduced to the general let-endlet construct of the language. Thus, let us assume that the library contains a list of declarations $\displaystyle d_{1},d_{2},\ldots,d_{n}$ already added by the `add` command. Then any query $q$ can use these declarations and thus can contain constants and query names which are not declared in $q$, but must be declared above in the library. In fact, any such query set query $q$; or boolean query $q$; is automatically transformed by the implemented query system, respectively, to the query $\displaystyle\texttt{set/boolean query let }d_{1},d_{2},\ldots,d_{n}\texttt{ in }q\texttt{ endlet;}$ (3.1) Then this query is checked to be well-formed and well-typed and then executed as it is discussed formally in Chapters 9 and 8. This way also the problem of dependency between library declarations $d_{1},d_{2},...,d_{n}$, whose order may be essential666 A declaration $d_{i}$ can depend only on $d_{j}$ with $j<i$. Even if $d_{i}$ calls a constant or query name declared by $d_{k}$ with $i<k$, appropriate (rightmost) $d_{j}$ with $j<i$ should be really found and used. But this does not require any special or additional care for the library declarations because the contextual analysis algorithm in Section 9.2 will guarantee this automatically under translation (3.1). , is resolved automatically. Also query declarations when added to the library are automatically checked simply by transforming them to the usual query $\displaystyle\texttt{set query let }d_{1},d_{2},\ldots,d_{n}\texttt{ in }\\{\\}\texttt{ endlet;}$ where the trivial version of $q=\\{\\}$ is used. Well-formedness and well- typedness of the latter query is considered, by definition, as well-formedness and well-typedness of the declarations in the library. #### 3.5 Example $\Delta$-queries Let us consider the following example queries based on the bibliographic WDB presented in [50] and similar to the example in [1]. This WDB is distributed (split into two fragments) as illustrated by the colouring of the graph in Figure 3.1. Each fragment is given by a subsystem of set equations represented practically as an XML-WDB file (see Chapter 10 for the technical details of the XML-WDB representation). These files can be examined in the Appendix A.2. Figure 3.1: Example distributed WDB of a small bibliographic database, distributed into two fragments. Let us consider the corresponding subsystems of set equations represented practically as XML-WDB files. Note that, full set names are denoted as the concatenation of URL, `#`, and simple set name; however, the URL and the delimiter `#` can be omitted for local set names. The subsystem of set equations represented by the XML-WDB file http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml is as follows: BibDB = { ’book’:b1, ’book’:b2, ’paper’:http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p1, ’paper’:http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p2, ’paper’:http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p3 } b1 = { ’refers-to’:http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#b2, ’refers-to’:p1 } b2 = { ’author’:"Jones", ’title’:"Databases" } The XML-WDB file http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml represents the subsystem p1 = { ’refers-to’:p2 } p2 = { ’author’:"Smith", ’title’:"Databases", ’refers-to’:p3 } p3 = { ’author’:"Jones", ’title’:"Databases" } Recall that single quotation marks are used to denote labels such as `’author’`, whereas double quotation marks denote atomic values which are, strictly speaking, special singleton sets, e.g. `"Jones"` means `{’Jones’:{}}`. ##### 3.5.1 Example of a non-well-typed query In our first example the query is non-well-typed because the identifiers `BibDB` and `b2` are formally undeclared within the following query, although intuitively corresponding to some graph nodes. The intended informal meaning of the query being: find all publications which refer to the book `b2`. set query collect { pub-type:pub where pub-type:pub in BibDB and exists ’refers-to’:ref in pub . ref=b2 }; The result of running this query is the error messages: Query is well-formed, but not well-typed Error at character 76, occurrence of identifier name BibDB not declared: set query collect { pub-type:pub where pub-type:pub in BibDB <------- and exists ’refers-to’:ref in pub . Error at character 127, occurrence of identifier name b2 not declared: and exists ’refers-to’:ref in pub . ref=b2 <------- }; Here well-typed would intuitively mean that all identifiers and their types (_set_ or _label_ , etc.) in the query are appropriately described by declarations, quantifiers, etc., and used in other places of the query accordingly. But unfortunately the error messages show that it is not the case. The corrected version of this query is presented in Section 3.5.2, where the identifiers `BibDB` and `b2` are appropriately related to the WDB considered. We will pay much more attention to well-typedness of queries in Chapter 9 which is highly important for the correct implementation of $\Delta$. ##### 3.5.2 Example of valid and executable query After correction of the above query we have: set query let set constant BibDB be http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#BibDB, set constant b2 be http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2 in collect { pub-type:pub where pub-type:pub in BibDB and exists ’refers-to’:ref in pub . ref=b2 } endlet; Evidently the result of this query contains the book `b1` (which refers to `b2`) and, not so obviously, the paper `p2` which refers to `p3`, the latter being formally bisimilar to `b2` with the same `title` and `author` elements. The result of the modified query is, Query is well-formed, well-typed and executable Result = { ’paper’:http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p2, ’book’:http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b1 } Finished in: 398 ms This result might seem strange, but formally it is correct taking into account our hyperset theoretic approach to WDB. The question here is to the designer(s) of this bibliographic database who overlooked that essentially the _same_ publication is presented in the database both as a book and as a paper. If these are really different publications then they should be represented in the database accordingly (as discussed in the considerations below). Note that the incoming edges labelled by book or paper do not count when determining bisimilarity of the nodes `p3` and `b2` — only outgoing edges play a role. Such fundamental flaws can be introduced accidentally when possibly many users create distributed WDB. Evidently, this WDB was poorly designed, therefore, better understanding of the structural design of WDB would make this process less error-prone. Anyway, even with the (traditional) relational approach database design is a crucial step. ###### 3.5.2.1 Query semantics versus WDB design If we really want to include only references to the book `b2` (without redesigning this WDB), then it might seem that the solution is to replace the equality `ref=b2` by the formula (ref=b2 and ’book’:ref in BibDB) in the above query. However, this would not really help because in any case `p3=b2` (these set names / graph nodes are bisimilar) in the above WDB. Equality of (hyper)sets is defined by their elements, elements of elements, etc., i.e. by outgoing edges, and not by incoming edges. So, after formally removing redundancies (say, omitting `p3`) we should have one joint node `b2` with two incoming edges `BibDB` $\stackrel{{\scriptstyle\texttt{book}}}{{\longrightarrow}}$ `b2` and `BibDB` $\stackrel{{\scriptstyle\texttt{paper}}}{{\longrightarrow}}$ `b2` (besides two more incoming `refers-to` edges from `b1` and `p2` and the evident two outgoing edges). This is probably not what the designer(s) of this distributed WDB had in mind. Anyway, we will continue using this example as a good and simple illustration of the (hyper)set theoretic approach. In principle, we could imagine that the creators of this WDB really wanted to have publications classified both as a book and a paper. This is not a contradiction, as anything is possible in semi-structured data. In fact, the problem is only to decide what we really want and whether this intuition is reflected correctly by the given WDB design. This example emphasises the real meaning of set theoretic versus pure graph approaches to semi-structured databases, and the role of removing redundancies on the level of the design. The right approach here should be based on a well- chosen discipline, for example: * (i) _Reconstruct_ this database by replacing labels `book` and `paper` by `publication` and adding outgoing edges from each publication showing its `type` (`’book’` or `’paper’`; see Figure 3.2 777 Strictly speaking, Figure 3.2 reflects this idea only partially because it is devoted to illustrate a related but formally different example of restructuring query in the $\Delta$-language. It still has a publication which is characterised as both book and a paper, however, this is more noticeable “locally” reducing accidental user error. ), or alternatively * (ii) Enforce some WDB _schema_ during the design of WDB e.g. requiring that there is only one `book` or `paper` edge from `BibDB` leading to any given publication considered up to bisimulation. Here the term “up to bisimulation” means that if two children of `BibDB` are bisimilar then they, in fact, have identical labelling. But it is not our goal here to go into details of such kind of discipline and consider WDB schemas. In any case, we should be precise and accurate with the design of WDB, and in formulating both formal and intuitive versions of our queries. The mathematical ground of hyperset theory is quite solid and sufficient for that. The main point is that any formal query has a unique (up to bisimulation) answer – in fact, either a hyperset or boolean value – and all the queries are _bisimulation invariant_ and can be computed in polynomial time (with respect to the size of WDB). Vice versa, any P-time computable and bisimulation invariant (and also “generic” [41, 57]) query is definable in $\Delta$. In fact, this also means that the language $\Delta$ has full P-time computable power of _restructuring_ , not only simple retrieval of already existing elements in the WDB. For example the query restructuring the `BibDB` database as is essentially described in (i) above could be written in $\Delta$ using the plan performance operator Dec. ##### 3.5.3 Restructuring query The ability to define queries arbitrarily restructuring any given data is the most essential requirement of any database query language. Here we will consider one simple example which could hopefully convince the reader that $\Delta$ has a very strong restructuring power. Firstly, let us recall the informal meaning of the following useful query declarations in the default library (with the formal meaning fully described in Section 3.4.2.1) and introduce semi-formally one more query CanGraph to be formally defined in Section 3.5.6: * • `Pair(x,y)` – denoting the ordered pair $\langle x,y\rangle$, in fact the two element set of the form `{’fst’:x,’snd’:y}` allowing to distinguish between the first and second elements. * • `First(p)` – first element of $p$ if $p$ is an ordered pair. * • `Second(p)` – second element of $p$ if $p$ is an ordered pair. * • `CanGraph(x)` – denoting the set of labelled pairs $l:\langle u,v\rangle$ where $l\\!:\\!v\in u$ holds in the transitive closure $\mbox{\sf TC}(x)$. Then the required restructuring query (described informally in (i) above) is defined as follows: set query let set constant BibDB = http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#BibDB, set constant restructuredBibDB be (U collect{ ’null’:if (L=’paper’ or L=’book’) then { ’publication’:X, ’type’:call Pair(call Second(X),{L:{}}), L:call Pair({L:{}}, {}) } else {L:X} fi where L:X in call CanGraph(BibDB) } ) in decorate ( restructuredBibDB, BibDB ) endlet; Here `CanGraph(BibDB)` is essentially the bibliography graph in Figure 3.1, but represented in the traditional set theoretic way as the set of labelled ordered pairs, each denoted in the query as L:X with L the label and X the ordered pair in question. The required restructuring in terms of ordered pairs consists in relabelling of labels `’book’` and `’paper’` as `’publication’`, and creating additional leaf edges with the publication type is done essentially by the following fragment ’null’:if (L=’paper’ or L=’book’) then { ’publication’:X, ’type’:call Pair(call Second(X),{L:{}}), L:call Pair({L:{}}, {}) } else {L:X} fi generating appropriate sets of labelled ordered pairs. Then these sets888 where the value of the label ’null’ is not important are collected, and taking the union gives rise to the required restructured set of labelled ordered pairs denoted as restructuredBibDB. But abstractly, we need a hyperset rather than this graph (a set of pairs). Thus, finally, the decoration operation applied to the graph `restructuredBibDB` and the vertex `BibDB` generates the required abstract hyperset (as described in general in Section 3.2.2.2). The result of this query is, Query is well-formed, well-typed and executable Result = { ’publication’:res2, ’publication’:res0, ’publication’:res1, ’publication’:{ ’type’:"book", ’refers-to’:res1, ’refers-to’:res2 } } res0 = { ’type’:"paper", ’author’:"Smith", ’title’:"Databases", ’refers-to’:res1 } res1 = { ’type’:"paper", ’type’:"book", ’author’:"Jones", ’title’:"Databases" } res2 = { ’type’:"paper", ’refers-to’:res0 } Finished in: 1646 ms (query execution is 1643 ms, and postprocessing time is 3 ms) As we discussed formerly, atomic values, strictly speaking, denote corresponding singleton sets, for example `"Smith"`, denotes `{’Smith’:{}}`. The (new) set names `res0`, `res1` and `res2` correspond, respectively, to the “restructured” publications p2’, p3’/b2’ and p1’. Note that, the query system replaces some set names on the right-hand side by the corresponding bracket expression where suitable, thereby presenting the result in a “nested” form. For example the publication b1’ is implicitly nested in the `Result` set equation. This result can be more conveniently visualised by Figure 3.2 with the set name Result replaced by BibDB’, and new set names replaced by corresponding names revelant to the restructured publications (as was discussed above). Figure 3.2: The result of the restructuring query. Note that the publication p3’/b2’999 denoted by the new set name res1 (see query result above) has both the type `book` and `paper`, and that this unusual feature is the result of the initial design of `BibDB` and not a failure of the above query. Anyway, in principle this graph suggests a potentially better (less semantically error prone) design for the bibliography database. ##### 3.5.4 Horizontal transitive closure Let us now consider the query which can generate the “horizontal” transitive closure101010 This should not be mixed with the set theoretic meaning of the $\Delta$-term operator transitive closure TC which can be understood intuitively as “vertical” transitive closure, that is $\mbox{\sf TC}(x)$ represents the set of (labelled) elements of element of elements, etc. of $x$ (including $x$ itself) as defined in Section 3.2.2. The point is that it is typically convenient to think of elements of a set as lying _under_ this set – hence _vertical_ view. of any graph $g$ (a set of ordered pairs). Consider the trivial example graph $g$ represented as the nodes $a,b,c$ with edges $\langle a,b\rangle$ and $\langle b,c\rangle$ depicted by solid black edges in Figure 3.3111111 We should not mix this graph, which is only a visual representation of a _set of ordered pairs_ , with any other graphs depicted before and having rather a visual representation of a _system of set equations_. . The result of applying horizontal transitive closure to the graph $g$ is shown by the original edges (in solid black) and the additional edges $\langle a,c\rangle$, $\langle a,a\rangle$, $\langle b,b\rangle$ and $\langle c,c\rangle$ highlighted in Figure 3.3 as red dashed edges. Figure 3.3: The result of “horizontal” transitive closure applied to the abstract graph $g$. The result is also a graph denoted as $g^{}$ which extends $g$ by new ordered pairs ($g\subseteq g^{}$) such that for each edge $\langle x,y\rangle\in g^{}$ there exists a path from $x$ to $y$ belonging to the original graph $g$, and vice versa. This can be recursively defined as follows: $\displaystyle\langle x,y\rangle\in g^{}\iff x=y\vee\exists z.(\langle x,z\rangle\in g^{}\wedge\langle z,y\rangle\in g)$ or as $\displaystyle g^{}=\\{\langle x,y\rangle\in\|g\|\mid x=y\vee\exists z\in\|g\|.(\langle x,z\rangle\in g^{}\wedge\langle z,y\rangle\in g)\\}$ (3.2) where $\|g\|$ is the set of all $g$-nodes. It is assumed that $g^{}$ is the least set of pairs satisfying the above equivalence. This operation could prove useful complementing “vertical” transitive closure $\mbox{\sf TC}(x)$ in the original $\Delta$-language, whose result is the set of elements of elements, etc. for any given set $x$ (including $x$ itself). Thus, let us implement $g^{}$ (denoted below as `HorizontalTC(g)`) in the following straightforward way based on the above formula (3.2). Firstly, let us add to the library the set query declaration Nodes(g) (formally described in Section 8.1.4.1), denoted above as $\|g\|$ and extracting from the set of ordered pairs g the set of elements participating in these ordered pairs. ###### Nodes: set query Nodes (set g) = union separate { m : p in g \| call isPair ( p ) } We will also need the ordinary and very important (not only for defining the horizontal transitive closure) set theoretic operations of ###### CartProduct and Square: set query CartProduct(set X,set Y) = U collect {’null’:collect {’null’:call Pair(x,y) where l:y in Y } where m:x in X } set query Square(set X) = call CartProduct(X,X) Finally, the set query `HorizontalTC(g)` can be easily defined using the recursion operator as follows. ###### HorizontalTC: set query HorizontalTC(set g) be recursion p { ’null’:pair in call Square(call Nodes(g)) where ( call First(pair)=call Second(pair) or exists m:z in call Nodes(g) . ( ’null’:call Pair(call First(pair),z) in p and ’null’:call Pair(z,call Second(pair)) in g ) ) } Let us now execute HorizontalTC applied to the graph g (see above), set query let set constant g be { ’null’:call Pair("a","b"), ’null’:call Pair("b","c") } in call HorizontalTC(g) endlet; and see that the result is as expected, although with many repetitions which witness that the implementation is currently not optimal. However, all the repetitions in the query result can be easily eliminated by _canonisation_ (to be discussed in Section 3.5.6 below). First note that the canonisation set query declaration (Can) is already added to the default library set query Can(set x) be decorate(call CanGraph(x),x) and that the above query can be rewritten using Can as follows: set query let set constant g be { ’null’:call Pair("a","b"), ’null’:call Pair("b","c") } in call Can(call HorizontalTC(g)) endlet; Now, by running the amended query, we see that all repetitions have been eliminated. ##### 3.5.5 Dealing with proper hypersets The hyperset theoretic approach to WDB can represent and query semi-structured databases possibly involving arbitrary cycles (see Chapter 2). For example let us consider the WDB graph in Figure 3.4 with the cycle between the vertices $a$ and $b$ (edges $a\longrightarrow b$ and $b\longrightarrow a$). Figure 3.4: WDB graph with cycle. It is easy to see that $a\approx b$ and $c\approx d$ are the only positive bisimulation facts, and hence $a$ and $b$, and also $c$ and $d$ actually denote the same hypersets (the latter two denote $\emptyset$). The strongly extensional version of this WDB with all redundancies removed is shown in Figure 3.5. Figure 3.5: Strongly extensional version of the WDB in Figure 3.4. Let us show how to define in $\Delta$ the hyperset denoted by the vertex $a$. It can be done with the help of decoration operation as follows: set query let set constant g = { ’null’:call Pair("a","b"), ’null’:call Pair("b","a"), ’null’:call Pair("a","c"), ’null’:call Pair("a","d"), ’null’:call Pair("b","d") } in decorate (g, "a") endlet; The result of this query exactly corresponds to the graph in Figure 3.4: Query is well-formed, well-typed and executable Result = { ’null’:{ ’null’:Result, ’null’:{} }, ’null’:{}, ’null’:{} } Finished in: 20 ms (query execution is 20 ms, and postprocessing time is 0 ms) In the next section we will show how the strongly extensional result (corresponding to Figure 3.5) can be obtained. In fact, without using decoration it would be impossible to define this cyclic set `Result` corresponding to the vertex $a$. Further, let us consider the query to compute equality (bisimulation) between the sets denoting the vertices $a$ and $b$ as boolean query let set constant g = { ’null’:call Pair("a","b"), ’null’:call Pair("b","a"), ’null’:call Pair("a","c"), ’null’:call Pair("a","d"), ’null’:call Pair("b","d") } in decorate (g, "a") = decorate (g, "b") endlet; where the evident result true of this query corresponds to the intuitive observation that, in fact, `"a"` and `"b"` denote bisimilar graph $g$-nodes. ##### 3.5.6 Query optimisation by removing redundancies The following example demonstrates the general task of removing redundancies by a particular set query Can (for “canonisation”) on the above graph in Figure 3.4 (in Section 3.5.5). Here we use our knowledge121212 This solution may not be so intuitively evident yet to those users who are unfamiliar with the set theoretic meaning of decoration and the details of _how_ this operation was implemented (see Section 8.1.4). But running queries with Can can nevertheless clearly demonstrate its usefulness. on the implementation of the decoration operation (see Section 8.1.4) to remove the redundancies in the original graph (see the result of the set query above) by applying the decoration operator to the canonical form of this graph (as a set of pairs representing graph edges) and the participating vertex $a$. First, let us define the set query declaration ###### LabelledPairs: set query LabelledPairs (set v) be collect { l:{ ’fst’:v , ’snd’:u } where l:u in v } with the result of `LabelledPairs(v)` being the set of labelled pairs $l\\!:\\!\langle v,u\rangle$ denoting labelled edges $v\stackrel{{\scriptstyle l}}{{\longrightarrow}}u$ corresponding to the set memberships `l:u` in the set `v`. This set query declaration participates in another important library set query ###### CanGraph: set query CanGraph(set x) be union collect { ’null’:call LabelledPairs ( v ) where m:v in TC(x) } whose output is the set of labelled pairs $l\\!:\\!\langle u,v\rangle$ corresponding to those labelled elements $l:v\in u$ with $u$ ranging over the elements of transitive closure $\mbox{\sf TC}(x)$. Here `’null’` is a label whose value is not important. Indeed, the `union` operation unifies the labelled pairs from `LabelledPairs(v)`. The third library query we need is the set query `Can(set x)` (invoking `CanGraph` above) which takes any set $x$ and returns the same abstract set $x$, but in its strongly extensional form. ###### Can: set query Can(set x) be decorate (call CanGraph(x), x) In fact, we should always have `Can(x)=x` because `CanGraph(x)` is evidently the canonical graph whose node `x` represents the set `x` itself, and, in this sense, the set query `Can` does nothing. It follows also that `Can` and `decorate` are essentially inverse operations. Thus, `Can` changes nothing in the abstract set theoretical sense. But due to applying decoration to get `Can(x)` and taking into account both strong extensionality of CanGraph(x) and the way decoration used in Can is implemented in Section 8.1.4, the resulting system of set equations generated by Can(x) is always non-redundant (strongly extensional). Therefore the result of `Can(a)` for the example in Figure 3.4 consists of one set equation for the node $a/b$ of the graph shown in Figure 3.5. Indeed, running the query: set query let set constant g = { ’null’:call Pair("a","b"), ’null’:call Pair("b","a"), ’null’:call Pair("a","c"), ’null’:call Pair("a","d"), ’null’:call Pair("b","d") } in call Can ( decorate (g, "a") ) endlet; gives the result: Query is well-formed, well-typed and executable Result = { ’null’:Result, ’null’:{} } Finished in: 35 ms (query execution is 35 ms, and postprocessing time is 0 ms) with the set `Result` denoting $a/b$. From the abstract hyperset view this is exactly the same result as without using Can, but represented in a better, non-redundant way. Note that `Can` can be used for the more general purpose of query optimisation (not only for optimisation of query results by removing redundancies). Of course, using `Can(t)` instead of `t` will require some time to compute TC(t) and then decoration (which in fact requires computation of many bisimulation facts). But the benefit is that `Can(t)` will be represented without any redundancies at all, in contrast to the set `t` which could contain a large number of equal elements due to possible redundancies and thus would be much smaller after eliminating them. Then, for example, `Square(t)` (the Cartesian product of `t`) would also be represented without any unnecessary repetitions, and thus possibly much smaller. In particular, if we want to have recursion over this Square (like in the case of recursive definition of `HorizontalTC`), it would be computed much more efficiently, also with smaller number of iteration steps, assuming `Can(t)` instead of `t`. In principle, we could extend the language by adding _literal equality_ eq(x,y) for set names (object identities). This, of course, would change the set theoretic character of the language as queries using such equality will not necessarily be bisimulation invariant. But if we would use this equality only over the elements of sets represented as `Can(t)`, then this can work as an additional optimisation. In principle, the query system could recognise the expressions `Can(t)` and automatically replace bisimulation over this set by literal equality. Finally, note that the above optimisation was given for the current implementation of the $\Delta$-language so that users can exploit canonisation to optimise some queries. In principle, this optimisation could be build into the implementation, so that, any possible redundancies are removed during query execution. In fact, the query system, while executing a query, supports a list of currently known positive bisimulation facts (see Chapter 4) which can be used in background time to remove at least some redundancies in set equations stored in local memory. #### 3.6 Imitating path expressions The ability to select nodes of a WDB graph to arbitrary depth can be elegantly achieved using path expressions. As shown in [61], the action of a rich class of path expressions is definable in the original $\Delta$, itself having no path expressions at all, with the help of TC and Rec. In spite of this fact, an important goal for the future work is to implement the extension of $\Delta$ by such user friendly path expressions like in the following example query131313 The keyword path is added to aid reading. (for simplicity only involving set constants for full set names from the bibliographic WDB): set query separate { pub-type:x in BibDB where exists path <b1>refers-to<x>refers-to<b2> . ’author’:"Smith" in x }; The result of this query would be: Result = { paper:p2 } Quantification goes over paths from `b1` to `b2` having an appropriate intermediate set (or node for a publication) `x` which is required to have the element `author:"Smith"`, but it appears that there does not exists such an explicit path. Nevertheless, the required path does exist, as shown in Figure 3.6 by the dashed edges labelled `refers-to` leading from `b1` to `p3`, where `p3` is equal (bisimilar) to `b2` (`p3`${}\approx{}$`b2`) as we already know. In strongly extensional graphs (where there are no bisimilar nodes) path expressions would be understood quite straightforwardly. Our hyperset approach leads to such kind of complications, but this is the compromise for having a natural language with clear semantics and strong (also precisely characterised) expressive power. Note that the result of the above query would be the empty set if the Kleene star “``” was removed from the path expression. Indeed, there are no paths of length two from b1 to b2, even up to bisimulation. Figure 3.6: Visualisation of the path expression <b1>refers-to<x>refers- to<b2> applied to the bibliographic WDB. The action of the path expression `<b1>refers-to<x>refers-to<b2>` can, in fact, be “rewritten” into $\Delta$ (in its present form) by the following steps. Firstly, consider the subexpression `<x>refers-to<b2>` denoting a path from the candidate publication `x` to `b2` labelled by `’refers-to’`. This can be expressed as the $\Delta$-formula: ’refers-to’:b2 in x where `b2` is set constant and `x` is set variable. Secondly, the subpath expression `<b1>refers-to<x>` denotes set of candidate publications `x` which can be reached from `b1` by navigating zero or more `refers-to` labelled edges. Thus, let us include in the library the general set query which will give the set of graph nodes (of a graph representing a hyperset `z`) reachable by navigating zero or more `l`-labelled edges. ###### TC_along_label: set query TC_along_label(label l, set z) be recursion p { k:x in TC(z) where ( ( x=z and k=’null’ ) or ( k=l and exists m:y in p . l:x in y ) ) }; Here p is a recursion set variable to representing the set `T=TC_along_label(l,z)` of nodes lying on potentially all the l-labelled paths outgoing from z. All elements of T are l-labelled, except possibly z. If l:z is in z then l:z will be added to T. But in any case ’null’:z will appear in T at the first stage of iteration. Hence the query call TC_along_label(’refers-to’, b1) represents the path expression `<b1>refers-to<x>` where `’refers-to’` is label value and `b1` is set constant. Finally the path expression `<b1>refers-to<x>refers-to<b2>`, understood as the set of all `x` lying on the paths matching this path expression, is expressed as: set query separate { n:xx in call TC_along_label(’refers-to’, b1) where ’refers-to’:b2 in xx }; Now, the fragment exists path <b1>refers-to<x>refers-to<b2> . ’author’:"Smith" in x of our path expression query can be rewritten as exists m:y in separate {n:xx in call TC_along_label(’refers-to’,b1) where ’refers-to’:b2 in xx } . (x=y and ’author’:"Smith" in x) so that we can insert it in the full query set query let set constant BibDB = http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#BibDB, set constant b1 = http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b1, set constant b2 = http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2 in separate { pub-type:x in BibDB where exists m:y in separate { n:xx in call TC_along_label(’refers-to’,b1) where ’refers-to’:b2 in xx } . ( x=y and ’author’:"Smith" in x) } endlet; and run it to see the required result: Query is well-formed, well-typed and executable Result = { ’paper’:http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p2 } Finished in: 5766 ms (query execution is 5764 ms, and postprocessing time is 2 ms) Despite this example of successfully imitating path expressions it would be more useful to also include path expressions directly within the implementation language. Although much more general path expressions can be imitated by $\Delta$-queries in the current version [61], this imitation can be quite complicated in general and is not a particularly efficient way of implementing and executing queries with path expressions. Anyway, the $\Delta$-language, as it is implemented now, is very expressive. #### 3.7 Linear ordering query The query example considered in this section has mainly theoretical interest, although it might be useful in practice. The point is that we can define in $\Delta$ linear ordering on the transitive closure of any hyperset by using the lexicographical linear ordering we have on labels. In fact, the resulting linear ordering on hypersets is itself, in a sense, lexicographical. Having defined linear ordering, we can further define any (“generic” polynomial-time) computable operation over hypersets by simulating any given Turing Machine (as shown in descriptive complexity theory [34, 37, 55, 74]). This is the key point of the main result in [57] (for well-founded sets) and in [58, 41, 43] (for hypersets) on the expressive power of $\Delta$ coinciding with polynomial time computability over (hyper)sets. (We omit precise formulation which is more subtle in the case of hypersets having labelled elements; see [57, 41]). Let us consider the set query declaration `StrictLinOrder_on_TC(set z)` (and other associated declarations) which can be found in Appendix A.3141414 It is based on formula (22) and Theorem 2 in [43]. We leave this for the reader to realise how this query below is related with this formula and why it gives a strict linear ordering (see [43]). . In fact, the rather complicated query `StrictLinOrder_on_TC` serves as additional witness demonstrating that everything is implemented correctly, and to check whether and where any optimisation of the implementation is required. Note that `StrictLinOrder_on_TC` invokes Can and without this canonisation the transitive closure TCPure(BibDB) participating in the query below (according to Appendix A.3) would have too many repetitions, and, hence, Square would have even more repetitions so that the recursion in the set query `StrictLinOrder_on_TC` over this Square would take many hours. Now let us run set query let set constant BibDB = http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#BibDB in call SuccessorPairs( call StrictLinOrder_on_TC(BibDB) ) endlet; Note that `SuccessorPairs` (defined in Appendix A.3) makes the result more concise. We see that our database `BibDB` becomes linear ordered (with corresponding simple set names from the bibliographic database substituted in the place of new set names generated by the query system): Query is well-formed, well-typed and executable Result = { ’null’:{’fst’:{}, ’snd’:"Databases"}, ’null’:{’fst’:"Databases",’snd’:"Jones"}, ’null’:{’fst’:"Jones", ’snd’:"Smith"}, ’null’:{’fst’:"Smith", ’snd’:BibDB}, ’null’:{’fst’:BibDB, ’snd’:p1}, ’null’:{’fst’:p1, ’snd’:b1}, ’null’:{’fst’:b1, ’snd’:b2/p3}, ’null’:{’fst’:b2/p3, ’snd’:p2} } p2 = {’author’:"Smith",’title’:"Databases",’refers-to’:b2/p3} b2/p3 = {’author’:"Jones",’title’:"Databases"} p1 = {’refers-to’:p2} b1 = {’refers-to’:b2/p3,’refers-to’:p1} BibDB = {’paper’:p1,’paper’:p2,’paper’:b2/p3,’book’:b1, ’book’:b2/p3} Finished in: 270500 ms (~ 4 minutes and 30 seconds) The correspondence of set names with those nodes in the graph in Figure 3.1 is explicitly shown in the above result. Thus, the resulting linear ordering on the transitive closure of `BibDB` is: {}, "Databases", "Jones", "Smith", BibDB, p1, b1, b2/p3, p2. Here it is important that recursion in `StrictLinOrder_on_TC` does not use bisimulation for comparison iteration steps (see Chapter 4). This crucially optimises recursion, and in particular the query `StrictLinOrder_on_TC` which also uses Can in its library declaration. Without the first optimisations this query would take about 30 minutes, and without also using Can even hours. Of course, several minutes for such a small database (with TC(BibDB) containing 9 sets) is also quite long, and thus the query system implementation needs to be further optimised. But the query is rather complicated (see Appendix A.3), and recursion actually uses $81=9^{2}$ steps of iteration if Can is involved. This means in the average about 3.3 seconds per iteration step. ### Chapter 4 Bisimulation Before discussing the theoretical and practical issues surrounding bisimulation, let us recall some relevant details of the hyperset approach to WDB. As previously described in Chapter 2 WDB is represented as a system of set equations $\bar{x}=\bar{b}(\bar{x})$ where $\bar{x}$ is a list of set names $x_{1},\ldots,x_{k}$ and $\bar{b}(\bar{x})$ is the corresponding list of bracket expressions (for simplicity, “flat” ones). Visually equivalent representation can be done in the form of labelled directed graph, where labelled edges $x_{i}\stackrel{{\scriptstyle label}}{{\longrightarrow}}x_{j}$ correspond to the set memberships $label\\!:\\!x_{j}\in x_{i}$ meaning that the equation for $x_{i}$ has the form $x_{i}=\\{\ldots,label\\!:\\!x_{j},\ldots\\}$. In this case we also call $x_{j}$ a child of $x_{i}$. Note that, our usage of the membership symbol ($\in$) as relation between set names or graph nodes is non-traditional but very close to the traditional set theoretic membership relation between abstract (hyper)sets. Of course this analogy is very important for us and it is indeed highly natural, hence we decided not to introduce a new kind of membership symbol here. For the purposes of our description below labels can be ignored, as inclusion of labels will not affect the nature of our discussion. We will also apply the transitive closure operator $\mbox{\sf TC}(x)$ to a set name $x$. The essential point is that in this context $\mbox{\sf TC}(x)$ is understood as a set of set names (or graph nodes) rather than of abstract sets denoted by these names. Again, we do not bother with introducing a new denotation for such TC. #### 4.1 Hyperset equality and the problem of efficiency One of the key points of our approach is the interpretation of WDB-graph nodes as set names $x_{1},\ldots,x_{k}$ where different nodes $x_{i}$ and $x_{j}$ can, in principle, denote the same (hyper)set, $x_{i}=x_{j}$. This notion of equality between nodes is defined by the bisimulation relation denoted also as $x_{i}\approx x_{j}$ (to emphasise that set names can be syntactically different, but denote the same set) which can be computed by the appropriate recursive comparison of child nodes or set names. Thus, in outline, to check bisimulation of two nodes we need to check bisimulation between some children, grandchildren, and so on, of the given nodes, i.e. many nodes could be involved. If the WDB is distributed amongst many WDB files and remote sites, downloading the relevant WDB files might be necessary in this process and will take significant time. There is also the analogous problem with the related transitive closure operator (TC) whose efficient implementation in the distributed case requires additional considerations not discussed here. So, in practice the equality relation for hypersets seems intractable, although theoretically it takes polynomial time with respect to the size of WDB. Nevertheless, we consider that the hyperset approach to WDB based on bisimulation relation is worth implementing because it suggests a very clear and mathematically well-understood view on semi-structured data and the querying of such data. Thus, the crucial question is whether the problem of bisimulation can be resolved in any reasonable and practical way. Some possible approaches and strategies related with the possible distributed nature of WDB and showing that the situation is manageable in principle are outlined below. Although for the general database perspective we should consider graphs with labels on edges and hypersets with labelled elements, the majority of our considerations in this chapter will be devoted to the pure case without any labels. Extension to the labelled case is quite straightforward and is not explicitly considered, except in Definition 2 (b). Of course, our implementation of bisimulation relation considers the labelled case. ##### 4.1.1 Bisimulation relation Equality between set names (or graph nodes) of any WDB is determined by bisimulation relation defined according to [3] (see also [48, 53]). ###### Definition 2. (a) _Bisimulation relation_ $\approx$ (or $\approx_{\rm WDB}$) on a WDB without labels (the pure case) is the largest one such that for all set names $x,y$ the following implication holds: $x\approx y\Rightarrow\forall x^{\prime}\in x\exists y^{\prime}\in y(x^{\prime}\approx y^{\prime})\;\&\;\forall y^{\prime}\in y\exists x^{\prime}\in x(x^{\prime}\approx y^{\prime}).$ (4.1) (b) In the general labelled case, it should satisfy the implication $\displaystyle x\approx y\Rightarrow$ $\displaystyle\;\forall l:x^{\prime}\in x\exists m:y^{\prime}\in y(l=m\wedge x^{\prime}\approx y^{\prime})\;\&\;$ $\displaystyle\;\forall m:y^{\prime}\in y\exists l:x^{\prime}\in x(l=m\wedge x^{\prime}\approx y^{\prime}).$ (4.2) It is well-known that the largest such relation does exist. Indeed, the class $\cal R$ of relations $R$ satisfying any of the above formulas (in place of $\approx$) is evidently closed under taking unions, so the union of all of them is the required largest one $\approx$. In fact, for $\approx$ the implication $\Rightarrow$ above can be replaced by $\iff$. Moreover, the class $\cal R$ evidently contains the identity relation $=$ and is closed under taking compositions $R\circ S$ and inverse relations $R^{-1}$. It follows that the largest such relation $\approx$ is reflexive, transitive and symmetric, that is, an equivalence relation. The bisimulation relation is completely coherent with hyperset theory as it is fully described in the books of Aczel [3], and Barwise and Moss [5] for the pure case, and this fact extends easily to the labelled case. It is by this reason that the bisimulation relation $\approx$ between set names can be considered as equality relation $=$ between corresponding abstract hypersets. So, we will not go into further general theoretical details concerning the bisimulation relation (except for the concept of local bisimulation in Chapter 6 below), paying the main attention to implementation aspects. #### 4.2 Computing bisimulation over WDB Bisimulation relation is computed in our implementation by recursively deriving bisimulation facts. Both positive ($\approx$) and negative ($\not\approx$) bisimulation facts can be derived with the following rules: $x\approx y\mathrel{:-}\forall x^{\prime}\in x\exists y^{\prime}\in y(x^{\prime}\approx y^{\prime})\;\&\;\forall y^{\prime}\in y\exists x^{\prime}\in x(x^{\prime}\approx y^{\prime}).$ (4.3) $x\not\approx y\mathrel{:-}\exists x^{\prime}\in x\forall y^{\prime}\in y(x^{\prime}\not\approx y^{\prime})\vee\exists y^{\prime}\in y\forall x^{\prime}\in x(x^{\prime}\not\approx y^{\prime}).$ (4.4) In principle, using the rule (4.3) for deriving positive facts is unnecessary. They will be obtained, anyway, at the moment of stabilisation in the derivation process by using only (4.4) (see below). Derivation of bisimulation facts using the above rules (4.3 and 4.4) occur after initial facts have been derived. The rules for deriving these initial facts are partial cases of the main rules (4.3 and 4.4): $x\approx y\mathrel{:-}(x=\emptyset\;\&\;y=\emptyset)$ (4.5) $x\not\approx y\mathrel{:-}(x=\emptyset\;\&\;y\not=\emptyset)\vee(y=\emptyset\;\&\;x\not=\emptyset)$ (4.6) $x\approx x$ (4.7) After the derivation of initial facts, rules 4.3 and 4.4 can be recursively applied. Since it is known that bisimulation is an equivalence relation, the following transitivity and symmetry rules can be used alongside the above rules: $x\approx z\mathrel{:-}x\approx y\;\&\;y\approx z$ (4.8) $x\approx y\mathrel{:-}y\approx x$ (4.9) All these rules should be applied until stabilisation, the stage when no more new $x\approx y$ or $x\not\approx y$ facts can be derived by the above rules. Evidentially, stabilisation is inevitable because there are only finitely many set names in the original WDB, i.e. in the corresponding system of set equations. All remaining non-resolved bisimulation questions ($x\stackrel{{\scriptstyle?}}{{\approx}}y$) can now be concluded as resolved positively as $x\approx y$. ##### 4.2.1 Implemented algorithm for computing bisimulation over distributed WDB The deeply recursive nature of the bisimulation algorithm seems to suggest that it maybe necessary to effectively compute the transitive closure of the two set names participating in any bisimulation question. For example in the case of the bisimulation question $x\stackrel{{\scriptstyle?}}{{\approx}}y$, stabilisation is sufficient to establish only for the facts between set names in $\mbox{\sf TC}(x)$ and $\mbox{\sf TC}(y)$. In general, it may happen that the full transitive closures will be involved. However, in an optimistic approach, derivation rules (described in Section 4.2) may be applied to the partial transitive closures, with a “progressive” transitive closures computed as necessitated by the derivation rules to facilitate the resolution of a bisimulation question. ##### Bisimulation algorithm $Bis(x,y)$: 1. START with resolving the bisimulation question $x\stackrel{{\scriptstyle?}}{{\approx}}y$. 2. 1. Create two (initially empty) lists $Q$ and and $Eq$. $Q$ will consist of bisimulation questions $u\stackrel{{\scriptstyle?}}{{\approx}}v$ or their possible answers, and $Eq$ of (downloaded) set equations. Note: _During the computation, some bisimulation questions $u\stackrel{{\scriptstyle?}}{{\approx}}v$ from the list $Q$ can be resolved – replaced by either $u\approx v$ (positive) or $u\not\approx v$ (negative) facts. Thereby $Q$ will contain both non-resolved questions, and positive or negative facts. The process will continue until $Q$ will stabilise_111In the case of using the Oracle, as described later in Chapter 5, the questions already asked to the Oracle should be appropriately labelled to avoid asking them again. . 3. 2. Initialise populating $Q$ by inserting the bisimulation question $x\stackrel{{\scriptstyle?}}{{\approx}}y$. 4. 3. Acquire set equations corresponding to those set names involved in all non- resolved bisimulation questions in $Q$ by downloading appropriate WDB files containing these equations. That is, for the question $u\stackrel{{\scriptstyle?}}{{\approx}}v$ in $Q$, download the uniquely defined WDB files (by full set names $u,v$) containing equations $u=\\{\ldots\\}$ and $v=\\{\ldots\\}$ (if they have not been downloaded yet). Add these equation into the (originally empty) list of set equations $Eq$ (acquired from the WDB). Extend $Q$ by all new bisimulation questions (more precisely, those not yet included in $Q$ neither as questions nor as positive or negative answers) for all set names participating in $Q$ plus set names in the right hand side of the (downloaded) set equations from $Eq$. Note: Not all the downloaded equations (from the downloaded files) will likely participate in $Eq$ and in the generation of transitive closure $\mbox{\sf TC}(x)\cup\mbox{\sf TC}(y)$ for the initial question $x\stackrel{{\scriptstyle?}}{{\approx}}y$, and in this case they may be ignored when generating new questions (to be added in $Q$). But they could probably be useful in future computations and could save time on downloading if some equations to be downloaded as prescribed by the current stage have been already downloaded earlier. Thus, all downloaded equations (in fact, WDB files) should be saved in a cache of WDB (in memory) for possible future use. Therefore, before making the quite expensive step of downloading a WDB file the system should check whether it has already been downloaded. This WDB cache should be initialised when beginning general query execution and used by both the general query evaluation procedure and algorithm described here for evaluating bisimulation (or equality) subqueries $u\stackrel{{\scriptstyle?}}{{\approx}}v$. Similarly to the cache of WDB, the current versions of $Q$ and $Eq$ should not be discarded from the memory till the end of executing a given query, involving the subquery $x\stackrel{{\scriptstyle?}}{{\approx}}y$ considered in the current algorithm, because some other bisimulation questions might be involved which could be easily answered with already known $Q$ and $Eq$. 5. 4. Iteratively apply derivation rules (4.3) and (4.4) (thereby resolving some questions in $Q$) until the initial bisimulation question $x\stackrel{{\scriptstyle?}}{{\approx}}y$ becomes a resolved fact or, otherwise, until exhaustion by using the currently downloaded (probably incomplete) list $Eq$ of set equations. Note: _Some enumerated in $Q$ questions could still remain unresolved._ 6. 5. Recursive jump: 1. (a) Is the initial bisimulation question $x\stackrel{{\scriptstyle?}}{{\approx}}y$ now a resolved fact in $Q$? Yes – The original bisimulation question has now been resolved (end of algorithm). No – Move to step 5b to continue trying to resolve initial bisimulation question and other non-resolved questions in $Q$. 2. (b) Are there set names $u$ participating in non-resolved questions in $Q$ for which set equations $u=\\{\ldots\\}$ have not yet been downloaded? Yes – Then move to step 3 by which further facts may be derived once the relevant set equations have been downloaded. No – Then the full transitive closure $\mbox{\sf TC}(x)\cup\mbox{\sf TC}(y)$ of the initial bisimulation question $x\stackrel{{\scriptstyle?}}{{\approx}}y$ has been completed, therefore there are no further possibilities to derive/resolve new facts, and stabilisation of the list $Q$ has been achieved. Postulate all non-resolved bisimulation questions as true facts. In particular, the original bisimulation question $x\stackrel{{\scriptstyle?}}{{\approx}}y$ has now been resolved positively as $x\approx y$ (end of algorithm). 7. END with the bisimulation question $x\stackrel{{\scriptstyle?}}{{\approx}}y$ resolved positively $x\approx y$ or negatively $x\not\approx y$. The essential point of the above algorithm for computing bisimulation is that downloading of WDB files is done in a “lazy” way – only when no derivation step is possible. This strategy is chosen because downloading WDB files is the most expensive process of the general implemented bisimulation algorithm. Therefore only in the worst case downloading all the necessary set equations (generating the full transitive closure of the original bisimulation question) will be necessary. Usually this should save a lot of time and memory. ## Part II Local/global approach to optimise bisimulation and querying ### Chapter 5 The Oracle #### 5.1 Computing bisimulation with the help of the Oracle The concept of the Oracle for Web-like databases is somewhat similar to that of an Internet search engine, such as Google, where the Oracle will attempt to provide bisimulation facts to the $\Delta$-query system when requested and thereby to facilitate the easier computation of set equality. Furthermore, the Oracle should work in background time independently (as well as by requests from the $\Delta$-query system) to derive bisimulation facts. We assume that to the bisimulation question $x\stackrel{{\scriptstyle?}}{{\approx}}y$ the Oracle should give one of three answers _“Yes”_ , _“No”_ or _“Unknown”_ 111More precisely, to know which question is answered, full answers should be given: “$x\approx y$”, “$x\not\approx y$” or “$x\stackrel{{\scriptstyle?}}{{\approx}}y$”. . In the latter case _“Unknown”_ should consequently be replaced by the Oracle (after resolving the question itself, probably resulting in some delay) with either _“Yes”_ or _“No”_. The answers _“Yes”_ or _“No”_ must be correct. In fact, asking the Oracle is a way to resolve bisimulation questions, just like applying derivation rules. However, it is likely that the Oracle only provides a partial bisimulation relation (depending on the current state of its work) because of possible updates to WDB forcing the Oracle to redo at least some of its work and the time required to compute bisimulation. Thus, those bisimulation questions answered _“Unknown”_ should invoke an initial attempt by the query system to resolve the question locally, hence downloading WDB files with those set equations corresponding to the set names participating in the question(s), etc., as in the algorithm of Section 4.2.1 above. If during the process of local computation the Oracle will replace _“Unknown”_ by _“Yes”_ or _“No”_ then this local attempt to resolve the bisimulation question will be automatically halted due to replacing this question by its answer, however, downloaded WDB files may prove to be useful in future derivation steps of other possible bisimulation questions and should not be discarded from the local cache. For example, let us consider the Oracle attempting to resolve a bisimulation question posed by the $\Delta$-query system as shown below: $\Delta$-query system: $x\stackrel{{\scriptstyle?}}{{\approx}}y$ (is the set name $x$ bisimilar to the set name $y$?). * Oracle: _“Unknown”_ (based on the current state of knowledge of the Oracle). * The Oracle works towards resolving various bisimulation questions, in particular $x\stackrel{{\scriptstyle?}}{{\approx}}y$. * 500ms later… * Oracle: _“No”_ ($x\not\approx y$ holds). #### 5.2 Imitating the Oracle for testing purposes As the first attempt, an Oracle which is able to answer bisimulation questions can be simulated with a single file containing a list of bisimulation facts with the states _“Yes”_ or _“No”_. Further, those bisimulation questions initially answered as _“Unknown”_ can be also simulated as delayed answers of _“Yes”_ and _“No”_ by associating each bisimulation fact with number of milliseconds delay. For the purposes of our preliminary implementation the trivial Oracle (simulated as a file instead of a special Internet server) was implemented as an XML file222 which should not be mixed with XML-WDB files used to represent set equations . The trivial Oracle (XML file) contains all the necessary information to simulate the behaviour of the Oracle: bisimulation facts corresponding to all possible bisimulation questions. Also, to simulate those questions initially answered _“Unknown”_ by the Oracle (such as in the example above) each bisimulation fact has an associated delay time. These XML files are generated by one of the programs belonging to our suite of tools from a given WDB in such a way that all _“Yes”_ /_“No”_ facts presented there are automatically true, that is the bisimulation relation is computed by this program and presented as an XML file. Furthermore, arbitrary delay times (useful for the purposes of testing) are added manually to those XML files generated by this program. Each bisimulation fact (in the trivial Oracle) is represented as an XML tag with `set_name`s, bisimulation `value` and `delay` times as mandatory attributes. For example, let us consider the bisimulation fact $y\not\approx z$ with no delay time represented in the trivial Oracle as, <facts set_name="y"> <fact set_name="z" value="no" delay="0" /> </facts> where bisimulation facts are grouped, inside `<facts>` and `<fact>` tags, according to those set name participating in the WDB. The grouping of facts is a feature of the implementation used to generate these XML files. Let us consider the trivial Oracle for the bibliographic WDB (considered in Section 3.5) represented as the XML file: <oracle> <facts set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#BibDB"> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b1" value="no"/> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2" value="no"/> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p1" value="no"/> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p2" value="no"/> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p3" value="no"/> </facts> <facts set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b1"> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2" value="no"/> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p1" value="no"/> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p2" value="no"/> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p3" value="no"/> </facts> <facts set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2"> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p1" value="no"/> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p2" value="no"/> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p3" value="yes"/> </facts> <facts set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p1"> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p2" value="no"/> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p3" value="no"/> </facts> <facts set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p2"> <fact delay="0" set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p3" value="no"/> </facts> <facts set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p3"> </facts> </oracle> Note that only one value `"yes"` appears above as it is already known concerning our bibliography database that only the set names `b2` and `p3` are bisimilar. Information encoded within the such an XML file simulates the responses of the Oracle, i.e. the responses to bisimulation questions. These responses, i.e. the desired bisimulation facts (possibly delayed with the immediate temporary answer _“Unknown”_) may assist the regular bisimulation algorithm. To simulate the Oracle, the bisimulation algorithm in Section 4.2.1 should be extended replacing step 3 as follows: * 3. Acquiring set equations $u=\\{\ldots\\}$ and $v=\\{\ldots\\}$ corresponding to all those unresolved questions $u\stackrel{{\scriptstyle?}}{{\approx}}v$ in $Q$ should now begin with asking the Oracle all these questions (which have not already been asked), and the necessary downloads should follow only in the case where the Oracle answers with _“Unknown”_. Note: _According to Footnote 1 (on page 1), the answer _“Unknown”_ , in fact, means that the Oracle returns back to the query system the question “$u\stackrel{{\scriptstyle?}}{{\approx}}v$”, and similarly for the answers _“Yes”_ and _“No”_ in which case the full answers “$x\approx y$” and “$x\not\approx y$”, respectively, should be returned. Otherwise, because of delays, the system will not know how to treat _“Yes”_ , _“No”_ and _“Unknown”_._ Evidentially, whilst resolving bisimulation questions (the modified version of) Step 2 will pose many bisimulation question to the Oracle, which will be answered either “Yes” ($u\approx v$) or “No” ($u\not\approx v$) possibly with delays. In fact, the behaviour of the modified bisimulation algorithm can be characterised as follows, depending on the Oracle’s responses: * • Bisimulation questions ($u\stackrel{{\scriptstyle?}}{{\approx}}v$) to the Oracle directly answered _“Yes”_ ($u\approx v$) or _“No”_ ($u\not\approx v$): In this case, the answer from the Oracle should immediately replace the unresolved question in $Q$, and the modified bisimulation algorithm will resume its work resolving other non-resolved bisimulation questions from $Q$. * • Bisimulation questions ($u\stackrel{{\scriptstyle?}}{{\approx}}v$) to the Oracle initially answered _“Unknown”_ ($u\stackrel{{\scriptstyle?}}{{\approx}}v$): In this case, the modified bisimulation algorithm will, in fact, resume its work resolving $u\stackrel{{\scriptstyle?}}{{\approx}}v$ and other non-resolved bisimulation questions from $Q$. Thus, the question will either be resolved locally or the Oracle will replace its answer _“Unknown”_ ($u\stackrel{{\scriptstyle?}}{{\approx}}v$) by either _“Yes”_ ($u\approx v$) or _“No”_ ($u\not\approx v$) possibly with some delay. Note that, if the Oracle answers the question positively or negatively before being resolved locally then this answer should replace the question in $Q$ and the modified bisimulation algorithm should continue its work (taking into account the newly resolved question – it does not matter in which way the question is resolved, by the Oracle or by the query system)333 A question answered _“Unknown”_ does not require asking the Oracle again. In general, Oracle (as a special Internet server) should remember all questions and reply to the appropriate client accordingly when the answer will be ready. . Note that, step 2 in the present modified form plays a crucial role in performance: resolution of bisimulation questions by the Oracle will save costly downloading of WDB files. #### 5.3 Empirical testing of the trivial Oracle In principle, with the help of the Oracle those $\Delta$-queries which involve set equality (bisimulation) should be executed quicker. The aim of the following empirical testing is to measure the improvement in query performance with the help of the Oracle, in addition to demonstrating the effects of delayed answers to bisimulation questions (those initially answered _“Unknown”_) by the Oracle.444Even more optimal would be to postpone local resolution of bisimulation questions in favour of some other independent subqueries of the given query with the hope that the Oracle will give a definite answer before starting local resolution. There are many ways to optimise our implementation, but we can consider only a limited range of such possibilities. The distributed bibliographic WDB considered in Section 3.5 (see Figure 3.1) is fragmented into two XML-WDB files, thus making computation of bisimulation more dependent on the time taken to download these files. The following example query (already considered in Section 3.5.2) involves set equality to determine which publications belonging to BibDB refer to the publication (possibly bisimilar to) b2. The requirement to compute bisimulation across the distributed bibliographic WDB makes this simple example particularly suitable for empirical testing of the Oracle: set query let set constant BibDB be http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#BibDB, set constant b2 be http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#b2 in collect { pub-type:pub where pub-type:pub in BibDB and exists ’refers-to’:ref in pub . ref=b2 } endlet; The execution time of this example query under various experimental conditions can be seen in the Table 5.1. The results suggest a marked improvement in performance with help of the Oracle, and only a slight improvement in performance when the Oracle returns an answer after delay 50ms or 75ms. However, when the Oracle provided a greatly delayed answer ($\geq$ 100ms) query execution occurs with no real help by the Oracle, and bisimulation is computed locally without any real help from the Oracle. Thus, under this circumstance, query execution time increases, and the optimal approach appears to be query execution without invoking the Oracle. This result may be explained by the numerous (and seemingly futile) bisimulation questions posed to the Oracle (all of which are answered _“Unknown”_ and never improved) which provide no real help. In summary, these results were based on experiments with the trivial Oracle (simulated as an XML file instead of an Internet server). Additionally, the example WDB is too small and, crucially, only distributed into two fragments. In principle, invoking the help of the Oracle should improve query performance considerably when the WDB is distributed into a large number of fragments. Strategy \| Query execution time [ms] ---\|--- Bisimulation algorithm without invoking the Oracle \| 588 with help of the Oracle (no delay time per question) \| 390 with help of the Oracle (50ms delay time per question) \| 500 with help of the Oracle (75ms delay time per question) \| 500 with help of the Oracle (100ms delay time per question) \| 608 with help of the Oracle (125ms delay time per question) \| 608 Table 5.1: Experimental results showing query execution time [ms] corresponding to each strategy for computing bisimulation. In a more realistic situation, the Oracle should be implemented as an Internet service (called the bisimulation engine) for large distributed WDB, working in background time to derive all possible bisimulation facts on the current state of WDB. The goal of the bisimulation engine consists in answering bisimulation questions $x\stackrel{{\scriptstyle?}}{{\approx}}y$ from the $\Delta$-query system (possibly with a delay555 In principle, the Oracle, when asked the question $x\stackrel{{\scriptstyle?}}{{\approx}}y$, could change its regular behaviour, and try to resolve such questions (with appropriate strategy of priority) from one or more querying clients. ). The Oracle should be based on the bisimulation algorithm described in Section 4.2.1 and, additionally, on the idea of local/global bisimulation considered in Chapter 6. We will consider implementation (still rather an imitation) of the Oracle in Chapter 7 and some further advanced experiments. ### Chapter 6 Local/global bisimulation Let a proper set111$L\neq\emptyset$ and $L\neq\textit{SNames}$ $L\subseteq\textit{SNames}$ of “local” vertices (set names) in a graph WDB (a system of set equations) be given, where SNames is the set of all WDB vertices (set names). Let us also denote by $L^{\prime}\supseteq L$ the set of all set names participating in the set equations for each set name in $L$ both from left and right-hand sides. Considering the graph as a WDB distributed among many _sites_ , $L$ plays the role of (local) set names defined by set equations in some (local) WDB files of one of these sites. Then $L^{\prime}\setminus L$ consists of non-local set names which, however, participate in the local WDB files, have defining equations in other (possibly remote) sites of the given WDB. Non-local (full) set names can be recognised by their URLs as different from the URL of the given site. Set names (or vertices) from $L^{\prime}$ can be reasonably called “almost local”. We will consider _derivation rules_ of the form $xRy\mathrel{:-}\ldots R\ldots$ for three relations over SNames: ${\approx^{L}_{-}}\subseteq{\approx}\subseteq{\approx^{L}_{+}}\quad\textrm{or, rather, their negations}\quad{\not\approx^{L}_{+}}\subseteq{\not\approx}\subseteq{\not\approx^{L}_{-}}$ defined on the whole WDB graph (however, we will be mainly interested in the behaviour of $\approx^{L}_{-}$ and $\approx^{L}_{+}$ on $L$). We will usually omit the superscript $L$ when it is clear from the context. In particular, this chapter deals mainly with one $L$, so no ambiguity can arise. #### 6.1 Defining the ordinary bisimulation relation $\approx$ Recall the derivation rule defining $\not\approx$: $x\not\approx y\mathrel{:-}\exists x^{\prime}\in x\forall y^{\prime}\in y(x^{\prime}\not\approx y^{\prime})\vee\exists y^{\prime}\in y\forall x^{\prime}\in x(x^{\prime}\not\approx y^{\prime}).$ (6.1) If $u\not\approx v$ is underivable for some vertices/set names $u,v$ then we assume $u\approx v$ to be true (indistinguishable sets are considered equal), and similarly in other cases below. Equivalently, $\not\approx$ is the least relation satisfying (6.1), and its positive version $\approx$ is the largest relation satisfying $x\approx y\Rightarrow\forall x^{\prime}\in x\exists y^{\prime}\in y(x^{\prime}\approx y^{\prime})\;\&\;\forall y^{\prime}\in y\exists x^{\prime}\in x(x^{\prime}\approx y^{\prime}).$ (6.2) The relation $\approx$ is called _bisimulation_ relation which is also known to be an equivalence relation on the whole graph. Below are defined its upper and lower (relativised to $L$) approximations $\approx_{+}$ and $\approx_{-}$. #### 6.2 Defining the local upper approximation $\approx^{L}_{+}$ of $\approx$ Let us define the relation ${\not\approx_{+}}\subseteq\textit{SNames}^{2}$ by derivation rule $x\not\approx_{+}y\mathrel{:-}x,y\in L\;\&\;[\exists x^{\prime}\in x\forall y^{\prime}\in y(x^{\prime}\not\approx_{+}y^{\prime})\vee\ldots].$ (6.3) Here and below “$\ldots$” represents the evident symmetrical disjunct (or conjunct). Thus the premise (i.e. the right-hand side) of (6.3) is a _restriction_ of that of (6.1). It follows by induction on the length of derivation of the $\not\approx_{+}$-facts that, $\displaystyle{\not\approx_{+}}\subseteq{\not\approx},\quad{\approx}\subseteq{\approx_{+}}$ (6.4) $\displaystyle x\not\approx_{+}y\Rightarrow x,y\in L$ (6.5) $\displaystyle x\not\in L\vee y\not\in L\Rightarrow x\approx_{+}y.$ (6.6) As $L\neq\textit{SNames}$, the set of all vertices, it follows from (6.6) that $\approx_{+}$ can be an equivalence relation on the whole graph _only_ if it is trivial, making all vertices equivalent. But we will show below that it is an equivalence relation locally, that is on $L$. Let us also consider another, “more local” version of the rule (6.3) $x\not\approx_{+}y\mathrel{:-}x,y\in L\;\&\;[\exists x^{\prime}\in x\forall y^{\prime}\in y(x^{\prime},y^{\prime}\in L\;\&\;x^{\prime}\not\approx_{+}y^{\prime})\vee\ldots].$ (6.7) It defines the same relation $\not\approx_{+}$ because in both cases (6.5) holds implying that the right-hand side of (6.7) is equivalent to the right- hand side of (6.3). The advantage of (6.3) is its formal simplicity whereas that of (6.7) is its “local” computational meaning. From the point of view of distributed WDB with $L$ one of its local sets of vertices/set names (corresponding to one of the sites of the distributed WDB), we can derive $x\not\approx_{+}y$ for local $x,y$ via (6.7) by looking at the content of local WDB files only. Indeed, participating URLs (full set names) $x^{\prime}\in x$ and $y^{\prime}\in y$, although likely non-local names ($\in L^{\prime}\setminus L$), occur in the locally stored WDB files with local URLs $x$ and $y\in L$. However, despite the possibility that $x^{\prime}$ and $y^{\prime}$ can be in general non-local, we will need to use in (6.7) the facts of the kind $x^{\prime}\not\approx_{+}y^{\prime}$ derived on the previous steps for local $x^{\prime},y^{\prime}\in L$ only. Therefore, ###### Note 1 (Local computability of $x\not\approx_{+}y$). For deriving the facts $x\not\approx_{+}y$ for $x,y\in L$ by means of the rule (6.3) or (6.7) we will need to use the previously derived facts $x^{\prime}\not\approx_{+}y^{\prime}$ for set names $x^{\prime},y^{\prime}$ from $L$ only, and additionally we will need to use set names from a wider set $L^{\prime}$ (available, in fact, also locally)222 This is the case when $y=\emptyset$ but there exists according to (6.7) an $x^{\prime}$ in $x$ which can be possibly in $L^{\prime}\setminus L$ (or similarly for $x=\emptyset$). When $y=\emptyset$ then, of course, there are no suitable witnesses $y^{\prime}\in y$ for which $x^{\prime}\not\approx_{+}y^{\prime}$ hold. Therefore, only the existence of some $x^{\prime}$ in $x$ plays a role here. . In this sense, the derivation of all facts $x\not\approx_{+}y$ for $x,y\in L$ can be done locally and does not require downloading of any external WDB files. (In particular, facts of the form $x\not\approx_{+}y$ or $x\approx_{+}y$ for set names $x$ or $y$ in $L^{\prime}\setminus L$ present no interest in such derivations.) The upper approximation $\approx_{+}$ (on the whole WDB graph) can be equivalently characterised as the largest relation satisfying any of the following (equivalent) implications for all graph vertices $x,y$: $\displaystyle x\approx_{+}y\Rightarrow x\not\in L\vee y\not\in L\vee[\forall x^{\prime}\in x\exists y^{\prime}\in y(x^{\prime}\approx_{+}y^{\prime})\;\&\;\ldots]$ $\displaystyle x\approx_{+}y\;\&\;x,y\in L\Rightarrow[\forall x^{\prime}\in x\exists y^{\prime}\in y(x^{\prime}\approx_{+}y^{\prime})\;\&\;\ldots]$ (6.8) The set of relations $R\subseteq\textit{SNames}^{2}$ satisfying (6.8), in place of $\approx_{+}$, evidently: (i) contains the identity relation $=$ and is closed under (ii) unions (thus the largest $\approx_{+}$ does exist), and (iii) taking inverse. Evidently, any ordinary (global) bisimulation relation $R\subseteq\textit{SNames}^{2}$ (that is, a relation satisfying (6.2)) satisfies (6.8) as well333This imples (6.4) again because $\approx_{+}$ is the largest relation satisfying (6.8). . For any $R\subseteq L^{2}$ the converse also holds: if $R$ satisfies (6.8) then it is actually a global bisimulation relation (and $R\subseteq{\approx}$). It is easy to check that (iv) relations $R\subseteq L^{2}$ satisfying (6.8) are closed under compositions. It follows from (i) and (iii) that $\approx_{+}$ is reflexive and symmetric. Over $L$, the relation $\approx_{+}$ (that is the restriction $\approx_{+}\upharpoonright L$) is also transitive due to (iv). Therefore, $\approx_{+}$ is an _equivalence relation_. (In general, as we noticed above, $\approx_{+}$ cannot be equivalence relation on the whole graph, due to (6.6).) Moreover, any $x\not\in L$ is $\approx_{+}$ to all vertices (including itself). #### 6.3 Defining the local lower approximation $\approx^{L}_{-}$ of $\approx$ Consider the derivation rule for the relation ${\not\approx_{-}}\subseteq\textit{SNames}^{2}$: $\displaystyle x\not\approx_{-}y$ $\displaystyle\mathrel{:-}$ $\displaystyle(x,y\not\in L\;\&\;x\neq y)\vee(x\in L\;\&\;y\not\in L)\vee(y\in L\;\&\;x\not\in L)\vee{}$ $\displaystyle\qquad\qquad[\exists x^{\prime}\in x\forall y^{\prime}\in y(x^{\prime}\not\approx_{-}y^{\prime})\vee\ldots].$ The following is an equivalent simplified rule: $\displaystyle x\not\approx_{-}y$ $\displaystyle\mathrel{:-}$ $\displaystyle((x\not\in L\vee y\not\in L)\;\&\;x\neq y)\vee$ (6.9) $\displaystyle\qquad\qquad[\exists x^{\prime}\in x\forall y^{\prime}\in y(x^{\prime}\not\approx_{-}y^{\prime})\vee\ldots]$ which can also be equivalently replaced by two rules: $\displaystyle x\not\approx_{-}y$ $\displaystyle\mathrel{:-}$ $\displaystyle(x\not\in L\vee y\not\in L)\;\&\;x\neq y\textrm{ -- ``a priori knowledge''},$ (6.10) $\displaystyle x\not\approx_{-}y$ $\displaystyle\mathrel{:-}$ $\displaystyle\exists x^{\prime}\in x\forall y^{\prime}\in y(x^{\prime}\not\approx_{-}y^{\prime})\vee\ldots\;.$ (6.11) Thus, in contrast to (6.3), this is a _relaxation_ , or, an _extension_ of the rule (6.1) for $\not\approx$. It follows that $\displaystyle{\not\approx}\subseteq{\not\approx_{-}}\ ({\approx_{-}}\subseteq{\approx}),$ $\displaystyle x\approx_{-}x\textrm{ for all }x\in\textit{SNames}\textrm{ --- reflexivity}.$ The former is trivial, and the latter means that $x\not\approx_{-}x$ is not derivable. (Indeed, $x\not\approx_{-}x$ can be derivable only if $x^{\prime}\not\approx_{-}x^{\prime}$ is derivable for some $x^{\prime}\in x$ on an earlier stage; thus, there cannot exists a first such derivable fact.) It is also evident that $\displaystyle\textrm{any }x\not\in L\textrm{ is }\not\approx_{-}\textrm{ to all vertices different from }x,$ $\displaystyle x\approx_{-}y\;\&\;x\neq y\Rightarrow(x,y\in L).$ The latter means that $\approx_{-}$ (which is an equivalence relation on SNames and hence on $L$ as it is shown below) is non-trivial only on the local set names. Again, like for $\not\approx_{+}$, we can conclude from the above considerations that, ###### Note 2 (Local computability of $x\not\approx_{-}y$). We can compute the restriction of $\not\approx_{-}$ on $L$ locally: to derive $x\not\approx_{-}y$ for $x,y\in L$ with $x\neq y$ (taking into account reflexivity of $\approx_{-}$) by (6.9) we need to use only $x^{\prime},y^{\prime}\in L^{\prime}$ (by $x^{\prime}\in x$ and $y^{\prime}\in y$) and already derived facts $x^{\prime}\not\approx_{-}y^{\prime}$ for $x^{\prime},y^{\prime}\in L,x\neq y$, as well as the facts $x^{\prime}\not\approx_{-}y^{\prime}$ for $x^{\prime}$ or $y^{\prime}\in L^{\prime}\setminus L$, $x^{\prime}\neq y^{\prime}$ following from the “a priori knowledge” (6.10). The lower approximation $\approx_{-}$ can be equivalently characterised as the largest relation satisfying $x\approx_{-}y\Rightarrow(x,y\in L\vee x=y)\;\&\;(\forall x^{\prime}\in x\exists y^{\prime}\in y(x^{\prime}\approx_{-}y^{\prime})\;\&\;\ldots).$ Evidently, $=$ (substituted for $\approx_{-}$) satisfies this implication. Relations $R$ satisfying this implication are also closed under unions and taking inverse and compositions. It follows that $\approx_{-}$ is reflexive, symmetric and transitive, and therefore an _equivalence relation over the whole WDB graph_ , and therefore _on its local part_ $L$. Finally, we summarise that both upper and lower approximations $\approx^{L}_{+}$ and $\approx^{L}_{-}$ to $\approx$ restricted to $L$ are computable “locally”. Each of them is defined in a trivial way outside of $L$, and the computation requires only knowledge at most on the $L^{\prime}$-part of the graph. In fact, only edges from $L$ to $L^{\prime}$ are needed, everything being available locally. #### 6.4 Using local approximations to aid computation of the global bisimulation The point of previous considerations of this chapter was that, given any set $L$ of “local” set names (or WDB graph vertices), we defined two (local to $L$) approximations $\approx^{L}_{+}$ and $\approx^{L}_{-}$ to the global bisimulation relation $\approx$. Now, assume that the set SNames of all set names (nodes) of a WDB is disjointly divided into a family of local sets $L_{i}$, for each “local” site $i\in I$ (so that SNames is the disjoint union $\textit{SNames}=\bigcup_{i\in I}L_{i}$). Then we have many local approximations $\approx^{L_{i}}_{+}$ and $\approx^{L_{i}}_{-}$ to the global bisimulation relation $\approx$. As we discussed above, these relations can be easily computed locally by each site $i$ using the derivation algorithms described in Notes 1 and 2, respectively. Now the problem is how to compute the global bisimulation relation $\approx$ with the help of many its local approximations $\approx^{L_{i}}_{+}$ and $\approx^{L_{i}}_{-}$ in all sites $i$. ##### 6.4.1 Granularity of sites However, for simplicity of implementation and testing the above idea (and also because it is more problematic to create many sites with their servers) we will redefine the scope of $i$ to a smaller granularity. Instead of taking $i$ to be a site, consisting of many WDB files, we will consider that each $i$ itself is a name of a single WDB file $\textit{file}_{i}$. More precisely, $i$ is considered as the URL of any such a file. This will not change the main idea of implementation of the Oracle on the basis of using local information for each $i$. That is, we reconsider our understanding of the term local – from being _local to a site_ to _local to a file_ 444 Moreover, this idea of locality to files (described below in detail) belonging to each such a site $i$ is useful for computing $i$-th site’s local upper and lower approximations of bisimulation as an intermediate step. Then these $i$-th approximations could be used in implementation of the global Oracle. That is, the idea of locality can be fruitfully used on various levels of granularity to optimise performance of the bisimulation engine (the Oracle). – as shown in Figure 6.1. Then $L_{i}$ is just the set of all (full versions of) set names defined in file $i$ (left-hand sides of all set equations in this file). Evidently, so defined sets $L_{i}$ are disjoint and cover the class SNames of all (full) set names from the WDB considered. Recall that $\approx^{L_{i}}_{+}$ and $\approx^{L_{i}}_{-}$ are formally defined on the whole WDB (not only on $L_{i}$). Their restrictions to $L_{i}$ are also equivalence relations (on $L_{i}$) denoted, for brevity and when it is clear from the context, also as $\approx^{L_{i}}_{+}$ and $\approx^{L_{i}}_{-}$. (a) Local to files (b) Local to sites Figure 6.1: Summary of a distributed WDB showing the difference between interpretation of local as: local to a file, or local to a site. The relations $\approx^{L_{i}}_{+}$ and $\approx^{L_{i}}_{-}$ should be automatically computed, saved as file and maintained as the current local approximations for each WDB file $i$. In principle a suitable tool is necessary for editting (and maintaining) WDB, which would save a WDB file $i$ and thereby generate the approximation relations $\approx^{L_{i}}_{+}$ and $\approx^{L_{i}}_{-}$ (file) automatically. ##### 6.4.2 Local approximations giving rise to global bisimulation facts We know that these approximations satisfy, ${\approx^{L_{i}}_{-}}\subseteq{\approx}\subseteq{\approx^{L_{i}}_{+}},$ or, equivalently, ${\not\approx^{L_{i}}_{+}}\subseteq{\not\approx}\subseteq{\not\approx^{L_{i}}_{-}}.$ It evidently follows that, * • each positive local fact of the form $x\approx^{L_{i}}_{-}y$ is a positive fact about $\approx$, i.e. gives rise to the fact $x\approx y$, and * • each negative local fact of the form $x\not\approx^{L_{i}}_{+}y$ is a negative fact about $\approx$, i.e. gives rise to the fact $x\not\approx y$. Let $\approx^{L_{i}}$ (without subscripts $+$ or $-$) denote the set of positive and negative facts for set names in $L_{i}$ on the global bisimulation relation $\approx$ obtained by these two clauses. This set of facts $\approx^{L_{i}}$ is called the _local simple approximation set_ to $\approx$ for the file (or site) $i$. Let the _local Oracle_ $LO_{i}$ just answer _“Yes”_ (“$x\approx y$”), _“No”_ (“$x\not\approx y$”) or _“Unknown”_ to questions $x\stackrel{{\scriptstyle?}}{{\approx}}y$ for $x,y\in L_{i}$ according to $\approx^{L_{i}}$. In the case of $i$ considered as a site (rather than a file) then $LO_{i}$ can have delays when answering _“Yes”_ (“$x\approx y$”) or _“No”_ (“$x\not\approx y$”) because $LO_{i}$ should rather compute $\approx^{L_{i}}$ itself and find out in $\approx^{L_{i}}$ answers to the questions asked which takes time. But, if $i$ is understood just as a file saved together with all the necessary information on local approximations at the time of its creation then $LO_{i}$ can submit the required answer and, additionally, all the other facts it knows at once (to save time on possible future communications). Therefore, a centralised Internet server (for the given distributed WDB) working as the (global) Oracle or _Bisimulation Engine_ , which derives positive and negative ($\approx$ and $\not\approx$) global bisimulation facts can do this by the algorithm of Section 4.2.1, in addition to asking (when required) various local Oracles $LO_{i}$ concerning $\approx^{L_{i}}$. That is, the algorithm from Section 4.2.1 extended to exploit local simple approximations $\approx^{L_{i}}$ should, in the case of the question $x\stackrel{{\scriptstyle?}}{{\approx}}y$ in $Q$ with $x,y\in L_{i}$ from the same site/WDB file $i$555$x,y\in L_{i}$ can be trivially checked by comparing the full set names $x,y$ with the URL $i$ , additionally ask the oracle $LO_{i}$ whether it already knows the answer (as described in the above two items). If the answer is known, the algorithm should just use it. Otherwise (if $LO_{i}$ does not know the answer or $x,y$ do not belong to one $L_{i}$ – that is, they are “remote” one from another), the global Oracle should work as described in Section 4.2.1 by downloading set equations, making derivation steps, etc. Thus, local approximations serve as auxiliary local Oracles $LO_{i}$ helping the global Oracle. ##### 6.4.3 Practical algorithm for computation of local approximations The derivations rules for computing local approximations (described above by rules 6.3, 6.9 together with Notes 1, 2) can be implemented in a very similar way to the practical algorithm for computing the global bisimulation described in Section 4.2. Given a WDB file $i$ as the input, the algorithm will generate _approximation files_ $i^{A}$ and $i^{SA}$ containing local approximations $\approx^{L_{i}}_{+}$, $\approx^{L_{i}}_{-}$ and, respectively, local simple approximation set $\approx^{L_{i}}$ (all three approximations restricted to $L_{i}$). The derivation rules (6.3, 6.9) were formulated to compute the relations $\approx^{L_{i}}_{+}$ and $\approx^{L_{i}}_{-}$ over all set names (both local and non-local). According to Notes 1, 2 on local computability of local approximations the computation of restricted relations can be also restricted to local set names in $L_{i}$ (or to slightly wider set $L^{\prime}_{i}$). Additionally, the two clauses in Section 6.4.2 should be used. Unlike the practical algorithm for computing global bisimulations, the computation of local approximations $\approx^{L_{i}}_{+},\approx^{L_{i}}_{-}$, and $\approx^{L_{i}}$ (creation of approximation files $i^{A}$ and $i^{SA}$) should be done after creating (and saving) WDB files $i$, therefore this operation does not require much attention towards optimisation. Local simple approximation files, $i^{SA}$, are represented as XML files (quite similar to those of the imitated Oracle; see Section 5.2) containing global bisimulation facts derived locally on the fragment $i$ ($\approx^{L_{i}}$). Each approximation fact is represented as an (XML) `fact` tag with boolean local approximation value and set name as mandatory attributes `value` and `set_name`. These approximation facts are grouped (inside `facts` tag) corresponding to all local set names in $L_{i}$666 This is quite similar to the previous implemented tool to generate the (trivial) Oracle XML files. . For example, let us consider the simple approximation file $i^{SA}$, corresponding to the local simple approximation set $\approx^{L_{i}}$, for one particular fragment of the bibliographic WDB (see Section 3.5) http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml: <simple-approximation> <facts set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#BibDB"> <fact set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b1" value="no"/> <fact set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2" value="no"/> </facts> <facts set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b1"> <fact set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2" value="no"/> </facts> <facts set_name="http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml#b2"> </facts> </simple-approximation> Note that all “no” values above correspond to negative bisimulation facts ($\not\approx$) resulting from the computation of the local simple approximation set $\approx^{L_{i}}$, where $i$ is the WDB file mentioned above. Simple approximation files are predictably named based on the name of the corresponding WDB file $i$ by concatenating the string “approximation” to the end of the WDB file name, for example the WDB file name “BibDB-f1.xml” will have corresponding simple approximation file with the name “BibDB-f1.approximation.xml”. ### Chapter 7 The Oracle based on the idea of local/global bisimulation #### 7.1 Description of the bisimulation engine (implementation of a more realistic Oracle) Empirical evidence from the implementation of the imitated Oracle in Section 5.3 concluded that a centralised service providing answers to bisimulation question would increase query performance (for those queries exploiting set equality) – this service could be named _bisimulation engine_. The goal of such bisimulation engine would be: * • Answer bisimulation queries – Answers bisimulation questions communicating via standardised protocol (as discussed in Section 5.1). * • Compute bisimulation – Derive bisimulation facts in background time, and strategically prioritise bisimulation questions posed by the $\Delta$-query system by temporary changing the fashion of the background time work in favour of resolving these questions111 Although due to limitations of time, the current implementation is more basic and does not adopt this strategy of prioritising. (See more in Section 7.1.1. . * • Exploit local approximations – Exploit those local approximations corresponding to WDB files to assist in the computation of bisimulation. * • Maintain cache of set equations – The Oracle (just like the $\Delta$-query system) should maintain a cache of the downloaded set equations in the previous steps. These set equations may later prove useful in deriving new bisimulation facts with saving time on downloading of already known equations. ##### 7.1.1 Strategies In principle, the bisimulation engine should give strategic prioritisation to resolving those bisimulation questions posed by clients – favouring resolution of these bisimulation questions over background tasks (resolving all other bisimulation questions). Moreover, it is reasonable to make the query system adopt a “lazy” strategy while working on a query $q$. This strategy consists of sending bisimulation subqueries of $q$ to the Oracle but not attempting to resolve them in the case of the Oracle’s answer “Unknown” (according to the standard algorithm). Instead of such attempts, the query system could try to resolve other subqueries of the given query $q$ until the resolution of the bisimulation question sent to the Oracle is absolutely necessary. The hope is that before this moment the bisimulation engine will have already given a definite answer. However these useful features have not yet been implemented. In the current version, we have only a simplified imitation of bisimulation engine which resolves all possible bisimulation questions for the given WDB in some predefined standard order without any prioritisation and answers these questions in a definite way when it has derived the required information. Thus the Oracle, while doing its main job in background time, should only remember all the pairs (client, question) for questions asked by clients and send the definite answer to the corresponding client when it is ready. ##### 7.1.2 Exploiting local approximations to aid in the computation of bisimulation For implementation of the Oracle we use again the algorithm for computing the bisimulation relation, as described in Section 4.2.1. But, this algorithm will be extended to exploit local approximations by adding an additional step after acquiring set equations (step 3). This additional step (step 3’) is detailed below: * 3’. Acquire local approximations by (i) downloading the local approximation set $\approx^{L_{i}}$ (consisting of some positive and negative bisimulation facts) represented as the simple approximations file $i^{SA}$ (cf. Section 6.4.3) for each WDB file $i$ retrieved during step 3, and (ii) adding all the positive and negative bisimulation facts from $i^{SA}$ to the list $Q$ of questions and answers (replacing those questions in $Q$ which were thereby answered positively or negatively). Additionally, while computing global bisimulation by exploiting local approximations, the Oracle should always be ready to receive questions $u\stackrel{{\scriptstyle?}}{{\approx}}v$ from various, possibly remote $\Delta$-query systems and answer them immediately that the result is yet unknown (if it is so) and, when the result will become known either as $u\approx v$ or $u\not\approx v$, sending it back to the corresponding $\Delta$-query system. #### 7.2 Empirical testing of the bisimulation engine Preliminary results from testing of the simulated Oracle (described in Section 5.3) indicated that, in principle, an Internet Service providing answers to bisimulation questions would decrease query execution time for those queries involving set equality. However, these preliminary tests were idealised situations and did not describe the _relationship_ between background work by the bisimulation engine and query performance. (In fact, the simulated Oracle did not work in background time, and only some intermediate result was represented.) Additionally, advantages of exploiting local approximations should be demonstrated. Let us consider empirical testing of the bisimulation engine by measuring the performance of the query client executing (with the help of the bisimulation engine) set equality queries of the form $x\stackrel{{\scriptstyle?}}{{\approx}}y$ where $x,y$ belong to a some suitable large WDB. To simplify our considerations on measuring efficiency and to demonstrate some desirable effects we will consider rather artificial examples of WDB. As for WDB size, we will try to determine a threshold where the execution time becomes either unrealistically long or sufficiently reasonable. Note that, labels are ignored with just one (identical) label on all graph edges, as labels typically allow the bisimulation algorithm (see Section 4.2.1) to derive more negative facts and, thus, possibly terminating too early (before the transitive closure of both set names involved in any bisimulation question will be fully explored). ##### 7.2.1 Determining the benefit of background work by the bisimulation engine on query performance The aim of this experiment is to demonstrate the relationship between query execution time $t$ by the query system, and background work by the bisimulation engine. Background work by the bisimulation engine is simulated by delay time $d$, summarised briefly as follows: 1. 1. The bisimulation engine should begin working with the goal of resolving all possible questions $u\stackrel{{\scriptstyle?}}{{\approx}}v$ for arbitrary set names of a given WDB. For the sake of the experiment, it should work uninterrupted (without being posed any questions by the query client) for the delay time $d$. 2. 2. The query client should start executing the test query $x\stackrel{{\scriptstyle?}}{{\approx}}y$ after the delay time $d$ has expired, attempting resolution of the test question (and possibly other bisimulation questions which may arise during this process) with the help of the bisimulation engine. The bisimulation engine should continue its work, but now communicating with the query client. Thus, the query execution time $t(d)$ by the query client (working with the bisimulation engine starting from the delay time $d$) depends on $d$, and it is this dependence which we want to investigate experimentally. Evidently, $t(d)$ should be a decreasing function: the later the client starts its work after the bisimulation engine, the more help it can provide, and the smaller should be the client’s working time $t(d)$. Note that this is still an idealised experiment, in practice, there could be many query clients communicating with the bisimulation engine at arbitrary times. ###### Note 3. It should be noted that the current implementation of the hyperset language $\Delta$ does not use yet any bisimulation engine. These experiments were implemented separately and only to demonstrate some potential strategies for more efficient implementation of the most crucial concept of bisimulation relation underlying the hyperset approach. In this experiment, the example WDB consists of 51 set names distributed over 10 WDB files, connected in chains as shown by the schematical graph in Figure 7.1. To increase the difficulty of computing bisimulation a copy WDB’ of this WDB was made, changing only the URL part of full set names. Thus, the experiment is done over WDB + WDB’. Bisimulation between corresponding set names in WDB and WDB’ under this circumstance is intuitively trivial (the answer being always “true”). However, it is a non-trivial task when calculated by our algorithm which has no advance knowledge that WDB and WDB’ are essentially identical (isomorphic). Further, our experimental procedure here was the measurement of execution time $t(d)$ by the query client executing the test query $x\stackrel{{\scriptstyle?}}{{\approx}}x^{\prime}$ where $x,x^{\prime}$ are corresponding set names in WDB and, its isomorphic copy, WDB’. Figure 7.1: Schematical WDB graph divided into WDB files as shown by the red dashed ovals. ###### 7.2.1.1 Experiment results On examination of the results graph in Figure 7.2 the trend curve suggests an exponential decay relationship between partial work of the bisimulation engine and query performance. Moreover, this qualitative assessment by inspection of the graph is confirmed by examining the experimental values in Table 7.1, which demonstrate that $t(d)$ approximately halves as $d$ increases by steps of 2500ms. Therefore, query performance benefits considerably even when the bisimulation engine has been working (in the background) for relatively short periods of time (say, 5 seconds or more), with an exponential decrease in $t(d)$ as $d$ increases. However, for sufficiently small delay time $d$, query performance suffers, as the bisimulation engine answers _“Unknown”_ to nearly all posed bisimulation questions. Thus, in this case, the bisimulation engine provides no real help, and the query client is forced to start resolving the bisimulation question itself. This suggests that in this circumstance that local computation of bisimulation by the query system without invoking the help of the bisimulation engine would be more efficient, as shown by the threshold on the graph (dashed horizontal line). In fact, here query execution time $t(d)$ with the help of the bisimulation engine is smaller than without the help of the bisimulation engine when delay $d$ is $>2000$ms. Figure 7.2: Graph of experimental results (cf. Table 7.1 below) showing the dependence of query execution time $t(d)$ [ms] on delay time $d$ [ms] Delay time $d$ [ms] \| Execution time $t(d)$ [ms] ---\|--- 0 \| 31050 2500 \| 16300 5000 \| 7930 7500 \| 4090 10000 \| 2040 12500 \| 1380 15000 \| 770 17500 \| 320 20000 \| 10 22500 \| 10 25000 \| 10 Table 7.1: Experimental results showing dependence of query execution time $t(d)$ [ms] on delay time $d$ [ms] ##### 7.2.2 Determining the benefit of exploiting local approximations by the bisimulation engine on query performance It seems plausible to expect that, in practice, each WDB file (or a group of closely related WDB files) should be sufficiently self-contained and have few links to the external files – relatively small dependence on the “external world”. Therefore, we should expect that the set of locally derived bisimulation facts should be sufficiently large (the majority of questions $x\stackrel{{\scriptstyle?}}{{\approx}}y$ for local set names should be resolved locally based on $\approx^{L}_{+}$ and $\approx^{L}_{-}$), and, hence, helpful for the work of bisimulation engine and improving its performance. Figure 7.3: Schematical WDB graph consisting of one WDB file as shown by the red dashed oval. Taking this into account, our alternative example WDB for testing consists of one WDB file containing a variable number $n$ of set names (our experimental parameter as described below) connected in one chain, as shown by the schematical graph in Figure 7.3. Also, like the previous experiment, a copy WDB’ of this WDB was made, changing only the URL part of full set name. Likewise, the experimental queries to follow are over WDB + WDB’, that is over two files. This example represents an extreme, idealised case when each of these two files is fully self-contained, i.e. has no links to the “external world”. As we wrote above, in more realistic situations we should rather expect a relatively small number of such external links. Recall that each of the WDB and WDB’ files has a corresponding local approximations file, as described in Section 6.4.3, containing, respectively, local sets $\approx^{L}$ and $\approx^{L^{\prime}}$ of (positive and negative) bisimulation facts which now will be available by demand to the bisimulation engine (as well as to the query system) which should considerably improve the performance. Thus, for our self-contained WDB file 1 (and similarly with its duplicate) the set of local set names is $L=\\{x_{1},\ldots,x_{n}\\}$ and the corresponding local facts $\approx^{L}$ and $\not\approx^{L}$ obtained from the local approximations $\approx^{L}_{+}$ and $\approx^{L}_{-}$ trivially coincide with those global bisimulation facts $\approx$ and $\not\approx$ restricted to the set of names $L$. The aim of the experiment is to determine the relationship between the size of WDB (input size based on the parameter $n$) and query performance time comparing the three strategies: (i) with the help of the bisimulation engine not exploiting local approximations; (ii) with the help of the bisimulation engine, exploiting local approximations; and (iii) without the help of the bisimulation engine222 That is, without the help of the bisimulation engine the query client running the test query is forced to compute bisimulation itself. . Similarly to the previous experiment we measure query performance for the test query $x_{1}\stackrel{{\scriptstyle?}}{{\approx}}x_{1}^{\prime}$ where $x_{1}$,$x_{1}^{\prime}$ are corresponding set names of the example WDB and its copy WDB’. But now there is _no delay time_ between the client and the bisimulation engine starting work. Delay time $d=0$ is the “worst case” for the bisimulation engine, as proved by the previous experiment. (The case of variable $d$ for a fixed $n$ will be considered in another experiment later.) ###### 7.2.2.1 Experiment results The graph in Figure 7.4 suggests a sufficiently close to linear trend between query performance and WDB size when the bisimulation engine exploits local approximations. Moreover, this looks almost like a horizontal line, and query execution seems practically viable ($\sim 41$ seconds for $n=70$; see Table 7.2). On the other hand, with help of the bisimulation engine not exploiting local approximations, as well as without help of the bisimulation engine at all, query performance with sufficiently large WDB ($n=70$) becomes intractable (more than one hour). In fact query performance improves at a threshold level of approximately $n=27$ (see Table 7.2) with the bisimulation engine exploiting local approximations, with significant improvement in query performance for larger $n$ compared to the bisimulation engine not exploiting local approximations or without using bisimulation engine at all. Moreover, the absence of hyperlinks to other WDB files in our example WDB gives local approximations facts that coincide with those global bisimulation facts restricted to the set names in $L$ or $L^{\prime}$. Thus, computing bisimulation requires fewer derivation steps, dramatically decreasing the time required to compute bisimulation. Furthermore, these results suggest that local approximations are more useful when the WDB is divided into larger almost self-contained fragments. The latter is definitely the case when local is understood as _local to a site_. However, in the latter case, local approximations to $\approx$ could take some time to compute at each site. This situation is somewhat different from saving a WDB file with its local approximation set $\approx^{L}$. Thus more experimentation is required. Figure 7.4: Graph of experimental results (cf. Table 7.2 below) showing the relationship between query execution time [ms] and size of WDB (based on the parameter $n$) – comparing the three strategies towards computing bisimulation It might seem unexpectedly, but is actually quite natural that the results of this experiment also demonstrate that query performance is worse with the help of the bisimulation engine not exploiting local approximations compared to without the help of the bisimulation engine. In fact, this experiment was conducted with no delay time ($d=0$), and we should recall the results of the experiments in Section 7.2.1 where a sufficiently small delay times decreased query performance with the help of the bisimulation engine (not exploiting local approximations) due to the additional expense of communication with the bisimulation engine. Note that the WDB considered in this and the following experiments was artificially created to make computation of bisimulation more difficult. In real situations, in particular where labels are used, it should be possible to derive non-bisimilarity of vertices without the need to go so deeply. However, only realistic application of the $\Delta$-query language can fully show its efficiency and where it should be improved. \| Query execution time (ms) ---\|--- Number of set names $n$ \| with bisimulation engine exploiting local approximations \| without bisimulation engine \| with bisimulation engine not exploiting local approximations 15 \| 3422 \| 1015 \| 1340 20 \| 4360 \| 1781 \| 2428 25 \| 5500 \| 3422 \| 4585 30 \| 7015 \| 7781 \| 10368 35 \| 8547 \| 19766 \| 26309 40 \| 10375 \| 48422 \| 64400 50 \| 20063 \| 746187 ($\sim 13$ mins) \| 989750 ($\sim 16$ mins) 60 \| 27516 \| 2113375 ($\sim 35$ mins) \| 2810800 ($\sim 47$ mins) 70 \| 40983 \| 5069797 ($\sim 84$ mins) \| 6742890 ($\sim 112$ mins) Table 7.2: Experimental results showing query execution time [ms] against WDB size (based on the parameter $n$) – comparing the three strategies towards computing bisimulation. ##### 7.2.3 Determining the benefits of background work by the bisimulation engine exploiting local approximations Now let us consider the realistic case where the bisimulation engine is working in background time, comparing both strategies of working by the bisimulation engine: (i) with exploitation of local approximations, and (ii) without exploitation of local approximations. We shall adopt the same method of testing as previously in Section 7.2.1 with the aim to determine the relationship between query execution time against partial background work333Recall that, in Section 7.2.1 the experimental parameter, delay time $d$, simulated partial background work by the bisimulation engine. by the bisimulation engine for both strategies. The example WDB used in this experiment is based on notions described in Section 7.2.2 that WDB files (or groups of WDB files) should be relatively self contained with few external links. Thus, here the experimental WDB consists of one (main) WDB file with hyperlinks to two other (auxiliary) WDB files, describing 61 set names in total, as shown by the schematical graph in Figure 7.5. Note that, like those previous experiments in Section 7.2.1 and 7.2.2, the following experimental queries are over WDB and its identical copy WDB’. The aim of this experiment is to measure query execution time $t(d)$ by the query client with the help of the bisimulation engine for the test query $x\stackrel{{\scriptstyle?}}{{\approx}}x^{\prime}$ where $x,x^{\prime}$ are corresponding “root” set names of the example WDB and its copy WDB’. Our experimental parameter is the delay time $d$, as detailed in the previous experiment Section 7.2.1. Figure 7.5: Schematical WDB graph divided into three WDB files as shown by the red dashed ovals. ###### 7.2.3.1 Experiment results The results of the experiment in Table 7.3 extend previous results in Section 7.2.2 which suggested that exploitation of local approximation by the bisimulation engine increases query performance. However, comparing the influence of partial background work by the bisimulation engine, for both strategies of working, is somewhat difficult due to the difference in magnitude between the results (see Figure 7.6a). In fact, exploitation of local approximations (by the bisimulation engine) reduces query execution time from minutes to seconds, and hours to minutes. Note that in the case of exploitation of local approximations, the process of derivation is preceded444 Downloading approximation files can occur at any stage whilst resolving some bisimulation question. by acquiring these approximations. The additional plot of data in Figure 7.6b shows threshold level, when $d$ is small, that background work by the bisimulation engine does not improve query performance whilst (the initial required) local approximations are being downloaded, as shown by the brown arrow in Figure 7.6b. Furthermore, when exploiting local approximations, a sufficiently large number of locally derived bisimulation facts (on the stage of creating WDB files) actually means in this example that fewer real derivation steps are required. (a) Comparison between bisimulation engines with and without exploiting local approximations (b) Bisimulation engine exploiting local approximations Figure 7.6: Graphs of experimental results demonstrating the relationship between query execution time [ms] and background work by the bisimulation engine simulated by delay time $d$ [ms] #### 7.3 Overall conclusion In summary, here two strategies were suggested towards improving the performance of queries involving set equality (bisimulation), these strategies are: (i) implementation of an Internet service, bisimulation engine, answering bisimulation questions; and (ii) exploitation of local approximations (by the bisimulation engine) to facilitate the quicker computation of bisimulation. It was shown empirically that for an artificial WDB that both strategies, and most dramatically (ii), improved query performance. In fact, the latter strategy demonstrates that querying of a medium sized example WDB could become practically viable. Note that other recent research into the efficient computation of the bisimulation relation was not considered here, for example the bisimulation algorithm described by Dovier et al [24] (which was intended to optimise the theoretical semi-structured query language G-log [19]). However, the point of the approach presented here was to demonstrate some strategies for computing bisimulation in the case of distributed semi-structured data, unlike that by Dovier et al which did not consider distribution. There was not enough time to consider all possibilities for optimisation, and here we concentrated on those most novel and appropriate to our approach. \| Query execution time with help of the bisimulation engine $t(d)$ (ms) ---\|--- Delay time $d$ [ms] \| exploiting local approximations \| not exploiting local approximations 0 \| 11546 \| 1340250 ($\sim 22$ mins, 20 secs) 2500 \| 11550 \| 1315269 ($\sim 21$ mins, 55 secs) 5000 \| 180 \| 1290715 ($\sim 21$ mins, 31 secs) 7500 \| 28 \| 1266620 ($\sim 21$ mins, 7 secs) 10000 \| 10 \| 1243000 ($\sim 20$ mins, 43 secs) 12500 \| 10 \| 1219769 ($\sim 20$ mins, 20 secs) 15000 \| 10 \| 1197025 ($\sim 19$ mins, 57 secs) 20000 \| 10 \| 1152728 ($\sim 19$ mins, 13 secs) 40000 \| 10 \| 1000520 ($\sim 17$ mins) 70000 \| 10 \| 790760 ($\sim 13$ mins) 100000 \| 10 \| 630772 ($\sim 11$ mins) 500000 ($\sim 8$ mins) \| 10 \| 28765 1000000 ($\sim 17$ mins) \| 10 \| 118 1250000 ($\sim 21$ mins) \| 10 \| 10 1500000 ($25$ mins) \| 10 \| 10 Table 7.3: Experimental results showing query execution time $t(d)$ [ms] against partial background work by the bisimulation engine simulated by delay time $d$ [ms] – comparing both strategies towards computing bisimulation, with and without exploiting local approximations. ##### 7.3.1 Claims and limitations The main conclusion from the above experiments is that, although bisimulation (crucial to the hyperset approach to WDB and the $\Delta$-query language) presents some difficulty in efficient and realistic implementation, this problem appears to be resolvable in principle. Moreover, this assertion is somewhat supported by the empirical testing of artificial WDB examples described in Sections 7.2.1–7.2.3. In particular, these artificial WDB were chosen to simulate some specific worst case structural features of WDB similarly to physicists conducting some very specific experiments allowing to understand the most fundamental laws of the nature instead of dealing with something complicated as in the real life. On the other hand, those artificial WDB example presented here are intrinsically limited by their small size555 with the largest WDB considered here involving only 70 set names and have restricted structural features666 which should involve not only nested chains but also nested tree structures , and, in principle, further comprehensive tests should be done to further characterise the usefulness of those practical strategies towards computing bisimulation suggested here. Also, empirical testing of some particular real-world WDB of sufficiently big size is important, but in this case a lot of further work should be done on optimisation of query execution which was outside of the scope of this work but deserves further investigation. We only considered one essential aspect of efficiency for the current version of the query system related with the idea of local/global bisimulation. However, in principle, the experiments done here suggest that these strategies show potential and merit further investigation. What has been demonstrated here is probably insufficient for a full-fledged implementation because in real-world circumstances using the $\Delta$-query language could be much more complicated. Anyway, only further work and practical experimentation can reveal problems with the current implementation, which is, of course, not fully perfect. However, it shows that the hyperset approach to databases looks promising and deserves further not only theoretical but also practical considerations – and this was actually our main goal, as well as the goal to create a working implementation available to a wider range of users to realise practically what is the hyperset approach to WDB or semistructured databases. ## Part III Implementation issues ### Overview of Part III In this part we discuss the most essential issues of implementing the $\Delta$-query language: (i) query execution (Chapter 8), (ii) syntactical aspects (Chapter 9), and (iii) XML representation of WDB (Chapter 10). These chapters can be read (almost) independently, however, logically their order should be the inverse. The chosen order rather reflects the importance of the material for the reader, who probably should be more interested in the principles of query execution than in the very technical details of implementation of the syntax (in particular related with the subtle points of well-formed vs. well-typed queries). But from the point of view of the actual implementation (including execution of queries) such syntactical aspects were very crucial and, in fact, such technical details serve as a guarantee that the whole implementation was done correctly. Indeed, the content of Chapter 9 arose to overcome the problems of ensuring well-formed/well-typed queries encountered during the first attempt at implementation [49]. Finally, Chapter 10 details the XML representation of WDB, and has quite a separate role. We think and work exclusively in terms of hypersets and set equations, and any WDB could be represented adequately and straightforwardly in the latter form. However, we have chosen XML form (XML-WDB format) as a representation of set equations to make our approach potentially more closely related to the existing practice of using XML for semistructured data. The reader should choose the level of details he/she needs from this chapter for understanding examples of XML-WDB files we use when running $\Delta$-queries. ### Chapter 8 $\Delta$ Query Execution #### 8.1 Implementation of $\Delta$-query execution by reduction process How to execute any $\Delta$-query was explained mostly in Section 3.3 as operational semantics (based on the general abstract mathematical approach described in [61]) and continued in Section 4.2 on computing bisimulation. Here we will finalise the operational semantics by considering the clauses omitted in Section 3.3 in the style more close to that of implementation. Recall that in this approach, any $\Delta$-term or $\Delta$-formula query $q$ should be equated, respectively, to a new set or boolean name $res$. Then this equation $res=q$ is reduced (in the context of all set equations of WDB) to an equation $res=V$, $res=q\rhd res=V,$ (8.1) where $V$ is, respectively, either a * • set value – flat bracket expression $\\{l_{1}:v_{1},\ldots,l_{n}:v_{n}\\}$ where $v_{i}$ are set names and $l_{i}$ label values, or * • boolean value – true or false. Note that this process of reduction can extend the original WDB by the auxiliary set equations $v_{i}=\\{\ldots\\}$ defining those set names $v_{i}$ participating in $V$ which were not the original set names in the WDB, and, possibly, many others participating in equations for $v_{i}$, and so on. Thus, strictly speaking, the reducibility statement (8.1) only partially reflects this process of reduction as the whole WDB extended by the equation $res=q$ can be involved. In the case of distributed WDB, over which some query $q$ should be executed, this process of reduction also tacitly assumes downloading the (remote) WDB files with those required set equations participating in this process. Implementation of the $\Delta$-language should evidently follow the operational semantics in [61] or in Section 3.3. In this chapter, we will give implementation details on four important $\Delta$-language constructs: separation, quantification, recursion, decoration and transitive closure. Equality (bisimulation) was already discussed in detail. Other cases are sufficiently evident or do not add much to the operational semantics and by this reason are omitted. ##### 8.1.1 Separation construct In the case of those queries which involve complex subqueries new equations will be created during the evaluation of the subquery (which was conceptually understood as the “splitting” rule; cf. Section 3.3). Consider the reduction process for $\Delta$-term separate $\\{l\\!:\\!x\in t\mid\varphi(l,x)\\}$: $\displaystyle res=\\{l\\!:\\!x\in t\mid\varphi(l,x)\\}\rhd$ $\displaystyle\;res=\\{l_{1}\\!:x_{1},...,l_{n}\\!:x_{n}\\}$ where $t$ is a set name with a flat set equation $t=\\{l_{1}\\!:x_{1},...,l_{m}\\!:x_{m}\\}$ in the current version of WDB (possibly extended locally by the query system). In reality $t$ could be a complicated $\Delta$-term, but we may assume that the “splitting” rule from Section 3.3 has already been applied so that we have here just a set name. In fact, $l_{1}\\!:x_{1},...,l_{n}\\!:x_{n}$ should be a sublist of $l_{1}\\!:x_{1},...,l_{m}\\!:x_{m}$ separated by the formula $\varphi(l,x)$ – for simplicity of denotation some initial sublist (so that $n\leq m$). Note that $l,x$ are label and set variables whereas $l_{i},x_{i}$ are label values and set names participating in the current extended version of WDB. (See also the $\Delta$-language syntax in Appendix A.1 on set names, and label and set variables.) The process of reduction is the quite evident iterative procedure, Separation algorithm: 1. START with the current version of WDB and the separation term $\\{l\\!:\\!x\in t\mid\varphi(l,x)\\}$ where $t$ is set name, and WDB contains flat set equation $t=\\{l_{1}\\!:\\!x_{1},...,l_{m}\\!:\\!x_{m}\\}$. 2. 1. Extend current version of WDB by the equation $res=\\{l\\!:\\!x\in T\mid\varphi(l,x)\\}$ where $res$ is a new set name. 3. 2. Create the new (temporary) set equation $res=\\{\\}$ (empty set) for the same set name $res$. (After populating the right-hand side by labelled set names, this equation will replace the above.) 4. 3. Iterate over the labelled elements $l_{i}\\!:\\!x_{i}$ of $t$ where $t=\\{l_{1}\\!:\\!x_{1},...,l_{m}\\!:\\!x_{m}\\}$. 1. (a) Call the corresponding reduction procedure for the $\Delta$-formula $\varphi(l_{i},x_{i})$, $res_{i}=\varphi(l_{i},x_{i})\rhd res_{i}=\ldots,$ for new set names $res_{i}$ resulting in the boolean equations $res_{i}=\mbox{\bf true}$ or $res_{i}=\mbox{\bf false}$.111As the $\Delta$-language is bounded (quantifiers and other variable binding constructs are bounded by appropriately restricting the range of variables explicitly required by the language syntax) any such reduction process will inevitably halt (in fact, in polynomial time). In the current case either true or false will be obtained. Does $res=\varphi(l_{i},x_{i})\rhd res_{i}=\mbox{\bf true}$? Yes – Amend the equation for $res=\\{\ldots\\}$ initiated in the step 2 as $res=\\{\ldots,l_{i}\\!:\\!x_{i}\\}$ by adding the labelled element $l_{i}\\!:\\!x_{i}$. Move back to step 3 (iterate over next labelled element, if one exists). No – Move back to step 3 (iterate over next labelled element, if one exists). 5. END with the (simplified) set equation $res=\\{l_{1}\\!:\\!x_{1},...,l_{n}\\!:\\!x_{n}\\}$ (with $res$ a subset of $t$). ##### 8.1.2 Quantification Consider, for example, the reduction process for the quantified formula $\exists l\\!:\\!x\in t.\varphi(l,x)$ where $t$ is (for simplicity) a set name with a flat set equation $t=\\{l_{1}\\!:\\!x_{1},...,l_{m}\\!:\\!x_{m}\\}$ (for $l_{i},x_{i}$ label values and set names, like above). It starts by replacing the bounded existential quantifier with the disjunction: $\displaystyle res=\exists l\\!:\\!x\in t.\varphi(l,x)\rhd res=\varphi(l_{1},x_{1})\vee...\vee\varphi(l_{m},x_{m})\rhd\ldots.$ By invoking the “splitting” rule it assumes the recursive subprocesses $res_{i}=\varphi(l_{i},x_{i})\rhd\ldots$ (with new boolean names $res_{i}$) leading to truth values for $res_{i}$ from which an appropriate truth value for $res$ can evidently be obtained. ##### 8.1.3 Recursive separation Consider the recursion query: $\displaystyle\mbox{\sf Rec}\;p.\\{l\\!:\\!x\in t\mid\varphi(x,l,p)\\}$ where, as above, $t$ is considered as set name with a flat set equation $t=\\{l_{1}\\!:\\!x_{1},...,l_{m}\\!:\\!x_{m}\\}$ for $l_{i},x_{i}$ label values and set names. To execute it, we should start by adding the set equation to the WDB with the new set name $res$, $\displaystyle res=\mbox{\sf Rec}\;p.\\{l\\!:\\!x\in t\mid\varphi(x,l,p)\\}.$ The set name $res$ denoting the result of the recursion query should represent a subset of $t$ where only some of $l_{i}\\!:\\!x_{i}$ will participate. It is computed iteratively as an increasing sequence $p_{k}$ of subsets of $t$: $\displaystyle p_{0}=\;$ $\displaystyle\\{\\}\;\;\mbox{(empty set)}$ $\displaystyle p_{1}=\;$ $\displaystyle p_{0}\cup\\{l\\!:\\!x\in t\mid\varphi(x,l,p_{0})\\}\rhd p_{1}=P_{1}$ $\displaystyle p_{2}=\;$ $\displaystyle p_{1}\cup\\{l\\!:\\!x\in t\mid\varphi(x,l,p_{1})\\}\rhd p_{2}=P_{2}$ $\displaystyle\ldots$ This sequence of equations with new set names $p_{k}$ (in fact, intermediate results) should be generated iteratively, with each new set equation generated after the previous one. Each of these complicated equations is reduced essentially by using the above process of reduction for the ordinary separation construct giving rise to a subset $P_{k}$ of $t$. As these subsets are inflating, and $t$ is finite, this process should be halted when $P_{k}=P_{k+1}$ (stabilisation). Note that checking equality between these sets does not require the computation of bisimulation as each iterative set $p_{k}$ is an “explicit” subsets of $t$ (elements of the bracket expression $P_{k}$ are exactly, i.e. not up to bisimulation, some of $l_{i}:x_{i}$ from the right-hand side of the equation for $t$). Now, simplify the initial equation $res=\mbox{\sf Rec}\;p.\\{...\\}$ by replacing it with $res=P_{k}$: $\displaystyle res=\mbox{\sf Rec}\;p.\\{l\\!:\\!x\in t\mid\varphi(x,l,p)\\}\rhd res=P_{k}.$ Note that the subprocesses of the above process $res_{ik}=\varphi(x_{i},l_{i},p_{k})\rhd\ldots$ (where $\varphi$ can be quite complicated formula involving complicated subterms) may introduce new set names with their corresponding set equations. Of course, they should also be considered as the part of the result of this computation (as soon as they are contained in the transitive closure of $res$). Thus, it has been demonstrated how to resolve the $\Delta$-term recursive separation. ##### 8.1.4 Decoration Although the decoration operator can be explained sufficiently easily on the intuitive level (see [3] and Section 3.2.2.2), its implementation should be done particularly carefully and precisely. To resolve the query $\displaystyle\mbox{\sf Dec}(g,v)$ over a WDB with $g$ and $v$ arbitrary set names, i.e. to simplify the equation $\displaystyle res=\mbox{\sf Dec}(g,v)\rhd res=\\{\ldots\\},$ let us firstly consider some auxiliary queries which deserve to be included as library query declarations and, most importantly, add an intermediate conceptual level of abstraction in the description of the operational semantics for the decoration operator. ###### 8.1.4.1 Auxiliary (library) queries useful for computing decoration Let us now define several auxiliary queries dealing with representation of graphs as sets of ordered pairs. ###### 8.1.4.1.1 Nodes: Now, consider a set name `g` with the flat222Recall that the query system considers WDB as a flat system of set equations, and all set equations it eventually produces are also flat. (Only at the very last step of outputting the query result will the system produce set equations with reasonably nested right-hand sides.) WDB-equation g = { ..., l:p, ...} with `l:p` any labelled set name appearing in the right-hand side (which can be a name of an ordered pair or just of an arbitrary set). The (abstract) set values `First(p)` and `Second(p)` are called _$g$ -nodes_333 Recall that First(p) and Second(p) are library queries defined in Section 3.4.2.1.3 and Appendix A.3. so that $\verb+First(p)+\stackrel{{\scriptstyle l}}{{\longrightarrow}}\verb+Second(p)+$ serves as an _$g$ -edge_, and therefore the (absolutely arbitrary) set $g$ plays the role of a _graph_. Alternatively, we could ignore those `p` in `g` which are not ordered pairs – the approach adopted below. Note that different set names may denote the same set, in particular, the same $g$-node, so that we will need to choose canonical `g`-node names in the algorithm considered below. The set of `g`-nodes can be formally defined in $\Delta$ as library query declaration set query Nodes (set g) = union separate { m : p in g \| call isPair ( p ) } The set `Nodes(g)` (the union of two element sets `p` in `g`) contains exactly all `g`-nodes, but, strictly speaking, each `g`-node in this set (being an element of some $p$ in $g$) has a label `fst` or `snd` and possibly appears twice, under both of these labels. However, this feature (which could be corrected by replacing these labels by the neutral “empty” label `null`) will play no role in the following considerations. On the other hand, preserving this information on the nodes in `Nodes(g)` might be useful in other examples of using this query declaration. ###### 8.1.4.1.2 Children: We also need the concept of _$g$ -children_ of a node $x$ in a graph $g$ (as a set of ordered pairs), which is essentially the set of all outgoing edges from $x$ in $g$. This can be defined set theoretically by the following library query declaration (with three occurrences of the `call` keyword omitted to simplify reading): set query Children(set x,set g)= collect {l:Second(p) where l:p in g and ( isPair(p) and First(p)=x ) } Evidently, if the set `x` is not the value of `First(p)` for some pair `p` as required in this declaration then `Children(x,g)={}` (the empty set). ###### 8.1.4.1.3 Regroup: Let us now define the set valued library operation `Regroup(g)` that can reorganise (without losing any essential information) any graph $g$ into something closely similar to the system of set equations represented by this graph. (For simplicity we again omit all `call` keywords.) Pay attention to the use of the label `null` which can be considered here as the “empty” label (some label is formally necessary according to the BNF of the language): set query Regroup(set g)= collect {’null’:Pair(x, Children(x,g)) where m:x in Nodes(g) } Informally, each pair `Pair(x,Children(x,g))` collected in `Regroup(g)` is considered as _abstractly_ representing a set equation, where: * • first element `x` of the pair (understood as the abstract set denoted by `x`) plays the role of a node of `g` or of an abstract set name – the left-hand side of the intended equation, and * • second element, set `Children(x,g)`, plays the role of the right-hand side of this equation – the evident bracket expression enumerating the labelled elements (`g`-nodes) of this set. It is crucial here that the set of ordered pairs `Regroup(g)` is _functional_ in the sense that for each (abstract set) `x` there exist at most one (abstract) pair `Pair(x,c)` in `Regroup(g)` with the first element `x` (and with `c` uniquely defined by `x` as `c=Children(x,g)`). In fact, `Regroup(g)` defines abstractly the correct system of set equations where each abstract set name (a set in `Nodes(g)`) has exactly one (abstract) equation with this name as the left-hand side. The operation `Regroup(g)` will make it easier extracting from `g` the required system of set equations, described in the main algorithm for computing decoration operation below. ###### 8.1.4.1.4 An assumption. _Now, let us assume that the fragment of the $\Delta$-language without decoration operation has already been implemented_. Then we can make calls to the above library queries applied to appropriate set name arguments in a given WDB, such as the set name `g` (representing a set of ordered pairs) in the call `Regroup(g)`. The latter call will be used in the implementation of decoration operator in the next section. As usually, when executed by the query system, these library operations generate new set names and set equations and add them to the WDB. In particular, considering set names generated by the query system, the result of `Regroup(g)` is, informally, a set of ordered pairs of the form `{’fst’:x,’snd’:Children_x}` where `x` and `Children_x` (denoted as `c` in the algorithm below) are now set names444In further detail, when executing the query Regroup(g), a new set name r and set equation r=Regroup(g) are generated. Then, the implemented reduction process ($\rhd$) executing this query will give rise to a flat equation r={..., ’null’:e, ...} with each set name e in the right-hand side having the equation e={’fst’:x,’snd’:Children___x}. . Moreover, according to the natural implementation of the declaration for the query `Children(x,g)`, the right- hand side of the equation for each set name `Children_x`, Children_x = { ..., l:y, ... }, contains labelled set names (in fact, `g`-node names) `l:y` for all (labelled) $g$-children of the $g$-node named by `x`. Note that the algorithm described in the next section operates with these $g$-node names. ###### 8.1.4.2 Algorithm for computing decoration We will show how the decoration operation `decorate(g,v)` can be implemented over a given WDB (with `g` and `v` any set names from the WDB) exploiting the above library query declarations. This can be done as follows: 1. START with the current version of WDB and the term $\mbox{\sf Dec}(g,v)$ for a given set names $g$ and $v$. 2. 1. Extend current version of WDB by the equation $res=\mbox{\sf Dec}(g,v)$ where $res$ is a new set name. 3. 2. Regroup `g` and canonise `g`-node names. 1. (a) Call the query `Regroup(g)`. This amounts to simplifying the extended system of set equations WDB + (`r=Regroup(g)`) for `r` a new set name, which results in some new (auxiliary) set names and flat set equations, including the flattened version `r={..., ’null’:e, ...}` of `r=Regroup(g)`, and, for each `e` in `r`, `e={’fst’:x,’snd’:c}`, `c={..., l:y, m:z, ...}`. 2. (b) Canonise `g`-node names: 1. i. Extract `g`-_node names_ (all `x`, `y`, `z`, `...`) from the result in (2a), 2. ii. Compare which of them, considered as sets, are equal between themselves (bisimilar as set names, represent the same abstract `g`-node). 3. iii. For each `g`-node name `u` find its canonical representative `Can_u` as the first in the lexicographical order `g`-node name bisimilar to `u`. (Thus, `u` is bisimilar to `Can_u`. Note that `Can_u` is not a new set name — just one of those extracted in the step 2(b)i.) 4. iv. In the resulting set equations in (2a) `e={’fst’:x,’snd’:c}`, `c={..., l:y, m:z, ...}` (for each `e` in `r`) replace `g`-node names `x` and `y`,$\ldots$, respectively, by `Can_x` and `Can_y`,$\ldots$, thereby transforming these equations to `e={’fst’:Can_x,’snd’:c}`, `c={.., l:Can_y, m:Can_z,..}`, $\ldots$. (The original versions of these equations should be deleted.) 5. v. If for another pair of such equations (for `e’` in `r`), `e’={’fst’:Can_x’,’snd’:c’}`, `c’={..., l’:Can_y’, ...}`, set names `Can_x` and `Can_x’` in `e` and `e’`, respectively, coincide then omit one of these pairs (does not matter which), and repeat this until no such coincidence of canonical node names will exist. 6. vi. Eliminate possible repetitions of labelled canonical node names `l:Can_y` in each `c` (which can arise, e.g. due to replacements in (2(b)iv) as `l:Can_y` can literally coincide with some `m:Can_z` in `c` for different `g`-node names `y` and `z`). From now on, these `Can_u` serve as _canonical_ `g`-_node names_. Only these node names will be used below as uniquely representing `g`-nodes. 4. 3. Does a canonical g-node name bisimilar to v exist? Find a canonical `g`-node name `w` bisimilar to set name `v` (or just coinciding with `v` if `v` is itself a canonical `g`-node name). Two answers are possible: No - The required canonical `g`-node name `w` bisimilar to `v` does not exist (and thus `v` can be treated as naming an isolated `g`-node): 1. (a) Simplify the equation `res = decorate(g,v)` to `res = {}` (empty set). Then move to END of the algorithm. Yes - The required canonical `g`-node name `w` does exist (and thus `v` can be treated as naming a proper `g`-node): 1. (a) Generate new set equations for duplicated canonical `g`-node names: 1. i. For each set name `s` which is a canonical `g`-node name create a new duplicate set name `Dupl_s` (in particular, `Dupl_w`, `Dupl_Can_x`, etc.). 2. ii. For the equations `e={’fst’:Can_x,’snd’:c}`, `c={...,l:Can_y,m:Can_z,...}`, obtained in (2(b)iv, 2(b)v, 2(b)vi) for each `e` in `r`, extend further the current extension of WDB by new set equations: `Dupl_Can_x = {..., l:Dupl_Can_y, m:Dupl_Can_z, ...}`, thereby constructing a system of set equations for duplicate names whose graph is isomorphic to the abstract graph `g`. In particular, this will add to the WDB the equation for `Dupl_w`: Dupl_w = W with the right-hand side a bracket expression `W` defined as described above (and involving only duplicated canonical `g`-node names). 2. (b) Simplify the equation, res=decorate(g,v) by replacing it with the (flat) equation res = W. (End of algorithm.) 5. END with the (simplified) set equation $res=\\{l_{1}\\!:\\!x_{1},...,l_{n}\\!:\\!x_{n}\\}$ (and the associated equations for set names in `W`, etc.). In the case of the query, $res=\mbox{\sf Dec}(G,V)$ where $G$ and $V$ are $\Delta$-terms and not just set names (as above), the “splitting” rule should be invoked first, which will result in three equations $g=G$, $v=V$ and $res=\mbox{\sf Dec}(g,v)$ for the new set names $g$ and $v$. Then these equation should be simplified, in particular, by using the above algorithm for the decoration. ##### 8.1.5 Transitive closure Let us now consider implementation of the transitive closure operation $\mbox{\sf TC}(a)$, where $a$ is considered as a set name with the flat equation $a=\\{l_{1}\\!:\\!x_{1},...,l_{m}\\!:\\!x_{m}\\}$ for $l_{i},x_{i}$ label values and set names, as the following (recursive) algorithm: 1. START with the current version of WDB and the transitive closure term $\mbox{\sf TC}(a)$ where $a$ is set name, and WDB contains flat set equation $a=\\{l_{1}\\!:\\!x_{1},...,l_{m}\\!:\\!x_{m}\\}$. 2. 1. Extend current version of WDB by the equation $res=\mbox{\sf TC}(a)$ where $res$ is a new set name. 3. 2. Replace the original set equation $res=\mbox{\sf TC}(a)$ by the new (temporary) set equation $res=\\{^{\prime}null^{\prime}\\!:\\!a\\}$ (singleton set) for the same set name $res$. (This will be further populated below.) 4. 3. Find the first labelled element $m\\!:\\!z$ of $res=\\{\ldots,m\\!:\\!z,\ldots\\}$ such that $z\not\subseteq res$. (Elements for which $z\subseteq res$ should be marked and put at the end of the current bracket expression for $res$ so that they will not be considered again and again. For efficiency, the bracket expression for $res$ can be organised as a directed “loop” structure with some point of entrance. Each time when $z\subseteq res$ holds at the entrance point then this point in the loop will be marked and the entrance point shifted to the next one to repeat the inclusion test.) If it does not exist (the currently observed element and hence all $m\\!:\\!z$ are marked), go to the END. Else replace the current equation $res=\\{\ldots,m\\!:\\!z,\ldots\\}$ with the $m\\!:\\!z$ found (at the current entrance point) by $res=\\{\ldots,m\\!:\\!z,\ldots\\}\cup(z\setminus res)$ (inserting elements of $z\setminus res$ in the loop immediately after $m\\!:\\!z$, then marking $m\\!:\\!z$ as now $z\subseteq res$ for the extended $res$ and shifting the entrance point from $m\\!:\\!z$ to the next point of so extended loop — the first element in $z\setminus res$). (Computing $z\setminus res$ can evidently also use the loop structure of $res$ with marking ignored.) Repeat 3. 5. END with the set equation for $res$. Note that in fact $\mbox{\sf TC}(a)=\bigcup\\{\\{a\\},a,\bigcup a,\bigcup\bigcup a,...\\}$. #### 8.2 Representation of query output Recall that the implemented query system works internally with (WDB represented as) a flat system of set equations, and produces query results in this flat form. The resulting set equations also use internally generated (local) set names having no mnemonics. It appears that some nesting in the outputted equations might be desirable which would simultaneously eliminate some internal set names by substituting them with bracket expressions. This substitution can be repeated giving rise to possibly deeply nested results. Consider, for example the result of the _restructuring query_ from Section 3.5.3 obtained after some such automatic substitutions: Query is well-formed, well-typed and executable Result = { ’publication’:res2, ’publication’:res0, ’publication’:res1, ’publication’:{ ’type’:"Book", ’refers-to’:res1, ’refers-to’:res2 } } res0 = { ’type’:"Paper", ’author’:"Smith", ’title’:"Databases", ’refers-to’:res1 } res1 = { ’type’:"Paper", ’type’:"Book", ’author’:"Jones", ’title’:"Databases" } res2 = { ’type’:"Paper", ’refers-to’:res0 } Finished in: 1866 ms (query execution is 1864 ms, and postprocessing time is 2 ms) Comment(s): Double quotation denotes atomic values like "atom" representing singleton sets "atom" = {’atom’:{}}, etc. Note that, in this example further substitutions could be made to eliminate even those few local names `res0`, `res1`, `res2`, so that there would be just one deeply nested equation `result={...}`. However, this would be a rather inconvenient form as set names to be substituted occur several times, and identical subexpressions could be repeated many times making the query result difficult to grasp. Thus, the system makes such suitable nesting to avoid multiple substitutions in the whole system of equations. Additionally, nested bracket expressions like `{Paper:{}}` which imitate atomic values in our approach are replaced, quite naturally, by `"Paper"`. Note that in the later case there may be multiple substitutions and replacements of the same expression. Similarly, set names for the empty set are always replaced by `{}`. In this way query results become sufficiently readable. Lastly, in the case of cycles substitutions could be infinitely repeated. To avoid this, the system should only substitute those set names $res_{i}$ with the corresponding bracket expression if $res_{i}\not\in\mbox{\sf TC}(res_{i})$ holds (in addition to the other rules for substitutions above). Also, the computation of transitive closure should be restricted to those new set names resulting from the execution of the query, thus, in principle, this can be done quickly on only local set names. However, any such postprocessing of the query result can sometimes lead to unnatural looking output, for example in the above query result there is some undesirable extra nesting for one of the publications. In other cases (such as showing a graph as a set of ordered pairs) such nesting appears more reasonable. Also atomic values and explicitly shown empty sets`{}` are very natural. Of course it would be better if the user could choose the preferred form, or the result could be optionally visualised as a graph. ### Chapter 9 $\Delta$ Query Syntax #### 9.1 Parsing (well-formed queries) ##### 9.1.1 Implemented $\Delta$-language grammar The syntax of the implemented language was discussed in Chapter 3, with the full syntax appearing in Appendix A.1. The implemented language is described as _Extended Backus-Naur form_ (EBNF or, shortened, BNF), defined as a set of production rules, with each production describing one syntactical category represented as a non-terminal. For example, the production rule <query> ::= "boolean query" <delta-formula> \| "set query" <delta-term> defines the `<query>` syntactical category (also called _non-terminal_) by stipulating in general that a terminal can be substituted by a sequence of _terminals_ such as `"boolean query"` and other non-terminals such as `<delta- formula>`. Here the symbol `\|` allows to describe alternative productions. (There are also other ways in the BNF to describe more complicated alternations in production rules.) Continuing such substitutions by using production rules for `<delta-formula>`, etc., a sequence consisting only of terminals can be obtained. Further, as terminals are strings of symbols, the final concatenation is also a string of symbols which, properly speaking, is called _well-formed query_ , provided it was generated starting from the non- terminal `<query>`. (Quite similarly we can consider well-formed _delta formulas_ , _delta terms_ , etc.) Thus, the BNF defines how to construct any query in $\Delta$. In fact, each $\Delta$-query, if well-formed, generates a parse tree (by using BNF-forks discussed below) which should be subsequently checked for well-typedness (see Section 9.2). ##### 9.1.2 BNF forking Firstly a general note on the BNF grammar. Each production rule from the BNF (except some auxiliary ones which can be eliminated as we will see below) can be represented as one, several, or even infinitely many alternative _forks_ `F1,F2,...` each having the same label (syntactical category or non-terminal) on the root of the fork. For example, the rule <A> ::=Ψ<B><C> \| <B><D><E> splits into two rules <A> ::=Ψ<B><C> <A> ::=Ψ<B><D><E>, evidently corresponding to two forks with the branching degree two and three, whose roots are labelled by `<A>` and leafs labelled, respectively, as `<B>`, `<C>` and `<B>`, `<D>`, `<E>`. Let us analogously consider the production rule <set constant declaration> ::= "set constant" <set constant> ("be"\|"=") <delta term> which generates two unique forks depending on whether `"be"` or `"="` is used – each fork has a branching degree of four. Thus whole BNF grammar can then be represented as a set of all such forks. In fact, the parse tree of a query is constructed of such forks. However, not all BNF production rules are so simple and literally split into forks as will be discussed below. ###### 9.1.2.1 Recursion by Kleene operators Recursive BNF rules using repetition by the Kleene star and plus (`` and `+`) operators generates an infinite set of forks; `` represents zero or more repetitions, and `+` represents one or more repetitions. For example the following rule represents a sequence of declarations: <declarations> ::= <declaration> ( "," <declaration> )* Each fork has a root labelled by `<declarations>` and any number of leaves labelled by `<declaration>`, separated by the terminal leaves labelled by `","`. Evidently, the branching of these forks have an arbitrary odd degree because of the separator `","` considered formally as a leaf. Analogously the following syntactic categories are also considered: <variables>, <parameters>, <multiple union>, <conjunction> <disjunction>, <quasi-implication>, <labelled terms> ###### 9.1.2.2 Identifier forks There is further simplification to the BNF forks and to parse trees by eliminating the “intermediate” `<identifier>` category playing rather an auxiliary role. Thus, we will replace corresponding production rules by those generating infinitely many simple (one child) forks: <boolean query name> ::=Ψ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ <set query name> ::=ΨΨ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ <label variable> ::=ΨΨ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ <label constant> ::=ΨΨ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ <set variable> ::=ΨΨ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ <set constant> ::=ΨΨ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ There are infinitely many of such identifier forks because there are infinitely many sequences of _alphanumeric characters_ (just those characters participating in the identifier forks) which can serve as a leaf label of a fork for each of the above syntactical categories. Root nodes of these forks of the corresponding nodes in a parse tree are called _Identifier Nodes_ (IN). In general, every occurrence of `<identifier>` in the right-hand sides of production rules in BNF is replaced by: ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ There is, however, restrictions on these alphanumeric strings: they should not coincide with keywords of $\Delta$ language. ###### 9.1.2.3 Set name forks Let us recall the production rules related with _full set names_ represented by the syntactical category `<set name>`. This important category, including some additional auxiliary productions, appears as follows: <set name> ::= <URI> "#" <simple set name> <URI> ::= ( <web prefix> \| <local prefix> ) <file path> <web prefix> ::= "http://" <host> "/" [ "~" <identifier> "/" ] <local prefix> ::= "file://" ( (A-Z) \| (a-z) ) ":/" <host> ::= <identifier> [ "." <host> ] <file path> ::= <identifier> ( "/" <file path> \| <extension> ) <extension> ::= ".xml" <simple set name> ::=Ψ<identifier> <identifier> ::=Ψ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ Here all the syntactical categories, besides `<set name>`, play an auxiliary role. Therefore, by composing them, all these production rules will produce two kind of one child forks for set names <set name> ::= "http://... " "#" ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ or <set name> ::= "file://... " "#" ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ Here `"http://..."` and `"file://..."` represent any string of symbols allowed by the `<URI>` production rule. Therefore, the production rule `<set name>` generates an infinite number of (one child) forks with the root `<set name>` and the leaf a string of characters as defined in the above productions. We will not consider other cases of defining BNF forks relying on the readers’ intuition which should be based on the above examples. Assertions 1-3 from the next section should summarise and give more understanding on the way which BNF forks are defined. ###### 9.1.2.4 Assertions on BNF forks After defining the set of forks of the BNF, we can make the following assertions. ###### Assertion 1. Only Identifier Nodes (IN) can have just one child leaf labelled by a sequence of alphanumeric characters. ###### Proof. Inspection of the whole BNF (and the definitions above) show that only IN can have just one child leaf labelled by a sequence of alphanumeric characters. ∎ Note that `<set name>` forks, although one child, have leafs containing non- alphanumeric characters ":", "/" and "#". ###### Assertion 2. In fact, parsing of any given query generates a corresponding query parse tree constructed from these forks connected in the evident way. Here it is assumed that all keywords like ”forall”, ”let”, etc are included in the parse tree as terminals (except they are not allowed to be leafs of identifier forks). ###### Assertion 3 (Uniqueness of forks111This assertion will be used in the syntactical category renaming algorithm in Section 9.2.3.2). Two different forks can have coinciding leaf labels (in the natural order) only if each of them is an identifier fork (see above). That is, if one of the two forks F1 and F2 is not an identifier fork and both forks have the same leaves then (their roots coincide and) F1 = F2. Or equivalently, the syntactic category of any fork, except for identifier forks, can be determined according to the syntactic categories of its children. ###### Proof. We should check all possible cases. Assuming that two forks F1 and F2 have the same leaves and one of them has the root labelled not as identifier fork, show that F1 = F2. _Example:_ If F1 or F2 has the root `<quantified formula>` then both have the same first leaf e.g. `<forall>` (or `<exists>`). Then, according to the BNF, another fork must also have the root `<quantified formula>` and therefore F1 = F2, as required. _Example:_ If F1 or F2 has the root `<forall>` then both have the same first leaf `"forall"` and the leaf `"in"` (or `"<-"`). Inspection of all BNF forks shows that any fork containing both these leafs must have the root `<forall>`. Therefore F1 = F2. _Example:_ If F1 or F2 has the root `<union>` then both have the same first leaf `"union"` (or `"U"`) and second leaf `<delta-term>`. Inspection of all BNF forks shows that any fork containing both these leafs must have the root `<union>`. Therefore F1 = F2. All other cases follow as above. ∎ ###### Note 4. Despite this Assertion which means a kind of unambiguity of parsing (actually only a conditional and partial unambiguity) we will see in Section 9.1.4 that parsing according to the BNF of $\Delta$ is actually quite ambiguous. This means that the same query can have parse trees of the same form, but with different labelling of nodes by syntactical categories. Later we will consider contextual analysis algorithm dealing with typing which will resolve this kind of ambiguity. ##### 9.1.3 Query parsing The parser for the BNF syntax of the language Delta can easily be implemented which can transform any query $q$ into parse tree. The process of parsing $q$ involves matching of BNF production rules (represented rather in the form of forks defined above) starting at the root production rule for `<top level command>` until all possibilities are exhausted. The output of parsing the query $q$ is the query parse tree $qt$. During the process of parsing, successful matching of production rules creates new nodes in the parse tree connected by fork edges from the previous node, except for the root production rule which itself has no parent node. Successful matching of terminals creates new nodes labelled by the sequence of matched characters. ###### 9.1.3.1 Example query parse tree Let us consider the simple example of query boolean query let label constant l=’Robert’ in l=’Rob’ endlet; and the corresponding query parse tree, Figure 9.1: Example parse tree Strictly speaking, some parts of this parse tree are omitted for brevity. Say, according to Section 9.1.2.1, between <declarations> and <label constant declaration> we should have a tree node <declaration>. ###### 9.1.3.2 Aims of query parsing Well-formedness of any query is determined according to the rules of the BNF grammar. However, when all possibilities for matching productions are unsuccessfully exhausted in any attempt to construct a parse tree then the query is considered as non-well-formed with appropriate error messages outputted. Moreover, to further aid contextual analysis (see Section 9.2) the parser should output, in addition to the parse tree of the query, the list of all Identifier Nodes (see Section 9.1.2.2) in the parse tree labelled by: <boolean query name>, <set query name>, <label variable>, <label constant>, <set variable>, <set constant>. ##### 9.1.4 Parsing ambiguities The syntax of the implemented $\Delta$-language (expressed as BNF) is intended for any user to understand the constructs of $\Delta$, and how to write valid $\Delta$-queries – well-formed and well-typed. However, the implemented parser alone cannot guarantee well-typedness of queries. Note that, well-typedness is checked by the contextual analysis algorithms described later in Section 9.2. The problem is that the grammar of our implemented $\Delta$-language is ambiguous concerning types as we briefly commented this in Note 4 above. Thus, the typing of identifiers, say as label constant or variable, or set constant or variable, etc., is actually decided from the context. For example, let us consider the equality query: boolean query a=b; Parsing of this query could realise two unique parse trees, where the statement `a=b` represents either `<label equality>` or `<set equality>`. Thus, the syntactical category of this statement depends wholly on the interpretation of the identifiers `a` and `b` as either, label constants or variables, or set constants or variables, respectively. The parse tree presented above in Figure 9.1 is also not unique one because the syntactic category `<label constant>` under `<label>` could be formally replaced according to syntax by `<label variable>`, however, intuitively contradicting the label constant declaration let label constant l = .... Furthermore, let us even strengthen the above example, boolean query let label constant l=’Robert’, label constant m=’John’ in l=m endlet; where the statement `l=m` intuitively represents the syntactic category `<label equality>` because according to the context the identifiers `l` and `m` are label constants. However, the BNF formally allows that `<label equality>` could be replaced with `<set equality>` and `l` and `m` are are taken as `<delta-term>`s, independently of the declarations that `l` and `m` are both label constants. Even the following query can be formally parsed, i.e. is well-formed, boolean query let label constant l=’Robert’, set constant m={} in l=m endlet; despite being evidently non-well-typed by equating label with set. Therefore, the syntax (expressed as BNF) alone is insufficient and requires guessing which rule to apply to make the parse tree (and to guarantee that the parsed query is) well-typed. Therefore, such guesses by the parser should be subsequently checked, to ensure no contradictions with the actual typing of identifiers. Moreover, the syntactic categories of all nodes, not just IN, should be checked and possibly renamed (according to the grammar) without changing the structure of the parse tree. Such renaming is done by the _contextual analysis algorithm_ , detailed in Section 9.2, whose role is to ensure query well-typedness and eliminate potential ambiguities, as above. ##### 9.1.5 Grammar classification Note that the syntax of $\Delta$-query language, fully presented as BNF in Appendix A.1, can be classified as _context-free grammar_ according to Chomsky’s definitions of formal languages. Taking the definition from the textbook about parsing [75], all production rules of a context free grammar have the form: $A\longrightarrow\gamma$ where $A$ represents a unique non-terminal, and $\gamma$ represents an ordered list of terminals and/or non-terminals (possibly empty). Context free grammars are those where each non-terminal $A$ can be transformed by a production rule into corresponding $\gamma$ without any additional criteria of context. Our grammar satisfies this property and therefore cannot grasp contexts which are necessary for correct typing of queries. Thus, an additional contextual analysis algorithm working jointly with the parser is required which we discuss in the following section. #### 9.2 Contextual analysis (well-typed queries) ##### 9.2.1 Aim of contextual analysis The aim of contextual analysis is to determine whether every _identifier occurrence_ in a query $q$ is declared222 An identifier occurrence in some expression $e$ (not necessary a full-fledged query; $e$ can be a fragment of a query $q$) which is non-declared inside $e$ can also be called _free_ in $e$, whereas those correctly declared inside $e$ identifier occurrences are called _closed_. Therefore the terms “declared” and “closed”, and “non-declared” and “free”, have the same meaning. (This agreement on terminology is, however, non-traditional in the particular case of (set or label) constants for which it is more habitual to use the terms “declared” or “non-declared” instead of “closed” or “free”.) We assume that each full-fledged query $q$ must be closed in this sense (all its identifiers must be declared inside $q$). , thereby having type, and whether the whole query is well-typed (all types are coherent). Each identifier occurrence should be appropriately typed as either: _set constant_ or _variable_ , _label constant_ or _variable_ or _query name_ of some type333 To simplify terminology, we consider _variable_ or _constant_ or _query name_ as typing information of some identifier, alongside the proper types _set_ or _label_ or _boolean_ or the complex type (9.1). . Note that query names can have more complicated types than variables or constants, $(type_{1},type_{2},...,type_{n}\longrightarrow type)$ (9.1) where each participating $type_{i}$ is either _set_ or _label_ , and $type$ after the arrow is either _set_ or _boolean_ 444 Note that, we formally have no queries or query names in $\Delta$ of the type _label_. However, label values can be represented in the same way as atomic values, i.e. as singleton sets of the form $\\{l:\emptyset\\}$. . Each $type_{i}$ is the expected type of $i$-th parameter of the query name $q$, and $n$ is the required number of parameters – according to the declaration of this query name. From this type it should be already clear that the identifier $q$ is a (set or boolean) query name, how many arguments it has, and the typing of each argument. Furthermore, an identifier occurrence is considered declared if it is contained within the scope of an appropriate identifier declaration, and well- typed if both the identifier occurrence and identifier declaration have the same types. Moreover, for query to be well-typed, coherence of typing (for equalities, as in the examples above, membership statements and query calls) should be additionally required. ###### 9.2.1.1 Strategies for computing contextual analysis In principle there are two possible algorithms for performing contextual analysis of any query $q$, both algorithms are named after the way in which they walk the parse tree of $q$: • Top-down contextual analysis – The parse tree is walked in breadth first manner starting at the root node, creating a list of the identifier declarations (called the context) which is used to check that all other identifier occurrences are closed and well-typed according to these declarations. * • Bottom-up contextual analysis – Walking of the parse tree starts from any identifier occurrence leaf $i$555 For example, the second leaf labelled by the identifier $l$ in Fig. 9.1 above ascending up the corresponding branch of the parse tree, searching for an identifier declaration which declares $i$666 In Fig. 9.1 above the corresponding node would be <delta-formula with declarations> having the declaration of the label constant $l$ under it. Note that quantifiers and other quantifier-like constructs, called binders (see Section 9.2.2), are also considered as identifier declarations. . The existence of a corresponding identifier declaration indicates that the identifier occurrence is declared. Moreover, the real types of all such $i$ can be extracted from the corresponding declarations and compared with syntactical categories of these nodes $i$ in the parse tree. In the case of coherence, the parse tree and hence the query is considered _well-typed_. Otherwise, syntactical categories of the parse tree nodes could be possibly corrected by (another bottom-up procedure of) renaming syntactical categories of some non-leaf nodes by the iterative algorithm described below in Section 9.2.3. If such a renaming is successful – giving rise to a correct parse tree according to both the BNF and the typing, then the resulting version of tree and the original query are also considered _well-typed_ , otherwise _non-well- typed_. ##### 9.2.2 Some useful definitions ###### Definition 1 (Identifier Node). _Identifier Nodes_ (IN) were introduced in Section 9.1.2.2, as those nodes in the parse tree labelled by one of the following syntactic categories: <boolean query name>, <set query name>, <label variable>, <label constant>, <set variable>, <set constant>. Additionally, let us define _Identifier Node Name_ (INN) as string of symbols labelling the unique child (in fact, a leaf called above as $i$) of the corresponding IN fork in the parse tree. ###### Definition 2 (Binder Node). _Binder (or binding) Nodes_ (BN) are those nodes in the parse tree labelled by one of the following syntactic categories: <delta-term with declarations>, <delta-formula with declarations>, <collect>, <separate>, <recursion>, <quantified formula>. Binder nodes can have appropriate declarations like `"let..."`, `"forall..."`, `"exists..."`, etc., as described in Definition 3, and thereby can _bind_ identifier occurrences (or IN). ###### Definition 3 (Identifier Declaration Node). Following from Definition 2 those declarations belonging to BN are called _identifier declarations nodes (IDN) of a BN_ and defined as follows. * • For BNs `<delta-formula with declarations>` with `"let"` declaration(s), and `<delta-term with declarations>` with `"let"` declaration(s) the IDNs are: * – `<set constant declaration>` grandchild of `<declarations>`, * – `<label constant declaration>` grandchild of `<declarations>`, * – `<set query declaration>` grandchild of `<declarations>`, and * – `<boolean query declaration>` grandchild of `<declarations>`. * • For BNs `<separate>` and `<collect>` the IDNs are: * – `<label variable>` grandchild of `<variable pair>`, and * – `<set variable>` grandchild of `<variable pair>`. * • For BN `<recursion>` the IDNs are: * – `<set variable>` child of `<recursion>`, * – `<label variable>` grandchild of `<variable pair>`, and * – `<set variable>` grandchild of `<variable pair>`. * • For BN `<quantified formula>` the IDNs are: * – `<label variable>` grandchild of `<variable pair>`, and * – `<set variable>` grandchild of `<variable pair>`. For example, Figure 9.2 depicts a fragment of a query parse tree, where the root node `<separate>` is a BN and the corresponding IDN nodes (described above) can be found by walking the paths from the `<separate>` node, * `<variable pair>` $\rightarrow$ `<variable pair label>` $\rightarrow$ `<label variable>` * `<variable pair>` $\rightarrow$ `<variable pair set>` $\rightarrow$ `<set variable>` All other cases follow as the above. Note that there may be many IDNs of a given BN. Any IDN declares one or more identifiers (IN) each of which has its name as a string of symbols (the leaf under IN). ###### Definition 4 (Bounding Term or Formula or Label Value Node ). * (a) Following from Definition 2, the _bounding_ term or formula or label value nodes (BTFLVN) of a BN <collect> <separate> <recursion> <quantified formula> <delta-term with declarations> <delta-formula with declarations> is defined, respectively, as * – a unique `<delta-term>` child of: * * `<collect>` or `<separate>` or `<recursion>` or * * `<forall>` child of `<quantified formula>` or * * `<exists>` child of `<quantified formula>` or * * any `<set constant declaration>` grandchild of `<delta-term with declarations>` or `<delta-formula with declarations>` or * * any `<set query declaration>` grandchild of `<delta-term with declarations>` or `<delta-formula with declarations>`, or * – a unique `<label value>` child of: * * any `<label constant declaration>` grandchild of `<delta-term with declarations>` or `<delta-formula with declarations>` or * – a unique `<delta-formula>` child of: * * any `<boolean query declaration>` child of <delta-term with declarations> or <delta-formula with declarations>. * (b) Each BTFLVN of a BN restricts the range of the value of some INs (variables, constants or query names) which BN binds777Which was briefly hinted in the Definition 2 and which we also call _bounded or restricted IN(s) by the BTFLVN_ 888 Moreover, the IN bounded by BTFLVN should not be free in the BTFLVN (i.e., if present in the BTFLVN, it should be declared inside this BTFLVN) as we will discuss later as one of the conditions to be checked by contextual analysis algorithm. This is the reason why we need Definition 4. . These INs are defined as follows: * – In the case of BNs <collect>, <separate>, <recursion> and <quantified formula>, the bounded INs are respectively <label variable> and <set variable> grandchildren of <variable pair>. * – Additionally, in the case of BN <recursion> one more bounded IN is its immediate <set variable> child. * – In the case of BNs <delta-formula with declarations> or <delta-term with declarations>, the bounded IN is either the declared <set constant> or <label constant>, or <set query name>, or <boolean query name>. For example, Figure 9.2 depicts the query parse tree for an expression $e$ (fragment of a query $q$), where the root node `<recursion>` is a BN and the corresponding BTFLVN and the bounded INs can be found by walking the paths, * `<recursion>` $\rightarrow$ `<delta-term>` (BTFLVN) * `<recursion>` $\rightarrow$ `<set variable>` (IN) * `<recursion>` $\rightarrow$ `<variable pair>` $\rightarrow$ ` <variable pair label>` $\rightarrow$ `<label variable>` (IN) * `<recursion>` $\rightarrow$ `<variable pair>` $\rightarrow$ ` <variable pair term>` $\rightarrow$ `<set variable>` (IN) whereas <label variable> ($l$) and <set variable> ($x$) are INs bounded by this `<delta-term>` (BTFLVN). Additional (recursion) <set variable> ($r$) is IN also bounded by `<delta-term>` (BTFLVN). Figure 9.2: Fragment of a query parse tree ##### 9.2.3 Bottom-up contextual analysis in detail As stated in the brief description in Section 9.2.1, contextual analysis should check that the given well-formed query (according to the parser) is also well-typed. To this end, the bottom-up contextual analysis algorithm, first of all, iteratively searches for the nearest identifier declaration for each identifier occurrence, i.e. each IN in the parse tree. We assume that before starting contextual analysis the parser generates a list of all INs (not those INs of the declarations in IDNs) along with their currently chosen typing (immediately seen from syntactical categories of these INs, say, `<set variable>`, etc.) during the parsing process. The parser outputs this list if the query is well-formed. ###### 9.2.3.1 Identifier declaration search (IDS) algorithm Single iteration of the search for the nearest identifier declaration of an IN is determined by the _Identifier Declaration Search_ (IDS) algorithm. The inputs to this algorithm is any $qt$ (query parse tree) and some IN in $qt$. The output of the IDS algorithm is the ordered triple $<BN,IDN,IN>$ (if the required one exists at all) consisting of: BN (Binding Node), IDN (Identifier Declaration Node) and the given IN. Note that, IDN contains typing information of the declared identifier (including the information whether it is a constant or variable, or a query name – also a kind of typing information). In fact, the IDN is recoverable from BN and IN in the parse tree, however, it is convenient to have IDN included in the triple obtained during this process. ###### Identifier Search Algorithm $IDS(qt,IN)$: 1. START with a given IN belonging to $qt$. 2. 1. Make this node (IN) the _current node_. 3. 2. Ascend from the _current node_ traversing up $qt$ to its unique parent node, making this node the _current node_. 4. 3. Is the _current node_ a BN? No – Move to step 4. Yes – Iterate from right to left through IDNs of the BN, searching for the first999 Formally, it is not forbidden that the same identifier name could be multiply declared even in the same binder, but only the right most one is that which binds the IN considered and which assigns a type to IN. suitable candidate identifier declaration whose declared identifier has the same name (INN) as the given IN. If a suitable candidate IDN exists then construct the ordered triple $<BN,IDN,IN>$ (end of algorithm), _otherwise_ move to step 4. 5. 4. Is the _current node_ the root node of $qt$? Yes – No suitable candidate identifier declaration could be found, and therefore, the IN is non-declared. Output ordered triple $<NULL,NULL,IN>$ (end of algorithm). No – Continue searching for a suitable identifier declaration by moving to step 2. 6. END with the ordered triple $<BN,IDN,IN>$ if a suitable identifier declaration exists, otherwise with $<NULL,NULL,IN>$. The IDS algorithm should iteratively generate the triples as above for all INs (actually, for those identifier occurrences not in a declaration) of the given parse tree $qt$. If all these are non-null triples then the query $q$ is considered as _closed_ (yet possibly not well-typed). Thus, any closed query $q$ has all INs declared with preliminary typing according to the declarations (IDN) from the corresponding triples. For non-closed query an error message should be generated by the implementation saying that the query has non- declared identifiers. Moreover, any closed query $q$ and its parse tree $qt$ are considered also well-typed if all identifiers have coherent typing both in respect to their corresponding declarations and syntactical categories of the parse tree $qt$. More precisely, this means that: 1. 1. Syntactical categories of IN (e.g. <set variable> or <boolean query name>, etc.) should be the same as declared in IDN (in corresponding triple), and 2. 2. Types of participating parameters in query calls should agree with types discovered from IDNs declaring corresponding query names. If these two clauses do hold then in other nodes the BNF itself supports correct typing and/or syntactical categories (such as <set equality> vs. <label equality>, etc.). Otherwise, an appropriate renaming of syntactical categories of the nodes in $qt$ should be tried (as detailed in the next section), based on the initial partial correcting only the discrepancies in the clauses (1) and (2), with the aim to recover well-typed version of $qt$ and conclude that the query $q$ is _well-typed_. If such a renaming is impossible, then $q$ is considered as _non-well-typed_. ###### 9.2.3.2 Syntactic category renaming (SCR) algorithm It is required that renaming should lead to a correct parse tree. This means that the _syntactic category renaming_ (SCR) algorithm, * • takes a parse tree with some already correctly renamed nodes (such as INs, by removing the discrepancies mentioned above, and may be some other nodes as we will see below) and formally marked as “correct”, and * • if necessary, attempts to rename other nodes ensuring that the parse tree remains faithful to the $\Delta$-language BNF syntax (well-formed). Thus, the _input_ is a given parse tree $qt$ with some (non-leaf) labels _already relabelled_ 101010 Note that, INs are formally non-leaf nodes, although neighboring to leafs. As we will see below in Section 9.2.3.3, not only INs should be initially relabelled in the input parse tree. These may be also query call <parameters> which, unlike INs, may be far away from leaves in the parse tree. and additionally marked as “correct”, with the output being either: (i) parse tree with all other nodes successfully relabelled ($q$ is well-formed), or (ii) an error state ($qt$ is inconsistent with the $\Delta$-language syntax, even after further relabelling). The procedure of relabelling starts from the leafs of the parse tree, and, while going bottom-up along the tree relabels according to the $\Delta$-language BNF syntax (if necessary) those nodes which have not already been relabelled. Newly relabelled nodes are additionally marked as “correct”, and visited nodes are marked also as “seen” as described formally below. At each stage of the computation some nodes are already marked by this procedure as “correct”, and only a node $N$ can be relabelled and then also marked as “correct” and “seen” which, (i) has not yet marked as “seen” (although probably marked as “correct” by the input marking), and (ii) all its children, $Children(N)$, have already marked as both “seen” and “correct”. ###### Syntactical renaming algorithm $SCR(qt)$: 1. START with parse tree $qt$. 2. 1. Initially mark some nodes as “seen” and “correct”. Mark all leaf nodes, INs, IDNs and `<set name>` nodes both as “seen” and “correct”111111 In fact, as we discussed above, INs and query call <parameters> are already marked as correct in the input parse tree $qt$. . Note: Syntactic categories of “correct” nodes will not be renamed by this algorithm. Furthermore, `<set name>` nodes should not be renamed (and thus, these are initial marked as “correct”) as they evidently have unambiguous type _set_ and definitely require no renaming. 3. 2. Find any node suitable for correcting. Find node $N$, which is not marked as “seen”, and whose all children are marked both as “correct” and “seen” (giving rise to a fork $N\longrightarrow Children(N)$ in $qt$). Does the required $N$ exist in $qt$? No – Therefore, by induction, all nodes in the tree are already marked as “correct”, (end of algorithm). Yes \- Check and (if necessary, and possible) rename according to BNF the syntactical category of $N$: 1. (a) Find a suitable fork $F$ in the BNF that matches the children of $N$. Find a fork $F$ from the BNF whose leaves match with $Children(N)$. As N is not an identifier node, it follows from Assertion 3 from Section 9.1.2 that there can exists only one such fork $F$, if any. If the required fork $F$ does not exist in the BNF, output error message “query is not well-typed” indicating the statement in the query $q$ corresponding to the node $N$ which “cannot be properly typed”, and halt (end of algorithm). Otherwise, if $F$ exists, move to step 2b or 2c depending on whether $N$ is already marked as “correct” or not. Note: The term ‘matching’ means that the branching degree should be the same and the matching children nodes (in the natural order) have the same labels. The labels of $N$ and the root of $F$ are not required to coincide for matching to be successful. 2. (b) _N_ is not marked as “correct” \- relabel syntactical category of $N$ exactly as the root of $F$, mark $N$ as “correct” and “seen”, and move to step 2. 3. (c) _N_ is marked as “correct” \- if the label of the root of $F$ _coincides_ with the label on $N$ then mark $N$ also as “seen” and move to step 2. However, if the label of the root of $F$ _differs_ from the label on $N$, generate the error message “query is not well-typed; conflicts with the expected syntax” and indicate which syntactic category name (and corresponding place in the query) requires renaming. (End of algorithm.) 4. END with either correctly relabelled parse tree, or an appropriate error state. The successful result of this algorithm would give us a full guarantee that the resulting relabelled tree is still the correct parse tree of the given query which is therefore well-formed. Most importantly121212 also, taking into account appropriate renaming of syntactical categories of query call <parameters> considered below , it will also guarantee that the query is well- typed: parse tree labelling is fully coherent, both with the typing and all other details in declarations of identifiers (such as to be a constant or variable or query name). ###### 9.2.3.3 Contextual analysis algorithm The complete algorithm for bottom-up contextual analysis consists of the following (macro) steps. The input is any query parse tree $qt$ and the list of INs (both obtained from the parser). The output being either: (i) correctly relabelled query parse tree ($q$ is well-typed), or (ii) an error message ($q$ is non-well-typed). Contextual analysis algorithm $CA(qt,\mathrm{the\ list\ of\ INs})$: 1. START with the list of INs of the query parse tree $qt$. 2. 1. Find suitable candidate declaration (BN and IDN) for each identifier occurrence (each IN). That is, iterate over the given list of INs calling IDS algorithm for each IN (see Section 9.2.3.1). The result of these identifier declaration searches is the list of declaration triples for all INs. For those INs for which the algorithm IDS outputs $<NULL,NULL,IN>$ the corresponding error messages “identifier non-declared” should be outputted concerning all such identifier occurrences in the query $q$ and additionally that the “query is not well typed”. If IDS outputted NULL triple for some IN then end of algorithm; otherwise move to step 2. 3. 2. Relabel syntactical categories of some parse tree nodes according to step 1. 1. (a) Relabel syntactical categories of identifier occurrences. Labels of nodes (i.e. syntactical categories) generated by the parser contain the preliminary information on the typing (assigned by the parser and possible contradicting the actual type). The real typing of any IN and, in fact, the real syntactical categories (the node labels) of the INs can be correctly determined using the IDN from the declaration triple of IN. The parse tree labelling for these INs should be updated accordingly (may be vacuously if the given IN, in fact, does not need updating according to the IDN) with marking these nodes as “correct”. This can be done straightforwardly for all INs (in particular for query names to be discussed below). Thus after relabelling, all INs will be actually marked as “correct”. 2. (b) Relabel syntactical categories of query call parameters131313 In some cases similar to query parameters the parser already assumes some typing. For example, in the membership statement $l:a\in b$ the syntactical categories of $l,a,b$ must be, respectively, <label>, <delta-term> and <delta-term>, according to the BNF. In the case of equality $a=b$, the expressions $a$ and $b$ must be of the same type according to BNF, although the choice of type is ambiguous as shown by those examples in Section 9.1.4. But, the case of query call parameters requires our special attention in the currently described algorithm. . In the case of INs which are query names in query calls some additional renaming of some (possibly) non-IN nodes (query parameters) is required as described below. If we have a query call $q(t_{1},...,t_{n})$ with the query name $q$ of the type $(type_{1},type_{2},...,type_{m}\longrightarrow type)$ obtained from the appropriate IDN by the algorithm IDS (where all participating $type_{i}$ are _set_ or _label_ , and the type after arrow is _set_ or _boolean_) then we should: 1. i. Check whether $m=n$; if not, the query is not well-typed, and the algorithm should halt with an appropriate error message. 2. ii. If $m=n$, rename (possibly vacuously) syntactical categories of parameter nodes $t_{i}$ (`<delta-term>` or `<label>`) according to the types $type_{i}$ (_set_ or _label_), and mark them as “correct”. 4. 3. Relabel syntactical categories of all other parse tree nodes. Apply SCR algorithm (Section 9.2.3.2) to the resulting partially relabelled parse tree. Thereby other nodes of the parse tree will also be potentially renamed. 1. (a) Were all other nodes successfully renamed? Yes \- If the SCR algorithm renamed and marked all nodes as “correct”, then move to Step 4 to check for additional requirement (that query is properly “bounded”). No \- Parsing agreeing with typing is impossible, and appropriate error messages from SCR algorithm should be outputted. End of algorithm. 5. 4. Additional requirements on bounding terms or formulas (BTFLVNs) 1. (a) Check that (the names of) bounded identifiers (INs) of: <separate>, <recursion>, <collect>, <delta-formula with declarations>, <delta-term with declarations>, and <quantified formula> have no non-declared occurrences inside the bounding term or formula (BTFLVN). For convenient implementation of this clause we assume additionally that the parser also generates for each bounding term or formula (BTFLVN) the sub-list of INs (from the list of all INs generated by the parser) lying _under_ BTFLVN in $qt$141414 If BTFLVN is LVN – a label value node – then this list is, of course, empty. . In other words, these are some of the identifiers occurring in the query $q$. This can be represented as lists (for each BTFLVN) of the form: $<BTFLVN,IN_{1},\ldots,IN_{k}>.$ Using the list of these INi under the given BTFLVN and the declaration triples of the form $<BN,IDN,IN>$ generated by the IDS algorithm, it should be checked that each INi from the above list whose name coincides with the name of some bounded IN by the given BTFLVN (see Definition 4 (b)) is declared in this BTFLVN. The latter means that such an IN has its own binding node BN (from the appropriate unique triple), and this BN lies under or coincides with the given BTFLVN. This should hold for each BTFLVN in $qt$. Otherwise contextual analysis should be aborted with corresponding error message. In particular, in the case or recursion, we should check that the recursion binding set variable, as well as variables from the binding variable pair, do not occur free in the bounding term. Also, each query name should not occur free in the defining term or formula, and set constant should not occur free (non-declared) in the defining term, etc. However, in the case of set constants and query names we need to add the following additional requirements. 2. (b) Check that for each <set constant declaration> the defining <delta-term> has all of its set or label _variables_ declared within this term. That is, intuitively, <delta-term> defining a set constant should have a constant value. However, constants and query names inside this <delta-term> may be declared in the query outside this term. To do this, use the list of INs of _variables_ lying under the node <delta- term> of <set constant declaration> and the identifier declaration triples of the form $<BN,IDN,IN>$ generated by the above IDS algorithm, and check that each BN of such a variable IN lies in the <delta-term> node subtree. Otherwise, such a variable IN of the <delta-term> is considered as free, and the contextual analysis should be aborted with the corresponding error message. 3. (c) Check that for each <set query declaration> the defining <delta-term> has all its set or label _variables_ declared (quantified, etc.) either inside this term or in the given <set query declaration> as <variables> parameters of the declared query. Constants, and query names inside this <delta-term> may be declared in the query outside this term. Quite similarly check for each <boolean query declaration> and corresponding <delta-formula>. 4. (d) The remaining check that `<label constant declaration>` uses closed `<label value>` is evidently vacuous, as actually there is nothing to check. 6. END with a correctly relabelled and well-typed and properly bounded parse tree (“query is well-formed and well-typed”), or a partially relabelled parse tree plus additional error messages (“query is well-formed but not well-typed”, etc.). ##### 9.2.4 Extension of contextual analysis to support libraries That the library declarations are well-formed and well-typed can be checked by reducing these declarations to the ordinary queries, as it was shown in Section 3.4.2.2, and applying parsing and contextual analysis algorithm described above to the resulting query. ### Chapter 10 XML Representation of Web-like Databases (XML-WDB Format) #### 10.1 Represention of WDB by graph or set equations As we discussed in Chapter 2 the (hyper)set theoretic approach [40, 41, 43, 56, 57, 61] to WDB is based on the concept of hereditary finite sets or, more generally, hyperset theory [3, 5]. Such semi-structured data is represented as abstract sets (of sets of sets, etc.) with the possibility for membership relation to form cycles. Figure 10.1: Example WDB representing a fictitious family For visualisation purposes hyperset databases are represented as _graphs_ (see Figure 10.1) where nodes correspond to set names and labelled edges to membership relation. When considering implementation (and also intuitively from the set theoretic view) it is far more appropriate to represent WDB as _system of set equations_. Each set equation consists of a _set name_ equated to a _bracket expression_ ; _labelled elements_ of such sets may be either atomic values, nested bracket expressions, or set names (described in some other equations). For example, system of _flat_ set equations corresponding to the WDB graph in Figure 10.1 looks as follows: bob = { name:"Bob", wife:alice } alice = { name:"Alice", husband:bob, pet:sam } sam = { name:"Sam", species:"cat" } or, equivalently, with the _nesting_ allowed: bob = { name:"Bob", wife:alice } alice = { name:"Alice", husband:bob, pet:{name:"Sam", species:"cat"} } In particular, this demonstrates that the specific form of set names (e.g. `bob`, `alice`, `sam`) however helpful intuitively are formally not important. They can always be renamed (say by numbered “object identities” e.g. `&23`, etc.) or substituted as above. In general, the role of set names in any system of set equations depends on its position. Those set names occurrences on the left-hand side of set equation (simple set names) are also called _defined_ set names, whereas, all other set name occurrences are called _referenced_ set names. Each referenced set name should be defined somewhere in the system, and only once. The implemented query system considers WDB as systems of flat set equations (without any nesting). As described below, WDB is represented practically as a system of XML files each containing a fragment of the whole system of set equations of the WDB, which proves convenient. From the perspective of any database designer, the informational content of WDB is carried by: * • Labels on WDB-graph edges e.g. `name`, `wife`, `husband`, etc. * • Atomic data (see Note 5) on leaves e.g. `"Bob"`, `"Alice"`, etc. * • Graph structure or, respectively, set-element nesting. ###### Note 5 (Atomic data). Atomic data is, in fact, treated as singleton sets consisting of a labelled empty set or, equivalently, as labels on additional leaf edges in the WDB graph. For example, the atomic value `"Bob"` from the above example is formally represented as {Bob:{}} or, respectively, as the labelled edge with the target node being a leaf, For example, taking into account the above description, the corresponding system of (almost) flat set equations (with atomic values simulated as labelled empty sets) representing the WDB graph depicted in Figure 10.1 should actually be: bob = { name:bob_name, wife:alice } bob_name = { Bob:{} } alice = { name:alice_name, husband:bob, pet:sam } alice_name = { Alice:{} } sam = { name:sam_name, species:cat_name } sam_name = { Sam:{} } cat_name = { cat:{} } To completely flatten this system we need to further replace all nested occurrences of `{}`, say, by the set name `empty` and add one more equation `empty = {}`. Of course, nesting is a reasonable notion, and atomic values are more user friendly from the external point of view. Thus, these concepts are included in the XML representation of WDB considered below, although the query system internally uses only completely flat set equations111 Note that WDB may (briefly) involve complicated equations, such as $res=q$ where $q$ is an arbitrarily complicated term or formula, during the execution of queries $q$ or after invoking the “splitting” rule during reduction. But, this extended system is, in fact, reduced to the flat form, and it is technically more convenient to work with other given WDB equations if they are presented in the flat form. . #### 10.2 Practical representation of WDB as XML Although set equations represent WDB in the most natural and intuitive way, directly suggesting that such data are hypersets, it makes sense to relate this approach to the popular XML representation of semi-structured data and use appropriate existing techniques. Thus, numerous and independently existing XML data can be treated by our approach, making its application considerably wider. Extensible Markup Language (XML) is popular model for ordered (typically) tree-like semi-structured data. The portability, scaleability and tree (but extendable to graph) structure of XML has given rise to its wide spread useage. As such, systems of set equations, possibly allowing deep nesting, although very intuitively appealing could be represented practically as XML documents also based on the idea of representation of nesting data. However, the primary goal of our approach is not the implementation of XML querying, as much research and practical work has already been devoted to the latter: _CDuce_ [7], _Lore_ [33], _Quilt_ [14] _XML-GL_ [13], and _XML-QL_ [23]; as well as the W3C standards _XSLT_ [15], _XPath_ [22], and _XQuery_ [8] (based on Quilt). The main idea of the proposed XML-WDB format is to represent WDB systems of set equations as XML documents of a special form, and the most essential step consists in recursively replacing any labelled bracket expression label : {...} by the XML element: <label>...</label> Additionally, XML-WDB documents require: (i) the special root element `<set:eqns>` which denotes system of set equations, and (ii) the nested elements `<set:eqn>` denoting particular set equations. Defined set names participate as values of the `set:id` attribute of `<set:eqn>` tags, and referenced set names as values of the `set:ref` attribute (and also `set:href` attribute discussed later) of any other tags. Note that, as stated above, XML represents _ordered_ tree-like semi-structured data, however, our set- theoretic approach to WDB ignores order. Thus, such XML documents are treated by our approach ignoring the order (and possible repetition) of elements. Let us consider the system of set equations (with nesting allowed) in Section 10.1 (depicted visually in Figure 10.1) and its representation as an XML document in XML-WDB file 1. The names of the special elements (`set:eqns` and `set:eqn`) and special attributes (`set:id`, `set:ref` and `set:href`) should appeal to the readers’ intuition that the XML-WDB document below corresponds to the above system of set equations. <?xml version="1.0"?> <set:eqns xmlns:set="http://www.csc.liv.ac.uk/~molyneux/XML-WDB"> <set:eqn set:id="bob"> <name>Bob</name> <wife set:ref="alice" /> </set:eqn> <set:eqn set:id="alice"> <name>Alice</name> <husband set:ref="bob" /> <pet> <name>Sam</name><species>cat</species> </pet> </set:eqn> </set:eqns> XML-WDB file 1 Family database (cf. Figure 10.1). Recall that atomic data such as `name:"Bob"` is interpreted as `name:{Bob:{}}`, and should therefore be translated into `<name><Bob></Bob></name>` or, equivalently, into `<name><Bob/></name>`. This might seem to contradict XML-WDB file 1 where rather `<name>Bob</name>` is used, but the inverse translation in Section 10.2.3 (Rule 2) shows that the empty element `<Bob></Bob>` or `<Bob/>` is treated equivalently as text data `Bob`. Here it appears as text data for the readers’ convenience. ##### 10.2.1 XML-WDB document format In general, an arbitrary XML-WDB document is defined as follows. ###### Definition 5 (XML-WDB; see also Section 10.2.4 for the corresponding XML schema). A well-formed and valid XML-WDB file is an XML document with the root element `<set:eqns>` containing possibly several `<set:eqn>` sub-elements. The `<set:eqns>` element should contain no attributes, whereas, the element `<set:eqn>` should contain the required `set:id` attribute only. The value of the attribute `set:id` should have a unique value (across the whole document) called the _defined set name_ and can only be be a string of symbols which is any _simple set name_ (according to the syntactical category `<simple set name>` in the BNF). The elements `<set:eqns>`, `<set:eqn>`, and the attribute `set:id` are not allowed to appear anywhere else in the document. The element `<set:eqn>` can contain possibly several arbitrary XML sub-elements. The attributes `set:ref` and `set:href` can appear (at any depth) in those arbitrary elements under `<set:eqn>`. The values of the attributes `set:ref` and `set:href` are called _referenced set names_ , and must correspond to some existing `set:id` value in the same XML-WDB document in the case of `set:ref`, or `set:id` value in some other XML-WDB document in the case of `set:href`. To this end, the value of the attribute `set:href` should be _full set name_ (as discussed in Section 10.2.2; cf. the syntactical category `<set name>` in the BNF) consisting of an (XML-WDB file) URL and simple set name defined in that file (delimited by #). Everything else allowed by XML standard, what is not forbidden by the above restrictions, is permitted in the XML-WDB format. ###### Note 6. The important feature of this definition is that XML-WDB documents can contain quite arbitrary XML elements under `<set:eqn>`, thus allowing to include arbitrary XML data with any nesting, any text data and any attributes222 In general, arbitrary attributes are treated by the Rule 1 in Section 10.2.3 below. (except `set:id`, and with restrictions on values of `set:ref` and `set:href`, as described above) into our hyperset approach to WDB. However, the order and repetitions of data will be irrelevant for our approach, and the usual XML attributes (except the attributes `set:ref` and `set:href` which have a special role, as described above) will be treated rather as tags which permit no further nesting. ##### 10.2.2 Distributed WDB Any WDB system of set equations may be divided into several subsystems (as XML-WDB files) with the possibility for the set names $s$ participating in one subsystem (XML-WDB file) to be defined by set equations $s=\\{\ldots\\}$ either in the same or in some other subsystems (XML-WDB files). Thus, strictly speaking, we should always consider the corresponding full versions of set names defined in set equations of distributed WDB, even when a simple set name is used for simplicity. That is, each simple set name occurring as a value of set:id or set:ref attributes within an WDB-XML file should be understood as full set name obtained from the URL of this file by concatenating it with the simple name using `#` to delimite these parts. Moreover, this technique allows to avoid unintended simple set name clashes without cooperation or collaboration between the authors of distributed WDB-XML files. (Unfortunately, unintended clashes for using the same label for different intuitive meanings is still possible, however, this is not formal contradiction in our approach. Here the well-known idea of namespaces in XML could be used.) Figure 10.2: Example distributed WDB representing two fictitious families, divided into two fragments represented as white and grey nodes Defined set names appearing in some XML-WDB file can participate as referenced set names in the same or other XML-WDB files. Those set names defined in the same XML-WDB file are referenced as simple set name values of the attribute `set:ref`, whereas, set names defined in some other XML-WDB file are referenced as full set name values of the attribute `set:href`. It is required that each full set name should refer to an existing XML-WDB file and the set equation within that file for the simple set name part (after the `#` symbol). Let us now consider an example of distributed WDB, representing two families (visualised in Figure 10.2) and the corresponding XML-WDB files `family1.xml` and `family2.xml` (XML files 2 and 3) appearing below. Both simple and full set names participate as referenced set names in this example distributed WDB. For example, take the labelled element `daughter:emma` represented in XML-WDB file `family1.xml` as <daughter set:ref="emma" /> where the attribute `set:ref` refers to simple set name `emma` defined within the same file. As an illustration of distribution, consider the labelled element `friend:mark` represented as <friend set:href="...family2.xml#mark" /> where the attribute `set:href` refers to set name `mark` defined in the file `family2.xml`. Note that, the URL in this example has shorted for the sake of simplicity. <?xml version="1.0"?> <set:eqns xmlns:set="http://www.csc.liv.ac.uk/~molyneux/XML-WDB"> <set:eqn set:id="bob"> <daughter set:ref="emma" /> </set:eqn> <set:eqn set:id="alice"> <daughter set:ref="emma" /> </set:eqn> <set:eqn set:id="emma"> <friend set:href="...family2.xml#mark" /> </set:eqn> </set:eqns> XML-WDB file 2 Family database fragment (cf. grey nodes Figure 10.2): family1.xml <?xml version="1.0"?> <set:eqns xmlns:set="http://www.csc.liv.ac.uk/~molyneux/XML-WDB"> <set:eqn set:id="paul"> <son set:ref="mark" /> </set:eqn> <set:eqn set:id="amy"> <son set:ref="mark" /> </set:eqn> <set:eqn set:id="mark"> <friend set:href="...family1.xml#emma" /> </set:eqn> </set:eqns> XML-WDB file 3 Family database fragment (cf. white nodes Figure 10.2): family2.xml The analogy of WDB with the WWW and, in particular possible distributed character of WDB does not imply it is necessarily so huge and unorganised as the WWW. It could be distributed between several sites, and supported by specialised WDB servers of some departments of an organisation owning this WDB and maintaining some specific structure of this WDB. Thus, WDB might, in fact, be much more structured than the WWW, however, the general approach imposes no restrictions. Therefore, the concept of WDB _schema_ or _typing_ relation between hypersets or graphs (much more flexible than for the relational databases and based on the notion of bisimulation or “one-way” simulation) relativised to some typing relation on labels/atomic values can be considered for such databases [9, 41, 57, 69]. Here we will not go into details of this important topic as our main concern is the straightforward implementation of querying WDB which does not take into account any such WDB schemas. ##### 10.2.3 Transformation rules from XML to systems of set equations Let us show how any XML-WDB document, as described above, can be treated as a system of set equations by using the following simple transformations (applicable, in fact, to arbitrary XML documents, but giving the desired system of set equations only for the XML-WDB documents). There are however currently some restrictions on XML-WDB in these transformation rules which can easily be relaxed, for example attributes having many values attr="value1 value2 ..." are not taken into account. ###### 10.2.3.1 Elimination of attributes and text data The first two transformation rules, applied recursively, will eliminate attributes and atomic (text) data from arbitrary XML element by treating them as tags. Rule 1 (Attribute elimination, except attributes `set:id`, `set:ref` and `set:href`). XML tags which have attributes, <tag attr="value" other-attributes> some-content </tag> transform to <tag other-attributes> <attr>value</attr> some-content </tag> where `attr` is restricted to be any attribute name except the distinguished attributes `set:id`, `set:ref` and `set:href` belonging to the `set` namespace which will be considered later. Additionally, `some-content` means arbitrary XML content of an XML element. In the case of empty element with attributes, <tag attr="value" other-attributes /> transformation quite analogously gives the similar result, <tag other-attributes> <attr>value</attr> </tag> This rule is applied until all attributes, except those attributes beglonging to the `set` namespace (`set:id`, `set:ref` and `set:href`), are eliminated. This way attributes are actually treated as tags. Rule 2 (Atomic data elimination). Text data with no white spaces any-text-data transforms to the empty XML element <any-text-data/> In the case of text data containing white characters (spaces, carriage- returns, tabs), any text data all white characters are ignored, and the result is the corresponding sequence of the empty elements, <any/><text/><data/> As our set theoretic approach ignores order and repetitions (in contrast with the ordinary XML approach) this, in fact, means that a sentence (any text data) is considered rather as an unordered set of words. This way text data are actually treated as tags. (An another alternative would be to replace all white characters by the underscore symbol, thus giving rise to `<any_text_data/>`, like above.) Iterated application of rules 1 and 2 eliminates all atomic (text) data and attributes except those attributes belonging to the `set` namespace (`set:id`, `set:ref` and `set:href`). ###### 10.2.3.2 Elimination of tags The remaining rules below allow transformation of XML elements with (simple) attributes and text data eliminated by the above rules into bracket expressions (possibly involving set names), and into set equations if there are tags set:eqns and set:eqn occurring as described in Definition 5. In the intermediate steps, the expression transformed will be in the mixed language. Rule 3 (Tag elimination, except the tags set:eqns and set:eqn). For arbitrary XML tags, except set:eqns and set:eqn, which have no attributes, <tag> some-content </tag> transforms into tag:{some-content}. Those possibly remaining tags in sub-elements of `some-content` will be eliminated recursively by application of transformation rules 3 and 4. Quite analogously for the case of the empty element, <tag/> transforms to tag:{} Rule 4 (Elimination of tags with `set:ref` and `set:href` attributes). <tag set:ref="set-name" /> transforms to the sequence tag:set-name Recall that other attributes were already eliminated by Rule 1. Furthermore, according to the definition of well-formed XML document an attribute name must only appear once in any tag, however, `set:ref` and `set:href` may participate together in any tag. The above elimination is considered as typical if only the attribute `set:ref` or `set:href` occurs. Additionally, we must consider the following more general, however unlikely case when some content is present: <tag set:ref="set-name1" set:href="set-name2"> some-content </tag> transforms to tag:set-name1, tag:set-name2, tag:{some-content}. However, to be consistent with the first version of Rule 4, if `some-content` is empty, then (as an exception) the result should not contain the labelled element, `tag:{}`. The above rules hold also for the case of the attribute `set:href`, or when both `set:ref` and `set:href` are present within a tag. Note that after applying Rule 4, the difference between these two attributes is not taken into account in generating the result. Recall that `set:ref` refers to a simple set name, whereas, `set:href` refers to a full set name which is actually an URL together with simple set name (see Section 10.2.2). Such syntax explicitly differentiating between simple and full set names is convenient for implementation. After applying this rule this feature will disappear, but the difference between the shapes of simple and full set names will remain, so that nothing essential will be lost. Rule 5 (Elimination of tags `set:eqn` and `set:eqns`). <set:eqn set:id="simple-set-name">some-content</set:eqn> is replaced by the equation, simple-set-name = {some-content} and, <?xml ... > <set:eqns>some-content</set:eqns> is replaced by some-content that is, by system of set equations (in the case of a well-formed XML-WDB document; cf. Definition 5 above). Note that, all the above rules can be applied in arbitrary order, leading to a unique system of set equations. ##### 10.2.4 XML schema for XML-WDB format A well-formed and valid XML-WDB document must conform to Definition 5. As our general goal is implementation, let us also present the XML schema333 also available at http://www.csc.liv.ac.uk/~molyneux/XML-WDB/schema/xml-wdb.xsd (at the end of this section) which corresponds to this definition almost completely (as XML schemes are, in fact, insufficiently expressible). First of all, the schema requires that all the declared elements `eqns` and `eqn`, and attributes `id`, `ref` and `href` are qualified under the namespace http://www.csc.liv.ac.uk/~molyneux/XML-WDB. In practice the author of any XML- WDB document can declare this namespace as the mnemonic `set`444 In fact, the namespace http://www.csc.liv.ac.uk/~molyneux/XML-WDB could be declared by any chosen mnemonic, let us say s. and use set:eqns instead of just eqns, etc. to emphasise these special elements/attributes are subject to the rules of this schema. The root element `eqns` of an XML-WDB document is declared in the schema as having the complex type `system_of_set_equations`, as follows, <xsd:element name="eqns" type="system_of_set_equations"/>. The complex type `system_of_set_equations` is defined as <xsd:complexType name="system_of_set_equations"> <xsd:sequence minOccurs="0" maxOccurs="unbounded"> <xsd:element name="eqn" type="set_equation"/> </xsd:sequence> </xsd:complexType> where an arbitrary number ($\geq 0$) of set equations can participate in any XML represented system of set equations. Note that, by definition only, `eqn` subelements can participate under an `eqns` element. Here, `eqn` elements represent set equations by the given complex type `set_equation`, which is defined by two elements: <xsd:sequence minOccurs="0" maxOccurs="unbounded"> <xsd:any namespace="##any" processContents="lax"/> </xsd:sequence> <xsd:attribute form="qualified" name="id" type="xsd:ID" use="required"/> Thus, any `eqn` element must contain the required attribute `id`, and may contain arbitrary XML sub-elements. Note that, by definition, only one attribute, `id`, must appear in `eqn` elements. The corresponding value of the `id` attribute must be unique over the entire XML-WDB document according the type `xsd:ID`. However, the schema only ensures the well-formedness with `lax` processing of arbitrary XML sub-elements, and therefore does not check that such elements are XML-WDB valid according to Definition 5. In particular this schema says nothing about `ref` and `href` attributes and how they can be used. Thus, our implementation additionally ensures the following: * • The elements `eqns` and `eqn` and attribute `id` qualified under the http://www.csc.liv.ac.uk/~molyneux/XML-WDB/ namespace can not participate in arbitrary XML sub-elements. * • The attribute `ref` must have simple set name value, defined by the `id` attribute in the same XML-WDB file. Furthermore, the attribute `href` must have full set name value whose simple set name part is defined in some other well-formed and valid XML-WDB file. Thus, any well-formed XML document is considered as valid XML-WDB document if it can be successfully validated against the above schema and conforms to these additional rules. However, our $\Delta$ language query implementation deals directly with systems of set equations, therefore it is necessary to rewrite from valid XML-WDB files into systems of set equations, by treating them with the rules from Section 10.2.3. The inverse transformation from systems of set equations to XML-WDB format is also implemented. <?xml version="1.0" encoding="UTF-8"?> <xsd:schema xmlns:xsd="http://www.w3.org/2001/XMLSchema" targetNamespace="http://www.csc.liv.ac.uk/~molyneux/XML-WDB" xmlns="http://www.csc.liv.ac.uk/~molyneux/XML-WDB" elementFormDefault="qualified" attributeFormDefault="unqualified"> <xsd:complexType name="system_of_set_equations"> <xsd:sequence minOccurs="0" maxOccurs="unbounded"> <xsd:element name="eqn" type="set_equation"/> </xsd:sequence> </xsd:complexType> <xsd:complexType name="set_equation"> <xsd:sequence minOccurs="0" maxOccurs="unbounded"> <xsd:any namespace="##any" processContents="lax"/> </xsd:sequence> <xsd:attribute form="qualified" name="id" type="xsd:ID" use="required"/> </xsd:complexType> <xsd:element name="eqns" type="system_of_set_equations"/> </xsd:schema> XML schema 1 XML-WDB file schema: xml-wdb.xsd ## Part IV Evaluation ### Chapter 11 Comparative analysis #### 11.1 Preliminary comparison There have been many proposed approaches for modelling and querying semi- structured data. Many of these approaches are based on the graph model, which has become the prevalent model for representation of semi-structured data. For example, the graphical Object Exchange Model (OEM) [51] was used in the integration of heterogeneous information sources in Tsimmis [31] and the semi- structured query language Lorel [2, 46]. Moreover, there has been some trend toward the XML document model, which is essentially the graph model restricted to ordered trees, but arbitrary graphs can be imitated by using the attributes `id` and `ref` to define links between tree branches. In fact, Lore (implementation of the Lorel language) was later migrated to XML [33]. The most natural and intuitive way of querying graphs employed in most approaches is path navigation by using path expressions. However, path expressions are evidently sufficiently complicated syntactical means to achieve expressive power in queries. This is practically very reasonable and means path expressions are a strong technical tool. But, on a logical level (in the wide sense of this word) such complicated things are always considered as definable in terms of some other more fundamental concepts. Thus, in foundation of mathematics such fundamental concepts are set, membership relation, logical quantifiers, etc. allowing to express all other concepts, constructions and proofs in mathematics and (theoretical) computer science. In a sense, the graph approach to semi-structured databases lacks natural logically fundamental concepts, and in these circumstances path expressions are included as the main tool for achieving expressive power. On the other hand, the set theoretic approach to semi-structured databases presented in this thesis does not require path expressions111besides the related classical operation of transitive closure of a set and a general recursion operator — classical inductive definitions to achieve high expressive power which in fact captures exactly all “generic” polynomial time computable operations over hypersets [41, 43, 56, 57]. Therefore, the language can be considered theoretically as having in this sense no “gaps”. But, from the point of view of practical usability and efficiency of implementation, path expressions should be eventually included in our implementation of the $\Delta$-language although not increasing its expressive power (see [61]). From the traditional theoretical point of view polynomial time computability of queries in $\Delta$ (which is usually theoretically considered as “feasible computability”) allows to consider $\Delta$ as computationally viable. However, in a practical sense, we cannot insist on this usage of the term “feasible” because polynomials can be of high degree and with huge coefficients. Also, this makes less sense in the context of those most expensive computational steps assuming downloading numerous files from the World-Wide Web. Thus, we rather consider this characteristic not as a witness of efficiency of $\Delta$ but as a good witness of expressive power of the language. Anyway, when comparing this approach with others, it can be considered as top-down from theory to practice. In particular, this explains again our attitude to not include path expressions in the main conceptual version of the $\Delta$-language, being a definable concept, and considering them only as technical “conservative” extension, although very important practically. Recall that hypersets representing WDB can be visualised as graphs, and thus, in principle, our approach can treat graph structured data from other approaches, but assuming that the order and repetition of such data does not matter. As the latter is not always the case, the precise comparison with other approaches is not so straightforward. Similarly, our implementation can query arbitrary XML elements, rewriting from XML-WDB to systems of set equations and ignoring order. Although the aim of the project was not XML querying, this accomplishment extends possible applicability of our implementation. Now, after these preliminary general comments, let us consider several known approaches to semi-structured databases and to set theoretic programming. #### 11.2 SETL An important practical predecessor of our work is the set theoretic programming language SETL [62, 63, 64] which deals with hereditarily-finite well-founded sets (without cycles) and tuples. (Note that tuples or, more generally, records $[a_{1}:x_{1},\ldots,a_{n}:x_{n}]$ can be trivially treated in our approach as sets $\\{a_{1}:x_{1},\ldots,a_{n}:x_{n}\\}$ in which all labels $a_{i}$ are different.) This general purpose programming language exploits the notion of set as fundamental data structure with its set theoretic style of constructs like collection in $\Delta$. It is, however, an imperative language using such traditional operators as the assignment operator, loops, etc. For example, let us consider the SETL program: A = {1,2,3,4,5}; B = { x: x in A \| x >= 3 }; print(B); where the statement on the second line reminds us of the $\Delta$-term collect. In fact, the result of executing this SETL program is the output set of `B`, which is, in fact, defined as those numbers `x` belonging to the set `A` such that the number `x` is greater than or equal to three, as follows: {3,4,5}. Furthermore, in SETL, equality between sets is understood as “deep” set equality implemented as the following (recursive) procedure taken from [62]: proc equal(S1,S2); if # S2 /= # S1 then return false; else (forall x in S1) if x notin S2 then return false; end if; end forall loop; return true; -- S1 and S2 are equal end if; end proc; That is, the two sets `S1` and `S2` are equal if they have the same cardinality and each element `x` of the set `S1` participates as a member in the set `S2`. In fact, this equality procedure will be called recursively for each membership test `notin` (where, like in our case, $x\in y\iff\exists x^{\prime}\in y\,.\,\texttt{Equal}(x,x^{\prime})$). Hence, `S1` and `S2` are equal if their elements are equal and their elements are also equal, and so on. This is similar to bisimulation equivalence which is an important concept in our hyperset theoretic approach. The use of cardinality operator $\\#$ either witnesses that hereditarily-finite sets are represented in SETL implementation in strongly extensional form and, anyway, assumes further recursive call of equality. In contrast to SETL, the implemented $\Delta$ language is actually a declarative query language to semi-structured or Web- like databases and, as such, is not intended to be a universal language. The degree of universality of $\Delta$ is characterised by its expressive power equivalent to polynomial time. Also, SETL does not have any construct similar to the decoration operator within the $\Delta$-language which allows for restructuring, but its universal character should allow to define decoration for acyclic graphs. In contrast to SETL, the main characteristic feature of $\Delta$ is the extension of the ideas of descriptive complexity theory [37, 38, 55, 74] (usually considered in connection with the relational approach to databases) from finite relational structures to hereditarily-finite (hyper)sets and, thereby, to semistructured databases. The most recent development on the SETL language was the implementation described in [4], which introduced Internet programming using sockets into the SETL language. In fact, these latest considerations further support that SETL is actually a general purpose programming language, and in this sense differs from $\Delta$ which is a query language. #### 11.3 UnQL The UnQL query language [10, 11] is closest to our approach as it is based on bisimulation, with its operators also being bisimulation invariant as in our case. However, despite considering bisimulation, UnQL is based on the graph model, and the op. cit. do not even mention hyperset theory. UnQL can also be characterised as a bottom-up approach from graphs to something reminding us of hypersets. Moreover, there is no operator for testing equality between graph vertices (neither literal nor based on bisimulation) in the UnQL language. However, bisimulation should be used in defining the semantics of path expressions (patterns in their terminology) in the UnQL language, as shown in [61] and in our example in Section 3.6, ensuring that its operations really are bisimulation invariant. Much of the UnQL approach is devoted to the rather complicated way in which they deal with graphs, which appears more technical compared to the intuitive denotational and operational semantics of the hyperset approach. In a sense, UnQL has defined only operational semantics over graphs, which is bisimulation invariant. No abstract concept like hyperset and corresponding (hyper)set theoretical style of thought is explicitly described. Moreover, operational semantics of the structural recursion operator is rather complicated by working with multiple “input” and “output” vertices considered as essential part of graphs to be queried by UnQL. Therefore, semi-structured data represented in UnQL does not exactly correspond to hypersets, although it can be imitated by hypersets as shown in [61]. Also, the UnQL language and related language UnCal were shown in [61] to be embeddable within $\Delta$, but, as reasonably conjectured, not vice versa. This embedding, although done in purely set theoretic terms, is based on the interpretation of arbitrary graphs as sets of ordered pairs. The bisimulation invariant operations on graphs of UnQL are defined set theoretically but as operations on graphs rather than as operations on abstract entities denoted by these graphs (with multiple “inputs” and “outputs”) considered up to simulation. In particular, the main structural recursion construct of UnQL is definable in $\Delta$ by manipulating graphs using recursive separation and concluded by applying decoration operation to get a hyperset imitating the result (with multiple “inputs” and “outputs”). In fact, many of the operations in UnQL are based on various ways of appending such kind of graphs (via “inputs” and “outputs”), including structural recursion, all of which may be considered as a special versions of the decoration operator. However, the full version of the powerful decoration operator (which is much simpler and logically more fundamental than its particular versions mentioned) is neither considered nor definable in UnQL (according to the conjecture in [61, page 813]). #### 11.4 Lore Lore (Lightweight Object REpository) [46] is the implementation of the Lorel query language [2] based on the OEM graph model [51]. Lorel is an extention of the Object Query Language (OQL) [12] and, in fact, statements written in the Lorel are translated to OQL. Moreover, additional features of Lorel (such as path expressions, and type coercion) are syntactical sugaring of OQL. The OEM model is similar to the data model used in UnQL, but unlike UnQL and also our approach, does not consider graphs up to bisimulation. Therefore, bisimulation invariance is not pursued in this approach, hence, in this way it is crucially different from UnQL and $\Delta$. In the OEM model equality is between graph nodes (OIDs) rather than value equality using bisimulation. Lorel also uses ordinary equality between sets of OIDs, which, however, is not the “deep” set equality assumed by bisimulation. Therefore, Lorel would treat some of our examples differently, and thus, only very informal and superficial comparison is possible, unlike the comparison with UnQL. However, the `select` operator of Lorel is very similar to our `collect` construct, as illustrated in the following example Lorel query: SELECT pub FROM pub in BibDB WHERE pub.author = "Smith" and the (strikingly similar) corresponding $\Delta$-query, set query collect { ’null’:pub where pub-type:pub in BibDB and author:"Smith" in pub } Note that only OIDs are `select`ed in Lorel, whereas in $\Delta$ (OIDs or) set names denote (hyper)sets which are, in fact (on the level of abstract semantics) `collect`ed. Note that, OIDs in Lorel denote just themselves and nothing more. Lorel can not express restructuring queries, unlike $\Delta$ which can perform restructuring queries with the decoration operation (at the final stage). Thus, informally (as formal comparison is impossible due to the above differences in data models – graphs vs. hypersets represented by graphs) Lorel (and also UnQL) can be said to be also strictly embeddable in $\Delta$222ignoring so called path variables which may potentially lead to exponential complexity and, for simplicity, some less essential aspects like typing and coercion . Finally, there is also no recursion operator (except for Kleenes star in path expressions) and nothing similar to decoration operator (important for deep restructuring). #### 11.5 Strudel Strudel is a Web site management system [26] for creating Web pages from heterogeneous data sources via the StruQL query language [27] (see also [1]). In particular, the `link` clause in StruQL is able to do simple restructuring. In fact, Strudel allows to generate real Web sites in a declarative way from a site graph (a graphical “plan” of a site) that encodes the Web site’s structure. The latter feature resembles the decoration construct although outside of hyperset approach. In Studel data is integrated from heterogeneous sources by mediators which rewrite from various data sources (such as XML files, bibtex files, etc.) to Strudel data graphs. StruQL queries over these data graphs, in fact, define the Web site structure creating Web pages and hyperlinks between Web pages. #### 11.6 G-Log G-Log [19] is another query language for semi-structured data represented as arbitrary labelled graphs. However, unlike the other approaches consider so far (Lorel, UnQL, $\Delta$) any query, as well as data, in G-log is represented graphically as a set of schematical red/green coloured “rule” graphs. Querying in G-log (in general, updating) is based on matching the query rule graph with the “concrete” black coloured data graph. This matching assumes one of three possible kinds of bisimulation (in particular, isomorphic embedding) of the red part of the rule with a subgraph of the black concrete data graph, and using the green part for updating the concrete data graph. This procedure is essentially non-deterministic and, in fact, can be executed in non-deterministic polynomial time (rather than polynomial time in the case of $\Delta$). The expressive power of G-log in its present form, or its potential extensions, is unclear, as well as precise comparison with $\Delta$. Granted, both are based on bisimulation but in a somewhat different way. The rule graphs of G-log can be described in some logical form, but it is unclear how to systematically relate this with the syntax of $\Delta$ to have a better comparison. In principle, extending $\Delta$ by quantification over the subset of a set, $\forall x\subseteq t,\exists x\subseteq t$, together with definability in $\Delta$ the necessary versions of bisimulation over graphs could make it possible to imitate matching of a rule graph with a subgraph of the data graph. But, it seems unclear whether there exists a natural unifying conceptual framework for both approaches. Furthermore, G-log is an open ended language with some ideas of its extension discussed in [19]. In any case, we can conclude that UnQL and even Lorel333 ignoring that Lorel does not consider bisimulation are syntactically, as well as in terms of operational semantics, much closer to $\Delta$ than G-log. However, matching with a subgraph is somewhat similar to the idea of path expressions which appear in both UnQL and Lorel, the latter being imitated in $\Delta$ as illustrated in Section 3.6. #### 11.7 Tree (XML) model approaches The XML data model is based on ordered trees, whereas the other approaches to querying semi-structured databases discussed so far deal with arbitrary graphs. (However, as we already mentioned, using attributes `id` and `ref` in XML allows imitate arbitrary graphs.) It might seem that querying XML data is formally outside of the (hyper)set theoretic view as the XML document model assumes a fixed order on the children of any node. Despite this our approach is able to query restricted XML documents (XML-WDB files which, however, can involve arbitrary nested XML elements) interpreted as systems of set equations. The following comparisons focus on three contemporary XML data model approaches, XSLT, XQuery and XPath, all of which were developed by W3C working groups. In fact, these languages are the successors to many other XML model approaches, for example, XQuery is based on the Quilt query language [14]. However, for brevity no comparisons will be made with these predecessors. ##### XSLT XSLT (eXtensible Stylesheet Language transformations) [15] is a rule based language for transforming the structure of an XML document, that is, XSLT rewrites an XML document to another XML document with different structure. Thus, XSLT does allow convenient manipulation of XML documents. XSLT rules are composed of template rules which _match_ attributes/elements using XPath-like expressions (discussed below) and create new XML elements/attributes or apply other template rules. This style of language and its operational semantics is rather different from the $\Delta$-query language. In particular XSLT is typically used to visualise XML documents by transforming them into HTML Web pages. ##### XQuery XQuery [8] is declarative query language for XML documents, and was derived from Quilt [14], Lorel [2] (described above) and XML-QL [23]. XQuery is, in fact, Turing complete and thus can be considered as more than just a query language but also, in a sense, as a general purpose programming language. ##### Path expressions (XPath) XQuery and XSLT include XPath path expressions in its syntax. XPath is a language especially created to express paths navigating over XML document trees, and, in fact, XPath itself can serve as a query language. Currently path expressions are not included in the implemented $\Delta$-query language, however, they were shown to be definable in the original language [61], and a simple example demonstrating how $\Delta$ could be extended syntactically to have path expressions and how it can define their meaning was shown in Section 3.6. Thus, our language is rich enough by fundamental operators over sets so that, at least theoretically, path expressions are unnecessary. Of course, practically they are very desirable and must be included in $\Delta$ to make it more practically convenient and user friendly. Moreover, path expressions, if implemented well, would make execution time of queries better than queries imitating path expressions in the current version of $\Delta$. In general, comparison of $\Delta$ with query languages for XML can be done only on a rather superficial level. In fact, they do not share a common data model and the levels of abstraction are so different that more detailed comparison in general terms is difficult. We can only repeat that the closest approach to ours is UnQL where comparisons can be done in quite precise mathematical formulations [61]. ### Chapter 12 Conclusion and future outlook In this thesis we explored the experimental implementation of the hyperset approach to semi-structured or Web-like databases and the query language $\Delta$ originally known only on a pure theoretical level. The primary goal was to demonstrate working practically with the $\Delta$-query language, and secondly, some considerations towards one crucial aspect of efficiency of such querying in the case of distributed WDB. The latter involves some theoretical considerations in Chapter 6 and empirical testing in Section 7.2. This chapter begins by reviewing the hyperset approach to semi-structured databases in the context of this thesis. In Section 12.2 we summarise the main results of our work which, in brief, consist in (i) the implementation of the query language $\Delta$ and (ii) development the concept of local/global bisimulation and running experiments demonstrating its fruitfulness in making query execution more efficient when equality (bisimulation) is involved. Some further simple optimisations used in our implementation are also discussed. Then we recapitulate briefly in Section 12.3 comparisons of $\Delta$ with other most close query languages. Finally, we conclude in Section 12.4 with some closing discussion towards possible future extensions and optimisations. #### 12.1 Hyperset approach to semi-structured databases First of all, the hyperset approach to semi-structured or Web-like databases and their querying was described in this thesis on the base of the earlier theoretical work done in [41, 57, 61]. This approach considers hypersets as the abstract data model for WDB where the concrete representation of hypersets is given by systems of set equations which can be saved either as plain text files or as XML-WDB files. Likewise in relational databases where the abstract data model is relations, our approach focuses on abstract hypersets and strongly distinguishes them from their concrete representations by set equations (or corresponding XML-WDB form). Set theory is known to play an extraordinary foundational role in mathematics, and here we wanted to demonstrate in a practical context that very general set theoretic approach towards semi-structured or Web-like databases is also quite reasonable. Systems of set equations can also be trivially represented as graphs where the latter, if considered literally, lead to the more traditional approach to semi-structured databases. To visualise our considerations we also use graphs, but they play only an auxiliary role. Abstractly, graph nodes as well as corresponding set names in set equations, denote hypersets. In fact, it is assumed that any user of our query system should mainly rely on pure set theoretic style of thought which is (mostly) simple and intuitive.111 The most subtle concept in our approach is the decoration operation. Otherwise it would not be so widely accepted both in the foundation of mathematics, and in everyday mathematical practice. As graphs or corresponding systems of set equations can involve cycles, their nodes or set names denote, in general, hypersets. They differ from the ordinary concept of sets in the fact that hypersets are not necessary well-founded. Based on well-developed and understood hyperset theory [3, 5], such sets pose no conceptual difficulty in our approach. This approach demonstrates on a practical level that hypersets are no more difficult than the usual concept of sets, and are quite useful by allowing arbitrary semi-structured data to be represented in a completely set theoretic manner. An additional feature of our data model is its distributed character, that is any system of set equations representing a WDB is allowed to be distributed, with set names used in one (XML-WDB) file possibly described by set equations in the others files. This leads to distinctions between simple set names described in the same file, and full set names involving also the URL of the file where this set name is described. This does not change the hyperset approach but extends its possible applicability. On the other hand, this distributed character of a WDB poses an additional challenge on how to check practically whether two set names (possibly described in remote files) denote the same abstract hyperset, i.e. whether two given set names or graph nodes are bisimilar. However, the problem of computing bisimulation in the distributed case was shown here to be, in principle, resolvable practically, as remarked later in Section 12.2.2. Respectively, the $\Delta$-query language considered here is set theoretic with the denotation $\Delta$ bearing from logic and set theory and traditionally emphasising its bounded character. The latter guarantees that all queries in $\Delta$ are computable in finite, in fact, polynomial time with respect to the the size of the input WDB. Moreover, it is known to have expressive power exactly corresponding to polynomial time (see [43, 57] and particularly [41, 57] for precise formulations of the labelled case considered here). #### 12.2 Novel contributions The main results of this work are the implementation of the hyperset approach to semi-structured databases and the query language $\Delta$, and, secondly, the local/global approach towards efficient computation of bisimulation in the case of distributed WDB. ##### 12.2.1 Implementation of the hyperset approach to semi-structured databases The implemented version of the language $\Delta$ is quite complex and even somewhat comparable with practical programming languages. In fact, there was not enough time to create the most optimal implementation. The general problem of efficiency is so difficult and involving so many various aspects (see e.g. [32]) that it is mostly outside the scope of this thesis (with one exception which is most essential to our hyperset approach; see Section 12.2.2). Taking this into account, the main criteria were correctness of the implementation and its user friendliness so that the language could be demonstrated to a more practically oriented, rather than just a mathematically inclined, audience. As far as we see, the implementation satisfies these criteria based on our testing and also writing and running the worked examples in Sections 3.5–3.7. This query system was also used by my supervisor, Vladimir Sazonov, as demonstration tool for undergraduate students. This initial practical goal of the project lead to the successful development of: * • Implementation of the $\Delta$-query language as a declarative language, based on those theoretical constructs in the original $\Delta$-language. Furthermore, for the convenience of writing queries some important features were included in the implemented language, such as _library declarations_ and _query declarations_ which, although very useful as the reader can see from the example queries, do not extend the theoretical expressive power of the language. * • Algorithms for checking the validity of queries to ensure both well-formedness and well-typedness. These algorithms add important low-level details for our implementation serving also as a sufficiently strong guarantee that the implementation was done correctly. The aim of the _parsing_ algorithm is to ensure well-formedness, according to the BNF grammar in Appendix A.1; whereas the aim of the _contextual analysis_ algorithm is to ensure well-typedness (which required considerable efforts to develop). The above syntactical considerations were highly important for implementation, and much time was dedication to ensuring these algorithms were described and implemented correctly. In fact, the following developments strongly rely on these algorithms: * • Implementation of operational semantics of $\Delta$ language according to reduction rules in [61] with some additional low-level descriptions for the operators recursion, decoration and TC also given here to aid implementation. * • XML representation of WDB by developing the XML-WDB format for systems of set equations and implementing algorithms rewriting from XML-WDB documents into systems of set equations, and vice versa. Currently we accepted this XML-WDB format as the standard way of representing WDB. These files can be saved on various sites and hyperlinked via full set names as we discussed above, and thus, WDB can be distributed (and queried) over the Internet. In fact, the XML-WDB format allows our approach to treat arbitrary nested XML elements within a WDB. The aim of this practical representation of WDB as XML is the ability, in principle, to query any existing XML data in our hyperset approach (assuming order and repetition in these data play no essential role). ##### 12.2.2 Local/global approach towards efficient implementation of bisimulation Bisimulation between WDB graph nodes or set names (i.e. whether they denote the same hypersets) is a crucial concept for the whole hyperset approach to WDB. The equality symbol (`=`) in our language means, abstractly, the identity between hypersets. But, from the point of view of implementation which deals with set names, rather than with abstract hypersets, the equality operator (`=`) means bisimulation which assumes sufficiently complicated computation. Thus, if we want to remain faithful to this approach and really value this set theoretic style then we should not only implement bisimulation, as it is described in Chapter 4, but also work towards optimising this expensive operation. It can be particularly expensive in the case of distributed WDB when computing bisimulation would assume potentially downloading lots of (possibly) remote WDB files, and we pay special attention to this challenge. The main idea of the local/global approach consists in computing the (global) bisimulation relation ($\approx$) on the whole distributed WDB from many couples of local approximation relations ($\approx^{L}_{+}$ and $\approx^{L}_{-}$) for each WDB site (or even for each WDB file), and that the latter relations are easily derivable locally. This way the global task is distributed between the main agent (Bisimulation Engine) and local agents (servers of WDB sites). Furthermore, empirical testing suggested that the exploitation of local approximations in the computation of global bisimulation relation $\approx$ can considerably improve performance. Also, the idea that the Bisimulation Engine is working in background time (similarly to Google) to compute the global bisimulation relation from local approximations was crucial in this performance improving strategy. Experiments described in Section 7.2 suggested that bisimulation, although a very challenging problem, especially in distributed case, is not so hopeless practically as it might seem. In particular, taking such optimisations into account the hyperset approach to WDB seems also potentially feasible practically. ##### 12.2.3 Further optimisation The work done on local/global bisimulation was the main focus of our attempts to optimise our implementation of the hyperset approach in the case of distributed WDB. Also, some additional consideration was given on writing more efficient queries in the current implemented version of $\Delta$, such as the removal of redundancies by using the so called canonisation query `Can(x)`. In fact, this query does not change its input (`Can(x)=x` as abstract hypersets) but transforms its representation into an equivalent strongly extensional (non-redundant) form. The effect of using `Can` in one particular example (in the query which linear orders any hyperset, Section 3.7) is quite impressive. Another general optimisation related with the recursion operator (and also crucially improving execution time of the linear ordering query mentioned above) is based on the possibility of replacing bisimulation to compare the iteration steps by simple comparison of participating set names only. Of course, further work on optimising the implementation of $\Delta$ (in comparison with writing optimal queries, for example exploiting `Can` above) remains to be done (see Section 12.4 below). #### 12.3 Comparisons with other approaches After considering various approaches in Chapter 11 we have found that the UnQL and Lorel query languages are closest to our approach. However conceptually, i.e., in fact, from the point of view of the hyperset approach, UnQL is the most close to $\Delta$. The implemented $\Delta$-language does not include yet path expressions typical for other approaches. But, this language is already a very expressive, and, in a sense, subsumes both the UnQL and (the main features of) Lorel languages. #### 12.4 Further work In short, the primary goal of implementation and attempts towards optimisation described in this thesis can be considered as successful. However, development of the implementation and the experiments was very time consuming, and there was insufficient time to implement all potential ideas. Many useful features have yet to be implemented, such as: * • Extending the implemented $\Delta$-query language to make it more user friendly with quantification over multiple variables. Also, similarly for the case of collection, separation and recursion constructs. * • Improving the library function, in particular to allow multiple or user defined libraries. * • Extending the implemented $\Delta$-query language to include path expressions which are typically included in other approaches towards semi-structured databases and, additionally, are very useful practically. In principle, path expressions could be implemented by rewriting them into $\Delta$-queries according to definitions in [61]. But, straightforward implementation should be more efficient. * • Extending the implemented $\Delta$-query language by update queries. * • More user friendly interface for inputting queries and WDB, as well as for outputting query results. In particular, the graphical visualisation of WDB and query results (developing a special WDB browser, as well as an editor for WDB files). Additionally, suitable techniques should be developed for creating WDB, taking into account its hyperset theoretic character: * • Using WDB schemas in the context of hyperset approach to impose restriction on the structure of WDB, just like in the relational approach but not necessarily so rigid. In fact, enforcing structure makes queries easier to write, and, additionally, can serve to eliminate possible unintended redundancies in set equations which could arise otherwise due to poor WDB design. Furthermore, although some suggestions towards efficiency were made here, there remains much work towards development of a practically efficient implementation: * • Adapting known and developing new optimisation techniques such as indexing, hashing and other data structures helping to implement efficient searching as described in [73] to the case of semi-structured data. Redundancies in set equations arising during computation should be regularly eliminated, thus allowing writing queries without explicit using the canonisation query. In this case equality between sets trivially becomes the identity relation rather than the bisimulation relation. Also, identical query calls should be executed only once. * • Dealing with redundancies in various circumstances by developing various techniques and methodology e.g. related with redundancies (bisimilarities) arising due to local updates in a WDB file (answering questions such as: are redundancies possibly arising in such local way easy to eliminate? under which conditions? etc.), or due to mirroring WDB sites, etc. * • Further improvements on the bisimulation engine transforming it from imitational to a more realistic version (Web service) assuming several levels (granularity) of locality (WDB-files, WDB-sites, the whole WDB) and extending the range of experiments with this engine. * • Adopting known [24, 25] and developing new techniques for optimisation of bisimulation which, for example, may take advantage of WDB scheme (see above). There is great scope for further theoretical and practical work. In summary, this could mean developing a full-fledged WDB management system and also WDB design techniques, and other methodologies based on the hypeset approach. Of course, the hyperset approach could be further evolved, e.g. it can be extended to also involve standard datatypes like integers, reals, strings as atomic data or label values with arithmetical and other operations over them (completely lacking in the current version of $\Delta$), etc. Also, multi- hypersets [44], records, lists, etc. could be allowed. Another version of the $\Delta$ language capturing LogSpace [40, 42] (currently for well-founded sets only) could be either implemented in its present form or, firstly, theoretically extended to the case of hypersets. Anyway, working on the theoretical level in various directions and simultaneously developing more practically oriented implementations, like in this thesis, seems a fruitful style of research. ### Appendix A Appendix #### A.1 Implemented BNF grammar of $\Delta$-query language The grammar of the implemented $\Delta$-language is represented by the metasyntax notation Extended Backus-Naur Form (EBNF) which allows for example to define the repetition of syntactical categories using `` or `+` (unlike regular BNF which does not have these features). For example, the EBNF production rule of `<declarations>` in Section A.1 defines an infinite number of possible forks, with any number of leaves labelled by `<declaration>` each separated by the terminal leaf labelled by `","`. The EBNF notation (used here to express the $\Delta$-language grammar) defines production rules as sequence of terminals (symbols) or non-terminals, `"xxx"` \| \- Terminal ---\|--- `<yyy>` \| \- Non-terminal where production rules are constructed (from those terminals or non-terminals) according to the following rules, Parentheses, `()` \| \- Grouping ---\|--- Vertical bar, `\|` \| \- Alternation Square brackets, `[]` \| \- Optional Kleene star, `` \| \- Repeat 0 or more times Kleene plus, `+` \| \- Repeat 1 or more times ##### Top level commands <top level command> ::= ( "library" <library command> \| <query> \| "exit" ) ";" <query> ::= "boolean query" <delta-formula> \| "set query" <delta-term> ##### Library commands <library command> ::= "add" <declarations> \| "list" [ "verbose" ] ##### Declarations <declarations> ::= <declaration> ( "," <declaration> )* <declaration> ::= <set constant declaration> \| <label constant declaration> \| <set query declaration> \| <boolean query declaration> <set constant declaration> ::= "set constant" <set constant> ("be"\|"=") <delta-term> <label constant declaration> ::= "label constant" <label constant> ("be"\|"=") <label value> <set query declaration> ::= "set query" <set query name> "(" <variables> ")" ("be"\|"=") <delta-term> <boolean query declaration> ::= "boolean query" <boolean query name> "(" <variables> ")" ("be"\|"=") <delta-formula> <variables> ::= <variable> ( "," <variable> )* <variable> ::= ( "set" <set variable> \| "label" <label variable> ) <parameters> ::= <parameter> ( "," <parameter> )* <parameter> ::= ( <delta-term> \| <label> ) <boolean query name> ::= <identifier> <set query name> ::= <identifier> ##### $\Delta$-terms <delta-term> ::= <set variable> \| <set constant> \| <set name> \| <atomic value> \| <enumerate> \| <union> \| "(" <multiple union> ")" \| <collect> \| <separate> \| <transitive closure> \| <recursion> \| <decoration> \| <if-else term> \| <set query call> \| <delta-term with declarations> <set name> ::= <URI> "#" <simple set name> <atomic value> ::= """ <identifier> """ <enumerate> ::= "{" <labelled terms> "}" <union> ::= ( "U" \| "union" ) <delta-term> <multiple union> ::= <delta-term> ( ( "U" \| "union" ) <delta-term> )* <collect> ::= "collect" "{" <labelled term> ( "where" \| "\|" ) <variable pair> ("in"\|"<-") <delta-term> [ "and" <delta-formula> ] "}" <separate> ::= "separate" "{" <variable pair> ("in"\|"<-") <delta-term> ( "where" \| "\|" ) <delta-formula> "}" <transitive closure> ::= ( "tc" \| "TC" \| "transitiveclosure" ) <delta-term> <recursion> ::= "recursion " <set variable> " {" <variable pair> (" in "\| "<-") <delta-term> ( "where" \| "\|" ) <delta-formula> "}" <decoration> ::= "decorate" "(" <delta-term> ", " <delta-term> ")" <if-else term> ::= "if" <delta-formula> "then" <delta-term> "else" <delta-term> "fi" <set query call> ::= "call" <set query name> "(" <parameters> ")" <delta-term with declarations> ::= "let " <declarations> "in" <delta-term> " endlet" <URI> ::= ( <web prefix> \| <local prefix> ) <file path> <web prefix> ::= "http://" <host> "/" [ "~" <identifier> "/" ] <local prefix> ::= "file://" ( (A-Z) \| (a-z) ) ":/" <host> ::= <identifier> [ "." <host> ] <file path> ::= <identifier> ( "/" <file path> \| <extension> ) <extension> ::= ".xml" <simple set name> ::= <identifier> ##### $\Delta$-formulas <delta-formula> ::=Ψ<atomic formula> \| "(" <conjunction> ")" \| "(" <disjunction> ")" \| "(" <quasi-implication> ")" \| <quantified formula> \| <negated formula> \| <if-else formula> \| <delta-formula with declarations> <atomic formula> ::= <equality> \| <label relationship> \| <membership> \| <boolean query call> \| "true" \| "false" <equality> ::= <set equality> \| <label equality> <set equality> ::= <delta-term> "=" <delta-term> <label equality> ::= <label> "=" <wildcard label> \| <wildcard label> "=" <label> <wildcard label> ::= [""] ( <label variable> \| <label constant> ) [""] \| "’" [""] <identifier> [""] "’" <label relationship> ::= <label> "<" <label> <label> ">" <label> <label> "<=" <label> <label> ">=" <label> <membership> ::= <labelled term> ("in"\|"<-") <delta-term> <boolean query call> ::= "call" <boolean query name> "(" <parameters> ")" <if-else formula> ::= "if" <delta-formula> "then" <delta-formula> "else" <delta-formula> "fi" <delta-formula with declarations> ::= "let" <declarations> "in" <delta-formula> "endlet" <conjunction> ::= <delta-formula> ( "and" <delta-formula> )* <disjunction> ::= <delta-formula> ( "or" <delta-formula> )* <quasi-implication> ::= <delta-formula> ( <quasi-implication connective> <delta-formula> )* <quasi-implication connective> ::= "<=" \| "=>" \| "implies" \| "iff" \| "<=>" <quantified formula> ::= <forall> <delta-formula> \| <exists> <delta-formula> \| <forall> ::= "forall" <variable pair> ("in"\|"<-") <delta-term> [ "." ] <exists> ::= "exists" <variable pair> ("in"\|"<-") <delta-term> [ "." ] <negated formula> ::= "not" <delta-formula> ##### Variables, constants, literals etc. <label> ::= <label variable> \| <label value> \| <label constant> <label variable> ::= <identifier> <label constant> ::= <identifier> <label value> ::= "’" <identifier> "’" <set variable> ::= <identifier> <set constant> ::= <identifier> <labelled terms> ::= <labelled term> ( "," <labelled term> )* <labelled term> ::= <label> ":" <delta-term> <variable pair> ::= <variable pair label> ":" <variable pair term> <variable pair label> ::= <label variable> \| <label value> <variable pair term> ::= <set variable> <identifier> ::= ( (A-Z) \| (a-z) \| (0-9) \| "_" \| "-" )+ #### A.2 Example XML-WDB files <?xml version="1.0"?> <set:eqns xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation= "http://www.csc.liv.ac.uk/~molyneux/XML-WDB/schema/xml-wdb.xsd" xmlns:set="http://www.csc.liv.ac.uk/~molyneux/XML-WDB"> <set:eqn set:id="BibDB"> <paper set:href= "http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p1"/> <paper set:href= "http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p2"/> <paper set:href= "http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p3"/> <book set:ref="b1"/> <book set:ref="b2"/> </set:eqn> <set:eqn set:id="b1"> <refers-to set:ref="b2"/> <refers-to set:href= "http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml#p1"/> </set:eqn> <set:eqn set:id="b2"> <author>Jones</author> <title>Databases</title> </set:eqn> </set:eqns> XML-WDB file 4 XML-WDB file http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f1.xml (cf. Section 3.5). <?xml version="1.0"?> <set:eqns xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation= "http://www.csc.liv.ac.uk/~molyneux/XML-WDB/schema/xml-wdb.xsd" xmlns:set="http://www.csc.liv.ac.uk/~molyneux/XML-WDB"> <set:eqn set:id="p1"> <refers-to set:ref="p2"/> </set:eqn> <set:eqn set:id="p2"> <author>Smith</author> <title>Databases</title> <refers-to set:ref="p3"/> </set:eqn> <set:eqn set:id="p3"> <author>Jones</author> <title>Databases</title> </set:eqn> </set:eqns> XML-WDB file 5 XML-WDB file http://www.csc.liv.ac.uk/~molyneux/t/BibDB-f2.xml (cf. Section 3.5). #### A.3 Predefined library queries set query Pair (set x,set y) be { ’fst’:x, ’snd’:y }, boolean query isPair (set p) be ( exists l: x in p . ( l=’fst’ and forall m:z in p . ( m=’fst’ => z=x ) ) and exists l:y in p . ( l=’snd’ and forall m:z in p .( m=’snd’ => z=y ) ) ), set query First (set p) be union separate { l:x in p where l=’fst’ }, set query Second (set p) be union separate { l:x in p where l=’snd’ }, set query CartProduct (set x,set y) be union collect { ’null’:collect { ’null’:call Pair ( xx, yy ) where l:yy in y } where m : xx in x }, set query Square (set z) be call CartProduct ( z, z ), set query LabelledPairs (set v) be collect { l:{ ’fst’:v, ’snd’:u } where l:u in v }, set query Nodes (set g) be union separate { m:p in g where call isPair ( p ) }, set query Children (set x,set g) be collect { l:call Second ( p ) where l:p in g and ( call isPair ( p ) and call First ( p ) = x ) }, set query Regroup (set g) be collect { ’null’:call Pair ( x, call Children ( x , g ) ) where m : x in call Nodes ( g ) }, set query CanGraph (set x) be union collect { ’null’:call LabelledPairs ( v ) where m:v in TC ( x ) }, set query Can (set x) be decorate ( call CanGraph ( x ), x ), set query TCPure(set x) be collect{ ’null’:v where l:v in TC ( x ) }, set query HorizontalTC (set g) be recursion p { ’null’:pair in call Square ( call Nodes ( g ) ) where ( call First ( pair ) = call Second ( pair ) or exists m:z in call Nodes ( g ) . ( ’null’:call Pair ( call First ( pair ), z ) in p and ’null’:call Pair ( z, call Second ( pair ) ) in g ) ) }, set query TC_along_label (label l,set z) be recursion p { k:x in TC ( z ) where ( ( x=z and k = ’null’ ) or ( k=l and exists m:y in p . l:x in y ) ) }, set query SuccessorPairs (set L) be separate { l:pair in L and not exists l:x in call Nodes(L) . ( ’null’:call Pair ( call First ( pair ),x ) in L and ’null’:call Pair ( x, call Second ( pair ) ) in L ) }, boolean query Precedes5(set R,label l,set x,label m,set y) be ( l < m or ( l=m and exists ’null’:p in R . ( ’fst’:x in p and ’snd’:y in p ) ) ), set query StrictLinOrder_on_TC (set z) be recursion R { ’null’:p_xy in call Square( call Can(call TCPure(z)) ) where ( ( not ’null’:p_xy in R and not exists ’fst’:xx in p_xy . exists ’snd’:yy in p_xy . exists ’null’:inv_p in R . ( ’fst’:yy in inv_p and ’snd’:xx in inv_p ) ) and exists ’snd’:yyy in p_xy . exists lu:u in yyy . ( exists ’fst’:xxx in p_xy . forall lv:v in xxx . ( call Precedes5(R,lu,u, lv,v) or call Precedes5(R,lv,v, lu,u) ) and forall fs:xy in p_xy . forall lw:w in xy . ( call Precedes5(R,lu,u, lw,w) => exists ’fst’:xxxx in p_xy . exists lp:p in xxxx . exists ’snd’:yyyy in p_xy . exists lq:q in yyyy . ( not call Precedes5(R,lp,p, lw,w) and not call Precedes5(R,lw,w, lp,p) and not call Precedes5(R,lq,q, lw,w) and not call Precedes5(R,lw,w, lq,q) ) ) ) ) } ### Bibliography * [1] Serge Abiteboul, Peter Buneman, and Dan Suciu. Data on the Web - From Relations to Semi-structured Data and XML. Morgan Kaufmann Publishers, San Francisco, CA, USA, 2000. * [2] Serge Abiteboul, Dallan Quass, Jason McHugh, Jennifer Widom, and Janet Lynn Wiener. The Lorel query language for semistructured data. International Journal on Digital Libraries, 1(1):68–88, 1997. * [3] Peter Aczel. Non-Well-Founded Sets. CSLI, Stanford, CA, USA, 1988. * [4] David Bacon. SETL for Internet Data Processing. PhD thesis, New York University, NY, USA, 2000. * [5] John Barwise and Lawrence Moss. Vicious circles: on the mathematics of non-well-founded phenomena. Center for the Study of Language and Information, 1996. * [6] Jon Barwise. Admissible Sets and Structures. Springer, Berlin, Germany, 1975. * [7] Véronique Benzaken, Giuseppe Castagna, and Alain Frisch. CDuce: an XML-centric general-purpose language. In Colin Runciman and Olin Shivers, editors, Proceedings of the Eighth ACM SIGPLAN International Conference on Functional Programming, ICFP 2003, Uppsala, Sweden, August 25-29, 2003, pages 51–63. ACM, 2003. * [8] Scott Boag, Donald Dean Chamberlin, Mary Fernández, Daniela Florescu, Jonathan Robie, and Jérôme Siméon. XQuery 1.0: An XML query language. W3C recommendation, W3C, January 2007. http://www.w3.org/TR/2007/REC-xquery-20070123/. * [9] Peter Buneman, Susan Davidson, Mary Fernández, and Dan Suciu. Adding structure to unstructured data. In ICDT ’97: Proceedings of the 6th International Conference on Database Theory, pages 336–350, London, UK, 1997. Springer-Verlag. * [10] Peter Buneman, Susan Davidson, Gerd Hillebrand, and Dan Suciu. A query language and optimization techniques for unstructured data. In SIGMOD ’96: Proceedings of the 1996 ACM SIGMOD international conference on Management of data, pages 505–516, Montreal, Quebec, Canada, 1996\. ACM. * [11] Peter Buneman, Mary Fernández, and Dan Suciu. UnQL: a query language and algebra for semistructured data based on structural recursion. The VLDB Journal, 9(1):76–110, 2000. * [12] Roderick Geoffrey Galton Cattell and Tom Atwood, editors. The Object Database Standard: ODMG-93. Series in Data Management Systems. Morgan Kaufmann, 1993. * [13] Stefano Ceri, Sara Comai, Ernesto Damiani, Piero Fraternali, Stefano Paraboschi, and Letizia Tanca. XML-GL: a graphical language for querying and restructuring XML documents. Computer Networks, 31(11-16):1171–1187, 1999. * [14] Donald Dean Chamberlin, Jonathan Robie, and Daniela Florescu. Quilt: An XML query language for heterogeneous data sources. In Dan Suciu and Gottfried Vossen, editors, The World Wide Web and Databases: Third International Workshop (WebDB 2000), Dallas, Texas, USA, May 18-19, 2000, Selected Papers, volume 1997 of Lecture Notes in Computer Science, pages 1–25. Springer-Verlag, 2001. * [15] James Clark. XSL transformations (XSLT) version 1.0. W3C recommendation, W3C, November 1999. http://www.w3.org/TR/1999/REC-xslt-19991116. * [16] Edgar Frank Codd. A relational model of data for large shared data banks. Communications of the ACM, 26(1):64–69, 1983. * [17] Thomas Connolly and Carolyn Begg. Database Systems: A Practical Approach to Design, Implementation and Management. Addison-Wesley Publishing Company, third edition, 2002. * [18] Mariano P. Consens and Alberto O. Mendelzon. GraphLog: a visual formalism for real life recursion. In Proceedings of the Ninth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, April 2-4, 1990, Nashville, Tennessee, pages 404–416. ACM Press, 1990. * [19] Agostino Cortesi, Agostino Dovier, Elisa Quintarelli, and Letizia Tanca. Operational and abstract semantics of the query language G-Log. Theoretical Computer Science, 275(1-2):521–560, 2002. * [20] Elias Dahlhaus and Johann Andreas Makowsky. The choice of programming primitives for SETL-like programming languages. In ESOP 86, European Symposium on Programming, Lecture Notes in Computer Science 213, pages 160–172. Springer, March 1986. * [21] Elias Dahlhaus and Johann Andreas Makowsky. Query languages for hierarchic databases. Information and Computation, 101:1–32, 1992. * [22] Steven DeRose and James Clark. XML path language (XPath) version 1.0. W3C recommendation, W3C, November 1999. http://www.w3.org/TR/1999/REC-xpath-19991116. * [23] Alin Deutsch, Mary Fernández, Daniela Florescu, Alon Levy, and Dan Suciu. A query language for XML. Computer Networks, 31(11-16):1155–1169, 1999. * [24] Agostino Dovier, Carla Piazza, and Alberto Policriti. An efficient algorithm for computing bisimulation equivalence. Theoretical Computer Science, 311(1-3):221–256, 2004. * [25] Jean-Claude Fernandez. An implementation of an efficient algorithm for bisimulation equivalence. Science of Computer Programming, 13(1):219–236, 1989. * [26] Mary Fernández, Daniela Florescu, Jaewoo Kang, Alon Levy, and Dan Suciu. Catching the boat with strudel: experiences with a web-site management system. In SIGMOD ’98: Proceedings of the 1998 ACM SIGMOD international conference on Management of data, pages 414–425, NY, USA, 1998. ACM. * [27] Mary Fernández, Daniela Florescu, Alon Levy, and Dan Suciu. A query language for a web-site management system. SIGMOD Record, 26(3):4–11, 1997. * [28] Marcelo Pablo Fiore, Achim Jung, Eugenio Moggi, Peter O’Hearn, Jon Riecke, Giuseppe Rosolini, and Ian Stark. Domains and denotational semantics: History, accomplishments and open problems. Bulletin of EATCS, 59:227–256, 1996. * [29] Marco Forti and Furio Honsell. Set theory with free construction principles. Annali Scuola Normale Superiore Pisa Classe di Scienza, 10(4):493–522, 1983. * [30] Robin Oliver Gandy. Set theoretic functions for elementary syntax. Proceedings of Symposia in Pure Mathematics, 13(2):103–126, 1974\. * [31] Hector Garcia-Molina, Yannis Papakonstantinou, Dallan Quass, Anand Rajaraman, Yehoshua Sagiv, Jeffrey Ullman, Vasilis Vassalos, and Jennifer Widom. The tsimmis approach to mediation: data models and languages. Journal of Intelligent Information Systems, 8(2):117–132, 1997\. * [32] Hector Garcia-Molina, Jeffrey David Ullman, and Jennifer Widom. Database Systems: The Complete Book. Prentice Hall Press, NJ, USA, 2008. * [33] Roy Goldman, Jason McHugh, and Jennifer Widom. From semistructured data to XML: Migrating the Lore data model and query language. In Sophie Cluet and Tova Milo, editors, Proceedings of the 2nd International Workshop on the Web and Databases (WebDB ’99), pages 25–30, Philadelphia, PA, USA, June 1999. * [34] Yuri Gurevich. Algebras of feasible functions. In In Proceedings of the 24th Annual Symposium on Foundations of Computer Science, pages 210–214. IEEE Computer Society Press, 1983. * [35] Vadim Guzev, Vladimir Sazonov, and Yuri Serdyuk. Distributed querying of web by using dynamically created mobile agents, http://www.csc.liv.ac.uk/~sazonov/papers/distributed_querying_of_web.p%df, 2002. Supported by an RDF grant from The University of Liverpool. * [36] Marc Gyssens, Jan Paredaens, Jan Van den Bussche, and Dirk Van Gucht. A graph-oriented object database model. IEEE Transactions on Knowledge Data Engineering, 6(4):572–586, August 1994. * [37] Neil Immerman. Relational queries computable in polynomial time. In Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, pages 147–152, San Francisco, CA, USA, May 1982. ACM. * [38] Neil Immerman. Descriptive Complexity. Texts in Computer Science. Springer-Verlag, 1999. * [39] Ronald Bjorn Jensen. The fine structure of the constructible hierarchy. Annals of Mathematics and Logic, 4:229–308, 1972. * [40] Alexander Leontjev and Vladimir Sazonov. $\Delta$: Set-theoretic query language capturing logspace. Annals of Mathematics and Artificial Intelligence, 33(2-4):309–345, 2001. * [41] Alexei Lisitsa and Vladimir Sazonov. Bounded hyperset theory and web-like data bases. In Proceedings of the Kurt Goedel Colloquium (KGC 1997), volume 1234, pages 178–188, 1997. * [42] Alexei Lisitsa and Vladimir Sazonov. $\Delta$-languages for sets and LOGSPACE computable graph transformers. Theoretical Computer Science, 175(1):183–222, 1997. * [43] Alexei Lisitsa and Vladimir Sazonov. Linear ordering on graphs, anti-founded sets and polynomial time computability. Theoretical Computer Science, 224(1–2):173–213, 1999. * [44] Alexei Lisitsa and Vladmir Sazonov. Bounded multi-hyperset theory and polynomial computability. Unpublished manuscript, 2007. * [45] Kenneth C. Louden. Compiler Construction: Principles and Practices. PWS Publishing Company/International Thomson Publishing, Boston, MA, USA, 1997. * [46] Jason McHugh, Serge Abiteboul, Roy Goldman, Dallan Quass, and Jennifer Widom. Lore: A database management system for semistructured data. SIGMOD Record, 26(3):54–66, 1997. * [47] Brett McLaughlin and Justin Edelson. Java and XML. O’Reilly Media, third edition, 2006. * [48] Robin Milner. A Calculus of Communicating Systems, volume 92 of Lecture Notes in Computer Science. Springer-Verlag, 1980. * [49] Richard Molyneux. Implementation of a hyperset $\Delta$-language as query language to web-like databases. Undergraduate dissertation, The University of Liverpool, May 2004. * [50] Richard Molyneux and Vladimir Sazonov. Hyperset/web-like databases and the experimental implementation of the query language delta - current state of affairs. In ICSOFT 2007 Proceedings of the Second International Conference on Software and Data Technologies, volume 3, pages 29–37. INSTICC, 2007. * [51] Yannis Papakonstantinou, Hector Garcia-Molina, and Jennifer Widom. Object exchange across heterogeneous information sources. In In Proceedings of the Eleventh International Conference on Data Engineering, pages 251–260, Taipei, Taiwan, 1995. * [52] Jan Paredaens, Paul De Bra, Marc Gyssens, and Dirk Van Gucht. The structure of the relational database model. Springer-Verlag New York, Inc., New York, NY, USA, 1989. * [53] David Park. Concurrency and automata on infinite sequences. In Proceedings of the 5th GI-Conference on Theoretical Computer Science, pages 167–183, London, UK, 1981. Springer-Verlag. * [54] Mark A. Roth, Herry F. Korth, and Abraham Silberschatz. Extended algebra and calculus for nested relational databases. ACM Transactions on Database Systems, 13(4):389–417, 1988. * [55] Vladimir Sazonov. Polynomial computability and recursivity in finite domains. Elektronische Informationsverarbeitung und Kybernetik, 16(7):319–323, 1980. * [56] Vladimir Sazonov. Bounded set theory, polynomial computability and $\Delta$-programming. In Lecture Notes in Computer Science, volume 278, pages 391–397, 1987. * [57] Vladimir Sazonov. Hereditarily-finite sets, data bases and polynomial-time computability. Theoretical Computer Science, 119(1):187–214, 1993. * [58] Vladimir Sazonov. A bounded set theory with anti-foundation axiom and inductive definability. In CSL ’94: Selected Papers from the 8th International Workshop on Computer Science Logic, pages 527–541, London, UK, 1995. Springer-Verlag. * [59] Vladimir Sazonov. On bounded set theory. In Invited talk on the 10th International, Congress on Logic, Methodology and Philosophy of Sciences, Florence, August 1995, in Volume I: Logic and Scientific Method, pages 85–103. Kluwer Academic Publishers, 1997\. * [60] Vladimir Sazonov. Using agents for concurrent querying of web-like databases via a hyperset-theoretic approach. In PSI ’02: 4th International Andrei Ershov Memorial Conference on Perspectives of System Informatics, pages 378–394, London, UK, 2001. Springer-Verlag. * [61] Vladimir Sazonov. Querying hyperset / web-like databases. Logic Journal of the IGPL, 14(5):785–814, 2006. * [62] J. T. Schwartz, Robert B. K. Dewar, E. Schonberg, and E. Dubinsky. Programming with sets: an introduction to SETL. Texts and Monographs in Computer Science. Springer-Verlag New York, Inc., New York, NY, USA, 1986. * [63] Jacob T. Schwartz. Set theory as a language for program specification and programming. Technical report, Courant Institute of Mathematical Sciences, New York University, NY, USA, 1970. * [64] Jacob T. Schwartz. On programming, an interim report on the setl project. Technical report, Courant Institute of Mathematical Sciences, New York University, NY, USA, 1973. * [65] Dana Stewart Scott and Christopher Strachey. Toward a mathematical semantics for computer languages. In Proceedings Symposium on Computers and Automata, volume 21 of Microwave Institute Symposia Series. Polytechnic Institute of Brooklyn, 1971. * [66] Yuri Serdyuk. Delta-language implementation, http://www.botik.ru/~logic/bst/delta_implementation.html, 1996. Supported by RBRF (project 96-01-01717) and INTAS (project 93-0972). * [67] Simon St.Laurent and Michael Fitzgerald. XML pocket reference. O’Reilly Media, third edition, 2005. * [68] Christopher Strachey. Fundamental concepts in programming languages. Lecture Notes, International Summer School in Computer Programming, Copenhagen, August 1967. Reprinted in Higher-Order and Symbolic Computation, 13(1–2), pp. 1–49, 2000. * [69] Dan Suciu. Typechecking for semistructured data. In Database Programming Languages, 8th International Workshop, DBPL 2001 Frascati, Italy, September 8 10, 2001 Revised Papers, volume 2397, pages 1–20, London, UK, 2002. Springer-Verlag. * [70] Robert Walker Taylor and Randall L. Frank. Codasyl data-base management systems. ACM Computing Survey, 8(1):67–103, 1976. * [71] Stan J. Thomas and Patrick C. Fischer. Nested relational structures. Advances in Computing Research, 3:269–307, 1986. * [72] Dennis Tsichritzis and Frederick Horst Lochovsky. Hierarchical data-base management: A survey. ACM Computing. Survey, 8(1):105–123, 1976. * [73] Jeffrey David Ullman. Principles of Database and Knowledge-Base Systems, volume 1 of Principles of Computer Science Series, 14. Computer Science Press, Rockville, MD, USA, 1988. * [74] Moshe Y. Vardi. The complexity of relational query languages. In Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, pages 137–146, San Francisco, CA, USA, 1982. ACM. * [75] Des Watson. High-Level Languages and their Compilers. Addison-Wesley Publishing Company, 1989. * [76] Reinhard Wilhelm and Dieter Maurer. Compiler Design. Addison-Wesley Publishing Company, 1995.	arxiv-papers	2009-02-15T18:53:19	2024-09-04T02:49:00.582218	{ "license": "Public Domain", "authors": "Richard Molyneux", "submitter": "Richard Molyneux", "url": "https://arxiv.org/abs/0902.2504" }
0902.2590	# The Case for Deep, Wide-Field Cosmology Ryan Scranton (UC Davis), Andreas Albrecht (UC Davis), Robert Caldwell (Darthmouth), Asantha Cooray (UC Irvine), Olivier Dore (CITA),Salman Habib (LANL), Alan Heavens (U. Edinburgh), Katrin Heitmann (LANL), Bhuvnesh Jain (U. Pennsylvania), Lloyd Knox (UC Davis), Jeffrey A. Newman (U. Pittsburgh), Paolo Serra (UC Irvine), Yong-Seon Song (U. Portsmouth), Michael Strauss (Princeton), Tony Tyson (UC Davis), Licia Verde (UAB & Princeton), Hu Zhan (UC Davis) The Case for Deep, Wide-Field Cosmology Ryan Scranton (UC Davis), Andreas Albrecht (UC Davis), Robert Caldwell (Dartmouth), Asantha Cooray (UC Irvine), Olivier Dore (CITA), Salman Habib (LANL), Alan Heavens (IfA Edinburgh), Katrin Heitmann (LANL), Bhuvnesh Jain (U. Pennsylvania), Lloyd Knox (UC Davis), Jeffrey A. Newman (U. Pittsburgh), Paolo Serra (UC Irvine), Yong-Seon Song (U. Portsmouth), Michael Strauss (Princeton), Tony Tyson (UC Davis), Licia Verde (UAB & Princeton), Hu Zhan (UC Davis) Contact: Ryan Scranton, Department of Physics, University of California, Davis, CA 95616 Contact: scranton@physics.ucdavis.edu, (530) 752-2012 Overview Much of the science case for the next generation of deep, wide-field optical/infrared surveys has been driven by the further study of dark energy. This is a laudable goal (and the subject of a companion white paper by Zhan et al.). However, one of the most important lessons of the current generation of surveys is that the interesting science questions at the end of the survey are quite different than they were when the surveys were being planned. The current surveys succeeded in this evolving terrain by being very general tools that could be applied to a number of very fundamental measurements. Likewise, the accessibility of the data enabled the broader cosmological and astronomical community to generate more science than the survey collaborations could alone. With that in mind, we should consider some of the basic physical and cosmological questions that surveys like LSST and JDEM-Wide will be able to address. * • With the level of precision available in these surveys, what can they tell us about fundamental physics? With the standard $\Lambda$CDM cosmology as determined by current surveys, we can use the precision available to next generation surveys to examine the foundations of particle physics and gravity. Is the current model of general relativity (GR) correct or are the effects that we have ascribed to the presence of dark energy actually a signal that GR is broken in some way? What can cosmology do to constrain extensions to the Standard Model of particle physics? * • What can a deep, wide-field survey tell us about the basic assumptions behind the standard cosmology? Now that the current generation of surveys have given us a stronger grasp on the basic cosmological model, we can begin to question its fundamental assumptions. Does the cosmological principle of isotropy and homogeneity hold true? Are the primordial perturbations that seeded structure formation Gaussian? Do we know enough about the intergalactic medium to trust measurements of background sources seen through foreground structure? * • What are the technical challenges to making these future surveys productive for the larger cosmological and astronomical community? Maximizing the science from these surveys will mean delving into the non-linear regime for many measurements and the data size and complexity will be considerably more daunting than current surveys. What improvements will need to be made to simulations to properly characterize these data sets? How will that analysis change when even the catalog data from these surveys is too large to transmit over the network? ## Physics Beyond the Standard Model ### Modifying General Relativity There is a possibility that the observed cosmic acceleration results from a new theory of gravity at cosmological length scales. While a compelling underlying theory is still lacking in the community, we can consider constraints on General Relativity (GR). The model-independent lingua franca is the relationship between the Newtonian ($\psi$) and longitudinal ($\phi$) gravitational potentials. The potentials, as defined through the perturbed Robertson-Walker metric $ds^{2}=a^{2}[-(1+2\psi)d\tau^{2}+(1-2\phi)d\vec{x}^{2}],$ (1) are most familiar for their roles in Newton’s equation, $\ddot{\vec{x}}=-\vec{\nabla}\psi$, and the Poisson equation, $\nabla^{2}\phi=4\pi Ga^{2}\rho_{m}\delta_{m}$, under GR. Figure 1: The projected 68% and 95% likelihood contours in the $\varpi_{0}-\Omega_{m}$ parameter space are shown. The blue contours are based on current WMAP 5-year CMB data alone. The red contours add current weak lensing and ISW-galaxy correlation data. The yellow contours are based on mock Planck data. The green contours add mock weak lensing data of the type expected for a 20,000 deg2 survey. The underlying model is assumed to be $\varpi_{0}=0$ with $\Omega_{m}=0.26$. The gravitational potentials are equal in the presence of non-relativistic stress-energy under GR, but alternate theories of gravity make no such guarantee and a slip between the two is expected such that $\phi\neq\psi$ in the presence of non-relativistic stress-energy. A possible parametrized-post- Friedmannian (PPF) description of this departure is the one discussed in [7] with $\psi=[1+\varpi(z)]\phi,~{}~{}\varpi(z)=\varpi_{0}(1+z)^{-3}.$ (2) The CMB probes primordial perturbations, while at late times ISW is a function of $\dot{\phi}+\dot{\psi}$ and weak lensing the sum $\phi+\psi$. Thus cosmological observations that combine CMB anisotropies with LSS data such as weak lensing can separate the $\phi$ and $\psi$ and put constraints on the PPF framework. In Figure 1, we show a summary of results comparing present-day constraints to those possible with Planck \+ LSST. In the latter case, it should be possible to determine $\varpi_{0}$ to within 10% at the 95% confidence level. Figure 2: The web of interconnected GR consistency tests. Alternatively, one could examine departures from GR in a model-independent way using consistency relations[23]. As seen in Figure 2, there are four fundamental equations governing the relationships between the energy and momentum perturbations ($\delta_{m}$ & $\Theta_{m}$, respectively) and the metric perturbations from Equation 1. From this basic set, we can form pairs of estimators, predicting the result of a measurement drawing from one side of the equation from another based on the opposite side. If these relations were found to be inconsistent, it would be a clear signal of a breakdown in GR. This sort of test is not prescriptive in the same way as the PPF treatment, but it is sensitive to any range of departures from standard GR. As an example, consider the Poisson equation given above. The left side of the equation is a function of the metric perturbation $\phi$. Weak gravitational lensing is generated by the gradient of $\phi$, making it a direct probe of those perturbations. For the right side of the equation, we need an estimator sensitive to $\delta_{m}$. This can be found directly from the pair-wise velocity dispersion, which generally requires a redshift survey. In the absence of such a survey to the depth of LSST or JDEM-Wide, we can obtain a similar quantity by cross-correlating the induced lensing shear with the projected galaxy density. There are potential complications due to non- linearities, but at large scales the combination of these two measurements gives us an estimator for deviations from the Poisson equation that should be detectable at the few percent level with these future surveys[23]. This approach remains model independent and does not rely on any specific parametrization, so it would apply just as readily to any theory for modified gravity that altered the Poisson equation. ### Massive Neutrinos Figure 3: Forecasted constraints in the context of what is known today from neutrino oscillations experiments. The narrow green band represents the normal hierarchy and the red band the inverted one. The light blue regions represent the $1-\sigma$ constraints for the combination Planck+LSST for the two fiducial models (massive and near-massless neutrinos) discussed in the text. The darker band shows the forecasted $1-\sigma$ constraint obtained in the context of a power-law $P(k)$, $\Lambda CDM$ \+ massive neutrinos model. (Figure courtesy of E. Fernandez-Martinez) The primary tool for constraining massive neutrinos with a large scale structure survey is measurement of the 3D cosmic shear (cf. [11]); the mass of the neutrinos can be inferred based on the suppression of growth in the matter power spectrum inferred from the cosmic shear. There is a degeneracy between this effect and dark energy parameters[10], which can be characterized using a Fisher matrix approach with a prior based on the expected results from the Planck CMB experiment. The following constraints[13] are obtained allowing for non-zero curvature and for a dark energy component with equation of state parameterization given by $w_{0},w_{a}$; all results on individual parameters are fully marginalized over all other cosmological parameters. By combining 3D cosmic shear constraints with Planck’s, the massive neutrino (fiducial values $m_{\nu}=0.66$eV ; $N_{\nu}=3$) parameters could be measured with marginal errors of $\Delta m_{\nu}\sim 0.03$ eV and $\Delta N_{\nu}\sim 0.08$, a factor of 4 improvement over Planck alone. If neutrinos are massless or have a very small mass (fiducial model $m_{\nu}=0$eV ; $N_{\nu}=3$) the marginal errors on these parameters degrade ($\Delta m_{\nu}\sim 0.07$ eV and $\Delta N_{\nu}\sim 0.1$), but remain an equal improvment over Planck alone. This degradation in the marginal error occurs because the effect of massive neutrinos on the matter power spectrum and hence on 3D weak lensing is non- linear. These findings are in good agreement with an independent analysis[8] and should not degrade by more than a factor of $\sqrt{2}$ due to systematic errors[12, 13]. Alternatively, the constraints could improve by as much as a factor of 2 if complementary data sets were used to break the degeneracies between $m_{\nu}$ and the running of the spectral index, $w_{a}$ and $w_{0}$[14]. Figure 3 shows these constraints in the context of what is known currently from neutrino oscillations experiments. Particle physics experiments which will be completed by the time LSST will start producing results do not guarantee a determination of the neutrino mass $m_{tot}$ if it is below $0.2$ eV. Neutrino-less double beta decay experiments will be able to constrain neutrino masses only if the hierarchy is inverted and neutrinos are Majorana particles. On the other hand, oscillations experiments will determine the hierarchy only if the the composition of electron flavor in all the neutrino mass states is large. Cosmological observations are sensitive to the sum of neutrino masses, offering the possibility to distinguish between normal and inverted hierarchy. Thus, this data set combination could offer valuable constraints on neutrino properties, highly complementary to particle physics parameters like $\theta_{13}$. These constraints can also be considered in terms of Bayesian evidence[13]. As introduced in the companion “dark energy” white paper (see references therein), the Bayesian factor is a prediction of an experiment’s ability to distinguish one model from another. The combination of Planck+LSST could provide strong evidence for massive neutrinos over models in which there are no massive neutrinos, and, if the neutrino mass is small $\delta m_{\nu}<0.1$ eV, there will be substantial evidence for these models. One could also decisively distinguish between models in which there are no massive neutrinos and those in which $N_{\nu}<3.00-0.40$ or $N_{\nu}>3.00+0.40$ and $m_{\nu}>0.25$ eV. ## Testing Cosmological Assumptions ### Universal Isotropy Figure 4: Detectable deviation between LSST measurements of dark energy parameter $w_{p}$ and error product as a function of the number of patches. While testing the homogeneity of the universe remains a very difficult task[16], a wide, deep survey like LSST or space-based mission with equivalent area would be in a prime position to check universal isotropy, specifically the isotropy of dark energy. There are two potential approaches: trying to measure the projected dark energy density quadrupole over the survey area or looking for variation in dark energy parameters in different patches of the sky. For the former, one could calculate the angular power spectrum of the luminosity fluctuations for the million SNe expected to be observed by LSST[5]. At large angles, this power spectrum would be sensitive to the projected inhomogenieties in the dark energy density. For an LSST-like survey, the quadrupole moment ($l=2$) of this measurement would be able to detect fractional dark energy density fluctuations as small as $2\times 10^{-4}$. Alternatively, one could take a divide-and-conquer approach: dividing the total survey area into a number of separate patches and measuring the scatter in dark energy parameters measured via weak-lensing (WL) and baryon acoustic oscilations (BAO) in each section. The expected results for such a test using LSST are shown in Figure 4, where $w_{a}$ is the linearly evolving dark energy EOS and $w_{p}$ is the EOS orthogonal to $w_{a}$. The constraints are marginalized over 9 other cosmological parameters including the curvature and over 140 parameters that model the linear galaxy clustering bias, photometric redshift bias, rms photometric redshift error and additive & multiplicative errors on the power spectrum[24]. Such a measurement should be able to constrain the product $\sigma(w_{a})\times\sigma(w_{p})$ to $<$ 0.04% in $<10$ patches over the sky. ### Primordial Perturbations One of the core predictions of inflationary cosmology is that the initial perturbations that seeded structure formation have a nearly Gaussian distribution. Measuring the deviation from this non-Gaussianity can provide us with strong clues as to the flavor of inflationary model that drove the expansion of the very early universe. In particular, curvaton or multi-field inflationary models can produce large values of $f_{NL}$, a parameter commonly used to describe the magnitude of the non-Gaussian contribution to the perturbations: $\Phi=\phi+f_{NL}(\phi^{2}-\langle\phi^{2}\rangle)$. Recently, it has been shown[6, 18] that primordial non-Gaussianity affects the clustering of dark matter halos, inducing a scale-dependent bias. This is in addition to the contribution to the standard halo bias arising even for Gaussian initial conditions. In this case, the non-Gaussian correction ($\Delta b^{f_{NL}}$) to the standard halo bias increases as $\sim 1/k^{2}$ at large scales and evolves over time as $\sim(1+z)$. This is detectable for a survey like LSST or JDEM-Wide through measurements of the galaxy power spectrum at large scales. This is a smooth feature on the power spectrum, so large photometric surveys are particularly well suited to study the effect. LSST should be able to detect even a value of $f_{NL}\lesssim 1$ at $1\sigma$[3]. While this error could be in principle reduced further if cosmic variance could be reduced (cf. [22, 21]) this limit of $\Delta f_{NL}\lesssim 1$ is particularly interesting for two reasons. First, it is comparable if not better than the limit achievable from an ideal CMB experiment, making this approach highly complementary with the CMB approach. Second, many well- motivated inflationary models yield $f_{NL}$ well above this threshold. Detecting $f_{NL}$ at this level of precision will be a critical test for these models. ### Universal Transparency Recent work[19] has revealed that the amount of dust in the intergalactic medium is roughly twice that of previous estimates. While the dust content of the universe remains small by mass ($\Omega_{dust}\sim 10^{-5}$), the physical extent of the dust around galaxies was found to far exceed that of the visible light, stretching to scales beyond 100 $h^{-1}$kpc. Preliminary calculations[20] also indicate that the extinction is large enough to bias cosmological parameter estimates from the $\sim 300$ “Union” supernovae[17], moving the values for $\Omega_{\rm M}$, $\Omega_{\rm B}$ & $w$ by $\sim 0.5\sigma$. With the next generation of wide, deep surveys, we should be able to make significant strides in understanding the nature and distribution of this intergalactic dust. One obvious motivation to do so would be to prevent it from acting as a significant source of systematic error on supernova magnitudes used as standard candles. Beyond its role as a source of error, however, detecting dust on these scales represents an intriguing glimpse into the history of star formation in and around galaxies. Current models for dust generation vary in their conclusions about how extended dust halos should be and how the halo is generated (in situ, as a result of dust outflows, galaxy interactions and so on). Likewise, the current measurements at SDSS wavelengths are unable to make any conclusions about the chemical composition of the dust or how the opacity of the universe has evolved, which would be a key indicator of whether the dust was generated by on-going processes or if it was a relic of the earliest days of star formation. By extending this measurement to higher redshifts and increasing the sensitivity, we should gain considerable insight into the star formation history of galaxies across a wide range of environments, types and luminosities as well as understanding more about the intergalactic medium. ## Data Challenges ### Next Generation Simulations In order to extract signatures of new physics beyond the Standard Model as detailed in the previous sections, a next-generation simulation and modeling capability is essential. Currently, all observations are described within the $\Lambda$CDM model at 10% error. The signatures of new physics will be subtle and to extract them from upcoming observations, the corresponding theoretical predictions must be obtained at unprecedented accuracies. The state of the art in modeling and simulation must improve by at least an order of magnitude in order to match the precision of the observations. Improvements are necessary in three areas. First, the dynamic range of the simulations has to increase – larger volumes and higher force and mass resolution are needed. The next-generation surveys will cover enormous volumes that the simulations must capture along with all the halos hosting galaxies within. To model a survey such as the LSST one would like to cover a (3Gpc)3 volume. To match the mass resolution of the “Millennium” run with a particle mass of $\sim 10^{9}M_{\circ}$ would require a trillion particle simulation. This will be possible on next-generation petaflop supercomputers, but will require major rewriting of current cosmology codes and a new paradigm for analyzing the large data volume (petabytes) that will be produced. First efforts in this direction are already underway[9]. Second, we have to include cosmological new physics in the simulations and extract its signatures on the large-scale distribution of galaxies. Precision is again key, as numerical errors can easily mimic effects at the several percent level. The simulations will be extremely important to help distinguish the detection of new and unexpected physics from systematic errors. They will also serve in their traditional role as a testbed for new ideas. And finally, we have to improve the treatment of gas physics and feedback effects. Currently, such treatments are accurate at most at the 10 - 20 % level. Here the key issue is not so much accuracy as fidelity. There are still astrophysical effects that remain to be properly understood and incorporated in the simulations. Such effects will be extremely important if we start beginning to explore smaller and smaller scales; extracting cosmological information from the non-linear regime from galaxy clustering, for example. Because these effects may never be incorporated at a first-principles level, it is imperative to develop a phenomenological approach that appropriately combines simulations with observations. At the same time we have to improve semi-analytic modeling as an attractive alternative to a full simulation. ### Data Size & Complexity As mentioned in the overview, one of the keys to the success of the current generation of cosmological surveys was their use by members of the astronomical community outside of the survey collaborations themselves. This brought in astronomers with a wider range of interests and skills and began a process of deep data mining that will continue for the next several years. For surveys like LSST and JDEM-Wide, this degree of access will be complicated by the sheer volume of the data involved (tens of petabytes for LSST) and the increase in complexity for both surveys. Both of these factors will push astronomical data analysis away from the current model where data is downloaded and processed through custom software packages like IRAF or IDL. Instead, these surveys will need to adopt a “cloud computing model”, creating a work environment at the survey data centers where astronomers can query and analyze the data remotely, downloading only the results of the job rather than the raw data. ## Conclusions Building the next generation of deep, wide-field surveys will profoundly increase our knowledge about the universe. They will yield not only a better insight into the nature of dark energy, but also allow us to examine physics on an incredible range of scales, from gigaparsec to sub-atomic. LSST and JDEM-Wide will test fundamental cosmological and physical models with unprecedented precision, probing the foundations of the theories that inform modern astrophysics. The technical challenges of turning these data sets into science are formidable, but surmountable, and the resulting insights into cosmology and fundamental physics will be well worth the effort. Further, the wide net cast over the skies by these surveys will serve as an invaluable resource for the broader astronomical community, driving advances in galaxy and stellar science as well as variability studies and solar system science. ## * [1] * [2] * Carbone et al. [2008] Carbone, C., Verde, L., & Matarrese, S. 2008, ApJ, 684, L1 * Castro et al. [2006] Castro, P. G.; Heavens, A. F.; Kitching, T. D.; 2005, PhRvD, 72, 3516 * Cooray et al. [2008] Cooray, A., Holz, D. E., & Caldwell, R. 2008, arXiv:0812.0376 * Dalal et al. [2008] Dalal, N., Doré, O., Huterer, D., & Shirokov, A. 2008, Phys. Rev. D, 77, 123514 * Daniel et al. [2009] Daniel, S. F., Caldwell, R. R., Cooray, A., Serra, P., & Melchiorri, A. 2009, arXiv:0901.0919 * Hannestad et al. [2006] Hannestad S.; Tu H.; Wong Y.; 2006, JCAP 0606, 025 * Heitmann et al. [2008] Heitmann, K., White, M., Wagner, C., Habib, S., & Higdon, D. 2008, arXiv:0812.1052 * Kiakotou et al. [2008] Kiakotou, A., Elgaroy, O., & Lahav, O. 2008, PRD, 77, 063005 * Kitching et al. [2007] Kitching, T. D.; Heavens, A. F.; Taylor, A. N.; Brown, M. L.; Meisenheimer, K.; Wolf, C.; Gray, M. E.; Bacon, D. J.; 2007, MNRAS, 376, 771 * Kitching et al. [2008a] Kitching, T. D.; Taylor, A. N.; Heavens, A. F.; 2008a, MNRAS, 389, 173 * Kitching et al. [2008] Kitching, T. D.; Heavens, A. F.; Verde, L.; Serra P.;Melchiorri A., 2008, PRD, 77, 10, 103008 * [14] T. Kitching, private communication * Koivisto & Mota [2008] Koivisto, T., & Mota, D. F. 2008, Journal of Cosmology and Astro-Particle Physics, 6, 18 * Kolb et al. [2005] Kolb, E. W., Matarrese, S., Notari, A., & Riotto, A. 2005, Phys. Rev. D, 71, 023524 * Kowalski et al. [2008] Kowalski, M., et al. 2008, ApJ, 686, 749 * Matarrese & Verde [2008] Matarrese, S., & Verde, L. 2008, ApJ, 677, L77 * Menard et al. [2009] Menard, B., Scranton, R., Fukugita, M., Richards, G., 2009, in preparation * Menard et al. [2009] Menard, B., Kilbinger, M., Scranton, R., 2009, in preparation * Seljak [2008] Seljak, U., eprint arXiv:0807.1770. * Slosar [2008] Slosar, A. 2008, arXiv:0808.0044 * Song and Dore [2008] Song, Y.-S., Doré, O., JCAP 1208 039 * Zhan [2006] Zhan, H. 2006, JCAP, 8, 8	arxiv-papers	2009-02-16T01:20:44	2024-09-04T02:49:00.607701	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ryan Scranton (UC Davis), Andreas Albrecht (UC Davis), Robert Caldwell\n (Dartmouth), Asantha Cooray (UC Irvine), Olivier Dore (CITA), Salman Habib\n (LANL), Alan Heavens (IfA Edinburgh), Katrin Heitmann (LANL), Bhuvnesh Jain\n (U. Pennsylvania), Lloyd Knox (UC Davis), Jeffrey A. Newman (U. Pittsburgh),\n Paolo Serra (UC Irvine), Yong-Seon Song (U. Portsmouth), Michael Strauss\n (Princeton), Tony Tyson (UC Davis), Licia Verde (UAB & Princeton), Hu Zhan\n (UC Davis)", "submitter": "Ryan Scranton", "url": "https://arxiv.org/abs/0902.2590" }
0902.2674	2010395-404Nancy, France 395 Lance Fortnow Jack H. Lutz Elvira Mayordomo # Inseparability and Strong Hypotheses for Disjoint NP Pairs L. Fortnow Northwestern University, EECS Department, Evanston, Illinois, USA. fortnow@eecs.northwestern.edu , J. H. Lutz Department of Computer Science, Iowa State University, Ames, IA 50011 USA. lutz@cs.iastate.edu and E. Mayordomo Departamento de Informática e Ingeniería de Sistemas, Instituto de Investigación en Ingeniería de Aragón, María de Luna 1, Universidad de Zaragoza, 50018 Zaragoza, SPAIN. elvira at unizar.es ###### Abstract. This paper investigates the existence of inseparable disjoint pairs of NP languages and related strong hypotheses in computational complexity. Our main theorem says that, if NP does not have measure 0 in EXP, then there exist disjoint pairs of NP languages that are P-inseparable, in fact TIME(2(n k))-inseparable. We also relate these conditions to strong hypotheses concerning randomness and genericity of disjoint pairs. ###### Key words and phrases: Computational Complexity, Disjoint NP-pairs, Resource-Bounded Measure, Genericity ###### 1991 Mathematics Subject Classification: F.1.3 Thanks: Fortnow’s research supported in part by NSF grants CCF-0829754 and DMS-0652521. Lutz’s research supported in part by National Science Foundation Grants 0344187, 0652569, and 0728806. Mayordomo’s research supported in part by Spanish Government MICINN Project TIN2008-06582-C03-02. ## 1\. Introduction The main objective of complexity theory is to assess the intrinsic difficulties of naturally arising computational problems. It is often the case that a problem of interest can be formulated as a decision problem, or else associated with a decision problem of the same complexity, so much of complexity theory is focused on decision problems. Nevertheless, other types of problems also require investigation. This paper concerns promise problems, a natural generalization of decision problems introduced by Even, Selman, and Yacobi [7]. A decision problem can be formulated as a set $A\subseteq\\{0,1\\}^{}$, where a solution of this problem is an algorithm, circuit, or other device that decides $A$, i.e., tells whether or not an arbitrary input $x\in\\{0,1\\}^{}$ is an element of $A$. In contrast, a promise problem is formulated as an ordered pair $(A,B)$ of disjoint sets $A,B\subseteq\\{0,1\\}^{}$, where a solution is an algorithm or other device that decides any set $S\subseteq\\{0,1\\}^{}$ such that $A\subseteq S$ and $B\cap S=\emptyset$. Such a set $S$ is called a separator of the disjoint pair $(A,B)$. Intuitively, if we are promised that every input will be an element of $A\cup B$, then a separator of $(A,B)$ enables us to distinguish inputs in $A$ from inputs in $B$. Since each decision problem $A$ is clearly equivalent to the promise problem $(A,A^{c})$, where $A^{c}=\\{0,1\\}^{}-A$ is the complement of $A$, promise problems are, indeed, a generalization of decision problems. A disjoint NP pair is a promise problem $(A,B)$ in which $A,B\in{\mathrm{NP}}$. Disjoint NP pairs were first investigated by Selman and others in connection with public key cryptosystems [7, 15, 26, 17]. They were later investigated by Razborov [25] as a setting in which to prove the independence of complexity-theoretic conjectures from theories of bounded arithmetic. In this same paper, Razborov established a fundamental connection between disjoint NP pairs and propositional proof systems. Propositional proof systems had been used by Cook and Reckhow [6] to characterize the NP versus co-NP problem. Razborov [25] showed that each propositional proof system has associated with it a canonical disjoint NP pair and that important questions about propositional proof systems are thereby closely related to natural questions about disjoint NP pairs. This connection with propositional proof systems has motivated more recent work on disjoint NP pairs by Glaßer, Selman, Sengupta, and Zhang [10, 9, 12, 13]. It is now known that the degree structure of propositional proof systems under the natural notion of proof simulation is identical to the degree structure of disjoint NP pairs under reducibility of separators [12]. Much of this recent work is surveyed in [11]. Goldreich [14] gives a recent survey of promise problems in general. Our specific interest in this paper is the existence of disjoint NP pairs that are P-inseparable, or even ${\mathrm{TIME}(2^{n^{k}})}$-inseparable. As the terminology suggests, if ${\mathcal{C}}$ is a class of decision problems, then a disjoint pair is ${\mathcal{C}}$-inseparable if it has no separator in ${\mathcal{C}}$. The existence of P-inseparable disjoint NP pairs is a strong hypothesis in the sense that (1) it clearly implies ${\mathrm{P}}\neq{\mathrm{NP}}$, and (2) the converse implication is not known (and fails relative to some oracles [17]). It is clear that ${\mathrm{P}}\neq{\mathrm{NP}}\cap\mathrm{co}{\mathrm{NP}}$ implies the existence of P-inseparable disjoint NP pairs, and Grollmann and Selman [15] proved that ${\mathrm{P}}\neq\mathrm{UP}$ also implies the existence of P-inseparable disjoint NP pairs. The hypothesis that NP is a non-measure 0 subset of EXP, written $\mu({\mathrm{NP}}\mid{\rm EXP})\neq 0$, is a strong hypothesis in the above sense. This hypothesis has been shown to have many consequences not known to follow from more traditional hypotheses such as ${\mathrm{P}}\neq{\mathrm{NP}}$ or the separation of the polynomial-time hierarchy into infinitely many levels. Each of these known consequences has resolved some pre-existing complexity-theoretic question in the way that agreed with the conjecture of most experts. This explanatory power of the $\mu({\mathrm{NP}}\mid{\rm EXP})\neq 0$ hypothesis is discussed in the early survey papers [23, 2, 24] and is further substantiated by more recent papers listed at [16] (and too numerous to discuss here). In several instances, the discovery that $\mu({\mathrm{NP}}\mid{\rm EXP})\neq 0$ implies some plausible conclusion has led to subsequent work deriving the same conclusion from some weaker hypothesis, thereby further illuminating the relationships among strong hypotheses. Our main theorem states that, if NP does not have measure zero in EXP, then, for every positive integer $k$, there exist disjoint NP pairs that are ${\mathrm{TIME}(2^{n^{k}})}$-inseparable. Such pairs are a fortiori P-inseparable, but the conclusion of our main theorem actually gives exponential lower bounds on the inseparability of some disjoint NP pairs. These are the lower bounds that most experts conjecture to be true, even though an unconditional proof of such bounds may be long in coming. The proof of our main theorem combines known closure properties of NP with the randomness that the $\mu({\mathrm{NP}}\mid{\rm EXP})\neq 0$ hypothesis implies must be present in NP to give an explicit construction of a disjoint NP pair that is ${\mathrm{TIME}(2^{n^{k}})}$-inseparable. (Technically, this is an overstatement. The last step of the “construction” is the removal of a finite set whose existence we prove, but which we do not construct.) The details are perhaps involved, but we preface the proof with an intuitive motivation for the approach. We also investigate the relationships between the two strong hypotheses in our main theorem (i.e., its hypothesis and its conclusion) and strong hypotheses involving the existence of disjoint NP pairs with randomness and genericity properties. Roughly speaking (i.e., omitting quantitative parameters), we show that the existence of disjoint NP pairs that are random implies both the $\mu({\mathrm{NP}}\mid{\rm EXP})\neq 0$ hypothesis and the existence of disjoint NP pairs that are generic in the sense of Ambos-Spies, Fleischhack, and Huwig [1]. We also show that the existence of such generic pairs implies the existence of disjoint NP pairs that are ${\mathrm{TIME}(2^{n^{k}})}$-inseparable. Taken together, these results give the four implications at the top of Figure 1. (The four implications at the bottom are well known.) We prove that three of these implications cannot be reversed by relativizable techniques, and we conjecture that this also holds for the remaining implication. ## 2\. Preliminaries We write $\mathbb{N}$ for the set of nonnegative integers and $\mathbb{Z}^{+}$ for the set of (strictly) positive integers. The Boolean value of an assertion $\phi$ is $[\\![\phi]\\!]=$ if $\phi$ then 1 else 0. All logarithms here are base-2. We write $\lambda$ for the empty string, $\|w\|$ for the length of a string $w$, and $s_{0},s_{1},s_{2},\ldots$ for the standard enumeration of $\\{0,1\\}^{}$. The index of a string $x$ is the value $\mathrm{ind}(x)\in\mathbb{N}$ such that $s_{\mathrm{ind}(x)}=x$. We write $\mathrm{next}(x)$ for the string following $x$ in the standard enumeration, i.e., $\mathrm{next}(s_{n})=s_{n+1}$. More generally, for $k\in\mathbb{N}$, we write $\mathrm{next}^{k}$ for the $k$-fold composition of next with itself, so that $\mathrm{next}^{k}(s_{n})=s_{n+k}$. A Boolean function is a function $f:\left\\{0,1\right\\}^{m}\to\\{0,1\\}$ for some $m\in\mathbb{N}$. The support of such a function $f$ is $\mathrm{supp}(f)=\left\\{{x\in\left\\{0,1\right\\}^{m}}\>\Big{\|}\>{f(x)=1}\right\\}$. We write $w[i]$ for the $i^{\mathrm{th}}$ symbol in a string $w$ and $w[i..j]$ for the string consisting of the $i^{\mathrm{th}}$ through $j^{\mathrm{th}}$ symbols. The leftmost symbol of $w$ is $w[0]$, so that $w=w[0..\|w\|-1]$. For (infinite) sequences $S\in\Sigma^{\infty}$, the notations $S[i]$ and $S[i..j]$ are defined similarly. A string $w\in\Sigma^{}$ is a prefix of a string or sequence $x\in\Sigma^{}\cup\Sigma^{\infty}$, and we write $w\sqsubseteq x$, if there is a string or sequence $y\in\Sigma^{}\cup\Sigma^{\infty}$ such that $wy=x$. A language, or decision problem, is a set $A\subseteq\\{0,1\\}^{}$. We identify each language $A$ with the sequence $A\in\\{0,1\\}^{\infty}$ defined by $A[n]=[\\![s_{n}\in A]\\!]$ for all $n\in\mathbb{N}$. If $A$ is a language, then expressions like $\lim_{w\to A}f(w)$ refer to prefixes $w\sqsubseteq A$, e.g., $\lim_{w\to A}f(w)=\lim_{n\to\infty}f(A[0..n-1])$. A martingale is a function $d:\\{0,1\\}^{}\to[0,\infty)$ satisfying $d(w)=\frac{d(w0)+d(w1)}{2}$ (2.1) for all $w\in\\{0,1\\}^{}$. Intuitively, $d$ is a strategy for betting on the successive bits of a sequence $S\in\\{0,1\\}^{\infty}$: The quantity $d(w)$ is the amount of money that the gambler using this strategy has after $\|w\|$ bets if $w\sqsubseteq S$. Condition (2.1) says that the payoffs are fair. A martingale $d$ succeeds on a language $A\subseteq\\{0,1\\}^{}$, and we write $A\in S^{\infty}[d]$, if $\limsup_{w\to A}d(w)=\infty$. If $t:\mathbb{N}\to\mathbb{N}$, then a martingale $d$ is (exactly) $t(n)$-computable if its values are rational and there is an algorithm that computes each $d(w)$ in $t(\|w\|)$ time. A martingale is p-computable if it is $n^{k}$-computable for some $k\in\mathbb{N}$, and it is ${{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}$-computable if it is $2^{(\log n)^{k}}$-computable for some $k\in\mathbb{N}$. ###### Definition 2.1. [22] Let $X$ be a set of languages, and let $R$ be a language. 1. (1) $X$ has p-measure 0, and we write $\mu_{\mathrm{p}}(X)=0$, if there is a p-computable martingale $d$ such that $X\subseteq S^{\infty}[d]$. The condition $\mu_{{{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}}(X)=0$ is defined analogously. 2. (2) $X$ has measure 0 in EXP, and we write $\mu(X\mid{\rm EXP})=0$, if $\mu_{{{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}}(X\cap{\rm EXP})=0$. 3. (3) $R$ is p-random if $\mu_{\mathrm{p}}(\\{R\\})\neq 0$, i.e., if there is no p-computable martingale that succeeds on $R$. Similarly, $R$ is $t(n)$-random if no $t(n)$\- computable martingale succeeds on $R$. It is well known that these definitions impose a nontrivial measure structure on EXP [22]. For example, $\mu({\rm EXP}\mid{\rm EXP})\neq 0$. We use the following fact in our arguments. ###### Lemma 2.2. [3, 18] The following five conditions are equivalent. 1. (1) $\mu({\mathrm{NP}}\mid{\rm EXP})\neq 0$. 2. (2) $\mu_{\mathrm{p}}({\mathrm{NP}})\neq 0$. 3. (3) $\mu_{{{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}}({\mathrm{NP}})\neq 0$. 4. (4) There exists a p-random language $R\in{\mathrm{NP}}$. 5. (5) For every $k\geq 2$, there exists an $2^{\log n^{k}}$-random language $R\in{\mathrm{NP}}$. Finally, we note that $\mu({\mathrm{P}}\mid{\rm EXP})=0$ [22], so $\mu({\mathrm{NP}}\mid{\rm EXP})\neq 0$ implies ${\mathrm{P}}\neq{\mathrm{NP}}$. ## 3\. Inseparable Disjoint NP Pairs and the Measure of NP This section presents our main theorem, which says that, if NP does not have measure 0 in EXP, then there are disjoint NP pairs that are P-inseparable. In fact, for each $k\in\mathbb{N}$, there is a disjoint NP pair that is ${\mathrm{TIME}(2^{n^{k}})}$-inseparable. It is convenient for our arguments to use a slight variant of the separability notion. ###### Definition 3.1. Let $(A,B)$ be a pair of (not necessarily disjoint) languages, and let ${\mathcal{C}}$ be a class of languages. 1. (1) A language $S\subseteq\\{0,1\\}^{}$ almost separates $(A,B)$ if there is a finite set $D\subseteq\\{0,1\\}^{}$ such that $S$ separates $(A-D,B-D)$. 2. (2) We say that $(A,B)$ is ${\mathcal{C}}$-almost separable if there is a language $S\in{\mathcal{C}}$ that almost separates $(A,B)$. If a pair $(A,B)$ is not ${\mathcal{C}}$-almost separable, then $(A-D,B-D)$ is ${\mathcal{C}}$-inseparable for every finite set $D$. Before proving our main theorem, we sketch the intuitive idea of the proof. We want to construct a disjoint NP pair $(A,B)$ that is P-inseparable. Our hypothesis, that NP does not have measure 0 in EXP, implies that NP contains a language $R$ that is p-random. Since we are being intuitive, we ignore the subtleties of p-randomness and regard $R$ as a sequence of independent, fair coin tosses (with the $n^{\mathrm{th}}$ toss heads iff $s_{n}\in R$) that just happens to be in NP. If we use these coins to randomly put strings in $A$ or $B$ but not both, we can count on the randomness to thwart any would-be separator in P. The challenge here is that, if we are to deduce $A,B\in{\mathrm{NP}}$ from $R\in{\mathrm{NP}}$, we must make the conditions “$s_{n}\in A$” and “$s_{n}\in B$” depend on the coin tosses in a monotone way; i.e., adding a string to $R$ must not move a string out of $A$ or out of $B$. This monotonicity restriction might at first seem to prevent us from ensuring that $A$ and $B$ are disjoint. However, this is not the case. Suppose that we decide membership of the $n^{\mathrm{th}}$ string $s_{n}$ in $A$ and $B$ in the following manner. We toss $2\log n$ independent coins. If the first $\log n$ tosses all come up heads, we put $s_{n}$ in $A$. If the second $\log n$ tosses all come up heads, we put $s_{n}$ in $B$. If our coin tosses are taken from $R$, which is in NP, then $A$ and $B$ will be in NP. Each string $s_{n}$ will be in $A$ with probability $\frac{1}{n}$, in $B$ with probability $\frac{1}{n}$, and in $A\cap B$ with probability $\frac{1}{n^{2}}$. Since $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges and $\sum_{n=1}^{\infty}\frac{1}{n^{2}}$ converges, the first and second Borel- Cantelli lemmas tell us that $A$ and $B$ are infinite and $A\cap B$ is finite. Since $A\cap B$ is finite, we can subtract it from $A$ and $B$, leaving two disjoint NP languages that are, by the randomness of the construction, P-inseparable. What prevents this intuitive argument from being a proof sketch is the fact that the language $R$ is not truly random, but only p-random. The proof that $A\cap B$ is finite thus becomes problematic. There is a resource-bounded extension of the first Borel-Cantelli lemma [22] that works for p-random sequences, but this extension requires the relevant sum of probabilities to be p-convergent, i.e., to converge much more quickly than $\sum_{n=1}^{\infty}\frac{1}{n^{2}}$. Fortunately, in this particular instance, we can achieve our objective without p-convergence or the (classical or resource-bounded) Borel-Cantelli lemmas. We do this by modifying the above construction. Instead of putting the $n^{\mathrm{th}}$ string into each language with probability $\frac{1}{n}$, we put each string $x$ into each of $A$ and $B$ with probability $2^{-\|x\|}$ so that $x$ is in $A\cap B$ with probability $2^{-2\|x\|}$. By the Cauchy condensation test, the relevant series have the same convergence behavior as those in our intuitive argument, but we can now replace slow approximations of tails of $\sum_{n=1}^{\infty}\frac{1}{n^{2}}$ with fast and exact computations of geometric series. We now turn to the details. ###### Construction 1. 1. (1) Define the functions $u,v:\\{0,1\\}^{}\to\\{0,1\\}^{}$ by the recursion $\begin{array}[]{l}u(\lambda)=\lambda,\\\ v(x)=\mathrm{next}^{\|x\|}(u(x)),\\\ u(\mathrm{next}(x)=\mathrm{next}^{\|x\|}(v(x)).\end{array}$ 2. (2) For each $x\in\\{0,1\\}^{}$, define the intervals $I_{x}=[u(x),v(x)),\ J_{x}=[v(x),u(next(x))).$ 3. (3) For each $R\subseteq\\{0,1\\}^{}$, define the languages $\begin{array}[]{l}A^{+}(R)=\left\\{{x}\>\Big{\|}\>{I_{x}\subseteq R}\right\\},\ B^{+}(R)=\left\\{{x}\>\Big{\|}\>{J_{x}\subseteq R}\right\\},\\\ A(R)=A^{+}(R)-B^{+}(R),\ B(R)=B^{+}(R)-A^{+}(R).\end{array}$ Note that each $\|I_{x}\|=\|J_{x}\|=\|x\|$. Also, $I_{\lambda}=J_{\lambda}=\emptyset$ (so $\lambda\in A^{+}(R)\cap B^{+}(R)$), and $I_{0}<J_{0}<I_{1}<J_{1}<I_{00}<J_{00}<I_{01}<\ldots,$ with these intervals covering all of $\\{0,1\\}^{}$. A routine witness argument gives the following. 1. (1) If $R\in{\mathrm{NP}}$, then $A^{+}(R),B^{+}(R)\in{\mathrm{NP}}$. 2. (2) If $R\in{\mathrm{NP}}$ and $\|A^{+}(R)\cap B^{+}(R)\|<\infty$, then $(A(R),B(R))$ is a disjoint NP pair. We now prove two lemmas about Construction 1. ###### Lemma 3.2. Let $k\in\mathbb{N}$. If $R\subseteq\\{0,1\\}^{}$ is $2^{(\log n)^{k+2}}$\- random, then $(A^{+}(R),B^{+}(R))$ is not ${\mathrm{TIME}(2^{n^{k}})}$-almost separable. ###### Lemma 3.3. If $R\subseteq\\{0,1\\}^{}$ is p-random, then $\|A^{+}(R)\cap B^{+}(R)\|<\infty$. We now have what we need to prove our main result. ###### Theorem 3.4. (main theorem) If NP does not have measure 0 in EXP, then, for every $k\in\mathbb{Z}^{+}$, there is a disjoint NP pair that is ${\mathrm{TIME}(2^{n^{k}})}$-inseparable, hence certainly P-inseparable. ###### Proof 3.5. Assume that $\mu({\mathrm{NP}}\mid{\rm EXP})\neq 0$, and let $k\in\mathbb{N}$. Then, by Lemma 2.2, there is a $2^{(\log n)^{k+2}}$-random language $R\in{\mathrm{NP}}$. By Lemma 3.2, the pair $(A^{+}(R),B^{+}(R))$ is not ${\mathrm{TIME}(2^{n^{k}})}$-almost separable. Since $R$ is certainly p-random, Lemma 3.3 tells us that $\|A^{+}(R)\cap B^{+}(R)\|<\infty$. It follows by Observation 3 that $(A(R),B(R))$ is a disjoint NP pair, and it follows by Observation 3 that $(A(R),B(R))$ is ${\mathrm{TIME}(2^{n^{k}})}$-inseparable. ## 4\. Genericity and Measure of Disjoint NP Pairs In this section we introduce the natural notions of resource-bounded measure and genericity for disjoint pairs and relate them to the existence of P-inseparable pairs in NP. We compare the different strength hypothesis on the measure and genericity of NP and disjNP establishing all the relations in Figure 1. Notation. Each disjoint pair $(A,B)$ will be coded as an infinite sequence $T\in\\{-1,0,1\\}^{\infty}$ defined by $T[n]=\left\\{\begin{array}[]{ll}1&\mathrm{if}\ s_{n}\in A\\\ -1&\mathrm{if}\ s_{n}\in B\\\ 0&\mathrm{if}\ s_{n}\not\in A\cup B\end{array}\right.$ We identify each disjoint pair with the corresponding sequence. Resource-bounded genericity for disjoint pairs is the natural extension of the concept introduced for languages by Ambos-Spies, Fleischhack and Huwig [1]. ###### Definition 4.1. A condition $C$ is a set $C\subseteq\\{-1,0,1\\}^{}$. A $t(n)$-condition is a condition $C\in\mathrm{DTIME}(t(n))$. A condition $C$ is dense along a pair $(A,B)$ if there are infinitely many $n\in\mathbb{N}$ such that $(A,B)[0..n-1]i\in C$ for some $i\in\\{-1,0,1\\}$. A pair $(A,B)$ meets a condition $C$ if $(A,B)[0..n-1]\in C$ for some $n$. A pair $(A,B)$ is $t(n)$-generic if $(A,B)$ meets every $t(n)$-condition that is dense along $(A,B)$. We first prove that generic pairs are inseparable. ###### Theorem 4.2. Every $t(\log n)$-generic disjoint pair is $\mathrm{TIME}(t(n))$-inseparable. We can now relate genericity in disjNP and inseparable pairs as follows. ###### Corollary 4.3. If disjNP contains a $2^{(\log n)^{k}}$-generic pair for every $k\in\mathbb{N}$, then disjNP contains a ${\mathrm{TIME}(2^{n^{k}})}$-inseparable pair for every $k\in\mathbb{N}$. Resource-bounded measure on classes of disjoint pairs is the natural extension of the concept introduced for languages by Lutz [22], and is defined by using martingales on a three-symbol alphabet as follows. ###### Definition 4.4. 1. (1) A pair martingale is a function $d:\\{-1,0,1\\}^{}\to[0,\infty)$ such that for every $w\in\\{-1,0,1\\}^{}$ $d(w)=\frac{1}{4}d(w0)+\frac{3}{8}d(w1)+\frac{3}{8}d(w(-1)).$ 2. (2) A pair martingale $d$ succeeds on a pair $(A,B)$ if $\limsup_{w\to(A,B)}d(w)=\infty$. 3. (3) A pair martingale $d$ succeeds on a class of pairs $X\subseteq\\{-1,0,1\\}^{\infty}$ if it succeeds on each $(A,B)\in X$. Our intuitive rationale for the coefficients in part 1 of this definition is the following. We toss one fair coin to decide whether $s_{\|w\|}\in A$ and another to decide whether $s_{\|w\|}\in B$. If both coins come up heads, we toss a third coin to break the tie. The reader may feel that some other coefficients, such as $\frac{1}{3},\frac{1}{3},\frac{1}{3}$ are more natural here. Fortunately, a routine extension of the main theorem of [5] shows that the value of $\mu(\mathrm{disjNP}\mid\mathrm{disjEXP})$ will be the same for any choice of three positive coefficients summing to 1. When restricting martingales to those computable within a certain resource bound, we obtain a resource-bounded measure that is useful within a complexity class. Here we are interested in the class of disjoint EXP pairs, disjEXP. ###### Definition 4.5. 1. (1) Let ${{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}$ be the class of functions that can be computed in time $2^{(\log n)^{O(1)}}$. 2. (2) A class of pairs $X\subseteq\\{-1,0,1\\}^{\infty}$ has ${{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}$-measure 0 if there is a martingale $d\in{{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}$ that succeeds on $X$. 3. (3) $X\subseteq\\{-1,0,1\\}^{\infty}$ has ${{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}$-measure 1 if $X^{c}$ has ${{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}$-measure 0. 4. (4) A class of pairs $X\subseteq\\{-1,0,1\\}^{\infty}$ has measure 0 in disjEXP, denoted $\mu(X\mid\mathrm{disjEXP})=0$, if $X\cap\mathrm{disjEXP}$ has ${{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}$-measure 0. 5. (5) $X\subseteq\\{-1,0,1\\}^{\infty}$ has measure 1 in disjEXP if $X^{c}$ has measure 0 in disjEXP. It is easy to verify that ${{\mathrm{p}}_{\thinspace\negthinspace{}_{2}}}$-measure is nontrivial on disjEXP (as proven for languages in [22]). In the following we consider the hypothesis that disjNP does not have measure 0 in disjEXP (written $\mu(\mathrm{disjNP}\mid\mathrm{disjEXP})\neq 0$). We start by proving that this hypothesis is at least as strong as the well studied $\mu({\mathrm{NP}}\mid{\rm EXP})\neq 0$ hypothesis. ###### Theorem 4.6. If $\mu(\mathrm{disjNP}\mid\mathrm{disjEXP})\neq 0$ then $\mu({\mathrm{NP}}\mid{\rm EXP})\neq 0$. We finish by relating measure and genericity for disjoint pairs. ###### Theorem 4.7. If $\mu(\mathrm{disjNP}\mid\mathrm{disjEXP})\neq 0$, then disjNP contains a $2^{(\log n)^{k}}$-generic pair for every $k\in\mathbb{N}$. Figure 1. Relations among some strong hypotheses. ## 5\. Oracle Results All the techniques in this and related papers relativize, that is they hold when all machines involved have access to the same oracle $A$. In this section we give relativized worlds where the converses of most of the results in this paper, as expressed in Figure 1, do not hold. Since the implications trivially all hold in any relativized world where ${\mathrm{P}}={\mathrm{NP}}$ [4], one cannot use relativizable techniques to settle these converses. We’ll work our way from the bottom up of Figure 1. ###### Theorem 5.1 (Homer-Selman [17], Fortnow-Rogers [8]). There exists oracles $A$ and $B$ such that • ${\mathrm{P}}^{A}\neq{\mathrm{NP}}^{A}$ and $\mathrm{disjNP}^{A}$ does not contain ${\mathrm{P}}^{A}$-inseparable pairs. * • ${\mathrm{P}}^{B}={\mathrm{NP}}^{B}\cap\mathrm{co}{\mathrm{NP}}^{B}=\mathrm{UP}^{B}$ and $\mathrm{disjNP}^{B}$ does contain ${\mathrm{P}}^{B}$-inseparable pairs. ###### Theorem 5.2. There exists an oracle $C$ such that ${\mathrm{P}}^{C}\neq\mathrm{UP}^{C}$ but ${\mathrm{NP}}^{C}$ is contained in $\mathrm{TIME}^{C}(n^{O(\log n)})$. In particular this means that relative to $C$, $\mathrm{disjNP}$ contains ${\mathrm{P}}$-inseparable pairs but there is a $k$ (and in fact any real $k>0$) such that $\mathrm{disjNP}$ has no $\mathrm{TIME}(2^{n^{k}})$-inseparable pairs. ###### Theorem 5.3. There exists a relativized world $D$, relative to which for all $k$, $\mathrm{disjNP}$ contains a $\mathrm{TIME}(2^{n^{k}})$-inseparable pair but $\mu({\mathrm{NP}}\|{\rm EXP})=0$ and $\mathrm{disjNP}$ does not contain a $2^{(\log n)^{k}}$-generic pair. ###### Theorem 5.4. There exists an oracle $E$ relative to which for all $k$, $\mathrm{disjNP}$ contains a $2^{(\log n)^{k}}$-generic pair but $\mu(\mathrm{disjNP}\|\mathrm{disjEXP})=0$. ###### Conjecture 5.5. There exists an oracle $H$ relative to which $\mu({\mathrm{NP}}\|{\rm EXP})\neq 0$ but $\mu(\mathrm{disjNP}\|\mathrm{disjEXP})=0$. Let $K$ be a ${\rm PSPACE}$-compete set, $R$ be a “random” oracle and let $H=K\oplus R=\\{\langle 0,x\rangle\ \|\ x\in K\\}\cup\\{\langle 1,y\rangle\ \|\ y\in R\\}.$ Kautz and Miltersen show in [20] that relative to $H$, $\mu({\mathrm{NP}}\|{\rm EXP})\neq 0$. Kahn, Saks and Smyth [19] combined with unpublished work of Impagliazzo and Rudich show that relative to $H$ there is a polynomial-time algorithm that solves languages in ${\mathrm{NP}}\cap\mathrm{co}{\mathrm{NP}}$ on average for infinitely-many lengths which would imply $\mu({\mathrm{NP}}\cap\mathrm{co}{\mathrm{NP}}\|{\rm EXP})=0$ relative to $H$. We conjecture that one can modify this proof to show $\mu(\mathrm{disjNP}^{H}\|\mathrm{disjEXP}^{H})=0$. ## References * [1] K. Ambos-Spies, H. Fleischhack, and H. Huwig. Diagonalizations over polynomial time computable sets. Theoretical Computer Science, 51:177–204, 1987. * [2] K. Ambos-Spies and E. Mayordomo. Resource-bounded measure and randomness. In A. Sorbi, editor, Complexity, Logic and Recursion Theory, Lecture Notes in Pure and Applied Mathematics, pages 1–47. Marcel Dekker, New York, N.Y., 1997. * [3] K. Ambos-Spies, S. A. Terwijn, and X. Zheng. Resource bounded randomness and weakly complete problems. Theoretical Computer Science, 172:195–207, 1997. * [4] T. Baker, J. Gill, and R. Solovay. Relativizations of the P =? NP question. SIAM Journal on Computing, 4:431–442, 1975. * [5] J.M. Breutzmann and J.H. Lutz. Equivalence of measures of complexity classes. SIAM Journal on Computing, 29(1):302–326, 1999. * [6] S. Cook and R. Reckhow. The relative efficiency of propositional proof systems. Journal of Symbolic Logic, 44:36–50, 1979. * [7] S. Even, A. L. Selman, and Y. Yacobi. The complexity of promise problems with applications to public-key cryptography. Information and Control, 61(2):159–173, 1984. * [8] L. Fortnow and J. Rogers. Separability and one-way functions. Computational Complexity, 11:137–157, 2002. * [9] C. Glaßer, A. L. Selman, and S. Sengupta. Reductions between disjoint NP-pairs. Information and Computation, 200:247–267, 2005. * [10] C. Glaßer, A. L. Selman, S. Sengupta, and L. Zhang. Disjoint NP-pairs. SIAM Journal on Computing, 33:1369–1416, 2004. * [11] C. Glaßer, A. L. Selman, and L. Zhang. Survey of disjoint NP-pairs and relations to propositional proof systems. In Theoretical Computer Science: Essays in Memory of Shimon Even, pages 241–253. Springer, 2006. * [12] C. Glaßer, A. L. Selman, and L. Zhang. Canonical disjoint NP-pairs of propositional proof systems. Theoretical Computer Science, 370:60–73, 2007. * [13] C. Glaßer, A. L. Selman, and L. Zhang. The informational content of canonical disjoint NP-pairs. In COCOON, LNCS, pages 307–317. Springer, 2007. * [14] O. Goldreich. On promise problems: A survey. In Theoretical Computer Science: Essays in Memory of Shimon Even, pages 254–290. Springer, 2006. * [15] J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. SIAM Journal on Computing, 11:309–335, 1988. * [16] J.M. Hitchcock. Resource-bounded measure bibliography. http://www.cs.uwyo.edu/~jhitchco/bib/rbm.shtml. * [17] S. Homer and A. L. Selman. Oracles for structural properties: The isomorphism problem and public-key cryptography. Journal of Computer and System Sciences, 44:287–301, 1992. * [18] D.W. Juedes and J.H. Lutz. Weak completeness in E and $\mathrm{E}_{2}$, 1995. * [19] J. Kahn, M. E. Saks, and C. D. Smyth:. A dual version of reimer’s inequality and a proof of rudich’s conjecture. In Proceedings of the 15th Annual IEEE Conference on Computational Complexity, pages 98–103, 2000. * [20] S. M. Kautz and P. B. Miltersen. Relative to a random oracle, NP is not small. Journal of Computer and System Sciences, 53:235–250, 1996. * [21] A.K. Lorentz and J.H. Lutz. Genericity and randomness over feasible probability measures. Theoretical Computer Science, 207(1):245–259, 1998. * [22] J. H. Lutz. Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences, 44(2):220–258, 1992. * [23] J. H. Lutz. The quantitative structure of exponential time. In L. A. Hemaspaandra and A. L. Selman, editors, Complexity Theory Retrospective II, pages 225–254. Springer-Verlag, 1997. * [24] J. H. Lutz and E. Mayordomo. Twelve problems in resource-bounded measure. In G. Păun, G. Rozenberg, and A. Salomaa, editors, Current Trends in Theoretical Computer Science, entering the 21st century, pages 83–101. World Scientific Publishing, 2001. * [25] A. Razborov. On provably disjoint NP pairs. Technical Report 94-006, ECCC, 1994. * [26] A.L. Selman. Complexity issues in cryptography. In Computational complexity theory (Atlanta, GA, 1988), volume 38 of Proc. Sympos. Appl. Math., pages 92–107. Amer. Math. Soc., 1989.	arxiv-papers	2009-02-16T12:27:54	2024-09-04T02:49:00.613998	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lance Fortnow, Jack H. Lutz, Elvira Mayordomo", "submitter": "Evira Mayordomo", "url": "https://arxiv.org/abs/0902.2674" }
0902.2712	# What can we learn from TMD measurements?111 Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce this manuscript for U.S. Government purposes. Alessandro Bacchetta ###### Abstract Transverse-momentum-dependent parton distribution and fragmentation functions describe the partonic structure of the nucleon in a three-dimensional momentum space. They are subjects of flourishing theoretical and experimental activity. They provide novel and intriguing information on hadronic structure, including evidence of the presence of partonic orbital angular momentum. ###### Keywords: parton distribution functions, semi-inclusive DIS, transverse momentum ###### : 12.38.-t, 13.60.-r, 13.88.+e TMDs is an acronym for Transverse Momentum Distributions or Transverse Momentum Dependent parton distribution functions, also called unintegrated parton distribution functions. Most of our knowledge of the inner structure of nucleons is encoded in parton distribution functions (PDFs). We introduce them to describe hard scattering processes involving nucleons. The presence of a hard probe in these processes — e.g., in DIS the photon with virtuality $Q^{2}$ — identifies a longitudinal direction, and a plane perpendicular to that, the transverse plane. Intuitively, standard collinear PDFs describe the probability to find in a fast-moving nucleon a parton with a specific fraction of the nucleon’s longitudinal momentum. TMDs describe also the probability that the parton has a specific transverse momentum. They are therefore a natural extension of standard PDFs from one to three dimensions in momentum space. Although useful from the intuition point of view, the probabilistic interpretation of PDFs and TMDs has some technical problems and is not strictly needed Collins (2003). What is essential is that PDFs and TMDs can be defined in a formally clear way, through the application of factorization theorems. They reveal crucial aspects of the dynamics of confined partons, they can be extracted from experimental data, and allow us to make prediction for hard-scattering experiments involving nucleons. In this sense, the information contained in TMDs is as important as that contained in standard PDFs. The main difference between collinear PDFs and TMDs is that the latter do not appear in totally inclusive processes. For instance, they do not appear in totally inclusive DIS, but they are needed when semi-inclusive DIS is studied and the transverse momentum of one outgoing hadron, $P_{h\perp}$, is measured. They are necessary to describe Drell–Yan processes when the transverse momentum of the virtual photon, $q_{T}$, is measured. Factorization for processes involving TMDs has been worked out explicitly at leading twist (twist 2) and one-loop order and argued to hold at all orders Collins and Soper (1981); Ji et al. (2005). For instance, in unpolarized semi- inclusive DIS we can measure the structure function $F_{UU,T}$, which in the region $P_{h\perp}^{2}\ll Q^{2}$ can be expressed as Bacchetta et al. (2008a) $\displaystyle F_{UU,T}$ $\displaystyle=\bigl{\|}H\bigl{(}x\zeta^{1/2},z^{-1}\zeta_{\smash{h}}^{1/2},\mu_{F}\bigr{)}\bigr{\|}^{2}\,\sum_{a}x\,e_{a}^{2}\int d^{2}\boldmath{p}_{T}\,d^{2}\boldmath{k}_{T}\,d^{2}\boldmath{l}_{T}\,$ $\displaystyle\times\delta^{(2)}\bigl{(}\boldmath{p}_{T}-\boldmath{k}_{T}+\boldmath{l}_{T}-\boldmath{P}_{h\perp}/z\bigr{)}\,f_{1}^{a}(x,p_{T}^{2};\zeta,\mu_{F})\,D_{1}^{a}(z,k_{T}^{2};\zeta_{h},\mu_{F})\,U(l_{T}^{2};\mu_{F})\,.$ (1) Apart from the transverse-momentum-dependent PDFs and fragmentation functions, the formula contains the soft factor $U$, a nonperturbative and process- independent object. For the specific case of unpolarized observables integrated over the azimuthal angle of the measured transverse momentum, the analysis is usually performed in $b$-space in the Collins–Soper–Sterman framework Collins et al. (1985). The region of $P_{h\perp}^{2}\gg M^{2}$, or $b^{2}\ll 1/M^{2}$, can be calculated perturbatively, but when $P_{h\perp}^{2}\approx M^{2}$ a nonperturbative component has to be introduced and its parameters must be fitted to experimental data. This component is usually assumed to be a flavor- independent Gaussian Landry et al. (2003). At present, especially for azimuthally-dependent structure functions, phenomenological analyses are often carried out using the tree-level approximated expression $\displaystyle F_{UU,T}$ $\displaystyle=\sum_{a}x\,e_{a}^{2}\int d^{2}\boldmath{p}_{T}\,d^{2}\boldmath{k}_{T}\,\delta^{(2)}\bigl{(}\boldmath{p}_{T}-\boldmath{k}_{T}-\boldmath{P}_{h\perp}/z\bigr{)}\,f_{1}^{a}(x,p_{T}^{2})\,D_{1}^{a}(z,k_{T}^{2})\,.$ (2) Also in this case, the transverse-momentum dependence of the partonic functions is assumed to be a flavor-independent Gaussian D’Alesio and Murgia (2004). The tree-level approximation and the Gaussian assumption are known to be inadequate at $P_{h\perp}^{2}\gg M^{2}$, but they could still effectively describe the physics at $P_{h\perp}^{2}\approx M^{2}$. Especially for low- energy experiments, this is where the bulk of the data is. The definition of quark TMDs is Collins (2003); Ji et al. (2005) (taking the example of the fully unpolarized distribution of a quark with flavor $a$) $f_{1}^{a}(x,p_{T}^{2};\zeta,\mu_{F})=\int\frac{d\xi^{-}d^{2}\boldmath{\xi}_{T}}{(2\pi)^{3}}\;e^{ip\cdot\xi}\,\langle P\|\bar{\psi}^{a}(0)\,{\cal L}^{v\dagger}_{(\pm\infty,0)}\,\gamma^{+}{\cal L}^{v}_{(\pm\infty,\xi)}\,\psi^{a}(\xi)\|P\rangle\bigg{\|}_{\xi^{+}=0}.$ (3) The Wilson lines, ${\cal L}$, guarantee the gauge invariance of the TMDs. They depend on the gauge vector $v$ and contain also components at infinity running in the transverse direction. A remarkable property of TMDs is that the detailed shape of the Wilson line is process-dependent. This immediately leads to the conclusion that TMDs are not universal. However, the situation is not hopeless and the predictive power of TMD factorization is not completely destroyed, for the following reasons * • For transverse-momentum-dependent fragmentation functions, the shape of the Wilson line appears to have no influence on physical observables Metz (2002); Collins:2004nx; Yuan:2008yv; Gamberg:2008yt; Meissner:2008yf. * • In SIDIS and Drell–Yan, the difference between the Wilson line consists in a simple direction reversal and leads to calculable effects, namely a simple sign reversal of all T-odd TMDs Collins (2002). * • In hadron-hadron collisions to hadrons, standard universality cannot be applied. It is however conceivable that only a manageable number of TMDs with distinct Wilson lines are needed, preserving part of the predictive power of the formalism Collins and Qiu (2007); Vogelsang:2007jk. • If we consider specific transverse-momentum-weighted observables instead of unintegrated observables, it should be possible to obtain factorized expressions in terms of transverse moments of TMDs multiplied by calculable, process-dependent factors Bacchetta et al. (2005); Bomhof:2006ra; Bacchetta:2007sz. Our understanding of TMDs and their extraction from data has made giant steps in the last years, thanks to new theoretical ideas and experimental measurements. In the near future, more experimental data are expected from HERMES, COMPASS, BELLE and JLab. When the spin of the nucleon and that of the quark are taken into account, eight twist-2 functions can be introduced. They are listed in Tab. What can we learn from TMD measurements?111 Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce this manuscript for U.S. Government purposes.. As with collinear PDFs, extracting TMDs calls for global fits to semi-inclusive DIS, Drell–Yan, and $e^{+}e^{-}$-annihilation data. Care has to be taken when considering the peculiar universality properties of TMDs. At the moment, we have some information only about the two functions in the first column of the table: $f_{1}$ (unpolarized function) and $f_{1T}^{\perp}$ (Sivers function). Twist-2 transverse-momentum-dependent distribution functions. The U,L,T correspond to unpolarized, longitudinally polarized and transversely polarized nucleons (rows) and quarks (columns). Functions in boldface survive transverse momentum integration. Functions in gray cells are T-odd. Ultimately, the knowledge of TMDs should allow us to build tomographic images of the inner structure of the nucleon in momentum space, complementary to the impact-parameter space tomography that can be achieved by studying generalized parton distribution functions (GPDs). An example of tomographical images of the nucleon based on a model calculation of TMDs Bacchetta et al. (2008b) is presented in Fig. 1. \| ---\|--- \| Figure 1: Momentum-space tomographic “images” of the up quarks in a nucleon obtained from a model calculation of TMDs Bacchetta et al. (2008b). The circle with the arrow indicates the nucleon and its spin orientation. The distortion in the lower panels is due to the Sivers function. In the future, it should be possible to reconstruct these images from experimental data. TMDs measurements should allow us to address some intriguing questions, e.g., * • Are there differences between the TMDs of different quark flavors (and of gluons)? We know that collinear PDFs are different, not only in normalization, but also in shape. We can expect that also the transverse momentum distribution is different. See Ref. Mkrtchyan et al. (2008) for an example of an experimental analysis of this issue. * • How does the transverse momentum dependence change with $x$? Such a dependence has already been introduced to describe data at low $x$ Landry et al. (2003). * • Does the transverse momentum dependence of fragmentation functions change for different quark flavors and different produced hadrons? * • Are there reasons to abandon a Gaussian ansatz? We know that this assumption fails at high transverse momentum, but there are no compelling reasons to take a Gaussian shape even for the low-transverse-momentum, nonperturbative region. The last item of the list is connected also to another fundamental issue that makes TMDs interesting, i.e., the observation of partonic orbital angular momentum. In nonrelativistic quantum mechanics, it is well known that wavefunctions with orbital angular momentum vanish at zero momentum. This is a general statement independent of the specific potential in which the wavefunction is computed. This feature is reflected also in TMDs: contributions from partons with nonzero angular momentum have to vanish at zero transverse momentum (and therefore cannot be described by a simple Gaussian). In general, a downturn of a TMD going to zero transverse momentum can signal the presence of nonzero orbital angular momentum. While this effects could barely be visible in unpolarized TMDs, certain combinations of polarized TMDs could filter out more clearly the configurations with nonzero orbital angular momentum. Fig. (2) shows an example of this phenomenon, using a model calculation for illustration purposes. Figure 2: An illustration of how the presence of orbital angular momentum can influence the shape of TMDs. The model calculation shows different combinations of the $f_{1}$ and $g_{1}$ TMDs for $u$ and $d$ quark at $x=0.02$. The downturns for $p_{T}^{2}\to 0$ are due to the presence of orbital angular momentum. Apart from the details of their shape, all the TMDs that are not boldface in Tab. What can we learn from TMD measurements?111 Authored by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177. The U.S. Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce this manuscript for U.S. Government purposes. vanish in the absence of orbital angular momentum due to angular momentum conservation. Measuring any one of them to be nonzero is already an unmistakable indication of the presence of partonic orbital angular momentum. We know already from other sources (in particular the measurement of nucleons’ anomalous magnetic moments) that partonic orbital angular momentum is not zero, however TMDs have the advantage that they can be flavor-separated and that they are $x$ dependent. Thus, they allow us to say if orbital angular momentum is present for each quark flavor and for gluons, and at each value of $x$. If stating that a fraction of partons have nonzero orbital angular momentum is relatively simple, it is not easy to make a quantitative estimate of the net partonic orbital angular momentum using TMDs. Any statement in this direction is bound to be model-dependent. Generally speaking, TMDs have to be computed in a model and the parameters of the model have to be fixed to reproduce the TMDs extracted from data. Then, the total orbital angular momentum can be computed in the model. Unfortunately, it is possible that two models describe the data equally well, but give two different values for the total orbital angular momentum. As an example of a procedure of this kind, let us take the measurement of the Sivers function. We know that the proper way to measure the quark total angular momentum is by measuring the combination Ji (1997) $J^{a}=\int_{0}^{1}dx\,x\,\Bigl{(}H^{a}(x,0,0)+E^{a}(x,0,0)\Bigr{)},$ (4) where the definition of the generalized parton distribution $E$ in terms of light-cone wavefunctions is $\begin{split}E(x,0,0)=\lim_{q_{T}\to 0}\biggl{(}-\frac{1}{q_{x}-iq_{y}}\int\frac{d^{2}p_{T}}{16\pi^{2}}\,\Bigl{[}&\psi^{+\ast}_{+}\bigl{(}x,p_{T}\bigr{)}\psi^{-}_{+}\bigl{(}x,p_{T}+(1-x)q_{T}\bigr{)}\\\ &+\psi^{+\ast}_{-}\bigl{(}x,p_{T}\bigr{)}\psi^{-}_{-}\bigl{(}x,p_{T}+(1-x)q_{T}\bigr{)}\Bigr{]}\biggr{)}.\end{split}$ (5) On the other hand, the definition of the Sivers function in terms of light- cone wavefunctions can be written as $f_{1T}^{\perp}(x,p_{T})=\frac{1}{16\pi^{3}}{\rm Im}\Bigl{[}\psi^{+\ast}_{+}\bigl{(}x,p_{T}\bigr{)}\psi^{-}_{+}\bigl{(}x,p_{T}\bigr{)}+\psi^{+\ast}_{-}\bigl{(}x,p_{T}\bigr{)}\psi^{-}_{-}\bigl{(}x,p_{T}\bigr{)}\Bigr{]}.$ (6) In spite of the similarities between the two expressions and the fact that the same light-cone wavefunctions are involved, in general there is no straightforward connection between the Sivers function and the GPD $E$ Meissner et al. (2007). Nevertheless, in a certain class of spectator models it turns out that Burkardt and Hwang (2004); Lu:2006kt $f_{1T}^{\perp a}(x)=-L(x)\,E^{a}(x,0,0).$ (7) Exploiting this very simple relation and using for illustration purposes the results of the Sivers function fit from Ref. Arnold et al. (2008) we obtain $\frac{E^{a}(x,0,0)}{E^{u}(x,0,0)}=\frac{f_{1T}^{\perp a}(x)}{f_{1T}^{\perp u}(x)}=\frac{A_{a}}{A_{u}}\,\frac{f_{1}^{a}(x)}{f_{1}^{u}(x)},$ (8) where (error estimates do not take into account parameter correlations) $\displaystyle\frac{A_{d}}{A_{u}}$ $\displaystyle=-1.8\pm 0.2,$ $\displaystyle\frac{A_{\bar{u}}}{A_{u}}$ $\displaystyle=-1.1\pm 0.1,$ $\displaystyle\frac{A_{\bar{d}}}{A_{u}}$ $\displaystyle=1.3\pm 0.2,$ $\displaystyle\frac{A_{s}}{A_{u}}=-\frac{A_{\bar{s}}}{A_{u}}$ $\displaystyle=-4.8.$ (9) Although assumption-based, the above analysis shows that the measurement of the Sivers function can be used to give interesting constraints on the GPD $E$ and ultimately on the amount of total orbital angular momentum for each flavor. In summary, TMDs open new dimensions in the exploration of the partonic structure of the nucleon. They require challenging extensions of the standard formalism used for collinear parton distribution functions, leading us to a deeper understanding of QCD. Among other things, they give evidence of the presence of partonic orbital angular momentum and, with model assumptions, they can help constraining its size. ## References Collins (2003) J. C. Collins, _Acta Phys. Polon._ B34, 3103 (2003) * Collins and Soper (1981) J. C. Collins, and D. E. Soper, _Nucl. Phys._ B193, 381 (1981) * Ji et al. (2005) X. Ji, J.-P. Ma, and F. Yuan, _Phys. Rev._ D71, 034005 (2005) * Bacchetta et al. (2008a) A. Bacchetta, D. Boer, M. Diehl, and P. J. Mulders, _JHEP_ 08, 023 (2008a) * Collins et al. (1985) J. C. Collins, D. E. Soper, and G. Sterman, _Nucl. Phys._ B250, 199 (1985) * Landry et al. (2003) F. Landry, R. Brock, P. M. Nadolsky, and C. P. Yuan, _Phys. Rev._ D67, 073016 (2003) * D’Alesio and Murgia (2004) U. D’Alesio, and F. Murgia, _Phys. Rev._ D70, 074009 (2004) * Metz (2002) A. Metz, _Phys. Lett._ B549, 139–145 (2002) * Collins and Metz (2004) J. C. Collins, and A. Metz, _Phys. Rev. Lett._ 93, 252001 (2004) * Yuan (2008) F. Yuan, _Phys. Rev._ D77, 074019 (2008) * Gamberg et al. (2008) L. P. Gamberg, A. Mukherjee, and P. J. Mulders, _Phys. Rev._ D77, 114026 (2008) * Meissner and Metz (2008) S. Meissner, and A. Metz, arXiv:0812.3783[hep-ph] * Collins (2002) J. C. Collins, _Phys. Lett._ B536, 43–48 (2002) * Collins and Qiu (2007) J. Collins, and J.-W. Qiu, _Phys. Rev._ D75, 114014 (2007) * Vogelsang and Yuan (2007) W. Vogelsang, and F. Yuan, _Phys. Rev._ D76, 094013 (2007) * Bacchetta et al. (2005) A. Bacchetta, C. J. Bomhof, P. J. Mulders, and F. Pijlman, _Phys. Rev._ D72, 034030 (2005) * Bomhof and Mulders (2007) C. J. Bomhof, and P. J. Mulders, _JHEP_ 02, 029 (2007) * Bacchetta et al. (2007) A. Bacchetta, C. Bomhof, U. D’Alesio, P. J. Mulders, and F. Murgia, _Phys. Rev. Lett._ 99, 212002 (2007) * Bacchetta et al. (2008b) A. Bacchetta, F. Conti, and M. Radici, _Phys. Rev._ D78, 074010 (2008b) * Mkrtchyan et al. (2008) H. Mkrtchyan, et al., _Phys. Lett._ B665, 20–25 (2008) * Ji (1997) X. Ji, _Phys. Rev. Lett._ 78, 610–613 (1997) * Meissner et al. (2007) S. Meissner, A. Metz, and K. Goeke, _Phys. Rev._ D76, 034002 (2007) * Burkardt and Hwang (2004) M. Burkardt, and D. S. Hwang, _Phys. Rev._ D69, 074032 (2004) * Lu and Schmidt (2007) Z. Lu, and I. Schmidt, _Phys. Rev._ D75, 073008 (2007) * Arnold et al. (2008) S. Arnold, A. V. Efremov, K. Goeke, M. Schlegel, and P. Schweitzer, arXiv:0805.2137[hep-ph]	arxiv-papers	2009-02-16T15:40:42	2024-09-04T02:49:00.619816	{ "license": "Public Domain", "authors": "Alessandro Bacchetta (JLab)", "submitter": "Alessandro Bacchetta", "url": "https://arxiv.org/abs/0902.2712" }
0902.2738	, , and # Fisher-based thermodynamics for scale-invariant systems: Zipf’s Law as an equilibrium state of a scale-free ideal gas A Hernando1, D Puigdomènech1, D Villuendas2 and C Vesperinas3 1 Departament ECM, Facultat de Física, Universitat de Barcelona. Diagonal 647, 08028 Barcelona, Spain 2 Departament FFN, Facultat de Física, Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain 3 Sogeti España, WTCAP 2, Plaça de la Pau s/n, 08940 Cornellà, Spain alberto@ecm.ub.es puigdomenech@ecm.ub.es diego@ffn.ub.es cristina.vesperinas@sogeti.com ###### Abstract We present a thermodynamic formulation for scale-invariant systems based on the principle of extreme information. We create an analogy between these systems and the well-known thermodynamics of gases and fluids, and study as a compelling case the non-interacting system —the _scale-free ideal gas_ — presenting some empirical evidences of electoral results, city population and total cites of Physics journals that confirm its existence. The empirical class of universality known as Zipf’s law is derived from first principles: we show that this special class of power law can be understood as the density distribution of an equilibrium state of the scale-free ideal gas, whereas power laws of different exponent arise from equilibrium and non-equilibrium states. We also predict the appearance of the log-normal distribution as the equilibrium density of a harmonically constrained system, and finally derive an equivalent microscopic description of these systems. ###### pacs: 89.70.Cf, 05.90.+m, 89.75.Da ## 1 Introduction The study of scale-invariant phenomena has unravelled interesting and somewhat unexpected behaviours in systems belonging to disciplines of different nature, from physical and biological to technological and social sciences [1]. Indeed, empirical data from percolation theory and nuclear multifragmentation [2] reflect scale-invariant behaviour, and so do the abundances of genes in various organisms and tissues [3], the frequency of words in natural languages [4], scientific collaboration networks [5], the Internet traffic [6], Linux packages links [7], as well as electoral results [8], urban agglomerations [9] and firm sizes all over the world [10]. The common feature in these systems is the lack of a characteristic size, length or frequency for an observable $k$ at study. This lack generally leads to a power law distribution $p(k)$, valid in most of the domain of definition of $k$, $p(k)\sim 1/k^{1+\gamma},$ (1) with $\gamma\geq 0$. Special attention has been paid to the class of universality defined by $\gamma=1$, which corresponds to Zipf’s law in the cumulative distribution or the rank-size distribution [2, 3, 4, 6, 7, 9, 10, 11]. Recently, Maillart et al. [7] have studied the evolution of the number of links to open source software projects in Linux packages, and have found that the link distribution follows Zipf’s law as a consequence of stochastic proportional growth. In its simplest formulation, the stochastic proportional growth model, or namely the geometric Brownian motion, assumes the growth of an element of the system to be proportional to its size $k$, and to be governed by a stochastic Wiener process. The class $\gamma=1$ emerges from the condition of stationarity, i.e., when the system reaches a dynamic equilibrium [11]. There is a variety of models arising in different fields that yield Zipf’s law and other power laws on a case-by-case basis [9, 11, 12]. In the context of complex networks, proportional growth processes known as preferential attachment [6] and competitive cluster growth [13] have been used to explain many of the properties of natural networks, from social to biological. The emergence of power laws in all these models is explained by W. J. Redd and B. D. Hughes [14], which have shown analytically that models based on stochastic processes with exponential growth —as the geometric Brownian motion, discrete multiplicative process, the birth-and-death process, or the Galton-Watson branching process— generate power laws in one of both tails of the statistical distributions. However, in spite of the success of these models, the intrinsic complexity involved makes the study at a macroscopic level difficult since a general formulation of the thermodynamics of scale-invariant physics is not established yet. Frieden et al. [15] have shown that equilibrium and non-equilibrium thermodynamics can be derived from the principle of extreme Fisher information. The information measure is done for the particular case of _translation families_ , i.e., distribution functions whose form does not change under translational transformations. In this case, Fisher measure becomes _shift-invariant_ [16], what yields most of the canonical Hamiltonians of theoretical physics [17]. Variations of the information measure lead to a Schrödinger equation [18] for the probability amplitude, where the ground state describes equilibrium physics and the excited states account for non- equilibrium physics. As for Hamiltonian systems [19], it has been recently shown that the principle of extreme physical information allows to describe the behaviour of complex systems, as the allometric or power laws found in biological sciences [20]. In this work we present a theoretical framework based on the principle of extreme physical information that aims to describe scale-invariant systems at a macroscopic level. We show that the thermodynamics for such systems can be formulated when the information measure is taken on distributions that do not change under _scale_ transformations. We show that proportional growth is intrinsic to this symmetry, and the processes that describe Zipf’s law, as the geometrical Brownian motion, are the equivalent microscopic description of these systems. This work is organized as follows. In Sec. 2 we present the Fisher information measure for a scale-invariant system. In Sec. 3 we derive the equilibrium state and non-equilibrium states of the most general case of non-interacting scale-invariant system: the _scale-free ideal gas_ (SFIG), and present some empirical evidences of its existence in electoral results and city population. In Sec. 4 we derive from first principles the special case of Zipf’s law, what we call the _Zipf regime_ of the SFIG, and study empirical data of the total number of cites of Physics journals to understand the conditions leading to its appearance. In Sec. 5 we constrain the system harmonically, finding that the equilibrium density follows a log-normal distribution. In Sec. 6 we derive from the SFIG the microscopic stochastic equation of motion, showing that the system can be described by geometrical Brownian walkers. Finally, in Sec. 7 we summarize our results and discuss some aspects of our work. In the Appendix we derive from the Fisher information the equations of the well-known translational-invariant ideal gas, which we use as analogy in the derivation of the SFIG. ## 2 The principle of extreme information for a scale-invariant system The Fisher information measure $I$ for a system of $N$ elements, described by a set of coordinates $\bi{q}$ and physical parameters $\bi{\theta}$, has the form [17] $I(F)=\int\rmd\bi{q}F(\bi{q}\|\bi{\theta})\sum_{ij}c_{ij}\frac{\partial}{\partial\theta_{i}}\ln F(\bi{q}\|\bi{\theta})\frac{\partial}{\partial\theta_{j}}\ln F(\bi{q}\|\bi{\theta}),$ (2) where $F(\bi{q}\|\bi{\theta})$ is the density distribution in configuration space ($\bi{q}$) conditioned by the physical parameters ($\bi{\theta}$). The constants $c_{ij}$ account for dimensionality, and take the form $c_{ij}=c_{i}\delta_{ij}$ if $q_{i}$ and $q_{j}$ are uncorrelated. Following the principle of extreme information (PEI), the state of the system extremizes $I$ subject to prior conditions, as the normalization of $F$ or any constraint on the mean value of an observable $\langle A_{i}\rangle$. The PEI is then written as a variation problem of the form $\delta\left\\{I(F)-\sum_{i}\mu_{i}\langle A_{i}\rangle\right\\}=0,$ (3) where $\mu_{i}$ are the Lagrange multipliers. In the Appendix, we derive from the PEI the density distribution in configuration space and the entropy equation of state for the well-known translational invariant ideal gas (IG) [21]. In analogy with this derivation, we follow here the same steps to obtain the SFIG density distributions and entropy equations of state. We consider a one-dimensional system with dynamical coordinates $\bi{q}=(k,v)$ where $\rmd k/\rmd t=v$. We define $k$ as a discrete variable, i.e. $k=k_{1},k_{2},\ldots,k_{M}$, where $k_{i}=i\Delta k$ and $M$, assumed to be large, is the total number of bins of width $\Delta k$. In order to address the scale-invariance behaviour of $k$ in the Fisher formulation, we change to the new coordinates $u=\ln k$ and $w=\rmd u/\rmd t$, and assume that $u$ and $w$ are canonical [22] and uncorrelated. This assumption leads to the proportional growth $\rmd k/\rmd t=v=kw.$ (4) For constant $w$ this equation yields an exponential growth $k=k_{0}\rme^{wt}$, which is a uniform linear motion for $u$: $u=wt+u_{0}$, with $u_{0}=\ln k_{0}$ 111This exponential growth allows to recognize the systems that we study in this work at the macroscopic level with those studied in [14].. It is easy to check that the scale transformation $k^{\prime}=k/\theta_{k}$ leaves invariant the coordinate $w$, whereas the coordinate $u$ transforms translationally $u^{\prime}=u-\Theta_{k}$, where $\Theta_{k}=\ln\theta_{k}$. If physics does not depend on scale, i.e., the system is translationally invariant with respect to the coordinates $u$ and $w$, the distribution of physical elements can be described by the monoparametric translation families $F(u,w\|\Theta_{k},\Theta_{w})=f(u^{\prime},w^{\prime})$. By analogy with the IG, we define the SFIG as a system of $N$ non-interacting elements for which the density distribution can be factorized as $f(u,w)=g(u)h(w)$. Taking into account that $u$ and $w$ are canonical and uncorrelated ($c_{ii}=c_{i}\neq 0$ and $c_{uw}=c_{wu}=0$), and that the Jacobian for the change of variables is $\rmd k\rmd v=\rme^{2u}\rmd u\rmd w$, the information measure $I=I_{u}+I_{w}$ can be obtained in the continuous limit as $\begin{array}[]{rl}I_{u}=&\displaystyle c_{u}\int\rmd u~{}\rme^{2u}g(u)\left\|\frac{\partial\ln g(u)}{\partial u}\right\|^{2}\\\ I_{w}=&\displaystyle c_{w}\int\rmd w~{}h(w)\left\|\frac{\partial\ln h(w)}{\partial w}\right\|^{2}.\end{array}$ (5) The constraints to the given observables $\langle A_{i}\rangle$ in the extremization problem determine the behaviour of the system. In the next sections we study three different cases: the general case of the scale-free ideal gas —the step-by-step analogy of the ideal gas—, a un-constrained gas or what we call the _Zipf regime_ , and the harmonically constrained gas. ## 3 The scale-free ideal gas For the general case, in the extremization of Fisher information we constrain the normalization of $g(u)$ and $h(w)$ to the total number of particles $N$ and to $1$, respectively $\int\rmd u~{}\rme^{2u}g(u)=N,\qquad\int\rmd w~{}h(w)=1.$ (6) In addition, we penalize infinite values for $w$ with a constraint on the variance of $h(w)$ to a given measured value $\int\rmd w~{}h(w)(w-\overline{w})^{2}=\sigma_{w}^{2},$ (7) where $\overline{w}$ is the average growth. The variation yields $\delta\left\\{c_{u}\int\rmd u~{}\rme^{2u}g\left\|\frac{\partial\ln g}{\partial u}\right\|^{2}+\mu\int\rmd u~{}\rme^{2u}g\right\\}=0$ (8) and $\delta\left\\{c_{w}\int\rmd w~{}h\left\|\frac{\partial\ln h}{\partial w}\right\|^{2}+\lambda\int\rmd w~{}h(w-\overline{w})^{2}+\nu\int\rmd w~{}h\right\\}=0,$ (9) where $\mu$, $\lambda$ and $\nu$ are Lagrange multipliers. Introducing $g(u)=\rme^{-2u}\Psi^{}(u)\Psi(u)$, and varying (8) with respect to $\Psi^{}$ leads to the Schrödinger equation $\left[-4\frac{\partial^{2}}{\partial u^{2}}+4+\mu^{\prime}\right]\Psi(u)=0,$ (10) where $\mu^{\prime}=\mu/c_{u}$. Analogously to the IG, we impose solutions compatible with a finite normalization of $g$ in the thermodynamic limit $N,\Omega\rightarrow\infty$ with $N/\Omega=\rho_{0}$ finite, where $\Omega=\ln(k_{M}/k_{1})=\ln M$ is the volume in $u$ space and $\rho_{0}$ is defined as the _bulk density_. Solutions compatible with the normalization of (6) are given by $\Psi(u)=A_{\alpha}\rme^{-\alpha u/2}$, where $A_{\alpha}$ is the normalization constant and $\alpha=\sqrt{4+\mu^{\prime}}$. In this general case, the density distribution as a function of $k$ takes the form of a power law: $g_{\alpha}(\ln k)=A^{2}/k^{2+\alpha}$. The equilibrium is defined by the ground state solution, which correspond the lowest allowed value $\alpha=0$. It can be show that it is just a uniform density distribution in $u$ space at the bulk density: $g(u)\rme^{2u}\rmd u=N/\Omega\rmd u=\rho_{0}\rmd u$. Introducing $h(w)=\Phi^{}(w)\Phi(w)$ and varying (9) with respect to $\Phi^{}$ leads to the quantum harmonic oscillator equation [18] $\left[-4\frac{\partial^{2}}{\partial w^{2}}+\lambda^{\prime}(w-\overline{w})^{2}+\nu^{\prime}\right]\Phi(w)=0,$ (11) where $\lambda^{\prime}=\lambda/c_{w}$ and $\nu^{\prime}=\nu/c_{w}$. The equilibrium configuration corresponds to the ground state solution, which is now a Gaussian distribution. Using (7) to identify $\|\lambda^{\prime}\|^{-1/2}=\sigma_{w}^{2}$ we get the Boltzmann distribution $h(w)=\frac{\exp\left[-(w-\overline{w})^{2}/2\sigma_{w}^{2}\right]}{\sqrt{2\pi}\sigma_{w}}.$ (12) The density distribution in configuration space $\widetilde{f}(k,v)\rmd k\rmd v=f(u,w)\rme^{2u}\rmd u\rmd w$ is then $\widetilde{f}(k,v)=\frac{N}{\Omega k^{2}}\frac{\exp\left[-(v/k-\overline{w})^{2}/2\sigma_{w}^{2}\right]}{\sqrt{2\pi}\sigma_{w}}.$ (13) If we define $H=\Delta k^{2}/\Delta\tau$ as the elementary volume in phase space, where $\Delta\tau$ is the time element, the total number of microstates is $Z=N!H^{N}\prod_{i=1}^{N}f_{1}(k_{i},v_{i})$, where $f_{1}=\widetilde{f}/N$ is the monoparticular distribution function and $N!$ counts all possible permutations for distinguishable elements. The entropy equation of state $S=-\kappa\ln Z$ reads $S=N\kappa\left\\{\ln\frac{\Omega}{N}\frac{\sqrt{2\pi}\sigma_{w}}{H^{\prime}}+\frac{3}{2}\right\\},$ (14) where $\kappa$ is a constant that accounts for dimensionality and $H^{\prime}=H/(k_{M}k_{1})=H/(M\Delta k^{2})=1/(M\Delta\tau)$. Remarkably, this expression has the same form as the one-dimensional IG ($D=1$ in (38)); instead of the thermodynamical variables $(N,V,T)$, here we deal with the variables $(N,\Omega,\sigma_{w})$, which make the entropy scale-invariant. Figure 1: (colour on-line) a, rank-size distribution of the cities of the province of Huelva, Spain (2008), sorted from largest to smallest, compared with the result of a simulation with Brownian walkers (green squares). b, rank-plot of the 2008 General Elections results in Spain. c, rank-plot of the 2005 General Elections results in the United Kingdom. (red dots: empirical data; blue lines: linear fitting). The total density distribution for $k$ is obtained integrating for all $v$ the density distribution in configuration space. Integrating (13) we get $\widetilde{f}(k)=\int\rmd v\widetilde{f}(k,v)=\frac{N}{\Omega}\frac{1}{k}=\frac{\rho_{0}}{k},$ (15) which corresponds for large $N$ to an exponential rank-size distribution $k(r)=k_{1}\exp\left[\Omega-\frac{r-1}{\rho_{0}}\right],$ (16) where $r$ is the rank. This behaviour, which corresponds to the class of universality $\gamma=0$ in (1), has been empirically found by Costa Filho et al. [8] in the distribution of votes in the Brazilian electoral results. We have found such a behaviour in the city-size distribution of small regions and electoral results, like the province of Huelva (Spain) [23], and the 2008 Spanish General Elections results [24], respectively. We show in figure 1a and 1b their rank-size distribution in semi-logarithmic scale, where a straight line corresponds to a distribution of type (16). Most of the distribution can be linearly fitted, with a correlation coefficient of $0.994$ and $0.998$ respectively. From these fits we have obtain a bulk density of $\rho_{0}=0.058$ for the General Elections results, and in the case of Huelva $\rho_{0}=17.1$ ($N=77$, $\Omega=4.5$). Using historical data for the latter [23], we have used the backward differentiation formula to calculate the relative growth rate of the $i$-th city as $w_{i}=\frac{\ln k_{i}^{(2008)}-\ln k_{i}^{(2007)}}{\Delta t}$ (17) where $k_{i}^{(2007)}$ and $k_{i}^{(2008)}$ are the number of inhabitants of the $i$-th city in $2007$ and $2008$ respectively and $\Delta t=1$ year. We have obtained $\overline{w}=0.012$ years-1 and $\sigma_{w}=0.032$ years-1. However, the regularities are not always obvious, as shown for the most voted parties in Spain’08 or the whole distribution of the 2005 General Elections results in the United Kingdom [25] (figure 1c). In both cases, the competition between parties seems to play an important role, and the assumption of non- interacting elements can be unrealistic 222The effects of interaction are studied in [26], where we go beyond the non-interacting system using a microscopic description based on complex networks.. ## 4 The Zipf regime In the previous subsection we considered that $N/\Omega$ remains finite even in the thermodynamic limit, i.e., the system reaches the bulk density $\rho_{0}$. However, if $N/\Omega\rightarrow 0$ as $\Omega\rightarrow\infty$, i.e., the system is exposed to an empty infinite volume, the normalization can not be achieved and the constraint has to be removed ($\mu=0$). We call this case the _Zipf regime_ , in order to distinguish it from the general. Considering only the $k$ coordinate in the domain $[k_{1},\infty)$, the information measure for the total density distribution $\widetilde{f}(k)\rmd k=f(u)\rme^{u}\rmd u=p(u)\rmd u$, reads $I_{u}=c_{u}\int\rmd u~{}\rme^{u}f(u)\left\|\frac{\partial\ln f(u)}{\partial u}\right\|^{2},$ (18) and the extremization problem $\delta\left\\{c_{u}\int\rmd u~{}\rme^{u}f\left\|\frac{\partial\ln f}{\partial u}\right\|^{2}\right\\}=0.$ (19) Introducing $f(u)=\rme^{-u}\Psi^{}(u)\Psi(u)$, and varying with respect to $\Psi^{}$ leads to the Schrödinger equation $\left[-4\frac{\partial^{2}}{\partial u^{2}}+1\right]\Psi(u)=0.$ (20) Taking the boundary conditions $\lim_{u\rightarrow\infty}f(u)=0$ and $f(u_{1})=C$ where $u_{1}=\ln k_{1}$ and $C$ is a constant, the solution to the equation is $\Psi(u)=C^{\prime}\rme^{-u/2}$, where $C^{\prime}=\sqrt{C}\rme^{u_{1}}$. It can be shown that this is just an exponential decay in $u$ space $f(u)\rme^{u}\rmd u=C\rme^{-u}\rmd u$. This solution leads to the total density distribution $\widetilde{f}(k)=\frac{C^{2}}{k^{2}}$ (21) with $C^{2}=Nk_{1}$ for a normalized density. It corresponds in the continuous limit to a rank-size distribution of the type $k(r)=\frac{Nk_{1}}{r},$ (22) which is the Zipf’s law (universal class $\gamma=1$) of [2, 3, 4, 6, 7, 9, 10, 11]. This result is remarkable: for the first time _Zipf’s law is derived from first principles_. In figure 2 we show the known behaviour [11] of the rank size distribution of the top 100 largest cities of the United States [27], which shows an slope near $-1$ ($\gamma=1$) in the logarithmic representation of the rank-plot. Figure 2: (colour on-line) Rank-plot of the 100 largest cities of the United States. The appearance of the bulk and the Zipf regime in a SFIG can be understood studying empirical data. We have studied the system formed by all Physics journals [28] ($N=310$) using their total number of cites as coordinate $k$. If a journal receives more cites due to its popularity, it becomes even more popular and therefore it will receive more cites. Under such conditions proportional growth and scale invariance are expected. Since we consider all fields of Physics, correlation effects are much lower than only consider journals of an specific field, so the non-interacting approximation seems realistic in this case. In figure 3 we show the rank-plot of the number of cites of Physic journals, where we have found a slope near $-1$ for the most- cited journals in the logarithmic representation (figure 3a) and an slope near $+1$ for the less-cited journals (figure 3b). For the central part of the distribution bulk density reaches a value of $\rho_{0}\sim 57$ (figure 3c). Figure 3: (colour on-line) a, rank-plot of the total number of cites of physics journal, from most-cited to less-cited, in logarithm scale. b, sorted from less-cited to most cited c, same as a, in semi-logarithm scale.(red dots: empirical data; blue line: linear fitting). This distribution shows an extraordinary symmetric behaviour under the change $k\rightarrow 1/k$ ($u\rightarrow-u$). We show in figure 4 the raw empirical data compared with the distribution obtained from the transformation $k^{\prime}=c/k$ ($u^{\prime}=-u+\ln c$), where $c=3.3\times 10^{6}$. The symmetry of this system is an important clue to understand both regimes, and represents a perfect example of the conditions needed to observe bulk and Zipf regimes in a non-interacting scale-invariant system. The main part of the density distribution reaches the bulk density obeying (15), whereas Zipf’s law emerge at the edges, obeying (21): following the analogy with the physics of gases and fluids, we can think of the system as a drop, where the Zipf regime is the sign of a _surface_ since it reproduces how the density exponentially falls from the bulk density to zero in $u$ space when the system is exposed to an infinite empty volume. This effect is clearly visible in figure 5, where the empirical density distribution $p(u)\rmd u$ in $u$ space is compared with the fitted density $p(u)=\left\\{\begin{array}[]{ll}\rho_{a}\rme^{u}&\mathrm{if~{}}u<u_{a}\\\ \rho_{0}&\mathrm{if~{}}u_{a}<u<u_{b}\\\ \rho_{b}\rme^{-u}&\mathrm{if~{}}u_{b}>u\end{array}\right.$ (23) where $\rho_{a}=0.1$, $\rho_{0}=57$, $\rho_{b}=4.3\times 10^{5}$, $u_{a}=5.2$ and $u_{b}=10$. These findings lead us to conclude that the system of Physics journals sorted by total number of cites is a perfect example of the scale- free ideal gas at equilibrium. Figure 4: (colour online) Rank-plot of the total number of cites of Physics journal, from most-cited to less-cited, compared with the distribution obtained from the inverse transformation $k^{\prime}=3.3\times 10^{6}/k$ where $k$ is the number of cites. Figure 5: (colour online) Empirical density distribution in $u$ space of the total number of cites of Physics journals, compared with (23). The bulk regime and the Zipf regime at the edges is clearly visible. ## 5 The harmonically constrained system We now consider a system with a constraint in a given observable $\langle A\rangle$ which locally depends on $k$, $A=A(k)$. The second order Taylor expansion with respect to $u=\ln k$ near a minimum is written as $\widetilde{A}(u)=A(\rme^{u})\simeq A_{0}+A_{2}/2(u-u_{m})^{2}$, where $A_{0}$, $A_{2}$ and $u_{m}$ are constants. Introducing this constraint and the normalization condition to the number of elements $N$ of the total density distribution, the extremization problem reads $\begin{array}[]{rl}\delta&\displaystyle\left\\{c_{u}\int\rmd u~{}\rme^{u}f\left\|\frac{\partial\ln f}{\partial u}\right\|^{2}+\mu\int\rmd u~{}\rme^{u}f\right.\\\ &\displaystyle\left.+\lambda\int\rmd u~{}\rme^{u}f\left[A_{0}+\frac{1}{2}A_{2}(u-u_{m})^{2}\right]\right\\}=0\end{array}$ (24) Introducing $f(u)=\rme^{-u}\Psi^{}(u)\Psi(u)$, and varying with respect to $\Psi^{}$ leads to the quantum harmonic oscillator equation $\left[-4\frac{\partial^{2}}{\partial u^{2}}+\lambda^{\prime}(u-u_{0})^{2}+\nu^{\prime}\right]\Psi(u)=0,$ (25) where now we have defined $\lambda^{\prime}=(\lambda A_{2})/2c_{w}$ and $\nu^{\prime}=(\nu+1+\lambda A_{0})/c_{u}$. The ground state solution is a gaussian distribution, which now yields a total density distribution of the form of a log-normal distribution $\widetilde{f}(k)=\frac{N}{k\sqrt{2\pi}\sigma_{u}}\exp\left(\frac{-(\ln k-u_{m})^{2}}{2\sigma_{u}^{2}}\right),$ (26) with $\|\lambda^{\prime}\|^{-1/2}=\sigma_{u}^{2}=\langle A\rangle-A_{0}$. Note that if $A_{0}=0$, the constraint can be also understood as a constraint in the variance of $u$. The log-normal distribution has been widely observed in a large number of scale-invariant systems [29]. In [30] S. Fortunato and C. Castellano found this behaviour in the electoral results of different countries and for different years. We can think of this constraint as the effect of polices or social factors: low popularity candidates are penalized since the party does not present them for the elections, and high popularity candidates are penalized by the competition in campaign. Both effects can be approximated to second order as a harmonic potential, however anharmonic effects are expected in a high order study. Defining $H^{\prime\prime}=1/\Delta\tau$ and $k_{m}=\ln u_{m}$ being $k_{m}=m\Delta k$, the entropy equation of state reads in this case $S=N\kappa\left\\{\ln\frac{2\pi\sigma_{u}\sigma_{w}m^{2}}{NH^{\prime\prime}}+2\right\\},$ (27) which maintains scale invariance. ## 6 The microscopic description The dynamics of the system can be microscopically described as a stochastic process using (4) and the density distribution (12). Treating $w$ as a random variable, the stochastic equation of motion is written as a geometrical Brownian motion $\rmd k=k\overline{w}\rmd t+k\sigma_{w}\rmd W,$ (28) where $\rmd W$ is a Wiener process. In the $u$ space, this equation reads $\rmd u=\overline{w}\rmd t+\sigma_{w}\rmd W,$ (29) which describes the well-known Brownian motion. (28) exactly describes the dynamical condition found empirically in [7] and also the stochastic proportional growth model used in [11] to obtain Zipf’s law. We can think of this sort of simulations as the equivalent of molecular dynamics simulations for gases and liquids [31]. Effectively, (29) implies that a uniform density in $u$ space of $N$ Brownian walkers moving in a fixed volume $\Omega$ —a model used in the literature to describe the IG [31]— describes the SFIG when we represent the system with the coordinates $(k,v)$. In figure 1a we show the rank-plot for a system of $N=78$ geometrical Brownian walkers with $\sigma_{w}=0.029$ in a volume of $\Omega=4.5$ and $k_{1}=200$ in reduced units, which nearly describes the distribution of the population of the province of Huelva. ## 7 Summary and Discussion We have shown that a thermodynamic description of scale-invariant systems can be formulated from the principle of extreme information, finding an analogy with the thermodynamics of gases and fluids. We have derived the density distribution in configuration space and the entropy equation of state of the scale-free ideal gas in the thermodynamic limit, and have found empirical evidences of its existence in city population, electoral results and cites to Physics journals. In this context, Zipf’s law emerges naturally as the equilibrium density of the non-interacting system when the volume grows to infinity, what we call the Zipf regime. Using empirical data we have seen that this regime can be understood as the density fall of a surface between the bulk and an empty volume. We have also studied the effect of a harmonic constraint, finding that in this case the density of the system follows a log- normal distribution, which has been empirically observed in electoral results and in many other scale-invariant systems [29]. Finally we have shown with a simulation of city population that a geometrical Brownian motion can describe the system at a microscopic level. It is well known that in real gases the most interesting situations emerge when interactions between particles become relevant, originating deviations from the equation of state of the IG, and making room for the appearance of, e.g., phase transitions [21]. Analogously, one should also expect this rich phenomenology to show up in scale-invariant real systems, which may explain deviations from Zipf’s law in empirical distributions. A study beyond the ideal gas is in progress, and further results will be reported [26]. We would like to thank M. Barranco, R. Frieden, A. Plastino, and B. H. Soffer for useful discussions. This work has been partially performed under grant FIS2008-00421/FIS from DGI, Spain (FEDER). ## Appendix A The translational invariant ideal gas In this appendix we derive from the principle of extreme information the density distribution in configuration space and the entropy equation of state of the translational invariant ideal gas (IG) [21]. The IG model describes non-interacting classical particles of mass $m$ with coordinates $\bi{q}=(\bi{r},\bi{p})$, where $m\rmd\bi{r}/\rmd t=\bi{p}$. We assume that these coordinates are canonical [22] and uncorrelated. This assumption is introduced in the information measure (2) as $c_{ij}=c_{i}\delta_{ij}$, where $c_{i}=c_{r}$ for space coordinates, $c_{i}=c_{p}$ for momentum coordinates, and $\delta_{ij}$ is the Kronecker delta. The density distribution can be factorized as $f(\bi{r},\bi{p})=\rho(\bi{r})\eta(\bi{p})$, and the information measure $I=I_{r}+I_{p}$ reads, if $D$ is the dimension of the space $\begin{array}[]{rl}I_{r}=&\displaystyle c_{r}\int\rmd^{D}\bi{r}~{}\rho(\bi{r})\left\|\bi{\nabla}_{r}\ln\rho(\bi{r})\right\|^{2}\\\ I_{p}=&\displaystyle c_{p}\int\rmd^{D}\bi{p}~{}\eta(\bi{p})\left\|\bi{\nabla}_{p}\ln\eta(\bi{p})\right\|^{2}.\end{array}$ (30) In the extremization of Fisher information we constrain the normalization of $\rho(\bi{r})$ and $\eta(\bi{p})$ to the total number of particles $N$ and to $1$, respectively $\int\rmd^{D}\bi{r}~{}\rho(\bi{r})=N,\qquad\int\rmd^{D}\bi{p}~{}\eta(\bi{p})=1.$ (31) In addition, we penalize infinite values for the particle momentum with a constraint on the variance of $\eta(\bi{p})$ to a given measured value $\int\rmd^{D}\bi{p}~{}\eta(\bi{p})(\bi{p}-\overline{\bi{p}})^{2}=D\sigma_{p}^{2},$ (32) where $\overline{\bi{p}}$ is the mean value of $\bi{p}$. For each degree of freedom it is known from the Virial theorem that the variance is related to the temperature $T$ as $\sigma_{p}^{2}=mk_{B}T$, being $k_{B}$ the Boltzmann factor. The variation yields $\displaystyle\delta\left\\{c_{r}\int\rmd^{D}\bi{r}~{}\rho\left\|\bi{\nabla}_{r}\ln\rho\right\|^{2}+\mu\int\rmd^{D}\bi{r}~{}\rho\right\\}=0$ (33) and $\delta\left\\{c_{p}\int\rmd^{D}\bi{p}~{}\eta\left\|\bi{\nabla}_{p}\ln\eta\right\|^{2}+\lambda\int\rmd^{D}\bi{p}~{}\eta(\bi{p}-\overline{\bi{p}})^{2}+\nu\int\rmd^{D}\bi{p}~{}\eta\right\\}=0,$ (34) where $\mu$, $\lambda$ and $\nu$ are Lagrange multipliers. Introducing $\rho(\bi{r})=\Psi^{}(\bi{r})\Psi(\bi{r})$ and varying (33) with respect to $\Psi^{}$ leads to the Schrödinger equation [18] $\left[-4\nabla_{r}^{2}+\mu^{\prime}\right]\Psi(\bi{r})=0,$ (35) where $\mu^{\prime}=\mu/c_{r}$. To fix the boundary conditions, we first assume that the $N$ particles are confined in a box of volume $V$, and next we take the thermodynamic limit (TL) $N,V\rightarrow\infty$ with $N/V$ finite. The equilibrium state compatible with this limit corresponds to the ground state solution, which is the uniform density $\rho(\bi{r})=N/V$. Introducing $\eta(\bi{p})=\Phi^{}(\bi{p})\Phi(\bi{p})$ and varying (34) with respect to $\Phi^{}$ leads to the quantum harmonic oscillator equation [18] $\left[-4\nabla_{p}^{2}+\lambda^{\prime}(\bi{p}-\overline{\bi{p}})^{2}+\nu^{\prime}\right]\Phi(\bi{p})=0,$ (36) where $\lambda^{\prime}=\lambda/c_{p}$ and $\nu^{\prime}=\nu/c_{p}$. The equilibrium configuration corresponds to the ground state solution, which is now a gaussian distribution. Using (32) to identify $\|\lambda^{\prime}\|^{-1/2}=\sigma_{p}^{2}$ we get the Boltzmann distribution, which leads to a density distribution in configuration space of the form $f(\bi{r},\bi{p})=\frac{N}{V}\frac{\exp\left[-(\bi{p}-\overline{\bi{p}})^{2}/2\sigma_{p}^{2}\right]}{(2\pi\sigma_{p}^{2})^{D/2}}.$ (37) If $H$ is the elementary volume in phase space, the total number of microstates is $Z=N!H^{DN}\prod_{i=1}^{N}f_{1}(\bi{r}_{i},\bi{p}_{i})$, where $f_{1}=f/N$ is the monoparticular distribution and $N!$ counts all possible permutations for distinguishable particles. The entropy $S=-k_{B}\ln Z$ is written as $S=Nk_{B}\left\\{\ln\frac{V}{N}\left(\frac{2\pi\sigma_{p}^{2}}{H^{2}}\right)^{D/2}+\frac{2+D}{2}\right\\},$ (38) where we have used the Stirling approximation for $N!$. This expression is in exact accordance with the known value of the entropy for the IG [21], which shows the predictive power of the Fisher formulation. ## References ## References * [1] Fractals in Physics, edited by Aharony A and Feder J 1989 Proc. Conf. in honor of Mandelbrot B B, Vence, France (North Holland, Amsterdam). * [2] Paech K, Bauer W and Pratt S 2007 Phys. Rev. C 76, 054603; Campi X and Krivine H 2005 Phys. Rev. C 72, 057602 ; Ma Y G et al. 2005 Phys. Rev. C 71, 054606. * [3] Furusawa C and Kaneko K 2003 Phys. Rev. Lett. 90, 088102. * [4] Zipf G K 1949 Human Behavior and the Principle of Least Effort (Addison-Wesley Press, Cambridge, Mass.); Kanter I and Kessler D A 1995 Phys. Rev. Lett. 74, 4559. * [5] Newman M E J 2001 Phys. Rev. E 64, 016131. * [6] Barabasi A L and Albert R 2002 Rev. Mod. Phys. 74, 47 * [7] Maillart T, Sornette D, Spaeth S and von Krogh G 2008 Phys. Rev. Lett. 101, 218701. * [8] Costa Filho R N, Almeida M P, Andrade J S and Moreira J E 1999 Phys. Rev. E 60, 1067. * [9] Malacarne L C, Mendes R S and Lenzi E K 2001 Phys. Rev. E 65, 017106; Marsili M and Zhang Yi-Cheng 1998 Phys. Rev. Lett. 80, 2741. * [10] Axtell R L 2001 Science 293, 1818. * [11] Gabaix X 1999 Quarterly Journal of Economics 114, 739. * [12] Kechedzhi K E, Usatenko O V and Yampol’skii V A 2005 Phys. Rev. E. 72, 046138; Ree S 2006 Phys. Rev. E. 73, 026115. * [13] Moreira A A, Paula D R, Costa Filho R N and Andrade J S 2006 Phys. Rev. E 73, 065101(R). * [14] Reed W J and Hughes B D 2002 Phys. Rev. E. 66, 067103. * [15] Frieden B R, Plastino A, Plastino A R and Soffer B H 1999 Phys. Rev. E 60, 48; 2002 Phys. Rev. E 66, 046128. * [16] Pennini F, Plastino A, Soffer B H and Vignat C 2009 Phys. Let. A 373, 817. * [17] Frieden B R and Soffer B H 1995 Phys. Rev. E 52, 2274; Frieden B R 1998 Physics from Fisher Information, 2nd Ed. (Cambridge Univ. Press, Cambridge); Frieden B R 2004 Science from Fisher Information (Cambridge Univ. Press, Cambridge). * [18] Cohen-Tannoudji C, Diu B and Laloe F 2006 Quantum Mechanics (Wiley-Interscience, New York). * [19] Pennini F and Plastino A 2006 Phys. Lett. A 349, 15. * [20] Frieden B R and Gatenby R A 2005 Phys. Rev. E 72, 036101. * [21] Zemansky M W and Dittmann R H 1981 _Heat and Thermodynamics_(McGraw-Hill, London). * [22] Goldstein H, Poole C and Safko J 2002 _Classical Mechanics_ 3rd Ed. (Addison Wesley, San Francisco). * [23] National Statistics Institute, Spain, www.ine.es. * [24] Ministry of the Interior, Spain, www.elecciones.mir.es * [25] Electoral Commission, Government of the UK, www.electoralcommission.org.uk. * [26] Hernando A, Villuendas D, Abad M and Vesperinas C (2009) arXiv:0905.3704v1 [cond-mat.stat-mech] * [27] Census bureau website, Government of the USA, www.census.gov. * [28] Journal Citation Reports (JCR) for 2007, Thomson Reuters * [29] Limpert E, Stahel W and Abbt M 2001 BioScience, 51, 341. * [30] Fortunato S and Castellano C 2007 Phys. Rev. Lett., 99, 138701. * [31] Gould H and Tobochnik J 1996 _An Introduction to Computer Simulation Methods: Applications to Physical Systems_ , 2nd Ed. (Addison-Wesley).	arxiv-papers	2009-02-16T16:51:21	2024-09-04T02:49:00.624693	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "A. Hernando, D. Puigdomenech, D. Villuendas and C. Vesperinas", "submitter": "Alberto Hernando", "url": "https://arxiv.org/abs/0902.2738" }
0902.2829	# On the residual effective potential within Global 1D Quantum Gravity Lukasz A. Glinka111Electronic address: laglinka@gmail.com _International Institute for Applicable_ _Mathematics & Information Sciences,_ _Hyderabad (India) and Udine (Italy),_ _B.M. Birla Science Centre, Adarsh Nagar,_ _Hyderabad - 500 063, Andra Pradesh, India_ ###### Abstract The conjecture on Global One–Dimensionality within Quantum General Relativity leads to the model of quantum gravity possessing nontrivial field theoretic content. This is a midisuperspatial model, which quantum mechanical part can be considered independently. The fragment, basing on the Dirac–Faddeev canonical primary quantization of Hamiltonian constraint, in fact constitutes minimal effective model within standard quantum geometrodynamics with potential different from the standard. It uses one–dimensional wave functions, where the (global) dimension is a volume form of a 3-embedding. In this paper some elements of the global 1D quantum mechanics are presented. We consider absence of matter fields. Generalized functional expansion in the global dimension of the effective potential is discussed. Finally, its residual approximation, the Newton–Coulomb type potential, realized by all embeddings being maximally symmetric 3-dimensional Einstein manifolds is studied. ## 1 Introduction As it was shown in the last topical papers [1, 2, 3, 4, 5, 6, 7] of the author, taking into account the Global One–Dimensionality supposition within Quantum General Relativity given by the Wheeler–DeWitt quantum geometrodynamics, according to the Dirac–Faddeev Hamiltonian approach, allows to consider this model of Quantum Gravity as a bosonic classical field theory. For this field theory quantization in the Fock space of creators and annihilators with stable Bogoliubov–Heisenberg vacuum state can be done standardly, and an adequate thermodynamics of macrostates – quantum states of 3-dimensional embedded induced space – can be constructed directly. This approach results in the model of Quantum Gravity with unique quantum–statistical content. However, the part of this full field-theoretic model of Quantum Gravity, strictly related to canonical primary quantization of the Hamiltonian constraint, can be reconsidered separately, as independent model of Quantum Gravity. In and of itself this small piece of the quantum-statistical field theory constitutes a (globally) one-dimensional quantum mechanics describing Quantum Gravity also related to any $3+1$ metric of General Relativity. The Quantum Mechanics looks like formally as radial-type Schrödinger wave equation, where the global dimension is generalized distance of common situation – determinant (volume form) of metric of 3-dimensional embedding. In this paper some elements of the one-dimensional quantum-mechanical construction are discussed. Maximally symmetric 3-dimensional Einstein manifolds, those are embeddings reconstructing the Newton–Coulomb type potential within the model of Quantum Gravity, are mainly considered. The content of this paper is as follows. First, we discuss in condensed way the standard way from the Einstein–Hilbert General Relativity with cosmological constant and the Hawking–Hartle nondynamical boundary term, by $3+1$ Dirac–ADM decomposition of metric, the DeWitt constraints algebra, and the Hamiltonian Dirac–Faddeev quantization of primary and secondary constraints resulting in the Wheeler–DeWitt evolution equation. For the obtained quantum geometrodynamical model of Quantum Gravity we apply the Global One–Dimensionality supposition and by global transformation of variables we reduce the Wheeler–DeWitt theory to one-dimensional quantum mechanics with an effective potential. Received model is related to 3-dimensional embeddings. We discuss some possible physical scenarios with respect to the effective potential. The crucial subject is discussing the situation, where the generalized Newton–Coulomb potential can be obtained. In the presented model this type effective theory is obtained by any maximally symmetric Einstein 3-manifolds. We discuss generalized boundary conditions for this case. ## 2 Global 1D Quantum Gravity ### 2.1 Quantum geometrodynamics Pseudo–Riemannian [8] manifold $(M,g)$ given by metric $g_{\mu\nu}$, coordinates $x^{\mu}$, affine connections $\Gamma^{\rho}_{\mu\nu}$, curvatures: Riemann $R^{\lambda}_{\mu\alpha\nu}$, Ricci $R_{\mu\nu}$, Ricci scalar $R$ $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ \Gamma_{\sigma\mu\nu}=\dfrac{1}{2}\left(g_{\mu\sigma,\nu}+g_{\sigma\nu,\mu}-g_{\mu\nu,\sigma}\right)\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ \Gamma^{\rho}_{\mu\nu}=g^{\rho\sigma}\Gamma_{\sigma\mu\nu}\leavevmode\nobreak\ ,$ (1) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!R^{\lambda}_{\mu\alpha\nu}=\Gamma^{\lambda}_{\mu\nu,\alpha}-\Gamma^{\lambda}_{\mu\alpha,\nu}+\Gamma^{\lambda}_{\sigma\alpha}\Gamma^{\sigma}_{\mu\nu}-\Gamma^{\lambda}_{\sigma\nu}\Gamma^{\sigma}_{\mu\alpha}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ R_{\mu\nu}=R^{\lambda}_{\mu\lambda\nu}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\ R=g^{\kappa\lambda}R_{\kappa\lambda}\leavevmode\nobreak\ ,$ (2) according to Einstein [9] is a solution of General Relativity field equations222In this article we use standardly the geometrized units system $8\pi G/3=c=\hbar=1$. $G_{\mu\nu}+\Lambda g_{\mu\nu}=3T_{\mu\nu}\qquad,\qquad G_{\mu\nu}\equiv R_{\mu\nu}-\dfrac{1}{2}Rg_{\mu\nu}\qquad,$ (3) where $\Lambda$ is cosmological constant, and $T_{\mu\nu}$ is stress-energy tensor, arise by Palatini [10] principle used to Hilbert–Hartle–Hawking [11, 12] action $S[g]\\!=\\!\int_{M}d\mu_{g}\left\\{-\dfrac{R}{6}+\dfrac{\Lambda}{3}+\mathcal{L}\right\\}-\dfrac{1}{3}\int_{\partial M}d\mu_{h}K\quad,$ (4) where $K$ is Gauss scalar curvature of spacelike boundary $(\partial M,h)$, $\mathcal{L}$ is Matter lagrangian, and $d\mu_{g}=d^{4}x\sqrt{-g}$, $d\mu_{h}=d^{3}x\sqrt{h}$ are invariant measures. Nash embedding theorem [13, 14, 15, 16] allows using $3+1$ Dirac–ADM decomposition [17, 18, 19], by embedding metric $h_{ij}$, lapse $N$ and shift $N_{i}$, $\displaystyle g_{\mu\nu}=\left[\begin{array}[]{cc}-N^{2}+h^{ij}N_{i}N_{j}&N_{j}\\\ N_{i}&h_{ij}\end{array}\right]\quad,\quad h_{ik}h^{kj}=\delta_{i}^{j}\quad,$ (7) and transforms the action (4) into the Hamiltonian form $\displaystyle S[g]=\int dt\int_{\partial M}d^{3}x\left\\{\pi\dot{N}+\pi^{i}\dot{N_{i}}+\pi^{ij}\dot{h}_{ij}-NH- N_{i}H^{i}\right\\},\leavevmode\nobreak\ \left(\dot{a}\equiv\dfrac{\partial a}{\partial t}\right).$ (8) By Gauss–Codazzi equations [20, 21, 22], nontrivial $\pi$’s, and $H$, $H^{i}$ are $\displaystyle\pi^{ij}$ $\displaystyle=$ $\displaystyle\sqrt{h}\left(K^{ij}-Kh^{ij}\right)\quad,$ (9) $\displaystyle H$ $\displaystyle=$ $\displaystyle\sqrt{h}\left\\{{{}^{(3)}\\!R}+K^{2}-K_{ij}K^{ij}-2\Lambda-6\varrho\right\\}\quad,\quad H^{i}=2\pi^{ij}_{\leavevmode\nobreak\ ;j}\quad,$ (10) where ${{}^{(3)}\\!R}$ is Ricci scalar of embedding, $\varrho=n^{\mu}n^{\nu}T_{\mu\nu}$ is stress-energy tensor projected onto normal vector field $n^{\mu}=[1/N,-N^{i}/N]$. Extrinsic curvature $K_{ij}$ ($\mathrm{Tr}K_{ij}\equiv K$) is constrained with $\dot{h}_{ij}$, $N$, and symmetrized intrinsic covariant derivative of $N_{(i\|j)}$ $\dot{h}_{ij}=2\left(NK_{ij}+N_{(i\|j)}\right).$ (11) According to DeWitt [23] $H^{i}$ are diffeomorphisms $\widetilde{x}^{i}=x^{i}+\delta x^{i}$ generators $\displaystyle i\left[h_{ij},\int_{\partial M}H_{a}\delta x^{a}d^{3}x\right]$ $\displaystyle=$ $\displaystyle-h_{ij,k}\delta x^{k}-h_{kj}\delta x^{k}_{\leavevmode\nobreak\ ,i}-h_{ik}\delta x^{k}_{\leavevmode\nobreak\ ,j}\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (12) $\displaystyle i\left[\pi_{ij},\int_{\partial M}H_{a}\delta x^{a}d^{3}x\right]$ $\displaystyle=$ $\displaystyle-\left(\pi_{ij}\delta x^{k}\right)_{,k}+\pi_{kj}\delta x^{i}_{\leavevmode\nobreak\ ,k}+\pi_{ik}\delta x^{j}_{\leavevmode\nobreak\ ,k}\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (13) where $H_{i}=h_{ij}H^{j}$. Dirac [24] time-preservation of the primary constraints $\pi\approx 0$ and $\pi^{i}\approx 0$ leads to secondary constraints - scalar and vector $\displaystyle H\approx 0\quad,\quad H^{i}\approx 0\quad,$ (14) which create nontrivial first-class type constraints algebra [23] $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!i\left[\int_{\partial M}H\delta x_{1}d^{3}x,\int_{\partial M}H\delta x_{2}d^{3}x\right]=\int_{\partial M}H^{a}\left(\delta x_{1,a}\delta x_{2}-\delta x_{1}\delta x_{2,a}\right)d^{3}x\quad,$ (15) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!i\left[H_{i}(x),H_{j}(y)\right]=\int_{\partial M}H_{a}c^{a}_{ij}d^{3}z\quad,\quad i\left[H(x),H_{i}(y)\right]=H\delta^{(3)}_{,i}(x,y)\quad,$ (16) where $c^{a}_{ij}=c^{a}_{ij}[x,y,z]=\delta^{a}_{i}\delta^{b}_{j}\delta^{(3)}_{,b}(x,z)\delta^{(3)}(y,z)-(i\leftrightarrow j,x\leftrightarrow y)$ are structure constants of diffeomorphism group, and all Lie’s brackets of $\pi$’s and $H$’s vanish. Scalar constraint determines dynamics, vector one merely reflects diffeoinvariance. By using of the conjugate momenta (9) the scalar constraint transforms into the Einstein–Hamilton–Jacobi equation widely famous in the last four decades literature [25]–[68] $H=G_{ijkl}\pi^{ij}\pi^{kl}+\sqrt{h}\left({}^{(3)}R-2\Lambda-6\varrho\right)\approx 0\quad,$ (17) where $G_{ijkl}\equiv(2\sqrt{h})^{-1}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right)$ is metric of the Wheeler–DeWitt superspace, a factor space of all $C^{\infty}$ Riemannian metrics on $\partial M$, and a group of all $C^{\infty}$ diffeomorphisms of $\partial M$ that preserve orientation [69]. The Dirac–Faddeev primary canonical quantization method [17, 70] $\displaystyle i\left[\pi^{ij}(x),h_{kl}(y)\right]$ $\displaystyle=$ $\displaystyle\dfrac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{i}\delta_{k}^{j}\right)\delta^{(3)}(x,y)\quad,$ (18) $\displaystyle i\left[\pi^{i}(x),N_{j}(y)\right]$ $\displaystyle=$ $\displaystyle\delta^{i}_{j}\delta^{(3)}(x,y)\quad,\quad i\left[\pi(x),N(y)\right]=\delta^{(3)}(x,y)\quad,$ (19) used for the Hamiltonian constraint (17) leads to the standard model of Quantum Gravity based on the Wheeler–DeWitt equation [71, 23] $\left\\{-G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta h_{kl}}-\sqrt{h}\left(-\leavevmode\nobreak\ {{}^{(3)}\\!R}+2\Lambda+6\varrho\right)\right\\}\Psi[h_{ij},\phi]=0\quad,$ (20) called quantum geometrodynamics, where $\phi$ are Matter fields. Other first class constraints $\pi\Psi[h_{ij},\phi]=0\quad,\quad\pi^{i}\Psi[h_{ij},\phi]=0\quad,\quad H^{i}\Psi[h_{ij},\phi]=0\quad,$ (21) merely reflect diffeoinvariance, and are not important in this model. ### 2.2 The Global Dimension The toy model of Quantum Gravity, Global One–Dimensionality supposition within the Quantum General Relativity determined by the Wheeler–DeWitt equation (20), arises from the assumption that Matter fields $\phi$ as well as the quantum- geometrodynamical wave function $\Psi[h_{ij},\phi]$ are functionals of only embedding 3-space volume form $h=\det h_{ij}=\dfrac{1}{3}\varepsilon^{ijk}\varepsilon^{abc}h_{ia}h_{jb}h_{kc}\quad,$ (22) where $\varepsilon^{ijk}$ is the Levi-Civita density. So, actually the following situation $\displaystyle\phi(x)$ $\displaystyle\rightarrow$ $\displaystyle\phi[h]\quad,$ (23) $\displaystyle\varrho(\phi)$ $\displaystyle\rightarrow$ $\displaystyle\varrho[h]\quad,$ (24) $\displaystyle\Psi[h_{ij},\phi]$ $\displaystyle\rightarrow$ $\displaystyle\Psi[h]\quad,$ (25) lies in the fundamentals of the model. Applying the transformation of variables $h_{ij}\rightarrow h$ in the Wheeler–DeWitt equation (20), _i.e._ putting into the differential operator the relation $\dfrac{\delta}{\delta h_{ij}}=\mathcal{J}\left(h_{ij},h\right)\dfrac{\delta}{\delta h}\quad,$ (26) where $\mathcal{J}\left(h_{ij},h\right)$ is formally the Jacobi matrix of variables transformation $\mathcal{J}\left(h_{ij},h\right)=hh^{ij}\quad,$ (27) and doing elementary algebraic manipulations, the reduction of full quantum geometrodynamics – (globally) one-dimensional quantum mechanical model of Quantum Gravity is received $\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+V_{eff}[h]\right)\Psi[h]=0.$ (28) Here $V_{eff}[h]$ is the effective potential $V_{eff}[h]\equiv V_{G}[h]+V_{C}[h]+V_{M}[h]\quad,$ (29) that is a simple algebraic sum of the three fundamental potential constituents $\displaystyle V_{G}[h]=\dfrac{2}{3}\dfrac{{{}^{(3)}\\!R}}{h}\quad,\quad V_{C}[h]=-\dfrac{4}{3}\dfrac{\Lambda}{h}\quad,\quad V_{M}[h]=-\dfrac{4}{h}\varrho[h]\quad,$ (30) related to pure geometry of 3-dimensional embedding space ($G$), cosmological constant ($C$), and Matter fields ($M$). On the one side, by the simple identification of the effective potential with the square of mass of the boson $V_{eff}[h]\equiv m^{2}[h]$, one can state that the quantum evolution (28) describes the model of Quantum Gravity in terms of classical theory of massive bosonic field $\Psi[h]$. It leads to construction of an adequate quantum field theory in the Fock space of static Bogoliubov–Heisenberg operator basis of creators and annihilators. One can do also some statistical nature conclusions on thermodynamics of quantum states related to any 3-dimensional embedding space. The meaningful part of this field-theoretic model was discussed in the previous papers of the author [2, 3, 4, 5, 6, 7], and is not the leading theme of the present paper. However, on the other side one can approve the nonrelativistic type interpretation of the one–dimensional quantum dynamics (28)-(30), and treat the received global model as some the effective one-dimensional Schrödinger quantum mechanics with a certain selected potential being a functional of volume form of 3-dimensional embedding space. In the spirit of this philosophy the potential $V_{eff}[h]$ has intriguing meaning – the equation (29) is the equality between any ”effective physics”, maybe given by other type considerations of particle physics or condensed matter physics, and three basic constituents related to an embedding 3-space – ”geometric”, ”cosmological”, and ”material” ones. Let us assume that concrete form of $V_{eff}$ can be established from any other theoretical digressions. In this case Ricci scalar curvature of a 3-dimensional embedding can be established as ${{}^{(3)}\\!R}=6\left(\varrho[h]+\dfrac{\Lambda}{3}+\dfrac{h}{4}V_{eff}[h]\right).$ (31) Immediately, however, the fundamental question suggests itself from this type construction: How to determine the potential $V[h]$ correctly? This theoretical problem is more sophisticated and can not be solved by direct simple way. Presently, one can list some possible physical scenarios in the one-dimensional model, with respect to the form of the potential $V[h]$. 1. 1. The case of constant non vanishing total potential $V_{eff}=V_{0}\neq 0$. For this situation the Ricci scalar curvature of an embedding and the one- dimensional wave equation are ${{}^{(3)}\\!R}=6\left(\varrho+\dfrac{\Lambda}{3}+\dfrac{V_{0}}{4}h\right)\quad,\quad\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+V_{0}\right)\Psi_{0}[h]=0.$ (32) Here $\Psi_{0}[h]$ is the wave function related to $V_{eff}=V_{0}$. 2. 2. The case of vanishing total potential $V_{eff}=0$. For this situation the Ricci scalar curvature of an embedding and the one-dimensional wave equation are ${{}^{(3)}\\!R}=6\left(\varrho+\dfrac{\Lambda}{3}\right)\quad,\quad\dfrac{\delta^{2}}{\delta{h^{2}}}\Psi_{F}[h]=0.$ (33) Here $\Psi_{F}$ is the "free" wave function related to $V_{eff}=0$. 3. 3. The case, when a sum of geometric and cosmological potential contributions vanishes $V_{G}+V_{C}=0$, but total potential is no zeroth $V_{eff}\neq 0$. For this situation the Ricci scalar curvature of an embedding and the one- dimensional wave equation are ${{}^{(3)}\\!R}=2\Lambda\quad,\quad\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+V_{M}[h]\right)\Psi_{M}[h]=0.$ (34) Here $\Psi_{M}$ is the "material" wave function related to $V_{M}\neq 0$. 4. 4. The case, when a sum of geometric and material potential contributions vanishes $V_{G}+V_{M}=0$, but total potential is no zeroth $V_{eff}\neq 0$. For this situation the Ricci scalar curvature of an embedding and the one- dimensional wave equation are ${{}^{(3)}\\!R}=6\varrho\quad,\quad\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+V_{C}[h]\right)\Psi_{C}[h]=0.$ (35) Here $\Psi_{C}$ is the "cosmological" wave function related to $V_{C}\neq 0$. 5. 5. The case, when a sum of cosmological and material potential contributions vanishes $V_{C}+V_{M}=0$, but total potential is no zeroth $V_{eff}\neq 0$. For this situation the matter fields energy density and the one-dimensional wave equation are $\varrho+\dfrac{\Lambda}{3}=0\quad,\quad\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+V_{G}[h]\right)\Psi_{G}[h]=0.$ (36) Here $\Psi_{G}$ is the "geometric" wave function related to $V_{G}\neq 0$. 6. 6. The other, more general, proposition can be application for the effective potential $V_{eff}[h]$ the formal (functional) Laurent series expansion in volume form $h$ in a infinitesimal neighborhood (a circle with a radius $h_{\epsilon}$) of the fixed initial value of volume form $h_{0}$ $V_{eff}[h]=\sum_{-\infty}^{\infty}a_{n}\left(h-h_{0}\right)^{n}\quad\mathrm{in}\quad C(h_{\epsilon})=\left\\{h:\|h-h_{0}\|<h_{\epsilon}\right\\},$ (37) where $a_{n}$ are series coefficients given by (classical) functional integral $a_{n}=\dfrac{1}{2\pi i}\int_{C(h_{\epsilon})}\dfrac{V_{eff}[h]}{\left(h-h_{0}\right)^{n+1}}\delta h.$ (38) In this case the Ricci scalar curvature of an embedding equals ${{}^{(3)}\\!R}=6\left(\varrho+\dfrac{\Lambda}{3}+\dfrac{1}{4}\sum_{-\infty}^{\infty}a_{n}h^{n+1}\right),$ (39) and the one-dimensional wave equation (28) yields $\left(\dfrac{\delta^{2}}{\delta{h^{2}}}+\sum_{-\infty}^{\infty}a_{n}h^{n}\right)\Psi[h]=0.$ (40) Naturally, there is many other opportunities for a form of the potential $V_{eff}[h]$. However, in this paper we will discuss only an especial case. ### 2.3 Digression on generalized dimensions Let us note that generally the quantum mechanical equation (28) cane be reduced by more general transformation of variables $h\rightarrow\xi[h],$ (41) where $\xi[h]$ is any functional in the global dimension $h$. In this case one can rewrite the one-dimensional equation (28) as $\left\\{\left(\dfrac{\delta\xi[h]}{\delta h}\right)^{2}\dfrac{\delta^{2}}{\delta{\xi[h]^{2}}}+V_{eff}\left[\xi[h]\right]\right\\}\Psi\left[\xi[h]\right]=0,$ (42) and if the functional derivative in the differential operator is non zeroth then one can rewrite this equation as $\left\\{\dfrac{\delta^{2}}{\delta{\xi^{2}}}+V[\xi]\right\\}\Psi\left[\xi\right]=0,$ (43) where the new potential $V[\xi]$ is scaled effective potential $V_{eff}$ expressed by the generalized dimension $\xi$ $V[\xi]=\left(\dfrac{\delta\xi[h]}{\delta h}\right)^{-2}V_{eff}\left[\xi[h]\right].$ (44) From this consideration the following choice of the ”gauge” $\xi[h]$ $\xi[h]\equiv h,$ (45) that leads to the quantum mechanics (28) is the minimal choice. The choice of the transformation of variables in the form (45) is the simplest transformation of the kind $h_{ij}\rightarrow\xi[\det h_{ij}]$ within the Wheeler–DeWitt theory. Other, more advanced constructions, can be generated directly from this basic case, and should be justified by some physical nature’s arguments. For example let us consider the following transformation of variables $\xi[h]=\sqrt{h},$ (46) which can be justified by the form of the invariant measure on an 3-dimensional embedding present in the action (4) with assumption that $h>0$. This change yields the equation (43) with the following modified effective potential $V[\xi]=4\xi^{2}V_{eff}[\xi].$ (47) It cancels the singularity $\dfrac{1}{h}$, but actually causes that $V[\xi]$ must be studied with respect to the generalized dimension $\xi$, not the global dimension $h$. Further aspects of similar considerations would be studied in further papers of the author. The very good a point of reference in searching for the generalized dimension $\xi$ is the normalization condition of the Schrödinger quantum mechanics, which for the considered situation takes the form of a classical functional integral $\int_{\Omega(h_{I},h)}\left\|\Psi\left[\xi[h]\right]\right\|^{2}\delta\xi[h]=1,$ (48) where $\Omega(h_{I},h)$ is some region of integrability in a space of all 3-dimensional embeddings with metric $h_{ij}$ and a volume form $h=\det h_{ij}$. In fact this is the main condition for possible solutions of the studied model: ###### Proposition. _Integrability of the wave functional $\Psi[\xi[h]]$ in the sense of functional integration in the normalization condition (48) determines the generalized dimension $\xi[h]$ in the Quantum Gravity model_. The generalized dimension $\xi[h]$ can be established in the region of integrability $\Omega(h_{I},h)$ as $\xi_{\Omega}[h_{I},h]$ by using of the functional integration formula $\xi_{\Omega}[h_{I},h]=\int_{\Omega(h_{I},h)}\delta\xi[h].$ (49) In this paper we will study further consequences of the simplest transformation (45). We will use standard argument which states that the normalization condition (48) establishes integrability constants of any quantum mechanical solution. ## 3 Maximally symmetric Einstein embeddings Let us consider the following case in the functional expansion of the effective potential (37) $a_{n}=\left\\{\begin{array}[]{cc}a_{-1}=const&\mathrm{for}\leavevmode\nobreak\ n=-1\\\ 0&\mathrm{for}\leavevmode\nobreak\ n\neq-1\end{array}\right.,$ (50) which follows the effective potential (29) in the Newton–Coulomb form $V_{eff}[h]=\dfrac{a_{-1}}{h}.$ (51) In this case the Ricci scalar curvature of a 3-dimensional embedding becomes ${{}^{(3)}\\!R}=6\left(\varrho+\dfrac{\Lambda}{3}+\dfrac{1}{4}a_{-1}\right),$ (52) and the effective potential as well as the evolution (28) take the form $\left(\dfrac{\delta^{2}}{\delta h^{2}}+\dfrac{a_{-1}}{h}\right)\Psi[h]=0.$ (53) In this paper we will consider the case of maximally symmetric embeddings _i.e._ the 3-dimensional manifolds with the vacuum condition $\varrho\equiv 0,\leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\ \mathrm{all}\leavevmode\nobreak\ \mathrm{values}\leavevmode\nobreak\ \mathrm{of}\leavevmode\nobreak\ h.$ (54) For this situation the wave function becomes ”geometric” $\Psi[h]\equiv\Psi_{G}[h]$, and the Ricci scalar curvature (52) simplifies to ${{}^{(3)}}R=\dfrac{3}{2}a_{-1}+2\Lambda\quad,$ (55) and by computation of the Ricci curvature tensor and comparison with a 3-dimensional Einstein manifold condition [72] $R_{ij}=\lambda h_{ij}\quad,$ (56) we obtain that the sign $\lambda$ of the considered Einstein manifolds equals $\dfrac{1}{2}a_{-1}+\dfrac{2}{3}\Lambda=\lambda.$ (57) In general case, one can consider classification of maximally symmetric 3-dimensional Einstein manifolds (56) with respect to its sign $\lambda$ (57). ###### Conclusion. In Global One–Dimensional Quantum Gravity model the embeddings which are maximally symmetric 3-dimensional Einstein manifolds with the sign (57), reconstruct the Newton–Coulomb potential $V_{eff}[h]=\dfrac{a_{-1}}{h}$. 1. 1. For the case of non vanishing sign $\lambda\neq 0$ and negative $a_{-1}=-\|\alpha\|$, the effective potential $V_{eff}[h]$ is Newtonian attractive potential. 2. 2. For the case of non vanishing sign $\lambda\neq 0$ and positive $a_{-1}=+\|\alpha\|$, the effective potential $V_{eff}[h]$ is Coulombic repulsive potential. In these cases, positiveness or negativeness of the sign $\lambda$ determines inequalities for cosmological constant $\Lambda$ as follows $\Lambda\gtrless\left\\{\begin{array}[]{rl}\dfrac{3}{4}\|\alpha\|&\mathrm{for\leavevmode\nobreak\ Newtonian\leavevmode\nobreak\ case}\vspace{10pt}\\\ -\dfrac{3}{4}\|\alpha\|&\mathrm{for\leavevmode\nobreak\ Coulombic\leavevmode\nobreak\ case}\end{array}\right.$ (58) where the inequality $\gtrless$ is directed according to value of sign of the Einstein manifold $\lambda\gtrless 0$. 3. 3. For vanishing sign $\lambda=0$, one determine uniquely $a_{-1}=\mp\|\alpha\|=-\dfrac{4}{3}\Lambda$. In this case, from the Newton law of gravitation and the Coulomb law of electrostatics we obtain the values of cosmological constant $\Lambda=\left\\{\begin{array}[]{rl}-\dfrac{9}{32\pi}m_{1}m_{2}&\mathrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\ Newton\leavevmode\nobreak\ law}\vspace{10pt}\\\ \dfrac{3}{16\pi}\dfrac{q_{1}q_{2}}{\epsilon_{0}}&\mathrm{for\leavevmode\nobreak\ the\leavevmode\nobreak\ Coulomb\leavevmode\nobreak\ law}\end{array}\right.$ (59) where geometrized units was used, $m_{1,2}$ are masses of bodies which interact gravitationally in vacuum, $\epsilon_{0}$ is the dielectric constant in vacuum, $q_{1,2}$ are values of charges interact electrically in vacuum. Note that in fact, by assuming the relation (38), the constant coefficient $a_{-1}$ is equal to the Cauchy residuum of the effective potential $V_{eff}[h]$ in any fixed point $h_{0}$ $a_{-1}=\dfrac{1}{2\pi i}\int_{C(h_{\epsilon})}V_{eff}[h]\delta h=Res\left[\dfrac{2}{3h}\left({{}^{(3)}\\!R}-2\Lambda-6\varrho\right),h=h_{0}\right]\quad,$ (60) and can be computed by elementary way $a_{-1}=\dfrac{2}{3}\left.\left({{}^{(3)}\\!R}-2\Lambda-6\varrho\right)\right\|_{h=h_{0}}=\dfrac{2}{3}{{}^{(3)}\\!R_{0}}-\dfrac{4}{3}\Lambda-4\varrho_{0}\quad,$ (61) where prefix ”$0$” on the LHS means value in the fixed initial value of volume form $h_{0}$. Application of the relation (61) in the constraint (57) leads to the relation $\dfrac{1}{3}{{}^{(3)}\\!R_{0}}=\lambda\quad,$ (62) where the initial assumption of maximality $\varrho_{0}\equiv 0$ was imputed. So, the studied approximation of the effective potential worthy of the title of _residual approximation_. Let us note that, if we want to associate the residual approximation $V_{eff}[h]=\dfrac{a_{-1}}{h}$ with any realistic quantized Kepler problem in Newtonian or Coulombic potentials, we should put by hands the identification $h\equiv r\quad,$ (63) where $r=\sqrt{x^{2}+y^{2}+z^{2}}$ is a space distance in harmonic coordinates. For this case, with the formal assumption $\delta=d$, the studied evolution equation (53) becomes more familiar equation $\left(\dfrac{d^{2}}{d{r^{2}}}+\dfrac{\mp\|\alpha\|}{r}\right)\Psi(r)=0\quad,$ (64) where the number $\|\alpha\|$ can be taken from the Newton law of gravitation or from the Coulomb law of electricity. Of course, the obtained equation (64) looks like formally as the radial-type Schrödinger wave equation [73] with classical Newton–Coulomb potential. The assumption $\delta=d$ is well established in the context of classical mechanics [74] and continuation of this idea into quantum mechanics is a question of an analogy only. However, there is many possible metrics $h_{ij}$ with the same determinant $r$, for example we have obviously $\displaystyle h_{ij}=r^{1/3}\delta_{ij}.$ (65) However, more generally, one can parameterize the relation (63) by $SO(3)$ group rotation matrix $r_{ij}$: $h_{ij}=r^{1/3}r_{ij}$, which allows use the Euler angles $(\theta,\varphi,\phi)$ as follows $r_{ij}(\theta,\varphi,\phi)\equiv r_{il}^{(3)}(\theta)r_{lk}^{(2)}(\varphi)r_{kj}^{(3)}(\phi)\quad,$ (66) where matrices $r_{ij}^{(p)}(\vartheta)$ are rotation matrices around the $p$-axis $\displaystyle r_{ij}^{(3)}(\vartheta)=\left[\begin{array}[]{ccc}\cos\vartheta&-\sin\vartheta&0\\\ \sin\vartheta&\cos\vartheta&0\\\ 0&0&1\end{array}\right],\qquad r_{ij}^{(2)}(\vartheta)=\left[\begin{array}[]{ccc}\cos\vartheta&0&\sin\vartheta\\\ 0&1&0\\\ -\sin\vartheta&0&\cos\vartheta\end{array}\right].$ (73) ## 4 Geometric wave functions Still we will consider solutions of the one-dimensional quantum mechanics (28) for the discussed residual approximation of the effective potential $V_{eff}[h]$. For the considered case the evolution is solved by two type geometric wave functions $\Psi_{G}[h]\equiv\Psi_{G}^{\mp}[h]$ $\left(\dfrac{\delta^{2}}{\delta{h^{2}}}\mp\dfrac{\|\alpha\|}{h}\right)\Psi_{G}^{\mp}[h]=0\quad,$ (74) where the attractive wave functions $\Psi_{G}^{-}[h]$ are associated with the sign ”$-$” in the potential (Newtonian case), and the repulsive ones $\Psi_{G}^{+}[h]$ are associated with the sign ”$+$” in the potential (Coulombic case). One treat the functional evolution (74) as a type of an order differential equation for wave functions $\Psi_{G}^{\mp}[h]$. General solution of this equation can be constructed directly in terms of the Bessel functions $J_{n}$ and $Y_{n}$ for the case of Newtonian attractive potential $\Psi_{G}^{-}[h]=\sqrt{\|\alpha\|h}\left[C_{1}^{-}J_{1}\left(2\sqrt{\|\alpha\|h}\right)+2iC_{2}^{-}Y_{1}\left(2\sqrt{\|\alpha\|h}\right)\right]\quad,$ (75) as well as in terms of the modified Bessel functions $I_{n}$ and $K_{n}$ for the case of Coulombic repulsive potential $\Psi_{G}^{+}[h]=-\sqrt{\|\alpha\|h}\left[C_{1}^{+}I_{1}\left(2\sqrt{\|\alpha\|h}\right)+2C_{2}^{+}K_{1}\left(2\sqrt{\|\alpha\|h}\right)\right]\quad,$ (76) where $C_{1}^{\pm}$ and $C_{2}^{\pm}$ are constants of integration, the Bessel functions of first and second kind, $J_{\alpha}(x)$ and $Y_{\alpha}(x)$, are $\displaystyle J_{\alpha}(x)$ $\displaystyle=$ $\displaystyle\dfrac{1}{\pi}\int_{0}^{\pi}dt\cos\left(x\cos t-\alpha t\right)\quad,$ (77) $\displaystyle Y_{\alpha}(x)$ $\displaystyle=$ $\displaystyle\dfrac{J_{\alpha}(x)\cos\left(\alpha\pi\right)-J_{-\alpha}(x)}{\sin\left(\alpha\pi\right)}\quad,$ (78) and the modified Bessel functions of first and second kind, $I_{\alpha}(x)$ and $K_{\alpha}(x)$, are $\displaystyle I_{\alpha}(x)$ $\displaystyle=$ $\displaystyle\dfrac{1}{\pi}\int_{0}^{\pi}dt\exp\left(x\cos t\right)\cos\left(\alpha t\right)\quad,$ (79) $\displaystyle K_{\alpha}(x)$ $\displaystyle=$ $\displaystyle\dfrac{\pi}{2}\dfrac{I_{-\alpha}(x)-I_{\alpha}(x)}{\sin\left(\alpha\pi\right)}.$ (80) Standardly, values of the second kind Bessel functions and modified ones for any integers $n$ can be received by employing of the limiting procedure $Y_{n}(x)=\lim_{\alpha\rightarrow n}Y_{\alpha}(x)$, $K_{n}(x)=\lim_{\alpha\rightarrow n}K_{\alpha}(x)$ [75]. The main subject of this section is studying of solutions of the quantum mechanical evolution (74) with respect to boundary conditions for the general solutions (75) and (76). ### 4.1 Boundary conditions I Let us consider the case of the quantum evolution (28) with the boundary values for some fixed $h=h_{I}$: $\Psi[h_{I}]=\Psi_{I}\quad,\quad\dfrac{\delta\Psi}{\delta h}[h_{I}]=\Psi^{\prime}_{I}.$ (81) With using of the regularized hypergeometric functions ${{}_{p}}\tilde{F}_{q}$ $\displaystyle{{}_{p}}\tilde{F}_{q}\left(\begin{array}[]{c}a_{1},\ldots,a_{p}\\\ b_{1},\ldots,b_{q}\end{array};x\right)$ $\displaystyle=$ $\displaystyle\dfrac{{{}_{p}}F_{q}\left(\begin{array}[]{c}a_{1},\ldots,a_{p}\\\ b_{1},\ldots,b_{q}\end{array};x\right)}{\Gamma(b_{1})\ldots\Gamma(b_{q})},$ (86) $\displaystyle{{}_{p}}F_{q}\left(\begin{array}[]{c}a_{1},\ldots,a_{p}\\\ b_{1},\ldots,b_{q}\end{array};x\right)$ $\displaystyle=$ $\displaystyle\sum_{r=0}^{\infty}\dfrac{(a_{1})_{r}\ldots(a_{p})_{r}}{(b_{1})_{r}\ldots(b_{q})_{r}}\dfrac{x^{r}}{r!},$ (89) $\displaystyle(a)_{r}$ $\displaystyle\equiv$ $\displaystyle\dfrac{\Gamma(a+r)}{\Gamma(a)},$ (90) one write the general solutions (75) and (76) for the considered boundary conditions (81) in the following form $\displaystyle\Psi_{G}^{-}=C^{-}_{1}\left(2\sqrt{{\|\alpha\|h}}\right)K_{1}\left(2\sqrt{{\|\alpha\|h}}\right)+C^{-}_{2}\left(2\sqrt{{\|\alpha\|h}}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h\right),$ (93) with constans $\displaystyle C^{-}_{1}$ $\displaystyle=$ $\displaystyle\Psi_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 1\end{array};\|\alpha\|h_{I}\right)-\Psi^{\prime}_{I}h_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h_{I}\right),$ (98) $\displaystyle C^{-}_{2}$ $\displaystyle=$ $\displaystyle\dfrac{1}{2}\left(\Psi_{I}K_{0}\left(2\sqrt{{\|\alpha\|h_{I}}}\right)+\Psi^{\prime}_{I}\sqrt{{\dfrac{h_{I}}{\|\alpha\|}}}K_{1}\left(2\sqrt{{\|\alpha\|h_{I}}}\right)\right),$ (99) for Newtonian case, and $\displaystyle\Psi_{G}^{+}=C^{+}_{1}\left(2\sqrt{{\|\alpha\|h}}\right)Y_{1}\left(2\sqrt{{\|\alpha\|h}}\right)+C^{+}_{2}\left(2\sqrt{{\|\alpha\|h}}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h\right),$ (102) with constans $\displaystyle C^{+}_{1}$ $\displaystyle=$ $\displaystyle\dfrac{\pi}{2}\left(\Psi^{\prime}_{I}h_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h_{I}\right)-\Psi_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 1\end{array};-\|\alpha\|h_{I}\right)\right),$ (107) $\displaystyle C^{+}_{2}$ $\displaystyle=$ $\displaystyle\dfrac{\pi}{2}\left(\Psi_{I}Y_{0}\left(2\sqrt{{\|\alpha\|h_{I}}}\right)-\Psi^{\prime}_{I}\sqrt{{\dfrac{h_{I}}{\|\alpha\|}}}Y_{1}\left(2\sqrt{{\|\alpha\|h_{I}}}\right)\right),$ (108) for Coulombic case. ### 4.2 Boundary conditions II The second case which we want to present, are the boundary conditions for 1st and 2nd functional derivatives $\dfrac{\delta\Psi}{\delta h}[h_{I}]=\Psi^{\prime}_{I}\quad,\quad\dfrac{\delta^{2}\Psi}{\delta h^{2}}[h_{I}]=\Psi^{\prime\prime}_{I}.$ (109) One more using hypergeometric functions, one can express the solution for attractive case as follows $\displaystyle\Psi_{G}^{-}=C^{-}_{1}\left(2\sqrt{{\|\alpha\|h}}\right)K_{1}\left(2\sqrt{{\|\alpha\|h}}\right)+C^{-}_{2}\left(2\sqrt{{\|\alpha\|h}}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h\right),$ (112) where $C^{-}_{1}$ and $C^{-}_{2}$ are constants defined as $\displaystyle C^{-}_{1}$ $\displaystyle=$ $\displaystyle- h_{I}\left(\Psi^{\prime}_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h_{I}\right)-\dfrac{\Psi^{\prime\prime}_{I}}{\|\alpha\|}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 1\end{array};\|\alpha\|h_{I}\right)\right),$ (117) $\displaystyle C^{-}_{2}$ $\displaystyle=$ $\displaystyle\dfrac{1}{2}\sqrt{\dfrac{h_{I}}{\|\alpha\|}}\left(\Psi^{\prime\prime}_{I}\sqrt{\dfrac{h_{I}}{\|\alpha\|}}K_{0}\left(2\sqrt{{\|\alpha\|h_{I}}}\right)+\Psi^{\prime}_{I}K_{1}\left(2\sqrt{{\|\alpha\|h_{I}}}\right)\right).$ (118) Similarly for repulsive one we obtain $\displaystyle\Psi_{G}^{+}=C^{+}_{1}\left(2\sqrt{{\|\alpha\|h}}\right)Y_{1}\left(2\sqrt{{\|\alpha\|h}}\right)+C^{+}_{2}\left(2\sqrt{{\|\alpha\|h}}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h\right),$ (121) with constans $\displaystyle C^{+}_{1}$ $\displaystyle=$ $\displaystyle\dfrac{\pi h_{I}}{2}\left(\Psi^{\prime}_{I}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h_{I}\right)+\dfrac{\Psi^{\prime\prime}_{I}}{\|\alpha\|}\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 1\end{array};-\|\alpha\|h_{I}\right)\right),$ (126) $\displaystyle C^{+}_{2}$ $\displaystyle=$ $\displaystyle\dfrac{\pi}{4}\sqrt{\dfrac{h_{I}}{\|\alpha\|}}\left(\Psi^{\prime\prime}_{I}\sqrt{\dfrac{h_{I}}{\|\alpha\|}}Y_{0}\left(2\sqrt{{\|\alpha\|h_{I}}}\right)+\Psi^{\prime}_{I}Y_{1}\left(2\sqrt{{\|\alpha\|h_{I}}}\right)\right).$ (127) ### 4.3 Boundary conditions III The last possible case of boundary conditions for the considered problem is $\Psi[h_{I}]=\Psi_{I}\quad,\quad\dfrac{\delta^{2}\Psi}{\delta h^{2}}[h_{I}]=\Psi^{\prime\prime}_{I}.$ (128) These boundaries are formally improper for the problem; they give singluar solutions. However, in this case one can present solutions in form with formally singular constans. For the Newtonian attractive potential the solution is $\Psi_{G}^{-}=C^{-}_{1}\left(2\sqrt{\|\alpha\|h}\right)K_{1}\left(2\sqrt{\left\|\alpha\right\|h}\right)+C^{-}_{2}\left(2\sqrt{\|\alpha\|h}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h\right),$ (129) with constans ($\epsilon\rightarrow 0$) $\displaystyle C^{-}_{1}$ $\displaystyle=$ $\displaystyle\dfrac{2}{\epsilon}\sqrt{\|\alpha\|h_{I}}\left(\Psi_{I}-\dfrac{h_{I}}{\left\|\alpha\right\|}\Psi^{\prime\prime}_{I}\right){{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h_{I}\right),$ (132) $\displaystyle C^{-}_{2}$ $\displaystyle=$ $\displaystyle\dfrac{1}{\epsilon}\left(\Psi_{I}-\dfrac{h_{I}}{\|\alpha\|}\Psi^{\prime\prime}_{I}\right)K_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right),$ (133) and for the Coulombic repulsive potential we have $\Psi_{G}^{+}=C^{+}_{1}\left(2\sqrt{\|\alpha\|h}\right)Y_{1}\left(2\sqrt{\left\|\alpha\right\|h}\right)+C^{+}_{2}\left(2\sqrt{\|\alpha\|h}\right)^{2}{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h\right),$ (134) with constants ($\epsilon\rightarrow 0$) $\displaystyle C^{+}_{1}$ $\displaystyle=$ $\displaystyle\dfrac{2}{\epsilon}\sqrt{\|\alpha\|h_{I}}\left(\Psi_{I}+\dfrac{h_{I}}{\left\|\alpha\right\|}\Psi^{\prime\prime}_{I}\right){{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h_{I}\right),$ (137) $\displaystyle C^{+}_{2}$ $\displaystyle=$ $\displaystyle\dfrac{1}{\epsilon}\left(\Psi_{I}+\dfrac{h_{I}}{\|\alpha\|}\Psi^{\prime\prime}_{I}\right)Y_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right).$ (138) However, when the following condition for initial data holds $\pm\dfrac{h_{I}}{\left\|\alpha\right\|}{\Psi^{\pm}_{I}}^{\prime\prime}+\Psi^{\pm}_{I}\equiv\epsilon f_{\pm}[h_{I},\|\alpha\|],$ (139) where $f_{\pm}[h_{I},\|\alpha\|]\neq 0$ is some (now unknown and arbitrary) nonsingular functional of $h_{I}$ and $\|\alpha\|$, the sign $+$ is related to the Newtonian case, and the sign $-$ to the Coulombic one, then solutions (129) and (134) are nonsingular. In this case initial value of the wave function $\Psi_{I}$ is for the attractive case $\displaystyle\Psi^{-}_{I}=-\|\alpha\|h_{I}\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h_{I}\right)\left[c^{-}_{1}+2\epsilon\sqrt{\|\alpha\|}\int_{1}^{h_{I}}\dfrac{dt}{\sqrt{t}}f_{-}[t,\|\alpha\|]K_{1}\left(2\sqrt{\|\alpha\|t}\right)\right]+$ (142) $\displaystyle+\,2\sqrt{\|\alpha\|h_{I}}K_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)\left[c^{-}_{2}+\epsilon\|\alpha\|\int_{1}^{h_{I}}dtf_{-}[t,\|\alpha\|]\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|t\right)\right],$ (145) and similarly for the repulsive one $\displaystyle\Psi^{+}_{I}=\|\alpha\|h_{I}\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h_{I}\right)\left[c^{+}_{1}-\epsilon\pi\sqrt{\|\alpha\|}\int_{1}^{h_{I}}\dfrac{dt}{\sqrt{t}}f_{+}[t,\|\alpha\|]Y_{1}\left(2\sqrt{\|\alpha\|t}\right)\right]+$ (148) $\displaystyle+\,2i\sqrt{\|\alpha\|h_{I}}Y_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)\left[c^{+}_{2}-\epsilon\dfrac{i\pi}{2}\|\alpha\|\int_{1}^{h_{I}}dtf_{+}[t,\|\alpha\|]\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|t\right)\right],$ (151) where $c^{\pm}_{1,2}$ are constants of integration. The functions $f_{\pm}[h_{I},\|\alpha\|]\neq 0$ can be established by using of the condition (139) in general solutions (129) and (134), it yields $\displaystyle\Psi_{I}^{-}$ $\displaystyle=$ $\displaystyle 8\|\alpha\|h_{I}K_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h_{I}\right)f_{-}[h_{I},\|\alpha\|],$ (154) $\displaystyle\Psi_{I}^{+}$ $\displaystyle=$ $\displaystyle 8\|\alpha\|h_{I}Y_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h_{I}\right)f_{+}[h_{I},\|\alpha\|].$ (157) Now by direct application of these equations into received equalities (145) and (151) one can obtain the following integral equations for the functions $f_{\pm}$. For the Coulombic situation we have $\displaystyle-\dfrac{c^{-}_{1}}{4}\left(2\sqrt{\|\alpha\|h_{I}}\right)\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h_{I}\right)+c^{-}_{2}K_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)+$ (160) $\displaystyle+$ $\displaystyle\epsilon\|\alpha\|\Bigg{\\{}K_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)\int_{1}^{h_{I}}dt\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|t\right)-$ (163) $\displaystyle-$ $\displaystyle\sqrt{h_{I}}\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h_{I}\right)\int_{1}^{h_{I}}\dfrac{dt}{\sqrt{t}}K_{1}\left(2\sqrt{\|\alpha\|t}\right)\Bigg{\\}}f_{-}[t,\|\alpha\|]=$ (166) $\displaystyle=$ $\displaystyle 2\left(2\sqrt{\|\alpha\|h_{I}}\right)K_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};\|\alpha\|h_{I}\right)f_{-}[h_{I},\|\alpha\|],$ (169) and for the Newtonian one we have $\displaystyle\dfrac{c^{+}_{1}}{4}\left(2\sqrt{\|\alpha\|h_{I}}\right)\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h_{I}\right)+ic^{+}_{2}Y_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)-$ (172) $\displaystyle-$ $\displaystyle\epsilon\dfrac{\pi}{2}\|\alpha\|\Bigg{\\{}\sqrt{h_{I}}\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h_{I}\right)\int_{1}^{h_{I}}\dfrac{dt}{\sqrt{t}}Y_{1}\left(2\sqrt{\|\alpha\|t}\right)+$ (175) $\displaystyle+$ $\displaystyle\,iY_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)\int_{1}^{h_{I}}dt\,{{}_{0}}{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|t\right)\Bigg{\\}}f_{+}[t,\|\alpha\|]=$ (178) $\displaystyle=$ $\displaystyle 2\left(2\sqrt{\|\alpha\|h_{I}}\right)Y_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)\,{{}_{0}}\tilde{F}_{1}\left(\begin{array}[]{c}-\\\ 2\end{array};-\|\alpha\|h_{I}\right)f_{+}[h_{I},\|\alpha\|].$ (181) However, in both cases the integral operators acting on the functions $f_{\pm}$ are nonsingular. In this situation one can put the formal limit $\epsilon\rightarrow 0$ in the equations (169) and (181), and by doing some elementary algebraic manipulations one can extract the searched functions. Finally, we obtain the results $\displaystyle f_{-}[h_{I},\|\alpha\|]$ $\displaystyle=$ $\displaystyle\dfrac{-c^{-}_{1}/8}{K_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)}+\dfrac{c^{-}_{2}/4}{I_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)},$ (182) $\displaystyle f_{+}[h_{I},\|\alpha\|]$ $\displaystyle=$ $\displaystyle\dfrac{c^{+}_{1}/8}{Y_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)}+\dfrac{ic^{+}_{2}/4}{J_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)}.$ (183) In this manner the initial data for the studied boundary conditions (128) can not be chosen arbitrary, but according to the rules $\displaystyle\Psi_{I}^{-}$ $\displaystyle=$ $\displaystyle\sqrt{\|\alpha\|h_{I}}\left[-c^{-}_{1}I_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)+2c^{-}_{2}K_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)\right],$ (184) $\displaystyle\Psi_{I}^{+}$ $\displaystyle=$ $\displaystyle\sqrt{\|\alpha\|h_{I}}\left[c^{+}_{1}J_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)+2ic^{+}_{2}Y_{1}\left(2\sqrt{\|\alpha\|h_{I}}\right)\right].$ (185) Of course, the supposed equation for boundary values(139) is here arbitrary, and can be replaced by other ones. However, the discussed case reflects some typical questions in the problem. ## 5 Vanishing sign Finally, let us discuss briefly the case of vanishing sign (57) for studied 3-dimensional Einstein manifolds $\lambda=\dfrac{1}{2}a_{-1}+\dfrac{2}{3}\Lambda\equiv 0.$ (186) For this situation we have of course $a_{-1}=-\dfrac{4}{3}\Lambda\equiv\pm\|\alpha\|.$ (187) So, for this case in absence of Matter fields the effective potential (29) becomes purely cosmological $V_{eff}[h]=V_{C}[h]=-\dfrac{4}{3}\dfrac{\Lambda}{h},$ (188) where the cosmological constant $\Lambda$ is established according to the relation (59). From the global one-dimensional quantum mechanics point of view, the our model of Quantum Gravity defines ”cosmological” wave function if and only if in the received solutions of the previous chapter we input the change $\|\alpha\|=\left\\{\begin{array}[]{cc}-\dfrac{4}{3}\Lambda&\mathrm{for}\leavevmode\nobreak\ \mathrm{Newtonian}\leavevmode\nobreak\ \mathrm{case}\vspace{10pt}\\\ +\dfrac{4}{3}\Lambda&\mathrm{for}\leavevmode\nobreak\ \mathrm{Coulombic}\leavevmode\nobreak\ \mathrm{case}\end{array}\right.,$ (189) so that the cosmological wave function is only one and can be determined by using geometric wave function with the identification (189) as follows $\Psi_{C}[h]=\Psi^{\pm}_{G}[h],$ (190) where the sign is chosen according to the sign of cosmological constant $\Lambda$. ## 6 Discussion We have presented the quantum mechanical point of view on the Global One–Dimensional supposition within Quantum General Relativity studied previously by the author in terms of quantum field theory [1, 2, 3, 4, 5, 6, 7]. The obtained model bases on the effective potential (29) being a simple algebraic sum of three fundamental constituents - geometric, cosmological, and material, with nontrivial change in potential behavior with respect to the entrance model that was the Wheeler–DeWitt theory (20) $V_{eff}[h]\rightarrow h^{-3/2}V_{WDW}[h].$ We have concentrated our attention on studying the elementary case, that we have called _residual approximation of the effective potential_ , that on some conventional level $h\rightarrow r$ can be identified with the attractive Newton gravitation or the repulsive Coulomb electrostatics. Studying of this case allowed to conclude that in the case of matter fields absence, in the Global One–Dimensional model of Quantum Gravity, the maximally symmetric 3-dimensional Einstein manifolds are crucial for the residual case. Finally, we have found some solutions of the Quantum Gravity model in the residual approximation. The digression about other possible transformations of variables $h\rightarrow\xi[h]$ (43) within the Global 1D quantum Gravity model leads to modification of the effective potential by the following way $V_{eff}[h]\rightarrow\left[\left(\xi[h]\right)^{-3/4}\dfrac{\delta\xi[h]}{\delta h}\right]^{2}V_{WDW}\left[\xi[h]\right],$ and shows that the studied case of $f\equiv h$, in some conventional sense, is the simplest and can be considered as minimal quantum mechanical model within the wider field theoretic model. ## References [1] L. A. Glinka, Macrostates thermodynamics and its stable classical limit in Global One-Dimensional Quantum General Relativity, 0809.5216[gr-qc] * [2] L. A. Glinka, On Global One-Dimensionality proposal in Quantum General Relativity, 0808.1035[gr-qc] * [3] L. A. Glinka, 1D Global Bosonization of Quantum Gravity, 0804.3516[gr-qc] * [4] L. A. Glinka, Quantum gravity as the way from spacetime to space quantum states thermodynamics, New Advances in Physics, Vol. 2, No. 1, 1 - 62, (2008), 0803.1533[gr-qc] * [5] L. A. Glinka, in Frontiers of Fundamental and Computational Physics. 9th International Symposium, Udine and Trieste, Italy 7–9 January 2008, p.94, eds. B. G. Sidharth, F. Honsell, O. Mansutti, K. Sreenivasan, and A. De Angelis. AIP Conf. Proc. 1018, American Institute of Physics, Melville, New York (2008), 0801.4157[gr-qc] * [6] L. A. Glinka, On quantum cosmology as field theory of bosonic string mass groundstate, 0712.1674[gr-qc] * [7] L. A. Glinka, Quantum Information from Graviton-Matter Gas. SIGMA 3, 087, (2007), 0707.3341[gr-qc] * [8] B. Riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen. Abh. Königl. Gesell. der Wissen. zu Göttingen, Band 13, 133 (1920). * [9] A. Einstein, Die formale Grundlage der allgemeinen Relativitätstheorie. Sitzungsber. Preuss. Akad. Wiss. Berlin 2, 1030 (1914); Prinzipielles zur verallgemeinerten Relativitätstheorie und Gravitationstheorie. Phys. Z. 15, 176 (1914); Zür allgemeinen Relativitätstheorie. Sitzungsber. Preuss. Akad. Wiss. Berlin 44, 778 (1915); Zür allgemeinen Relativitätstheorie (Nachtrag). Sitzungsber. Preuss. Akad. Wiss. Berlin 46, 799 (1915); Die Feldgleichungen der Gravitation. Sitzungsber. Preuss. Akad. Wiss. Berlin 48, 844 (1915); Die Grundlage der allgemeinen Relativitätstheorie. Ann. Phys. 49, 769 (1916). * [10] A. Palatini, Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton. Rend. Circ. Mat. Palermo 43, 203 (1919). * [11] D. Hilbert, Die Grundlagen der Physik. Konigl. Gesell. d. Wiss. Göttingen, Nachr., Math.-Phys. Kl. 27, 395 (1915); Die Grundlagen der Physik (Zweite Mitteilung). Konigl. Gesell. d. Wiss. Göttingen, Nachr., Math.-Phys. Kl. 61, 53 (1917). * [12] J. B. Hartle and S. W. Hawking, Wave function of the Universe. Phys. Rev. D 28, 2960 (1983). * [13] J. F. Nash, The imbedding problem for Riemannian manifolds. Ann. Math. 63, 20 (1956). * [14] M. Günther, On the perturbation problem associated to embeddings of Riemannian manifolds. Ann. Global Anal. Geom. 7, 69 (1989), Isometric embeddings of Riemannian manifolds. Proceedings of the International Congress of Mathematicians (Kyoto, 1990), pp. 1137-1143, Mathematical Society of Japan (1991). * [15] A. Kowalczyk, Whithey’s and Nash’s Embeddings Theorems for Differential Spaces. Bull. Acad. Polon. Sci. Ser. Sci. Math. 28, 385 (1981). * [16] S. Masahiro, Nash Manifolds. Springer–Verlag, Berlin (1987). * [17] P. A. M. Dirac, The theory of gravitation in Hamiltonian form. Proc. Roy. Soc. Lond. A 246, 333 (1958); Fixation of coordinates in the Hamiltonian theory of gravitation. Phys. Rev. 114, 924 (1959); Energy of the Gravitational Field. Phys. Rev. Lett. 2, 368 (1959); Generalized Hamiltonian dynamics. Proc. Roy. Soc. Lond. A 246, 326 (1958); Generalized Hamiltonian dynamics. Can. J. Math. 2, 129 (1950). * [18] R. Arnowitt, S. Deser and Ch.W. Misner, The dynamics of general relativity, in Gravitation: An Introduction to Current Research, ed. by L. Witten, p. 227. Wiley, New York (1962). * [19] B. DeWitt, The Global Approach to Quantum Field Theory, Vol. 1,2. Int. Ser. Monogr. Phys. 114, Clarendon Press, Oxford (2003). * [20] K. F. Gauss, Disquisitiones generales circa superficies curvas. Gottingae: Typis Di eterichiansis, (1828). * [21] D. Codazzi, Sulle coordinate curvilinee d’una superficie dello spazio. Ann. math. pura applicata 2, 101, (1868-1869). * [22] A. Hanson, T. Regge, and C. Teitelboim, Constrained Hamiltonian Systems. Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e loro Applicazioni, n. 22, Accademia Nazionale dei Lincei, Roma (1976). * [23] B. S. DeWitt, Quantum Theory of Gravity. I. The Canonical Theory. Phys. Rev. 160, 1113 (1967); Quantum Theory of Gravity. II. The Manifestly Covariant Theory. Phys. Rev. 162, 1195 (1967); Quantum Theory of Gravity. III. Applications of the Covariant Theory. Phys. Rev. 162, 1239 (1967). * [24] P. A. M. Dirac, Lectures on Quantum Field Theory, Belfer Graduate School of Science, Yeshiva University, New York (1966). * [25] F. A. E. Pirani and A. Schild, On the Quantization of Einstein’s Gravitational Field Equations. Phys. Rev. 79, 986 (1950). * [26] P. G. Bergmann, Introduction of ”true observables” into the quantum field equations. Nuovo Cim. 3, 1177 (1956); Summary of the Chapel Hill Conference. Rev. Mod. Phys. 29, 352 (1957); Observables in General Relativity. Rev. Mod. Phys. 33, 510 (1961); Hamilton–Jacobi and Schrödinger Theory in Theories with First-Class Hamiltonian Constraints. Phys. Rev. 144, 1078 (1966). * [27] P. G. Bergmann and A. B. Komar, in Recent Developments in General Relativity, p. 31, Pergamon Press, Inc., New York, (1962). * [28] J. A. Wheeler, in Battelle Rencontres: 1967 Lectures in Mathematics and Physics, eds. C. M. DeWitt and J. A. Wheeler, pp. 242. W.A. Benjamin, New York (1968). * [29] P. W. Higgs, Integration of Secondary Constraints in Quantized General Relativity. Phys. Rev. Lett. 1, 373 (1958); Integration of Secondary Constraints in Quantized General Relativity. Phys. Rev. Lett. 3, 66 (1959). * [30] J. L. Anderson, Factor Sequences in Quantized General Relativity. Phys. Rev. 114, 1182 (1959). * [31] R. Arnowitt, S. Deser, and Ch. W. Misner, Dynamical Structure and Definition of Energy in General Relativity. Phys. Rev. 116, 1322 (1959); Canonical Variables for General Relativity. Phys. Rev. 117, 1595 (1960); Energy and the Criteria for Radiation in General Relativity. Phys. Rev. 118, 1100 (1960); Gravitational-Electromagnetic Coupling and the Classical Self-Energy Problem. Phys. Rev. 120, 313 (1960); Canonical Variables, Expression for Energy, and the Criteria for Radiation in General Relativity. Nuovo Cim. 15, 487 (1960); Finite Self-Energy of Classical Point Particles Phys. Rev. Lett. 4, 375 (1960); Consistency of the Canonical Reduction of General Relativity. J. Math. Phys. 1, 434 (1960); Note on positive-definiteness of the energy of the gravitational field. Ann. Phys. 11, 116, (1960); Wave Zone in General Relativity. Phys. Rev. 121, 1556 (1961); Coordinate Invariance and Energy Expressions in General Relativity. Phys. Rev. 122, 997 (1961). * [32] A. Peres, On the Cauchy problem in general relativity. Nuovo Cim. 26, 53 (1962). * [33] R. F. Beierlein, D. H. Sharp, and J. A. Wheeler, Three–Dimensional Geometry as Carrier of Information about Time. Phys. Rev. 126, 1864 (1962). * [34] H. Leutwyler, Gravitational Field: Equivalence of Feynman Quantization and Canonical Quantization. Phys. Rev. 134, B1155 (1964). * [35] A. B. Komar, Hamilton–Jacobi Quantization of General Relativity. Phys. Rev. 153, 1385 (1967); Gravitational Superenergy as a Generator of Canonical Transformation. Phys. Rev. 164, 1595 (1967). * [36] B. S. DeWitt, Quantum theories of gravity. Gen. Rel. Grav. 1, 181 (1970). * [37] D. R. Brill and R. H. Gowdy, Quantization of general relativity. Rep. Prog. Phys. 33, 413 (1970). * [38] V. Moncrief and C. Teitelboim, Momentum Constraints as Integrability Conditions for the Hamiltonian Constraint in General Relativity. Phys. Rev. D 6, 966 (1972). * [39] A. E. Fischer and J. E. Marsden, The Einstein equations of evolution - A geometric approach. J. Math. Phys. 13, 546 (1972). * [40] C. Teitelboim, How commutators of constraints reflect the spacetime structure. Ann. Phys. NY 80, 542 (1973). * [41] A. Ashtekar and R. Geroch, Quantum theory of gravitation. Rep. Progr. Phys. 37, 1211 (1974). * [42] T. Regge and C. Teitelboim, Improved Hamiltonian for general relativity. Phys. Lett. B 53, 101 (1974); Role of surface integrals in the Hamiltonian Formulation of General Relativity. Ann. Phys. NY 88, 286, (1974). * [43] R. Geroch, Structure of the Gravitational Field at Spatial Infinity. J. Math. Phys. 13, 956 (1972). * [44] K. Kucha$\mathrm{\check{r}}$, Ground State Functional of the Linearized Gravitational Field. J. Math. Phys. 11, 3322 (1970); Canonical Quantization of Cylindrical Gravitational Waves. Phys. Rev. D 4, 955 (1971); A Bubble-Time Canonical Formalism for Geometrodynamics. J. Math. Phys. 13, 768 (1972); Geometrodynamics regained: A Lagrangian approach. J. Math. Phys. 15, 708 (1974); General relativity: Dynamics without symmetry. J. Math. Phys. 22, 2640 (1981); Dirac constraint quantization of a parametrized field theory by anomaly-free operator representations of spacetime diffeomorphisms. Phys. Rev. D 39, 2263 (1989). * [45] M. A. H. MacCallum, in Quantum Gravity, Oxford Symposium, eds. C. J. Isham, R. Penrose, and D. W. Sciama. Clarendon Press, Oxford (1975); * [46] C. J. Isham, in Quantum Gravity, Oxford Symposium, eds. C. J. Isham, R. Penrose, and D. W. Sciama. Clarendon Press, Oxford (1975); Canonical groups and the quantization of general relativity. Nucl. Phys. B Proc. Suppl. 6, 349, (1989). * [47] C. J. Isham and A. C. Kakas, A group theoretical approach to the canonical quantisation of gravity: I. Construction of the canonical group. Class. Quantum Grav. 1, 621 (1984); A group theoretical approach to the canonical quantisation of gravity. II. Unitary representations of the canonical group. Class. Quantum Grav. 1, 633 (1984). * [48] C. J. Isham and K. V. Kucha$\mathrm{\check{r}}$, Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories. Ann. Phys. 164, 288 (1985); Representations of spacetime diffeomorphisms. II. Canonical geometrodynamics. Ann. Phys. 164, 316 (1985). * [49] S. A. Hojman, K. Kucha$\mathrm{\check{r}}$, and C. Teitelboim, Geometrodynamics regained. Ann. Phys. NY 96, 88 (1976). * [50] G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity. Phys. Rev. D 15, 2752, (1977). * [51] D. Christodoulou, M. Francaviglia, and W. M. Tulczyjew, General relativity as a generalized Hamiltonian system. Gen. Rel. Grav. 10, 567 (1979). * [52] M. Francaviglia, Applications of infinite-dimensional differential geometry to general relativity. Riv. Nuovo Cim. 1, 1303 (1978). * [53] J. A. Isenberg, in Geometrical and topological methods in gauge theories. Lect. Notes Phys. 129, eds. J. P. Harnad and S. Shnider, Springer–Verlag Berlin Heidelberg New York, New York (1980). * [54] J. A. Isenberg and J. M. Nester, in General Relativity and Gravitation. One Hundred Years After the Birth of Albert Einstein., p.23, ed. A. Held, Plenum Press, NewYork and London (1980). * [55] Z. Bern, S. K. Blau, and E. Mottola, General covariance of the path integral for quantum gravity. Phys. Rev. D 33, 1212 (1991). * [56] P. O. Mazur, Quantum gravitational measure for three-geometries. Phys. Lett. B 262, 405 (1991). * [57] C. Kiefer and T. P. Singh, Quantum gravitational corrections to the functional Schrödinger equation. Phys. Rev. D 44, 1067 (1991). * [58] M. Ferraris, M. Francaviglia, and I. Sinicco, Covariant ADM formulation applied to general relativity. Nuovo Cim. B 107, 11 (1992). * [59] N. Pinto-Neto and A. F. Velasco, The search for new representations of the Wheeler–DeWitt equation using the first order formalism. Gen. Rel. Grav. 25, 10, 991 (1993). * [60] C. Kiefer, in Canonical Gravity: From Classical to Quantum, eds. J. Ehlers and H. Friedrich. Springer, Berlin (1994), arXiv:gr-qc/9312015 * [61] D. Giulini and C. Kiefer, Consistency of semiclassical gravity. Class. Quantum Grav. 12, 403 (1995). * [62] N. Pinto-Neto and E. S. Santini, Must quantum spacetimes be Euclidean? Phys. Rev. D 59, 123517 (1999). * [63] N. Pinto-Neto and E. S. Santini, The Consistency of Causal Quantum Geometrodynamics and Quantum Field Theory. Gen. Rel. Grav. 34, 505 (2002). * [64] M. J. W. Hall, K. Kumar, and M. Reginatto, Bosonic field equations from an exact uncertainty principle. J. Phys A: Math. Gen. 36, 9779 (2003). * [65] C. Rovelli, Quantum gravity. Cambridge University Press, Cambridge (2004). * [66] N. Pinto-Neto, The Bohm Interpretation of Quantum Cosmology. Found. Phys. 35, 577 (2005). * [67] M. J. W. Hall, Exact uncertainty approach in quantum mechanics and quantum gravity. Gen. Rel. Grav. 37, 1505 (2005). * [68] R. Carroll, Metric fluctuations, entropy, and the Wheeler–DeWitt equation. Theor. Math. Phys. 152, 904 (2007). * [69] A. E. Fischer, in Relativity (Proceedings of the Relativity Conference in the Midwest, held in Cincinatti, Ohio, June 2-6, 1969), p.303, eds. M. Carmeli, S. I. Fickler, and L. Witten. Plenum Press, New York (1970); Unfolding the singularities in superspace. Gen. Rel. Grav. 15, 1191 (1983); Resolving the singularities in the space of Riemannian geometries. J. Math. Phys. 27, 718 (1986). * [70] L. D. Faddeev, The energy problem in Einstein’s theory of gravitation (Dedicated to the memory of V. A. Fock). Usp. Fiz. Nauk 136, 435 (1982). * [71] J. A. Wheeler, On the Nature of Quantum Geometrodynamics. Ann. Physics 2, 604 (1957). * [72] A. L. Besse, Einstein manifolds. Springer–Verlag Berlin Heidelberg (2008). * [73] L. I. Schiff, Quantum Mechanics. McGraw-Hill Book Company, Inc., New York (1949). * [74] H. Goldstein, Ch. Poole, and J. Safko, Classical Mechanics (3rd ed.). Addison Wesley, San Francisco (2000). * [75] G. E. Andrews, R. Askey, and R. Roy, Special functions. Cambridge University Press, Cambridge (1999).	arxiv-papers	2009-02-17T04:13:15	2024-09-04T02:49:00.631812	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lukasz Andrzej Glinka", "submitter": "Lukasz Andrzej Glinka", "url": "https://arxiv.org/abs/0902.2829" }
0902.2870	11institutetext: Department of Physics, Anyang Normal University, Anyang, 455000, China, 11email: cuiht@aynu.edu.cn # Pairwise Entanglement and Geometric Phase in High Dimensional Free-Fermion Lattice Systems H. T. Cui Y. F. Zhang (Received: / Revised version: ) ###### Abstract The pairwise entanglement, measured by concurrence and geometric phase in high dimensional free-fermion lattice systems have been studied in this paper. When the system stays at the ground state, their derivatives with the external parameter show the singularity closed to the phase transition points, and can be used to detect the phase transition in this model. Furthermore our studies show for the free-fermion model that both concurrence and geometric phase show the intimate connection with the correlation functions. The possible connection between concurrence and geometric phase has been also discussed. ###### pacs: 03.65.Vf Phases: geometric; dynamic or topological; 03.65.Ud Entanglement and quantum nonlocality; and 05.70.Fh Phase transitions: general studies ## 1 introduction The understanding of quantum many-body effects based on the fundamentals of quantum mechanics, has been raising greatly because of the rapid development in quantum information theoryafov07 . Encouraged by the suggestion of Preskillpreskill , the connection between the quantum entanglement and quantum phase transition has been demonstrated first in 1D spin-$1/2$ $XY$ modeoo02 , and then was extended to more other spin-chain systems and fermion systems (see Ref afov07 for a review). Furthermore the decoherence of a simple quantum systems coupled with the quantum critical environment has been shown the significant features closed to the critical points ycw06 ; quan . Regarding these findings, the fidelity between the states across the transition point has also been introduced to mark the happening of the phase transitions zanardi . These intricate connections between quantum entanglement and phase transition in many-body systems have sponsored great effort devoted to the understanding of many-body effects from quantum information pointafov07 . In general quantum entanglement as a special correlation, is believed to play an essential role for the many-body effects since it is well accepted that the non-trivial correlation is at the root of many-body effects. Although the ambiguity existsyang , quantum entanglement provides us a brand-new perspective into quantum many-body effects. However the exact physical meaning of quantum entanglement in many body systems remains unclearvedral07 . Although the entanglement witnesses has been constructed in some many-body systemswvb05 , a general and physical understanding of quantum entanglement in many-body systems is still absent. On the other hand, the geometric phase, which was first studied systemically by Berryberry and had been researched extensively in the past 20 yearsgp , recently has also been shown the intimate connection to quantum phase transitionscp05 ; zhu ; hamma ; cui06 ; plc06 ; cui08 ; hkh08 (or see a recent review Ref.zhu08 ). This general relation roots at the topological property of the geometric phase, which depicts the curvature of the Hilbert space, and especially has direct relation to the property of the degeneracy in quantum systems. The degeneracy in the many-body systems is critical in our understanding of the quantum phase transition sachdev . Thus the geometric phase is another powerful tool for detecting the quantum phase transitions. Moreover recently geometric phase has been utilized to distinguish different topological phases in quantum Hall systemsshen , in which the traditional phase transition theory based on the symmetry-broken theory is not in functionSenthil . Hence it is very interesting to discuss the possible connection between entanglement and geometric phase, since both issues show the similar sensitivity to the happening of quantum phase transition. Recently the connection between the entanglement entropy and geometric phase has first been discussed with a special model in strongly correlated systems; the geometric phase induced by the twist operator imposed on the filled Fermi sphere, was shown to present a lower bound for the entanglement entropyrh06 . This interesting result implies the important relation between quantum entanglement and geometric phase, and provides an possible understanding of entanglement from the topological structure of the systems. In another way the two-particle entanglement was also importantoo02 . Especially in spin-chain systems two- particle entanglement is more popular and general because of the interaction between spins, and furthermore the quantum information transferring based on spin systems are generally dependent on the entanglement between two particlesss05 . So it is a tempting issue to extend this discussion to the universal two-particle entanglement situation. For this purpose the pairwise entanglement and geometric phase are studied systemically in this paper. Our discussion focuses on nearest-neighbor entanglement in the ground state in free-Fermion lattice systems because of the availability of the exact results. By our own knowledge, this paper first presents the exact results of entanglement and geometric phase in higher dimensional systems. In Sec.2 the model will be provided, and the entanglement measured by Wootter’s concurrence is calculated by introducing pseudospin operators. Furthermore the geometric phase is obtained by imposing a globe rotation, and its relation with concurrence are also discussed generally. In Sec.3, we discussed respectively the concurrence and geometric phase in 2D and 3D cases. Finally, the conclusion is presented in Sec.4. ## 2 Model The Hamiltonian for spinless fermions in lattice systems reads $H=\sum_{\mathbf{ij}}^{L}c_{\mathbf{i}}^{\dagger}A_{\mathbf{ij}}c_{\mathbf{j}}+\frac{1}{2}(c_{\mathbf{i}}^{\dagger}B_{\mathbf{ij}}c_{\mathbf{j}}^{\dagger}+\text{h.c.}),$ (1) in which $c_{\mathbf{i}}^{(\dagger)}$ is fermion annihilation(creation) operator and $L$ is the total number of lattice sites. The hermitity of $H$ imposes that matrix $A$ is Hermit and $B$ is an anti-symmetry matrix. The configuration of lattice does not matter for Eq. (1) since our discussion focuses on the general case and available exact results. This model obviously is solvable exactly and can be transformed into the free Bogoliubov fermionic model. So it is also called free-fermion model. By Jordan-Wigner transformationjw28 one can convert the spin-chain systems into spinless fermions systems, in which the physical properties can be readily determined. Therefore an alternative approach is necessary by which one can treat solvable fermion systems of arbitrary size. The model Eq. (1) serves this purpose. Without the loss of generality we assume $A$ and $B$ to be reallsm61 . An important property of Eq. (1) is $[H,\prod_{\mathbf{i}}^{L}(1-2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}})]=0.$ (2) This symmetry would greatly simplify the consequent calculation of the reduced density matrix for two fermions. One can diagonalize Eq. (1) by introducing linear transformation with real $g_{\mathbf{ki}}$ and $h_{\mathbf{ki}}$lsm61 $\eta_{\mathbf{k}}=\frac{1}{\sqrt{L}}\sum_{\mathbf{i}}^{L}g_{\mathbf{ki}}c_{\mathbf{i}}+h_{\mathbf{ki}}c_{\mathbf{i}}^{\dagger},$ (3) in which the normalization factor $1/\sqrt{L}$ have been included to ensure the convergency under the thermodynamic limit. After some algebra, the Hamiltonian Eq. (1) becomes $H=\sum_{\mathbf{k}}\Lambda_{\mathbf{k}}\eta_{\mathbf{k}}^{\dagger}\eta_{\mathbf{k}}+\text{const}.$ (4) in which $\Lambda_{\mathbf{k}}^{2}$ is the common eigenvalue of the matrices $(A-B)(A+B)$ and $(A+B)(A-B)$ with the corresponding eigenvectors $\phi_{\mathbf{ki}}=g_{\mathbf{ki}}+h_{\mathbf{ki}}$ and $\psi_{\mathbf{ki}}=g_{\mathbf{ki}}-h_{\mathbf{ki}}$ respectively (see Ref.lsm61 for details). The ground state is defined as $\|g\rangle$, which satisfies the relation $\eta_{\mathbf{k}}\|g\rangle=0$ (5) With respect to fermi operator $\eta_{\mathbf{k}}$, one has relations $\displaystyle\frac{1}{L}\sum_{\mathbf{i}}g_{\mathbf{ki}}g_{\mathbf{k^{\prime}i}}+h_{\mathbf{ki}}h_{\mathbf{k^{\prime}i}}$ $\displaystyle=$ $\displaystyle\delta^{(3)}_{\mathbf{k^{\prime}k}}$ $\displaystyle\frac{1}{L}\sum_{\mathbf{i}}g_{\mathbf{ki}}h_{\mathbf{k^{\prime}i}}+h_{\mathbf{ki}}g_{\mathbf{k^{\prime}i}}$ $\displaystyle=$ $\displaystyle 0$ (6) Furthermore the requirement that $\\{\phi_{k},\forall k\\}$ and $\\{\psi_{k},\forall k\\}$ be normalized and complete, reinforce the relations lsm61 $\displaystyle\frac{1}{L}\sum_{\mathbf{k}}g_{\mathbf{ki}}g_{\mathbf{kj}}+h_{\mathbf{ki}}h_{\mathbf{kj}}$ $\displaystyle=$ $\displaystyle\delta_{\mathbf{ij}}$ $\displaystyle\frac{1}{L}\sum_{\mathbf{k}}g_{\mathbf{ki}}h_{\mathbf{kj}}+h_{\mathbf{ki}}g_{\mathbf{kj}}$ $\displaystyle=$ $\displaystyle 0$ (7) With the help of these formula above, one obtains $c_{\mathbf{i}}=\frac{1}{\sqrt{L}}\sum_{\mathbf{k}}g_{\mathbf{ki}}\eta_{\mathbf{k}}+h_{\mathbf{ki}}\eta_{\mathbf{k}}^{\dagger},$ (8) which would benefit our calculation for the correlation functions. ### 2.1 Concurrence The concurrence, first introduced by Wootterswootters for the measure of two- qubit entanglement, is defined as $c=\max\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\},$ (9) in which $\lambda_{i}(i=1,2,3,4)$ are the square roots of eigenvalues of matrix $R=\rho(\sigma^{y}\otimes\sigma^{y})\rho(\sigma^{y}\otimes\sigma^{y})$ with decreasing order. Then the critical step is to determine the two-body reduced density operator $\rho$. The reduced density operator $\rho_{\mathbf{ij}}$ for two spin-half particles labeled $\mathbf{i,j}$ can be written generally as, $\rho_{\mathbf{ij}}=\text{tr}_{\mathbf{ij}}\rho=\frac{1}{4}\sum_{\alpha,\beta=0}^{4}p_{\alpha,\beta}\sigma^{\alpha}_{\mathbf{i}}\otimes\sigma^{\beta}_{\mathbf{j}},$ (10) in which $\rho$ is the density matrix for the whole system and $\sigma^{0}$ is the $2\times 2$ unity matrix and $\sigma^{\alpha}(\alpha=1,2,3)$ are the Pauli operators $\sigma^{x},\sigma^{y},\sigma^{z}$, which also the generators of $SU(2)$ group. $p_{\alpha\beta}=\text{tr}[\sigma^{\alpha}_{\mathbf{i}}\sigma^{\beta}_{\mathbf{j}}\rho_{\mathbf{ij}}]=\langle\sigma^{\alpha}_{\mathbf{j}}\sigma^{\beta}_{\mathbf{j}}\rangle$ is the correlation function. With the symmetry Eq. (2), one can verify that only $p_{00},p_{03},p_{30},p_{11},p_{22},p_{33},p_{12},p_{21}$ are not vanishing. After some efforts, one obtain $c=\max\\{0,c_{I},c_{II}\\},$ (11) in which $\displaystyle c_{I}$ $\displaystyle=$ $\displaystyle\frac{1}{2}[\sqrt{(p_{11}+p_{22})^{2}+(p_{12}-p_{21})^{2}}$ $\displaystyle-\sqrt{(1+p_{33})^{2}-(p_{30}+p_{03})^{2}}]$ $\displaystyle c_{II}$ $\displaystyle=$ $\displaystyle\frac{1}{2}[\|p_{11}-p_{22}\|-\sqrt{(1-p_{33})^{2}-(p_{30}-p_{03})^{2}}].$ (12) In order to obtain the reduced density operator for two fermions, it is crucial to construct $SU(2)$ algebra for the fermions in lattice systems. In 1D case, the Jordan-Wigner (JW) transformation is availablejw28 ; cp05 ; zhu ; lsm61 . For higher dimension cases the JW-like transformation has been constructed by different methodsjw . However the transformation is very complex and the calculation is difficult. Hence instead of a general calculation, we focus on the nearest neighbor two lattices in this paper. In this situation, the $SU(2)$ algebra can be readily constructed $\displaystyle\sigma_{\mathbf{i}}^{+}=(\sigma_{\mathbf{i}}^{x}+i\sigma_{\mathbf{i}}^{y})/2=c^{\dagger}_{\mathbf{i}}$ $\displaystyle\sigma_{\mathbf{i}}^{-}=(\sigma_{\mathbf{i}}^{x}-i\sigma_{\mathbf{i}}^{y})/2=c_{\mathbf{i}}$ $\displaystyle\sigma_{\mathbf{i}}^{z}=2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}-1$ $\displaystyle\sigma_{\mathbf{i}+1}^{+}=(\sigma_{\mathbf{i}+1}^{x}+i\sigma_{\mathbf{i}+1}^{y})/2=(2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}-1)c^{\dagger}_{\mathbf{i}+1}$ $\displaystyle\sigma_{\mathbf{i}+1}^{-}=(\sigma_{\mathbf{i}+1}^{x}-i\sigma_{\mathbf{i}+1}^{y})/2=(2c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}-1)c_{\mathbf{i}+1}$ $\displaystyle\sigma_{\mathbf{i}+1}^{z}=2c_{\mathbf{i}+1}^{\dagger}c_{\mathbf{i}+1}-1$ (13) in which $\mathbf{i}+1$ denotes the nearest neighbor lattice for site $\mathbf{i}$. This point can be explained as the following. The difficulty for the JW transformation in higher dimension case comes from the absence of a natural ordering of particles. However when one focuses on the nearest neighbored particle, this difficulty does not appear since for a definite direction the nearest neighbor particle is unique (for non-nearest neighbored case one have to consider the effect from the other particles). Then the correlation functions for the ground state are in this case $\displaystyle p_{00}$ $\displaystyle=$ $\displaystyle 1,p_{30}=1-\frac{2}{L}\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2};p_{03}=1-\frac{2}{L}\sum_{\mathbf{k}}h_{\mathbf{k(i+1)}}^{2};$ $\displaystyle p_{11}$ $\displaystyle=$ $\displaystyle\frac{2}{L}\sum_{\mathbf{k}}(h_{\mathbf{ki}}-g_{\mathbf{ki}})(h_{\mathbf{k(i+1)}}+g_{\mathbf{k(i+1)}});$ $\displaystyle p_{22}$ $\displaystyle=$ $\displaystyle\frac{2}{L}\sum_{\mathbf{k}}(h_{\mathbf{ki}}+g_{\mathbf{ki}})(h_{\mathbf{k(i+1)}}-g_{\mathbf{k(i+1)}})$ $\displaystyle p_{33}$ $\displaystyle=$ $\displaystyle(1-\frac{2}{L}\sum_{\mathbf{k}}h^{2}_{\mathbf{ki}})(1-\frac{2}{L}\sum_{\mathbf{k}}h^{2}_{\mathbf{k(i+1)}})$ $\displaystyle+\frac{4}{L^{2}}\sum_{\mathbf{k,k^{\prime}}}h_{\mathbf{ki}}h_{\mathbf{k(i+1)}}g_{\mathbf{k^{\prime}i}}g_{\mathbf{k^{\prime}(i+1)}}-h_{\mathbf{ki}}g_{\mathbf{ki}}h_{\mathbf{k^{\prime}(i+1)}}g_{\mathbf{k^{\prime}(i+1)}}$ $\displaystyle p_{12}$ $\displaystyle=$ $\displaystyle p_{21}=0$ (14) ### 2.2 Geometric Phase Following the method in Refs.cp05 ; zhu , one can introduce a globe rotation $R(\phi)=\exp[i\phi\sum_{i}c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}}]$ to obtain the geometric phase(GP). Then we have Hamiltonian with parameter $\phi$ $H(\phi)=\sum_{\mathbf{ij}}^{L}c_{\mathbf{i}}^{\dagger}A_{ij}c_{\mathbf{j}}+\frac{1}{2}(c_{\mathbf{i}}^{\dagger}B_{\mathbf{ij}}c_{\mathbf{j}}^{\dagger}e^{2i\phi}+\text{h.c.}),$ (15) and the ground state becomes $\|g(\phi)\rangle=R(\phi)\|g\rangle$. GP is defined as berry $\displaystyle\gamma_{g}$ $\displaystyle=$ $\displaystyle-i\int d\phi\langle g(\phi)\|\frac{\partial}{\partial\phi}\|(\phi)\rangle$ (16) $\displaystyle=$ $\displaystyle\frac{\phi}{L}\sum_{\mathbf{i}}\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2}$ Regarding to Eq.(15), one only require $\phi=\pi$ for a cycle evolution. Hence one has $\gamma_{g}=\frac{\pi}{L}\sum_{i}\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2}=\frac{1}{L}\sum_{\mathbf{i}}\gamma_{g\mathbf{i}}$. ### 2.3 GP vs. Concurrence At a glance of Eq.(2.1) and Eq.(16), GP and concurrence both are related directly to correlation functions. Hence it is tempting to find the relation between the two quantities, which would benefit to the understanding of the physical meaning of concurrence. According to Eqs.(2.1) and (2.1), the following inequality can be obtained (see Appendix for details of calculations) $\displaystyle c_{I}$ $\displaystyle\leq$ $\displaystyle\frac{1}{L\pi}(\gamma_{g\mathbf{i}}+\gamma_{g(\mathbf{i+1})})-\sqrt{(1+p_{33})^{2}-(p_{30}+p_{03})^{2}}$ $\displaystyle c_{II}$ $\displaystyle\leq$ $\displaystyle 1+\frac{1}{L\pi}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})-\frac{1}{2L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})^{2}$ (17) For the first inequality, a much tighter bound is difficult to find. While if the average of $c_{II}$ over all site $\mathbf{i}$ is considered, $c_{II}\leq 1-\frac{1}{2L^{3}\pi^{2}}\sum_{i}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})^{2}$. Fortunately in the following examples $c_{I}$ is always negative. Although the existence of this defect, in our own points, the relation between GP and concurrence have been displayed genuinely from the inequality above. ## 3 GP and Concurrence in Higher Dimensional $XY$ model The previous section presents the general discussion of GP and concurrence in free fermion lattice system Eq.(1). In this section a concrete model would be checked explicitly, of which the Hamiltonian is $H=\sum_{\langle\mathbf{i,j}\rangle}[c_{\mathbf{i}}^{\dagger}c_{\mathbf{j}}-\gamma(c_{\mathbf{i}}^{\dagger}c_{\mathbf{j}}^{\dagger}+\text{h.c.})]-2\lambda\sum_{\mathbf{i}}c_{\mathbf{i}}^{\dagger}c_{\mathbf{i}},$ (18) in which $\langle\mathbf{i,j}\rangle$ denotes the nearest-neighbor lattice sites and $c_{\mathbf{i}}$ is fermion operator. This Hamiltonian, first introduced in Ref.li06 , depicts the hopping and pairing between nearest- neighbor sites in hypercubic lattice systems, in which $\lambda$ is the chemical potential and $\gamma$ is the pairing potential. Eq.(18) could be considered as a $d$-dimensional generalization of 1D XY model. However for $d>1$ case, this model shows different phase features li06 . The Hamiltonian can be diagonalized by introducing the $d$-dimensional Fourier transformation with periodic boundary condition in momentum space li06 $H=\sum_{\mathbf{k}}2t_{\mathbf{k}}c_{\mathbf{k}}^{\dagger}c_{\mathbf{k}}-i\Delta_{\mathbf{k}}(c_{\mathbf{k}}^{\dagger}c_{-\mathbf{k}}^{\dagger}-\text{h.c.}),$ (19) in which $t_{\mathbf{k}}=\sum_{\alpha=1}^{d}\cos k_{\alpha}-\lambda$ and $\Delta_{\mathbf{k}}=\gamma\sum_{\alpha=1}^{d}\sin k_{\alpha}$. With the help of Bogoliubov transformation, one obtains $H=\sum_{\mathbf{k}}2\Lambda_{\mathbf{k}}\eta_{\mathbf{k}}^{\dagger}\eta_{\mathbf{k}}+\text{const}.$ (20) in which $\Lambda_{\mathbf{k}}=\sqrt{t_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2}}$. Based on the degeneracy of the eigenenergy $\Lambda_{\mathbf{k}}=0$, the phase diagram can be determined clearlyli06 ; When $d=2$, the phases diagram should be identified as two different situations; for $\gamma=0$, the degeneracy of the ground state occurs when $\lambda\in[0,2]$, whereas the gap above the ground state is non-vanishing for $\lambda>2$. However for $\gamma\neq 0$ three different phases can be identified as $\lambda=0$, $\lambda\in(0,2]$ and $\lambda>2$. The first two phases correspond to case that the energy gap above the ground state vanishes, whereas not for $\lambda>2$. One should note that $\lambda=0$ means a well-defined Fermi surface with $k_{x}=k_{y}\pm\pi$, whose symmetry is lowered by the presence of $\lambda$ term. For $d=3$ two phases can be identified as $\lambda\in[0,3]$ with the vanishing energy gap above the ground state and $\lambda>3$ with a non-vanishing energy gap above ground state. In a word the critical points can be identified as $\lambda_{c}=d(d=1,2,3)$ for any anisotropy of $\gamma$, and $\lambda=0$ for $d=2$ with $\gamma\neq 0$. One should note that since the $\gamma^{2}$ dependence of $\Lambda_{\mathbf{k}}$, the sign of $\gamma$ does not matter. Hence the plots below are only for positive $\gamma$. The correlation functions between nearest-neighbor lattice sites would play a dominant role in the transition between different phases because of the nearest-neighbor interaction, similar to the case in XY model oo02 . Then it is expected that the pairwise entanglement is significant in this model. In the following, concurrence for the nearest-neighbor sites of ground state is calculated for $d=2,3$ respectively. The geometric phase of ground state is also calculated by imposing a globe rotation $R(\phi)$. our calculation shows that both quantities show interesting singularity closed to the boundary of different phases. ### 3.1 Concurrence For $d>1$ case, the nearest-neighbor lattice sites appear in different directions. In order to eliminate the dependence of orientations, the calculation of correlation functions Eqs.(2.1) is implemented by averaging in all directions. With the transformation Eq.(2.1), one can determine under the thermodynamic limit $\displaystyle p_{11}$ $\displaystyle=$ $\displaystyle\frac{1}{d(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha}^{d}dk_{\alpha}(\Delta_{k}\sum_{\alpha=1}^{d}\sin k_{\alpha}-t_{k}\sum_{\alpha=1}^{d}\cos k_{\alpha})/\Lambda_{k}$ $\displaystyle p_{22}$ $\displaystyle=$ $\displaystyle-\frac{1}{d(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha}^{d}dk_{\alpha}(\Delta_{k}\sum_{\alpha=1}^{d}\sin k_{\alpha}+t_{k}\sum_{\alpha=1}^{d}\cos k_{\alpha})/\Lambda_{k}$ $\displaystyle p_{12}$ $\displaystyle=$ $\displaystyle p_{21}=0$ $\displaystyle p_{03}$ $\displaystyle=$ $\displaystyle p_{30}=p_{3}=\frac{1}{(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha}^{d}dk_{\alpha}\frac{t_{k}}{\Lambda_{k}}$ $\displaystyle p_{33}$ $\displaystyle=$ $\displaystyle p_{3}^{2}-(\frac{p_{11}+p_{22}}{2})^{2}+(\frac{p_{11}-p_{22}}{2})^{2}$ (21) $d=2$ Our calculation shows that $c_{I}$ is negative. So in Fig. 1, only $c_{II}$ and its derivative with $\lambda$ are numerically illustrated. In order to avoid the ambiguity because of the cutoff in the definition of concurrence, the derivative of $c_{II}$ with $\lambda$ is depicted in all region whether $c_{II}$ positive or notyang . Obviously the singularity for $\partial c_{II}/\partial\lambda$ can be found at the point $\lambda=0,2$ respectively, which are consistent with our knowledge about phase transitions. $d=3$ Similar to the case of $d=2$, our calculation shows $c_{I}<0$. Only $c_{II}$ and its derivative with $\lambda$ are numerically displayed in Fig.2. Different from the case of $d=2$, no singularity of the first derivative of $c_{II}$ with $\lambda$ is found at $\lambda=3$. While a cusp appears at $\lambda=1$. A further calculation demonstrates that the second derivative of $c_{II}$ is divergent genuinely at exact $\lambda=3$, as shown in Figs.2(c). which means the phase transition at this points. Furthermore our numerical calculations show that $\partial^{2}c_{II}/\partial\lambda^{2}$ is finite at $\lambda=1$, as shown in Figs.2(b). Hence one cannot attribute this feature to the phase transition. The similar feature has been found in the previous studies oo02 ; yang ; gu . However the underlying physical reason is unclear in general. But this special feature is not unique for concurrence; van Hove singularity in solid state physics displays the similar feature, which is because of the vanishing of the moment-gradient of the energy. Although we cannot established the direct relation between these two issues because of the bad definition of the moment-gradient of the energy when degeneracy happening, we affirm that this feature is not an accident and the underlying physical reason is still to be found. In a word the discussion above first demonstrates the exact connection between concurrence and quantum phase transitions in high-dimensional many body systems. However a question is still open; what the physical interpretation of concurrence is in many-body systems. In this study, we includes the negative part of $c_{II}$ to identify the phase diagram in free-fermion systems. In general, it is believed that the negative $c_{II}$ means no entanglement between two particles and then include no any useful information about state. But from the discussion one can note that the omission of the negative part of $c_{II}$ would lead to incorrect results. Moreover, for $\gamma=0$, our calculations show that $c_{I},c_{II}$ always are zero, and so one cannot obtain any the phase transition information from pair wise entanglement in this case. Further discussions will be presented in the final part of this paper. ### 3.2 Geometric Phase Geometric phase manifests the structure of Hilbert in the system and has intimate relation to the degeneracy. GP, defined in Eq. (16) by imposing a globe rotation $R(\phi)$ on ground state $\|g\rangle$ is calculated in this section. After some algebra, one obtains $\gamma_{g}=\frac{\pi}{2(2\pi)^{d}}\int_{-\pi}^{\pi}\prod_{\alpha=1}^{d}dk_{\alpha}(1-\frac{t_{k}}{\Lambda_{k}}).$ (22) $d=2$ In Fig.3, $\gamma_{g}$ and its derivative with $\lambda$ are displayed explicitly. Obviously one notes that $\partial\gamma_{g}/\partial\lambda$ shows the singularity closed to $\lambda=0,2$, which are exactly the phase transition points of Hamiltonian Eq.(18). An interesting observation is that closed to these points, both GP and concurrence $c_{II}$ show the similar behaviors. $d=3$ GP and its derivative are plotted explicitly in Fig.(4). One should note that there is a platform below $\lambda=1$ for $\partial\gamma_{g}/\partial\lambda$, as shown in Fig.4(a), but a further calculation shows that $\partial^{2}\gamma_{g}/\partial\lambda^{2}$ is continued (Fig.4(b)) and $\partial\gamma_{g}/\partial\lambda$ has no divergency at this point. This phenomena is very similar to the case of concurrence (see Fig.2(b, c)). As expected, $\partial^{2}\gamma_{g}/\partial\lambda^{2}$ is divergent at exact $\lambda=3$, which means a phase transition happens at this point. Together with respect of the case of $d=2$, it makes us a suspect that GP and concurrence in our model have the same physical origination. Furthermore for $\gamma=0$, GP fails to mark the phase transition too. This is similar to the case of concurrence, but has different physical reason. The further discussion is presented in the next section. ## 4 Discussion and Conclusions The pairwise entanglement and geometric phase for ground state in $d$-dimensional ($d=2,3$) free-fermion lattice systems are discussed in this paper. By imposing the transformation Eq.(2.1), the reduce two-body density matrix for the nearest neighbor particles can be determined exactly for any dimension, and the concurrence is also calculated explicitly. Furthermore geometric phase for ground state, obtained by introducing a globe rotation $R(\phi)$, has also been calculated. Given the known results for XY model oo02 ; cp05 ; zhu , our calculations show again that both GP and concurrence display intimate connection with the phase transitions. Moreover an inequality relation between concurrence and geometric phase is also presented in Eq. (2.3). The similar scaling behaviors at the transition point $\lambda=3$ has also been shown in Figs. 5. These facts strongly mean the intimate connection between the two items. This point can be understand by noting that both of them are connected to the correlation functions, as shown in Eqs. (2.1) and (16). An interesting point in our study is that in order to obtain all information of phase diagram in model Eq.(18), the negative part of $c_{II}$ has to be included to avoiding the confusion because of the mathematical cutoff in the definition of concurrenceyang . In general, it is well accepted that the negative part of $c_{II}$ gives no any information of quantum pairwise entanglement, and then is considered to be meaningless. However, in our calculation, the negative part of $c_{II}$ appears as an indispensable consideration to obtain the correct phase information. This point means that the pairwise entanglement does not provide the all information about the system since the two-body reduced density operator throw away much information. As for the geometric phase, defined in Eq. (16), it is obvious that $\gamma_{g}$ can tell us the happening of phase transition at the point, where $\gamma_{g}$ display some kinds of singularity. However it cannot distinguished the degenerate region from the nondegenerate, as shown in Figs. 3 and 4. Recently GP imposing by the twist operator in many-body systems is introduced as an order parameter to distinguish the phases cui08 ; hkh08 . For the free-fermion lattice system, this GP have also calculated and shows the intimate connection with the vanishing of energy gap above the ground state. However the boundary between the two different phases becomes obscure with the increment of dimensionality in that discussion cui08 , and moreover it cannot distinguish the phase transition not come from the degeneracy of the ground- state energy. While the geometric phase imposing by the globe rotation $R(\phi)$ clearly demonstrate the existence of this kind of phase transition, as shown in Fig.3, whether originated from the degeneracy or not. In fact this point can be understood by noting the intimate relation between $\gamma_{g}$ and correlation functions. It maybe hint that one has to find different methods for different many-body systems to identify the phase diagram. Although the intimate relationship of concurrence and GP with phase transitions in the model Eq.(18), a exceptional happens when $\gamma=0$, in which $c_{I},c_{II}$ are zero and GP is a constant independent of $\lambda$. From Eq.(18), $\gamma=0$ means the hopping of particles is dominant, and the position of particle becomes meaningless. Since the calculation of concurrence depend on the relative position of lattice site, the pairwise entanglement is disappearing. However one could introduce the spatial entanglement to detect the phase transition in this casehav07 . For GP, $\gamma=0$ means the emergency of new symmetry. One can find $[\sum_{\mathbf{i}}c^{\dagger}_{\mathbf{i}}c_{\mathbf{i}},H]=0$ in this case, which leads to the failure of $R(\phi)$ for construction of nontrivial GP. Finally we try to transfer two viewpoints in this paper. One is that concurrence and geometric phase can be used to mark the phase transition in many-body systems since both of them are intimately connected to the correlation functions. The other is that concurrence and the geometric phase are connected directly by the inequality Eq. (2.3). Then it is interesting to extend this relation to multipartite entanglement in the future works, which would be helpful to establish the physical understanding of entanglement. ###### Acknowledgements. The author (Cui) would appreciate the help from Dr. Kai Niu (DLUT) and Dr. Chengwu Zhang (NJU) in the numerical calculations and permission of the usage of their powerful computers. We also thank greatly the enlightening discussion with Dr. Chong Li (DLUT). Especially we thank the first referee for his/her important hint for the van Hove singularity. This work is supported by the Special Foundation of Theoretical Physics of NSF in China, Grant No. 10747159\. ## APPENDIX For the first inequality, one should note $\displaystyle\|p_{11}+p_{22}\|$ (23) $\displaystyle=$ $\displaystyle\frac{4}{L}\|\sum_{\mathbf{k}}h_{\mathbf{ki}}h_{\mathbf{k(i+1)}}\|\leq\frac{4}{L}\sum_{\mathbf{k}}\|h_{\mathbf{ki}}h_{\mathbf{k(i+1)}}\|$ $\displaystyle\leq$ $\displaystyle\frac{2}{L}\sum_{\mathbf{k}}(h^{2}_{\mathbf{ki}}+h^{2}_{\mathbf{k(i+1)}})=\frac{2}{L\pi}(\gamma_{g\mathbf{i}}+\gamma_{g(\mathbf{i+1})}).$ From inequality $\sqrt{x^{2}-y^{2}}\geq\|x\|-\|y\|(\|x\|>\|y\|)$, one reduces $\sqrt{(1+p_{33})^{2}-(p_{30}+p_{03})^{2}}]\geq\|1+p_{33}\|-\|p_{30}+p_{03}\|.$ (24) Then one obtains $c_{I}\leq\frac{1}{L\pi}(\gamma_{g\mathbf{i}}+\gamma_{g(\mathbf{i+1})})+\frac{1}{2}(\|p_{30}+p_{03}\|-\|1+p_{33}\|).$ (25) However a much tighter bound is difficult to decide because of the complexity of $p_{33}$. For the second inequality, it can be obtained easily by observing $\displaystyle p_{33}\leq 1-\frac{1}{L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}^{2}-\gamma_{g(\mathbf{i+1})}^{2}),$ (26) in which we have used the relation $2ab\leq a^{2}+b^{2}$. Then $1-p_{33}$ is non-negative and $\displaystyle c_{II}$ $\displaystyle=$ $\displaystyle\frac{2}{L}\sum_{\mathbf{k}}\|h_{\mathbf{ki}}g_{\mathbf{k(i+1)}}\|+\frac{p_{33}-1}{2}$ (27) $\displaystyle\leq$ $\displaystyle\frac{1}{L}\sum_{\mathbf{k}}(h^{2}_{\mathbf{ki}}+g^{2}_{\mathbf{k(i+1)}})-\frac{1}{2L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}-\gamma_{g(\mathbf{i+1})})^{2}$ $\displaystyle\leq$ $\displaystyle 1+\frac{1}{L\pi}(\gamma_{g\mathbf{i}}-\gamma_{g\mathbf{(i+1)}})-\frac{1}{2L^{2}\pi^{2}}(\gamma_{g\mathbf{i}}-\gamma_{g(\mathbf{i+1})})^{2}$ in which $1/L\sum_{\mathbf{k}}g_{\mathbf{ki}}^{2}=1-1/L\sum_{\mathbf{k}}h_{\mathbf{ki}}^{2}$ is used. ## References * (1) L. Amico, R. Fazio, A. Osterloh, V. Vedral, Rev. Mod. Phys. 80, 517 (2008) and available at arXiv: quant-ph/0703044(2007). * (2) J. Preskill, J. Mot. Opt. 47, (2000)127. * (3) A. Osterloh, L. Amico, G. Falci, R. Fazio, Nature, 416, 6(2002)08; T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, (2002)032110. * (4) X. X. Yi, H. T. Cui and L. C. Wang, Phys. Rev. A 74, (2006)054102. * (5) H.T. Quan, Z. Song, X.F. Liu, P. Zanardi, C.P. Sun, Phys. Rev. Lett. 96, (2006)140604. * (6) P. Zanardi and N Paunković, Phys. Rev. E 74, 031123 (2006); P. Zanardi, P. Giorda, M. Cozzini, Phys. Rev. Lett. 99, (2007)100603; Shi-Jian Gu, e-print available at arXiv: 0811.3127. * (7) M. F. Yang, Phys. Rev. A 71, (2005)030302. * (8) V. Vedral, J. Mod. Opt. 54, 2185(2007). * (9) M. R. Dowling, A. C. Doherty, and S. D. Bartlett, Phys. Rev. A 70, (2004)062113; Wieśniak, V. Vedral, and časlav Brukner, New. J. Phys. 7, 2005258. * (10) M. V. Berry, Proc. R. Soc. London A 392, (1984)45. * (11) A. Shapere and F. Wilczek, Geometric Phase in Physics (World Scientific, 1989); A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, J. Zwanziger, The Geometric Phase in Quantum System(Springer, 2003). * (12) Angelo C. M. Carollo, J. K. Pachos, Phys. Rev. Lett. 95, (2005)157203;J. K. Pachos, Angelo C. M. Carollo, Phil. Trans. R. Soc. A 364, 3463(2006). * (13) S. L. Zhu, Phys. Rev. Lett. 96, (2006)077206. * (14) A. Hamma, arXiv: quant-ph/0602091. * (15) H.T. Cui, K. Li, and X.X. Yi, Phys. Lett. A 360, (2006)243. * (16) F. Plastina, G. Liberti, A.Carollo, Europhys.Lett. 76, (2006)182. * (17) H.T. Cui, J. Yi, Phys. Rev. A 78, (2008)022101. * (18) T. Hirano, H. Katsura, and Y. Hatsugai, Phys. Rev. B 77, (2008)094431; Phys. Rev. B 78, (2008)054431. * (19) S.L. Zhu, Int. J. Mod. Phys. B 22, (2008)561. * (20) Subir Sachdev, Quantum Phase Transition(Cambridge University Press, Cambridge, 1999). * (21) S. Q. Shen, Phys. Rev. B 70, (2004)081311; M. C. Chang , Phys. Rev. B 71 , (2005)085315; T.W. Chen, C.M. Huang, G. Y. Guo, Phys. Rev. B, 73, (2006)235309; D. N. Sheng, Z. Y. Weng, L. Sheng, F. D. M. Haldane, Phys. Rev. Lett.97, (2006)036808; B Zhou, C.X. Liu, S.Q. Shen, Europhys. Lett. 79, (2007)47010. * (22) T. Senthil, Proceedings of conference on ‘Recent Progress in Many-Body Theories’, Santa Fe, New Mexico (USA, 2004). * (23) S. Ryu, Y. Hstsugai, Phys. Rev. B 73, (2006)245115. * (24) S. Bose, Phys. Rev. Lett. 91, (2003)207901; Z. Song and C.P. Sun, Low Temperature Physics, 31, (2005)8. * (25) E. Lieb, T. Schultz and D. Mattis, Ann. Phys. 16, (1961)407. * (26) W. K. Wootters, Phys. Rev. Lett. 80, (1998)2245. * (27) P. Jordan and E. Wigner, Z. Physik 47, (1928)631. * (28) E. Fradkin, Phys. Rev. Lett. 63, 322(1989); Y.R. Wang, Phys. Rev. B, 43, (1991)3786; L. Huerta and J. Zanelli, Phys. Rev. Lett. 71, (1993)3622; Shaofeng Wang, Phys. Rev. E, 51, (1995)1004; C.D. Batista and G. Ortiz, Phys. Rev. Lett, 86, (2001)1082; Adv. in Phys. 53, (2004)1. * (29) W.F. Li, L.T. Ding, R. Yu, T. Roscide, S. Haas, Phys. Rev. B 74, (2006)073103. * (30) S.J. Gu, G.S. Tian, H.Q. Lin, Phys. Rev. A 71, (2004)052332; Chin. Phys. Lett. 24, (2007)2737. * (31) P. Zanardi, Phys. Rev. A 65, 042101(2002). \begin{overpic}[width=170.71652pt]{1.eps} \put(-5.0,40.0){\large$c_{II}$} \put(50.0,0.0){\large$\lambda$}\put(97.0,50.0){\large\begin{rotate}{-90.0}$\partial c_{II}/\partial\lambda$\end{rotate}} \end{overpic} Figure 1: $c_{II}$ ($\bigcirc$) and its derivative with $\lambda$ ($\triangle$) vs. $\lambda$ when $d=2$. We have chosen $\gamma=1$ for this plot. \begin{overpic}[width=170.71652pt]{2a.eps} \put(15.0,60.0){(a)}\put(-5.0,40.0){\large$c_{II}$} \put(45.0,0.0){\large$\lambda$} \put(94.0,50.0){\large\begin{rotate}{-90.0}$\partial c_{II}/\partial\lambda$\end{rotate}} \end{overpic} \begin{overpic}[width=128.0374pt]{2b.eps} \put(18.0,78.0){(b)} \put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}c_{II}/\partial\lambda^{2}$\end{rotate}} \put(40.0,0.0){\large$\lambda$} \end{overpic}\begin{overpic}[width=136.5733pt]{2c.eps} \put(22.0,78.0){(c)}\put(40.0,0.0){\large$\lambda$} \end{overpic} Figure 2: $c_{II}$ ($\bigcirc$) and its derivative with $\lambda$ ($\triangle$) (a) vs. $\lambda$ when $d=3$. We have chosen $\gamma=1$ for this plot. The second derivative of $c_{II}$ with $\lambda$ are also displayed in this plot and focus the points closed to $\lambda=1$ (b) and $\lambda=3$ (c). \begin{overpic}[width=170.71652pt]{3.eps} \put(0.0,35.0){\large\begin{rotate}{90.0}$\gamma_{g}/\pi$\end{rotate}} \put(50.0,0.0){\large$\lambda$}\put(98.0,53.0){\large\begin{rotate}{-90.0}$\partial\gamma_{g}/\pi\partial\lambda$\end{rotate}} \end{overpic} Figure 3: $\gamma_{g}$ ($\bigcirc$) and its derivative with $\lambda$ ($\triangle$) vs. $\lambda$ when $d=2$. We have chosen $\gamma=1$ for this plot. \begin{overpic}[width=170.71652pt]{4a.eps} \put(13.0,60.0){(a)}\put(0.0,35.0){\large\begin{rotate}{90.0}$\gamma_{g}/\pi$\end{rotate}} \put(50.0,0.0){\large$\lambda$} \put(93.0,52.0){\large\begin{rotate}{-90.0}$\partial\gamma_{g}/\pi\partial\lambda$\end{rotate}} \end{overpic}\begin{overpic}[width=113.81102pt]{4b.eps} \put(13.0,77.0){(b)} \put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}\gamma_{g}/\pi\partial\lambda^{2}$\end{rotate}} \put(35.0,0.0){\large$\lambda$} \end{overpic}\begin{overpic}[width=130.88284pt]{4c.eps} \put(22.0,78.0){(c)}\put(42.0,0.0){\large$\lambda$} \end{overpic} Figure 4: $\gamma_{g}$ ($\bigcirc$) and its derivative with $\lambda$ ($\triangle$) (a) vs. $\lambda$ when $d=3$. We have chosen $\gamma=1$ for this plot. The second derivative‘ of $\gamma_{g}$ with $\lambda$ are also displayed in this plot and focus the points closed to $\lambda=1$ (b) and $\lambda=3$ (c). \begin{overpic}[width=125.19194pt]{5a.eps} \put(55.0,78.0){(a)} \put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}c_{II}\partial\lambda^{2}$\end{rotate}} \put(50.0,0.0){\large$\log\|\lambda-\lambda_{c}\|/\lambda_{c}$} \end{overpic}\begin{overpic}[width=130.88284pt]{5b.eps} \put(57.0,78.0){(b)}\put(5.0,35.0){\large\begin{rotate}{90.0}$\partial^{2}\gamma_{g}/\pi\partial\lambda^{2}$\end{rotate}} \end{overpic} Figure 5: The scaling of GP and concurrence for 3D case closed to the critical point $\lambda_{c}=3$. We have chosen $\gamma=1$ for this plot.	arxiv-papers	2009-02-17T09:08:11	2024-09-04T02:49:00.638808	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "H.T. Cui, Y.F. Zhang", "submitter": "Hai-tao Cui Dr.", "url": "https://arxiv.org/abs/0902.2870" }
0902.2995	ASF+ — eine ASF-”ahnliche Spezifikationssprache Rüdiger Lunde, Claus-Peter Wirth Searchable Online Edition December 22, 1994 SEKI-WORKING-PAPER SWP–94–05 (SFB) Fachbereich Informatik, Universität Kaiserslautern, D–67663 Kaiserslautern Zusammenfassung: Ohne auf wesentliche Aspekte der in [Bergstra&al.89] vorgestellten algebraischen Spezifikationssprache ASF zu verzichten, haben wir ASF um die folgenden Konzepte erweitert: W”ahrend in ASF einmal exportierte Namen bis zur Spitze der Modulhierarchie sichtbar bleiben m”ussen, erm”oglicht ASF+ ein differenziertes Verdecken von Signaturnamen. Das fehlerhafte Vermischen unterschiedlicher Strukturen, welches in ASF beim Import verschiedener Aktualisierungen desselben parametrisierten Moduls auftritt, wird in ASF+ durch eine ad”aquatere Form der Parameterbindung vermieden. Das neue Namensraum-Konzept von ASF+ erlaubt es dem Spezifizierer, einerseits die Herkunft verdeckter Namen direkt zu identifizieren und anderseits beim Import eines Moduls auszudr”ucken, ob dieses Modul nur benutzt oder in seinen wesentlichen Eigenschaften ver”andert werden soll. Im ersten Fall kann er auf eine einzige global zur Verf”ugung stehende Version zugreifen; im zweiten Fall mu”s er eine Kopie des Moduls importieren. Schlie”slich erlaubt ASF+ semantische Bedingungen an Parameter und die Angabe von Beweiszielen. Diese Arbeit ist aus einer von Klaus Madlener und Claus-Peter Wirth betreuten Projektarbeit R”udiger Lundes hervorgegangen und wurde gef”ordert von der Deutschen Forschungsgemeinschaft, SFB 314 (D4-Projekt). Abstract: Maintaining the main aspects of the algebraic specification language ASF as presented in [Bergstra&al.89] we have extend ASF with the following concepts: While once exported names in ASF must stay visible up to the top the module hierarchy, ASF+ permits a more sophisticated hiding of signature names. The erroneous merging of distinct structures that occurs when importing different actualizations of the same parameterized module in ASF is avoided in ASF+ by a more adequate form of parameter binding. The new “Namensraum”-concept of ASF+ permits the specifier on the one hand directly to identify the origin of hidden names and on the other to decide whether an imported module is only to be accessed or whether an important property of it is to be modified. In the first case he can access one single globally provided version; in the second he has to import a copy of the module. Finally ASF+ permits semantic conditions on parameters and the specification of tasks for a theorem prover. ###### Contents 1. 1 Einleitung 2. 2 Das Konzept, erkl”art anhand von Beispielspezifikationen 1. 2.1 Bottom-Up-Spezifikationen 2. 2.2 Parametrisierte Module 3. 2.3 Das Namensraumkonzept 4. 2.4 Explizites Renaming 5. 2.5 Parameterbindungen 3. 3 Strukturdiagramme 4. 4 Semantik hierarchischer Konzepte 1. 4.1 Der “benutzende” Import 2. 4.2 Der “kopierende” Import 3. 4.3 Abh”angigkeiten zwischen Namensr”aumen 4. 4.4 Verdeckte Namen 5. 4.5 Overloading 5. 5 Syntax 6. 6 Die Normalform-Prozedur 1. 6.1 Datenstrukturen 2. 6.2 Der Algorithmus 1. 6.2.1 Globale Hilfsfunktionen f”ur Sichtbarkeits”anderungen 2. 6.2.2 Kombination von Modulen 3. 6.2.3 Modulmodifikationen in Importbefehlen 4. 6.2.4 Die Normalisierungsfunktionen NF und NormalForm 3. 6.3 Ein Beispiel f”ur ein normalisiertes Modul 7. 7 Abschlie”sende Zusammenfassung ## 1 Einleitung Mit steigender Leistungsf”ahigkeit moderner automatischer Beweissysteme w”achst auch die Komplexit”at der mit ihnen zu bearbeitenden Problemstellungen. Auf der Suche nach Konzepten zur logisch strukturierten Formulierung derartiger Probleme haben sich in der Entwicklung von Spezifikationssprachen Modularisierungsans”atze herausgebildet. Eine Spezifikation besteht danach aus mehreren Modulen, die mit Hilfe von Importbefehlen aufeinander Bezug nehmen. Besonders in umfangreichen Spezifikationen erweisen sich modulare Repr”asentationen von Spezifikationen als vorteilhaft. Die Verst”andlichkeit wird durch die Zerlegung in einzelne, durch exakt definierte Schnittstellen (Importkonstrukte) miteinander verbundene Teilspezifikationen gesteigert. Au”serdem k”onnen h”aufig verwendete Strukturen (beispielsweise die Datenstruktur Boolean) in Bibliotheken abgelegt werden, was den Spezifikationsaufwand reduziert. Verschiedene M”oglichkeiten, Module miteinander zu kombinieren, werden in dieser Arbeit diskutiert. Das Hauptinteresse gilt der Entwicklung einer Sprache f”ur modulare Spezifikationen mit positiv/negativ bedingten Gleichungen. Ausgehend von der in [Bergstra&al.89] vorgestellten Sprache ASF, die bereits ”uber ein recht differenziertes Modularisierungskonzept verf”ugt, wird eine Erweiterung ASF+ vorgestellt, welche die im ersten und vorvorletzten Punkt von “1.4.1. Known defects and limitations of ASF” in [Bergstra&al.89] genannten M”angel von ASF behebt. ASF+ unterst”utzt: * • Import und Parametrisierung von Modulen * • ”Uberladen von Funktionsnamen * • Infix-Operatoren * • differenziertes Verdecken von Funktions- und Sortennamen * • positiv/negativ bedingte Gleichungen * • rudiment”are Verwaltung von Beweiszielen Als Semantik wird, analog zu [Bergstra & al. 89], semi-formal eine Normalisierungsprozedur angegeben, welche die Modulhierachie einer komplexen Spezifikation in eine flache Spezifikation (ohne Importe) umwandelt. Von zentraler Bedeutung ist in diesem Zusammenhang die Originfunktion, die jedem in der Spezifikation auftretenden Namen einen Informationsblock zuweist. Dieser enth”alt f”ur den Normalisierungsproze”s wichtige Informationen ”uber den Kontext der Namensdefinition, beispielsweise den Namen des Definitionsmoduls. Neben der Originfunktion verwaltet die Normalisierungsprozedur aus ASF+ eine Dependenzfunktion. Sie spielt bei expliziten Umbenennungen und Parameterbindungen eine wichtige Rolle und tr”agt der hierarischen Struktur der Spezifikation Rechnung. Neu in ASF+ ist auch, da”s bei der Kombination von Modulen das Umbenennen von verdeckten Namen nicht ausschlie”slich durch Konfliktfreiheit definiert wird. Jeder verdeckte Name beinhaltet in ASF+ unter anderem das K”urzel des Herkunftsmoduls, was zum einen Konfliktfreiheit garantiert, zum andern auch modulare Information sichtbar macht und damit der ”Ubersicht dient. ## 2 Das Konzept, erkl”art anhand von Beispielspezifikationen Um mit der Syntax von ASF+ vertraut zu werden und ein erstes intuitives Verst”andnis der neuen Sprache zu gewinnen, bietet es sich an, zun”achst einige Beispielspezifikationen zu betrachten. Die hier angegebenen Module Booleans, Naturals und Sequences entsprechen im wesentlichen den gleichnamigen Modulen aus [Bergstra & al. 89], Kapitel 1.1.2., was einen direkten Vergleich erlaubt. ### 2.1 Bottom-Up-Spezifikationen module Booleans short Bo { add signature { public: sorts BOOL constructors true, false : -> BOOL non-constructors and, or : BOOL # BOOL -> BOOL private: non-constructors not : BOOL -> BOOL } variables { non-constructors x,y : -> BOOL } equations { macro-equation and(x,y) { case { ( x @ true ) : y ( x @ false ): false } } macro-equation not(x) { case { ( x @ true ) : false ( x @ false ): true } } [e1] or(x, y) = not(and(not(x), not(y))) } } /* Booleans / Jedes Modul einer Spezifikation beginnt mit dem Schl”usselwort module, gefolgt vom Modulnamen, dem optionalen short-Konstrukt und einem Block. Das short- Konstrukt stellt ein Modulnamenk”urzel zur Verf”ugung, das beim Umbenennen verdeckter Namen Verwendung findet und zumindest bei langen Modulnamen nicht fehlen sollte. Fehlt die Angabe des Modulk”urzels, so wird der Modulname selbst ersatzweise als sein eigenes K”urzel verwendet. Die K”urzel werden global zur Bezeichnung der Module herangezogen und m”ussen daher innerhalb der Spezifikation eindeutig sein. Alle nicht importierten Teile der Signatur werden mit dem add signature- Konstrukt zur internen Signatur zusammengefa”st. Sie umfa”st einen nach au”sen sichtbaren (public) und einen nur innerhalb des Moduls zug”anglichen (private) Bereich, in denen Sorten- und Funktionsnamen deklariert werden k”onnen. Da der Spezifikationssemantik ein konstruktorbasierter Ansatz zu Grunde liegt (vergleiche etwa [Wirth&Gramlich93] oder [Wirth&Gramlich94]), wird zwischen constructors und non-constructors unterschieden. Im Beispiel sind die Sorte BOOL, die Konstanten true, false und die (Pr”adikats-) Funktionen and, or nach au”sen sichtbar (k”onnen also von anderen Modulen importiert werden). not wird zu Illustrationszwecken nicht exportiert, und kann infolgedessen nur innerhalb des Moduls referenziert werden. Im Beispiel folgt eine Variablenvereinbarung, die jeder im Gleichungsblock verwendeten Variable eine Sorte zuweist. Die Overloadingf”ahigkeit von ASF+ (d.h. die M”oglichkeit namensgleiche Funktionen mit verschiedenen Argumentsorten zu unterscheiden) macht eine Deklaration aller Variablen zwingend notwendig. ASF+ unterscheidet zwischen Konstruktor- und Non- Konstruktor-Variablen, die durch die Schl”usselw”orter constructors und non- constructors gekennzeichnet werden. Defaultwert ist constructors. Werden nur Konstruktor-Variablen verwendet, so kann deshalb das Schl”usselwort (wie in den folgenden Beispielen) entfallen. ASF+ unterst”utzt Spezifikationen mit positiv/negativ bedingten Gleichungen. Sie k”onnen im Gleichungsblock entweder explizit angegeben werden (im Beispiel die Zeile mit Marke e1) oder mit Hilfe des macro-equation-Konstrukts erzeugt werden. Das macro-equation-Konstrukt geht aus dem macro-rule-Konstrukt aus [Wirth&Lunde94] hervor und unterscheidet sich nur durch die C-”ahnliche Syntax. Seine Semantik ist durch Makro-Expansion in positiv/negativ bedingte Gleichungen gegeben. Eine wichtige Rolle spielen sogenannte match-conditions (Symbolisiert durch @), mit deren Hilfe Gleichungen, deren linke Seiten mit dem gleichen Funktionssymbol beginnen, zusammengefa”st werden k”onnen. Im Beispiel f”uhrt die Makro-Expansion zu den vier Gleichungen [me-and1] and(true, y) = y [me-and2] and(false, y) = false [me-not1] not(true) = false [me-not2] not(false) = true Bei umfangreichen Funktionsdefinitionen bietet die Darstellung als macro- equation gro”se Vorteile, weil durch verschachtelte case-Konstrukte zahlreiche Wiederholungen von Bedingungen eingespart werden k”onnen. F”ur die genaue Bedeutung der Makros @, case, if und else sei auf [Wirth&Lunde94] verwiesen. Alle verwendeten Variablen und Marken werden semantisch wie private- deklarierte Signaturnamen behandelt und m”ussen nur innerhalb des Moduls eindeutig sein. module Naturals short Nat { import Booleans { public: BOOL, true, false } add signature { public: sorts NAT constructors 0 : -> NAT s : NAT -> NAT non-constructors _ + _ : NAT # NAT -> NAT eq : NAT # NAT -> BOOL } variables { x,y,u : -> NAT } equations { macro-equation (x + y) { case { ( y @ 0 ) : x ( y @ s(u) ) : s(x + u) } } macro-equation eq(x,y) { if ( x = y ) true else false } } } / Naturals / Das Modul Naturals importiert das Modul Booleans. Der Block, der dem Importbefehl folgt, tr”agt der Forderung nach einem flexiblen Lokalit”atsprinzip Rechnung. Er sorgt daf”ur, da”s nur die im Block aufgef”uhrten Namen im Modul zug”anglich sind. Im Beispiel sind die Sorte BOOL und die Konstanten true und false innerhalb des Moduls Naturals sichtbar und werden auch von ihm exportiert. Die von Booleans exportierten, aber im Importkonstrukt nicht aufgef”uhrten Funktionen and und or und die nicht exportierte Funktion not k”onnen innerhalb von Naturals nicht referenziert werden. Ihre Namen gelten als verdeckt (hidden). Unter den im add signature-Konstrukt deklarierten Funktionssymbolen befindet sich auch der Infix-Operator “+”. Seine Deklarationssyntax wurde, wie auch die der Pr”afix-Operatoren, aus ASF ”ubernommen. module OrdNaturals short ONat { import Booleans { public: BOOL, true; private: or } import Naturals { public: NAT, 0, s, eq, false } add signature { public: non-constructors greater, geq: NAT # NAT -> BOOL } variables { x,y,u,v : -> NAT } equations { macro-equation greater(x,y) { case { ( x @ 0 ) : false ( x @ s(u), y @ 0 ) : true ( x @ s(u), y @ s(v) ) : greater(u,v) } } [e1] geq(x,y) = or(greater(x,y), eq(x,y)) } goals { [irref] greater(x, x) --> [trans] greater(x, u), greater(u, y) --> greater(x, y) [total] --> greater(x, y), greater(y, x), x = y } } / OrdNaturals / OrdNaturals spezifiziert eine irreflexive Ordnung greater und eine reflexive Ordnung geq f”ur Elemente des Typs NAT. Der doppelte Import des Moduls Booleans (direkt und indirekt ”uber Naturals) demonstriert, da”s die Sichtbarkeit von Namen eines importierten Moduls im allgemeinen nicht von einem Importbefehl allein abh”angt. So w”are es falsch, aus dem Fehlen des Namens false im ersten Importblock abzuleiten, da”s false innerhalb von OrdNaturals verdeckt sein mu”s. Der goals-Block am Ende von OrdNaturals erm”oglicht es dem Spezifizierer, Beweisziele anzugeben. Jede Beweisaufgabe besteht aus einer in eckigen Klammern eingefa”sten Marke, gefolgt von einer Gentzenklausel. Syntaktisch handelt es sich dabei um eine Folge von durch Kommas getrennte Gleichungen, gefolgt von einem Pfeil und einer weiteren Folge von Gleichungen. Semantisch ist die Gentzenformel $e_{1},\ldots,e_{n}$ \--> $e_{n+1},\ldots,e_{n+m}$ equivalent zu $e_{1}\wedge\ldots\wedge e_{n}\longrightarrow e_{n+1}\vee\ldots\vee e_{n+m}$. Gleichungen der Form $P(x_{1},\ldots,x_{n})$ = true k”onnen wie im Beispiel durch $P(x_{1},\ldots,x_{n})$ abgek”urzt werden. Syntaktisch korrekt ist eine solche abgek”urzte Gleichung jedoch nur dann, wenn true innerhalb des Moduls sichtbar und sortengleich mit der Zielsorte von $P$ ist. In ASF+ werden alle Beweisziele exportiert. Auf Flags zur Beschr”ankung der Sichtbarkeit, wie sie in ART [Eschbach94] Verwendung finden, wird verzichtet. ASF+ versteht sich als Eingabeschnittstelle zu einem Beweiser, nicht als Ausgabeschnittstelle. Deshalb wird auch auf solche Flags verzichtet, die Auskunft dar”uber geben, welche der Klauseln als bewiesen gelten d”urfen und welche nicht. Der Stempel “proved” ohne einen Verweis auf den Beweis, ist ohnehin von zweifelhaftem Wert, zumal kaum ”uberpr”uft werden kann, ob die Spezifikation nach setzen des Flags vom Benutzer ver”andert wurde. Es wird davon ausgegangen, da”s der Beweiser f”ur die bearbeitete Spezifikation eine Datei anlegt, die Informationen ”uber die Spezifikation enth”alt (zum Beispiel den Namen des Top-Moduls und Datum+Zeit der letzten Spezifikationsmodifikation) und neben allen bewiesenen Theoremen auch Referenzen auf die Beweise beinhaltet. ### 2.2 Parametrisierte Module Die bisher eingef”uhrten Konstrukte erscheinen ausreichend f”ur Bottom-Up- Spezifikationen. W”unschenswert sind jedoch auch Mechanismen, die es gestatten, Freir”aume innerhalb eines Modules zu erhalten, die erst sp”ater (z. B. beim Import des Moduls in ein weiteres) mit konkretem Inhalt gef”ullt werden m”ussen. Das Parameterkonzept von ASF+ gestattet es, Sorten und Funktionen in ein parametrisiertes Modul nachtr”aglich durch Parameterbindung zu “implantieren”. Als Beispiel betrachten wir das Modul Sequences, in dem Sequenzen von nicht n”aher spezifizierten Elementen definiert werden. Als Konstruktoren dienen nil (erzeugt die leere Sequenz) und cons (f”ugt ein Element an eine Sequenz an). module Sequences <(ITEMpar)> short Seq { add signature { parameters: ( sorts ITEMpar ) public: sorts SEQ constructors nil : -> SEQ cons : ITEMpar # SEQ -> SEQ } } / Sequences / In ASF+ m”ussen alle formalen Parameter (ob importiert, oder wie im Beispiel im add signature-Konstrukt deklariert) an prominenter Stelle direkt hinter dem Modulnamen in spitzen Klammern angegeben werden. Beim Auftreten mehrerer Parameter kann mit Hilfe der runden Klammern die Zahl der m”oglichen Parameterbindungen eingeschr”ankt werden. Alle Parameter eines durch runde Klammern eingefa”sten Tupels d”urfen nur an Namen desselben Moduls gebunden werden. Auch OrdSequences (unten) spezifiziert Sequenzen ”uber eine durch Bindung des Parameters ITEMpar zu pr”azisierende Sorte von Elementen. In Frage kommen hier jedoch nur Sorten, f”ur die eine irreflexive Ordnung spezifiziert wurde. Mit Hilfe dieser Ordnung wird eine lexikographische Ordnung auf Sequenzen definiert. module OrdSequences <(ITEMpar, ordpar)> short OSeq { import Booleans {public: BOOL, true, false} add signature { parameters: ( sorts ITEMpar non-constructors ordpar : ITEMpar # ITEMpar -> BOOL conditions [irref] ordpar(i1,i1) --> [trans] ordpar(i1,i2), ordpar(i2,i3) --> ordpar(i1,i3) [total] --> ordpar(i1,i2), ordpar(i2,i1), i1 = i2 ) public: sorts SEQ constructors nil : -> SEQ cons : ITEMpar # SEQ -> SEQ non-constructors greater : SEQ # SEQ -> BOOL } variables { i1, i2, i3 : -> ITEMpar seq1, seq2, s1, s2 : -> SEQ } equations { macro-equation greater(seq1, seq2) { / lex-order on sequences / case { ( seq1 @ nil ) : false ( seq1 @ cons(i1, s1), seq2 @ nil ): true ( seq1 @ cons(i1, s1), seq2 @ cons(i2, s2) ): if ( ordpar(i1, i2) ) true else if ( i1 = i2 ) greater(s1, s2) else false } } } } / OrdSequences / ASF+ verzichtet im Gegensatz zu ASF auf die Einf”uhrung eines formalen Parameters f”ur den Modulnamen, an den ein Parameter-Tupel gebunden wird. Als Parameter werden in ASF+ statt dessen die Sorten- und Funktionsnamen innerhalb der Parameter-Tupel bezeichnet. Die Gruppierung der Parameter in Bl”ocke (hier Tupel, dargestellt durch runde Klammern) wird jedoch beibehalten, weil sie sich bei der Formulierung semantischer Bedingungen als vorteilhaft erweist. Funktionsparameter k”onnen nicht ”uberladen werden. Unter “semantischen Bedingungen” verstehen wir in ASF+ Gentzenklauseln, die in der Definition eines Parameter-Tupels im Parameterteil des add signatur- Konstrukts angegeben werden k”onnen (hier irref, trans und total). Die Zul”assigkeit der Bindung eines Parameter-Tupels an Namen eines Moduls $M_{ACT}$ h”angt nun davon ab, ob die aus der Bindung hervorgehenden Gentzenklauseln innerhalb von $M_{ACT}$ gelten oder nicht. Da dieses Problem im allgemeinen unentscheidbar ist wird zus”atzlich gefordert, da”s $M_{ACT}$ Beweisziele enth”alt, die sich nur durch die Marken- und Variablennamen von den Bedingungsklauselinstanzen unterscheiden und f”ur die bereits Beweise existieren. Mit Bedingungen verkn”upfte Parameter-Tupel k”onnen nur an Namen solcher Module gebunden werden, die keine ungebundenen Parameter mehr enthalten, weil f”ur Module mit freien Parametern (bisher) keine Semantik innerhalb des ASF-Ansatzes existiert. Mit dem Konzept der semantischen Bedingungen werden vor allem zwei Ziele verfolgt: Einerseits werden semantisch unsinnige Parameterbindungen schon in der Akzeptanzphase der Spezifikation erkannt, au”serdem k”onnen diese Bedingungen in parametrisierten Beweisen als Lemmas von Bedeutung sein, weil sie die “wesentlichen” Eigenschaften der Parameter enthalten. Beim Import eines parametrisierten Moduls sind alle Parametertupel hinter dem Modulnamen in eckigen Klammern aufzuf”uhren: import OrdSequences <(ITEMpar, ordpar)> { public: SEQ, nil, cons } Da Parameter nicht verdeckt werden k”onnen, entspricht diese Syntax dem Grundsatz, da”s alle innerhalb eines Moduls sichtbaren Namen dort auch angegeben werden m”ussen. ### 2.3 Das Namensraumkonzept Eine Grundidee des flexiblen Verdeckungsmechanismus aus ASF+ ist die eindeutige Zuordnung von Namen zu Namensr”aumen. Im wesentlichen beschreibt der Namensraum das Modul, in dem der Name zum ersten Mal in Erscheinung tritt (im Folgenden als Definitionsmodul bezeichnet). Im Beispiel Naturals geh”oren unter anderem NAT und 0 dem Namensraum Naturals und BOOL und true dem Namensraum Booleans an. Der Name x kommt in beiden Namensr”aumen als Variable in unterschiedlicher Bedeutung vor. Ein Namensraum umfa”st also alle innerhalb eines Moduls eingef”uhrten Namen (einschlie”slich Marken) abz”uglich der importierten. Die Namen eines Moduls geh”oren im allgemeinen also verschiedenen Namensr”aumen an. Wir bezeichnen den Namensraum, dem die im Modul definierten Namen angeh”oren, als den moduleigenen Namensraum (er erbt auch den Namen des Moduls), alle anderen hei”sen importierte Namensr”aume. Namen aus verschiedenen Importbefehlen k”onnen nur dann miteinander identifiziert werden, wenn sie dem gleichen Namensraum angeh”oren, was bei mehrfachem Import desselben Moduls der Fall sein kann. Was geschieht aber, wenn Namen Namensr”aumen von Modulen angeh”oren, die durch Renaming oder Parameterbindung beim Import “manipuliert” wurden? ASF+ l”ost das Problem durch Schaffung neuer Namensrauminstanzen, die Kopien der urspr”unglichen Namensr”aume repr”asentieren. Das Kopieren einzelner Namen aus ASF wird durch gruppenweises Kopieren ersetzt, deren kleinste Einheiten die Namensr”aume bilden. Die schwerwiegenden Gr”unde f”ur diese konzeptionelle Entscheidung werden im Kapitel 4.2 diskutiert. ### 2.4 Explizites Renaming Unter explizitem Renaming verstehen wir in ASF+ das Umbenennen von Signatur- und Parameternamen aus importierten Modulen mit Hilfe des renamed to- Konstrukts. module Integers short Int { import Naturals[Int1] { public: NAT renamed to INT, 0, s, +, eq } add signature { public: constructors p : INT -> INT } variables { x, y : -> INT } equations { [e1] s(p(x)) = x [e2] p(s(x)) = x [e3] p(x) + y = p(x + y) } } / Integers / Integers spezifiziert den Datentyp der ganzen Zahlen unter Verwendung der nat”urlichen Zahlen. In ASF+ wird erwartet, da”s jeder Importbefehl, in dem ein explizites Renaming oder eine Parameterbindung vorgenommen wird, eine innerhalb der Spezifikation eindeutige (m”oglichst kurze) Instanzbezeichnung (im Beispiel Int1) beinhaltet. Sie wird gebraucht, um Namen unterschiedlich instanziierter Namensr”aume zu unterscheiden. Im letzten Beispiel geh”oren u. a. INT, 0 und s dem neuen Namensraum Naturals[Int1] an. Naturals[Int1] ist dabei eine Instanz (bzw. Kopie) des Namensraumes Naturals, die durch das explizite Renaming im Importbefehl geschaffen wurde. Nat”urlich k”onnen auch instanziierte Namesr”aume bei einem weiteren Import manipuliert werden: import Integers[Int2]{ public : INT renamed to INTnew } INTnew geh”ort, wie auch beispielsweise der hier nicht mehr sichtbare Name 0, nun dem Namensraum Naturals[Int1,Int2] an. Die hierachische Struktur einer Spezifikation bedingt Abh”angigkeiten zwischen Namensr”aumen. Im Beispiel f”uhrt die Umbenennung von INT des Namensraumes Naturals[Int1] nach INTnew auch zu einer ”Anderung der Konstruktordeklaration f”ur p des Namensraumes Integers (Definitions- und Wertebereich werden ge”andert), der Namensraum Booleans bleibt dagegen unbeeinflu”st. ASF+ tr”agt diesem Umstand Rechnung, indem der Konstruktor p und die Variablen x und y aus Integers dem neuen Namensraum Integers[Int2] zugeordnet werden. BOOL geh”ort nach wie vor dem Namensraum Booleans an. Allgemein h”angt ein moduleigener Namensraum von allen importierten Namensr”aumen ab, was bei der Modifikation von Namen aus indirekt importierten Modulen zur Instanziierung mehrerer Namensr”aume f”uhrt. Jede Instanzbezeichnung darf innerhalb einer Spezifikation nur ein einziges mal verwendet werden. Da zwischen Modulk”urzeln und Instanzbezeichnungen keine Verwechselungsgefahr besteht, bietet es sich an, das Modulk”urzel als Instanzbezeichnung wiederzuverwenden, sofern im Modul nur ein instanziierender Import vorgenommen wird. ### 2.5 Parameterbindungen module OrdNatSequences short ONSeq { import OrdSequences[ONSeq] <(ITEMpar bound to NAT, ordpar bound to greater) of OrdNaturals > { public: SEQ renamed to NSEQ, nil renamed to Nnil, cons, greater, BOOL, true, false } import OrdNaturals { public: NAT, greater, 0, s } } Analog zu ASF werden Parameter blockweise an ein Modul gebunden. Semantisch gesehen bedeutet die Bindung von Parametern eines Moduls $M_{FORM}$ (im Beispiel OrdSeqences) an Namen eines Moduls $M_{ACT}$ (im Beispiel OrdNaturals) einerseits, da”s Parameternamen aus $M_{FORM}$ durch Namen aus $M_{ACT}$ ersetzt werden. Letztere k”onnen entweder exportierbare Signaturnamen oder Parameter sein. Da sie jedoch nur im Kontext des Moduls $M_{ACT}$ eine Bedeutung besitzen, m”ussen andererseits beide Module miteinander kombiniert werden. Der Importblock, der der Parameterbindung folgt, bestimmt ausschlie”slich die Sichtbarkeit der Signaturnamen des Moduls $M_{FORM}$. Explizites Renaming ist zul”assig. Die Signaturnamen des Moduls $M_{ACT}$ (auch die aktuellen Parameter selbst, sofern sie nicht wieder Parameter sind) gelten im bindenden Modul (im Beispiel OrdNatSequences) als verdeckt, es sei denn, ein weiterer (direkter) Import nimmt wie im Beispiel Einflu”s auf die Sichtbarkeit einzelner Namen. Explizites Renaming ist in diesem Falle jedoch kaum sinnvoll, weil sonst die Signaturnamen des zus”atzlich importierten Moduls aufgrund der unterschiedlichen Namensrauminstanzen nicht mit denen aus $M_{ACT}$ identifiziert werden. Genau wie das explizite Renaming f”uhrt auch das Binden von Parametern zur Instanziierung der direkt betroffenen und aller davon abh”angigen Namensr”aume. Um auch ohne den direkten Import von $M_{ACT}$ eine vollst”andige Signatur zu garantieren, sorgt die Semantik der Parameterbindung daf”ur, da”s neben dem Modul $M_{FORM}$ auch automatisch $M_{ACT}$ (verdeckt) in das bindende Modul importiert wird — ein Vorgang, der im folgenden als impliziter Import bezeichnet wird. greater kann als Demonstrationsbeispiel f”ur eine ”uberladene Funktion gesehen werden. In OrdNatSequences referenziert der Name sowohl eine irreflexive Ordnung auf den nat”urlichen Zahlen als auch auf Sequenzen. Da jede Parameterbindung zu einer Instanziierung des Namensraumes der zu bindenden Parameter und aller davon abh”angigen Namensr”aume bis hin zum Modul $M_{FORM}$ f”uhrt (diese Namensr”aume fallen zusammen, falls wie in unseren Beispielen $M_{FORM}$ die Parameter selbst definiert, also nicht importiert), ist es auch m”oglich, Module “an sich selbst” zu binden, ohne da”s es zu einer unerw”unschten Vermischung der dort eingef”uhrten Strukturen kommt. SeqOfSeq spezifiziert Sequenzen von Elementen, die selbst Sequenzen sind. Um Namenskollisionen zwischen Signaturnamen der Module $M_{FORM}$ und $M_{ACT}$ zu vermeiden ist eine die Umbenennung aller Sorten und Konstanten, deren Sichtbarkeit im bindenden Modul erw”unscht ist (im Beispiel SEQ und nil) zwingend notwendig. Die sowohl aus $M_{FORM}$ als auch aus $M_{ACT}$ importierten Konstruktoren cons unterscheiden sich in ihren Argumentsorten und d”urfen daher ”uberladen werden. module SeqOfSeq <(ITEMpar)> short SOS { import Sequences[SOS] <( ITEMpar bound to SEQ ) of Sequences <(ITEMpar)> > { public: SEQ renamed to SEQ1, nil renamed to nil1, cons } import Sequences <(ITEMpar)> { public: SEQ, nil, cons } } ## 3 Strukturdiagramme Die modulare Struktur von ASF+-Spezifikationen kann mit Hilfe von Strukturdiagrammen veranschaulicht werden. Alle Namen innerhalb eines importfreien Moduls geh”oren demselben (moduleigenen) Namensraum an. Er wird durch ein Rechteck, genannt Namensraumbox dargestellt, in dem zentriert unter der Oberkante die Namensraumbezeichnung (= Modulname) steht. (160,0)(160,35)(0,35) (0,0)(160,0) Booleans Enth”alt das darzustellende Modul Importbefehle, so kann der Import durch ineinander verschachtelte Boxen dargestellt werden. Sie symbolisieren die hierarchische Struktur der Namensr”aume, die im Modul, dessen moduleigener Namensraum durch die ”au”serste Box gegeben ist, eine Rolle spielen. Ein Namensraum ist von allen Namensr”aumen abh”angig, die seine Box umschlie”st. (165,5)(165,40)(5,40) (5,5)(165,5) Booleans(170,0)(170,60)(0,60) (0,0)(170,0) Naturals Im add signatur–Konstrukt eines Moduls enthaltene Parametertupel werden oberhalb der Boxen f”ur importierte Namensr”aume in Sechsecken aufgef”uhrt. (5,50)(15,40)(155,40) (165,50)(155,60)(15,60) (5,50)(5,50) (165,5)(165,30)(5,30) (5,5)(165,5) (170,0)(170,80)(0,80) (0,0)(170,0) ITEMpar, ordparOrdSequencesBooleans Werden beim Import Namen eines Moduls ge”andert oder Parameter gebunden, f”uhrt das in ASF+ dazu, da”s alle direkt betroffenen, sowie die davon abh”angigen Namensr”aume mit der Instanzbezeichnung des Importbefehls instanziiert werden. Eine fehlerhafte Identifikation von Namen aus diesen manipulierten R”aumen mit den “Originalen” ist dadurch ausgeschlossen. (170,10)(170,40)(10,40) (10,10)(170,10) (175,5)(175,60)(5,60) (5,5)(175,5) (180,0)(180,80)(0,80) (0,0)(180,0) BooleansIntegersNaturals[Int1] Das Binden von Parametern eines Moduls $M_{FORM}$ (im Bsp. OrdSequence) an Namen eines weiteren Moduls $M_{ACT}$ (im Beispiel OrdNaturals) wird durch einen Pfeil angedeutet. Die Richtung des Pfeils verdeutlicht die Abh”angigkeit zwischen den Namensr”aumen der Module $M_{ACT}$ und $M_{FORM}$. Der instanziierte Namensraum der zu bindenden Parameter sowie alle von ihm abh”angigen Namensr”aume h”angen von jedem in $M_{ACT}$ enthaltenen Namensraum ab. (190,45)(190,70)(30,70) (30,45)(190,45) (195,40)(195,90)(25,90) (25,40)(195,40) (190,10)(190,35)(30,35) (30,10)(190,10) (200,5)(200,110)(20,110) (20,5)(200,5) OrdNaturalsNaturalsBooleansBooleans(190,160)(190,185)(30,185) (30,160)(190,160) (195,155)(195,205)(25,205) (25,155)(195,155) (190,125)(190,150)(30,150) (30,125)(190,125) (200,120)(200,225)(20,225) (20,120)(200,120) BooleansNaturalsOrdNaturalsBooleans(40,240)(180,240)(190,250) (180,260)(40,260)(30,250)(40,240) (200,235)(200,285)(20,285) (20,235)(200,235) (205,0)(205,310)(15,310) (15,0)(205,0) (39.000,246.000)(55.000,250.000)(39.000,254.000) (55,250)(0,250)(0,225)(10,225) ITEMpar, ordparOrdSequences[ONSeq]OrdNatSequences Da neben dem impliziten, durch die Parameterbindung verursachten Import von $M_{ACT}$ ein zus”atzlicher (direkter) Import erforderlich ist, um Signaturnamen aus $M_{ACT}$ f”ur das bindende Modul sichtbar zu machen f”uhren wir eine kompaktere Darstellung ein, in der wir den impliziten und den direkten Import (falls vorhanden) zu einer Box zusammenfassen. (190,45)(190,70)(30,70) (30,45)(190,45) (195,40)(195,90)(25,90) (25,40)(195,40) (190,10)(190,35)(30,35) (30,10)(190,10) (200,5)(200,110)(20,110) (20,5)(200,5) BooleansNaturalsOrdNaturalsBooleans(40,125)(180,125)(190,135) (180,145)(40,145)(30,135)(40,125) (200,120)(200,170)(20,170) (20,120)(200,120) (39.000,131.000)(55.000,135.000)(39.000,139.000) (55,135)(0,135)(0,110)(10,110) (205,0)(205,195)(15,195) (15,0)(205,0) ITEMpar, ordparOrdSequences[ONSeq]OrdNatSequences Erweitert werden k”onnen die ASF+-Strukturdiagramme durch Hinzunahme der Signatur. Jedes Modul zerf”allt zun”achst in zwei Bereiche. Links stehen die sichtbaren, rechts die verdeckten Signaturnamen. Der linke Bereich der sichtbaren Namen zerf”allt seinerseits in zwei Sichtbarkeitsstufen: Neben den public–deklarierten Namen, die vom betreffenden Modul exportiert werden k”onnen (auf die also importierende Module zugreifen k”onnen), gibt es noch die private–deklarierten Namen, welche nur innerhalb des Moduls sichtbar sind und auch nur dort referenziert werden k”onnen. Insgesamt existieren also die drei Bereiche “public”, “private” und “hidden”, die durch zwei gepunktete senkrechte Trennungslinien dargestellt werden k”onnen. NAT0seqgreatergeqandorandfalseBOOLtrue(390,195)(390,245)(10,245) (10,195)(390,195) (395,115)(395,265)(5,265) (5,115)(395,115) (390,45)(390,100)(10,100) (10,45)(390,45) (400,0)(0,0)(0,285) (400,285)(400,0) 5(310,245)(310,195) 5(350,245)(350,195) 5(350,100)(350,45) 5(310,100)(310,45) 5(240,195)(240,115) 5(200,195)(200,115) 5(200,240)(200,250) 5(240,265)(240,245) 5(120,265)(120,285) 5(80,265)(80,285) 5(120,100)(120,115) 5(80,100)(80,115) 5(120,45)(120,0) 5(80,45)(80,0) OrdNaturalsNaturalsBooleansfalsenotBooleansBOOLtruenot+or Im Beispiel sind innerhalb von Naturals die Namen NAT, 0, s, eq, + und die importierten Namen false, BOOL, true sichtbar. Nach dem Import in OrdNaturals bleiben davon zun”achst lediglich die Namen NAT, 0, s, eq und false ”ubrig. +, BOOL, und true werden hier hingegen nicht sichtbar. Der zweite Import des Moduls Booleans sorgt daf”ur, da”s auch f”ur OrdNaturals BOOL und true sichtbar sind. Hauptzweck dieses Imports ist es jedoch, die Referenzierbarkeit von or f”ur OrdNaturals zu erreichen, was beim indirekten Import ”uber Naturals nicht m”oglich war. Am Beispiel wird deutlich, da”s bei mehrfachem Import desselben Moduls ein Name in unterschiedlichen Sichtbarkeitsstufen auftreten kann und wird. Die Sichtbarkeit im importierenden Modul richtet sich bei ASF+ in diesem Fall nach der gr”o”sten importierten Sichtbarkeit (Auftreten am weitesten links im Strukturdiagramm). Gleichnamige Parameter, gleichnamige Sorten sowie gleichnamige Funktionen mit gleichen Argumentsorten, die innerhalb eines Moduls sichtbar sind und unterschiedlichen Namensr”aumen angeh”oren, stellen einen Namenskonflikt, also einen Spezifikationsfehler, dar. ## 4 Semantik hierarchischer Konzepte F”ur hierarchische Konzepte algebraischer Spezifikationssprachen sind grunds”atzlich zwei Semantikans”atze denkbar: • Jedes Modul erh”alt eine Semantik. Die Semantik einer hierarchisch modularisierten Spezifikation errechnet sich aus den einzelnen Modulsemantiken. * • Nur f”ur elementare (flache) Spezifikationen wird eine algebraische Semantik definiert. Hierarchischen Spezifikationen wird mit Hilfe eines Normalform- Algorithmus eine elementare Spezifikation zugewiesen, deren Semantik die Semantik der hierarchischen Spezifikation definiert. Die Bedeutung der Importkonstrukte ist hier eine auf der Syntax von Spezifikationsmodulen und nicht auf deren Semantiken definierte Funktion. Obwohl hinsichtlich der Modularisierung von Beweisen die erste Variante interessante Perspektiven bietet, f”allt unsere Wahl aufgrund der hohen Komplexit”at und der vielen offenen Fragen in bezug auf praktische Ad”aquatheit einer geeigneten Modulsemantik auf die zweite. Ein Vorteil dieser auch bei ASF angewandten Vorgehensweise ist die gute Operationalisierbarkeit. Von zentraler Bedeutung ist die Normalisierungsprozedur, da mit ihr (indirekt) die Semantik der einzelnen Importbefehle festgelegt wird. Im folgenden sollen grunds”atzliche M”oglichkeiten beleuchtet, Schwachstellen der ASF-Semantik erl”autert und Alternativen aufgezeigt werden. ### 4.1 Der “benutzende” Import Lassen wir zun”achst das Verdeckungskonzept au”ser acht und verzichten au”serdem auf die M”oglichkeit, Funktionen zu ”uberladen. Dann kann man sich die Bedeutung eines renamingfreien Importbefehls ohne Parameterbindungen in erster N”aherung als eine “komponentenweise” Vereinigung des importierten Moduls mit dem importierenden Modul vorstellen. Die Sortennamenmenge des resultierenden Moduls ergibt sich als Vereinigungsmenge der Sortennamen des importierten und des importierenden Moduls. Gleiches gilt f”ur Konstruktor- und Non-Konstruktor-Funktionsdeklarationen, Parametertupel, Variablendeklarationen, Gleichungen, Beweisziele und, mit Ausnahme des gerade ausgewerteten Importbefehls (der nun gel”oscht werden kann), auch f”ur die Importbefehle. Der Modulname des resultierenden Moduls ist durch die komponentenweise Vereinigung nicht festgelegt. Die Normalform einer mit Hilfe solcher Importbefehle hierarchisch strukturierten Spezifikation berechnet sich dann als komponentenweise Vereinigung aller direkt und indirekt importierten Module mit dem Top-Modul. Die Reihenfolge, mit der die Importbefehle eleminiert werden, spielt dabei f”ur das resultierende Normalformmodul keine Rolle. Ein Spezifikationsfehler liegt vor, wenn bei der Vereinigung ein inkorrektes Modul erzeugt wird. Alternativ k”onnen die Importbefehle eines Moduls auch in zwei Schritten eleminiert werden: Zun”achst werden die importierten Module untereinander und danach das Zwischenresultat mit dem importierenden Modul “vereinigt”. Dieses Vorgehen liefert bei der bisher betrachteten eingeschr”ankten Form von Importbefehlen das gleiche Resultat. Die Vereinigung der Importbefehlmenge mu”s in diesem Fall sinngem”a”s modifiziert werden: Bei der komponentenweisen Vereinigung im ersten Schritt (wir schreiben $\bigsqcup$) m”ussen alle Importbefehle der vereinigten (importierten) Module im Zwischenresultat ber”ucksichtigt werden, w”ahrend im zweiten Schritt (hier schreiben wir $\sqcup$) nur die Importbefehle des Zwischenresultats (und nicht die des importierenden Moduls) in das Resultat ”ubernommen werden d”urfen. Dies erlaubt nun die folgende Operationalisierung der Vereinigungssemantik, welche den Vorteil hat, da”s alle Zwischenergebnisse Normalformen sind, was bei der Behandlung von verdeckten Namen von Vorteil ist. * • Die Normalform eines importfreien Moduls ist das Modul selbst. * • Die Normalform eines Moduls $M$, welches $M_{1},\ldots,M_{n}$ importiert ergibt sich aus der komponentenweisen Vereinigung der Normalformen von $M_{1},\ldots,M_{n}$ und $M$. Wir schreiben: NF($M$) := $\left\\{\begin{array}[]{ll}M&\mbox{falls $M$ importfrei}\par\\\ M\sqcup\ \displaystyle\bigsqcup_{i=1}^{n}{\it NF\/}(M_{i})&\mbox{falls $M_{1},\ldots,M_{n}$ von $M$ importiert werden}.\end{array}\right.$ Die Vereinigungssemantik ist invariant gegen”uber mehrfachem Import des gleichen Moduls, auch ist die Reihenfolge der Importe ohne Bedeutung. Entscheidend bleibt lediglich, welche Module importiert werden. Diese Eigenschaften sind typisch f”ur eine bestimmte Art von Importen, die wir “benutzende Importe” nennen. Die Einfachheit der Vereinigungssemantik wird jedoch mit einem schweren Defekt erkauft. Sie identifiziert Sorten und Funktionsdeklarationen aus verschiedenen Herkunftsmodulen im Falle zuf”alliger syntaktischer Gleichheit, auch wenn sie nichts miteinander zu tun haben. Nur wenige Konflikte zwischen Modulen, die den gleichen Namen in unterschiedlicher Bedeutung benutzen, werden erkannt. Eine modifizierte Version der “Vereinigungssemantik” sollte also pr”ufen, ob es innerhalb der Spezifikation einen Namen gibt, der in zwei Modulsignaturen unterschiedlich definiert wird. In diesem Fall (wir gehen von sichtbaren, nicht ”uberladbaren Namen aus) liegt ein Namenskonflikt vor. Diese Modifikation kann auf die rekursive Variante nicht ohne weiteres ”ubertragen werden, da den Signaturnamen der Normalformen der zu importierenden Module nicht direkt angesehen werden kann, welchem Modul sie ihre Entstehung verdanken. (85,100)(85,140)(5,140) (5,100)(85,100) (85,55)(85,95)(5,95) (5,55)(85,55) (90,50)(90,160)(0,160) (0,50)(90,50) (290,75)(290,140)(200,140) (200,75)(290,75) (285,80)(285,120)(205,120) (205,80)(285,80) (295,0)(295,160)(195,160) (195,0)(295,0) (290,5)(290,70)(200,70) (200,5)(290,5) (285,10)(285,50)(205,50) (205,10)(285,10) 5(75,50)(75,160) 5(65,50)(65,160) 5(275,0)(275,160) 5(265,0)(265,160) M1M2M3M1’M2’M3’M4’M4’AAAA In obigem Beispiel tritt der Sortenname A sowohl links in den Signaturen der Normalformen von $M_{2}$ und $M_{3}$ (sie sind bereits in Normalform) als auch rechts in den Signaturen der Normalformen von $M_{2}^{\prime}$ und $M_{3}^{\prime}$ auf. W”ahrend dies in $M_{1}$ zu einem Namenskonflikt f”uhrt, k”onnen in $M_{1}^{\prime}$ beide Namen identifiziert werden, da sie aus der gleichen Definition in $M_{4}^{\prime}$ hervorgegangen sind. Den Normalformmodulen ist das jedoch nicht mehr zu entnehmen. ASF l”ost das Problem durch Einf”uhrung einer Originfunktion. Sie weist jedem Signaturnamen einen Informationsblock (Origin) zu, der u. a. den Namen des Moduls enth”alt, welches f”ur die Definition des Namens verantwortlich ist. Tritt in zwei zu importierenden Normalformmodulen der gleiche Signaturname auf, kann anhand der zugeordneten Origins entschieden werden, ob es sich um einen Namenskonflikt handelt oder nicht. Ein modifizierter Normalformalgorithmus k”onnte folgenderma”sen aussehen: Sei $\\{M_{i}\quad\|\quad 1\leq i\leq n\\}$ die Menge der vom Spezifizierer erzeugten Module einer Spezifikation, $\mbox{\it modn}_{i}$ der Name des Moduls $M_{i}$ und $\\{\mbox{\it sign}_{i,j}\quad\|\quad 1\leq j\leq m_{i}\\}$ die Menge aller Signaturnamen des Moduls $M_{i}$. Wir definieren zu jedem Modul eine Originfunktion $\begin{array}[]{lclcl}\mbox{\it Ur}_{i}&:&\\{\mbox{\it sign}_{i,j}\quad\|\quad 1\leq j\leq m_{i}\\}&\longrightarrow&\\{\mbox{\it modn}_{i}\\}\\\ \mbox{\it Ur}_{i}&:=&\\{\mbox{\it sign}_{i,j}\quad\|\quad 1\leq j\leq m_{i}\\}&\times&\\{\mbox{\it modn}_{i}\\}.\end{array}$ Die Normalform eines Moduls $M_{i}$ errechnet sich rekursiv wie folgt: ${\it NF\/}(M_{i})\ :=\ \left\\{\begin{array}[]{ll}(M_{i},\mbox{\it Ur}_{i})&\mbox{falls $M_{i}$ importfrei}\\\ (M_{i}\sqcup\displaystyle\bigsqcup_{k=1}^{p_{i}}M^{\prime}_{i^{\prime}_{k}},\ \mbox{\it Ur}_{i}\cup\bigcup_{k=1}^{p_{i}}o^{\prime}_{i^{\prime}_{k}})&\begin{array}[t]{@{}l@{}}\mbox{falls $M_{i}$ die Module $M_{i^{\prime}_{k}}$ importiert und}\\\ \mbox{$(M^{\prime}_{i^{\prime}_{k}},o^{\prime}_{i^{\prime}_{k}})={\it NF\/}(M_{i^{\prime}_{k}})$ gilt $(1\leq k\leq{p_{i}})$.}\end{array}\end{array}\right.$ Ein Namenskonflikt liegt genau dann vor, wenn $(\displaystyle\mbox{\it Ur}_{i}\cup\bigcup_{k=1}^{p_{i}}o^{\prime}_{i^{\prime}_{k}})$ keine Funktion ist. Der angegebene Algorithmus liefert genau die Semantik renamingfreier Importe ohne Parameterbindung f”ur sichtbare nicht ”uberladene Namen aus ASF bzw. ASF+. ### 4.2 Der “kopierende” Import Besonders in gro”sen Spezifikationen wird es h”aufig zu Namenskonflikten kommen, weil die Zahl der Namen mit jedem neuen Modul w”achst. W”urde das Aufl”osen solcher Konflikte das Edieren der verantwortlichen Module erzwingen, z”oge das gleichzeitig Namens”anderungen in allen Modulen nach sich, die auf das edierte Modul zugreifen. Der Spezifizierer h”atte bei der Erstellung eines neuen Moduls darauf zu achten, da”s alle neu eingef”uhrten Namen in keinem anderen bisher vorhandenen Modul verwendet werden, was der Konzeption des modularen Spezifizierens nicht entspricht. Deshalb stellt ASF ein Renamingkonstrukt zur Verf”ugung, welches das Umbenennen von Namen beim Import erm”oglicht. Leider f”uhrt jedoch die ASF-Bedeutung dieses Konstrukts zum Vermischen unterschiedlicher Strukturen, wie das folgende ASF-Beispiel zeigt: module exA begin exports begin sorts A functions mk_A : -> A end end exA Die Anweisung “ imports exA { renamed by [mk_A -> make_A] } ” bedeutet in ASF den Import eines Moduls namens exA, das sich vom Original exA dadurch unterscheidet, da”s jedes Auftreten vom Signaturnamen mk_A durch make_A ersetzt wurde. Das erscheint sinnvoll, solange innerhalb einer Spezifikation nur mit einer Version des Moduls gearbeitet wird. ”Au”serst unsch”on erweist sich die Semantik jedoch beim Import mehrerer Varianten eines Moduls: Module Murks begin imports exA, exA { renamed by [mk_A -> make_A] } end Murks Die Semantik von ASF kann zwischen beiden Instanzen des importierten Moduls exA nicht unterscheiden, was dazu f”uhrt, da”s Murks ”uber zwei Konstruktoren f”ur die Sorte A verf”ugt. Das namenweise Kopieren kann in gr”o”seren Spezifikationen leicht dazu f”uhren, da”s Namen, die nicht direkt am expliziten Renaming beteiligt sind, f”alschlich miteinander identifiziert werden. Zu derartig unmotivierten Namensidentifikationen kommt es in ASF auch beim Import verschiedener, durch Parameterbindungen aktualisierter Versionen des gleichen Moduls. Als Demonstrationsbeispiel untersuchen wir Sequenzen ”uber nat”urlichen Zahlen und Boole’schen Werten in ASF: module Sequences begin parameters Items begin sorts ITEM end Items exports begin sorts SEQ functions nil : -> SEQ cons: ITEM # SEQ -> SEQ end end Sequences module Auwei begin imports Sequences { Items bound by [ITEM -> NAT] to Naturals }, Sequences { Items bound by [ITEM -> BOOL] to Booleans } end Auwei Die Module Naturals und Booleans seien sinngem”a”s (analog zu den gleichnamigen ASF+-Modulen) definiert. Das Modul Auwei importiert zwei verschiedene Arten von Sequenzen. Beide Arten tragen jedoch den gleichen Sortennamen SEQ, was eigentlich einen Namenskonflikt erwarten lie”se. Statt dessen werden jedoch von ASF beide Sorten miteinander identifiziert, was dazu f”uhrt, da”s cons(s(0), cons(true, nil)) als wohlsortierter Term der Sorte SEQ akzeptiert wird. Dies entspricht sicherlich nicht den Vorstellungen des Spezifizierers! Lassen wir weiterhin verdeckte Namen und Overloading au”ser acht, dann kann das Renaming aus ASF als Erweiterung der modifizierten Vereinigungssemantik gesehen werden: ${\it NF\/}(M_{i}):=\left\\{\begin{array}[]{ll}(M_{i},\mbox{\it Ur}_{i})&\mbox{falls $M_{i}$ importfrei}\\\ (M_{i}\sqcup\displaystyle\bigsqcup_{k=1}^{p_{i}}R_{i^{\prime}_{k}}(M^{\prime}_{i^{\prime}_{k}}),\ \mbox{\it Ur}_{i}\cup\bigcup_{k=1}^{p_{i}}R_{i^{\prime}_{k}}(o^{\prime}_{i^{\prime}_{k}}))&$\begin{tabular}[t]{@{}l@{}}falls $M_{i}$ die Module $M_{i^{\prime}_{k}}$ impor-\\\ tiert und $(M^{\prime}_{i^{\prime}_{k}},o^{\prime}_{i^{\prime}_{k}})={\it NF\/}(M_{i^{\prime}_{k}})$\\\ gilt $(1\leq k\leq{p_{i}})$.\end{tabular}$\end{array}\right.$ Hier ist $R_{i^{\prime}_{k}}$ eine Funktion, die Signaturnamen des zu importierenden Moduls nach Ma”sgabe des Renamingkonstrukts (falls vorhanden) durch andere ersetzt, und auf Module und Originfunktionen angewendet werden kann. Falls der Importbefehl f”ur das Modul $M_{i^{\prime}_{k}}$ kein Renamingkonstrukt enth”alt, ist $R_{i^{\prime}_{k}}$ die Identit”at. Unver”andert bleiben in dieser Erweiterung (wie auch bei der hier nicht formalisierten Erweiterung f”ur Parameterbindungen) die Modulnamen im Wertebereich der Originfunktionen. So ist es zwar einerseits m”oglich, vorhandene Module zu modifizieren, andererseits k”onnen diese verschiedenen Aktualisierungen dann nicht unterschieden werden, was bei Mehrfachimporten zu ungew”unschter Vermischung der Strukturen f”uhrt. ASF+ geht hier einen anderen Weg. Die Modulnamen im Wertebereich der Originfunktion werden als Namensraumbezeichnungen interpretiert. Manipulationen wie explizites Renaming oder das Binden von Parametern stellen einen schwerwiegenden Eingriff in die den beteiligten Namen zugeordneten Namensr”aume dar. Um sicher zu stellen, da”s Namen aus den ver”anderten Namensr”aumen nicht mit Namen des urspr”unglichen Namensraumes identifiziert werden, ordenet ASF+ den ver”anderten Namensr”aumen neue Bezeichnungen zu. Diese setzen sich aus den alten Bezeichnungen und den Instanzbezeichnungen der instanziierenden Importbefehle zusammen. Wir sagen: Die Namensr”aume werden instanziiert. Ein interessanter Fall tritt ein, wenn durch Renaming oder Parameterbindung ein Modul ver”andert wird, das selbst weitere Module importiert, dessen (Signatur-) Namen also verschiedenen Namensr”aumen angeh”oren. Ein undifferenziertes Instanziieren aller Namensr”aume w”urde zu zahlreichen ”uberfl”ussigen Namenskonflikten f”uhren. Beispielsweise beeinflu”st das Binden des Parametertupels von OrdSequences (siehe Seite 2.2) beim Import das indirekt importierte Modul Booleans in keinster Weise, so da”s der Identifikation der Sorte BOOL mit dem Orginal (welches m”oglicherweise mittels weiterer Befehle importiert wird) nichts entgegen steht. Andererseits k”onnen Manipulationen, die beim kopierenden Import vorgenommen werden auch indirekt importierte Teilsignaturen betreffen. In diesem Fall gen”ugt es nicht, nur die Namensr”aume der direkt betroffenen Signaturnamen zu instanziieren. Vielmehr m”ussen ebenfalls alle Namensr”aume, die von den instanziierten Namensr”aumen abh”angen bis hin zum Namensraum des direkt importierten Moduls instanziiert werden. Allgemein f”uhrt das Manipulieren von indirekt importierten Modulsignaturen zur Instanziierung mehrerer Namensr”aume. Zur Illustration betrachten wir ein ASF+-Beispiel: module exA { add signature{ public: sorts A } } module exAB { import exA { public: A } add signature{ public: sorts B } } module exABC { import exAB { public: A, B } add signature{ public: sorts C } } module CopyDemo { import exABC[Copy]{ public: A, B renamed to Bnew, C } import exABC { public: A } import exABC { public: C } } (115,65)(115,105)(15,105) (15,65)(115,65) (120,40)(120,120)(10,120) (10,40)(120,40) (125,15)(125,135)(5,135) (5,15)(125,15) exABCexABexACAB(115,200)(115,240)(15,240) (15,200)(115,200) (120,175)(120,255)(10,255) (10,175)(120,175) (125,150)(125,270)(5,270) (5,150)(125,150) exABCexABexAABC(115,335)(115,375)(15,375) (15,335)(115,335) (120,310)(120,390)(10,390) (10,310)(120,310) (125,285)(125,405)(5,405) (5,285)(125,285) exAexAB[Copy]exABC[Copy]BnewAC(130,0)(130,430)(0,430) (0,0)(130,0) 5(105,135)(105,15) 5(95,135)(95,15) 5(105,270)(105,150) 5(95,270)(95,150) 5(105,405)(105,285) 5(95,375)(95,285) 5(55,415)(55,405) 5(55,285)(55,270) 5(55,150)(55,135) 5(55,15)(55,0) 5(65,415)(65,405) 5(65,285)(65,270) 5(65,150)(65,135) 5(65,15)(65,0) CopyDemo Der erste Importbefehl von CopyDemo manipuliert die Signatur des (indirekt) importierten Moduls exAB. Dies f”uhrt zu einer Instanziierung des zugeordneten Namensraumes — Bnew geh”ort nun dem neuen Namensraum exAB[Copy] an. Auf die Signatur des ebenfalls (indirekt) importieren Moduls exA hat das keinen Einflu”s, daher k”onnen die Sorten A aus den ersten beiden Importbefehlen identifiziert werden und nach wie vor dem Namensraum exA angeh”oren. Eine Manipulation in der Signatur von exAB hat Einflu”s auf die Signatur des exAB importierenden Moduls exABC, weil hier die ver”anderten Namen sichtbar sind und im allgemeinen auch in den Funktionsdeklarationen auftreten werden. ASF+ ordnet dem Sortennamen C im ersten Importbefehl den instanziierten Namensraum exABC[Copy] zu. Im dritten Importbefehl geh”ort C dagegen dem (nicht modifizierten) Namensraum exABC an. Die Identifikationsregel sieht darin einen Namenskonflikt und wird die vorliegende Spezifikation nicht akzeptieren. ASF hingegen w”urde die beiden Sorten C identifizieren, was im allgemeinen die weiter oben bereits aufgezeigten Probleme bereitet. In ASF+ bleibt der Namenskonflikt auch dann bestehen, wenn der erste Importbefehl durch import exABC[Copy]{ public: A, B renamed to B, C } ersetzt wird. Der Ausdruck $name_{1}$ renamed to $name_{2}$ hat im Kontext eines ASF+-Imports also zwei verschiedene Auswirkungen. Neben der Zugriffs”anderung bewirkt er auch eine Instanziierung eines oder mehrerer Namensr”aume. Ist man nur an letzterer Wirkung interessiert, kann ein Ausdruck $name$ renamed to $name$ sinnvoll sein, was in ASF+ auch als copy of $name$ geschrieben werden kann. In beiden F”allen l”ost sich der Namenskonflikt auf, wenn der dritte Importbefehl in CopyDemo entfernt wird. Die Realisation der hier vorgestellten Semantik erfordert zwei ”Anderungen in der modifizierten Vereinigungssemantik. Zun”achst mu”s der Wertebereich der Originfunktionen auf Namensraumbezeichnungen ausgedehnt werden. Sie setzen sich aus Modulnamen und Instanzbezeichnungen zusammen. Die Originfunktion des normalisierten Moduls exABC kann beispielsweise folgenderma”sen dargestellt werden: {(A,exA), (B,exAB), (C,exABC)}. Neben der Umbenennung von B zu Bnew ver”andert das explizite Renaming auch den Wertebereich der Originfunktion: {(A,exA), (Bnew,exAB[Copy]), (C,exABC[Copy])}. Um ermitteln zu k”onnen, welche Namensr”aume instanziiert werden m”ussen, wird au”serdem Information ”uber den hierarchischen Aufbau der Spezifikation ben”otigt. Aus diesem Grund f”uhren wir zur Erkl”arung der Semantik von ASF+ eine zus”atzliche Funktion namens Dependenzfunktion ein, die jedem innerhalb eines Moduls auftretenden Namensraum die Menge aller Namensr”aume zuordnet, die von ihm abh”angen. Sie wird im folgenden Abschnitt 4.3 diskutiert. Die hier vorgestellte Sorte von Importen bezeichnen wir als kopierende Importe. Ihre Verwendung ist immer dann sinnvoll, wenn “wesentliche” Eigenschaften der importierten Struktur ge”andert werden sollen. Was aber sind “wesentliche” Eigenschaften? Neben den bereits diskutierten Signaturmanipulationen (explizites Renaming und Parameterbindung) k”onnen im importierenden Modul auch neue Konstruktoren und Funktionen zu einer importierten Sorte bereitgestellt und im Gleichungsblock neue Beziehungen zwischen Elementen der importierten Struktur definiert werden (z. B. zwecks Erweiterung partiell definierter Funktionen). All dies hat Einflu”s auf die G”ultigkeit von Klauseln. W”urde man in allen F”allen kopierenden Import verlangen, h”atte das zur Folge, da”s bewiesene Beweisziele eines Moduls beim benutzenden Import des Moduls in ein anderes ihre G”ultigkeit behalten w”urden — ein denkbar einfacher Beweismodularisierungsansatz. Formal erscheint diese “seiteneffektfreie” Semantik des benutzenden Imports optimal. Auch kann die Einhaltung der Restriktionen vom Normalformalgorithmus syntaktisch gepr”uft werden. Andererseits scheinen die Bedingungen f”ur praktischen Gebrauch zu restriktiv, weil unn”otig viele Namen und Instanzbezeichnungen den Blick auf das Wesentliche versperren. ASF+ schreibt den kopierenden Import nur bei Manipulationen durch Renaming und Parameterbindung vor und ”uberl”a”st in allen anderen F”allen dem Spezifizierer die Wahl des Importtyps. Die folgende Spezifikation einer zyklischen Gruppe mit drei Elementen Nat3 kann als Anschauungsbeispiel daf”ur dienen, wie der kopierende Import auch ”uber Renaming und Parameterbindung hinaus als Spezifikationshilfsmittel sinnvoll eingesetzt werden kann: module Nat3 { import Naturals[Nat3] { public: copy of NAT, 0, s, + } variables { x: -> NAT } equations { [e1] s(s(s(x))) = x } } Die Gleichung e1 nimmt destruktiven Einflu”s auf die importierte Datenstruktur. W”urden hier die Namen NAT und s mit den Originalen aus Naturals identifiziert, so st”ande das unverf”alschte Original f”ur die gesamte Spezifikation nicht mehr zur Verf”ugung. Mit Einf”uhrung zus”atzlicher Restriktionen (z. B. “Verbot des Auftretens von Termen aus ausschlie”slich benutzend importierten Funktionssymbolen als linke Seite einer Gleichung im equations–Block.”) k”onnte der kopierende Import von Naturals erzwungen werden. ### 4.3 Abh”angigkeiten zwischen Namensr”aumen Die hierarchische Struktur der Spezifikation bedingt Abh”angigkeiten zwischen den erzeugten Namensr”aumen. Die Semantik von ASF+ wird ihnen durch Einf”uhrung einer sogenannten Dependenzfunktion gerecht, welche der Bezeichnung jedes Namensraumes die Menge der Bezeichnungen aller von ihm abh”angigen Namensr”aume zuweist. Diese Dependenzfunktion soll hier diskutiert werden. Ein importfreies Modul $M$ namens modn enth”alt nur einen Namensraum, n”amlich den moduleigenen. Seine Bezeichnung stimmt mit dem Modulnamen ”uberein, Abh”angigkeiten zu anderen Namensr”aumen bestehen nicht. Die zugeh”orige Dependenzfunktion lautet also ${\it depf\/}\quad:=\quad\\{(\mbox{\it modn},\emptyset)\\}$. Der moduleigene Namensraum eines nicht importfreien Moduls $M$ namens modn ist von allen importierten Namensr”aumen abh”angig. Die zugeh”orige Dependenzfunktion ${\it depf\/}$ kann aus den Dependenzfunktionen, die sich aus den einzelnen Importbefehlen ergeben, berechnet werden. Zu diesem Zweck definieren wir eine Hilfsfunktion CombineDependencies, die eine Menge von Dependenzfunktionen zu einer Funktion zusammenfa”st. ${\it CombineDependencies\/}(\\{{\it depf\/}_{i}\quad\|\quad i\in A\\})\quad:=$ --- $\\{\ ({\it modinst},\displaystyle\bigcup_{i\in B}{\it depf\/}_{i}({\it modinst}))\quad\|\quad$ \| ${\it modinst}\in\displaystyle\bigcup_{i\in A}\mbox{\sf Dom}({\it depf\/}_{i})\quad\wedge$ \| $B=\\{i\in A\ \|\ {\it modinst}\in\mbox{\sf Dom}({\it depf\/}_{i})\\}\ \\}$ Mit Hilfe von CombineDependencies kann nun die Dependenzfunktion ${\it depf\/}$ f”ur beliebige, nicht notwendigerweise importfreie Module definiert werden: ${\it depf\/}\ :=\ $ \| $\\{\ (\mbox{\it modn},\emptyset)\ \\}\quad\cup$ ---\|--- \| { \| $({\it modinst},{\it modinstances\/}\cup\\{\mbox{\it modn}\\})\quad\|\quad({\it modinst},{\it modinstances\/})$ \| \| $\in{\it CombineDependencies\/}(\\{\mbox{\it depf-imp-const\/}_{i}\ \|\ 1\leq i\leq l\\})\ \\}$ wobei $\mbox{\it depf-imp-const\/}_{i}$ die zum $i$-ten Importbefehl des Moduls $M$ zugeh”orige Dependenzfunktion beinhaltet ($1\leq i\leq l$). Wie die zu einem Importbefehl zugeh”orige Dependenzfunktion depf-imp-const aus der Dependenzfunktion depf-imp-mod des importierten Moduls zu berechnen ist h”angt vom Importtyp ab und wird im folgenden erkl”art. Handelt es sich um einen benutzenden Import des Moduls M-imp und ist depf-imp- mod die Dependenzfunktion des Moduls, so gilt $\mbox{\it depf-imp- const\/}:=\mbox{\it depf-imp-mod\/}$. Handelt es sich dagegen um einen kopierenden Import von M-imp, in dem explizites Renaming durchgef”uhrt wird, dann geht depf-imp-const aus depf-imp- mod dadurch hervor, da”s jedes Auftreten von Bezeichnungen der vom Renaming direkt betroffenen Namensr”aume sowie der von diesen bez”uglich depf-imp-mod abh”angigen Namensr”aume durch die mit der Instanzbezeichnung des Importbefehls instanziierten Namensraumbezeichnung ersetzt wird. Werden formale Parameter an Namen aus $k$ aktuellen Modulen M-actj ($1\leq j\leq k$) gebunden, so sind zus”atzlich die Namensraumbezeichnungen aller formalen Parameter, an die aktuelle Parameter gebunden werden, sowie alle bez”uglich depf-imp-mod von ihnen abh”angigen Namensr”aume in depf-imp-mod zu instanziieren. Die resultierende Funktion nennen wir $\mbox{\it depf-imp- mod\/}^{\prime}$. Des weiteren sind die Namensraumabh”angigkeiten der implizit importierten aktuellen Module $\mbox{\it depf-act-mod\/}_{j}$ zu ber”ucksichtigen. Sei analog zum expliziten Import $\mbox{\it depf-imp-const\/}\ :=\ {\it CombineDependencies\/}(\begin{array}[t]{@{}l}\\{\mbox{\it depf-imp- mod\/}^{\prime}\\}\ \cup\\\ \\{\mbox{\it depf-act-mod\/}^{\prime}_{j}\ \|\ 1\leq j\leq k\\}).\end{array}$ W”urde man hier $\mbox{\it depf-act-mod\/}^{\prime}_{j}$ mit $\mbox{\it depf- act-mod\/}_{j}$ gleichsetzen, so entspr”achen die aus implizitem Import resultierenden Abh”angigkeiten innerhalb des bindenden Moduls denen eines benutzenden Imports. Unber”ucksichtigt blieben dabei jedoch die Beziehungen zwischen dem Modul der formalen Parameter (hier M-imp) und den Modulen der aktuellen Parameter (hier M-actj). Dies ist jedoch erforderlich: Werden beispielsweise bei einem sp”ateren Import des bindenden Moduls (hier $M$) aktuelle Parameter aus der Bindung umbenannt, so hat dies auch Einflu”s auf die Namen des Moduls der formalen Parameter. Allgemein definieren wir daher: $\mbox{\it depf-act-mod\/}^{\prime}_{j}\quad:=\quad\\{\ $ \| $({\it modinst},{\it modinstances\/}\ \cup\ $ \| $\\{{\it paradefmodinst\/}_{j}\\}\ \cup$ ---\|---\|--- \| \| $\mbox{\it depf-imp-mod\/}^{\prime}({\it paradefmodinst\/}_{j})\quad\|$ \| $({\it modinst},{\it modinstances\/})\in\mbox{\it depf-act-mod\/}_{j}\ \\}$, wobei ${\it paradefmodinst\/}_{j}$ der Namensraum der an Namen des Moduls M-actj zu bindenden formalen Parameter des Moduls M-imp ist. Im folgenden Beispiel werden die Abh”angigkeiten zwischen den Namensr”aumen aus OrdNatSequences durch die zugeh”orige Dependenzfunktion dargestellt. Sie kann auch aus dem in Kapitel 3 vorgestellten Strukturdiagramm gewonnen werden. $\\{\ $ \| $({\tt Booleans},\quad\\{\ {\tt Naturals},{\tt OrdNaturals},{\tt OrdSequences[ONSeq]},{\tt OrdNatSequences}\ \\}),$ ---\|--- \| $({\tt Naturals},\quad\\{\ {\tt OrdNaturals},{\tt OrdSequences[ONSeq]},{\tt OrdNatSequences}\ \\}),$ \| $({\tt OrdSequences[ONSeq]},\quad\\{\ {\tt OrdNatSequences}\ \\})\ \\}$ ### 4.4 Verdeckte Namen Im Prinzip k”onnten alle Namenskonflikte, die beim Import von Modulen auftreten, durch “explizite” Umbenennungen (s. o.) aufgel”ost werden. Allerdings erfordert dies vom Spezifizierer einen ”Uberblick ”uber alle eingef”uhrten Namen, was mit zunehmender Spezifikationskomplexit”at immer schwieriger wird. Um den Spezifizierer vom Umbenennen “unwichtiger” Namen zu entlasten, unterscheidet ASF sichtbare und verdeckte Namen. W”ahrend die Konfliktl”osung zwischen sichtbaren Namen weiterhin in der Verantwortung des Spezifizierers liegt, werden Konflikte zwischen verdeckten Namen vom Normalformalgorithmus durch automatisches Umbenennen (implizites Renaming) aufgel”ost. ASF beschr”ankt die Referenzierbarkeit verdeckter Namen jeweils auf das definierende Modul, was f”ur Variablen (sie k”onnen in diesem Sinne als verdeckt betrachtet werden) ausreichend ist. Um die Zahl der Konflikte zwischen Sorten- und Funktionsnamen wirksam zu reduzieren, erscheint diese Einschr”ankung jedoch zu restriktiv. W”unschenswert w”are ein Mechanismus, der es erlaubt, Namen, die in der jeweiligen Spezifikationsebene nicht mehr gebraucht werden, “auszublenden”. Modulare Programmiersprachen stellen zu diesem Zweck Ex- und Importlisten zur Verf”ugung. ASF+ ”ubernimmt die Importlisten (alle weiterhin sichtbaren Namen m”ussen im Importkonstrukt aufgef”uhrt werden). Das Exportverhalten von Namen wird dagegen direkt in der Definition bzw. beim Import durch die Schl”usselworte private und public festgelegt. Dies reduziert den Code und dient der ”Ubersicht. Werden Namen beim Import verdeckt, so ersetzt der Normalformalgorithmus alle diese Namen durch neue, innerhalb der gesamten Spezifikation eindeutige Namen. Zu diesem Zweck wird dem alten vom Spezifizierer vereinbarten Namen der (abgek”urzte) Name des entsprechenden Namensraumes gefolgt von einem Bindestrich vorangestellt. Beispielsweise werden die Namen and, or und not aus Booleans beim verdeckten Import in das Modul Naturals durch Bo-and, Bo-or und Bo-not ersetzt. Neben der Trennungsfunktion zwischen Namensraum und urspr”unglichem Namen garantiert der Bindestrich die Konfliktfreiheit zwischen verdeckten und sichtbaren Namen, da er in letztgenannten nicht zugelassen ist. Besonders n”utzlich erweist sich das Instrument der Namensverdeckung in Verbindung mit dem kopierenden Import, bei dem zahlreiche Namensumbenennungen erforderlich werden, da Signaturnamen aus verschiedenen Instanzen eines Moduls nicht miteinander identifiziert werden d”urfen. ASF+ erledigt das f”ur die verdeckten Namen automatisch, der Spezifizierer mu”s sich lediglich um die sichtbaren, ihn interessierenden Namen k”ummern. ### 4.5 Overloading Bisher gingen wir davon aus, da”s jeder Signaturname genau ein Signaturobjekt (Sorte oder Funktion) spezifiziert. In der Praxis ist es jedoch sehr n”utzlich, wenn verschiedene Objekte mit dem gleichen Namen referenziert werden k”onnen. Beispielsweise schreibt man gew”ohnlich die Summe zweier Zahlen $x$ und $y$ als $(x+y)$, egal, ob es sich bei $x$ und $y$ um nat”urliche, ganze oder rationale Zahlen handelt. Die tats”achliche Bedeutung des Namens “$+$” ergibt sich aus dem Kontext. Das aus ASF ”ubernommene Overloading gestattet es, Funktionsnamen zu ”uberladen, wenn diese sich in ihren Argumentsorten unterscheiden. Die Restriktion erlaubt es, durch Bottom-Up-Sortenpr”ufung jedem Funktionsnamen innerhalb eines Terms eine eindeutige Funktion zuzuordnen. Um ”uberladene Funktionen behandeln zu k”onnen, m”ussen wir im Definitionsbereich der Originfunktion zu disambiguierten Namen ”ubergehen. Dabei handelt es sich um Tupel (specname, sortvector) bestehend aus dem Signaturnamen und einem (f”ur n-stellige Funktionen n-dimensionalen) Sortenvektor. Jede Funktionsdeklaration im add signature-Konstrukt definiert genau einen neuen disambiguierten Namen. Der Import eines Funktionsnamens zieht im allgemeinen den Import mehrerer disambiguierter Namen nach sich. Ist als Ergebnis der Normalisierung eine ”uberladungsfreie Spezifikation gew”unscht, kann dies erreicht werden, indem alle Funktionsnamen des Normalformmoduls durch eine geeignete Repr”asentation ihrer disambiguierten Namen (z. B. +[NAT,NAT]) ersetzt werden. ## 5 Syntax Die Syntax von ASF+ ist gegeben durch folgende kontextfreie Grammatik111Wir kennzeichenen Terminale durch Anf”uhrungszeichen und Typewriterfont und Nichtterminale durch spitze Klammern ($<$$\ldots$$>$). $x$* bedeutet null, eine oder mehrere und $x$+ eine oder mehrere Wiederholungen von $x$, ($x$ “$ts$”)* und ($x$ “$ts$”)+ stehen f”ur Wiederholungen von $x$, getrennt durch das Terminalsymbol $ts$. Optionale Zeichenketten sind in eckige Klammern ([$\ldots$]) eingefa”st.: $<$specification$>$ \| ::= \| $<$module$>$+ ---\|---\|--- $<$module$>$ \| ::= \| “module” $<$module-name$>$[ “<” $<$parameter-block$>$+ “>” ] \| \| [ “short” $<$short-module-name$>$ ] \| \| “{” \| $<$import$>$* --- [ $<$add-signature$>$ ] [ $<$variables$>$ ] [ $<$equations$>$ ] [ $<$goals$>$ ] “}” $<$parameter-block$>$ \| ::= \| “(” ($<$sort-or-func-name$>$ “,”)+ “)” $<$sort-or-func-name$>$ \| ::= \| $<$sort-name$>$ $\|$ $<$function-name$>$ $<$import$>$ \| ::= \| “import” $<$module-name$>$[ “[”$<$instance-name$>$“]” ] \| \| [ “<” $<$ext-para-block$>$+“>” ] \| \| [ $<$import-block$>$ ] $<$ext-para-block$>$ \| ::= \| “(”($<$name-with-ren$>$ “,”)+ “)” \| $\|$ \| “(”($<$sort-or-func-name$>$ “bound to” $<$sort-or-func-name$>$ “,”)+ \| \| “)” “of” $<$module-name$>$[ “<” $<$parameter-block$>$+ “>” ] $<$name-with-ren$>$ \| ::= \| $<$sort-or-func-name$>$ [ “renamed to” $<$sort-or-func-name$>$ ] \| $\|$ \| “copy of” $<$sort-or-func-name$>$ $<$import-block$>$ \| ::= \| “{” \| [ “public:” \| ($<$name-with-ren$>$ “,”)+ ] ---\|--- [ “private:” \| ($<$name-with-ren$>$ “,”)+ ] “}” $<$add-signature$>$ \| ::= \| “add signature” \| \| “{” \| [ “parameters:” \| $<$para-block-sig$>$+ ] ---\|--- [ “public:” \| $<$signature$>$ ] [ “private:” \| $<$signature$>$ ] “}” $<$para-block-sig$>$ \| ::= \| “(” \| $<$signature$>$ --- [ “conditions” $<$clause$>$+ ] “)” $<$signature$>$ \| ::= \| [ “sorts” ($<$sort-name$>$ “,”)+ ] \| \| \| [ “constructors” \| $<$function-dec$>$+ ] ---\|--- [ “non-constructors” \| $<$function-dec$>$+ ] $<$function-dec$>$ \| ::= \| ($<$ext-func-name$>$ “,”)+ “:” ($<$sort-name$>$ “#”)* \| \| “->” $<$sort-name$>$ $<$ext-func-name$>$ \| ::= \| $<$function-name$>$ [ “_” ] \| $\|$ \| “_” $<$function-name$>$ “_” $<$clause$>$ \| ::= \| “[” $<$label$>$“]” ($<$eq$>$ “,”)* “\-->” ($<$eq$>$ “,”)* $<$eq$>$ \| ::= \| $<$term$>$ [ “=” $<$term$>$ ] $<$term$>$ \| ::= \| [ $<$term$>$ $<$function-name$>$ ] $<$primary$>$ ---\|---\|--- $<$primary$>$ \| ::= \| $<$function-name$>$[ “(” ($<$term$>$ “,”)+ “)” ] \| $\|$ \| $<$variable-name$>$ \| $\|$ \| “(”$<$term$>$“)” \| $\|$ \| $<$function-name$>$ $<$primary$>$ $<$variables$>$ \| ::= \| “variables” \| \| “{” \| [ [ “constructors” ] \| $<$variable-dec$>$+ ] ---\|--- [ “non-constructors” \| $<$variable-dec$>$+ ] “}” $<$variable-dec$>$ \| ::= \| ($<$variable-name$>$ “,”)+ “:” “->” $<$sort-name$>$ $<$equations$>$ \| ::= \| “equations” \| \| “{” $<$equation$>$+ “}” $<$equation$>$ \| ::= \| “[” $<$label$>$“]” $<$eq$>$ [ “if” ($<$eq$>$ “,”)+ ] \| $\|$ \| $<$macro-equation$>$ $<$goals$>$ \| ::= \| “goals” \| \| “{” $<$clause$>$+ “}” Lexikalisch gelten in der Syntax von ASF+ folgende Konventionen: * • Als Trennzeichen zwischen den einzelnen lexikalischen Token sind erlaubt: Leerzeichen, horizontaler Tabulator, carriage return, Zeilen- und Seitenvorschub sowie jede Kombination dieser Zeichen. * • Modulnamen, -k”urzel, Instanzbezeichnungen, Marken- und Sortennamen (also $<$module-name$>$, $<$short-module-name$>$, $<$instance-name$>$, $<$label$>$ und $<$sort-name$>$) bestehen aus einer beliebigen Folge von Zahlen, Buchstaben, Apostroph “’”) und Unterstrich (“_”). Jedoch darf der Unterstrich weder am Anfang, noch am Ende eines Namens stehen. * • In Funktonsnamen ($<$function-name$>$), die hier auch die Operatoren aus ASF beinhalten, sind zus”atzlich folgende ASCI-Zeichen zul”assig: “!”, “$”, “%”, “&”, “$+$”, “$$”, “;”, “?”, “$\sim$”, “$\backslash$”, “$\|$”, “/”, “.”. • Die Schl”usselworte “if”, “equation”, “else”, “case”, “renamed”, “bound”, “sorts” und “constructors” stehen als Namen nicht zur Verf”ugung. Man beachte, da”s in benutzerdefinierten Modulen Sorten-, Funktions- und Markennamen keinen Bindestrich (“-”) enthalten d”urfen. Andernfalls w”aren Namenskonflikte zwischen benutzerdefinierten und verdeckten, vom Normalformalgorithmus erzeugten Namen nicht auszuschlie”sen. ## 6 Die Normalform-Prozedur Im Mittelpunkt dieses Kapitels steht der Algorithmus, mit dessen Hilfe beliebige ASF+-Spezifikationen, bestehend aus einem Topmodul und einer Folge von direkt, indirekt und implizit importierten Modulen, in flache, importfreie Spezifikationen umgewandelt werden k”onnen. Besonderen Wert wurde auf die m”oglichst konsequente Verwendung disambiguierter Namen gelegt. Die Formalisierung des ASF zugrunde liegenden Algorithmus in [Bergstra&al.89], Kapitel 1.3.2, l”a”st hier einige Fragen offen222 Beispielsweise ist der zweite Wert eines RENAMING-Tupels ($x$,$y$) im allgemeinen kein Element aus SFV. Trotzdem wird ihm in der Beschreibung von rename_visibles ein Origin zugeordnet.. Schwerwiegender ist dagegen das (nicht dokomentierte) Fehlverhalten des ASF-Normalformalgorithmus bei mehrfachem Import namensgleicher Sorten und Funktionen mit unterschiedlicher Sichtbarkeit: module exhiddenA begin sorts A end exhiddenA module exA begin exports begin sorts A end end exA module Certain-Clash begin imports exhiddenA, exA end Certain-Clash Da”s die Normalisierung von ASF hier einen Namenskonflikt ausgibt, erscheint genauso unverst”andlich wie die Tatsache, da”s er sich durch ”Anderung der Importreihenfolge in Certain-Clash beheben l”a”st. Zwar wird in der Beschreibung der Hilfsfunktion combine333 Siehe [Bergstra&al.89], Absatz 1.3.2.2.3 darauf hingewiesen, da”s verdeckte Namen des ersten Arguments mit sichtbaren Namen des zweiten Arguments kollidieren k”onnen, ein Hinweis auf die kaum akzeptablen Auswirkungen auf die Kombination mehrerer zu importierender Module (im Beispiel exhiddenA und exA) fehlt jedoch v”ollig. Der gleiche Fehler f”uhrt zusammen mit dem nur unpr”azise formalisierten impliziten Renaming sogar dazu, da”s Namenskonflikte zwischen Namen, die durch die Normalisierung ”uberhaupt erst erzeugt wurden, nicht auszuschlie”sen sind: module exAhiddenA begin exports begin sorts A end imports exhiddenA end exAhiddenA module exB begin sorts B end exB module Possible-Clash begin imports exAhiddenA, exB end Possible-Clash Im Zuge der Normalisierung wird zun”achst exAhiddenA in Normalform gebracht. Die dabei notwendige Umbenennung der verdeckt importierten Sorte A erledigt die Funktion rename_hiddens. Da sie keine Kenntnis ”uber das Modul exB hat, steht einer Ersetzung des Namens A durch B aus Sicht des Algorithmus nichts im Wege. In diesem Fall aber liefert die Normalisierung von Possible-Clash wieder einen Namenskonflikt (gleiche Situation wie oben). Grund f”ur die Namenskonflikte beider Beispiele ist die Asymetrie der Hilfsfunktion combine, die beim kombinieren zweier Module zwecks Konfliktvermeidung nur Umbenennungen innerhalb eines Modules vornehmen darf und sowohl bei der Kombination von Importen untereinander, als auch mit dem importierenden Modul selbst Verwendung findet.444 Siehe [Bergstra&al.89], Absatz 1.3.2.3, 4. Schritt des Algorithmus Wir ersetzten combine durch zwei verschiedene Varianten: CombineImports kombiniert zwei importierte Module untereinander. Ihre Argumente (zwei Module in Normalform) werden gleich behandelt, somit ist die Reihenfolge der Importanweisungen belanglos. CombineWithImports entspricht in etwa combine aus ASF — sie kombiniert das importierende Modul mit der Kombination aller Importe. ### 6.1 Datenstrukturen Bevor der Normalformalgorithmus vorgestellt werden kann, m”ussen zun”achst die Daten erl”autert werden, auf denen er operiert. Als Basistyp beschr”anken wir uns auf Zeichenketten. Sie werden in Mengen- und Strukturtypen, die wir als Tupel mit unterschiedlichen Komponententypen darstellen werden, zu komplexeren Datenstrukturen zusammen gesetzt. Funktionen werden als Mengen repr”asentiert: $f=\\{(x,y)\quad\|\quad y=f(x)\\}$. $\cal P$($X$) bezeichnet die Potenzmenge von $X$, also die Menge aller Teilmengen. Ziel der Normalisierung ist die Transformation einer ASF+-Spezifikation, bestehend aus einzelnen ASF+-Modulen, in eine neue importfreie ASF+-Spezifikation. Neben den Typen ASF-MODULE und ASF-SPEC werden f”ur die Eingabeschnittstelle der Normalisierungsprozedur auch Informationen ”uber bereits gef”uhrte Beweise ben”otigt. Sie werden im Typ PROVE-DB zusammengefa”st. * • ASF-MODULE ist die Menge aller Zeichenfolgen, die syntaktisch korrekte ASF+-Module darstellen. * • ASF-SPEC := ASF-MODULE $\times$ $\cal P$(ASF-MODULE) ASF+-Spezifikationen bestehen aus einem Topmodul und einer Menge von Modulen, die mindestens alle vom Topmodul direkt, indirekt und implizit importierten Module enthalten mu”s. * • PROVE-DB ist eine nicht n”aher konkretisierte Wissensbasis f”ur gelungene Beweise. Mit ihrer Hilfe wird die G”ultigkeit von semantischen Bedingungen f”ur Parameterbindungen gepr”uft. Allerdings eignet sich die Repr”asentation eines ASF+-Moduls als unstrukturierte Zeichenkette kaum zur ad”aquaten Beschreibung der f”ur die Transformation notwendigen Operationen (z. B. Kombination mehrerer Module). Wir f”uhren daher einen strukturierten Datentyp MODULE ein, der es erm”oglicht, auf einzelne Teile eines repr”asentierten Moduls (z. B. auf die Importbefehle) direkt zuzugreifen. Die kleinsten logischen Einheiten eines Moduls bestehen aus Namen, die in Abh”angigkeit vom Kontext ihres Auftretens als Modulnamen oder -k”urzel, als Marken, Instanzbezeichnungen, Variablen-, Sorten- oder als Funktionsnamen dienen. Unter den Sorten- und Funktionsnamen besitzen wiederum in einer Parametersignatur definierten Namen einen Sonderstatus, sie hei”sen Sorten- und Funktionsparameter. Einige Namen werden w”ahrend der Normalisierung ver”andert oder zur Ver”anderung anderer Namen gebraucht. Um die Zahl der Namenstypen m”oglichst ”uberschaubar zu halten, fassen wir Namen, auf denen die gleichen Operationen ausgef”uhrt werden, gruppenweise zusammen: * • MODULE-NAME ist Menge aller Modulnamen. * • SHORT-MODULE-NAME ist Menge aller abgek”urzten Modulnamen. Sie enth”alt alle Modulk”urzel sowie die Namen der Module, f”ur die kein K”urzel angegeben worden ist. Unter dem “abgek”urzten Namen eines Moduls” verstehen wir das im Modul vereinbarte K”urzel, oder (falls nicht vorhanden) den Modulnamen selbst. * • INST-NAME ist Menge aller Instanzbezeichnungen. * • USER-NAME ist die Menge aller dem Spezifizierer zur Verf”ugung stehenden Namen f”ur Parameter, Sorten, Funktionen, Variablen und Marken. Neben den vom Spezifizierer erzeugten Namen generiert der Normalformalgorithmus auch selbstst”andig Namen, die sich (im Gegensatz zu ASF) aus den vom Spezifizierer vorgegebenen Namen und K”urzeln zusammensetzen. Das modulare Konzept aus ASF+ basiert wesentlich auf der Zuordnung von Namen zu Namensr”aumen. Eine Namensraumbezeichnung besteht aus einem Modulnamen und einer gegebenenfalls leeren Liste von Instanzbezeichnungen, welche Auskunft dar”uber gibt, um welche Version des Namensraumes es sich handelt. Auch Namensraumbezeichnungen k”onnen mit Hilfe der Modulk”urzel abgek”urzt werden. * • MODINST-NAME enth”alt alle Namensraumbezeichnungen. Syntaktisch kann MODINST- NAME durch eine Grammatik-Produktionsregel wie folgt beschrieben werden: MODINST-NAME \| ::= \| MODULE-NAME ---\|---\|--- \| $\|$ \| MODULE-NAME“[”(INST-NAME “,”)+“]” * • SHORT-MODINST-NAME enth”alt alle abgek”urzten Namensraumbezeichnungen. SHORT-MODINST-NAME \| ::= \| SHORT-MODULE-NAME ---\|---\|--- \| $\|$ \| SHORT-MODULE-NAME“[”(INST-NAME “,”)+“]” Wird beim Import ein Name verdeckt, so ersetzt der ASF+-Normalforalgorithmus den Namen durch einen neuen, innerhalb der gesamten Spezifikation eindeutigen Namen, indem er dem alten Namen eine abgek”urzte Namensraumbezeichnung gefolgt von einem Bindestrich voranstellt. Der so erzeugte Name ist kein USER-NAME, kann also mit keinem vom Spezifizierer eingef”uhrten Namen in Konflikt geraten. * • SPEC-NAME umfa”st alle Parameter-, Sorten-, Funktions-, Variablen- und Markennamen, die nach der Normalisierung in der Spezifikation auftreten k”onnen. Zur Charakterisierung der Syntax geben wir wieder eine Produktionsregel an: SPEC-NAME ::= USER-NAME $\quad\|\quad$ SHORT-MODINST-NAME“-”USER-NAME ASF+ gestattet es, Funktionsnamen zu ”uberladen. Um eine spezielle Funktion identifizieren zu k”onnen ist deshalb die Kenntnis der Argumentsorten erforderlich. Dies f”uhrt uns zu disambiguierten Namen: * • SORT-VECTOR ist eine Menge von Listen, deren Komponenten Sortennamen ($\in$ SPEC-NAME) sind. * • DISAMB-SPEC-NAME := SPEC-NAME $\times$ SORT-VECTOR umfa”st die Menge der disamiguierten Namen. Disambiguierte Namen sind Tupel (name, sortv). Falls name ein Sorten-, Marken-, Variablen oder Konstantenname ist, ist der Sortenvektor sortv leer. Handelt es sich dagegen um einen Funktionsnamen (bzw. Funktionsparameter) enth”alt er die Namen der Argumentsorten. Die Datenstruktur f”ur die Importe nimmt alle, aus den Importkonstrukten hervorgehenden Informationen auf, gruppiert sie nach den Erfordernissen der sequenziellen Auswertung jedoch neu: * • VISIBILITY-FUNC := USER-NAME $\longrightarrow$ {“public”, “private”} Sie bestimmt die Sichtbarkeit von Signaturnamen beim Import eines Moduls. Da ASF+ verlangt, da”s alle nach dem Import sichtbaren Namen im Importkonstrukt aufgef”uhrt werden m”ussen, kann sie direkt aus der Importanweisung bestimmt werden. Namen aus dem importierten Modul, die keine Parameter sind und denen keine Sichtbarkeitsstufe $\in$ {“public”, “private”} zugewiesen wird, werden beim Import verdeckt. * • RENAMING-FUNC := USER-NAME $\longrightarrow$ SPEC-NAME Werden Sorten- und Funktionsnamen durch eine Funktion aus RENAMING-FUNC auf andere Sorten- und Funktionsnamen abgebildet, so beschreibt diese Funktion explizites Renaming. Handelt es sich dagegen im Definitionsbereich ausschlie”slich um Parameter, so kann mit ihrer Hilfe eine Parametertupelbindung beschrieben werden: * • BINDING-BLOCK := RENAMING-FUNC $\times$ MODULE-NAME Tupel (binding, modn) dieses Typs repr”asentieren einen Block des Importbefehls, der ”uber binding das Binden von Parametern eines Tupels an Signaturnamen eines Moduls namens modn beschreibt. * • IMPORT := MODULE-NAME $\times$ INST-NAME $\times$ VISIBILITY-FUNC $\times$ RENAMING-FUNC $\times$ $\cal P$(BINDING-BLOCK) Elemente dieses Typs repr”asentieren Importbefehle. Es handelt sich hier also um eine strukturierte Repr”asentation einer Zeichenkette, die den Import in ASF+ beschreibt. Wird ein benutzender Import dargestellt, ist die zweite Komponente leer. Zur Veranschaulichung sei als Beispiel folgender Importbefehl gegeben: import Sequences[NSeq] <(ITEMpar bound to NAT) of Naturals> { public: SEQ renamed to NSEQ private: nil renamed to nnil, cons } Wir erhalten folgende Tupeldarstellung: ( \| “Sequences”, “NSeq”, ---\|--- \| {(“SEQ”, “public”), (“nil”, “private”), (“cons”, “private”)}, \| {(“SEQ”, “NSEQ”), (“nil”, “nnil”)}, \| {({(“ITEMpar”, “NAT”)}, “Naturals”)} ) W”ahrend Importkonstrukte nur Teil einer nicht normalisierten Spezifikation sind, ist das Auftreten von Signaturen, Variablenvereinbarungen, Klauseln und Gleichungen unabh”angig vom Grad der Normalisierung: * • SIG := $\cal P$(SPEC-NAME) $\times$ $\cal P$(DISAMB-SPEC-NAME $\times$ SPEC- NAME)2 Dieser Datentyp repr”asentiert eine Teilsignatur eines Moduls. Teilsignaturen bestehen aus einer Menge von Sortennamen und je einer Mengen von Deklarationen f”ur Konstruktoren und Non-Konstruktoren. Jede Deklaration wird durch einen disambiguierten Namen (Funktionsname + Argumentsorten) und die zugeh”origen Zielsorte repr”asentiert. * • VAR-SORT-FUNC := SPEC-NAME $\longrightarrow$ SPEC-NAME Die Variablenvereinbarung eines Moduls beschreibt eine Funktion, die jedem Variablennamen eine Sorte zuweist. * • CLAUSE * • EQUATION Mit Hilfe der so definierten Strukturen kann nun ein ASF+-Modul wie folgt als 9-Tupel repr”asentiert werden: * • MODULE := MODULE-NAME $\times$ $\cal P$(IMPORT) $\times$ $\cal P$(SIG $\times$ $\cal P$(CLAUSE)) $\times$ SIG2 $\times$ VAR-SORT-FUNC2 $\times$ $\cal P$(EQUATION) $\times$ $\cal P$(CLAUSE) Dem Modulnamen folgen die Importe, eine Menge von Parametersignaturen (mit Bedingungsklauseln), die exportierte (public) und die nur innerhalb des Moduls sichtbare (private) Signatur, zwei Funktionen, die den Konstruktor- bzw. Non- Konstruktor-Variablen ihre jeweilige Sorte zuweisen, eine Menge von spezifizierenden Gleichungen und schlie”slich eine Menge von Beweiszielen. Wir k”onnen nun pr”azisieren, was wir im Folgenden unter der komponentenweisen Vereinigung einer Menge von Modulen verstehen werden: Sei $\\{{\it module\/}_{i}\quad\|\quad i\in A\\}$ eine Menge von Modulen und es gelte f”ur $i\in A$ modulei = \| ( \| modinamei, importsi, parametersi, ---\|---\|--- \| \| (sorts${}_{\mbox{\tiny pub},i}$, const${}_{\mbox{\tiny pub},i}$, non-const${}_{\mbox{\tiny pub},i}$), \| \| (sorts${}_{\mbox{\tiny pri},i}$, const${}_{\mbox{\tiny pri},i}$, non-const${}_{\mbox{\tiny pri},i}$), \| \| varsortfunc${}_{\mbox{\tiny const},i}$, varsortfunc${}_{\mbox{\tiny non-const},i}$, \| \| equationsi, goalsi ). $\displaystyle\bigsqcup_{i\in A}{\it module\/}_{i}:=$ \| ( \| $\emptyset,\displaystyle\bigcup_{i\in A}{\it imports\/}_{i},\displaystyle\bigcup_{i\in A}{\it parameters\/}_{i},$ \| \| $(\displaystyle\bigcup_{i\in A}{\it sorts}_{\mbox{\tiny pub},i},\displaystyle\bigcup_{i\in A}{\it const\/}_{\mbox{\tiny pub},i},\displaystyle\bigcup_{i\in A}\mbox{\it non-const\/}_{\mbox{\tiny pub},i}),$ \| \| $(\displaystyle\bigcup_{i\in A}{\it sorts}_{\mbox{\tiny pri},i},\displaystyle\bigcup_{i\in A}{\it const\/}_{\mbox{\tiny pri},i},\displaystyle\bigcup_{i\in A}\mbox{\it non-const\/}_{\mbox{\tiny pri},i}),$ \| \| $\displaystyle\bigcup_{i\in A}{\it varsortfunc\/}_{\mbox{\tiny const},i}$, $\displaystyle\bigcup_{i\in A}{\it varsortfunc\/}_{\mbox{\tiny non-const},i}$, \| \| $\displaystyle\bigcup_{i\in A}{\it equations\/}_{i},\displaystyle\bigcup_{i\in A}{\it goals\/}_{i}\ )$ Im Zuge der Importelemination geht Information ”uber den hierachischen Aufbau der Spezifikation und die Herkunft der Signaturnamen verloren. Um dieses Wissen der Normalisierungsprozedur zug”anglich zu machen, werden eine Origin- und eine Dependenzfunktion eingef”uhrt: * • ORIGIN := USER-NAME $\times$ MODINST-NAME $\times$ {“label”, “variable”, “sort”, “function”} $\times$ {“parameter”, “public”, “private”, “hidden”} Im Prinzip w”urden Origins, die Auskunft ”uber den Namensraum eines Bezeichners geben, ausreichen, um die angestrebte Semantik zu realisieren. Zur Formulierung des Normalformalgorithmus erweist es sich jedoch als zweckm”a”sig, weitere redundante Informationen, beispielsweise aus der Signatur aufzunehmen. In ASF+ werden Origins als Viertupel erkl”art. Die vier Komponenten eines Origins (uname, defmodiname, symboltype, visibility) sind wie folgt definiert: * – uname enth”alt den Namen ($\in$ USER-NAME), der vom Spezifizierer f”ur die spezifizierte Sorte, Funktion, Variable oder Marke (im folgenden als das spezifizierte Objekt bezeichnet) eingef”uhrt wurde. Explizites Renaming ver”andert nicht nur den Namen selbst sondern auch den Eintrag uname des zugeordneten Origins. * – modiname gibt Auskunft ”uber den Namensraum, dem der Name angeh”ort. * – symboltype: Namens”anderungen (sowohl implizites als auch explizites Renaming) werden im Normalformalgorithmus von ASF+ in zwei Stufen durchgef”uhrt. Zun”achst werden alle Sorten, Variablen und Marken umbenannt (f”ur zugeh”orige Origins gilt symboltype $\in$ {“label”, “variable”, “sort”}, danach folgt die Umbenennung der Funktionen. Diese Reihenfolge operationalisiert die durch Overloading bedingte rekursive Struktur der Identifikationsregel aus ASF+. * – visibility: F”ur Sorten und Funktionen gibt es drei Sichtbarkeitsstufen: * * “public”: innerhalb des Moduls sichtbar und exportf”ahig. * * “private”: innerhalb des Moduls sichtbar, jedoch nicht exportf”ahig. * * “hidden”: innerhalb des Moduls verdeckt und nat”urlich auch nicht exportf”ahig. Marken und Variablen gelten innerhalb des Moduls, in dem sie definiert werden als “private”, beim Import des Moduls werden sie verdeckt. Parameter geh”oren dem Sonderstatus “parameter” an und k”onnen nicht verdeckt werden. * • ORIGIN-FUNC := DISAMB-SPEC-NAME $\longrightarrow$ ORIGIN Funktionen dieses Typs weisen (disambiguierten) Namen aus der Spezifikation Origins zu. * • DEPENDENCY-FUNC := MODINST-NAME $\longrightarrow$ $\cal P$(MODINST-NAME) Dependenzfunktionen beschreiben die Abh”angigkeiten zwischen Namensr”aumen einer Spezifikation. Jeder Namensraumbezeichnung aus dem Definitonsbereich wird die Menge der Bezeichnungen aller abh”angigen Namensr”aume zugewiesen. Der rekursive Normalformalgorithmus operiert auf einer Datenstruktur, die wir “general forms” (GF) nennen: * • GF := MODULE $\times$ ORIGIN-FUNC $\times$ DEPENDENCY-FUNC General forms bestehen aus der (internen) Repr”asentation eines Moduls, einer Originfunktion und einer Dependenzfunktion. Origin- und Dependenzfunktion k”onnen partiell sein in dem Sinn, da”s Namen aus zu importierenden Modulen zun”achst unber”ucksichtigt bleiben. * • NF $\subseteq$ GF Als Normalformen bezeichenen wir alle general forms, die ein importfreies Modul, eine auf den im Modul vorkommenden Namen ($\in$ DISAMB-SPEC-NAME) totale Originfunktion und eine auf den Bezeichnungen aller im Modul enthaltenen Namensr”aume totale Dependenzfunktion beinhalten. Eine Normalform repr”asentiert also nicht nur ein normalisiertes Modul sondern auch den Bauplan der Spezifikation, der das Modul seine Erzeugung verdankt. Schlie”slich wird f”ur die Normalisierungsprozedur noch eine Funktion ben”otigt, die Namensumbenennungen einzelner ggf. ”uberladener Signaturnamen eindeutig beschreibt. Da sich das Exportverhalten ”uberladener Funktionsnamen unterscheiden kann, ist eine Differenzierung nach den Argumentsorten erforderlich: * • DISAMB-RENAMING-FUNC := DISAMB-SPEC-NAME $\longrightarrow$ SPEC-NAME Dieser Datentyp beschreibt Umbenennungen, die aufgrund von ”Anderungen der Sichtbarkeit einzelner Namen beim Import erforderlich werden. Es ist zu beachten, da”s die Durchf”uhrung von Funktionsumbenennungen dieser Art in Gleichungen das Disambiguieren der Funktionssymbole jedes einzelnen Terms erfordert. Hierzu wird die Signatur des Moduls gebraucht. ### 6.2 Der Algorithmus #### 6.2.1 Globale Hilfsfunktionen f”ur Sichtbarkeits”anderungen Das dynamische Verdeckungsprinzip von ASF+ erfordert bei der Kombination verschiedener Module zahlreiche Signaturnamensumbenennungen. Jede Namens”anderung zieht im allgemeinen Ver”anderungen in fast allen Teilen des Moduls und der Originfunktion nach sich. Der Normalformalgorithmus erledigt dies in zwei Schritten. Zuerst wird die 4. Komponente visibility der Origins aller Namen auf die Sichtbarkeitsstufe gesetzt, die die jeweiligen Namen zuk”unftig haben sollen. Die sich daraus ergebenden Umbenennungen im Modul und dem Definitionsbereich der Originfunktion, sowie die Neuordnung der Signatur werden dann von der Funktion MakeConsistent erledigt. Die Vorgehensweise des hier vorgestellten Algorithmus nutzt die in der Originfunktion enthaltene Redundanz aus: Jedes Origin origin beschreibt den ihm zugeordneten (nicht disambiguierten) Namen GetSpecName(origin) eindeutig. GetSpecName: $\begin{array}[t]{l}\mbox{\sf ORIGIN}\\\ \longrightarrow\mbox{\sf SPEC-NAME}\end{array}$ GetSpecName((uname, modiname, , visibility)) berechnet aus der ersten, zweiten und vierten Komponente eines Origins den zugeordneten (nicht disambiguierten) Namen ($\in$ SPEC-NAME). falls visibility $\in$ {“parameter”, “public”, “private”} * Setze specname := uname. falls visibility = “hidden” * Setze shortmodiname gleich dem abgek”urzten Modulinstanznamen von modiname. specname := shortmodiname“-”uname R”uckgabewert: specname Manipulationen der Sichtbarkeitskomponente im Wertebereich einer Originfunktion f”uhren im allgemeinen dazu, da”s die Namen des Definitionsbereichs nicht mehr zu den zugeordneten Origins passen. Sei ((namei, sortvi), origini) Element einer Originfunktion, dann entspricht GetSpecName(origini) dem “Sollwert” von namei. Zur Namensaktualisierung im Modul und im Definitionsbereich der Originfunktion dient eine “Istwert- Sollwert”-Liste, die von GetRenaming erzeugt wird: GetRenaming: $\begin{array}[t]{l}\mbox{\sf ORIGIN-FUNC}\ \times\ \mbox{$\cal P$}(\\{\mbox{``{\tt label}''},\mbox{``{\tt variable}''},\mbox{``{\tt sort}''},\mbox{``{\tt function}''}\\})\\\ \longrightarrow\mbox{\sf DISAMB- RENAMING-FUNC}\end{array}$ GetRenaming(originf, symboltypes) errechnet ein renaming f”ur disambiguierte Namen. Mit dessen Hilfe kann innerhalb der Originfunktion sowie eines Normalformmoduls ein konsistenter Zustand hergestellt werden. symboltypes bestimmt, welche Namenstypen in das renaming aufgenommen werden sollen. * ${\it renaming\/}\ :=\ \\{\ \begin{array}[t]{l}(({\it name\/},{\it sortv\/}),{\it name\/}^{\prime})\quad\|\\\ (u,n,{\it symboltype\/},v)={\it originf\/}(({\it name\/},{\it sortv\/}))\quad\wedge\\\ {\it symboltype\/}\in{\it symboltypes\/}\quad\wedge\\\ {\it name\/}^{\prime}={\it GetSpecName\/}((u,n,{\it symboltype\/},v))\quad\wedge\\\ {\it name\/}^{\prime}\neq{\it name\/}\ \\}\end{array}$ R”uckgabewert: renaming Bei der Beschreibung von Umbenennungen ”uberladbarer Signaturnamen ist zu ber”ucksichtigen, da”s Sortenumbenennungen auch die Sortenvektoren der umzubenennenden (disambiguierten) Funktionsnamen beeinflussen. Aus Gr”unden der ”Ubersichtlichkeit verzichten wir auf “simultanes” Umbenennen von Sorten und Funktionen, MakeConsistent behandelt Sorten und Funktionen nacheinander: MakeConsistent: $\begin{array}[t]{l}\mbox{\sf MODULE}\ \times\ \mbox{\sf ORIGIN-FUNC}\\\ \longrightarrow\mbox{\sf MODULE}\ \times\ \mbox{\sf ORIGIN- FUNC}\end{array}$ MakeConsistent(module, originfunc) erh”alt ein (normalisiertes) Modul module und eine Originfunktion originfunc deren Wertebereich zwecks Durchf”uhrung von Verdeckung oder ”Anderung des Exportverhaltens von Namen manipuliert wurde. Unter Zuhilfename der Funktionen GetSpecName und GetRenaming berechnet sie ein konsistentes Tupel (module′′′, originfunc′′). Durchgef”uhrt werden Umbenennung von Namen ($\in$ SPEC-NAME) in module und im Definitionsbereich von originfunc, sowie der Austausch von Sortennamen und Funktionsdeklarationen zwischen der public\- und private-Signatur. * renaming := GetRenaming(originfunc, {“label”, “variable”, “sort”}) Berechne module′ durch Ersetzen der Sorten-, Variablen- und Markennamen in module nach Ma”sgabe von renaming. Berechne originfunc′ durch Ersetzen der Sorten-, Variablen- und Markennamen im Definitionsbereich von originfunc nach Ma”sgabe von renaming. Betroffen sind insbesondere auch die Sortenvektoren der disambiguierten Funktionsnamen. renaming′ := GetRenaming(originfunc′, {“function”}) Berechne module′′ und originfunc′′ durch Ersetzen der Funktionsnamen in module′ und im Definitionsbereich der Originfunktion originfunc′ nach Ma”sgabe von renaming′. module′′′ entsteht aus module′′ durch Aktualisierung der public\- und private- Signatur. Namen mit Sichtbarkeitsstufe “private” oder “hidden” sind nicht exportf”ahig und geh”oren in die private-Signatur. Solche mit Sichtbarkeitsstufe “public” hingegen geh”oren in die public-Signatur. Parameter bleiben wo sie sind, n”amlich in der parameter-Signatur. R”uckgabewert: (module′′′, originfunc′′) #### 6.2.2 Kombination von Modulen Der Import sowie das Binden von Parametern an Signaturnamen eines Moduls f”uhrt bei der Normalisierung dazu, da”s mehrere general forms zu einer neuen general form zusammengefa”st werden m”ussen. Diese Aufgabe erledigen die drei Funktionen CombineImports, CombineWithImports und CombineWithActModule. Als Hilfsfunktion greifen sie auf CombineDependencies und AdaptVisibility, welche die notwendigen Sichtbarkeitsanpassungen vornimmt, zu. AdaptVisibility kann als Identifikationsregel gelesen werden, die beim Kombinieren mehrerer Module festlegt, wann (disambiguierte) Namen miteinander identifiziert werden d”urfen und unter welchen Umst”anden es zu Namenskonflikten kommt. Wegen der Overloading-F”ahigkeit ist hierbei von Wichtigkeit, da”s zuerst alle Sortenidentifikationen vorgenommen werden (Aufruf von AdaptVisibility mit symboltypes := {“sort”}). Erst danach k”onnen die Funktionsidentifikationen korrekt durchgef”uhrt werden (symboltypes := {“function”}). Die Zweistufigkeit reduziert den Sortenvektortest auf syntaktische Gleichheit. Andernfalls m”u”sten beim Test auf Identifizierbareit die Argumentsorten der (disambiguierten) Funktionsnamen komponentenweise (rekursiv) auf Identifizierbarkeit gepr”uft werden. Analog zur sogenannten “Originrule” aus ASF k”onnen wir die ASF+ zugrundeliegende Identifikationsregel folgenderma”sen beschreiben: ##### Identifikationsregel: Die (disambiguierten) Namen (name1, sortv1) und (name2, sortv2) aus zwei zu kombinierenden Modulen sind genau dann zu identifizieren, wenn * • die ihnen zugeordneten Origins in den ersten drei Komponenten ”ubereinstimmen, * • die 4. Komponenten der Origins ”ubereinstimmen oder eine 4. Komponente den Wert “hidden”, die andere 4. Komponente dagegen “private” oder “public” enth”alt und * • die in sortv1 und sortv2 enthaltenen Argumentsorten (nur bei Funktionsnamen relevant) miteinander identifiziert werden k”onnen. Man beachte, da”s diese Definition nicht eigentlich rekursiv ist, da der R”uckbezug nicht wiederum selbst r”uckbez”uglich ist. Zwischen den disambiguierten Namen $({\it name\/}_{1},{\it sortv\/}_{1})$ und $({\it name\/}_{2},{\it sortv\/}_{2})$ aus zwei zu kombinierenden Modulen kommt es genau dann zum Konflikt, wenn * • sie nicht miteinander identifiziert werden k”onnen, obwohl * • ${\it name\/}_{1}$ mit ${\it name\/}_{2}$ ”ubereinstimmt und * • die in sortv1 und sortv2 enthaltenen Argumentsorten miteinander identifiziert werden k”onnen. Wir f”uhren noch eine Sprechweise ein, die sich bei der Behandlung von Parameterbindungen als n”utzlich erweisen wird. Seien ${\it originf\/}_{1}$ und ${\it originf\/}_{2}$ zwei Originfunktionen. Sei $({\it name\/}_{1},{\it sortv\/}_{1})$ aus dem Definitionsbereich von ${\it originf\/}_{1}$ und $({\it name\/}_{2},{\it sortv\/}_{2})$ aus dem Definitionsbereich von ${\it originf\/}_{2}$. Wir definieren: $({\it name\/}_{1},{\it sortv\/}_{1})$ referenziert bez”uglich ${\it originf\/}_{1}$ dasselbe Objekt wie $({\it name\/}_{2},{\it sortv\/}_{2})$ bez”uglich ${\it originf\/}_{2}$ (im Zeichen: $({\it name\/}_{1},{\it sortv\/}_{1})/{\it originf\/}_{1}\approx({\it name\/}_{2},{\it sortv\/}_{2})/{\it originf\/}_{2}$) genau dann, wenn * • die ihnen zugeordneten Origins in den ersten drei Komponenten ”ubereinstimmen und * • die in den Komponenten von sortv1 und sortv2 enthaltenen Argumentsorten (nur bei Funktionsnamen relevant) jeweils dasselbe Signaturobjekt referenzieren. Die Identifikationsregel aus ASF+ identifiziert also Namen, die das gleiche Signaturobjekt referenzieren und deren Exportverhalten in den Importbefehlen nicht widerspr”uchlich festgelegt wird. AdaptVisibility: $\begin{array}[t]{l}\mbox{$\cal P$}(\mbox{\sf NF})\ \times\ \mbox{$\cal P$}(\\{\mbox{``{\tt label}''},\mbox{``{\tt variable}''},\mbox{``{\tt sort}''},\mbox{``{\tt function}''}\\})\\\ \longrightarrow\mbox{$\cal P$}(\mbox{\sf NF})\end{array}$ AdaptVisibility(normalforms, symboltypes) sorgt f”ur die Angleichung der Sichtbarkeit von Sorten- (“sort” $\in$ symboltypes) und Funktionsnamen (“function” $\in$ symboltypes) aus verschiedenen (NF-) Modulen. Gleichzeitige public\- und private-Importe eines Namens weisen auf einen Spezifikationsfehler hin, weil ein Name entweder exportierbar oder nicht- exportierbar sein kann aber nicht beides gleichzeitig. * Sei $\\{({\it mod\/}_{i},{\it originf\/}_{i},{\it depf\/}_{i})\ \|\ 1\leq i\leq p\\}\ =\ {\it normalforms\/}$ F”ur $i:=1$ bis $p$ wiederhole * F”ur $j:=i+1$ bis $p$ wiederhole * F”ur alle ((namei, sortvi), (unamei, modinamei, symboltypei, visibilityi)) $\in$ originfi wiederhole * F”ur alle ((namej, sortvj), (unamej, modinamej, symboltypej, visibilityj)) $\in$ originfj wiederhole * /* Alle Origins aller ”ubergebenen Originfunktionen originfi werden mit allen Origins aller anderen ”ubergebenen Originfunktionen originfj verglichen. / Falls (symboltypei $\in$ symboltypes) und sortvi = sortvj und unamei = unamej Falls modinamei = modinamej * Falls ${\it symboltype\/}_{i}\neq{\it symboltype\/}_{j}$ * SPEZIFIKATIONSFEHLER /* Beide disambiguierten Namen verdanken ihre Existenz derselben Definition / Falls (visibilityi = “hidden” und visibilityj $\in$ {“public”, “private”}) Setze (mit ”Anderung von originfi) visibilityi := visibilityj Sonst falls (visibilityj = “hidden” und visibilityi $\in$ {“public”, “private”}) * Setze (mit ”Anderung von originfj) visibilityj := visibilityi Sonst falls visibilityi $\neq$ visibilityj * EXPORTIERBARKEITS-KONFLIKT Sonst falls namei = namej * /* Der disambiguierte Name $({\it name\/}_{i},sortv_{i})$ tritt in beiden Normalformen mit unterschiedlicher Bedeutung auf. / NAMENSKONFLIKT F”ur alle $i\in\\{1,\ldots,p\\}$ $({\it mod\/}^{\prime}_{i},{\it originf\/}^{\prime}_{i},{\it depf\/}^{\prime}_{i}):={\it MakeConsistent\/}({\it mod\/}_{i},{\it originf\/}_{i},{\it depf\/}_{i})$ R”uckgabewert: $\\{({\it mod\/}^{\prime}_{i},{\it originf\/}^{\prime}_{i},{\it depf\/}^{\prime}_{i})\quad\|\quad 1\leq i\leq p\\}$ CombineDependencies: $\begin{array}[t]{l}\mbox{$\cal P$}(\mbox{\sf DEPENDENCY- FUNC})\\\ \longrightarrow\mbox{\sf DEPENDENCY-FUNC}\end{array}$ ${\it CombineDependencies\/}(\\{{\it depf\/}_{j}\ \|\ j\in A\\})$ erzeugt aus den Dependenzfunktionen mehrerer zu kombinierender general forms eine neue Dependenzfunktion ${\it depf\/}^{\prime}$. * /* Siehe Seite 4.3. / R”uckgabewert: ${\it depf\/}^{\prime}$ Importe werden in ASF+ eleminiert, indem zun”achst die Normalformen der importierten Module berechnet werden. Diese werden nach Ma”sgabe der Importbefehle modifiziert und instanziiert (siehe dazu den folgenden Abschnitt 6.2.3) und anschlie”send untereinander kombiniert. Daf”ur zust”andig ist die Funktion CombineImports: CombineImports: $\begin{array}[t]{l}\mbox{$\cal P$}(\mbox{\sf NF})\\\ \longrightarrow\mbox{\sf NF}\end{array}$ CombineImports(normalforms) kombiniert mehrere Normalformen. ${\it normalforms\/}^{\prime}\ :={\it AdaptVisibility\/}({\it normalforms\/},\\{\mbox{``{\tt label}''},\mbox{``{\tt variable}''},\mbox{``{\tt sort}''}\\})$ ${\it normalforms\/}^{\prime\prime}\ :={\it AdaptVisibility\/}({\it normalforms\/}^{\prime},\\{\mbox{``{\tt function}''}\\})$ Sei $\\{({\it mod\/}_{i},{\it originf\/}_{i},depf_{i})\ \|\ i\in A\\}={\it normalforms\/}^{\prime\prime}$. ${\it mod\/}^{\prime}\ :=\displaystyle\bigsqcup_{i\in A}{\it mod\/}_{i}$ ${\it originf\/}^{\prime}\ :=\displaystyle\bigcup_{i\in A}{\it originf\/}_{i}$ Falls ${\it originf\/}^{\prime}$ keine Funktion * NAMECLASH ${\it depf\/}^{\prime}\ :={\it CombineDependencies\/}(\\{{\it depf\/}_{i}\ \|\ i\in A\\})$ R”uckgabewert: $({\it mod\/}^{\prime},{\it originf\/}^{\prime},{\it depf\/}^{\prime})$ Mit Hilfe von CombineImports wird eine Normalform erzeugt, die alle importierten Module in sich vereint. Sie wird anschlie”send durch Anwendung der Funktion CombineWithImports mit der general form des importierenden Moduls kombiniert. Hier sind keine Sichtbarkeitsanpassungen mehr notwendig: CombineWithImports: $\begin{array}[t]{l}\mbox{\sf GF}\ \times\ \mbox{\sf NF}\\\ \longrightarrow\mbox{\sf NF}\end{array}$ ${\it Combine\\-With\\-Imports\/}(({\it mod\/},{\it originf\/},{\it depf\/}),({\it mod\/}_{\mbox{\tiny imp}},{\it originf\/}_{\mbox{\tiny imp}},{\it depf\/}_{\mbox{\tiny imp}}))$ kombiniert die general form einer Modulinstanz mit einer Normalform, die aus allen von ihr importierten Modulen errechnet worden ist. * mod′ geht aus mod durch L”oschen aller Importkonstrukte hervor. ${\it mod\/}^{\prime\prime}\ :=\ {\it mod\/}^{\prime}\sqcup{\it mod\/}_{\mbox{\tiny imp}}$ Der Modulname von mod′′ (erste Komponente) wird auf den f”ur die Normalform von mod vorgesehenen Namen gesetzt. Dieser kann beispielsweise aus dem Modulnamen von mod durch Anh”angen der Extension “.nf” gewonnen werden. ${\it originf\/}^{\prime}\ :={\it originf\/}\ \cup{\it originf\/}_{\mbox{\tiny imp}}$ Falls originf′ keine Funktion: NAMECLASH Sei nun modname der Modulname (1. Komponente) von mod. ${\it depf\/}^{\prime}\ :=\ $ \| $\\{\ $ \| $({\it modname\/},\emptyset)\ \\}\ \cup$ ---\|---\|--- \| $\\{\ ({\it modiname\/},{\it modinames\/}\cup\\{{\it modname\/}\\})\ \|$ \| \| $({\it modiname\/},{\it modinames\/})\in{\it depf\/}_{\mbox{\tiny imp}}\ \\}$ R”uckgabewert: (mod′′, originf′, depf′) Wird in einem Importbefehl die Bindung eines Parametertupels aus dem importierten Modul modFORM an Namen eines Moduls modACT vorgenommen, so erfordert die Auswertung das Kombinieren der zugeh”origen Normalformen. Dieser implizite Import des Moduls modACT in das Modul modFORM unterscheidet sich von gew”ohnlichen Importen, weil hierdurch ein Modul “nachtr”aglich” in eine bereits bestehende Modulhierarchie eingepflanzt wird. CombineWithActModule: $\begin{array}[t]{l}\mbox{\sf NF}\ \times\ \mbox{\sf MODINST-NAME}\ \times\ \mbox{\sf NF}\\\ \longrightarrow\mbox{\sf NF}\end{array}$ ${\it CombineWithActModule\/}(({\it mod\/}_{\mbox{\tiny FORM}},{\it originf\/}_{\mbox{\tiny FORM}},{\it depf\/}_{\mbox{\tiny FORM}}),$ paradefmod, (mod${}_{\mbox{\tiny ACT}}$, originf${}_{\mbox{\tiny ACT}}$, depf${}_{\mbox{\tiny ACT}}$)) “implantiert” die Normalform (mod${}_{\mbox{\tiny ACT}}$, originf${}_{\mbox{\tiny ACT}}$, depf${}_{\mbox{\tiny ACT}}$) in die Normalform (mod${}_{\mbox{\tiny FORM}}$, originf${}_{\mbox{\tiny FORM}}$, depf${}_{\mbox{\tiny FORM}}$). Dabei wird eine Abh”angigkeit zwischen den Namensr”aumen des Moduls mod${}_{\mbox{\tiny ACT}}$ und dem Namensraum der formalen Parameter paradefmod aus mod${}_{\mbox{\tiny FORM}}$ hergestellt. Es wird davon ausgegangen, da”s bereits alle Renamings in der Normalform des formalen Moduls und die Sichtbarkeitsanpassungen zwischen den Namen beider Normalformen durchgef”uhrt worden sind. * ${\it mod\/}\ :=\ {\it mod\/}_{\mbox{\tiny ACT}}\sqcup{\it mod\/}_{\mbox{\tiny FORM}}$ Der Modulname modname von mod${}_{\mbox{\tiny FORM}}$ (erste Komponente) wird in mod ”ubernommen. ${\it originf\/}\ :=\ {\it originf\/}_{\mbox{\tiny FORM}}\cup{\it originf\/}_{\mbox{\tiny ACT}}$ Falls originf keine Funktion: NAMECLASH ${\it depf\/}^{\prime}_{\mbox{\tiny ACT}}\ :=\ \\{\ \begin{array}[t]{@{}l}({\it modiname\/},{\it modinames\/}\ \cup\\{{\it paradefmod\/}\\}\cup{\it depf\/}_{\mbox{\tiny FORM}}({\it paradefmod\/})\ \|\\\ ({\it modiname\/},{\it modinames\/})\in{\it depf\/}_{\mbox{\tiny ACT}}\ \\}\end{array}$ ${\it depf\/}\ :=\ {\it CombineDependencies\/}({\it depf\/}_{\mbox{\tiny FORM}},{\it depf\/}^{\prime}_{\mbox{\tiny ACT}})$ R”uckgabewert: (mod, originf, depf) #### 6.2.3 Modulmodifikationen in Importbefehlen Werden in einem Importbefehl Namen umbenannt, Parameter gebunden oder die Sichtbarkeit von Signaturnamen ver”andert, so f”uhrt das semantisch dazu, da”s die Normalformen der zu importierenden Module modifiziert werden m”ussen, bevor sie zu einer einzigen Normalform zusammengefa”st werden k”onnen. Diese Aufgabe ”ubernehmen die Funktionen Hide, Rename und Bind mit den Hilfsfunktionen InstanciateModInstName, Instanciate, SeparateParaBlock, GetParameterRenamings und CheckSemanticConditions. Ein wesentlicher Teil eines jeden Importbefehls sind die den Schl”usselworten “private:” und “public:” folgenden Listen von Signaturnamen. Sie geben Auskunft ”uber die Sichtbarkeit der vom importierten Modul exportierten Signaturnamen. Mit Hilfe der Funktion Hide werden alle nicht exportierten Namen verdeckt und die Sichtbarkeit der exportierten Signaturnamen den Vorgaben des Importbefehls angepa”st. Hide: $\begin{array}[t]{l}\mbox{\sf NF}\ \times\ \mbox{\sf VISIBILITY-FUNC}\\\ \longrightarrow\mbox{\sf NF}\end{array}$ Hide((mod, originf, depf), visibilityf) verdeckt alle Namen mit Sichtbarkeitsstufe “private”. Namen mit Sichtbarkeitsstufe “public” werden auf die in visibilityf angegebene Sichtbarkeitsstufe gesetzt; ist keine Angabe vorhanden, erhalten sie die Sichtbarkeitsstufe “hidden”. * ${\it originf\/}^{\prime}\ :=\ \\{\ $ \| $(\mbox{\it dis-name},({\it uname\/},{\it modinst},{\it symboltype\/},{\it visibility\/}^{\prime}))\quad\|$ ---\|--- \| $(\mbox{\it dis-name},({\it uname\/},{\it modinst},{\it symboltype\/},{\it visibility\/}))\in{\it originf\/}$ \| $\wedge\ ($ \| $({\it visibility\/}\in\\{\mbox{``{\tt hidden}''},\mbox{``{\tt parameter}''}\\}\ \wedge$ ${\it visibility\/}={\it visibility\/}^{\prime})\ \vee$ \| \| $({\it visibility\/}=\mbox{``{\tt private}''}\wedge{\it visibility\/}^{\prime}=\mbox{``{\tt hidden}''})\ \vee$ \| \| $({\it visibility\/}=\mbox{``{\tt public}''}$ \| \| $\wedge\ ($ \| $({\it uname\/}\notin\mbox{\sf Dom}({\it visibilityf\/})$ $\wedge\ {\it visibility\/}^{\prime}=\mbox{``{\tt hidden}''})\ \vee$ \| \| \| $({\it uname\/}\in\mbox{\sf Dom}({\it visibilityf\/})$ $\wedge\ {\it visibility\/}^{\prime}={\it visibilityf\/}({\it uname\/})))))\ \\}$ $({\it mod\/}^{\prime},{\it originf\/}^{\prime\prime}):={\it MakeConsistent\/}({\it mod\/},{\it originf\/}^{\prime})$ R”uckgabewert: $({\it mod\/}^{\prime},{\it originf\/}^{\prime\prime},{\it depf\/})$ Der kopierende Import aus ASF+ basiert auf der Zuordnung der zu kopierenden (Signatur-) Namen zu neuen Namensr”aumen. Zu diesem Zweck werden neue Namensraumbezeichnungen generiert, die sich aus den alten Bezeichnungen und der Instanzbezeichnung des Importbefehls zusammensetzen. InstanciateModInstName: $\begin{array}[t]{l}\mbox{\sf MODINST-NAME}\ \times\ \mbox{\sf INST-NAME}\\\ \longrightarrow\mbox{\sf MODINST-NAME}\end{array}$ InstanciateModInstName(modiname, iname) instanziiert die Namensraumbezeichnung modiname mit der Instanzbezeichnung iname. Wurde modiname bereits mit iname instanziiert, so liegt ein Spezifikationsfehler vor. * Falls modiname $\in$ MODULE-NAME /* erste Instanziierung / imodiname := modiname“[”iname“]” Sonst * Sei modname“[”oldinames“]” = modiname Falls iname in oldinames enthalten ist * SPEZIFIKATIONSFEHLER! imodiname := modname“[”oldinames“,”iname“]” R”uckgabewert: imodiname Eine Normalform repr”asentiert nicht nur ein normalisiertes Modul; Origin- und Dependenzfunktion erlauben die Rekonstruktion der gesamten zugrundeliegenden Modulhierarchie. Werden Teile einer Normalform durch explizites Renaming oder Parameterbindung modifiziert, k”onnen die erforderlichen Instanziierungen auf die direkt betroffenen und die davon abh”angigen Namensr”aume begrenzt werden. Instanciate: $\begin{array}[t]{l}\mbox{\sf NF}\ \times\ \mbox{\sf RENAMING- FUNC}\ \times\ \mbox{$\cal P$}(\mbox{\sf BINDING-BLOCK})\ \times\ \mbox{\sf INST-NAME}\\\ \longrightarrow\mbox{\sf NF}\end{array}$ Instanciate((mod, originf, depf), renaming, bindingblocks, iname) instanziiert Namensraumbezeichnungen in der Normalform (mod, originf, depf) mit der Instanzbezeichnung iname. Instanziiert werden die Bezeichnungen aller vom expliziten Renaming renaming und von den Parameterbindungen bindingblocks direkt betroffenen Namensr”aume, sowie alle bez”uglich depf von ihnen abh”angigen Namensr”aume. * Sei $\\{({\it binding\/}_{i},{\it modname\/}_{i})\ \|\ i\in A\\}={\it bindingblocks\/}$ ${\it toinst\/}\ :=\ \\{\ {\it modiname\/}\ \|\ \begin{array}[t]{l}(({\it name\/},),(,{\it modiname\/},,))\in{\it originf\/}\ \wedge\\\ {\it name\/}\in\mbox{\sf Dom}({\it renaming\/})\cup\displaystyle\bigcup_{i\in A}\mbox{\sf Dom}({\it binding\/}_{i})\ \\}\end{array}$ ${\it toinst\/}^{\prime}\ :=\ {\it toinst\/}\ \cup\ \\{{\it depf\/}({\it modinst})\ \|\ {\it modinst}\in{\it toinst\/}\\}$ Berechne $({\it mod\/}^{\prime},{\it originf\/}^{\prime},{\it depf\/}^{\prime})$ durch Ersetzen jedes Auftretens einer Namensraumbezeichnung modiname $\in$ toinst′ * – in mod (”uberall dort, wo sie Teil eines verdeckten Namens ist), * – in originf (Im Definitionsbereich ”uberall dort, wo sie Teil eines verdeckten Namens ist und in der 2. Komponente der Origins des Wertebereichs) und * – in depf (wo immer sie auftritt) durch InstanciateModInstName(modiname, iname). R”uckgabewert: (mod′, originf′, depf′) Rename: $\begin{array}[t]{l}\mbox{\sf NF}\ \times\ \mbox{\sf RENAMING-FUNC}\\\ \longrightarrow\mbox{\sf NF}\end{array}$ Rename((mod, originf, depf), renaming) f”uhrt explizites Renaming durch. renaming enth”alt die Umbenennungen aller Renaminganweisungen des Importbefehls. * Sei ${\it ren\/}(x):=\ \left\\{\begin{array}[]{ll}y&$falls$\ (x,y)\in{\it renaming}\/\\\ x&$sonst$\end{array}\right.$ und ren′ die Erweiterung von ren auf Sortenvektoren: ${\it ren\/}^{\prime}(({\it sortn\/}_{1},\ldots,{\it sortn\/}_{n})):=({\it ren\/}({\it sortn\/}_{1}),\ldots,{\it ren\/}({\it sortn\/}_{n}))$ ${\it mod\/}^{\prime}$ wird aus ${\it mod\/}$ durch syntaktisches Ersetzen aller Signaturnamen ${\it name\/}$ durch ${\it ren\/}({\it name\/})$ erzeugt. Man beachte da”s ren nur Einflu”s auf sichtbare Namen ($\in$ USER-NAME) hat. SPEZIFIKATIONSFEHLER falls ${\it mod\/}^{\prime}$ keine korrekte Signatur enth”alt. /* Ursache kann hier ein fehlerhafter Renamingbefehl sein, der dazu f”uhrt, da”s urspr”unglich verschiedene Namen des gleichen Namensraumes nach Durchf”uhrung des Renamings zusammenfallen. Renamings dieser Art k”onnen Funktionen mit gleichen disambiguierten Namen aber unterschiedlichen Zielsorten erzeugen. / ${\it originf\/}^{\prime}\ :=\\{\ \begin{array}[t]{l}({\it ren\/}({\it name\/}),{\it ren\/}^{\prime}({\it sortv\/})),({\it uname\/}^{\prime},{\it modiname\/},{\it symboltype\/},{\it visibility\/}))\quad\|\\\ (({\it name\/},{\it sortv\/}),({\it uname\/},{\it modiname\/},{\it symboltype\/},{\it visibility\/}))\in{\it originf\/}\ \wedge\\\ (\begin{array}[t]{@{}l}({\it visibility\/}=\mbox{``{\tt hidden}''}\wedge\ {\it uname\/}^{\prime}={\it uname\/})\ \vee\\\ ({\it visibility\/}\neq\mbox{``{\tt hidden}''}\wedge\ {\it uname\/}^{\prime}={\it ren\/}({\it uname\/})))\ \\}\end{array}\end{array}$ R”uckgabewert: $({\it mod\/}^{\prime},{\it originf\/}^{\prime},{\it depf\/})$ Alle folgenden Funktionen dieses Abschnitts behandeln die Auswertung einer Parametertupelbindung. Der Trivialfunktion SeparateParaBlock und der (etwas technischen) Hilfsfunktion GetParameterRenamings folgen die Hauptfunktionen CheckSemanticConditions und Bind. Die Komplexit”at der Funktionen folgt aus der Tatsache, da”s es sich bei jeder Parametertupelbindung um einen impliziten Import (also einen Import im Import) handelt und neben den schon betrachteten Operationen (z. B. Verdecken von Namen, Instanziieren von Namensr”aumen) im Zuge des Testens semantischer Bedingungen und des Implantierens eines aktuellen Moduls in die bereits bestehende Modulhierarchie eines formalen Moduls eine Vielzahl neuer Rechenschritte erforderlich sind. SeparateParaBlock: $\begin{array}[t]{l}\mbox{\sf NF}\times\ \mbox{$\cal P$}(\mbox{\sf SPEC-NAME})\\\ \longrightarrow\mbox{\sf NF}\times\ (\mbox{\sf SIG}\times\ \mbox{$\cal P$}(\mbox{\sf CLAUSE}))\times\ \mbox{\sf MODINST- NAME})\end{array}$ SeparateParaBlock((mod, originf, depf), parameters) extrahiert aus der internen Moduldarstellung mod die Parametersignatur und -bedingungen der in parameters enthaltenen Parameter eines Tupels. Die Parameter werden aus dem Definitionsbereich der Originfunktion entfernt und paradefmod der Namensraum zugewiesen, dem die Parameter angeh”oren. SeparateParaBlock ist Hilfsfunktion von Bind. Seien ${\it sig\/}_{p}$ = die zu extrahierende Parametersignatur, conditions = die zu ${\it sig\/}_{p}$ geh”orenden Bedingungsklauseln und ${\it mod\/}^{\prime}$ = das Modul, das nach Entfernen von (${\it sig\/}_{p}$, conditions) aus mod entsteht. SPEZIFIKATIONSFEHLER, wenn keine Parametersignatur in mod enthalten ist, die genau alle Namen aus parameters enth”alt. Sei parameter $\in$ parameters (, paradefmod, , ) := originf(parameter) / Welcher Parameter genommen wird, hat keinen Einflu”s auf paradefmod / ${\it originf\/}^{\prime}\ :=\ \\{\ \begin{array}[t]{l}(({\it name\/},{\it sortv\/}),{\it origin\/})\quad\|\\\ (({\it name\/},{\it sortv\/}),{\it origin\/})\in{\it originf\/}\ \wedge\ {\it name\/}\notin{\it parameters\/}\ \\}\end{array}$ R”uckgabewert: $(({\it mod\/}^{\prime},{\it originf\/}^{\prime},{\it depf\/}),({\it sig\/}_{p},{\it conditions\/}),{\it paradefmod\/})$ GetParameterRenamings: $\begin{array}[t]{l}\mbox{\sf SIG}\ \times\ \mbox{\sf RENAMING-FUNC}\ \times\ \mbox{\sf ORIGIN-FUNC}^{2}\\\ \longrightarrow\mbox{\sf RENAMING-FUNC}\end{array}$ ${\it GetParameterRenamings\/}(({\it sorts}_{p},\mbox{\it cons- decs}_{p},\mbox{\it ncons-decs}_{p}),{\it binding\/},{\it originf\/}_{\mbox{\tiny ACT}},$ ${\it originf\/}_{\mbox{\tiny ACT-AV}})$ berechnet die jenigen Namen, durch welche die nach der Vorschrift binding an Namen eines aktuellen Moduls zu bindenden formalen Parameter syntaktisch ersetzt werden m”ussen. Die errechneten Namen sind im allgemeinen nicht mit denen aus Ran(binding) identisch, weil alle Namen aus dem aktuellen Modul beim impliziten Import verdeckt werden, sofern sie nicht bereits im formalen Modul sichtbar sind. Als Argumente werden die Signatur des zu bindenden Parametertupels $({\it sorts}_{p},\mbox{\it cons-decs}_{p},\mbox{\it ncons- decs}_{p})$, die Bindungsvorschrift binding, die Originfunktion des aktuellen Moduls ${\it originf\/}_{\mbox{\tiny ACT}}$ und eine weitere Originfunktion ${\it originf\/}_{\mbox{\tiny ACT-AV}}$, die aus ${\it originf\/}_{\mbox{\tiny ACT}}$ durch Setzen aller Namen auf die beim impliziten Import angestrebte Sichtbarkeitsstufe hervorgeht, ”ubergeben. Falls $\\{({\it binding\/}(sortpar),\emptyset)\ \|\ {\it sortpar}\in{\it sorts}_{p}\\}\ \not\subseteq\ \mbox{\sf Dom}({\it originf\/}_{\mbox{\tiny ACT}})$ * SPEZIFIKATIONSFEHLER /* Sortenparameterbindung fehlerhaft. Aktuelle Sorten existieren nicht im aktuellen Modul / $\mbox{\it sortpar-renaming}\ :=\\\ \mbox{\hskip 30.00005pt}\\{\ ({\it sortpar},{\it name\/})\ \|\ \begin{array}[t]{@{}l}{\it sortpar}\in{\it sorts}_{p}\ \wedge\\\ ({\it name\/},\emptyset)\in\mbox{\sf Dom}({\it originf\/}_{\mbox{\tiny ACT-AV}})\ \wedge\\\ ({\it name\/},\emptyset)/{\it originf\/}_{\mbox{\tiny ACT-AV}}\approx({\it binding\/}({\it sortpar}),\emptyset)/{\it originf\/}_{\mbox{\tiny ACT}}\ \\}\end{array}$555Zur Definition von $\approx$ siehe Seite 6.2.2 Sei ${\it sorts}_{\mbox{\scriptsize np}}$ die Menge aller Sortennamen, die in den Deklarationen $\mbox{\it cons-decs}_{p}\cup\mbox{\it ncons-decs}_{p}$ auftreten, aber nicht in ${\it sorts}_{p}$ enthalten sind. Falls $\\{({\it sort},\emptyset)\ \|\ {\it sort}\in{\it sorts}_{\mbox{\scriptsize np}}\\}\ \not\subseteq\mbox{\sf Dom}({\it originf\/}_{\mbox{\tiny ACT-AV}})$ SPEZIFIKATIONSFEHLER /* Da Funktionsparameter nur an aktuelle Funktionsnamen gleicher Deklaration gebunden werden k”onnen, m”ussen die Nicht-Parameter-Sortennamen der Funktionsparameterdeklarationen nicht nur im formalen, sondern auch im aktuellen Modul auftreten. / ${\it renaming\/}\ :=\\\ \mbox{\hskip 30.00005pt}\\{\ ({\it sortpar},{\it binding\/}({\it sortpar}))\ \|\ {\it sortpar}\in{\it sorts}_{p}\ \\}\ \cup\ \\\ \mbox{\hskip 30.00005pt}\\{\ ({\it sort},{\it name\/})\ \|\ \begin{array}[t]{@{}l}{\it sort}\in{\it sorts}_{\mbox{\scriptsize np}}\ \wedge\\\ ({\it name\/},\emptyset)\in\mbox{\sf Dom}({\it originf\/}_{\mbox{\tiny ACT}})\ \wedge\\\ ({\it name\/},\emptyset)/{\it originf\/}_{\mbox{\tiny ACT}}\approx({\it sort},\emptyset)/{\it originf\/}_{\mbox{\tiny ACT-AV}}\ \\}\end{array}$ Berechne $\mbox{\it cons-decs}^{\prime}_{p}$ und $\mbox{\it ncons- decs}^{\prime}_{p}$ aus $\mbox{\it cons-decs}_{p}$ und $\mbox{\it ncons- decs}_{p}$ durch Umbenennen aller Sorten nach Ma”sgabe von ${\it renaming\/}$. Sei ${\it disfuncs}_{p}\ =\ \\{\ ({\it funcpar},{\it sortv\/})\ \|\ (({\it funcpar},{\it sortv\/}),{\it sort})\in\mbox{\it cons- decs}^{\prime}_{p}\cup\mbox{\it ncons-decs}^{\prime}_{p}\ \\}$ Falls $\\{({\it binding\/}({\it funcpar}),{\it sortv\/})\ \|\ ({\it funcpar},{\it sortv\/})\in{\it disfuncs}_{p}\\}\ \not\subseteq\ \mbox{\sf Dom}({\it originf\/}_{\mbox{\tiny ACT}})$ SPEZIFIKATIONSFEHLER /* Bindung der Funktionsparameter fehlerhaft, kein “wohlsortierter” aktueller Parameter vorhanden / $\mbox{\it funcpar-renaming}\ :=\\\ \mbox{\hskip 30.00005pt}\\{\ \begin{array}[t]{@{}l}({\it funcpar},{\it name\/})\ \|\\\ ({\it funcpar},{\it sortv\/})\in{\it disfuncs}_{p}\ \wedge\\\ ({\it name\/},{\it sortv\/}^{\prime})\in\mbox{\sf Dom}({\it originf\/}_{\mbox{\tiny ACT-AV}})\ \wedge\\\ ({\it name\/},{\it sortv\/}^{\prime})/{\it originf\/}_{\mbox{\tiny ACT-AV}}\approx({\it binding\/}({\it funcpar}),{\it sortv\/})/{\it originf\/}_{\mbox{\tiny ACT}}\ \\}\end{array}$ $\mbox{\it par-renaming}\ :=\ \mbox{\it sortpar-renaming}\cup\mbox{\it funcpar-renaming}$ R”uckgabewert: par-renaming CheckSemanticConditions: $\begin{array}[t]{l}\mbox{$\cal P$}(\mbox{\sf CLAUSE})\ \times\ \mbox{\sf MODULE}^{2}\ \times\ \mbox{\sf NF}\ \times\ \mbox{\sf PROVE-DB}\\\ \longrightarrow-\end{array}$ ${\it CheckSemanticConditions\/}({\it conditions\/},{\it mod\/}_{\mbox{\tiny FORM}},{\it mod\/}_{\mbox{\tiny ACT-AV}},{\it nform}_{\mbox{\tiny ACT}},\mbox{\it prove-db\/})$ pr”uft, ob die semantischen Bedingungen, die an die Bindung von Parametern aus ${\it mod\/}_{\mbox{\tiny FORM}}$ an Namen des in ${\it nform}_{\mbox{\tiny ACT}}$ enthaltenen aktuellen Moduls gekn”upft wurden, erf”ullt sind. conditions ist eine Menge von Gentzen-Klauseln, die aus den semantischen Bedingungen nach Ersetzen der formalen durch die aktuellen Parameter hervorgegangen ist. ${\it mod\/}_{\mbox{\tiny ACT-AV}}$ ist eine Variante des aktuellen Moduls, in der die Sichtbarkeit der Signaturnamen an die Sichtbarkeit innerhalb des formalen Moduls angepa”st worden ist. Setze $(,,,,,{\it varsortfunc\/}_{\mbox{\tiny const,FORM}},{\it varsortfunc\/}_{\mbox{\tiny non-const,FORM}},,):={\it mod\/}_{\mbox{\tiny FORM}}$ Setze $(,,,,,{\it varsortfunc\/}_{\mbox{\tiny const,ACT-AV}},{\it varsortfunc\/}_{\mbox{\tiny non-const,ACT-AV}},,{\it goals\/}_{\mbox{\tiny ACT-AV}}):={\it mod\/}_{\mbox{\tiny ACT-AV}}$ $\begin{array}[]{@{}l@{}l@{}l}{\it varsortfunc\/}^{\prime}_{\mbox{\tiny FORM}}&:={\it varsortfunc\/}_{\mbox{\tiny const,FORM}}&\cup\ {\it varsortfunc\/}_{\mbox{\tiny non-const,FORM}}\par\\\ {\it varsortfunc\/}^{\prime}_{\mbox{\tiny ACT-AV}}&:={\it varsortfunc\/}_{\mbox{\tiny const,ACT-AV}}&\cup\ {\it varsortfunc\/}_{\mbox{\tiny non-const,ACT-AV}}\end{array}$ F”ur alle Gentzenklauseln ${\it condition\/}\in{\it conditions\/}$ Falls es nicht eine Gentzenklausel ${\it goal\/}\in{\it goals\/}_{\mbox{\tiny ACT-AV}}$ und eine Variablensubstitution sub gibt mit: * * $\mbox{\it sub}({\it goal\/})={\it condition\/}$ (Die Marken werden hier nicht ber”ucksichtigt), * * sub ist “sortenrein”, d h. f”ur alle $(x,y)\in\mbox{\it sub}$ gilt ${\it varsortfunc\/}^{\prime}_{\mbox{\tiny ACT-AV}}(x)={\it varsortfunc\/}^{\prime}_{\mbox{\tiny FORM}}(y)$, * * sub substituiert Konstruktor-Variablen mit Konstruktor-Variablen und Non- Konstruktor-Variablen mit Non-Konstruktor-Variablen, d. h. f”ur alle $(x,y)\in\mbox{\it sub}$ gilt $x\in\mbox{\sf Dom}({\it varsortfunc\/}_{\mbox{\tiny const,ACT-AV}})\Longleftrightarrow y\in\mbox{\sf Dom}({\it varsortfunc\/}_{\mbox{\tiny const,FORM}})$ und * * das zu goal korrespondierende Beweisziel in ${\it nform}_{\mbox{\tiny ACT}}$ (kann mit Hilfe des Markennamens bestimmt werden) gilt dort als bewiesen, dh. es gibt einen entsprechenden Beweis in prove-db. SEMANTIC ERROR: Bevor die Spezifikation akzeptiert werden kann mu”s (falls noch nicht vorhanden) ein entsprechendes Beweisziel in das aktuelle Modul eingef”ugt und dessen G”ultigkeit bewiesen werden. R”uckgabewert: - Bind: $\begin{array}[t]{l}\mbox{\sf NF}\ \times\ \mbox{\sf RENAMING-FUNC}\ \times\ \mbox{\sf NF}\ \times\ \mbox{\sf PROVE-DB}\\\ \longrightarrow\mbox{\sf NF}\end{array}$ ${\it Bind\/}(({\it nform}_{\mbox{\tiny FORM}},{\it binding\/},({\it mod\/}_{\mbox{\tiny ACT}},{\it originf\/}_{\mbox{\tiny ACT}},{\it depf\/}_{\mbox{\tiny ACT}})$, prove-db) f”uhrt die Bindung eines Parametertupels durch. ${\it nform}_{\mbox{\tiny FORM}}$ enth”alt das normalisierte, parametrisierte Modul, dessen Parameter nach der Vorschrift binding an Namen des normalisierten Moduls ${\it mod\/}_{\mbox{\tiny ACT}}$ gebunden werden sollen. * /* Zun”achst werden alle Namen aus $({\it mod\/}_{\mbox{\tiny ACT}},{\it originf\/}_{\mbox{\tiny ACT}},{\it depf\/}_{\mbox{\tiny ACT}})$ verdeckt. / ${\it nform}^{\prime}_{\mbox{\tiny ACT}}\ :=\ {\it Hide\/}(({\it mod\/}_{\mbox{\tiny ACT}},{\it originf\/}_{\mbox{\tiny ACT}},{\it depf\/}_{\mbox{\tiny ACT}}),\emptyset)$ /* AdaptVisibility ”andert die Sichtbarkeit von Namen aus verschiedenen Modulen nach dem Prinzip der “maximalen” Sichtbarkeit. Angewandt auf ${\it nform}_{\mbox{\tiny FORM}}$ und ${\it nform}^{\prime}_{\mbox{\tiny ACT}}$ bleibt ${\it nform}_{\mbox{\tiny FORM}}$ unver”andert, weil es dort keinen Signaturnamen gibt, der in ${\it nform}^{\prime}_{\mbox{\tiny ACT}}$ sichtbar ist. / ${\it nforms}\ :=\ {\it AdaptVisibility\/}(\\{{\it nform}^{\prime}_{\mbox{\tiny ACT}},{\it nform}_{\mbox{\tiny FORM}}\\},\\{\mbox{``{\tt label}''},\mbox{``{\tt variable}''},\mbox{``{\tt sort}''}\\})$ $\\{({\it mod\/}_{\mbox{\tiny ACT-AV}},{\it originf\/}_{\mbox{\tiny ACT- AV}},{\it depf\/}_{\mbox{\tiny ACT-AV}})\\}\ :=$ ${\it AdaptVisibility\/}({\it nforms},\\{\mbox{``{\tt function}''}\\})\quad\backslash\quad\\{{\it nform}_{\mbox{\tiny FORM}}\\}$ $(({\it mod\/}_{\mbox{\tiny FORM}},{\it originf\/}_{\mbox{\tiny FORM}},{\it depf\/}_{\mbox{\tiny FORM}}),(sig_{p},{\it conditions\/}),{\it paradefmod\/})\ :=$ ${\it SeparateParaBlock\/}({\it nform}_{\mbox{\tiny FORM}},\mbox{\sf Dom}({\it binding\/}))$ $\mbox{\it par-renaming}:={\it GetParameterRenamings\/}({\it sig\/}_{p},{\it binding\/},{\it originf\/}_{\mbox{\tiny ACT}},{\it originf\/}_{\mbox{\tiny ACT-AV}})$ Berechne ${\it mod\/}^{\prime}_{\mbox{\tiny FORM}},{\it originf\/}^{\prime}_{\mbox{\tiny FORM}}$ und ${\it conditions\/}^{\prime}$ aus ${\it mod\/}_{\mbox{\tiny FORM}},{\it originf\/}_{\mbox{\tiny FORM}}$ und ${\it conditions\/}$ durch Ersetzen der formalen Parameter nach Ma”sgabe von par-renaming. SPEZIFIKATIONSFEHLER falls ${\it mod\/}^{\prime}_{\mbox{\tiny FORM}}$ keine korrekte Signatur enth”alt. /* Eine fehlerhafter Parameterbindung kann dazu f”uhren, da”s Funktionen mit gleichen disambiguierten Namen und unterschiedlichen Zielsorten erzeugt werden. / ${\it CheckSemanticConditions\/}(\begin{array}[t]{@{}l}{\it conditions\/}^{\prime},mod^{\prime}_{\mbox{\tiny FORM}},{\it mod\/}_{\mbox{\tiny ACT-AV}},\\\ ({\it mod\/}_{\mbox{\tiny ACT}},{\it originf\/}_{\mbox{\tiny ACT}},{\it depf\/}_{\mbox{\tiny ACT}}),\mbox{\it prove-db\/})\end{array}$ ${\it nform}_{\mbox{\tiny result}}\ :=\ $ ${\it CombineWithActModule\/}(\begin{array}[t]{@{}l}({\it mod\/}^{\prime}_{\mbox{\tiny FORM}},{\it originf\/}^{\prime}_{\mbox{\tiny FORM}},{\it depf\/}_{\mbox{\tiny FORM}}),{\it paradefmod\/},\\\ ({\it mod\/}_{\mbox{\tiny ACT-AV}},{\it originf\/}_{\mbox{\tiny ACT-AV}},{\it depf\/}_{\mbox{\tiny ACT-AV}}))\end{array}$ R”uckgabewert: ${\it nform}_{\mbox{\tiny result}}$ #### 6.2.4 Die Normalisierungsfunktionen NF und NormalForm Ziel dieses Abschnitts ist die Vorstellung einer Funktion NormalForm, die eine gegebene hierarchische ASF+-Spezifikation in eine flache, nur aus einem Topmodul bestehende ASF+-Spezifikation transformiert. NormalForm besteht im wesentlichen aus einem Aufruf der rekursiven Funktion NF. NF ist die f”ur das Verst”andnis des Algorithmus grundlegende Funktion. Weiterhin werden die Trivialfunktionen ModuleText, MakeGF und ExternModRep ben”otigt. ModuleText: $\begin{array}[t]{l}\mbox{\sf MODULE-NAME}\ \times\ \mbox{\sf ASF- SPEC}\\\ \longrightarrow\mbox{\sf ASF-MODULE}\end{array}$ ModuleText(modname, spec) sucht ein Modul asf-module namens modname in spec. Falls kein solches Modul existiert: SPEZIFIKATIONSFEHLER! ModuleText ist Hilfsfunktion von NF. Weitere Formalisierung entf”allt. R”uckgabewert: asf-module MakeGF: $\begin{array}[t]{l}\mbox{\sf ASF-MODULE}\\\ \longrightarrow\mbox{\sf GF}\end{array}$ MakeGF(asf-module) berechnet aus einem isolierten nicht notwendig importfreien ASF+-Modul einer Spezifikation eine general form (mod, originf, depf). MakeGF ist Hilfsfunktion von NF. * Der Wert von mod wird direkt aus dem ASF-Modul ermittelt, es handelt sich hier lediglich um eine andere Repr”asentationsform. Jedem (disambiguierten) Sorten- und Funktionsnamen aus der Signatur, jedem (disambiguierten) Parameternamen aus einer der Parametersignaturen und jedem, innerhalb des Moduls auftretenen (disambiguierten) Variablen- und Markennamen wird vermittels originf ein Origin zugeordnet.666Siehe dazu Seite 6.1 originf ist zun”achst partiell in dem Sinn, da”s importierte (Teil-) Signaturen noch nicht in Dom(originf) enthalten sind. ${\it depf\/}\ :=\emptyset$ R”uckgabewert: (mod, originf, depf) NF: $\begin{array}[t]{l}\mbox{\sf MODULE-NAME}\ \times\ \mbox{\sf ASF-SPEC}\ \times\ \mbox{\sf PROVE-DB}\\\ \longrightarrow\mbox{\sf NF}\end{array}$ NF(modname, spec, prove-db) berechnet rekursiv die Normalform ${\it nform}_{\mbox{\tiny result}}$ der zum Modul namens modname zugeh”origen general form. * $\mbox{\it importing-gf}\ :=\ {\it MakeGF\/}({\it ModuleText\/}({\it modname\/},{\it spec\/}))$ Setze $((,{\it imports\/},\ldots),,):=\mbox{\it importing-gf}$ Sei $\\{({\it modname\/}_{i},{\it iname\/}_{i},{\it visibilityf\/}_{i},{\it renaming\/}_{i},{\it bindingblocks\/}_{i})\quad\|\quad i\in A\\}\ =\ {\it imports\/}$ F”ur alle $i\in A$ $\begin{array}[]{@{}ll}{\it nform}_{i}&:=\ {\it NF\/}({\it modname\/}_{i},{\it spec\/},\mbox{\it prove-db\/})\\\ {\it nform}^{\prime}_{i}&:=\ {\it Hide\/}({\it nform}_{i},{\it visibilityf\/}_{i})\end{array}$ Falls ${\it iname\/}_{i}=\emptyset\ \wedge\ ({\it renaming\/}_{i}\neq\emptyset\ \vee\ {\it bindingblocks\/}_{i}\neq\emptyset)$ * SPEZIFIKATIONSFEHLER Falls ${\it iname\/}_{i}\neq\emptyset$ * $\begin{array}[]{@{}ll}{\it nform}^{\prime\prime}_{i}&:=\ {\it Instanciate\/}({\it nform}^{\prime}_{i},{\it renaming\/}_{i},{\it bindingblocks\/}_{i},{\it iname\/}_{i})\\\ {\it nform}^{\prime\prime\prime}_{i}&:=\ {\it Rename\/}({\it nform}^{\prime\prime}_{i},{\it renaming\/}_{i})\end{array}$ F”ur alle $({\it binding\/},{\it modname\/}_{\mbox{\tiny ACT}})\in{\it bindingblocks\/}_{i}$ wiederhole * $\begin{array}[]{@{}ll}{\it nform}_{\mbox{\tiny ACT}}&:=\ {\it NF\/}({\it modname\/}_{\mbox{\tiny ACT}},{\it spec\/},\mbox{\it prove-db\/})\\\ {\it nform}^{\prime\prime\prime}_{i}&:=\ {\it Bind\/}({\it nform}^{\prime\prime\prime}_{i},{\it binding\/},{\it nform}_{\mbox{\tiny ACT}},\mbox{\it prove-db\/})\end{array}$ ${\it nform}_{\mbox{\tiny result}}:={\it Combine\\-With\\-Imports\/}(\mbox{\it importing-gf},{\it CombineImports\/}(\\{{\it nform}^{\prime\prime\prime}_{i}\ \|\ i\in A\\}))$ R”uckgabewert: ${\it nform}_{\mbox{\tiny result}}$ ExternModRep: $\begin{array}[t]{l}\mbox{\sf MODULE}\\\ \longrightarrow\mbox{\sf ASF-MODULE}\end{array}$ ExternModRep(module) berechnet die ASF+-Darstellung asf-module des Moduls module. Diese Funktion kann mit einer Option ausgestattet werden, die es erlaubt ”uberladene Funktionsnamen durch eindeutige Repr”asentationen ihrer disambiguierten Namen zu ersetzen. ExternModRep ist Hilfsfunktion von NormalForm. * Weitere Formalisierung entf”allt! R”uckgabewert: asf-module NormalForm: $\begin{array}[t]{l}\mbox{\sf ASF-SPEC}\ \times\ \mbox{\sf PROVE- DB}\\\ \longrightarrow\mbox{\sf ASF-SPEC}\end{array}$ NormalForm(spec, prove-db) berechnet aus einer modularen ASF+-Spezifikation eine Spezifikation, bestehend aus einem einzigen (importfreien) Modul asf- module. Die Wissensbasis prove-db beinhaltet Informationen ”uber gelungene Beweise und wird f”ur die ”Uberpr”ufung von semantischen Bedingungen gebraucht. * Sei modname der Name des Topmoduls aus spec. (mod, originf, depf) := NF(modname, spec, prove-db) asf-module := ExternModRep(mod) R”uckgabewert: (asf-module, $\emptyset$) ### 6.3 Ein Beispiel f”ur ein normalisiertes Modul Um die Arbeitsweise des Normalformalgorithmus zu veranschaulichen geben wir schlie”slich noch das importfreie, durch Normalisierung erzeugte Modul OrdNatSequences.nf an. module OrdNatSequences.nf { add signature { public: sorts BOOL, NAT, NSEQ constructors true, false : -> BOOL 0 : -> NAT s : NAT -> NAT Nnil : -> NSEQ cons : NAT # NSEQ -> NSEQ non-constructors greater : NAT # NAT -> BOOL greater : NSEQ # NSEQ -> BOOL private: non-constructors Bo-and, Bo-or : BOOL # BOOL -> BOOL Bo-not : BOOL -> BOOL _ Nat-+ _ : NAT # NAT -> NAT Nat-eq : NAT # NAT -> BOOL ONat-geq : NAT # NAT -> BOOL } variables { constructors Nat-x, Nat-y, Nat-u, ONat-x, ONat-y, ONat-u, ONat-v, OSeq-i1, OSeq-i2, OSeq-i3 : -> NAT OSeq-seq1, OSeq-seq2, OSeq-s1, OSeq-s2 : -> NSEQ non-constructors Bo-x, Bo-y : -> BOOL } equations { macro-equation Bo-and(Bo-x,Bo-y) { case { ( Bo-x @ true ) : Bo-y ( Bo-x @ false ): false } } macro-equation Bo-not(Bo-x) { case { ( Bo-x @ true ) : false ( Bo-x @ false ): true } } [Bo-e1] Bo-or(Bo-x, Bo-y) = Bo-not(Bo-and(Bo-not(Bo-x), Bo-not(Bo-y))) macro-equation (Nat-x Nat-+ Nat-y) { case { ( Nat-y @ 0 ) : Nat-x ( Nat-y @ s(Nat-u) ) : s(Nat-x Nat-+ Nat-u) } macro-equation Nat-eq(Nat-x, Nat-y) { if ( Nat-x = Nat-y ) true else false } } macro-equation greater(ONat-x, ONat-y) { case { ( ONat-x @ 0 ) : false ( ONat-x @ s(ONat-u), ONat-y @ 0 ) : true ( ONat-x @ s(ONat-u), ONat-y @ s(ONat-v) ): greater(ONat-u,ONat-v) } } [ONat-e1] ONat-geq(ONat-x,ONat-y) = Bo-or(greater(ONat-x,ONat-y), eq(ONat-x,ONat-y)) macro-equation greater(OSeq-seq1, OSeq-seq2) { /* lex-order of sequences / case { ( OSeq-seq1 @ Nnil ) : false ( OSeq-seq1 @ cons(OSeq-i1, OSeq-s1), OSeq-seq2 @ Nnil ): true ( OSeq-seq1 @ cons(OSeq-i1, OSeq-s1), OSeq-seq2 @ cons(OSeq-i2, OSeq-s2) ): if ( greater(OSeq-i1, OSeq-i2) ) true else if ( OSeq-i1 = OSeq-i2 ) greater(OSeq-s1, OSeq-s2) else false } } } goals { [ONat-irref] greater(ONat-x, ONat-x) --> [ONat-trans] greater(ONat-x, ONat-u), greater(ONat-u, ONat-y) --> greater(ONat-x, ONat-y) [ONat-total] --> greater(ONat-x, ONat-y), greater(ONat-y, ONat-x), ONat-x = ONat-y } } / OrdNatSequences.nf */ ## 7 Abschlie”sende Zusammenfassung Mit ASF+ ist es gelungen, eine algebraische Spezifikationssprache zu entwickeln, die neue Konzepte wie beispielsweise das differenzierte Verdecken von Signaturnamen, semantische Bedingungen an Parameter und die Angabe von Beweiszielen in sich vereint, ohne dabei auf wesentliche Elemente der bereits existierenden Sprache ASF verzichten zu m”ussen. Hierbei konnte die Syntax von ASF sogar noch vereinfacht werden. ASF+ ist jedoch mehr als eine nur um zus”atzliche Konstruke erweiterte Version von ASF. Grunds”atzliche Untersuchungen (wie in Kapitel 4 dargestellt) deckten Fehler in der Semantik von ASF auf und f”uhrten zu den Begriffsbildungen “benutzender” und “kopierender Import”. W”ahrend der benutzende Import aus ASF ”ubernommen wurde, verhindern in ASF+ von den kopierenden Importbefehlen zur Verf”ugung gestellte Instanzbezeichnungen Namensverwechselungen zwischen dem manipulierten Modul und seinem Original. Wesentlicher Bestandteil von ASF+ ist das Namensraumkonzept, welches jedem Signaturnamen bei seiner Definition den Modulnamen zuordnet. W”ahrend beim benutzenden Import der Namensraum unver”andert bleibt, f”uhrt der kopierende Import eines Namens zur Instanziierung des zugeordneten Namensraumes. Bei der Kombination mehrerer Module zu einem Normalformmodul werden nur solche Namen identifiziert, die dem gleichen Namensraum angeh”oren. Das Namensraumkonzept spielt auch in der Semantik verdeckter Namen eine wichtige Rolle. Jedem zu verdeckenden Namen wird im Zuge der Normalisierung die (abgek”urzte) Namensraumbezeichnung vorangestellt. Dies erh”oht die Verst”andlichkeit des erzeugten Normalformmoduls und macht den modularen Aufbau der Spezifikation sichtbar. Der Preis f”ur die Verbesserungen ist jedoch eine gewisse Verkomplizierung der Normalisierungsprozedur, was beim Vergleich des im Kapitel 6 vorgestellten Algorithmus mit dem aus [Bergstra&al.89] (Seite 23-28) deutlich wird. Schlie”slich erlauben die von uns entwickelten Strukturdiagramme eine ebenso informative wie leicht verst”andliche Darstellung von ASF+-Spezifikationen. Diese Strukturdiagramme eignen sich dar”uberhinaus auch dazu, ein korrektes intuitives Verst”andnis f”ur die wesentlichen Konzepte der Normalisierungsprozedur — wie Originfunktion, Dependenzfunktion, Sichtbarkeitsanpassung, Renaming, Parameterbindung, Namensrauminstanziierung, etc. — zu vermitteln. Literatur [Bergstra&al.89] \| J. A. Bergstra, J. Heering, P. Klint (1989). ---\|--- \| Algebraic Specification. \| ACM Press. [Eschbach94] \| Robert Eschbach (1994). \| ART — Modularisierung von \| Induktionsbeweisen ”uber Gleichungsspezifikationen. \| SEKI-WORKING-PAPER SWP–94–03 (SFB), \| Fachbereich Informatik, Universität Kaiserslautern, \| D–67663 Kaiserslautern. [Hendriks91] \| P. R. H. Hendriks (1991). \| Implementation of Modular Algebraic Specifications. \| PhD. Thesis, \| CWI (Centrum voor Wiskunde en Informatica), Amsterdam. [Wirth&Gramlich93] \| Claus-Peter Wirth, Bernhard Gramlich (1993). \| A Constructor-Based Approach for \| Positive/Negative-Conditional Equational Specifications. \| 3${}^{\mbox{\tiny rd}}$ CTRS 1992, LNCS 656, Seiten 198-212, Springer- Verlag. \| ”Uberarbeitete und erweiterte Version in: \| J. Symbolic Computation (1994) 17, Seiten 51-90, \| Academic Press. [Wirth&Gramlich94] \| Claus-Peter Wirth, Bernhard Gramlich (1994). \| On Notions of Inductive Validity \| for First-Order Equational Clauses. \| 12${}^{\mbox{\tiny th}}$ CADE 1994, LNAI 814, Seiten 162-176, Springer- Verlag. [Wirth&Lunde94] \| Claus-Peter Wirth, R”udiger Lunde (1994). \| Writing Positive/Negative-Conditional Equations \| Conveniently. \| SEKI-WORKING-PAPER SWP–94–04 (SFB), \| Fachbereich Informatik, Universität Kaiserslautern, \| D–67663 Kaiserslautern.	arxiv-papers	2009-02-17T20:52:11	2024-09-04T02:49:00.650752	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ruediger Lunde, Claus-Peter Wirth", "submitter": "Claus-Peter Wirth", "url": "https://arxiv.org/abs/0902.2995" }
0902.3122	# Soft interaction at high energy and N=4 SYM E. Levina, and I. Potashnikovab a) Department of Particle Physics, School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Science Tel Aviv University, Tel Aviv, 69978, Israel E-mail address: b) Departamento de Física y Centro de Estudios Subatómicos, Universidad Técnica Federico Santa María, Avda. España 1680, Casilla 110-V, Valparaíso, Chile E-mail address: leving@post.tau.ac.il irina.potashnikova@usm.cl ###### Abstract: In this paper we show that the N=4 SYM total cross section violates the Froissart theorem, and in the huge range of energy this cross section is proportional to $s^{1/3}$. The graviton reggeization will change this increase to the normal logarithmic behavior $\sigma\propto\ln^{2}s$. However, we demonstrated that this happens at ultra high energy, much higher than the LHC energy. In the region of accessible energy we need to assume that there is a different source for the total cross section, with the value of the cross section about 40 mb. With this assumption we successfully describe $\sigma_{tot},\sigma_{el}$ and $\sigma_{diff}$ for the accessible range of energy from the fixed target Fermilab to the Tevatron energies. It turns out that the N=4 SYM mechanism can be responsible only for a small part of the inelastic cross section for this energy region (about $2mb$). However, at the LHC energy the N=4 SYM theory can describe the multiparticle production with $\sigma_{in}\approx 30\,mb$. The second surprise is the fact that the total cross section and the diffraction cross section can increase considerably from the Tevatron to the LHC energy. The bad description of $B_{el}$ gives the strong argument that the non N=4 SYM background should depend on energy. We believe that we have a dilemma: to find a new mechanism for the inelastic production in the framework of N=4 SYM other than the reggeized graviton interaction, or to accept that N=4 SYM is irrelevant to any experimental data that has been measured before the LHC era. N=4 SYM, graviton reggeization, eikonal approach ††preprint: TAUP 2891/08 ## 1 Introduction At the moment N=4 SYM is the unique theory which allows us to study theoretically the regime of the strong coupling constant [1] . Therefore, in principle, considering the high energy scattering amplitude in N=4 SYM, we can guess which physics phenomena could be important in QCD, in the limit of the strong coupling. The attractive feature of this theory, is that N=4 SYM with small coupling leads to normal QCD like physics (see Refs. [2, 3]) with OPE and linear equations for DIS as well as the BFKL equation for the high energy amplitude. The high energy amplitude reaches the unitarity limit: black disc regime, in which half of the cross section stems from the elastic scattering and half relates to the processes of the multiparticle production. However, the physical picture in the strong coupling region turns out to be completely different [4, 5, 6, 6, 8, 9, 10], in the sense that there are no processes of the multiparticle production in this region, and the main contribution stems from elastic and quasi-elastic ( diffractive) processes when the target (proton) either remains intact, or is slightly excited. Such a picture not only contradicts the QCD expectations [11, 12, 13, 14, 15, 16], but also contradicts available experimental data. On the other hand, the main success of N=4 SYM has been achieved in the description of the multiparticle system such as the quark-gluon plasma and/or the multiparticle system at fixed temperature [17, 18, 19, 20]. Therefore, we face a controversial situation: we know a lot about something that cannot be produced. The goal of this paper is to evaluate the scale of the disaster, comparing the predictions of the N=4 SYM with the experimental data. We claim that at least half of the total cross section at the Tevatron energy has to stem from a different source than the N=4 SYM. Before discussing predictions of the N=4 SYM for high energy scattering, we would like to draw the reader’s attention that there exists two different regions of energy that we have to consider in N=4 SYM: $(2/\sqrt{\lambda})\,\alpha^{\prime}s<1$ and $(2/\sqrt{\lambda})\,\alpha^{\prime}s>1$ ($\lambda=\,4\pi g_{s}N_{c}$ where $g_{s}$ is the string coupling and $N_{c}$ is the number of colors). In the first region, the multiparticle production has a very small cross section, and it can be neglected. However, in the second region the graviton reggeization leads to the inelastic cross section that is rather large, and at ultra high energies the scattering amplitude reveals all of the typical features of the black disc regime: $\sigma_{el}=\sigma_{tot}/2$ and $\sigma_{in}=\sigma_{tot}/2$. Therefore, the formulation of the main result of this paper is the following: at the accessible energies the amplitude is in the first region, and at least half of the total cross section at the Tevatron energy has to stem from a different source than the N=4 SYM. However, at the LHC energy the N=4 SYM mechanism can be responsible for about 2/3 of the total cross section and, perhaps, at the LHC the final states will be produced with the typical properties of the N=4 SYM. ## 2 High energy Scattering in N=4 SYM ### 2.1 Eikonal formula The main contribution to the scattering amplitude at high energy in N=4 SYM, stems from the exchange of the graviton**Actually, the graviton in this theory is reggeized [5], but it is easy to take this effect into account (see Refs. [5, 7, 4]) and Eq. (2.9) below. . The formula for this exchange has been written in Ref.[6, 8, 10]. In $AdS_{5}=AdS_{d+1}$ space this amplitude has the following form (see Fig. 1) $A_{1GE}(s,b;z,z^{\prime})\,\,=\,\,g^{2}_{s}\,\,T_{\mu\nu}\left(p_{1},p_{2}\right)G_{\mu\nu\mu^{\,\prime}\nu^{\,\prime}}\left(u\right)\,T_{\mu^{\,\prime}\nu^{\,\prime}}\left(p_{1},p_{2}\right)\,\,\xrightarrow{s\gg\mu^{2}}g^{2}_{s}\,s^{2}z^{2}z^{\prime 2}\,G_{3}\left(u\right)$ (2.1) Figure 1: The one graviton (1GE) exchange. where $T_{\mu,\nu}$ is the energy-momentum tensor, and $G$ is the propagator of the massless graviton. The last expression in Eq. (2.1), reflects the fact that for high energies, $T_{\mu,\nu}=p_{1,\mu}p_{1,\nu}$ and at high energies the momentum transferred $q^{2}\,\to q^{2}_{\perp}$ which led to $G_{3}\left(u\right)$ (see Refs.[6, 10]). In $AdS_{5}$ the metric has the following form $ds^{2}\,\,=\,\,\frac{L^{2}}{z^{2}}\,\left(\,dz^{2}\,\,+\,\,\sum^{d}_{i=1}dx^{2}_{i}\right)\,=\,\frac{L^{2}}{z^{2}}\,\left(\,dz^{2}\,+\,d\vec{x}^{2}\right)$ (2.2) and $u$ is a new variable which is equal to $u\,\,=\,\,\frac{(z-z^{\prime})^{2}+(\vec{x}-\vec{x}^{\prime})^{2}}{2\,z\,z^{\prime}}\,\,=\,\,\frac{(z-z^{\prime})^{2}+b^{2}}{2\,z\,z^{\prime}}\,\,$ (2.3) and $G_{3}\left(u\right)\,\,=\,\,\frac{1}{4\pi}\,\frac{1}{\left\\{1+u+\sqrt{u(u+2)}\right\\}^{2}\,\sqrt{u(u+2)}}$ (2.4) where $b$ is the impact parameter (see Fig. 1). As one can see from Eq. (2.1) the one graviton exchange amplitude is real. As has been discussed [5] the graviton reggeization leads to a small imaginary part, and the amplitude can be re-written in the form [5, 10] $\tilde{A}_{1GE}(s,b;z,z^{\prime})\,\,\equiv\,\,\frac{A_{1GE}(s,b;z,z^{\prime})}{s}\,\,=\,\,g^{2}_{s}\,\left(1+i\rho\right)\,s\,z\,z^{\prime}\,G_{3}\left(u\right)$ (2.5) where $\rho\,=\,2/\sqrt{\lambda}\,\ll\,1$. $\tilde{A}_{1GE}$ steeply increases with energy $s$ and has to be unitarized using the eikonal formula [6, 7, 10] $A_{eikonal}\left(s,b;z,z^{\prime}\right)\,\,=\,\,i\left(\,1\,\,\,-\,\,\,\exp\left(i\,\tilde{A}_{1GE}\left(s,b;{Eq.~{}(\ref{N42})}\right)\right)\right)$ (2.6) In Ref. [10] it was argued that AdS/CFT correspondence leads to the corrections to Eq. (2.6) which are small $(\propto 2/\sqrt{\lambda})$. The unitarity constraints for Eq. (2.6) has the form $2\,\mbox{Im}{\cal A}_{eikonal}\left(s,b;z,z^{\prime}\right)\,\,\,=\,\,\,\|A_{eikonal}\left(s,b;z,z^{\prime}\right)\|^{2}\,\,\,+\,\,{\cal O}\left(\frac{2}{\sqrt{\lambda}}\right)$ (2.7) Figure 2: The diagrams for nucleon-nucleon interaction in N=4 SYM. Fig. 2-a and Fig. 2-b show the exchange of one and two gravitons that are included in the eikonal formula of Eq. (2.6), while other diagrams give the examples of corrections to the eikonal formula. The eikonal formula of Eq. (2.6) as well as the unitarity constraint of Eq. (2.7) are illustrated in Fig. 2. One can see that the diagrams shown in this figure have the following contributions: $A\left({Fig.~{}\ref{din4}}-a\right)\,\propto\,g^{2}_{s}\,s\,\,\approx\,\frac{s}{N^{2}_{c}}\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,A\left({Fig.~{}\ref{din4}}-b\right)\,\propto\,\left(g^{2}_{s}\,s\right)^{2}\,\,\approx\,\left(\frac{s}{N^{2}_{c}}\right)^{2}\,;$ (2.8) $A\left({Fig.~{}\ref{din4}}-c\right)\,\propto\,g^{2}_{s}\,\left(g^{2}_{s}\,s\right)^{2}\,\,\approx\,\frac{1}{N^{2}_{c}}\left(\frac{s}{N^{2}_{c}}\right)^{2}\,;\,\,\,\,\,\,\,\,A\left({Fig.~{}\ref{din4}}-d\,\,\mbox{and}\,\,{Fig.~{}\ref{din4}}-e\right)\,\propto\,\frac{2}{\sqrt{\lambda}}\,\left(\frac{s}{N^{2}_{c}}\right)^{2}.$ Therefore, the contributions that lead to a violation of the eikonal formula are small, at least as small as $2/\sqrt{\lambda}$. It is interesting to notice that actually they stem from the processes of the diffraction dissociation (see Fig. 2-e rather than from the processes of the multiparticle productions (see Fig. 2-c). Eq. (2.5) provides the simple method to take into account the reggeization of the graviton, in order to understand the main property of the scattering amplitude. However in our description of the experimental data, we will use the exact form of the amplitude for the exchange of the reggeized graviton (see Refs. [5, 10]), namely, $\tilde{A}_{1GE}(s,b;z,z^{\prime})\,\,=\,\,g^{2}_{s}\,(1+i\rho)\,\,\frac{1}{4\pi}\,\frac{\,\left(z\,z^{\prime}s\right)^{1-\rho}}{\sqrt{u(u+2)}}\,\sqrt{\frac{\rho}{\pi\,\ln\left(s\,z\,z^{\prime}\right)}}\,\exp\left(-\frac{\ln^{2}\left(1+u+\sqrt{u(u+2)}\right)}{\rho\,\ln\left(s\,z\,z^{\prime}\right)}\right)$ (2.9) Eq. (2.9) gives the description of one reggeized graviton in the limit $s\to\infty$ with $\lambda\,\gg\,1$ while the simple formula of Eq. (2.5) describes the one graviton exchange for $\lambda\to\infty$ but $s\gg 1/\alpha^{\prime}$. ### 2.2 Nucleon-nucleon high energy amplitude Discussing the hadron interaction at high energy, we need to specify the correct degrees of freedom that diagonalize the interaction matrix. We assume that a nucleon consists of $N_{c}$ quarks ($N_{c}$ colorless dipoles) that interact with each other with the eikonal formula of Eq. (2.6), namely, $\displaystyle A_{NN}\left(s,b\right)\,\,\,$ $\displaystyle=$ $\displaystyle\,\,\,\int\,dz\,dz^{\prime}\,\prod^{N_{c}}_{i=1}\,d^{2}r_{i}\prod\,\|\Psi\left(r_{i},z\right)]^{2}\,\,\prod^{N_{c}}_{i=1}\,d^{2}r^{\prime}_{i}\prod\,\|\Psi\left(r_{i},z^{\prime}\right)]^{2}\,$ (2.10) $\displaystyle\times$ $\displaystyle\,\,\,i\,\left(1\,\,-\,\,\,\,\,\exp\left(i\,N^{2}_{c}\,\tilde{A}_{1GE}\left(s,b;z,z^{\prime}\|{Eq.~{}(\ref{N42})}\right)\right)\right)$ $\displaystyle=$ $\displaystyle\,\,i\,\int\,dz\,dz^{\prime}\,\Phi\left(z\right)\,\Phi\left(z^{\prime}\right)\,\,\left(1\,\,-\,\,\exp\left(i\,g^{2}N^{2}_{c}\,(1+i\rho)\,z^{2}\,z^{\prime 2}\,G_{3}\left(u\right)\right)\right)$ where $\Phi\left(z\right)\,\,=\,\,\int\,d^{2}r\prod\,\|\Psi\left(r,z\right)\|^{2}\,\,$ (2.11) and $\rho=2/\sqrt{\lambda}$. In Eq. (2.10) the only unknown ingredient is $\Psi\left(r_{i},z\right)$. We can reconstruct this wave function using the Witten formula [21], namely, $\displaystyle\Psi\left(r,z\right)\,\,\,=$ (2.12) $\displaystyle\frac{\Gamma\left(\Delta\right)}{\pi\,\Gamma\left(\Delta-1\right)}\,\,\int\,d^{2}r^{\prime}\,\left(\frac{z}{z^{2}\,\,+\,\,(\vec{r}\,-\,\vec{r}^{\prime})^{2}}\right)^{\Delta}\,\,\Psi\left(r^{\prime}\right)\,\,\,\mbox{with}\,\,\,\,\Delta_{\pm}\,\,=\,\,\frac{1}{2}\left(d\,\,\pm\,\,\sqrt{d^{2}+4\,m^{2}}\right)$ where $\Psi\left(r^{\prime}\right)$ is the wave function of the dipole inside the proton on the boundary. For simplicity and to make all calculations more transparent, we choose $\Psi\left(r^{\prime}\right)=K_{0}\left(Qr^{\prime}\right)$. The value of the parameter $Q$ can be found from the value of the electromagnetic radius of the proton ($Q\,\approx 0.3\,GeV^{-1}$). In this presentation, we follow the formalism of Ref. [10], namely using the formulae 3.198, 6.532(4), 6.565(4) and 6.566(2) from the Gradstein and Ryzhik Tables, Ref. [22]. Introducing the Feynman parameter ($t$), we can rewrite Eq. (2.12) in the form $\displaystyle\Psi\left(r,z\right)\,\,\,=\,\,\frac{\Gamma\left(\Delta\right)}{\pi\,\Gamma\left(\Delta-1\right)}\,\int\,\xi\,d\xi\,d^{2}\,r^{\prime}\frac{J_{0}\left(Q\,\xi\right)}{\xi^{2}\,+\,r^{\prime 2}}\,\left(\frac{z}{z^{2}\,\,+\,\,(\vec{r}\,-\,\vec{r}^{\prime})^{2}}\right)^{\Delta}\,\,=\,\frac{\Gamma\left(\Delta+1\right)}{\pi\,\Gamma\left(\Delta-1\right)}\,\frac{1}{B\left(1,\Delta\right)}$ $\displaystyle\times\,\,\int\xi\,d\xi\,d^{2}\,r^{\prime}\int^{1}_{0}\,\frac{dt}{z}\,t^{\Delta-1}\,(1-t)\,\,J_{0}\left(Q\,\xi\right)\,\left(\frac{z}{t\,z^{2}\,\,+\,\,t\,(\vec{r}\,-\,\vec{r}^{\prime})^{2}\,\,+\,\,(1-t)\,r^{\prime 2}\,+\,(1-t)\,\xi^{2}}\right)^{\Delta+1}\,\,$ $\displaystyle=\,\,\frac{\Gamma\left(\Delta+1\right)}{\pi\,\Delta\,\Gamma\left(\Delta-1\right)}\,\,\,z^{\Delta}\,\,\int\tilde{\xi}\,d\tilde{\xi}\,\int^{1}_{0}\,dt\,\frac{1}{(1-t)^{\Delta}}\,\,J_{0}\left(Q\,\sqrt{\frac{t}{1-t}}\,\tilde{\xi}\right)\,\left(\frac{1}{\,r^{2}\,\,\,+\,\,\kappa\left(t,z,\tilde{\xi}\right)}\right)^{\Delta}$ (2.13) with $\kappa\left(t,z,\xi\right)\,=\,\left(t\,z^{2}\,+\,\tilde{\xi}^{2}\right)/\left(1-t\right)$ and $\tilde{\xi}=\xi\left(\sqrt{1-t}/\sqrt{t}\right)$. The amplitude $\tilde{A}_{1GE}\left(s,b;z,z^{\prime}{Eq.~{}(\ref{N42})}\right)$ depends only on $z$ and $z^{\prime}$, and we need to find $\int\|\Psi\left(r,z\right)]^{2}d^{2}r$. From Eq. (2.2), one can see that we have to evaluate the integral $\displaystyle\pi\int\,dr^{2}\,\left(\frac{1}{\,r^{2}\,\,\,+\,\,\kappa\left(t,z,\tilde{\xi}\right)}\right)^{\Delta}\,\left(\frac{1}{\,r^{2}\,\,\,+\,\,\kappa\left(t^{\prime},z,\tilde{\xi}^{\prime}\right)}\right)^{\Delta}\,\,=$ $\displaystyle\,\,\pi\,B\left(1,2\Delta-1\right){}_{2}F_{1}\left(1,\Delta,2\Delta-1,1-\frac{\kappa\left(t,z,\tilde{\xi}\right)}{\kappa\left(t^{\prime},z,\tilde{\xi}^{\prime}\right)}\right)$ $\displaystyle\approx\,\,\pi\,\frac{1}{2\Delta-1}\,\frac{\kappa\left(t,z,\tilde{\xi}\right)}{\left(\kappa\left(t,z,\tilde{\xi}\right)\,\kappa\left(t^{\prime},z,\tilde{\xi}^{\prime}\right)\right)^{\Delta}}$ (2.14) where we used ${\bf 3.197}$ of Ref. [22]. In the last equation we assumed that $\kappa\left(t,z,\xi\right)/\kappa\left(t^{\prime},z,\xi^{\prime}\right)$ is close to unity, since the integral has a symmetry with respect to $\xi\to xi^{\prime}$, and $t\to t^{\prime}$. The simplified form allows us to reduce the integral for $\Phi(z)$ (see Eq. (2.11)), to the form $\displaystyle\Phi\left(z\right)\,\,$ $\displaystyle=$ $\displaystyle\,\,z^{2\Delta}\,\,\left(\frac{\Gamma\left(\Delta+1\right)}{\pi\Gamma\left(\Delta-1\right)}\right)^{2}\,\,\frac{\pi}{2\Delta-1}\,\int\,\tilde{\xi}\,d\tilde{\xi}\,J_{0}\left(Q\,\sqrt{\frac{t}{1-t}}\,\tilde{\xi}\right)\,\,d\,t\,\,\frac{1}{\left(t\,z^{2}\,+\,\,\tilde{\xi}^{2}\right)^{\Delta}}\,\,$ (2.15) $\displaystyle\times$ $\displaystyle\int\,\tilde{\xi^{\prime}}\,d\tilde{\xi}^{\prime}\,J_{0}\left(Q\,\sqrt{\frac{t^{\prime}}{1-t^{\prime}}}\tilde{\xi}^{\prime}\right)\,\,d\,t^{\prime}\,\,(1-t^{\prime})\,\frac{1}{\left(t^{\prime}\,z^{2}\,+\,(1-t)\,\tilde{\xi}^{\prime 2}\right)^{\Delta-1}}\,\,$ $\displaystyle=$ $\displaystyle\,\,\,\,\left(\frac{2\alpha^{\prime}}{\rho}\right)^{\Delta-2}\,\,\frac{(\Delta-1)^{2}}{\Gamma\left(\Delta\right)\,\Gamma\left(\Delta-1\right)\,\pi}\,\,\frac{2\,Q^{2}\,2^{3-2\Delta}\,z^{3}}{2\Delta-1}\,\,\,\,\int^{1}_{0}dt\,\left(\sqrt{\frac{t}{1-t}}\,Q\right)^{\Delta-1}\,K_{\Delta-1}\left(\sqrt{\frac{t}{1-t}}\,Q\,z\right)\,$ $\displaystyle\times$ $\displaystyle\,\int^{1}_{0}\,dt^{\prime}\,\left(\sqrt{\frac{t^{\prime}}{1-t^{\prime}}}\,Q\right)^{\Delta-2}K_{\Delta-2}\left(\sqrt{\frac{t^{\prime}}{1-t^{\prime}}}\,Q\,z\right)$ In the last equation we included the factor $\left(\frac{2\alpha^{\prime}}{\rho}\right)^{2\Delta-4}$, which recovers the correct dimension of the wave function. The origin of this factor is simple: we assumed for simplicity in all our previous calculations, that $L=1$ in $AdS_{5}$. Since $L^{2}=\alpha^{\prime}\sqrt{\lambda}=\alpha^{\prime}2/\rho$, this factor is the way to take into account the fact that $L^{2}\neq 1$. ### 2.3 Qualitative features of high energy scattering From Eq. (2.10) one can see that $A_{NN}\left(s,b\right)$ tends to 1 in the region of $b$ from $b=0$ to $b=b_{0}(s)$. Since $G_{2}(u)\to 1/b^{6}$ at large $b$, we can conclude that $A_{NN}\left(s,b\right)\,\,\propto s/b^{6}$ at $b\gg z^{2}$. Therefore, $b^{2}_{0}\,\propto s^{1/3}$ and the nucleon-nucleon scattering amplitude generates the total cross section $\sigma_{tot}\,\,=\,\,2\int\,d^{2}b\,ImA_{NN}\left(s,b\right)\,\,\propto\,s^{1/3}$ (2.16) in obvious violation of the Froissart theorem [23]. As has been shown in Refs. [5, 10] the Froissart theorem can be restored if we consider the string theory which leads to N=4 SYM in the limit of the weak graviton interaction. In this string theory the graviton with positive $t$ ($t$ is the momentum transferred along the graviton in Fig. 1) lies on the Regge trajectory with the intercept $\alpha^{\prime}/2$ which corresponds to the closed string. On the other hand in $AdS_{5}$ the Einstein equation has the form $R_{\mu,\nu}\,\,-\,\,\frac{1}{2}\,R\,g_{\mu,\nu}\,=\,\,\frac{6}{L^{2}}\,\,g_{\mu,\nu}\,\,\,=\,\,\frac{6}{\sqrt{\lambda}\,\alpha^{\prime}}\,g_{\mu,\nu}$ (2.17) where $R_{\mu,\nu}$ is the Ricci curvature tensor and $R$ is the Ricci curvature. In consequence of Eq. (2.17) the graviton has the mass[24] $m^{2}_{graviton}\,=\,4/\sqrt{\lambda}\alpha^{\prime}\,=\,2\,\rho/\alpha^{\prime}$ and the intercept $2-m^{2}_{graviton}(\alpha^{\prime}/2)\,=\,2-2/\sqrt{\lambda}\,\,=\,\,2-\rho$. The fact that the graviton has mass results in the different behavior of the gluon propagator at large $b$, namely, at large $b$ it shows the exponential decrease $G(b)\to\exp\left(-m_{graviton}\,b\right)=\exp\left(-\sqrt{2\,\rho\,b^{2}}\right)$. Such behavior restores the logarithmic dependence of the cross section at high energy, in the agreement with the Froissart theorem but nevertheless we expect a wide range of energies where the cross section behaves as $s^{1/3}$. Experimentally, the total cross section in the energy range from fixed target experiment at FNAL to the Tevatron energy, has $\sigma_{tot}\propto s^{0.1}$. Therefore, we expect that the cross section cannot be described by Eq. (2.10). We replace $G_{3}(u)$ in Eq. (2.10) and in Eq. (2.9) by $\tilde{A}_{1GE}(s,b;z,z^{\prime})\,\,\,\longrightarrow\,\,\,\,\tilde{A}_{1GE}(s,b;z,z^{\prime})\,\,e^{-\sqrt{2\rho/\alpha^{\prime}}\,b}$ (2.18) to take into account the effect of the graviton reggeization. Introducing this equation we are able to specify the kinematic energy range, where we expect the $s^{1/3}$ behavior of the total cross section. ## 3 The comparison with the experimental data As we have discussed, we face two main difficulties in our attempts to describe the experimental data in N=4 SYM: the small value of the cross section of the multiparticle production and the violation of the Froissart theorem. The scale of both phenomena is given by the value of $2/\sqrt{\lambda}$ (see Eq. (2.7) and Eq. (2.18)) and, if this parameter is not small, we, perhaps, have no difficulties at all. On the other hand, the N=4 SYM could provide the educated guide only for $2/\sqrt{\lambda}\,\ll\,1$ since it has an analytical solution for such $\lambda$. Therefore, the goal of our approach is to describe the experimental data assuming that $2/\sqrt{\lambda}\,$ is reasonably small (say $2/\sqrt{\lambda}\,\leq\,0.3$), and to evaluate the scale of the cross section for the multiparticle production. As has been mentioned, the multiparticle production can be discussed in N=4 SYM since the confinement of the quarks and gluon, we believe , is not essential for these processes. We use Eq. (2.7) with Eq. (2.15) to calculate the physical observables, namely, $\displaystyle\sigma_{tot}\,\,$ $\displaystyle=$ $\displaystyle\,\sigma_{0}\,+\,\,2\,\int d^{2}b\,\,Im\,A\left(s,b\right)\,\,$ $\displaystyle=$ $\displaystyle\,\,\sigma_{0}\,+\,\frac{4}{\rho\alpha^{\prime}}\,\int d^{2}b\,\,\int\Phi(z)\,\Phi(z^{\prime})\,dz\,dz^{\prime}\,Re\left\\{1\,\,-\,\,\exp\left(iN^{2}_{c}\,\tilde{A}_{1GE}\left(s,b,z,z^{\prime}\|{Eq.~{}(\ref{EQ})}\right)\right)\,\right\\}\,$ $\displaystyle\sigma_{el}$ $\displaystyle=$ $\displaystyle\int d^{2}b\,\,\|A_{0}(b)\,+\,A\left(s,b\right)\|^{2}\,\,$ $\displaystyle=$ $\displaystyle\,\,\int d^{2}b\,\,\|A_{0}(b)\,+\,\int\Phi(z)\,\Phi(z^{\prime})\,dz\,dz^{\prime}\,i\left\\{1\,\,-\,\,\exp\left(iN^{2}_{c}\,\tilde{A}_{1GE}\left(s,b,z,z^{\prime}\|{Eq.~{}(\ref{EQ})}\right)\right)\,\right\\}\|^{2};$ $\displaystyle B_{el}$ $\displaystyle=$ $\displaystyle\,\,\frac{\int d^{2}b\,\,b^{2}\,\,\|A_{0}(b)\,+\,\int\Phi(z)\,\Phi(z^{\prime})\,dz\,dz^{\prime}\,i\left\\{1\,\,-\,\,\exp\left(iN^{2}_{c}\,\tilde{A}_{1GE}\left(s,b,z,z^{\prime}\|{Eq.~{}(\ref{EQ})}\right)\right)\,\right\\}\|^{2}}{\int d^{2}b\,\,\|A_{0}(b)\,+\,\int\Phi(z)\,\Phi(z^{\prime})\,dz\,dz^{\prime}\,i\left\\{1\,\,-\,\,\exp\left(iN^{2}_{c}\,\tilde{A}_{1GE}\left(s,b,z,z^{\prime}\|{Eq.~{}(\ref{EQ})}\right)\right)\,\right\\}\|^{2}}\,;\,\,$ (3.3) As has been expected, it turns out that in the experimental accessible region of energies, the cross section given in Eq. (2.10) shows the $s^{1/3}$ behavior for a wide range of parameters: $g^{2}=0.01\div 1$, $Q=0.2\div 1\,GeV^{-1}$ and $\rho=0\div 0.3$. Our choice of the parameters reflects the theoretical requirements for N=4 SYM, where we can trust this approach, namely, $g_{s}\ll 1$ while $g_{s}\,N_{c}>1$. The values of $\sigma_{tot}$ from Eq. (2.10) with $\tilde{A}$ from Eq. (2.9) are small for $W=\sqrt{s}=20GeV$ but it increases and becomes about $20-30\,mb$ at the Tevatron energy. Facing the clear indication that we need an extra contribution to the total cross section in Eq. (3)-Eq. (3.3), we introduce the contribution of the non N=4 SYM origin ($\sigma_{0}$ and the amplitude $A_{0}(b)$ ). It should be mentioned that we have also a hidden parameter $\Delta$ in the wave function of the proton. At the moment theoretically we know only that $\Delta>2$. This constraint stems from the convergence of the integral for the norm of the proton wave function (see Refs.[21, 4, 9, 25]). We have tried several values of $\Delta$ and $\Delta=3$ is our best choice (see Fig. 8). For a purely phenomenological background $A_{0}(b)$ we wrote the simplest expression $A_{0}(b)\,\,=\,\,i\,\frac{\sigma_{0}}{4\,\pi\,B_{0}}\,\exp\left(-b^{2}/2B_{0}\right)$ (3.4) where $B_{0}$ is the slope for the elastic cross section. With these two new parameters $\sigma_{0}$ and $B_{0}$, we tried to describe the data. The results are shown in Fig. 3,Fig. 4 and Fig. 5. Figure 3: The description of the total cross section $\sigma_{tot}=(\sigma_{tot}(pp)+\sigma_{tot}(p\bar{p}))/2$ with $Q=0.35GeV$, $g\,=\,g^{2}_{s}\,N^{2}_{c}=0.1$, $\rho=0.25$ , $\Delta=3$ and with $\sigma_{0}=37.3\,mb$. From these pictures one can see that for the total and elastic cross section, we obtain a good agreement with the experimental data, whereas for the elastic slope ($B_{el}$), the description is in contradiction with the experimental data. First we would like to understand the main ingredients of the total cross section. For doing so we need to estimate the cross section of the diffractive dissociation. In the N=4 SYM approach; $\displaystyle\sigma_{diff}\,\,$ $\displaystyle=$ $\displaystyle\,\,\frac{2}{\rho\alpha^{\prime}}\,\int d^{2}b\,\,\int\Phi(z)\,\Phi(z^{\prime})\,dz\,dz^{\prime}\,\|1\,\,-\,\,\exp\left(iN^{2}_{c}\,\tilde{A}_{1GE}\left(s,b,z,z^{\prime}\|{Eq.~{}(\ref{EQ})}\right)\right)\|^{2}$ (3.5) $\displaystyle-$ $\displaystyle\,\,\|\frac{2}{\rho\alpha^{\prime}}\,\int d^{2}b\,\,\int\Phi(z)\,\Phi(z^{\prime})\,dz\,dz^{\prime}\,\|1\,\,-\,\,\exp\left(iN^{2}_{c}\,\tilde{A}_{1GE}\left(s,b,z,z^{\prime}\|{Eq.~{}(\ref{EQ})}\right)\right)\|\|^{2}$ In Eq. (3.5) $\sigma_{diff}=\sigma_{sd}+\sigma_{dd}$ where $\sigma_{sd}$ and $\sigma_{dd}$ are cross sections of single and double diffraction respectively. Our predictions for $\sigma_{diff}$ have been plotted in Fig. 6, where curve 1 is the result of the calculation using Eq. (3.5), and curve 2 is the same except for the addition of $4mb$ from the diffractive cross section, which is of non N=4 SYM origin. In Table1, we compare our predictions with the phenomenological models that do not take into account the N=4 SYM physics. The result of this comparison is interesting, since our simple estimates show that the cross section of the diffractive production, could considerably grow from the Tevatron to the LHC energy. We want to recall that the unitarity constraints of Eq. (2.7), lead to $\|A(s,b;z,z^{\prime})\|\,\leq 2$ and $\sigma_{tot}=\sigma_{el}$. As far as the inelastic cross section is concerned, one can see that the inelastic cross section of the N=4 SYM origin $\sigma\left(\mbox{N=4 SYM}\right)\,\,=\,\,\sigma_{tot}-\sigma_{el}-\sigma_{diff}-\sigma_{0,in}$ is about 2 $mb$ both for RHIC and the Tevatron energy, and grows to 30 $mb$ at the LHC energy. Therefore, we can observe some typical features of the N=4 SYM theory, which only start at the LHC energy. The above estimates are based on the background that does not depend on energy. However, Fig. 4 illustrates that the non N=4 SYM background should also depend on energy. In Fig. 4 (the upper curve) we plot the elastic slope for the background of Eq. (3.4) but with $B_{0}=12.37+2\alpha^{\prime}_{P}\ln(s/s_{0})$. This amplitude corresponds to the exchange of the Pomeron with intercept 1 which generates the constant cross section but leads to a shrinkage of the diffraction peak. One can see that we are able to describe the slope in such a model. Figure 4: The description of the energy behavior of the elastic slope with the same set of parameters as in Fig. 3 and with $B_{0}=12.37\,GeV^{-2}$ (solid curve) and $B_{0}=12.37+2\,\alpha^{\prime}_{P}\ln(s/s_{0})$ ($\alpha^{\prime}_{P}=0.1\,GeV^{-2}$(dashed curve) and $\alpha^{\prime}_{P}=0.2\,GeV^{-2}$ (dotted curve)) Figure 5: The description of the energy behavior of the elastic cross section with the same set of parameters as in Fig. 3 and $B_{0}=12.37\,GeV^{-2}$. \| Tevatron \| LHC ---\|---\|--- \| GLMM KMR LP \| GLMM KMR LP $\sigma_{tot}$( mb ) \| 73.29 74.0 83.2 \| 92.1 88.0 124.9 $\sigma_{el}$(mb) \| 16.3 16.3 17.5 \| 20.9 20.1 24.4 $\sigma_{sd}\,+\,\sigma_{dd}$(mb) \| 15.2 18.1 24.4 \| 17.88 26.7 42.3 $\left(\sigma_{el}+\sigma_{sd}+\sigma_{dd}\right)/\sigma_{tot}$ \| 0.428 0.464 0.504 \| 0.421 0.531 0.536 Table 1: Comparison of the GLMM ([26]) and KMR[27] models and our estimates (LP). In Fig. 7 and Fig. 8 we plot the dependence of $\sigma_{tot}$ and $\sigma_{el}$ on the parameters of our approach to illustrate the sensitivity of our descriptions of the experimental data to their values. Figure 6: The description of the energy behavior of the diffraction production cross section $\sigma_{diff}\,=\,\sigma_{sd}\,+\,\sigma_{dd}$ with the same set of parameters as in Fig. 5. $\sigma_{sd}$ and $\sigma_{dd}$ are cross sections of single and double diffraction production respectively. The curve 2 shows the N=4 SYM contribution to the diffraction production while the curve 1 corresponds to the N=4 SYM prediction plus $4\,mb$ for the cross section of a different source than N=4 SYM. The data are only for single diffraction production. In curve 3 we plot the estimates of Ref.[26] for $\sigma_{diff}$ Figure 7: The dependence of the description on $Q$ and $g=N^{2}_{c}\,g^{2}$. Figure 8: The dependence of the description on $\rho$ and $\Delta$. The results of our calculation show that in the large range of energies, the N=4 SYM scattering amplitude behaves as $s^{1/3}$ with a rather small coefficient in front. The graviton reggeization that will stop the anti- Froissart behavior at ultra high energies, does not show up at the accessible range of energy from the fixed target Fermilab energy, until the Tevatron energy. This reggeization can be measured, perhaps, only at the LHC energy. ## 4 Conclusion In this paper we show that the N=4 SYM total cross section violates the Froissart theorem, and in the huge range of energy this cross section is proportional to $s^{1/3}$. The graviton reggeization will change this increase to the normal logarithmic behavior $\sigma\propto\ln^{2}s$. However, we demonstrated that this happens at ultra high energy, much higher than the LHC energy for reasonably low $2/\sqrt{\lambda}\approx 0.25$. We need to assume that there is a different source for the total cross section, with the value of the cross section about 40 mb. With this assumption we successfully describe $\sigma_{tot},\sigma_{el}$ and $\sigma_{diff}$ for the accessible range of energy from the fixed target Fermilab to the Tevatron energies. The N=4 SYM mechanism is responsible only for a small part of the inelastic cross section for this energy region (about $2mb$). However, at the LHC energy the N=4 SYM theory can lead to a valuable contribution to the inelastic cross section, namely, $\sigma_{in}\approx 30\,mb$ which is about a quarter of the total inelastic cross section. The second surprise is the fact that the total cross section and the diffraction cross section can increase considerably from the Tevatron to the LHC energy. The bad description of $B_{el}$ gives the strong argument that the non N=4 SYM background should depend on energy. It means that at RHIC energies, the N=4 SYM part of the inelastic cross section is negligible and the quark-gluon plasma is created by the mechanism outside of N=4 SYM. For the LHC energy, we can expect that N=4 SYM is responsible for the inelastic cross section of about $\sigma_{in}(N=4\,\,SYM)=30\,\,mb$ out of $\sigma_{tot}=121.9\,mb$. We believe that we have a dilemma: to find a new mechanism for the inelastic production in the framework of N=4 SYM other than the reggeized graviton interaction, or to accept that N=4 SYM is irrelevant to description of any experimental data that have been measured before the LHC era, with a chance that even at the LHC it will be responsible only for a quarter (or less) of the total cross section. Deeply in our hearts, we believe in the first way out, and we hope that this paper will draw attention to this challenging problem: searching for a new mechanism for multiparticle production in N=4 SYM. We wish to draw your attention to the fact that the scattering amplitude can change considerably from the Tevatron to LHC energy (see Table 1). Therefore, all claims that we can give reliable predictions for the values of the cross sections at the LHC energy and even of the survival probability for the diffractive Higgs production [27] looks exclusively naive and reflects our prejudice rather than our understanding. ## Acknowledgments We thank Boris Koppeliovich for fruitful discussion on the subject of the paper. Our special thanks go to Miguel Costa and Jeremy Miller for their careful reading of the first version of this paper and useful discussions. E.L. also thanks the high energy theory group of the University Federico Santa Maria for the hospitality and creative atmosphere during his visit. This work was supported in part by Fondecyt (Chile) grants, numbers 1050589, 7080067 and 7080071, by DFG (Germany) grant PI182/3-1 and by BSF grant $\\#$ 20004019. ## References [1] J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200]; S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428 (1998) 105 [arXiv:hep-th/9802109]; E. Witten, Adv. Theor. Math. Phys. 2 (1998) 505 [arXiv:hep-th/9803131]. * [2] J. Polchinski and M. J. Strassler, JHEP 0305 (2003) 012 [arXiv:hep-th/0209211]; Phys. Rev. Lett. 88 (2002) 031601 [arXiv:hep-th/0109174]. * [3] A. V. Kotikov, L. N. Lipatov, A. I. Onishchenko and V. N. Velizhanin, Phys. Lett. B 595 (2004) 521 [Erratum-ibid. B 632 (2006) 754] [arXiv:hep-th/0404092]; A. V. Kotikov and L. N. Lipatov, Nucl. Phys. B 661 (2003) 19 [Erratum-ibid. B 685 (2004) 405] [arXiv:hep-ph/0208220]; A. V. Kotikov and L. N. Lipatov, Nucl. Phys. B 582 (2000) 19 [arXiv:hep-ph/0004008]. A. V. Kotikov, L. N. Lipatov and V. N. Velizhanin, Phys. Lett. B 557 (2003) 114 [arXiv:hep-ph/0301021]; J. R. Andersen and A. Sabio Vera, Nucl. Phys. B 699 (2004) 90 [arXiv:hep-th/0406009]; Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, Phys. Rev. D 75 (2007) 085010 [arXiv:hep-th/0610248]; Z. Bern, L. J. Dixon and V. A. Smirnov, Phys. Rev. D 72 (2005) 085001 [arXiv:hep-th/0505205]; * [4] Y. Hatta, E. Iancu and A. H. Mueller, JHEP 0801 (2008) 026 [arXiv:0710.2148 [hep-th]]. * [5] R. C. Brower, J. Polchinski, M. J. Strassler and C. I. Tan, JHEP 0712 (2007) 005 [arXiv:hep-th/0603115]. * [6] R. C. Brower, M. J. Strassler and C. I. Tan, arXiv:0707.2408 [hep-th]. * [7] R. C. Brower, M. J. Strassler and C. I. Tan, JHEP 0806 (2008) 048 [arXiv:0801.3002 [hep-th]]. * [8] L. Cornalba and M. S. Costa, Phys. Rev. D 78, (2008) 09010, arXiv:0804.1562 [hep-ph]; L. Cornalba, M. S. Costa and J. Penedones, JHEP 0806 (2008) 048 [arXiv:0801.3002 [hep-th]]; JHEP 0709 (2007) 037 [arXiv:0707.0120 [hep-th]]. * [9] B. Pire, C. Roiesnel, L. Szymanowski and S. Wallon, Phys. Lett. B 670, 84 (2008) [arXiv:0805.4346 [hep-ph]]. * [10] E. Levin, J. Miller, B. Z. Kopeliovich and I. Schmidt, JHEP 0902 (2009) 048; arXiv:0811.3586 [hep-ph]. * [11] L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rep. 100, 1 (1983). * [12] A. H. Mueller and J. Qiu, Nucl. Phys. B 268 427 (1986) . * [13] L. McLerran and R. Venugopalan, Phys. Rev. D 49,2233, 3352 (1994); D 50,2225 (1994); D 53,458 (1996); D 59,09400 (1999). * [14] E. A. Kuraev, L. N. Lipatov, and F. S. Fadin, Sov. Phys. JETP 45, 199 (1977); Ya. Ya. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. 28, 22 (1978). * [15] I. Balitsky, [arXiv:hep-ph/9509348]; Phys. Rev. D60, 014020 (1999) [arXiv:hep-ph/9812311] Y. V. Kovchegov, Phys. Rev. D60, 034008 (1999), [arXiv:hep-ph/9901281]. * [16] J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, Phys. Rev. D59, 014014 (1999), [arXiv:hep-ph/9706377]; Nucl. Phys. B504, 415 (1997), [arXiv:hep-ph/9701284]; J. Jalilian-Marian, A. Kovner and H. Weigert, Phys. Rev. D59, 014015 (1999), [arXiv:hep-ph/9709432]; A. Kovner, J. G. Milhano and H. Weigert, Phys. Rev. D62, 114005 (2000), [arXiv:hep-ph/0004014] ; E. Iancu, A. Leonidov and L. D. McLerran, Phys. Lett. B510, 133 (2001); [arXiv:hep-ph/0102009]; Nucl. Phys. A692, 583 (2001), [arXiv:hep-ph/0011241]; E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran, Nucl. Phys. A703, 489 (2002), [arXiv:hep-ph/0109115]; H. Weigert, Nucl. Phys. A703, 823 (2002), [arXiv:hep-ph/0004044]. * [17] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 94 (2005) 111601 [arXiv:hep-th/0405231]. * [18] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz and L. G. Yaffe, JHEP 0607 (2006) 013 [arXiv:hep-th/0605158]. * [19] A. H. Mueller, Phys. Lett. B 668 (2008) 11 [arXiv:0805.3140 [hep-ph]]; F. Dominguez, C. Marquet, A. H. Mueller, B. Wu and B. W. Xiao, Nucl. Phys. A 811 (2008) 197 [arXiv:0803.3234 [nucl-th]]; Y. Hatta, E. Iancu and A. H. Mueller, JHEP 0805 (2008) 037 [arXiv:0803.2481 [hep-th]] * [20] P. M. Chesler and L. G. Yaffe, Phys. Rev. D 78 (2008) 045013 [arXiv:0712.0050 [hep-th]]; A. Yarom, Phys. Rev. D 75 (2007) 105023 [arXiv:hep-th/0703095]. * [21] E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253 [arXiv:hep-th/9802150]. * [22] I. Gradstein and I. Ryzhik, “ Tables of Series, Products, and Integrals”, Verlag MIR, Moskau,1981 * [23] M. Froissart, Phys. Rev. 123 (1961) 1053; A. Martin, “Scattering Theory: Unitarity, Analitysity and Crossing.” Lecture Notes in Physics, Springer-Verlag, Berlin-Heidelberg-New-York, 1969\. * [24] J. Naf, P. Jetzer and M. Sereno, Phys. Rev. D 79 (2009) 024014 [arXiv:0810.5426 [astro-ph]] ; L. Liu, arXiv:gr-qc/0411122 ;M. Novello and R. P. Neves, Class. Quantum Grav. 20 (2003) 67, arXiv:gr-qc/0210058. * [25] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, Nucl. Phys. B 562 (1999) 330 [arXiv:hep-th/9902042]; E. D’Hoker and D. Z. Freedman, Nucl. Phys. B 550 (1999) 261 [arXiv:hep-th/9811257]; D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, Nucl. Phys. B 546 (1999) 96 [arXiv:hep-th/9804058]; E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, Nucl. Phys. B 562 (1999) 353 [arXiv:hep-th/9903196]; H. Liu, Phys. Rev. D 60 (1999) 106005 [arXiv:hep-th/9811152]. * [26] E. Gotsman, E. Levin, U. Maor and J. S. Miller, Eur. Phys. J. C 57 (2008) 689 [arXiv:0805.2799 [hep-ph]]. * [27] M. G. Ryskin, A. D. Martin and V. A. Khoze, Eur. Phys. J. C54 (2008) 199 [arXiv:0710.2494 [hep-ph]].	arxiv-papers	2009-02-18T15:24:52	2024-09-04T02:49:00.674321	{ "license": "Public Domain", "authors": "E. Levin (Tel Aviv Un.) and I. Potashnikova (USM)", "submitter": "Eugene Levin", "url": "https://arxiv.org/abs/0902.3122" }
0902.3174	Oviedo, Asturias, Spanien Prof. Dr. F. Kneer Prof. Dr. W. Kollatschny Januar 2008 15 Februar, 2008 2008 978-3-936586-81-7 # Observations, analysis and interpretation with non-LTE of chromospheric structures of the Sun Bruno Sánchez-Andrade Nuño _For my parents Conchita and Julio, my sister Deva… _ _…and all those who shall learn something from this work._ ###### Contents 1. Summary 2. 1 Introduction 1. 1.1 The Sun 2. 1.2 The chromosphere 3. 1.3 Aim and outline of this work 3. 2 Spectral lines 1. 2.1 Radiative transfer and spectral line formation 2. 2.2 Hydrogen Balmer-$\alpha$ line (H$\alpha$) 3. 2.3 He i 10830 Å multiplet 4. 3 Observations 1. 3.1 Angular resolution and _Seeing_ 2. 3.2 Telescope 1. 3.2.1 Kiepenheuer Adaptive Optics System 3. 3.3 High spatial resolution 1. 3.3.1 Instrument 2. 3.3.2 Observations 3. 3.3.3 Data reduction 4. 3.4 Infrared spectrometry 1. 3.4.1 Instrument 2. 3.4.2 Observations 3. 3.4.3 Data reduction 5. 4 High resolution imaging of the chromosphere 1. 4.1 Dark clouds 2. 4.2 Fast events and waves 1. 4.2.1 Observations and data reduction 2. 4.2.2 Physical parameters 3. 4.2.3 Fast events in H$\alpha$ 4. 4.2.4 Magnetoacoustic waves 5. 4.2.5 Summary on some properties of the active chromosphere 3. 4.3 Comparison between speckle interferometry and blind deconvolution 6. 5 Spicules at the limb 1. 5.1 Spicule emission profiles observed in He i 10830 Å 1. 5.1.1 Observational intensity profiles and intensity ratio 2. 5.1.2 Results 3. 5.1.3 Conclusions 2. 5.2 High resolution imaging of spicules 7. 6 Conclusions and outlook 8. Publications 9. Acknowledgements 1. Lebenslauf ## Summary This thesis is based on observations performed at the _Vacuum Tower Telescope_ at the _Observatorio del Teide_ , Tenerife, Canary Islands. We have used an infrared spectropolarimeter (Tenerife Infrared Polarimeter – TIP) and a Fabry- Perot spectrometer (“Göttingen” Fabry-Perot Interferometer – G-FPI). Observations were obtained during several campaigns from 2004 to 2006. We have applied methods to reduce the atmospheric distortions both during the observations and afterwards in the case of the G-FPI data using image processing techniques. We have studied chromospheric dynamics inside the solar disc. The G-FPI provides means to obtain very high spatial, spectral and temporal resolution. We observe at several wavelengths across the H$\alpha$ line. With different post-processing techniques, we achieve spatial resolutions better than $0\farcs 5$. We present results from the comparison of the different image reconstruction methods. A time series of 55 min duration was taken from AR 10875 at $\vartheta\approx 36\degr$. From the wealth of structures we selected areas of interest to further study in detail some ongoing processes. We apply non-LTE inversion techniques to infer physical properties of a recurrent surge. We have studied the occurrence of simultaneous sympathetic mini-flares. Using temporal frequency filtering on the time series we observe waves along fibrils. We study the implications of their interpretations as wave solutions from a linear approximation of magneto-hydrodynamics. We conlude that a linear theory of wave propagation in straight magnetic flux tubes is not sufficient. Furthermore, emission above the solar limb is investigated. Using infrared spectroscopic measurements in the He i 10830 Å multiplet we have studied the spicules outside solar disc. The analysis shows the variation of the off-limb emission profiles as a function of the distance to the visible solar limb. The ratio between the intensities of the blue and the red components of this triplet $({\cal R}=I_{\rm blue}/I_{\rm red})$ is an observational signature of the optical thickness along the light path, which is related to the intensity of the coronal irradiation. The observable ${\cal R}$ as a function of the distance to the visible limb is given. We have compared the observational ${\cal R}$ with the intensity ratio obtained from Centeno (2006), using detailed radiative transfer calculations in semi-empirical models of the solar atmosphere assuming spherical geometry . The agreement is purely qualitative. We argue that this is a consequence of the limited extension of current models. With the observational results as constraints, future models should be extended outwards to reproduce our observations. To complete our analysis of spicules we report observational properties from high-resolution filtergrams in the H$\alpha$ spectral line taken with the G-FPI. We find that spicules can reach heights of 8 Mm above the limb. We show that spicules outside the limb continue as dark fibrils inside the disc. One and a half centuries after the hand-drawings by Secchi, the chromosphere is still a source of unforeseen and exciting new discoveries. ## Chapter 1 Introduction This thesis deals with the chromosphere of the Sun. To give some insight to the readers which are not familiar with the topics of this work we introduce in Section 1.1 the main characteristics of the Sun with a short general description. This will elucidate the position of the chromosphere in the solar structure and its role for the outer solar atmosphere. In the subsequent Section 1.2, those aspects of the chromosphere which are treated in the present work are specified. Finally Section 1.3 indicates the structure of this thesis work. ### 1.1 The Sun _It is just a ball of burning gas …right? _ The Sun is the central object of the Solar System, which also contains planets and many other bodies such as planetoids (small planets), comets, meteoroids and dust particles. However, the Sun on its own harbors 99.8% of the total mass of the system, so all other objects orbit around it. The Sun itself orbits the center of our Galaxy, the _Milky Way_ , with a speed of $217$ km/s. The period of revolution is $\sim 230$ million years (the last time the Sun was on this part of the Galaxy was the time the Dinosaurs appeared). Compared to the population of stars in our galaxy, the Sun is a middle-aged, middle-sized, common type star. In astrophysicist’s language it is of spectral type _G2_ and of luminosity class _V_ , located on the main sequence of stars in the Hertzsprung-Russell diagram. According to our understandings derived from models, it has been on the main sequence for $5\,000$ million years and it will remain there for another $5\,000$ million years before starting the giant phase. The Sun is the closest star to us, the next one being $250\,000$ times further away, but still light from the Sun’s surface takes around 8 minutes to reach the Earth. It is the only star from where we get enough energy to study its spectrum in great detail and with short temporal cadence. With indirect methods, we can produce images of the surface structuring on other nearby starts. But on the Sun, with current telescopes and techniques, we resolve structures down to 100 km size on its surface, which represents approximately the resolution limit in this thesis work. We can also investigate the structure of its atmosphere and the effects of its magnetism. Actually, we are embedded in the solar wind that has its origin in the outer solar atmosphere, the corona of the Sun. Thus, we can make _in-situ_ measurements. With special techniques and models, we can reconstruct the properties of its interior. Figure 1.1: The apparent size of the Sun on the sky is $\sim 32^{\prime}$, a little bit larger than one half degree. The Sun is the most brilliant object in the sky, 12 orders of magnitude brighter than the second brightest object, the full Moon, which actually only reflects the sunlight. Its light warms the surface of the Earth and is used by plants to grow. Its radiation is the input for the climate. The solar wind separates us from the interstellar medium. The magnetism of the Sun protects us from cosmic high-energy radiation and it influences the climate on Earth. Violent events in the solar ultraviolet radiation and the solar wind can also disrupt radio communications. The Sun possesses a complex structure. Essentially, it can be described as a giant conglomerate of Hydrogen and Helium ($\sim 74$% and $\sim 24$% of the mass, respectively) and traces of many other chemical elements. Due to its big mass the self-gravitation keeps the structure as a sphere. From the weight of the outer spherical gas shells the pressure increases towards the center of the sphere. During the gravitational contraction of the pre-solar nebula towards its center, i.e. when forming a protostar, the gas has heated up by converting potential energy into thermal (kinetic) energy. This produces, together with a high gas density, a high pressure, which prevents the sphere from collapsing further inwards. Eventually, near the center, temperature and pressure are high enough to ignite nuclear reactions. ##### Structure At the core of the Sun the density and temperature (of the order of 13 million Kelvin, or $13\cdot 10^{6}$ K) are high enough to fuse hydrogen and burn it into helium. This process also produces energy in the form of high-energy photons. This continuous, long-lasting energy output from the nuclear reactions keeps the core of the Sun at high temperature to sustain the gravitational load from the outer gas shells. Due to the high density, the photons are continuously absorbed and re-emitted by nearby ions, and in this way the big energy output is slowly _radiated_ outwards, while, towards the surface of the Sun, the density decreases exponentially, along with the temperature. Photons reaching today the Earth’s surface were typically generated on the early times of _Homo Sapiens_ , as the typical travel time is $\sim 170\,000$ years (Mitalas and Sills, 1992). At a distance from the center of approximately 70% of the solar radius, the radiation process is not efficient enough to transport the huge amount of energy produced in the core. There the gas is heated up, and expands, it becomes buoyant and rises. This creates _convection_ cells in which hot material is driven up by buoyancy while cool gas sinks to the bottom of the cells, where it is heated again. These gas flows transport the energy to the outer part of the Sun, where the temperature is measured to be $\sim 5\,700\,$K and the density is low enough that the photons can escape without much further absorption. The outer region from where we receive most of the optical photons can be called the surface of the Sun, although it is not a layer in the solid state. It is called the _photosphere_ (sphere of light). Most of the photons we receive come from this layer are in the _visible_ part of the spectrum: light. This is why Nature favored in the late evolution process the development of vision instruments that are more sensitive in the spectral region in which most emission from the Sun occurs. Figure 1.2: High resolution image of the “surface” (photosphere) of the Sun with a resolution of $\sim 140$ km. Granules are seen all around the photosphere outside the dark areas. They form the uppermost layers of the convection zone, in which the energy is transported from deep down outwards via gas motions. At the top, the gas cools down by radiating photons into space. Localized strong magnetic fields can also emerge and are seen as dark areas, the sunspots, which are a consequence of the less efficient energy transport. Further out of this layer the atmosphere of the Sun extends radially, with decreasing density. In this outer part, with its low density, magnetic fields rooted inside the Sun cease to be pushed around by gas flows. This transition occurs together with a still not completely understood increase of temperature up to several million degrees. Therefore, there must be a layer with a minimum temperature. Standard average models place it at a height of about 500 km with a temperature of about 4000 K, which is low enough to allow the formation of molecules like CO or water vapor. Beyond this layer the temperature rises outward. Again in standard models, the layer following the temperature minimum has an extent of about 1 500 km and its temperature rises to 8 000 – 10 000 K. This layer is called the _chromosphere_. The present work deals with some of its properties. Outside the chromosphere, the temperature rises abruptly within the _transition region_. The outermost part of the atmosphere, called _corona_ , drives a permanent outwards flow of particles moving along the magnetic field lines. This _solar wind_ extends to $100\,000$ times the solar radius, far beyond Pluto’s orbit, to the outer border of our Solar System, the _heliopause_. There the interaction with the interstellar medium creates a shock front, which is being measured these years by the Voyager 1 and Voyager 2 probes. Beyond this layered structure, the Sun is far more complex. Some other properties, which we describe shortly, are: \- The Sun vibrates. As a self gravitating compressible sphere, it vibrates. Pressure and density fluctuations mainly generated by the turbulent convection, are propagated through the Sun. Waves with frequencies and wavelengths close to those of the many normal modes of vibration of the Sun add up to a characteristic pattern of constructive interference. This vibration, although of low amplitude with few 100 m/s in the photosphere, can be measured and decomposed into eigenmodes by means of Doppler shifts and observations of long duration. The propagation of the waves depends on the properties of the medium. It is possible then to infer these properties from the measured vibration patterns. Some waves propagate only close to the surface, but others can propagate through the entire Sun. These latter waves provide means to infer some structural properties, such as temperature, of the solar interior and test models of the Sun. _Global Helioseismology_ provides means to infer the global properties of the interior of the Sun studying the vibration pattern, while _local helioseismology_ can depict the surroundings of the local perturbations. \- The Sun rotates. The conservation of angular momentum of a slowly rotating cloud that will form a star result, upon contraction, a rapid rotation. It is commonly accepted that most of the Sun’s angular momentum was removed during the first phases of the life of the Sun by braking via magnetic fields anchored in the surrounding interstellar medium and by a strong wind. The remaining angular momentum leads to today’s solar rotation period. But being the Sun not a rigid body this rotation varies from layer to layer and with latitude. Gas at the equator rotates at the surface with a period of 27 days, faster than at the poles where the rotation period is approximately 32 days. Using helioseismology observations we know that this differential rotation continues inside the Sun, until a certain depth, from which on the inner part rotates like a rigid sphere with a period of that at middle latitudes on the surface. This region corresponds to the layer where the convection starts, at around $0.7$ solar radii, and is called the _tachocline_. The differential rotation creates meridional flows of gas directed towards the poles near the surface and towards the equator near the bottom of the convection zone. \- The Sun shows (complex) magnetic activity. The Sun possesses a very weak overall magnetic dipole field. However, the solar surface can host very strong and tremendously complicated magnetic structures, which can be seen through their effects on the solar plasma, e.g. less efficient energy transport (that leads to dark sunspots). All matter in the Sun is in the form of plasma, due to the high temperature. The high mobility of charges that characterizes the plasma state, makes it highly conductive, causing magnetic field lines to be "frozen" into it. Provided that the gas pressure is much higher than the magnetic pressure, the magnetic field lines follow generally the dynamics of the plasma. The source of these localized strong magnetic fields is still to be understood. The dynamo theory addresses this problem suggesting that the weak dipolar magnetic field is amplified at the bottom of the convection zone by the stochastic mass motion and shear produced by the convection and the differential rotation. \- The Sun has cycles. The Sun suffers fluctuations in time. Changes occur in the total irradiance, in solar wind and in magnetic fields. They happen in approximately regular cycles, like the 11 years sunspot cycle, and aperiodically over extended times, like the Maunder Minimum (a period of 75 years in the XVII century when sunspots were rare, and which coincided with the coldest part of the _Little Ice Age_). These fluctuations modulate the structure of the Sun’s atmosphere, corona and solar wind, the total irradiance, occurrence of flares and coronal mass ejections and also indirectly the flux of incoming high-energy cosmic rays. None of these variations are fully understood and their effect on the Sun itself or Earth is still under debate. The generally accepted idea about the cyclic and more aperiodic fluctuations is that they are caused by variable magnetic fields. These are generated by dynamo mechanisms. \- The Sun evolves. The Sun is now in its main-sequence phase, where the main source of energy is the nuclear fusion of hydrogen to helium. After the initial phase of accretion of mass, a self gravitating star enters this phase, which lasts for most of its life. In the case of the Sun this phase will continue for approximately another five million years, after which the later evolution stages include a complex variation of the radius, with burning of helium as the source of energy in a later red giant phase. After this stage, the mass of the Sun is believed to be not large enough to undergo further fusion stages, and the Sun will slowly faint as a white dwarf star. Readers can find further general information about the Sun in e.g. Wikipedia (Sun); Stix (2002) and many references therein. ### 1.2 The chromosphere In our short description of the Sun’s structure we stated that the atmosphere of the Sun comprises a layer above the photosphere in which the temperature begins to rise again until the transition region where an abrupt increase of temperature, from approximately 10 000 K to 1 million K, occurs. This first layer above the photosphere is called _chromosphere_. The name comes from the greek of “color sphere”, as it can be seen as a ring of vivid red color around the Sun during total solar eclipses111The apparent size of the Sun on the sky happens to be very similar to the apparent size of the Moon, leading to annular or total solar eclipses, during which the red ring can be seen.. The boundaries of the chromospheric layer are very rugged, resembling more cloud structures than a spheric surface. Above quiet Sun regions the chromosphere can be about 2 000 km thick, but some structures seen in typical chromospheric lines can reach to much higher altitudes, like filaments (that can reach heights of $350\,000$ km). The solar chromosphere is a highly dynamic atmospheric layer. At most wavelengths in the optical range, it is transparent due to the fact that its density is low, much lower than in the photosphere below it. Nevertheless, in strong lines like H$\alpha$ (at 6563 Å) or Ca II K and H (at 3934 Å and 3969 Å, respectively) we have strong absorption (and re-emission) which allows direct studies about its peculiar characteristic, like bright plages around sunspots, dark filaments across the disk, as well as spicules and prominences above the limb. Indeed, recent works, e.g. Tziotziou et al. (2003), suggest that many of these chromospheric features could all have the same physical properties but within different scenarios. Figure 1.3: High resolution filtergram taken in the center of the H$\alpha$ spectral line, showing the chromosphere of the Sun with an image resolution of $\sim 150$ km. The same field of view as image 1.2. The localized strong magnetic fields causing sunspots in the photosphere are seen now as fibrils around the sunspots. Given the low $\beta$ parameter, the plasma is forced to follow the magnetic lines, providing visible tracers and the variety of structures seen in the chromosphere. In the image we can see a carpet of spicules, plage region and a top view of a rising twisted magnetic flux tube above the active region. This image corresponds to the dataset “sigmoid” studied in Chapter 4. The temporal evolution of the chromospheric structures is complex. The dynamics of a magnetised gas depends on the ratio of the gas pressure $P_{\mathrm{gas}}$ to the magnetic pressure $P_{\mathrm{mag}}$, i.e. the plasma $\beta$ parameter, $\beta$ = $P_{\mathrm{gas}}/P_{\mathrm{mag}}$, with $P_{\mathrm{mag}}$ = $B^{2}/(8\pi)$ and $B$ the magnetic field strength 222It is very common in astrophysics, specially in solar physics, to use magnetic field strength synonymously with magnetic flux density. The reason is that in most astrophysical plasmas B=H in Gaussian units. We follow this use in this thesis.. From the low chromosphere into the extended corona, this plasma parameter decreases from values $\beta>1$, where the magnetic lines follow the motion of the plasma (as in the photosphere and solar interior) to a low-beta regime, $\beta\ll 1$, where the plasma motions are magnetically driven, and the plasma follows the magnetic field lines, creating visible tracers of the magnetism. These effects give rise to a new variety of energy transport and phenomena, like magnetic reconnection, filaments standing high above the chromosphere or erupting prominences. ### 1.3 Aim and outline of this work Since the discovery of the chromosphere and since the hand-drawings of Secchi (1877) we have been able to observe this solar atmospheric layer in much detail. Many theoretical models have been proposed to understand its peculiar characteristics. But, only in the last recent years we have been able to address the problem with fine spectropolarimetry and high spatial resolution. We can study the fine details and resolve small structures, following their dynamics in time. Within these recent advances it has been possible both to test current theories and to observe new unexpected phenomena. This work thus aims at contributing to the understanding of the solar chromosphere. This first Chapter provided a broad introduction to the context of this work. We have briefly presented some general properties of the Sun and the chromosphere. In the following pages, throughout Chapter 2, we summarize some theoretical concepts of radiative transfer and spectral line formation needed for this work. We also present general characteristics of the two spectral lines studied: H$\alpha$ and He i 10830 Å. Chapter 3 presents in detail the observations. There we also summarize the characteristics of the used telescope and optical instruments, as well as the data reduction and post- processing methods applied to achieve spatial resolutions better than $0\farcs 5$. Next, in Chapter 4, we discuss results from data on the solar disc, dealing with the chromospheric dynamics and fast events observed in our data. We present the observations of magnetoacustic waves as well as other fast events. Chapter 5 is devoted to the spicules above the solar limb. The analysis of the spectroscopic intensity profiles from spicules in the infrared spectral range can be used to compare current theoretical models with observations. Further, we present high resolution images in H$\alpha$ of spicules. Finally, the concluding Chapter 6 of this thesis summarizes the main conclusions and gives an outlook for future work. ## Chapter 2 Spectral lines Most of the information from the extraterrestrial cosmos, also from the Sun, arrives as radiation from the sky. It comes encoded in the dependence of the intensity on direction, time and wavelength. Also, the polarization state of the light contains information. These characteristics of the light we observe from any object have their origin in the interaction of atoms and photons under the local properties (temperature, density, magnetic field, radiation field itself, …). To extract this encoded information from the recorded intensities it is important to understand how the radiation is created and transported in the cosmic plasmas and released into the almost empty space. This Chapter describes in the following sections the basis of radiative transfer and spectral line formation. We continue discussing the special properties of the spectral lines used in this work: the hydrogen Balmer-$\alpha$ line (named H$\alpha$ for short) at 6563 Å, and the He i 10830 Å multiplet. ### 2.1 Radiative transfer and spectral line formation Light, consisting of photons, interacts with the gas (of the solar atmosphere, in our case) via absorption and emission. Let $I_{\lambda}(\vec{r},t,\vec{\Omega})$ be the specific intensity (irradiance) at the point $\vec{r}$ in the atmosphere, at time $t$, and into direction $\vec{\Omega}$, with $\|\vec{\Omega}\|=1$. We further denote by $\kappa_{\lambda}$ and $\epsilon_{\lambda}$ as the absorption and emission coefficients, respectively. Along a distance $\mathrm{d}s$ in the direction $\vec{\Omega}$, the change of $I_{\lambda}$ is given by $\mathrm{d}\,I_{\lambda}=-\kappa_{\lambda}I_{\lambda}\mathrm{d}s+\epsilon_{\lambda}\mathrm{d}s\,,$ (2.1) or $\frac{\mathrm{d}\,I_{\lambda}}{\mathrm{d}s}=-\kappa_{\lambda}I_{\lambda}+\epsilon_{\lambda}\,.$ (2.2) We define also the optical thickness between some points $1$ and $2$ in the atmosphere by $\displaystyle\mathrm{d}\,\tau_{\lambda}=-\kappa_{\lambda}\mathrm{d}s$ ; $\displaystyle\tau_{\lambda,{1}}-\tau_{\lambda,{2}}=-\int_{2}^{1}\kappa_{\lambda}\mathrm{d}s\,,$ (2.3) and the source function $S_{\lambda}$ of the radiation field as $S_{\lambda}=\frac{\epsilon_{\lambda}}{\kappa_{\lambda}}\,.$ (2.4) In the solar atmosphere, absorption and emission are usually effected by transitions between atomic or molecular energy levels, i.e. by bound-bound, bound-free and free-free transitions. If collisions among atoms and with electrons occur much more often than the radiative processes, the atmospheric gas attains statistical thermal properties such as Maxwellian velocity distributions and the population and ionization ratios according to the Boltzamnn and Saha formulae. These properties define locally a temperature $T$. It can be shown (e.g. Chandrasekhar 1960) that in these cases, called _Local Thermodynamic Equilibrium_ (LTE), the source function is given by the Planck function or black body radiation $S_{\lambda}=B_{\lambda}=\frac{2hc^{2}}{\lambda^{5}}\frac{1}{e^{hc/\lambda kT}-1}\,.$ (2.5) $S_{\lambda}$ varies much more slowly with wavelength than the absorption/emission coefficients across a spectral line. Thus, within a spectral line, $S_{\lambda}$ can be considered independent of $\lambda$. Generally, LTE does not hold, especially in regions with low densities (thus with only few collisions relative to radiation processes) and near the outer boundary of the atmosphere from where the radiation can escape into space. The solar chromosphere is a typical atmospheric layer where non-LTE applies. In this case, the population densities of the atomic levels for a specific transition depend on the detailed processes and routes leading to the involved levels. Equation 2.2 has the following formal solution $I(\tau_{2})=I(\tau_{1})e^{-(\tau_{1}-\tau_{2})}+\int_{\tau_{2}}^{\tau_{1}}S(\tau^{\prime})e^{-(\tau^{\prime}-\tau_{2})}\,d\tau^{\prime}\,,$ (2.6) or, for the case when $\tau_{1}\rightarrow\infty$ (optically very thick atmosphere) and $\tau_{2}=0$ ($I(\tau_{2}=0)\Rightarrow$ emergent intensity), then $I_{\lambda}(\tau_{\lambda}=0)=\int_{0}^{\infty}S_{\lambda}(\tau^{\prime}_{\lambda})e^{-\tau^{\prime}_{\lambda}}\,d\tau_{\lambda}^{\prime}\,.$ (2.7) A second order expansion of $S(\tau_{\lambda})$ leads to the Eddington-Barbier relation $I_{\lambda}(\tau_{\lambda}=0)\approx S_{\lambda}(\tau_{\lambda}=1)\,.$ (2.8) This says that the observed intensity $I_{\lambda}$ at a wavelength $\lambda$ is approximately given by the source function at optical depth $\tau_{\lambda}=1$ at this same wavelength. In LTE, the intensity then follows the Planck function $B_{\lambda}(T(\tau_{\lambda}=1))$. In spectral lines, the opacity is much increased over the continuum opacity. Since the temperature decreases with height in the solar photosphere the intensity in spectral lines is decreased, and we probe higher and cooler layers. This explains the formation of absorption lines in LTE. In non-LTE, when collisional transitions between atomic levels occur seldom and near the outer atmospheric border, photons can escape and are thus lost for the build-up of a radiation field in the specific transition. Then, the upper level of the transition becomes underpopulated and the source function has decreased below the Planck function at the local temperature. It follows that, even for constant temperature atmospheres, a strong absorption line can be observed. Outside the solar limb, in the visible spectral range, one observes spectral lines (and very weak continua) in emission. In spectral lines, high chromospheric structures are seen in front of a dark background. ### 2.2 Hydrogen Balmer-$\alpha$ line (H$\alpha$) H$\alpha$ at $6563$ Å is a strong absorption line in the solar spectrum for two reasons: 1) hydrogen is the most abundant element in the Sun, and in the Universe. 2) The Sun, as a G2 $\mathrm{V}$ star, has the appropriate effective temperature $T_{eff}\approx 5\,800$ K to have the second level of hydrogen populated and thus to make absorptions in H$\alpha$ possible. As all strong lines, H$\alpha$ possesses a so-called Doppler core and damping wings. The Doppler core of H$\alpha$ and of other Balmer lines is much broader than of other strong lines from metals (atomic species with $Z>2$). The reason is the large thermal velocity of hydrogen compared to that of metals, thus leading to large Doppler widths $\Delta\lambda_{D}=\frac{\lambda_{0}}{c}\sqrt{\frac{2\,\mathcal{R}\,T}{\mu}}\,,$ (2.9) where $\lambda_{0}$ is the rest wavelength, $c$ the speed of light, $\mathcal{R}$ the universal gas constant, $T$ the temperature and $\mu$ the atomic weight (H has the minimum value among the chemical elements of $\mu=1.008$). An eventual “microturbulent” broadening has been omitted in Eq. 2.9. Another property of the Balmer transitions between the according hydrogen levels is the following: Chromospheric lines such as the Ca ii H and K and the Mg i h and k lines are weakly coupled to the local temperature through collisional transitions, effected by electrons, between the involved energy levels. Thus, these lines still contain information about the temperature of the electrons, although only in a “hidden” manner. However, for the Balmer lines of hydrogen and here especially for H$\alpha$, there exist also the routes for level populations through radiative ionization to the continuum and radiative recombination. These routes are taken much more often than the collisional transitions between the involved levels. The ionizing radiation fields, i.e. the Balmer and Paschen continua, originate in the lower to middle photosphere and are fairly constant, irrespective of the chromospheric dynamics. Only when many high-energy electrons, as during a flare, are injected into the chromosphere the H$\alpha$ line reacts to temperature and gets eventually into emission. Nonetheless, the chromosphere observed in H$\alpha$ exhibits rich structuring, due to absorption by gas ejecta, due to Doppler shifts of the H$\alpha$ profile in fast gas flows along magnetic fields, and due to channeling of photons around absorbing features. ### 2.3 He i 10830 Å multiplet Helium is the second most abundant element in the Universe, also in the Sun. It was first discovered in the Sun in 1868 (from where it was named after the greek word of Sun). At the typical chromospheric temperatures there is not enough energy to excite electrons to populate the upper levels from where these transitions occur. In coronal holes the helium lines are substantially weaker compared with the quiet Sun outside the limb. More information about recent advances in measuring chromospheric magnetic fields in the He I 10830 Å line can be found in Lagg (2007). The energy levels that take part in the transitions of the He i 10830 Å multiplet are basically populated via an ionization-recombination process (Avrett et al., 1994). The much hotter corona irradiates at high energies both outwards to space and inwards, to the chromosphere. The EUV coronal irradiation (CI) at wavelengths lambda $\lambda<504$ Å ionizes the neutral helium, and subsequent recombinations of singly ionized helium with free electrons lead to an overpopulation of the upper levels of the He i 10830 multiplet. Alternative theories suggest other mechanisms that may also contribute to the formation of the helium lines via the collisional excitation of the electrons in regions with higher temperature (e.g. Andretta and Jones, 1997). Figure 2.1: Schematic Grotrian diagram for the He i 10830 Å multiplet emission lines. The He i 10830 Å multiplet consists of the three transitions of the orthohelium (total spin of the electrons $S$=1) energy levels, from the upper term with angular momentum $L=1$ to the lower with $L=0$, in particular from 3P2,1,0, which has three sublevels ($J=2,1,0$), to the lower metastable term 3S1, which has one single level ($J=1$) (see Fig. 2.1). The two transitions from the J=2 and J=1 upper levels appear mutually blended, i.e. as merely one line, at typical chromospheric temperatures, and form the so-called red component, at 10830.3 Å. The two red transitions are only 0.09 Å apart. The blue component, at 10829.1 Å, corresponds to the transition from the upper level with J=0 to the lower level with J=1. The formation height of these lines is believed to be between 1 500 and 2 000 km, (e.g. Centeno 2006) although, as we already mentioned, the chromosphere is strongly rugged. The Landé factors of the lines are not zero, meaning that they are sensitive to external magnetic fields. A more detailed description about the properties of the He i 10830 multiplet, in particular related to the emission profiles observed in spicules above the limb is given in Chapter 5. ## Chapter 3 Observations For the present work we used data from two different instruments, both mounted on the same telescope, the _Vacuum Tower Telescope_ (VTT, Sec. 3.2) in Tenerife. One of the instruments, the _Göttingen Fabry-Perot Interferometer_ (G-FPI, Sec. 3.3.1) is able to achieve very high spatial resolution while the other, the _Tenerife Infrared Polarimeter_ (TIP, Sec. 3.4.2), is able to obtain full Stokes spectropolarimetric data with very high spectral resolution. Both instruments, in combination with the _Kiepenheuer Adaptive Optics System_ (KAOS, Sec. 3.2.1), provided the data for this work. In this Chapter we will describe the telescope, the instrumentation, the observations, and the reduction techniques. The latter are aimed at removing as many instrumental effects as possible. ### 3.1 Angular resolution and _Seeing_ When using any kind of an optical imaging system, the angular resolution in the focal plane is limited by diffraction at the aperture of the instrument. For circular apertures the image of a point source (the PSF) is an Airy function with a certain Full Width at Half Maximum (FWHM). Two point sources closer than the FWHM of a certain instrumental PSF are difficult to distinguish. If one considers diffraction of a telescope with a circular aperture of diameter $D$, the angular resolution limit is, in the usual Rayleigh definition, $\alpha_{min}=1.22\,\frac{\lambda}{D}\,.$ (3.1) The factor 1.22 is approximately the first zero divided by $\pi$ of the Bessel function involved in the Airy function. In the focal plane of such a telescope with a focal length $f$ the spatial resolution is therefore $d=1,22\,\lambda\,f/D$. For good sampling this should correspond to, or even be larger than, the resolution element of the detector (2 pixels). In the case of the VTT, with a main mirror of $D=70$ cm, the diffraction limited resolution is $0\farcs 24$ at $6563$ Å (H$\alpha$) and $0\farcs 39$ at $10830$ Å (He i triplet). In solar observation, it is common to use as the diffraction limit simply $\alpha_{min}=\lambda/D$. At this angular distance the modulation transfer function (MTF) has become zero. Unfortunately all imaging systems on the ground are subject to aberrations that degrade the image quality, resulting in a much lower spatial resolution than the diffraction limit. The light we observe from the Sun travels unperturbed along approximately 150 million km, but during the last few microseconds before detection it becomes distorted due to its interaction with the Earth’s atmosphere and our optical instrument. The refraction index of the air is very close to 1 at optical wavelengths, but depends on the local pressure and temperature. Their fluctuations in space and time produce aberrations of the wavefronts from the object to be observed111The local values of the temperature and pressure depend on the complicated turbulent dynamics of the atmosphere. This includes friction and heating of the Earth’s irregular surfaces, condensations and formation of clouds, shears produced by strong winds, …For more information we refer to e.g. Saha 2002.. Since the time scale of the variation of the aberrations of $\approx 10$ ms is usually smaller than the integration time, it also produces smoothing of the image details. Thus, the information at small scales is lost. Further, the turbulent state of the air masses through which the light is passing varies on small angular scales. This produces an anisoplanatism of the wavefronts arriving at the telescope, with angular sizes of the isoplanatic patches not larger than approximately $10\arcsec$. Beside the atmospheric factors, the final quality of the image is influenced by local factors like the aerodynamical shape of the telescope building or convection around the building and the dome. Finally, the internal _seeing_ of the telescope plays an important role for the image quality. Convection along the light path in the telescope triggered by heated optical surfaces can be avoided by allowing air flowing freely through the structure or, quite the contrary, by evacuating the telescope. In solar physics we usually measure the average image quality of the observations estimating the diameter of a telescope that would produce, from a point source, an image with the same diffraction-limited FWHM as the atmospheric turbulence or internal _seeing_ would allow even with a much larger telescope aperture. This is called the Fried parameter ($r_{0}$). Typically, upper limits for the “Observatorio del Teide” are $r_{0}\approx 15$ cm during night-time and $r_{0}\approx 7$ cm during day. Besides these structural requirements for best _seeing_ conditions, there are nowadays methods for correcting the images for seeing distortions to obtain near diffraction limited resolution. In this thesis we have used various methods: We correct partially the aberrations in real time using adaptive optics (Sec. 3.2.1) which can increase the $r_{0}$ around the center of the field of view up to $r_{0}\sim 25$ cm and we also apply post-processing methods of image reconstruction (Sec. 3.3.3) to approach the upper limit of $r_{0}\lesssim 70$ cm. ### 3.2 Telescope The _Vacuum Tower Telescope_ (VTT, Soltau, 1985, Fig. 3.1) is located at the Spanish “Observatorio del Teide” (2400 m above sea level, 1630’ W, 2818’ N) in Tenerife, Canary Islands. It is operated by the Kiepenheuer-Institut für Sonnenphysik, Freiburg, with contributions from the Institut für Astrophysik in Göttingen, the Max-Planck-Institut for Sonnensystemforschung, Katlenburg- Lindau, and the Astrophysikalisches Institut Potsdam. Figure 3.1: Building which houses the solar _Vacuum Tower Telescope_. The VTT optical setup is depicted in Fig. 3.2. At the top platform of the building, a coelostat achieves to follow the path of the Sun on the sky, by means of two flat mirrors of very high optical quality. The primary coelostat mirror rotates clockwise (seen pole-on) about an axis which is contained in the mirror surface and is parallel to the Earth’s rotation axis. It reflects the sunlight towards the secondary mirror. The latter redirects the beam towards the fixed telescope in the tower. The telescope is an off-axis system. It consist of a slightly aspherical main mirror of 70 cm diameter and a focal length of 46 m, and of a folding flat mirror. The free aperture of the circular entrance pupil with D=70 cm gives the telescopic diffraction limit for the angular resolution of $\alpha_{min}=\lambda/D\approx 0\farcs 16$ for $\lambda$ in the visible spectral range. To avoid turbulent air flows inside the telescope caused by heated surfaces, the telescope is mounted in a tank that is evacuated to 1 mbar. The vacuum tank has high quality transparent entrance and exit windows located below the coelostat and close to the primary focus, respectively. Shortly after the entrance window, a small part of the sunlight is reflected out to a second imaging device. This uses a quadrant cell to track the image of the solar disc and to correct slow image motions, e.g. due to a non-perfect hour drive of the coelostat. Telescope pointing to a target inside and near the solar disc is achieved by moving this tracking device as a whole in the image plane. The imbalanced illumination of the quadrant cell is transformed to a tip-tilt motion of the secondary coelostat mirror. After the main vacuum tank, the adaptive optics (Sec.3.2.1) device is located. This optical system is able to correct in real time the low order aberrations of the incoming wavefronts of the light beam caused by the turbulence in the Earth’s atmosphere. After the adaptive optics system, which can optionally be moved in or out of the path, the light path continues to the vertical slit spectrograph or to a folding mirror that can be used to direct the light to different other available science instruments. Figure 3.2: Optical setup of the VTT. The coelostat (mirrors _m1,m2_) follows the path of the Sun on the sky and directs the light to the entrance window of the vacuum tank (blue shaded). Mirror _m3_ takes out a small amount of the light and feeds the guiding telescope mounted outside the vacuum tank. The collimating mirror _m5_ produces, together with the flat mirror _m6_ , the solar image in the primary focal plane behind the exit window of the vacuum tank. There, a flat mirror can be mounted under $45^{\circ}$ to the vertical (not shown) to feed post-focus instruments in optical laboratories. The adaptive optics system is located below the exit window, and it is used optionally. #### 3.2.1 Kiepenheuer Adaptive Optics System As mentioned in the beginning of this Chapter (Section 3.1) the atmosphere of the Earth degrades the quality of the images during observations. KAOS (Kiepenheuer Adaptive Optics System, von der Lühe et al., 2003; Berkefeld, 2007) is a realtime correction device that calculates and corrects the instantaneous aberrations of the wavefront using special deformable mirrors. Figure 3.3: Scheme of typical AO. Inside the closed loop, a fraction of the incoming light is directed to the KAOS camera (semitransparent mirror _m1_), where a lenslet array (_ll_) produces many subfield images with light from different parts of the pupil. The calculated instantaneous aberration is compensated using the two (tip&tilt and deformable) mirrors, every 0.4 ms. The optical scheme of a typical adaptive optics (AO) system is shown in Fig. 3.3. By means of a dichroic semitransparent beam splitter, part of the light entering the system is directed to the wavefront sensor. The latter, a Shack- Hartmann sensor, consists of a lenslet array positioned in an image of the entrance pupil and a fast CCD detector. Each lenslet, cutting out a subaperture of the pupil image, produces an image of a small area on the Sun on a subarea of the CCD. Using a good, i.e. as sharp as possible, subimage of the present scenery on the Sun and with a correlation algorithm, it is possible to compute the displacement of each subimage and to estimate from this the aberrations of the wavefront. Every aberration can be expressed by a sum of adequate polynomials (for example Zernike polynomials) with appropriate coefficients. Each polynomial represents a specific wavefront aberration, e.g. tilt, defocus, astigmatismus …The AO is able to correct the low orders of the aberration, that is those with the largest scales. For this purpose it has two active optical surfaces (both of them in the main lightbeam, so the correction is done in a closed loop). In the case of KAOS the first element is the tip- tilt mirror that is able to displace the whole image in two perpendicular directions, thus tracking on the reference image. The second optical element is a bymorphous deformable mirror with 35 actuators. With appropriate voltages, the surface of this mirror obtains a shape that corrects the aberrations of the incoming wavefront up to the $27^{th}$ Zernike polynomial. This correction is done in a fast closed loop at 2100 Hz. The bandwidth of KAOS is 100 Hz. It thus operates at timescales comparable to that of the variation of the turbulence in the atmosphere. As already mentioned, the aberration of the wavefront is not constant, i.e. not isoplanatic across the whole field of view (FoV). The wavefront camera has a restricted FoV of $12\arcsec\times 12\arcsec$ where the assumption of isoplanatism is approximately valid. The center of this subfield of AO correction is called _lockpoint_. The restricted area of isoplanatism is one of the main limitations of current AO systems. The corrections are calculated for the lockpoint feature we are tracking on and applied to the whole FoV of the telescope. Therefore the correction becomes increasingly inaccurate with increasing distance from the lockpoint. The quality of the image is degraded outwards from the center of the FoV, where the _lockpoint_ is usually located. Fortunately this can be taken into account using post factum image reconstruction like speckle interferometry and blind deconvolution. In night-time astronomy, AO systems lock on a star image, so the displacements of the subfields imaged by the lenslet are easily calculated. In solar observations, the image used by the AO comes always from an extended source, making the calculations of the displacements much more demanding. In solar AOs, a reference image is taken and updated regularly during operation, and correlations between this image and the subfield images are used. For well defined maxima of the correlation functions we need features with sufficient contrast inside the FoV to lock on with the algorithm, e.g. a pore or the granulation pattern. Moreover, the wavefront sensor can only work with a high light level, e.g. integrated over some wavelength. So it is not possible to lock for example on features within the H$\alpha$ line with low intensity. Also, as we will explain in Sec. 3.4.2, near or off-limb observations are difficult as the AO algorithm is not able to track on that kind of references, as the one-dimensional limb image. ### 3.3 High spatial resolution For our study of the dynamics of chromospheric structures, we are interested in observations with the highest possible spatial resolution222It has become a widespread custom in solar observations to use “spatial resolution” synonymously with “angular resolution”., with the highest achievable temporal cadence, and with as much spectral information as possible. For that purpose we used the “Göttingen” Fabry-Perot Interferometer (G-FPI). Here, the designation FPI stands as _pars pro toto_ , for the whole post-focus instrument, a two-dimensional spectrometer based on wavelength scanning Fabry- Perot etalons. It was developed at the Universitäts-Sternwarte Göttingen (Bendlin et al., 1992; Bendlin, 1993; Bendlin and Volkmer, 1995). Subsequently, it had undergone several upgrades (Koschinsky et al., 2001; Puschmann et al., 2006; Bello González and Kneer, 2008). For the present work, the G-FPI with the high-efficiency performance described by Puschmann et al. (2006) was employed. Basically, this instrument was able, at the time the data for this study were taken, to produce an image from a selected wavelength range with a narrow passband of 45 mÅ FWHM at 6563 Å (H$\alpha$). A recent upgrade has reduced the FWHM. The spectrometer also can be tuned to almost any desired wavelength, being able to scan a spectral line, producing 2D filtergrams (images) at, e.g., 20 spectral position along a line. If we scan iteratively one spectral line we obtain a time sequence of very high spatial resolution, at several spectral positions and with a cadence which would be the time required to scan the full line, which is typically in the order of 20 seconds for our data. The main limitation of this kind of observational procedure is that the images corresponding to a single scan are not obtained simultaneously, as they are taken consecutively. This is of special importance when we compare the images in the two wings of a spectral line, as the small-scale solar structure under study may have changed during the time needed to scan between these positions. This should be taken into account when studying features whose typical timescale of variation is comparable to the scanning time. In Sec. 4.2.1, we will see that this limitation can partly be compensated when we have a long temporal series. #### 3.3.1 Instrument The Göttingen Fabry-Perot Interferometer (Bendlin and Volkmer, 1995; Volkmer et al., 1995; Koschinsky et al., 2001; Puschmann et al., 2006) is a speckle- ready two-dimensional (2D) spectrometer. It is able to scan a spectral line producing a set of speckle images at several spectral position with a narrow spectral FWHM, while taking simultaneous broadband images, needed for the post factum image reconstruction. ##### Fabry-Perot interferometer (FPI) A Fabry-Perot interferometer, or etalon, is an interference filter possessing two plane-parallel high-reflectance layers of high quality ($\sim\lambda/100$). Light entering the filter is many times reflected between the plane-parallel reflecting surfaces. These reflections will produce destructive interference for transmitted light at all wavelengths but the ones for which two times the spacing $d$ of the plates is very close to a multiple of the wavelength. This effect gives rise to a final Airy intensity function (Born and Wolf, 1999): $I=I_{max}\frac{1}{1+\frac{4R}{(1-R)^{2}}\sin^{2}\frac{\delta}{2}}\,,$ (3.2) where the maximum intensity $I_{max}=\frac{T^{2}}{(1-R)^{2}}$ , $T$ is the transmittance, $R$ is the reflectance ($R=1-T$ if absorption is negligible), and the dependence on wavelength $\lambda$, angle of incidence $\Theta$, and refractive index $n$ of the material between the surfaces is $\delta=\frac{4\pi}{\lambda}nd\,\cos\Theta\,.$ (3.3) (a) (b) Figure 3.4: Example of the narrow-band scanning with the G-FPI. Left: One narrow-band frame from a two-dimensional spectrometric scan through the hydrogen Balmer-$\alpha$ line (H$\alpha$). Right: H$\alpha$ line; _solid black_ from the Fourier Transform Spectrometer (FTS) atlas (Brault & Neckel, quoted by Neckel 1999); _blue_ : FTS profile convolved with the Airy transmission function of the FPIs; _dashed_ average $H\alpha$ profile observed with the spectrometer at 21 wavelength position (_rhombi_) with steps of 100 mÅ. The _red_ line is the Airy transmission function, positioned at the wavelength in which the image in the left panel was taken, and re-normalized to fit on the plot.. The narrow transmittance of the filter can be tuned to any desired wavelength by changing the spacing $d$ (or the refractive index $n$, for pressure controlled FPIs). One single FPI produces a channel spectrum according to the interference condition, i.e. for normal incidence ($\Theta=0^{\circ}$) and assuming $n$=1, $m\lambda=2d$ (3.4) with $m$ being the order. From here, the distance to the next transmission peak, or _free spectral range (FSR)_ , follows as $\emph{FSR}=\frac{\lambda^{2}}{2d}\,.$ (3.5) To suppress all but the desired transmission, the G-FPI has a second Fabry- Perot etalon with different spacing, i.e. different _FSR_. Both Fabry-Perot etalons need to be synchronized when scanning in order to keep the desired central transmittance peaks coinciding. The combination of two FPI with different _FSR_ removes effectively the undesired transmission peaks from other orders. An additional interference filter ($FWHM\approx 8\,\AA$) is used to reduce the incoming spectral range to the spectral line under observation. The combination of these three elements produces a single narrow central peak, as depicted in Fig. 3.5. The FP etalons are mounted close to an image of the telescope’s entrance pupil in the collimated, i.e. parallel, beam. On the one hand, this avoids the “orange peel” pattern in the images, which one obtains with the telecentric mounting near the focus and which arises from tiny imperfections of the etalon surfaces. On the other hand, in the collimated mounting one has to deal with the fact that the wavelength position of the maximum transmission depends on the position in the FoV. This can be seen from Eq. 3.3 where the angle of incidence $\Theta$ changes with position in the FoV. For the post factum image reconstruction (Sec. 3.3.3) we have to acquire simultaneously short-exposure images from the narrow-band FPI spectrometer and broadband images. The latter are taken through a broadband interference filter ($FWHM\approx 50\,\AA$) at wavelength close to the one observed with the spectrometer. Two CCD detectors, one for each channel, with high sensitivity and high frame rates were used which allow a high cadence of short exposures. All processes (simultaneous exposures, synchronous FPI scanning and observation parameters) are controled by a central computer. The imaging on the two CCDs is aligned with special mountings and adjusted to have the same image scale on the two detectors. The optical setup is shown schematically in Fig. 3.6. From the focal plane following KAOS the image from the region of interest on the Sun is transferred via a $1:1$ re-imaging system into the optical laboratory housing the FPI spectrometer. In front of the focus at the spectrometer entrance, a beam splitter directs 5% of the light into the broadband channel. The latter contains a focusing lens, the broadband interference filter (IF1), a filter blocking the infrared light (KG1, from _Kaltglas_ = “cold glass”, notation by Schott AG), a neutral density filter to reduce the broadband light level, and a detector CCD1. Most of the light (95 %), enters the narrow-band channel of the spectrometer through a field stop at the entrance focus. After the field stop follow: an infrared blocking filter (KG2), the narrow interference filter (IF2), a collimating lens giving parallel light, the two Fabry-Perot etalons (FPI-B and FPI-N), a camera lens focusing the light on the detector CCD2. Figure 3.4 gives an example of the type of observation one can obtain with this narrow- band spectrometer. The instrument has additional devices for calibration and adjustment: a feed of laser light, facilities to measure with a photomultiplier and to aid identifying the spectral line to be observed, and a feed of continuum light for various purposes, e.g. co-aligning the transmission maxima of the etalons or measuring the transmission curve of the pre-filter IF2. Figure 3.5: Transmission functions for the narrow-band channel of the G-FPI with the H$\alpha$ setup. The periodic Airy function of the narrow-band FPI (dashed line) coincides in the central wavelength with that of the broadband FPI (strong dashed green line). The global transmission of both FPIs has one single strong and narrow peak at the central wavelength (purple strong line). An additional interference filter (red line) is mounted to restrict the light to the scanned spectral line. Figure 3.6: Schema of the “Gottingen” Fabry- Perot interferometer optical setup. After KAOS, the light is transferred from the telescope’s primary focus to the spectrometer. A beam splitter BS directs 5% of the light into the broadband channel consisting of a focusing lens L1, a broadband interference filter IF1 ($FWHM\approx 50\,\AA$), an infrared blocking filter KG1 (“Kaltglas”), a neutral density filter ND, and the CCD1 detector. 95% of the light enter the spectrometer through a field stop at the entrance focus. Then follow: infrared blocking filter KG2, interference (pre-) filter IFII ($FWHM\approx 6\AA\dots 10\AA$, depending on the spectral line and wavelength range), collimating lens L2, the two FPI etalons FPI B and FPI N ($FWHM\approx 45m\AA$ at H$\alpha$), the focusing camera lens L3 and the CCD2 detector. CCD1 and CCD2 take short-exposure (3-20 ms) images strictly simultaneously. #### 3.3.2 Observations For the study of the chromospheric dynamics on the basis of high resolution observations we have used three data sets. Table 3.1 lists the details for each data set: * • Dataset mosaic focuses on the study of a large active solar region, where we find fast moving dark clouds, as we will discuss in Sec. 4.1. These data were obtained before the instrument upgrading in 2005 (Puschmann et al., 2006) with the old cameras. The exposure time was six times longer than with the new CCDs and the FoV of a single frame is one fourth of that of the new version of the G-FPI. The observers of these data were Mónica Sánchez Cuberes, Klaus Puschmann and Franz Kneer. * • Dataset sigmoid uses the improvements of the instrument from 2005 and was obtained during excellent seeing conditions from a very active region. During the time span of our observations at least one flare was recorded from this region in our FoV. Our focus with these data is the study of fast events and magnetoacustic waves (Sec. 4.2.4) with the original intention to detect Alfvén waves. Examples of these data were also used to compare the results from different methods of _post factum_ image reconstruction, as we will show in Sec. 4.3. * • With dataset limb and in Sec. 4.3 we apply blind deconvolution methods for image reconstruction (see Sec. 3.3.3). The observations were taken with the G-FPI, renewed in 2005, to study with very high spatial resolution the evolution of spicules as seen in the H$\alpha$ line. Data set name \| “mosaic” \| “sigmoid” \| “limb” ---\|---\|---\|--- Date \| May,31st,2004 \| April,26th,2006 \| May,4th, 2005 Object \| AR0621 \| AR10875 \| limb Heliocentric angle \| $\mu=0.68$ \| $\mu=0.59$ \| $\mu=0$ Scans # \| 5 \| 157 \| 5 Cadence \| 45 s \| $\sim 22$ s (see Sec. 4.2.1) \| $\sim 19$ s Time span \| 4 min \| 55 min \| 2 min Line positions # \| 18 \| 21 \| 22 FWHM \| 50 Å broadband / 45 mÅ narrow-band Broadband filter \| 6300 Å Stepwidth \| 125 mÅ \| 100 mÅ \| 93 mÅ Exposure time \| 30 ms \| 5 ms Seeing condition \| good \| $r_{0}\approx 32$ cm \| $r_{0}\approx 20$ cm KAOS support \| yes Image reconstruction \| speckle \| AO ready speckle \| MFMOBD Field of view \| 33$\times$ 23(total 103$\times$94) \| 77$\times$ 58 Table 3.1: Characteristics of the data sets taken with the G-FPI used in this work. #### 3.3.3 Data reduction After the recording of the data, several processing steps have to be carried out in order to minimize the instrumental effects. These are mainly to take into account the differential sensitivity of the CCDs from one pixel to another or the fixed imperfections on the optical surfaces positioned close to one of the focal planes. This concerns for example dust on the beam splitter, on the infrared blocking filters and interference filters and the CCDs. In this step we also remove an imposed bias signal applied electronically to every frame. This is the usual treatment of any CCD data. For this purpose we take flat fields, dark, continuum and target images (see Fig. 3.7). (a) Broad band raw frame (b) Flat field frame (c) Dark frame (d) Reduced frame Figure 3.7: Example of the standard data reduction process. Every frame taken with the CCD (a) includes instrumental artifacts like shadows from dust particles on the CCD chips or the filters near the focus (Fig. b) and the intrinsic differential response of each pixel (c). Subtracting the dark frame and dividing by the flat response provides a clean frame (d). _Target_. A target grid is located in front of the instrument, in the primary focal plane. Target frames therefore display in both channels a grid of lines that are used to focus and align the cameras in both channels. This is crucial for the image reconstruction. _Continuum_ data are taken with the same scanning parameters as with sunlight but using a continuum source, so we can test the transmission of the scanning narrow-band channel. _Dark_ frames are taken with the same integration time but blocking the incident light. These frames have information of the differential and total response of the CCD array without light, in order to remove this effect from the scientific data. _Flat fields_ are frames with the same scanning parameters and with sunlight, but without solar structures. In this way we can see the imperfections and dust on the optical surfaces fixed on every frame taken with the instrument, and remove them dividing our science data by these flat frames. To avoid signatures from solar structures in the flat frames, the telescope pointing is driven to make a random path around the center of the solar disc far from active regions. Thus, to reduce the instrumental effects we use the following formula, for each channel and for each spectral position independently: $reducedframe=\frac{raw\,frame-mean\,dark}{mean\,flatfield-mean\,dark}\,.$ (3.6) Our instruments produce data sets that can be subject to _post factum_ image reconstruction. We have applied speckle and blind deconvolution methods to minimize the wavefront aberrations and to achieve spatial resolution close to the diffraction limit imposed by the aperture of the telescope. The aberrations are changing in time and space. In a long exposure image, the temporal dependence will produce the summation of different aberrations, blurring the small details of the image. Therefore, for post-processing, all image reconstruction methods need input _speckle_ frames with integration times shorter than the typical timescale of the atmospheric turbulence. With this condition fulfilled, the images appear distorted and speckled but not blurred, and still contain the information on small-scale structures. Another common characteristic of speckle methods is the way to address the field dependence of the aberrations. In a wide FoV each part of the frame is affected by different turbulences. That is, inside the atmospheric column affecting the image, there are spatial changes of the wavefront aberration. Therefore, the FoV is divided into a set of overlapping subfields smaller than the typical angular scale of change of the aberrations (5– 8), the isoplanatic patch. Speckle interferometry denotes the interference of parts of a wavefront from different sub-apertures of a telescope. This results in a speckled image of a point source, e.g. of a star. The effect is used for “speckle interferometric” techniques of postproccesing. They are able to remove the atmospheric aberrations of the wavefronts that degrade the quality of the images. In the following Sections we introduce the basic background of the methods used and provide some examples and further reference. ##### Speckle interferometry of the broadband images This method is based on a statistical approach to deduce the influence of the atmosphere. It was developed following the ideas of Fried (1965); Labeyrie (1970); Korff (1973); Weigelt (1977); von der Lühe (1984) . The code used for our data was developed at the Universitäts-Sternwarte Göttingen (de Boer, 1996) . The _sigmoid_ dataset uses the latest improvements to take into account the field dependence of the correction from the AO systems (Puschmann and Sailer, 2006). In what follows we present a brief overview of the method: The observed image (_i_) is the convolution ($\star$) of the true object (_o_) with the _Point Spread function ( $PSF$)_. The $PSF$ is the intensity distribution in the image plane from a point source with intensity normalized to one, i.e. $\int\int PSF(x,y)dxdy=1\,,$ (3.7) where the integration is carried out in the image plane. The $PSF$ depends on space, time and wavelength. Its Fourier transform ($\mathscr{F}$) is the _OTF, Optical Transfer Function_ $\mathscr{F}(i)=\mathscr{F}(o\star PSF)\hskip 14.22636pt\rightarrow\hskip 14.22636ptI=O\cdot OTF\,.$ (3.8) A normal long exposure image would be just the summation of N speckle images: $\sum^{N}_{i=1}I_{i}=O\cdot\sum^{N}_{i=1}OTF_{i}\,.$ (3.9) The $OTF_{i}$ are continuously changing in time, which leads to a loss of information. The temporal phase change of the $OTF_{i}$ will, upon this summation, reduce strongly or even cancel the complex amplitudes at high wavenumbers. Labeyrie (1970) proposed to use the square modulus, to avoid cancellations: $\frac{1}{N}\sum^{N}_{i=1}\|I_{i}\|^{2}=\|O\|^{2}\cdot\frac{1}{N}\sum^{N}_{i=1}\|OTF_{i}\|^{2}=\|O\|^{2}\cdot STF\,.$ (3.10) Yet this procedure also removes the phase information on $o$. Thus, the phases have to be retrieved afterwards. _STF_ is the _Speckle Transfer Function_ , it contains the information on the wavefront aberrations during N speckle images. To deduce this STF is therefore one of the aims of the speckle method. On the Sun, point sources do not exist. It is thus not a trivial task to determine the $STF$. There are, however, models of $STF$ for extended sources from the notion that they depend only on the seeing conditions, through the _Fried_ parameter $r_{0}$ (Korff, 1973). This parameter can be calculated _statistically_ using the spectral ratio method (von der Lühe, 1984). As this is a statistical approach, a minimum number of speckle frames must be used, more than 100. To recover the phases of the original object the code uses the speckle masking method (Weigelt, 1977; Weigelt and Wirnitzer, 1983). It recursively recovers the phases from low to high wavenumbers. Finally a noise filter is applied, zeroing all the amplitudes at wavenumbers higher than a certain value, which depends on the quality of the data. (a) Average of 330 speckle images (total exposure time $\sim 1,6$ s). (b) Single speckle frame, 5 ms exposure time. (c) Reconstructed broadband image, using 330 speckle frames. Figure 3.8: Example of improvement of broadband images with the speckle reconstruction. The size of the image is $\sim$ 34$\times$ 19\. The achieved spatial resolution is close to the diffraction limit, $0\farcs 22$, with the diffraction limit $\alpha_{min}=\lambda/D\,\hat{=}\,0\farcs 19$ at $\lambda=6563$ Å (H$\alpha$) and telescope aperture $D=70$ cm. Figure 3.9: Power spectra showing the influence of the _post factum_ reconstruction. Ordinate is the relative power on logarithmic scale, and abscissa is the spatial frequency, from the largest scales near the origin to the smallest scales at the Nyquist limit, corresponding to two pixels. A long exposure image (_black dotted line_), taking the average of all speckle images, has very low noise, but the power is also low at all frequencies $\geqslant 0.8$ Mm-1 (blurring effect). A single speckle frame (_dashed blue line_) has more power at all frequencies, but also much more noise (more than two order of magnitude). The speckle reconstructed frame (_red solid line_) keeps the noise low while it possesses higher power at all frequencies, where the spatial information on small-scale structures is stored. ##### Influence of the AO on the speckle interferometry As explained in Sec. 3.2.1 the AO systems provide a realtime correction of the low order aberrations (up to a certain order of Zernike polynomials). Nonetheless, given the anisoplanatism of the large field of view, the corrections are calculated for the lock point and applied to the whole frame, resulting in a degradation of the image correction from the lock point outwards. The problem arises from the different atmospheric columns traversed by the light from different parts in the FoV. This creates, after the AO correction, an annular dependence of the correction about the lock point and therefore an annular dependence of the $STF$s when processing the data. Puschmann and Sailer (2006) provided a modified version of the reconstruction code that computes different $STF$s for annular regions around the lock point, providing a more accurate treatment over the field of view. The _sigmoid_ dataset was reduced using this last version of the code, improving substantially the quality of the results. Both AO and speckle interferometry work best with good seeing, and this data set was recorded under very good seeing conditions. ##### Speckle reconstruction of the narrow-band images The narrow-band channel scans the selected spectral line, taking several ($\sim 20$) images per spectral position. The statistical approach as for the broadband data can not be applied given the low number of frames per spectral position. To reconstruct these images from this channel we use a method proposed by Keller and von der Lühe (1992) and implemented in the code by Janssen (2003). For each narrow-band frame, there is a frame taken simultaneously in the broadband channel, which is degraded by the same wave aberrations. The images in the broadband channel were taken at 6300 Å, i.e. at a wavelength 260 Å shorter than that of H$\alpha$. We neglect the wavelength dependence of the aberration. For each position in the spectral line, for each subfield, we have a set of pairs of simultaneous speckle images from the narrow- and broadband channel, with a common $OTF_{i}$ for each realization in both channels: $I_{Broad_{i}}=O_{Broad}\cdot OTF_{i}$ (3.11) $I_{Narrow_{i}}=O_{Narrow}\cdot OTF_{i}$ (3.12) Using Equation 3.11 in 3.12, the reconstructed narrow-band image $O_{Narrow}$ is obtained from the minimization of the error metric $E=\sum_{i=1}^{N}\Big{\|}O_{Narrow}\cdot\frac{I_{Broad_{i}}}{O_{Broad_{i}}}-I_{Narrow_{i}}\Big{\|}^{2}\,,$ (3.13) where $N$ is the number of images taken at one wavelength position. Minimization of $E$ with respect to $O_{Narrow}$ yields $O_{Narrow}=H\cdot\frac{\sum_{i=1}^{N}I_{Narrow_{i}}\cdot I_{Broad_{i}}^{}}{\sum_{i=1}^{N}\|I_{Broad_{i}}\|^{2}}\cdot O_{Broad_{i}}\,.$ (3.14) Here we have included a noise noise filter ($H$) to remove the power at spatial frequencies higher than a certain threshold above which the noise dominates. The noise power is obtained from the flat field data. ##### Multi object multi frame blind deconvolution (MOMFBD) The speckle interferometry method presented above relies on a statistically average influence of the wavefront aberration. In this Section we shortly present another approach that we have also used in this work. It is based on the simultaneous estimation of the object and the aberrations in a maximum likelihood sense using different simultaneous channels and several speckle frames. For more information see e.g. (Löfdahl, 2002; van Noort et al., 2005; Löfdahl et al., 2007). The method used is called _Multi Object Multi Frame Blind Deconvolution_ (MOMFBD), which historically is a modification of the “Joint Phase Diverse Speckle” image restoration. The original method is based on the possibility of separating the aberrations from the object if we observe simultaneously in two channels introducing a known aberration, like defocussing the image, in one of them. Mathematically, both phase diversity and multi-object methods are particularizations from the “Multi Frame Blind Deconvolution”. Using a model of the optics, including its unknown pupil image, it is possible to jointly calculate the unaberrated object and the aberration, in a maximum likelihood sense. Coming back to Eq. 3.8 for a single isoplanatic speckle subfield, the Optical Transfer Function (OTF) is the Fourier transform of the Point Spread Function (PSF), which is the square modulus of the Fourier transform of the pupil function (P), that can be generalized with an expression like $P=A\cdot exp(i\phi)\,,$ (3.15) where $A$ stands for the geometrical extent of the pupil (A$=1$ inside pupil, A$=0$ outside). This unknown phase $\phi$ can be then parametrized using a polynomial expansion: $\phi=\sum_{m\in M}\alpha_{m}\psi_{m}\,,$ (3.16) where $\psi_{m},m\in M$, is a subset of a certain basis functions. The MOMFBD uses a combination of Zernike polynomials (Noll, 1976) for tilt aberrations and Karhunen-Loève for blurring effects, as they are optimal for atmospheric blurring effects (Roddier, 1990) . The $\\{\alpha_{m}\\}$ coefficients have therefore the information of the instantaneous wavefront aberration, whether it comes from seeing conditions, telescope aberrations or AO influence. It is interesting to note that the expansion of the phase aberration is therefore finite ($m\in M$) in our calculation, that leads to a systematic underestimation of the wings of the PSF (van Noort et al., 2005) For the calculation of the solution, the MOMFBD code uses a metric quantity that depends only on the $\\{\alpha_{m}\\}$ parameters and is expressed as the least square difference between the $j$ speckle data frames, $D_{j}$, and the present estimated synthesized data frame, obtained by convolving the present estimation of $PSF$ and object. $L(\\{\alpha_{m}\\})=\sum_{u}\Big{[}\sum_{j}^{J}\|D_{j}\|^{2}-\frac{\|\sum_{j}^{J}O^{}_{mj}\widehat{OTF}_{mj}\|^{2}}{\sum_{j}^{J}\|\widehat{OTF}_{mj}\|^{2}+\gamma}\Big{]}$ (3.17) where the $\gamma$ term accounts for the noise and corresponds to an optimum low pass filter (Löfdahl, 2002) and the $u$ index for the spatial index in the Fourier domain. This mathematical expression, from Paxman et al. (1996), to solve the blind deconvolution problem depends on the noise model used. In our case the MOMFBD assumes additive Gaussian statistics, which gives the simplest form of $L$ and the fastest code, and turns to be appropriate for low contrast objects. The solution of the problem of image reconstruction is to find the set of $\\{\alpha_{m}\\}$ that minimizes the metric $L(\\{\alpha_{m}\\})$, providing an estimation of the OTF, and from there the new estimation of the objects. Details on the process and optimization used can be found in Löfdahl (2002). The final converging solution provides thus the real object and instantaneous aberration simultaneously. With only one channel the $\\{\alpha_{m}\\}$ are independent, but if we can specify linear equality constraints (LEC) to these parameters we can reduce the number of unknown coefficients for multiple channels. The Phase Diversity method is one example of LEC. By defocussing one of the cameras on a simultaneous channel we introduce a known phase contribution in the expansion of Eq. 3.16. This creates a set of related pairs of $\\{\alpha_{m}\\}$. Typically, 10 or even less realizations of such pairs of images are enough for a good restoration. Different channels observing simultaneously in different, yet close, wavelengths can be used also to constrain the $\\{\alpha_{m}\\}$, as the instantaneous aberration can be considered the same for all channels. In our case we have several speckle images per position and two simultaneous channels. The broadband channel and the narrow-band channel scanning the spectral line at 21 positions with 20 frames per position. We have therefore a set of 21 pairs of 2 simultaneous objects, with 20 frames for each object and channel. One interesting outcome of this multi object approach is that, if the observed data frames are previously aligned using a grid pattern, the resulting images are then perfectly aligned between simultaneous channels, which greatly reduces possible artifacts on derived quantities as Dopplergrams or magnetograms. In this work we have used this MOMFBD approach to process the data where our usual speckle interferometry method was not applicable. This mainly applies for on-limb observations, as the limb darkening gradient on the field of view influences the statistics. Also, with the actual presence of the off-limb sky, the data are not suitable for the narrow-band speckle reconstruction, as we don’t have a broadband counterpart for the emission features present off the limb. The _limb_ data set was reduced using this code (see Sec. 5.2), as well as some other data frames for comparison purposes with the speckle interferometry (Sec. 4.3). The MOMFBD code was implemented by van Noort et al. (2005) and was made freely available at `www.momfbd.org`. Given the high processing power needed it is written and greatly optimized in `C++`. It is developed to run in a multithreaded clustering environment, where the work is split in workunits and sent back from the slave machines to the master once the processing is done. A typical run with one of our H$\alpha$ scans in broad and narrow-band channel, reconstructing the first 50 Karhunen-Loève modes, takes $\sim 7$ hours to process with 20 CPU cores of $3.2$ GHz. ### 3.4 Infrared spectrometry For this work we have also used spectroscopic data in the infrared region, to study the spicular emission in the He i 10830 Å multiplet. For this purpose we used the echelle spectrograph of the VTT and the Tenerife Infrared Polarimeter (TIP). In this Section we summarize the instrument characteristics, the optical setup and the observations performed for the study of the emission profiles of spicules, which will be presented in Chapter 5. #### 3.4.1 Instrument TIP was developed at the Instituto de Astrofísica de Canarias (Martínez Pillet et al., 1999) and recently upgraded with a new, larger infrared CCD detector (Collados et al., 2007). It is able to record simultaneously all four Stokes components with very high spectral resolution in the infrared region from $1\mu m$ to $2.3\mu m$, with a fast cadence and very high spatial resolution along the slit. The optical setup of the instrument is shown in Fig. 3.10. After the main tank and the AO system, a narrow ($\sim 100\,\mu$m wide) slit is mounted in the plane of the prime focus of the telescope. The light reflected from the slit jaws enters a camera system to provide images, to point the telescope and to have the region of interest imaged onto the slit. The small fraction of light entering the slit goes through the polarimeter, where the Stokes components are modulated. Then, the predisperser and spectrograph decompose the light into its spectral components. At the end of the optical path the detector is mounted, a CCD cooled below 100 K to reduce the thermal excitation of electrons in the CCD pixels. Figure 3.10: Optical schema of the Tenerife Infrared Polarimeter (TIP) with slit jaw camera, predisperser and spectrograph of the VTT. After the AO correction, the light from the prime focus of the telescope enters the instrument through the slit. The light reflected from the slit jaws is recorded with video cameras to create context frames. After the slit, the polarimeter with the ferroelectric liquid crystals modulates the polarization of the light beam. The predisperser selects, with mask (p1), the spectral region to observe, and the spectrograph disperses the light into its spectral components. The nitrogen-cooled CCD detector records the modulated polarization of the spectra. d1 and d2 are the diffraction gratings. ##### The polarimeter TIP is able to obtain simultaneously the full set of the four Stokes parameters that determines the polarization of the light, from each point in the slit. However, this work concentrates only on the intensity measurements. The polarization measurement is performed by means of two ferroelectric liquid crystals (FLC). These are electro-optic materials with fixed optical retardation, whose axis can be switched between two orientations by applying voltages of approximately $\pm$ 10V. This amplitude of the rotation of the retardation axis is somewhat dependent on the temperature, and is $\sim 45^{\circ}$ at $20-25$C. With two FLCs, with two possible states each, we can create four different combinations of modulation of the incident light. The four modulated intensities are four different linear combinations of {I,Q,U,V} with different weights on each parameter. With four consecutive measurements we can therefore retrieve the four components of the Stokes vector. Thus, TIP is able to obtain simultaneously the four components of the polarization for each full cycle of the polarimeter. Although TIP makes a full cycle of the FLCs in less than one second, we have to accumulate several spectrograms in order to increase the signal to noise ratio, especially when measuring weak signals like the polarization of spicules outside the solar limb. In the sequence following the light path, the physical setup of the polarimeter consists of a UV-blocking filter to protect the FLCs from intense high energy radiation at short wavelength. Then, the first FLC with a retardation of $\lambda/2$ and the second FLC with $\lambda/4$ follow. The retardances of $\lambda/2$ and $\lambda/4$ are nominal values. The actual retardances differ from these values and depend on wavelength. Finally a Savart plate splits the light into two orthogonal linearly polarized beams. As part of the instruments we need a calibration optic subsystem (see explanation in Sec. 3.4.3) to account for the influence of the mirrors following the telescope. For this reason, in front of the AO system, there is a polarization calibration unit (PCU) that can be moved into the light path. It is composed of a retarder with nominal retardance of $\lambda/4$ in the optical spectral range, and a fixed linear polarizer. The retarder rotates a full cycle with measurements taken every 5 degrees, creating a set of 73 modulations of the light beam that are used to model the influence of the optics behind the telescope, but including AO, till the detector. The influence of the coelostat mirrors and the telescope proper on the polarization state are taken into account with a polarization model of these parts by Beck et al. (2005). #### 3.4.2 Observations Table 3.2 summarizes the details of the observing campaign for the course of this work. It focuses on studying the emission profiles observed in spicules in the He i 10830 Å multiplet. The strong darkening close to the solar limb and the presence of the limb make it difficult to use KAOS for off-limb observations, since the correlation algorithm of KAOS was not developed for this kind of observations. We scanned the full height of the spicule extension, starting inside the disc. We made a single spatial scan with long integration time per position. As the _lock point_ of the AO was placed on a nearby facula inside the disc was chosen. Apart from the facula used for AO tracking, it was a quiet Sun region. In the present work we study only the intensity component of the Stokes vector (see definition in e.g. Chandrasekhar, 1960; Wikipedia, Stokes parameters). Date \| Dec,4th,2005 ---\|--- Location \| NE limb Spectral sampling # \| 10.9 mÅ/px Time span \| 1 scan in 66 min. Slit \| 40$\times$ 05 Integration time \| 5$\times$2.5 s Step size \| 035 Max. height off-limb \| 7 Seeing condition ($r_{0}$) \| $\sim 7$cm (max 12 cm) KAOS support \| yes Table 3.2: Characteristics of the data taken with TIP used in this work. $r_{0}$ is the Fried parameter. #### 3.4.3 Data reduction As for the G-FPI case, the data reduction process aims to remove the instrumental effects as well as the atmospheric influence. For TIP data this involves three steps. The first is common to all CCD observations and consists in removing instrumental effects, the second is the polarimetric calibration of the signal, and the third is the spectrosposcopic calibration. ##### Reduction of CCD effects (a) Frame of raw data (b) Flat field (c) Dark frame (d) Reduced frame Figure 3.11: Examples of the standard data reduction process for spectral data. The Flat field frame (b) is calculated dividing average flat field data by the mean spectra of the average. This processing is basically the same for all CCD observations: removal of dark counts and correction for differential sensitivity of the pixel matrix with the gain table (using the flat fields). The only difference to G-FPI data reduction is when creating the flat fields. The mean flat field frame is not _flat_. Although being a spatial average, it still contains spectral information. To retain only the gain table information we divide the flat field by the mean spectrogram, so that only the differential response of the pixels is left (see Fig. 3.11). The mean spectrogram is obtained by averaging the flat field spectrograms over the spatial coordinate. ##### Polarimetric calibration The signals recorded with the CCD are not directly the Stokes parameters (see description in e.g. Chandrasekhar, 1960) . With two FLCs we have four different combinations in one full cycle. For each configuration in the cycle, we measure intensities as a particular linear combination of {I,Q,U,V} with different weights, so we can solve the ensuing system of equations. Also, in each CCD frame, we measure light of two orthogonal linearly polarized beams (see Sec. 3.4.1). An important problem in polarimetric observations is that each reflecting surface of the telescope changes the polarization state of the incoming light. So the optical path, with all the reflecting surfaces from the coelostat to the CCD, introduces a complex modulation of the incoming polarization. At the VTT there is a polarization calibration unit (PCU) mounted in front of the AO system. This device feeds the subsequent optical components with light of well defined polarization states. So, once we have a set of Stokes parameters from different configurations of the PCU, we can obtain the modulation induced by the optical path, the Mueller matrix $\mathbb{M}$, from the PCU to the polarimeter: $\left(\begin{array}[]{c}I\\\ Q\\\ U\\\ V\end{array}\right)_{polarimeter}=\mathbb{M}\cdot\left(\begin{array}[]{c}I\\\ Q\\\ U\\\ V\end{array}\right)_{input}$ (3.18) The inverse matrix of $\mathbb{M}$ will therefore relate the polarization state of the light that reaches the polarimeter with the light arriving at the PCU position. However, the light path from the coelostat to the PCU (in front of the AO) cannot be calibrated with this system, so the reduction routines use a theoretical model of this part of the telescope. This process is already implemented with available reduction pipelines. Further investigation of _crosstalk_ or other additional polarimetric reduction are needed to reduce the instrumental effect in our data. However, this is not necessary for our case, since this work concentrates only on the intensity component. ##### Spectroscopic reduction The last type of reduction procedure is related to the nature of spectroscopic data and consists of the calibration in wavelength, the continuum correction and a low pass filtering to remove noise. To calibrate our spectrograms in wavelength we make use of the two telluric lines present in our spectral range of the TIP data. Solar lines are subject to Doppler shifts from local flows and solar rotation. Yet, telluric absorption lines are formed in the atmosphere of the Earth. Therefore, they are always narrow due to only small Doppler broadening and are located at fixed wavelength. This provides a fixed reference coordinate that we use with the FTS atlas (Neckel, 1999). Comparing both spectra we can accurately measure the spectral sampling which is for all data sets $10.9$mÅ/pixel . See wavelength scale abscissa of Fig. 3.12. The transmission of the filters is not a constant in the transmitted wavelength range, so this creates an intensity variation curve in all our spectrograms. For normalization, we have to find the correct level of the continuum intensities of the spectrograms observed on the disc. For this, we use several spectral positions between spectral lines and calculate the ratio between the observed data and the values from the FTS atlas. We interpolate to create the continuum correction (see green dashed line on Fig. 3.12). An electronic signal was also found in some observed spectrograms with a frequency higher than those containing information on the solar spectrogram. We used for all data a low-pass filter which removes the power at all frequencies higher than a certain threshold, preserving the spectral line information. Once we have filtered and corrected the signal for all instrumental effects we have to remove finally the scattered light. We define the position of the solar limb as the height of the first scanning position (counting from inside the limb outwards), where the helium line appears in emission. For increasing distances to the solar limb a decreasing amount of sunlight is added to the signal by scattering in the Earth’s atmosphere and by the telescope’s optical surfaces. Since the true off-limb continuum must be close to zero, i.e. below our detection limit, the observed continuum signal measures the spurious light. Therefore, we removed the spurious continuum intensity level by using the information given by a nearby average disc spectrogram. This first subtraction estimates the continuum level on a region 6 Å away from the He i 10830 Å emission lines. After this correction with a coarse estimate of the spurious light, a second correction was applied to remove the residual continuum level seen around the emission lines. This was needed since the transmission curve of the used prefilter is not flat but variable with wavelength. Figure 3.12: Example of intensity calibrated spectra on the disc near the limb. Raw spectrogram (blue line) has to be corrected for the continuum level to agree with the values in the FTS atlas (Neckel, 1999, black line). Using the continuum at several positions we can estimate the continuum correction (green dashed line). The corrected data (not filtered) are shown in orange. For the wavelength calibration we use the two telluric H2O lines (labeled in the figure). The region of the He i 10830 Å multiplet is also labeled, as well as some other lines in the range (Si, Ca i , Na i). ## Chapter 4 High resolution imaging of the chromosphere111Contents from this Chapter have been partially published as Sánchez-Andrade Nuño et al. (2005, 2007) Since the discovery of the chromosphere 150 years ago, it has remained a lively and exciting field of research. Especially the chromosphere of active regions exhibits a wealth of dynamic interaction of the solar plasma with magnetic fields. The literature on the solar chromosphere, and on stellar chromospheres, is numerous. We thus restrict here citations to the monographs by Bray and Loughhead (1974) and Athay (1976) and to the more recent proceedings from the conferences Chromospheric and Coronal Magnetic Fields (Innes et al., 2005) and The Physics of Chromospheric Plasmas (Heinzel et al., 2007). With the latest technological advances we are able to scrutinize this atmospheric layer in great detail. The G-FPI in combination with post- processing techniques used in this work aims for the study of the temporal evolution of the chromospheric dynamics with very high spatial, spectral and temoral resolution. In this Chapter we present our investigations with the G-FPI inside the solar disc. The first Section focusses on data set “mosaic” and the presence of fast moving clouds. The subsequent Section presents the results of the investigation of fast events and waves from dataset “sigmoid”. Finally we make a comparison between SI+AO and BD methods. ### 4.1 Dark clouds As already noted in Sec. 1.2, the chromosphere is highly dynamic. Within and in the vicinity of active regions the interaction of the plasma with the strong magnetic fields gives rise to specially complex phenomena with fast flows. As an example we refer to a recent observation of fast downflows from the corona, observed in the XUV and in H$\alpha$ by Tripathi et al. (2007). Fast horizontal, apparent displacements of small bright blobs with velocities of up to 240 km s-1 were observed in H$\alpha$ by van Noort and Rouppe van der Voort (2006). ##### Observations and data processing In this Section we use the data set “mosaic” (See Table 3.1) recorded on May, 31, 2004 by K. G. Puschmann, M. Sánchez Cuberes and F. Kneer. It consists of a wide mosaiqued FoV around the active region AR0621. For each single FoV a series of five consecutive scans was performed, spanning a total of 4 min to study the temporal evolution. The FoV of a single exposure was $\sim$ 33${}^{\prime\prime}\times$23\. To study a wide area the telescope was pointed consecutively to 13 overlapping contiguous areas. The resulting mosaic covers a wide region with a total FoV of $\sim$ 103$\times$94\. In Figs. 4.2 and 4.2 we present the broadband image and narrow-band line core filtergram, respectively. In all mosaics, both in broadband and in all the narrow-band images there is a blank central area, that just corresponds to a small non- covered area. After dark subtraction and flat fielding, the data were processed using the SI approach (see Sec. 3.3.3). Figure 4.1: Mosaic of speckle reconstructed broadband images of the active region NOAA AR0621, at $\mu$ = 0.68. The achieved high resolution by means of the adaptive optics and post factum reconstruction is $\sim 0.2\arcsec$. The total area covered is $\sim$ 103$\times$94\. Limb is located to the left lower corner. Figure 4.2: H$\alpha$ line center filtergram. It corresponds to one of the 18 reconstructed images along the spectral line. The resolution in these narrow- band images is $<$ 0.5. One notes the various chromospheric features: ubiquitous short fibrils with different orientation, a wide bright plage region full of facular grains on the lower central part, and dark fibrils packed together outlining the magnetic field lines between sunspots around the central data gap. White arrow indicates position and direction of the dark cloud in Fig. 4.3 After the SI reconstruction, we have applied a destreching algorithm between the consecutive broadband images to remove residual _seeing_ effects. The deformation matrix for the destreching was calculated for the broadband channel using a mean image as reference. The same deformation matrix was then applied to the narrow-band spectrograms. To constrain the different frames of the mosaic of the broadband data, i.e. for connecting the individual subfields, a cross-correlation algorithm has been developed. The frames were smoothed by a boxcar of 5$\times$5 pixels to take into account only large structures for the destreching and to reduce noise. The overlapping regions between the individual subfields have been used to scale the intensities and the several areas have been connected after proper apodisation. The arrangement of the individual subfields inside the broadband mosaic have been directly applied to the narrow-band data. ##### Data analysis and interpretation We report the observations of numerous fast moving dark clouds in the FoV. Dopplergrams reveal that these clouds correspond to downward motion. Here we show a particular fast dark cloud. Neither the continuum image nor the line center exhibit strong activity. However, if we study the filtergram taken in the red wing of the H$\alpha$ line, a group of dark features becomes apparent (see panel 1 of Fig. 4.3). Successive spectrograms every 45 s of the same region (panels 2 to 5 of Fig. 4.3 ) reveal a fast differential motion of this dark cloud. The position and direction is marked by the white arrow in Fig. 4.2. A horizontal surface velocity of $\sim 90$ km/s is measured. Interestingly, the cloud has suddenly disappeared and was not longer seen in the last two observed frames. In Fig. 4.3 we display the corresponding spectral profiles for the central part of one of the cloud members (marked by white crosses in Fig. 4.3) at different times. We interpret the observed dark cloud, seen as a line depression in the red wing of the H$\alpha$ line, as a signature of the Doppler shifts related to the fast movement of the dark cloud. From the spectral distance between the line core of H$\alpha$ and the minimum position of the line depression we estimate a LOS downflow speed of $\sim$ 51 km/s. This, in combination with the observed horizontal velocity leads to an approximate total speed of $\sim 103$ km/s directed downwards. Further, the sudden disappearance of the cloud from the last 2 frames could be explained with a very strong related Doppler shift, thus the position of the line depression is displaced outside the scanned wavelength range. Figure 4.3: _Left_ : Motion of dark feature seen in H$\alpha$ at +1 Å off line center, presented in false color to increase contrast. Vertical red lines are separated by 3.15($\sim$ 2280 km). Time step between consecutive images $\sim$ 45 s. Horizontal tiles represent consecutive frames from the time sequence (from top to bottom). _Right_ : Spectral profiles, normalized to the quiet Sun spectrum at 6562 Å, of the central part of one of the cloud members, marked by white crosses on the left image. Black solid line is the mean profile of the surrounding quiet Sun. ### 4.2 Fast events and waves We continue investigating the active chromosphere on the disc of the Sun. We report on fast phenomena and waves observed in the H$\alpha$ line with high spatial, temporal, and wavelength resolution. Figure 4.4: Broadband image of part of the active region AR10875 on April 26, 2006 at heliocentric angle $\vartheta=36\degr$. The rectangles, denoted by A, B, B, C, and D, are the areas of interest (AOIs) to be analyzed and discussed below. #### 4.2.1 Observations and data reduction Figure 4.5: Narrow-band image corresponding to Fig. 4.4 in H$\alpha$ at +0.5Å off line center. The same areas of interest are indicated as in Fig. 4.4 by the rectangles. The observations correspond to dataset “sigmoid” in Table 3.1. They consist of a time sequence of 55 min duration of H$\alpha$ scans with a mean cadence of 22 sec from the active region AR 10875. The observations were supported by the Kiepenheuer Adaptive Optics system (KAOS, von der Lühe et al. 2003) under extremely good seeing conditions. Due to a technical problem, an increasing delay between successive scans was noticed during the observations. When the accumulative delay reached around seven seconds a new scanning procedure was restarted to avoid higher gaps between frames. This operation needs around one minute. During the 55 minutes of this series, such an interrupt occurred twice, at 08:10:19 UT and 08:29:46 UT. This programming bug was corrected afterwards for future observations. The reduction process with SI+AO is explained in Sec. 3.3.3. We achieve a spatial resolution of $\sim$025 for the broadband images at 630 nm and better than 05 for each of the 21 narrow-band filtergrams. Further, to follow the temporal evolution in time, both broadband and narrow-band images where cropped to the same common FoV, removing overall image shifts due to residual seeing effects. Afterwards, the speckle reconstructed broadband images were co-aligned to spatially and temporally smoothed images via a destretching code provided by Yi et al. (1992). The destretching matrix from the broadband image was also applied to the simultaneous narrow-band scan. To minimize the effects of the irregular sampling rate, the time sequences were interpolated to equidistant times with the cadence that leads to a minimum shift in time for each frame. This corresponds to a regular time step of 22 s. The data gaps at the times when observation was interrupted were filled by linear interpolation between closest observed images. Figures 4.4 and 4.5 give the broadband scenery at $t=40.9$ minutes during the series and the associated H$\alpha$ image at $+$0.5 Å off line center, respectively. The whole region was very active with a flare during the observation of the time sequence (Sánchez-Andrade Nuño et al., 2007b). The data set is certainly rich of information on the dynamics of the active chromosphere, especially since the spatial resolution is high throughout the sequence. For the present study, we restrict further analyses and discussions to few regions. The areas of interest (AOIs) are indicated by rectangles and denoted by A, B, B, C, and D. In the presentations below the images from the AOIs were rotated to have their long sides parallel to the spatial co-ordinate in space-time images. AOI A contains a region where a long fibril developed twice during our observations. It has the appearance of a small surge (Tandberg-Hanssen, 1977). AOIs B and B show a simultaneous fast event, possibly ‘sympathetic’ mini-flares with strong, small-scale brightenings in the H$\alpha$ line core which last only few tens of seconds. AOIs C and D, with their long fibrils, are suitable for the study of magnetoacoustic waves along magnetic field lines. Area C contains in its right part a region from which H$\alpha$ fibrils stretch out to both sides and which, at the beginning of the time sequence, contained a small pore that disappeared in the course of the observations. Note also from Fig. 4.4 that the fibrils on the upper left side of area D originate in the penumbra of a small sunspot. #### 4.2.2 Physical parameters The possibility to extract information from a good part of the H$\alpha$ line profile in two dimensions and along the time series is highly valuable. We are interested in the physical parameters of the H$\alpha$ structures. The line- of-sight velocities $v_{\mathrm{LOS}}$ can be retrieved using the lambdameter method, while also many other parameters like the temperature and the mass density can be inferred by means of the cloud model. The lambdameter method (Tsiropoula et al., 1993) is a common procedure to measure line of sight (LOS) velocities. It compares the Doppler shift of a spectral line with the position of the quiet Sun profile. We measure the profile bisector at several line widths. The method consists in measuring the displacement between the bisectors of the spectral profile and the reference quiet Sun profile. As pointed out by Alissandrakis et al. (1990) the resulting velocities give systematically lower LOS velocities than the cloud model (see below) by a factor of approximately $3$. However, the qualitative behavior of both methods are the same. The lambdameter method is therefore a fast method for a qualitative description of the velocity pattern of a region. The cloud model yields a non-LTE inversion technique. The formation of line profiles is the result of a complex interaction between the plasma and the radiation. Among others, the formation of a spectral line depends on the local temperature, velocity, chemical composition, magnetic fields, radiation fields, …Inversion techniques aim at retrieving this set of parameters from a given spectral profile. These techniques modify, in an iteration scheme, the starting guesses of parameters based on certain assumptions until a converged solution, with modeled radiation and observations in close agreement, is obtained. We assume then that the calculated parameters producing the synthetic profile are the same as in the observed structure, as long as the assumptions are considered valid. Figure 4.6: Geometry of the cloud model. The cloud model allows the application of an inversion technique in cases when one can describe the radiation transfer through structures located high above the unperturbed solar photosphere. This method was first described by Beckers (1964) and has been extensively used afterwards, e.g. by Tsiropoula and Schmieder (1997); Tsiropoula and Tziotziou (2004); Tziotziou et al. (2004); Al et al. (2004). See also the recent review by Tziotziou (2007). Figure 4.6 depicts the geometry of the cloud model. The considered “cloud” is located above the underlying photosphere at a height $H$ and moving at a speed $\vec{V}$. From the observer’s position we can measure the projected proper motion relative to the background and the LOS velocity ($V_{los}$ in Fig. 4.6) as Doppler shifts. The observed intensity $I(\Delta\lambda)$ is the combination of the absorption of the background intensity $I_{0}(\Delta\lambda)$ with the emission from the cloud, dependent on the optical thickness of the cloud $\tau(\Delta\lambda)$: $I(\Delta\lambda)=I_{0}(\Delta\lambda)\cdot e^{-\tau(\Delta\lambda)}+\int_{0}^{\tau(\Delta\lambda)}S_{t}e^{-t(\Delta\lambda)}dt\,,$ (4.1) where $S$ is the source function, which depends on the optical thickness along the cloud. In the model we make the following assumptions: 1. 1. The structure is well above the underlying unperturbed chromosphere. 2. 2. Within the cloud, the source function $S$, velocity and Doppler width are constant along the LOS. 3. 3. The background intensity profiles entering the cloud from below and in the surroundings are the same. These assumptions simplify Eq. 4.1 to $I(\Delta\lambda)=I_{0}(\Delta\lambda)\cdot e^{-\tau(\Delta\lambda)}+S\,(1-e^{-\tau(\Delta\lambda)})\,.$ (4.2) This, in terms of the _contrast profile_ , $C(\Delta\lambda):=\frac{I(\Delta\lambda)-I_{0}(\Delta\lambda)}{I_{0}(\Delta\lambda)}\,,$ (4.3) can be rewritten as $C(\Delta\lambda)=\Bigg{(}\frac{S}{I_{0}(\Delta\lambda)}-1\Bigg{)}\,\Big{(}1-e^{-\tau(\Delta\lambda)}\Big{)}\,.$ (4.4) Further, neglecting collisional and radiative damping of the H$\alpha$ absorption profile within the cloud, the optical depth can be given by a Gaussian profile, i.e. $\tau(\Delta\lambda)=\tau_{0}\,e^{-\Big{(}\frac{\Delta\lambda-\Delta\lambda_{I}}{\Delta\lambda_{D}}\Big{)}^{2}}\,,$ (4.5) where $\tau_{0}$ is the line center optical thickness. Also, the central wavelength of the profile can be displaced due to a LOS velocity $v$ of the cloud with a Doppler shift, $\Delta\lambda_{I}=\lambda_{0}v/c$, where $\lambda_{0}$ is the rest central wavelength and $c$ is the speed of light. The width of the profile $\Delta\lambda_{D}$ depends on the temperature $T$ and the microturbulent velocity $\xi_{t}$ trough the relation $\Delta\lambda_{D}=\frac{\lambda_{0}}{c}\sqrt{\frac{2kT}{m}+\xi_{t}^{2}}\,,$ (4.6) where $m$ is the atom rest mass. With these assumptions we end up with an inversion problem with 4 parameters: $S$, $\Delta\lambda_{D}$, $\tau_{0}$ and $v_{LOS}$. $\Delta\lambda_{D}$ is, in turn, the combination of 2 physical parameters (temperature and microturbulent velocity). More complex cloud models have recently been developed. These mainly focus on the nature of the source function $S$, allowing the parameters to vary along the LOS or multi-cloud models. However, as pointed out by Alissandrakis et al. (1990), a simple Beckers cloud model like the one described above and used here provides useful, reasonable estimates for a large number of optically not too thick structures, $\tau_{0}\lesssim 1$, for which the assumptions are adequate. In this work we also used the inversion in H$\alpha$ structures where possible. The undisturbed reference profile $I_{0}(\lambda)$ is taken from a nearby area with low activity outside the FoV shown in Fig. 4.4. As the region under study was ‘clouded out’ in H$\alpha$, i.e. covered with structures to a large extent, the cloud model inversion failed often. In these latter cases, instead, the LOS velocity maps were determined with the lambdameter method or from difference images at H$\alpha\pm$0.5 Å off line center with appropriate scaling. Calibration curves to estimate from such Doppler-grams the true velocities were calculated by Georgakilas et al. (1990). From these we obtained that the velocities from the difference images were lower by a factor 2–4 than the true velocities, in agreement with those parts in the FoV where the cloud model inversion was successfully applied and with the results by Tziotziou et al. (2004). With the application of the cloud model and the inferred values of $S$, $\Delta\lambda_{D}$, $\tau_{0}$ and $v_{LOS}$ we can derive other physical parameters. Following the approach by e.g. Tsiropoula and Schmieder (1997) we can calculate the population densities of the hydrogen levels 1, 2, 3 ($N_{1}$,$N_{2}$,$N_{3}$), the total hydrogen density ($N_{H}$, including protons), the electron density ($N_{e}$), the total particle density ($N_{t}$) the electron temperature ($T_{e}$), gas pressure ($p_{g}$), total column mass ($M$), mass density ($\rho$) and degree of ionization of hydrogen ($x_{H}$): $\displaystyle N_{1}=$ $\displaystyle\frac{N_{t}-(2+\alpha)N_{e}}{1+\alpha}$ (4.7) $\displaystyle N_{2}=$ $\displaystyle 7.26\leavevmode\nobreak\ 10^{7}\frac{\tau_{0}\Delta\lambda_{D}}{d}$ $\displaystyle\mbox{ cm}^{-3}$ (4.8) $\displaystyle N_{3}=$ $\displaystyle\frac{g_{3}}{g_{2}}N_{2}\Big{(}{\frac{2h\nu^{3}}{Sc^{2}}+1}\Big{)}^{-1}$ (4.9) $\displaystyle N_{e}=$ $\displaystyle 3.2\leavevmode\nobreak\ 10^{8}\sqrt{N_{2}}$ $\displaystyle\mbox{ cm}^{-3}$ (4.10) $\displaystyle N_{H}=$ $\displaystyle 5\leavevmode\nobreak\ 10^{8}\sqrt{N_{2}}$ (4.11) $\displaystyle N_{t}=$ $\displaystyle N_{e}+(1+\alpha)N_{H}$ (4.12) $\displaystyle p_{g}=$ $\displaystyle kN_{t}T_{e}$ (4.13) $\displaystyle M=$ $\displaystyle(N_{H}m_{H}+0.0851N_{H}\cdot 3.97m_{H})d$ (4.14) $\displaystyle\rho=$ $\displaystyle M/d$ (4.15) $\displaystyle x_{H}=$ $\displaystyle N_{e}/N_{H}$ (4.16) where $d$ is the path length along the LOS through the structure, $\alpha$ is the abundance ratio of helium to hydrogen ($\approx 0.0851$), $g_{2},g_{3}$ are the statistical weights of the hydrogen levels 2 and 3 respectively, $h$ is the Planck constant, $\nu$ is the frequency of H$\alpha$, $c$ the speed of light and $k$ the Boltzmann constant. Table 4.1 summarizes average results from the cloud model and derived quantities for the long fibril in Fig. 4.7: Parameter \| Av. value \| Parameter \| Av. value ---\|---\|---\|--- v [km/s] \| 11.7 \| $\Delta\lambda_{D}$ [Å] \| 0.34 $S/I_{c}$ \| 0.154 \| $\tau$ \| 1.05 N2 [cm-3] \| $4.5\cdot 10^{4}$ \| Ne [cm-3] \| $6.8\cdot 10^{10}$ NH [cm-3] \| $1.1\cdot 10^{11}$ \| N1 [cm-3] \| $3.8\cdot 10^{10}$ N3 [cm-3] \| $4.2\cdot 10^{2}$ \| $T_{e}$ [K] \| $1.51\cdot 10^{4}$ $p$ [dyn cm-2] \| $0.38$ \| M [g cm-2] \| $1.39\cdot 10^{-4}$ $\rho$ [g cm-3] \| $2.3\cdot 10^{-13}$ \| $x_{H}$ \| 0.64 $c_{s}[km/s]$ \| 14.4 \| \| Table 4.1: Several derived parameters from the cloud model for the lower half section of the long fibril in Fig. 4.7 at $t=25$ min. We assume a LOS thickness equal to the width of the fibril (cylindrical shape) of $590$ km and a micro-turbulent velocity of $10$ km/s. First two rows result from the inversion technique while the others are parameters derived from them. #### 4.2.3 Fast events in H$\alpha$ Figure 4.7: Space-time image of surge in AOI A at H$\alpha$ \+ 0.5 Å off line center, starting at 14.7 min after the beginning of the sequence. The spatial axis runs along the minima of the surge intensities at this wavelength. ##### Small recurrent surge Ejecta from low layers of active regions, called surges, have been observed in time sequences of H$\alpha$ filtergrams since many decades (e.g., Tandberg- Hanssen, 1977). In AOI A, a small surge occurred during the observed time series. It started near the pore at the upper right end of region A (cf. Figs. 4.4 and 4.5). It was straight and thin, with a projected length at its maximum extension of at least 15 Mm and with widths of approximately 2 at its mouth and 1 at its end. Figure 4.7 shows the temporal evolution of the surge in H$\alpha$ +0.5Å off line center. The space-time image starts 14.7 min after the beginning of the series and goes to the end of it. Along the spatial axis in Fig. 4.7, the minimum intensities along the surge are represented. The surge consisted of very thin fibrils, at the resolution limit $<0$5, being ejected in parallel. It started with several small elongated clouds lasting for 1–2 min. Afterwards, it rose, reaching a projected length of around 14 Mm, and fell back after $\sim 7$min. Then it suddenly rose again after two min reaching lengths out of the FoV (more than 15 400 km) and lasted another five min before retreating again. And finally, the process recurred a third time, yet with lower amplitude in extension and velocity than for the first two times. The (projected) proper motion of the tip of the surge reaches a maximum velocity of approximately 100 km s-1, for both the ascent and the descent phases. Especially the second rise and fall showed large velocities. It is unlikely that the rapid rise and appearance of the surge in H$\alpha$ are caused by cooling of coronal gas to chromospheric temperatures. The cooling times are much too long, of the order of hours (Hildner, 1974). Thus, the proper motions represent gas motions. The LOS velocities measured from Doppler-grams and corrected with the calibrations described above in Sect. 4.2.2, amounted to +15 km s-1 during the ascent of the surge and reached $-$45 km s-1 at the mouth during retreat. These latter velocities are lower than the proper motions. It thus appears that the chromospheric gas is ejected obliquely into the direction towards the limb. Average physical parameters in the surge obtained with the cloud model inversion are listed in Table 4.1. They are very similar to those of other chromospheric structures (see e.g. Tsiropoula and Schmieder, 1997). Surges are known to show a strong tendency for recurrence, but on time scales of $\sim$1 h. Sterling and Hollweg (1989) have treated numerically rebound shocks in chromospheric fibrils and presented results in which a single impulse at the base of the involved magnetic flux tube drives a series of shocks on time scales of approximately 5 min. This appears to be a viable mechanism for the small surge observed here, apart from the initial conditions. The small ‘firings’ at the beginning of this surge suggest magnetic field dynamics that ultimately do cause a strong impulsive force, after some minor events. Figure 4.8: Simultaneous flash event on AOIs B and B with projected distance $\approx$13.7 Mm. A pair of simultaneous, short, brightening was recorded at $t=52.2$ min. Top row from B, bottom row from B. The tiles from left to right correspond to two successive H$\alpha$ scans. Upper x-axis is scaled to the wavelength of each 2-D filtergram tiles. Scanning time is numbered on the lower x-axis. $t=0$ corresponds to the beginning of the scan at 08:44 UT. The integration time for each spectral position is $\approx 1$s, while the delay between two scans is $\approx 3$s (vertical dashed line). Each spectrogram on B is normalized with the background profile (see Fig. 4.9) to emphasize the flash event. Neither the previous nor the following scan to the two presented exhibited any emission. The second scan (right half size of the figure) still shows some emission on the same positions. White arrows correspond to the position of the three different profiles in Fig. 4.9. ##### Synchronous flashes In the AOI pair (B, B) with a projected distance of $\sim$14 Mm, brightenings occurred 52.2 min after the start of the series in both sites at least as simultaneously as we can detect with the observational mode of scanning the H$\alpha$ line. AOI B is located in the umbra of a small spot with a complex penumbra and AOI B next to a pore. In between the two AOIs the sigmoidal filament ended while more structures of the extended and active filament system crossed the region between the two AOIs. Figure 4.8 shows the temporal evolution of the brightenings. The upper row of this figure is from AOI B, the lower from B. Two scans through the H$\alpha$ profile are presented, of course without interpolation of the images to an identical time. The horizontal axes contain the run in both time and wavelength. The flash-like brightenings lasted only for less than 45 s, they were present neither in the scans before nor after the two scans shown in Fig. 4.8. The simultaneity of the two flashes, or mini-flares, suggests a relation between them. Possibly, one sees here a kind of sympathetic flares. These were discussed earlier in the context of synchronous flares excited by activated filaments (Tandberg-Hanssen, 1977). Another interpretation is that one sees a mini-version of two-ribbon flares with a common excitation in the corona above them and simultaneous injection of electrons into the chromosphere. In AOI B, the flash exhibited sub-structure and apparently moved during the first presented scan with speeds up to 200 km s-1. This strong brightening between 15 and 22 s has disappeared in the following scan. Figure 4.9 depicts the recorded H$\alpha$ profiles at the positions of the flash in AOI B, as indicated by the arrows on the left side of Fig. 4.8. The profiles are compared with those from the quiet Sun and from the average background. The profile from the isolated bright blob at 3.7 Mm (see inset in Fig. 4.9) shows a blue shifted emission above the background profile. This emission is still present in the following scan. At 2.8 Mm the line core is filled resulting in a contrast profile with strong emission (cf. Eq. 4.4). The profile at 1.1 Mm exhibits a strong emission beyond the continuum intensity in the red wing while the whole profile is enhanced above the background profile. the position of the emission peak would indicate a down flow with LOS velocity of 35 km s-1. It was shown by Al et al. (2004) that such emission (contrast) profiles can be understood if one assumes an injection, likely from the corona, of much energy and electrons to obtain a response of the H$\alpha$ line to temperature. These last two emissions at 2.8 Mm and 1.1 Mm have disappeared at the time of the following scan. Obviously, such fast events as in AOIs B and B lie beyond the observing capabilities of our consecutive scanning method. We could however retrieve high spatial resolution filtergrams at several wavelengths to follow the temporal evolution at time scales of few seconds. With the present data set at our hand, we cannot decide whether the apparent proper motion of the flashing structure in AOI B is indeed as high as 200 km s-1 or whether the temporal resolution is too fast for the consecutive scanning. For example, the H$\alpha$ profile from 1.1 Mm could have been in emission over the whole profile, but only for few seconds. It is, however, possible to design adequate observing sequences with duration of few seconds per scan, on the expense of taking filtergrams at fewer wavelength positions. Figure 4.9: H$\alpha$ profiles from the flash event. Each profile corresponds to an average over three pixels around the three selected points where the emission is highest in the blue wing, at central wavelength, and in the red wing respectively, corresponding to the white arrows in Fig. 4.8 (left half) at x=[3.7, 2.8, 1.1] Mm, respectively. For comparison the quiet Sun profile is also shown. Background profile corresponds to the mean from previous and following scans, where no brightening was found. The emission profile at 1.1 Mm reaches an intensity of 1.1 of the quiet Sun continuum intensity. #### 4.2.4 Magnetoacoustic waves In this Section, for the investigation waves in the chromosphere, we first refer to previous observations of magnetoacoustic waves and then outline the magnetohydrodynamic (MHD) approximation (Ferraro and Plumpton, 1966; Kippenhahn and Möllenhoff, 1975; Priest, 1984). From this, dispersion relations for sound waves, atmospheric waves, Alfvén waves, and magnetosonic, or magnetoacustic, waves are derived by linearization. We continue discussing the observations of waves in long H$\alpha$ fibrils. Finally we try an interpretation in the framework of waves in thin magnetic flux tubes. ##### Previous observations of magnetoacoustic waves Apart from oscillations in sunspot umbrae (Beckers and Tallant, 1969; Wittmann, 1969) and running penumbral waves (e.g., von Uexküll et al., 1983, and references therein), waves in the chromosphere were observed by many authors. E.g., Giovanelli (1975) describes waves along H$\alpha$ mottles and fibrils with speeds of 70 km s-1 and interprets them as Alfvén waves in magnetic flux tubes with approximately 10 Gauss field strength. Kukhianidze et al. (2006), from time sequences in H$\alpha$ off the limb, found kink waves in spicules with periods of 35–70 s. Hansteen et al. (2006), and Rouppe van der Voort et al. (2007) observed spicules and fibrils in the quiet Sun and in active regions with high spatial and temporal resolution observations. They succeeded via numerical simulations in explaining the dynamics of these chromospheric small-scale structures by magnetoacoustic shocks, excited mainly by the solar 5-min oscillations (see also the simulations by Sterling and Hollweg 1989 and the review by Carlsson and Hansteen 2005). Waves in the corona have as well been observed: E.g., Robbrecht et al. (2001) report on slow magnetoacoustic waves in coronal loops observed in high-cadence images from SoHO/EIT and TRACE. The speeds amount to 100 km s-1. De Moortel et al. (2002), also from high-cadence 171 Å TRACE images, find that 3- and 5-min oscillations are common in coronal loops. They are also interpreted as magnetoacoustic waves. Tothova et al. (2007), from SoHO/SUMER data, study Doppler shift oscillations identified as slow mode standing waves in hot coronal loops. Fast-mode, transverse, incompressible Alfvén waves, with speeds of 2 Mm s-1, in the solar corona were reported by Tomczyk et al. (2007). ##### Magnetohydrodynamic (MHD) approximation We use here the Gauss system of units. The MHD approximation is obtained from Maxwell’s equations and the equation of mass conservation, the equation of motion, and an equation of state, under the following conditions: 1. 1. The gas velocities $v$ are small compared to the speed of light $c$, $v\ll c$. 2. 2. Any changes are slow, such that phase velocities $v_{ph}=L/t\ll c$, where $L$ is a typical length scale and $t$ a typical time scale. 3. 3. The electrical conductivity $\sigma$ is always very high such that the electric field is very small compared to the magnetic flux density, $\|\vec{E}\|\ll\|\vec{B}\|$. 4. 4. One usually adopts in addition, to a good approximation, $\vec{D}=\vec{E}$ and $\vec{B}=\vec{H}$, i.e. magnetic flux density and magnetic field have the same strength, in Gauss units. With $\vec{j}$ electrical current density, Maxwell’s equations are then reduced to $\nabla\times\vec{B}={4\pi\over c}\vec{j}\,,$ (4.17) $\nabla\times\vec{E}=-{1\over c}{\partial\vec{B}\over\partial t}\,,$ (4.18) $\nabla\cdot\vec{B}=0\,.$ (4.19) Ohm’s law, conservation of mass, and the equation of motion read as $\vec{j}=\sigma\left(\vec{E}+{1\over c}\vec{v}\times\vec{B}\right)\,,$ (4.20) ${\partial\rho\over\partial t}=\nabla\cdot\left(\rho\vec{v}\right)\,,$ (4.21) $\rho{\partial\vec{v}\over\partial t}+\rho\left(\vec{v}\cdot\nabla\right)\vec{v}=\rho\vec{g}-\nabla p+{1\over c}\vec{j}\times\vec{B}\,.$ (4.22) Here, $\rho$ is the mass density, $p$ the gas pressure, and $\vec{g}$ the gravitational acceleration vector. The viscous, centrifugal, and Coriolis forces were omitted in the equation of motion, Eq. 4.22. The equation of state relating the gas pressure $p$ with mass density $\rho$ and temperature $T$, is $p=p\left(\rho,T\right)\,.$ (4.23) Furthermore, we assume for simplicity adiabatic motion ${\mathrm{d}\over\mathrm{d}t}\left({p\over\rho^{\gamma}}\right)=0\,,$ (4.24) with the ratio of specific heats $\gamma=5/3$ for monoatomic gases. The current density $\vec{j}$ can be eliminated by means of Ohm’s law, Eq. 4.20 which yields the equation of motion $\rho{\partial\vec{v}\over\partial t}+\rho\left(\vec{v}\cdot\nabla\right)\vec{v}=\rho\vec{g}-\nabla p+{1\over 4\pi}\left(\nabla\times\vec{B}\right)\times\vec{B}\,,$ (4.25) and the induction equation ${\partial\vec{B}\over\partial t}=-\nabla\times\left({1\over\sigma}\nabla\times\vec{B}\right)+\nabla\times\left(\vec{v}\times\vec{B}\right)\,,$ (4.26) where the last term on the rhs of Eq. 4.25 contains Maxwell’s stress tensor. In an atmosphere with constant temperature and with the gravitational acceleration opposite to the vertical direction ($\vec{g}=-g\vec{e}_{z}$, $\vec{e}_{z}$ unity vector into $z$ direction, $\|\vec{e}_{z}\|=1$) the hydrostatic equilibrium is ${\mathrm{d}p_{0}\over\mathrm{d}z}=-\rho_{0}g\,,$ (4.27) with the solution $p_{0}=const1\cdot\mathrm{e}^{-z/\Lambda}\,;\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \rho_{0}=const2\cdot\mathrm{e}^{-z/\Lambda}\,,$ (4.28) where the scale height is $\Lambda=p_{0}/(\rho_{0}\cdot g)$. ##### Magnetoacoustic gravity waves We consider now, to arrive at a dispersion relation for magnetoacoustic gravity waves, small perturbations from the equilibrium $\vec{B}=\vec{B}_{0}+\vec{B}_{1}\,;\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \vec{E}=\vec{E}_{0}+\vec{E}_{1}\,;\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \vec{j}=\vec{j}_{0}+\vec{j}_{1}\,;$ (4.29) $p=p_{0}+p_{1}\,;\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \rho=\rho_{0}+\rho_{1}\,;\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \vec{v}=\vec{v}_{0}+\vec{v}_{1}\,.$ Assuming further that $\vec{B}_{0}$ is homogeneous, $\vec{v}_{0}=0$, and the conductivity is infinite, $\sigma\rightarrow\infty$, one obtains $\nabla\times\vec{B}_{0}={4\pi\over c}\vec{j}_{0}=0\,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \sigma\vec{E}_{0}=0\,,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{i.e.}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \vec{E}_{0}=0\,.$ (4.30) Inserting then the perturbed quantities from Eqs. 4.29 and 4.2.4 into the MHD equations and neglecting quadratic terms and terms of higher order we obtain the linearised MHD equations ${\partial\rho_{1}\over\partial t}+\left(\vec{v}_{1}\cdot\nabla\right)\rho_{0}+\rho_{0}\left(\nabla\cdot\vec{v}_{1}\right)=0\,,$ (4.31) $\rho_{0}{\partial\vec{v}_{1}\over\partial t}=-\nabla p_{1}+{1\over 4\pi}\left(\nabla\times\vec{B}_{1}\right)\times\vec{B}_{0}-\rho_{1}g\vec{e}_{z}\,,$ (4.32) ${\partial p_{1}\over\partial t}+\left(\vec{v}_{1}\cdot\nabla\right)p_{0}-c_{s}^{2}\left[{\partial\rho_{1}\over\partial t}+\left(\vec{v}_{1}\cdot\nabla\right)\rho_{0}\right]=0\,,$ (4.33) $\rho_{0}{\partial\vec{B}_{1}\over\partial t}=\nabla\times\left(\vec{v}_{1}\times\vec{B}_{0}\right)\,,$ (4.34) $\nabla\cdot\vec{B}_{1}=0\,.$ (4.35) with the sound speed $c_{s}=[(\gamma p_{0})/\rho_{0}]^{1/2}$. From these one arrives, after some algebra, at the wave equation for the velocity $\displaystyle\frac{\partial^{2}\vec{v}_{1}}{\partial t^{2}}=$ $\displaystyle c_{s}^{2}\nabla\left(\nabla\cdot\vec{v}_{1}\right)-\left(\gamma-1\right)g\,\vec{e}_{z}\left(\nabla\cdot\vec{v}_{1}\right)-g\nabla v_{1,z}$ (4.36) $\displaystyle+{1\over\rho_{0}}\left[\nabla\times\\{\nabla\times\left(\vec{v}_{1}\times\vec{B}_{0}\right)\\}\right]\vec{B}_{1}/\left(4\pi\right)\,.$ (4.37) ##### Wave modes With Eq. 4.37 we make the ansatz $\vec{v}_{1}(\vec{r},t)=\vec{v}_{1}\exp\left[i\left(\,\vec{k}\cdot\vec{r}-\omega t\right)\right]\,,$ (4.38) with wavevector $\vec{k}$. For $\vec{B}_{0}=0$ and $\vec{g}=0$ one gets pure sound waves with phase velocity $v_{ph}=\omega/k=c_{s}$. With $\vec{B}_{0}=0$ and $g>0$ one obtains atmospheric waves (Bray and Loughhead, 1974). When the gas pressure is negligible, $p=0$, and with $g=0$, but $\|\vec{B}_{0}\|>0$, the dispersion relation results $\omega^{2}\vec{v}_{1}/v_{A}^{2}=k^{2}\cos^{2}\alpha\,\vec{v}_{1}-(\vec{k}\cdot\vec{v}_{1})\,k\cos\alpha\,\vec{\hat{B}}_{0}+\left[(\vec{k}\cdot\vec{v}_{1})-k\cos\alpha\,(\vec{\hat{B}}_{0}\cdot\vec{v}_{1})\right]\vec{k}\,.$ (4.39) Here, $\vec{\hat{B}}_{0}$ is a unity vector parallel to $\vec{{B}}_{0}$, $\alpha$ is the angle between the wavevector $\vec{k}$ and $\vec{B}_{0}$, and $v_{A}$ is the Alfvén velocity with $v_{A}^{2}={B_{0}^{2}\over 4\pi\rho_{0}}=2{P_{m,0}\over\rho_{0}}\,,$ (4.40) where the magnetic pressure is $P_{m}=B^{2}/(8\pi)$. Scalar multiplication of Eq. 4.39 with $\vec{\hat{B}}_{0}$ shows that $\vec{\hat{B}}_{0}\cdot\vec{v}_{1}=0$. This means that the (perturbed) velocity is perpendicular to $\vec{B}_{0}$ (since the Lorentz force on the perturbed gas is perpendicular to $\vec{B}_{0}$). Scalar multiplication of Eq. 4.39 with $\vec{k}$ yields $\left(\omega^{2}-k^{2}v_{A}^{2}\right)\left(\vec{k}\cdot\vec{v}_{1}\right)=0\,.$ (4.41) From this equation one can derive two magnetic wave modes: * (1) Assuming $\nabla\cdot\vec{v}_{1}=0$ gives the so-called incompressible mode and from the ansatz Eq. 4.38 one gets $\vec{k}\cdot\vec{v}_{1}=0$. Thus, this mode is a transversal mode with the velocity perpendicular to the direction of propagation. From Eq. 4.39 we have $\omega=\pm\cos\alpha\,v_{A}$. The waves are also called shear Alfvén waves. For $\alpha=0\degr$ one derives that $\vec{B}_{1}$ and $\vec{v}_{1}$ are parallel and the propagation is along $\vec{B}_{0}$. * (2) Another solution of Eq. 4.41 is $\omega=kv_{A}$, independent of $\alpha$. These waves are compressional Alfvén waves, and for $\alpha=90\degr$ the velocity $\vec{v}_{1}$ is parallel to $\vec{k}$, i.e. we have longitudinal waves. Finally, admitting that the gas pressure is not negligible, $p>0$, the phase velocity comes out as $v_{ph}={\omega\over k}=\left[{1\over 2}\left(c-s^{2}+v_{A}^{2}\right)\pm\left(c_{s}^{4}+v_{A}^{4}-2c_{s}v_{a}^{2}\cos 2\alpha\right)^{1/2}\right]^{1/2}\,.$ (4.42) The ‘+’ sign above gives the so-called ‘fast magnetoacoustic waves’ and the ‘$-$’ sign the ‘slow magnetoacoustic waves’. Their phase speeds depend on $\alpha$ (see the hodographs in Ferraro and Plumpton, 1966; Kippenhahn and Möllenhoff, 1975; Priest, 1984). ##### Observational results Figure 4.10: Average temporal power spectra of velocity from AOI C in Figs. 4.4 and 4.5 before filtering (dashed) and after pass-band filtering (solid). The LOS velocities of the structures contain variations on long time scales of 10 min and longer as well as fluctuations with shorter time scales. To distill the latter, among them possibly magnetoacoustic waves, we applied a high-pass temporal filter and removed some high-frequency noise at the same time. The quantities then fluctuate about zero. Figure 4.10 depicts the average power spectra of the LOS velocities in AOI C in Fig. 4.5 before and after filtering. We note that the 5-min oscillations are filtered out, while some oscillations at the acoustic cutoff (corresponding to periods of approximately 200 s) are partially retained. Yet the unfiltered and filtered power spectra in Fig. 4.10 do not show any predominant period. Figures 4.11–4.14 show examples of space-time slices from AOIs C and D. The fluctuations of several quantities are shown: 1. 1. LOS velocities determined from differences of H$\alpha$ intensities at $\pm$0.5 Å off line center, henceforth referred to as Doppler-gram slices (bright indicates velocity towards observer); 2. 2. H$\alpha$ line center intensities, henceforth LC slices; 3. 3. in Figs. 4.11–4.13 differences of intensities at +0.5 Å off line center $I_{0.5}(t_{i+1})-I_{0.5}(t_{i})$ with cadence $\Delta t=t_{i+1}-t_{i}$ = 22 s, henceforth referred to as $\Delta I_{0.5}$ slices; 4. 4. in Fig. 4.14 differences of intensities at line center $I_{LC}(t_{i+1})-I_{LC}(t_{i})$, henceforth referred to as $\Delta I_{LC}$ slices. Time runs from bottom to top with $t=0$ at the start of the series. The interruptions/interpolations at $t\approx$ 18.0–19.6 min and 37.5–38.5 min are obvious. Figure 4.11: Example of space-time slices, of 11 width, from AOI C in Figs. 4.4 and 4.5. From left to right: LOS velocity, H$\alpha$ line center intensity, and intensity differences at H$\alpha$ +0.5Å off line center: $I_{0.5}(t_{i+1})-I_{0.5}(t_{i})$ with cadence of $\Delta t=t_{i+1}-t_{i}$ = 22 s. The intensity differences in the right column are shifted up by 11 s. They are referred to as $\Delta I_{0.5}$ slices in the text. Figure 4.12: Example of space-time slices, of 11 width, from AOI D. Same ordering as in Fig. 4.11. Figure 4.13: Selected part of space-time slices from AOI D with slice width of 22. Same order as in Fig. 4.11. Figure 4.14: Selected part of space-time slices from AOI D with slice width of 22. Left column: H$\alpha$ line center intensity fluctuations; right column: intensity differences at H$\alpha$ line center: $I_{LC}(t_{i+1})-I_{LC}(t_{i})$ with cadence of $\Delta t=t_{i+1}-t_{i}$ = 22 s, referred to as $\Delta I_{LC}$ slices in the text. We focus attention to the oblique stripes in Figs. 4.11–4.14. These are the signatures of magnetoacoustic waves. From their slopes we can measure phase velocities projected on the plane perpendicular to the LOS. In Fig. 4.11 from AOI C, the waves appear to originate near the right edge of the AOI. This is one side at which the fibrils are rooted. Presumably, the waves are excited by the buffeting of motions at the photospheric foot points of the magnetic fields. As seen especially well in the Doppler-gram slices of Fig. 4.11, but also in the LC slices, steep stripes originate from both sides of this region. The projected phase speeds are of the order of 8 km s-1. The stripes are often bent in the course of the temporal evolution, e.g. the wave parallel to the dashed line ‘1’ in the $\Delta I_{0.5}$ slices of Fig. 4.11. This wave starts off with a phase velocity of 14 km s-1 and speeds up to approximately 40 km s-1, one of the highest velocities measured. A prominent period is not detected. Sometimes, the waves appear to be repetitive, with two or three, at most, wave trains in sequence with periods between 90 s and 180 s. An example of consecutive wave trains is indicated by the three dashed lines ‘2’ in the LC slices of Fig. 4.11. Yet most time, the waves are solitary, with one single wave package traveling across the FoV. Many of the waves appear to spread out along the direction of propagation and to fade after having traveled a distance of 5–10 Mm. The amplitudes of the LOS velocities in the Doppler-gram slices are measured to approximately 1 km s-1, be it in the waves with low phase speeds or in those with high phase speeds. With the calibration discussed above in the context of the cloud model (see Sect. 4.2.2) these amplitudes have to be multiplied with a factor of approximately 3. The resulting amplitudes are thus of the order of 3 km s-1, which is not a small perturbation compared with the sound speed (c.f. below the discussion on the magnetoacoustic waves). Figure 4.12 from AOI D shows similar space-time slices as those from AOI C in Fig. 4.11. Yet here, the waves are excited at both sides and travel into the AOI, sometimes crossing from left and right and possibly colliding as in the example parallel to the dashed lines ‘1’ in the Doppler-gram. The long lasting (more than 7 min), solitary wave train (parallel to dashed line ‘2’ in Fig. 4.12) has a phase velocity of approximately 13 km s-1, a typical speed of the ‘slow’ waves in this AOI. A correction for foreshortening, i.e. that we see only the projection of the phase speed on the plane perpendicular to the LOS is to be excluded since the fibrils in AOI D, as well as those in AOI C, are oriented almost perpendicularly to the direction to the limb (see Fig. 4.5), thus perpendicularly to the LOS. The wave parallel to the dashed line ‘3’ in the $\Delta I_{0.5}$ slices of Fig. 4.12 gives a phase speed of 30 km s-1, again not to be corrected for foreshortening. This is a typical phase speed of the fast waves. Figure 4.13 gives another example of space-time slices from AOI D, with wider slices of 22 width and shorter time span of 13 min duration than in Figs. 4.11 and 4.12. The long lasting wave train in the Doppler-gram slices (parallel to dashed line ‘1’) gives again the typical phase speed of 13.3 km s-1 with (calibrated) LOS velocity amplitudes of approximately 2 km s-1. These LOS velocities are transversal, in the sense that they are perpendicular to the propagation and to the H$\alpha$ fibrils. The ‘fast’ wave in the $\Delta I_{0.5}$ slices (parallel to dashed line ‘2’) exhibits also the typical phase speed of 32 km s-1 with calibrated LOS velocities of approximately 1.5 km s-1. Figure 4.14 gives a 7.25 min long section of the temporal development of fluctuations in AOI C with slice widths of 22, but this time the LC slices and the $\Delta I_{LC}$ slices only. Note that dark and bright features in H$\alpha$ LC indicate increased and decreased absorption, respectively, not enhanced and reduced temperature (see Al et al., 2004; van Noort and Rouppe van der Voort, 2006). The two solitary waves between the pairs of horizontal dashed lines (a, a) and (b, b) have phase speeds of approximately 25 km s-1. Inspection of the LC slices shows that the waves consist of elongated, thin blobs with length of 1–2 and width of approximately 05. Apparently, the waves do not travel in the spatial direction along straight lines, but along sinuous lines with deviations from straight lines of approximately 05 in amplitude. This suggests that on these small scales the magnetic field is not straight and homogeneous but entangled. The presentation of the difference slices $\Delta I_{LC}$ in Fig. 4.14 is prepared to study temporal displacements of absorption features. These are only seen if the displacements have a strong component perpendicular to the LOS. The bright and dark small-scale features, lying parallel and next to each other, in the upper part of the $\Delta I_{LC}$ slices, between the dashed line pair (b, b) are suggestive of such displacements perpendicular to the direction of propagation. We summarize in short the observational findings on magnetoacoustic waves: 1. 1. Generally, we find two kinds of waves: slow waves with phase velocities of 12–14 km s-1 and fast waves with phase velocities of 25–33 km s-1 (maximum velocity found 42 km s-1). The waves appear to develop from low phase speed to high phase speed waves and vanish after having traveled a distance of 5–10 Mm. 2. 2. Irrespectively of the wave mode, the LOS gas velocities are of the order of 2–4 km s-1. 3. 3. The waves are mainly solitary. They consist of short (1–2) and thin ($\approx 0\farcs 5)$ blobs of compressed gas. 4. 4. The waves appear to follow wiggly, entangled magnetic field lines with possible lateral displacements. ##### Interpretation – waves in thin magnetic flux tubes For the interpretation of the observations from AOI C and D, we adopt the picture of waves in thin magnetic flux tubes, whose radius is small compared to the pressure scale height. The propagation of waves in magnetic flux tubes were treated by, among others, Defouw (1976), Wentzel (1979), Spruit (1982), and recently by Musielak et al. (2007). Spruit assumes a thin, cylindrical magnetic flux tube parallel to the $z$ axis, with radius $R$, magnetic field along the tube of strength $B$, gas pressure $p$, mass density $\rho$, and temperature $T$. The gravity is neglected. The tube is embedded in an external medium with properties $B_{e}$, $p_{e}$, $\rho_{e}$, and $T_{e}$. Inside and outside the tube the magnetic and atmospheric parameters are constant. In Spruit’s (1982) work, the MHD equations are linearized and a mode analysis is performed, with proper conditions at the interface between flux tube and surrounding medium. Incompressible Alfvén waves ($\nabla\cdot\vec{v}_{1}=0$, with small velocity perturbation $v_{1}$) are also possible in flux tubes. They are torsional Alfvén waves. The compressive solutions lead to $\nabla\cdot\vec{v}_{1}=A\,{\cal B}_{m}(nr)\exp[i\,(\omega t+m\phi+kz)]\,,$ (4.43) with amplitude $A$, ${\cal B}_{m}(nr)$ Bessel functions of order $m$, $r$ the distance from the axis of the tube, and $\phi$ the azimuthal angle. Inside the tube, the waves propagate along the $z$ direction. For $n$ the relation holds $n^{2}=(\omega^{2}-v_{A}^{2}k^{2})\,(\omega^{2}-c_{s}^{2}k^{2})/[(\omega^{2}-c_{t}^{2}k^{2})\,(v_{A}^{2}+c_{s}^{2})]\,.$ (4.44) Here, the tube speed $c_{t}$ is introduced with $c_{t}^{2}={v_{A}^{2}c_{s}^{2}\over v_{A}^{2}+c_{s}^{2}}\,,$ (4.45) which shows that the tube speed is smaller than both the Alfvén and the sound velocity. Spruit (1982) showed that in the limit $k\,R\rightarrow 0$ the mode with $m=0$ is a longitudinal mode with $v_{ph}=c_{t}$, which is approximately the sound speed $c_{s}$ for $v_{A}\gg c_{s}$. This mode is often referred to as the ‘sausage mode’, with velocity inside the tube parallel to the magnetic field. In the same limit and for $m>0$ one obtains the so-called ‘kink waves’, with phase speeds related to the magnetic fields and densities through $v_{ph}^{2}={\rho v_{A}^{2}+\rho_{e}v_{A,e}^{2}\over\rho+\rho_{e}}={1\over 4\pi}\cdot{B^{2}+B_{e}^{2}\over\rho+\rho_{e}}\,.$ (4.46) These waves are transversal waves, and Spruit’s (1982) analysis takes into account the dragging by the ambient medium. The phase speeds are obviously $v_{ph}^{2}=v_{A}^{2}$ for $\rho_{e}=\rho,\,\,B_{e}=B$; $v_{ph}^{2}=v_{A}^{2}/2$ for $\rho_{e}=\rho,\,\,B_{e}=0$, and $v_{ph}^{2}=2\cdot v_{A}^{2}$ for $\rho_{e}=0,\,\,B_{e}=B$. We now compare the observations of waves with the expectation from this linear wave theory. We adopt that the waves propagate along the magnetic field and that the influence of gravity on the wave properties is negligible. The period at the acoustic cutoff of 200 s is longer than the periods, actually seen only rarely, in our data. Likewise, the period for the cutoff of kink waves (Spruit, 1981; Choudhuri et al., 1993) is approximately 400 s, for small plasma $\beta$, which is the ratio of gas pressure to magnetic pressure, $\beta=(8\pi p)/B^{2}$. With the parameters in Table 4.1 for the surge discussed above in Sect. 4.2.3, i.e. with gas pressure $p=0.38$ dyn cm-2 and mass density $\rho=2.3\times 10^{-13}$ g cm-3, the sound velocity is $c_{s}=16.6$ km s-1. From the determination of parameters in a wide range of chromospheric H$\alpha$ structures by Tsiropoula and Schmieder (1997), Tsiropoula (2000), and Tsiropoula and Tziotziou (2004) we obtain values of the sound speed in the range of 13.5–16.7 km s-1. The widely found temperatures of $T=10^{4}$ K and the mean molar mass of 0.8 from the ionization equilibrium of hydrogen found from Table 4.1 and from the above works give a sound velocity of 14.4 km s-1. The phase velocities of the slow waves observed here are compatible with these values, if one accounts for possible small projection effects and for a small reduction for the velocity of tube waves (cf. Eq. 4.45). De Pontieu et al. (2004) adopted magnetic field strengths of the order of 100 Gauss in the chromosphere of active regions. With this value and the commonly found mass densities of 0.8–2.3$\times 10^{-13}$ g cm-3, the Alfvén velocity is $v_{A}=1\,000\dots 600$ km s-1, much higher than the velocities of the fast waves in the present observations. We believe, that 100 Gauss is an upper limit of the field strengths in the chromosphere of AOIs C and D. From high spatial resolution (approximately 035) data from a plage region by Bello González and Kneer (2008) we find an average field strength in the photosphere of 60–90 Gauss. This may possibly be reduced by a factor of 2 in chromospheric fibrils as in AOIs C and D by spreading out of the field lines over areas which possess little field in the photosphere. Otherwise the fibrils would not be so elongated. Yet still this yields to Alfvén velocities of $v_{A}\approx 200$ km s-1, as a minimum value. Giovanelli (1975) has measured velocities of 70 km s-1 in chromospheric H$\alpha$ structures. With a magnetic field strength of 10 Gauss and with reasonable particle densities he arrived at the Alfvén velocity in agreement with these measured phase velocities. In the present work, one would need field strengths as low as 5 Gauss for an Alfvén velocity of 32 km s-1 as observed. We note that even with 5 Gauss the motions are still dominated by the magnetic field, i.e. $\beta\ll 1$ holds. We estimate the maximum phase speed measurable from our data to 250–300 km s-1. Such velocities would still be detectable. The phase speeds found here are in the range 25–35 km s-1. (The highest measured speed amounts to 42 km s-1). These are obviously incompatible with Alfvén waves in a homogenous magnetic field with 30–100 Gauss. We mention several possibilities to reconcile our measurements with the picture of fast mode magnetoacoustic waves along the magnetic field, i.e. of Alfvén waves. 1. 1. The magnetic field strength in the fibrils of AOIs C and D is indeed as low as 5 Gauss, which is not very probable considering the very high activity in the whole area observed. AOIs C and D are not especially located at the outskirts of this activity. 2. 2. Propagation of a fast mode wave in a flux tube surrounded by a medium with low or zero field strength but with high gas density would reduce the phase speed (cf. Eq. 4.46). 3. 3. Apparently, the waves start as slow mode waves with phase velocities of the order of 10–14 km s-1 and then are transformed into fast mode waves propagating with Alfvén velocity. Yet the transformation does not occur immediately. Examples are seen in Fig. 4.11. While the solitary waves evolve into fast mode waves their wave packages get dispersed and they decay by spreading out along the direction of propagation. 4. 4. We do not measure phase velocities but group velocities of solitary wave packages. We have calculated for the slow mode the cusped surface of the wave front according to Ferraro and Plumpton (1966, cf. their Fig. 13), which is rotationally symmetric about the direction of the magnetic field. The adopted Alfvén and sound velocities were 200 km s-1 and 16 km s-1, respectively. The maximum velocity of this surface is only marginally larger than the sound speed by 3.2%, and the maximum deviation from the direction of the magnetic field is 001. Thus, the propagation of such slow mode pulses is practically along the magnetic field with the sound velocity. 5. 5. The picture is actually more complicated: The waves with low phase speed seen here are not pure longitudinal waves. The gas velocities of the waves have a strong transversal component of the order of 3 km s-1. Furthermore, the propagation of the fast waves deviates from straight lines, their motion appears more wiggly, possibly because the magnetic fields are entangled. Under the aspect of these observations the linear theory of small perturbations of straight flux tubes appears to be not sufficient. #### 4.2.5 Summary on some properties of the active chromosphere Thanks to the good resolution, we could follow the evolution of small-scale chromospheric structures of an active region. From the rich dynamical processes in the observed, very active, flaring region some areas were selected for detailed investigation in the present work: 1. 1. A small surge: It showed repetitive occurrence with a rate of some 10 minutes. The surge developed from initial small active fibrils to a straight, thin stucture of approximately 15 Mm length, then retreated back to its mouth to reappear again two times. The gas velocities reach approximately 100 km s-1. The rebound shock model by Sterling and Hollweg (1989) seems to be a viable explanation. 2. 2. Two small-scale, synchronous, possibly sympathetic flashes, or mini-flares: In a pair of small areas, two brightenings occurred simultaneously and disappeared during two H$\alpha$ scans with total duration of 45 s. Presumably, the evolutionary time scale is much shorter, few to 10 s. Yet we could follow the evolution with a temporal resolution of 2 s by analysing H$\alpha$ filtergrams at different wavelengths. One of the two flashes showed an apparent proper motion with a speed up to 200 km s-1, while it was developping a high emission, above the continuum intensity, in the red part of the H$\alpha$ profile. However, the cadence of the scanning was too slow to decide whether the temporal evolution consisted in a rapid horizontal proper motion with a final fast down flow or in a rapid change of emission at fixed local postitions. 3. 3. Magnetoacoustic waves in long fibrils: In two areas with long fibrils, the structures exhibited many magnetoacoustic waves running parallel to the fibrils, thus presumably also parallel to the magnetic field. The waves are mostly solitary. Few times, two or three repetitive wave trains could be seen with periods of 100–180 s. The waves start at the footings of the fibrils with a speed of 12–14 km s-1, which is not much lower than the sound speed estimated for such structures and similar to the tube speed. Most of the waves get accelerated to reach phase speeds of approximately 30 km s-1. Then they spread out along the fibrils and fade. The final phase speed is much lower than the Alfvén speed of $\geq 200$ km s-1, estimated from reasonable magnetic field strengths in the active region chromosphere of 30–100 Gauss and reasonable mass densities in the fibrils of 2$\times 10^{-13}$ g cm-3. Furthermore, we observe that the slow waves have strong transversal (LOS) velocity components with $\sim$3 km s-1, i.e. are not purely longitudinal, and that the fast waves consist of short (1–2), thin ($\sim$05) blobs and apparently move along sinuous lines. We conlude from these findings that a linear theory of wave propagation in straight magnetic flux tubes is not sufficient. ### 4.3 Comparison between speckle interferometry and blind deconvolution In Sec. 3.1 we introduced the image degradation problem due to atmospheric distortions for all ground based solar observations and astronomical observations on general. In Sec. 3.2.1 we explained the adaptive optics approach used at the VTT to reduce the image distortions in real time. Finally, in Sec. 3.3.3 we described the basis of two different _post factum_ image reconstruction techniques, the Speckle Imaging (SI) and one type of Blind Deconvolution (BD) with simultaneous Multiple Objects and Multiple Frames (MOMFBD). In the case of data from the G-FPI, the SI approach reconstructs separately the broadband images and uses both the original recorded frames (after dark subtraction and flat fielding) and the reconstructed image from this channel to obtain the reconstructed narrow-band images at the various positions along the spectral line. The MOMFBD code applied to our G-FPI data uses at the same time, for each spectral position, the various (15) pairs of simultaneously recorded broadband and narrow-band frames. Thus, at 21 wavelength positions along the spectral line, two different objects were observed and their images were reconstructed with 15 frames per object. In this thesis work we used the SI method for fields of view on the solar disc. The last version of the code takes into account the field dependence of the PSF around the AO lockpoint, so we will refer to this version as SI+AO. However, data frames on and off the limb cannot be reconstructed with the current code. Near the limb the contrast is lower than near disc center, which makes any reconstruction more difficult. Moreover, KAOS can lock on the low- contrast structures near the limb only under very good seeing conditions. Also, the limb darkening at large heliocentric angles makes it difficult to determine the STF on the rings around the lockpoint for the SI+AO. Beyond the problems inside the limb, off-the-limb emission features seen in the narrow- band images lack broadband counterparts, and therefore there exist no simultaneous data from which we could apply the second part of the SI to reconstruct the off-limb parts of the images. Dataset _limb_ (results in Sec. 5.2) was recorded under very good seeing conditions, with KAOS locked on a nearby facula correcting 27 (Zernike polynomial) modes most of the time. For the post factum image reconstruction we used the MOMFDB, for which the limb darkening presents no problems. Spicules above the limb do not posses a simultaneously observed broadband object, so it is expected that their spatial resolution is lower, since there are no multiple objects, just the multiple frames for each spectral position in the narrow-band channel. In order to compare results from both approaches we have reconstructed with both methods the same field of view on the disc, a frame of the _sigmoid_ data set. In this Section we present the comparison. As we show, for the BD case we made two different reconstructions with different limits of the expansion of the aberration in Karhunen-Loeve (K-L) modes. In the first case the expansion of the wavefront aberration is done until de $17^{th}$ K-L polynomial, while on the second run we expanded to the first 100 modes. The running time of the code is highly sensitive to the number of modes, being slower with more modes. Figure 4.15: Results for different image reconstruction techniques. Axes are in arcseconds. White rectangles enclose the region where the RMS contrast is calculated. _Upper left_ : Best speckle raw image, contrast is 5.9%. _Upper right_ : SI+AO result, contrast is 11.1%. _Lower left_ : MOMFBD running 17 modes, contrast is 6.7%. _Lower right_ : MOMFBD running 100 modes, contrast is 10.0%. Figure 4.16: Comparison of the power spectra (in arbitrary units) of the same image for different image reconstruction techniques. Figure 4.17: Results for different image reconstruction techniques for the line center narrow-band filtergram. Scales on the axes are in arcseconds. _Upper left_ : Best speckle raw image. _Upper right_ : SI+AO result. _Lower left_ : MOMFBD running 17 modes. _Lower right_ : Image difference. In this case the differences reach 45% of the fluctuations in the reconstructed frames. Figure 4.18: Comparison of the power spectra of the same image for different image reconstruction techniques. ##### Broadband Figure 4.16 compares the full FoV image, while Fig. 4.16 shows the corresponding power spectra. The speckle frame corresponds to the one with highest rms contrast (5.9% inside the white rectangle). The SI+AO reconstruction shows a much higher resolution, with more power at all frequencies than the speckle frame for angular scales larger than $\sim 0\farcs 32$. The reconstructed image has also less noise than the speckle frame (small-scale end of the power spectra). The rms contrast of granulation for this image is 11.1%. In the case of the MOMFBD with 17 modes the power of the reconstructed image is significantly lower than for SI+AO, albeit having a lower noise level (comparable even with the burst average). We have to run up to 100 modes to arrive at a similar contrast as for SI+AO. The noise threshold for the last run, coincides with that of the SI+AO approach. The rms contrast of granulation for these images are 6.7% with 17 modes and 10.0 % with 100 modes. Figure 4.16 shows also the power spectrum of the difference between SI+AO and MOMFBD100 (green line), which is many orders magnitude lower than one of the power spectra themselves. Only at scales smaller than $\sim 0\farcs 4$, the difference becomes of the same order as the power spectra. Taking the differences of the reconstructed images shows that $99.8\%$ of the pixels in the FoV have intensity fluctuations lower than $15\%$ of the intensities in the images themselves. ##### Narrow-band Figure 4.19: Close-up subfield from the narrow-band spectrogram at the H$\alpha$ core. Axes are in arcseconds. The narrow-band images have lower intensities than the broadband images, especially at the core of the H$\alpha$ line. Much less images are used for reconstruction, so a lower resolution is expected. Figure 4.18 compares the images at the H$\alpha$ line center, while Fig. 4.18 shows the corresponding power spectra. The SI + AO reconstruction shows a higher resolution than the speckle frame, with more power at scales larger than $\sim 0\farcs 5$. At smaller scales, the speckle frame is dominated by noise. The MOMFBD with only 17 modes gives already a similar resolution than the SI+AO and better treatment of the noise. Fig 4.19 shows a close-up region where the better noise treatment of the MOMFBD is clearly visible. The difference between the methods is bigger than in the broadband case, as expected since the intensity and resolution are lower. Nonetheless the agreement is very high, 92.8% of the pixels in the difference image have amplitudes smaller than $0.15$ of the average intensity in the reconstructed images (similar results are found for other spectral position, reaching 99% in the wings, at wavelengths $\pm 1$ Å off the line center). ##### Conclusions In this Section we have shown the good convergence of both post-processing approaches. Using different techniques we arrive at similar results and spectral profiles. The amplitudes of the difference images are lower than $0.15$ of the average image intensity in more than 99% of the broadband and above $\sim 90$% for the narrow-band images. In the case of the broadband reconstruction it was necessary to use 100 modes for the MOMFBD method to reach similar results as for SI+AO, while, in the narrow-band case, already with 17 modes the MOMFBD gives better images than SI+AO. The main disadvantage of BD methods is the computational load. The reconstruction of the single data set from broad and narrow-band and only 17 modes takes $\sim 7$ hours to process with 20 CPU cores of $3.2$GHz. For the 100 modes run, given the limited resources, we only used the broadband frames (Multi Frame BD). If the data set “sigmoid” were reconstructed with BD methods, even with only 17 modes, it would have taken around 130 days on our computing resources. The main advantage of the BD is its ability to reconstruct an image even with only few frames. This is of special importance when observing fast evolving targets. The SI needs much more frames. The property of reconstructing _simultaneously_ recorded images from different “objects” (e.g. broadband and the H$\alpha$ narrow-band) leads to a perfect sub-alignment of the results, which avoids spurious signals in derived quantities. Note however that, not simultaneously observed objects, like in the several consecutive scans with the G-FPI, are not aligned since they are not recorded under identical _seeing_ conditions. The SI+AO method is considerably much faster, around 10 and 15 minutes for the broadband and narrow-band images, respectively, with the same computers used for the MOMFBD reconstruction and gives better results for the broadband reconstruction, even using 100 modes in the latter method. However, with MOMFBD, the resolution and treatment of the noise is better in the narrow-band case. The main current advantage of the BD methods for our work and data is the possibility of reducing narrow-band limb and off-the-limb data scans. Anisoplanatism is an issue common to both approaches. In both cases the large FoV is divided into smaller subfields, where the assumption of isoplanatism is valid. It is therefore important to address this point for both cases. The image difference does not show any subfield pattern. However, there can still be some small effects. For this reason we have used the integrated contrast profile of the difference, defined as $\mathbb{CI}=\sum_{\lambda}\Big{\|}\frac{I_{SI+AO}(\lambda)-I_{MOMFBD_{17}}(\lambda)}{I_{SI+AO}(\lambda)}\Big{\|}\,,$ (4.47) where $I_{SI+AO}(\lambda)$ and $I_{MOMFBD_{17}}(\lambda)$ correspond to the reconstructed images using the SI+AO method, and to the images using MOMFBD with 17 modes, respectively. $\mathbb{CI}$ qualitatively measures the total difference between the profiles. If they were equal, then $\mathbb{CI}$ would be 0, while an increasing difference in the profiles increases the value of $\mathbb{CI}$. Since the subfield locations are the same for all the spectral positions, this information is added along the scan, while the intensities of the structures at each position are essentially subtracted out. The subfield pattern does not disappear with the subtraction of images reconstructed with different methods since they do not necessarily coincide. Figure 4.20 shows the calculated $\mathbb{CI}$. The weak subfield pattern is revealed, especially in regions where the difference is low (dark background). The mean edge length of the squares is approximately around 32 pixels. The amplitude of the grid pattern is very low, only revealed after the calculation of $\mathbb{CI}$. Presumably this comes from the apodization. When joining common regions on overlapping subfields, the common parts are overlaid in the final image. This, while preserving the structures, reduces the noise, which leads to slightly smaller noise levels in these overlapping lanes. This grid is common for all wavelength positions. The difference between the methods is low enough to reveal this small decrease of the noise level (leading to darker areas in Fig. 4.20) when the total effect is calculated, by using the $\mathbb{CI}$ parameter. Therefore, regions with more contrast, where the difference between SI and BD is bigger, the presence of this pattern is masked, as shown in the figure. Figure 4.20: Isoplanatic subfield array pattern when calculating $\mathbb{CI}$. The mean edge length of the squares is approximately 32 pixels. Axis scale is in arcseconds. ## Chapter 5 Spicules at the limb 111Contents from this Chapter have been partially published as Sánchez-Andrade Nuño et al. (2007a) Spicules, known for more than 130 years (see the hand drawings by Secchi, 1877), represent a prominent example of the dynamic chromosphere. We refer the reader to reviews by Beckers (1968, 1972) and to the paper by Wilhelm (2000) on UV properties. According to these works, spicules are seen at and outside the limb of the Sun as thin, elongated features. They develop speeds, measured from both proper motion and Doppler shifts, of 10–30 km s-1 and reach heights of 5–9 Mm on average, during their lifetimes of 3–15 minutes. Recent observations from HINODE (Kosugi et al., 2007) and own results presented below in Sec. 5.2 have changed the traditional picture. Some spicules live for only few seconds, and spicules may be much more inclined with respect to the vertical than adopted hitherto. As pointed out by Sterling (2000), a key impediment to develop a satisfactory understanding has been the lack of reliable observational data. Many theoretical models have been proposed to understand the nature of spicules, using a wide variety of motion triggers and driving mechanisms. In this Chapter we focus on the He i 10830 Å triplet emission line (see Sec. 2.3), using recent technical improvements in observational facilities, and on the results from the limb observations in H$\alpha$. ### 5.1 Spicule emission profiles observed in He i 10830 Å The energy levels that take part in the He i 10830 Å triplet are basically populated via an ionization-recombination process (Avrett et al., 1994). The EUV coronal irradiation (CI) at wavelengths $\lambda<504$ Å ionizes the neutral helium, and subsequent recombinations of singly ionized helium with free electrons lead to an overpopulation of all ortho-helium levels. Alternative theories suggest other mechanisms that might also contribute to the formation of the helium lines relying on the collisional excitation of the electrons in regions with higher temperature (e.g., Andretta and Jones, 1997). We are able to provide observational evidence of the link between the corona and the infrared emission of this line, in the frame of the current theoretical models of the solar atmosphere. Centeno (2006) modelled the ionisation and recombination processes using various amounts of CI, non-LTE radiative transfer, and different atmospheric models (see also Centeno et al., 2007). They have simulated limb observations for different heights, obtaining synthetic emission profiles in spherically symmetric models of the solar atmosphere. One conclusion of their study is that the ratio of intensities $({\cal R}=I_{\rm blue}/I_{\rm red})$ of the ‘blue’ to the ‘red’ components of the He i 10830 Å emission is a very good candidate for diagnosing the CI. The population of the metastable level depends on optical thickness, whose variation with height governs the change in the ratio $\cal R$ as a function of the distance to the limb. Trujillo Bueno et al. (2005) measured the four Stokes parameters of quiet-Sun chromospheric spicules and could show evidence of the Hanle effect by the action of inclined magnetic fields with an average strength of the order of 10 G. They modelled the He i 10830 Å profiles assuming the medium along the integrated line of sight as a slab of constant properties and with its optical thickness as a free parameter. Trujillo Bueno et al. (2005) showed that the observed intensity profiles and their ensuing $\cal R$ values can be reproduced by choosing an optical thickness significantly larger than unity. Centeno (2006) demonstrated that this optical thickness is related to the coronal irradiance (through the ratio $\cal R$), thus providing a physical meaning to the free parameter in the slab model (see also Centeno et al. 2007). Figure 5.1: Measured He i 10830 Å emission profiles for increasing distances to the solar limb, scanning a broad range of the height extension of the spicules. Each profile is the average of the 312 pixels along the slit (which was always kept parallel to the limb). #### 5.1.1 Observational intensity profiles and intensity ratio We present novel observations showing the spectral emission of He i 10830 Å and its dependence on the height of the spicules above a quiet region. We compare the deduced observational $\cal R$ with that obtained from detailed non-LTE numerical calculations using available atmospheric profiles. These data correspond to the data set described in Table 3.2. After the standard reduction process (Sec. 3.4.3 ) we obtain 21 intensity profiles above the infrared limb, with a step size of $0\farcs 35$. Figure 5.1 shows the emission profiles of the He i 10830 Å (after the reduction process) for different heights above the limb. Figure 5.2 illustrates this in three dimensions, as a function of wavelength and the distance to the solar limb, clearly showing a change in the intensity ratio of the blue and red components of the multiplet $({\cal R}=I_{\rm blue}/I_{\rm red})$ with height. Figure 5.2: 3D representation of the measured He i 10830 Å emission profiles for increasing distances to the solar visible limb. Note that the x-axis is wavelength, the y-axis the height above solar limb and the z-axis the intensity normalised to the maximum emission in the line center of the red component. For the calculation of $\cal{R}$ we need to determine the amplitudes of the blue and red components of the emission profile (as shown in Fig. 5.3). To determine the core wavelength of the red component of the triplet we fitted a Gaussian profile to its core, in a 1.3 Å range around the maximum. After symmetrising the observed profile around this maximum, using the values on the red side of the red component, we fitted another Gaussian function to the resulting symmetric profile. Subtraction of the fitted symmetric profile from the data leaves the emission profile of the blue component, which was also approximated by a Gaussian to determine its central wavelength. Our tests trying to fit directly both profiles using two Gaussians failed in a number of cases, probably due to the following reasons: (a) the red component is in fact the result of two blended lines, (b) the much weaker blue component was almost hidden in the broadened red component, and (c) the presence of noise. Our technique determines first the red component and then, after subtraction of the fitted profile, the blue one. We have thus separated the helium emissions into their red and blue components assuming only that both are present and that they are both symmetric. We can now measure their widths and intensities and also check that the line core positions coincide with the theoretical ones. After the fitting process the residuals between measured and observed profiles were small, the largest errors occurring in the determination of the core intensities of the red line. This happens because the red component consists of two blended lines (with a separation of 0.09 Å), a fact that flattens the emission profile near the core as opposed to a more peaked Gaussian function. Nevertheless, the differences between fitted profiles and data are only significant in the red core and are always lower than +0.08 of the maximum normalised intensity, with a mean difference of $\sim$0.02. To avoid systematic errors, we used the observational values for the center of the red component when calculating $\cal R$. Figure 5.3: Determination of the blue and red components of the He i 10830 Å triplet from the observed emission profiles. In this example the slit was placed at 14 off the solar visible limb. See text for details. The solid line represents the average emission profile. The dotted line is the Gaussian fit to the symmetrised red component. Subtraction of this from the observed profile leaves the blue component, which is also fitted by a Gaussian profile (thin solid line). The sum of both Gaussians (dashed line) gives the fit to the observed profile. #### 5.1.2 Results The chromospheric temperature and density are too low to populate the ortho- helium levels via collisions (Avrett et al., 1994). The EUV irradiation from the corona (CI) ionises the para-helium, and the subsequent recombinations lead to an overpopulation of all the ortho-helium levels, in particular those involved in the 10830 Å transitions. Centeno (2006) and Centeno et al. (2007) have modelled the off-the-limb emission profiles and concluded that the ratio $\cal{R}$ = $I_{blue}/I_{red}$ is a function of the height and a direct tracer of the amount of CI. Here we compare the results from the theoretical modelling with observations. Trujillo Bueno et al. (2005) modelled their spectropolarimetric observations assuming a slab with constant physical properties with a given optical thickness. In the optically thin regime $\cal R$ $\sim$ 0.12, which is the ratio of the relative oscillator strengths of the triplet. As the optical thickness (at the line-center of the red blended component) grows, this ratio also increases until it reaches a saturation value slightly larger than 1 for $\tau\sim 10$. (This type of calculation can be done and improved as explained in Trujillo Bueno & Asensio Ramos 2007). To reproduce the observed emission profile Trujillo Bueno et al. (2005) had to choose $\tau\sim 3$. Interestingly, the values of $\tau$ yielded by this modelling strategy are consistent with the more realistic approach of Centeno (2006), where non-LTE radiative transfer calculations in semi-empirical models of the solar atmosphere are presented, using spherical geometry and taking into account the ionising coronal irradiation. With our data we are able to test such theoretical calculations by comparing the measured values of $\cal R$ with those resulting from various chromospheric models. This way we may eventually trace the amount of CI inciding on the spicules. The analysis described above yielded the values of $\cal R$ for the observed profiles. The resulting dependence on the distance to the solar limb, for each pixel along the slit and each position of the slit above the limb, are presented in Fig. 5.4. The solid black line gives the average value of $\cal R$. Figure 5.4: Measured ratio $\cal R$ = $I_{blue}/I_{red}$ as a function of distance to the solar limb. Thin lines come from each pixel along the slit. The thick solid line represents the average and the dashed line the value of the optically thin regime. The dependence of $\cal R$ with height can be understood in a qualitative way as follows: In the outer layers of the chromosphere the density is so low that the transitions occur in the optically thin regime. With decreasing altitude the ratio $\cal R$ increases (proportionally with density) until a maximum optical thickness is reached. At even lower layers, although the density still continues to rise, the extinction of the coronal irradiance leads to a reduction in the number of ionizations, which results in a decrease of the optical thickness in the core wavelength of the red component, and thus in a decrease of $\cal R$. For a quantitative comparison with theoretical modelling we have used the results from Centeno (2006) and Centeno et al. (2007) where they calculated the ratios $\cal R$ for different standard model atmospheres: FAL-C and FAL-P (Fontenla et al., 1991) and FAL-X (Avrett, 1995). The FAL-C and FAL-X models may be considered as illustrative of the thermal conditions in the quiet Sun, while the FAL-P model of a plage region. The FAL-X model has a relatively cool atmosphere in order to explain the molecular CO absorption at 4.6 $\mu$. The comparison is shown in Fig. 5.5. We notice that the modelled height variations of $\cal{R}$ agree only in a qualitative manner with what is found in our observations. However, the calculations from different models of the solar atmosphere are unable to reproduce the measured ratio. Higher values of the coronal irradiance lead to an increase of the optical thickness (at the line centers of the He i multiplet) and an upward shift in the run of $\cal{R}$ vs. height. Yet the shape of the height dependence is mainly given by the atmospheric density profile and the attenuation of the ionising radiation as it reaches the lower layers of the chromosphere. It is also clear from Fig. 5.5 that the models do not extend high enough. Figure 5.5: Observed (average) vs. theoretical variation of the ratio $\cal R$$=I_{blue}/I_{red}$ with height. #### 5.1.3 Conclusions The theoretical behaviour of the ratio $\cal R$ agrees qualitatively with observations. Yet, a quantitative comparison shows poor agreement. Also, the simulated ratios are highly model dependent. As already explained, the failure to reproduce the observed profiles is very likely due to the density stratification not being adequate for spicule modelling and to the limited vertical extension of the atmospheric models. Modelling of the intensity ratio $\cal R$ in the He i infrared triplet should account for the fact that the solar chromosphere is inhomogeneous on small scales and that the spicules are small-scale intrusions of chromospheric matter into the hot corona. ### 5.2 High resolution imaging of spicules De Pontieu et al. (2007) recently published high resolution observations of spicules with the Solar Optical Telescope on board Hinode (Kosugi et al., 2007) in the Ca II H line at 3968 Å. They find at least two types of spicules that dominate the structure of the magnetic solar chromosphere: Type I with 3-7 minute timescales that correspond to the hitherto known spicules, and the new Type II spicules, developing in $\sim 10$ s, and lasting 10-150 s. These are also very thin, with widths down to the spatial resolution (120 km). Also, Trujillo Bueno et al. (2005) used spectropolarimetric observations and a theoretical modeling accounting for radiative transfer effects. They find that the magnetic field in spicules is aligned with the visible structures and measure field strengths of up to 40 G with an inclination of 35∘ with respect to he local vertical. This Section compares these observational and theoretical properties with our high spatial resolution observations with the G-FPI. We use the dataset “limb”. It consists of several consecutive H$\alpha$ scans with a field of view that includes the limb and a region outside the disc up to a height where no emission from spicules in the H$\alpha$ core is observed. The _seeing_ conditions were extremely good during the observations and the AO system could lock on a nearby facula. After usual dark subtraction and flat fielding we have used the BD method (see Sec. 3.3.3) to achieve highest spatial resolution. We were observing the limb near both poles. In Figures 5.6 and 5.7 we present some examples of the reduced data. Our time series of four minutes duration already shows a wide range of dynamics. We observe both long lasting spicules and fast evolving phenomena. Measuring the inclination of the projected spicules to the local vertical we find angles up to 30 for the north pole, as it has been reported (Beckers, 1968) . The projected height above the limb varies from the wings to the core, from 2770 to 3750 km at $\pm 0.5$ and $\pm 1$Å respectively. Near the south pole we find, however, much stronger emission and higher inclinations. The maximum angle is close to 70 from the local vertical, while the maximum height reaches up to 8250 km. We also find one horizontal fibril/spicule, as well as the presence of kinks or bends in some spicules. The width of single resolved spicules varies from a maximum width of 1 000 km at the spicule footing to a minimum size of 250 km, almost down to the resolution limit of the images, both in faint spicules and in others with strong emission. We also can retrieve the spectral profile at each pixel. Figure 5.6 demonstrates an important contribution to the understanding of spicules. It solves the long-standing question about the counterparts of spicules on the disc (Grossmann-Doerth and Schmidt, 1992). The first and last four filtergrams of the scan across H$\alpha$ in this figure show that spicules outside the limb continue as dark fibrils inside the disc. In Fig. 5.8 we show the mean variation from the disc to the limb of the intensity in H$\alpha$ around the north pole. Further, Fig. 5.9 presents mean intensity variations from the disc to the limb for several wavelengths around the H$\alpha$ line center. The emission at the line center is almost constant from the disc up to a height of around 5 above the limb, where the intensity starts to decrease. Figure 5.6: Reconstructed narrow-band scan observed near the solar north pole. The size of each image is $56\farcs 1\times 19\farcs 1$. The wavelength of the filtergrams increases by $0.1$Å from top left to bottom right row by row. The third image in the third row is closest to the center of the mean line profile. The images have been rotated to have the limb parallel to the horizontal axis. Figure 5.7: Limb H$\alpha$ 2D filtergram at $\lambda_{0}+1.1$ Å near the south pole, where a coronal hole was present. This region shows much stronger emission and more variation of spicule width, height and inclination as Fig. 5.6. A background of thin vertical spicules can be seen overlaid with wider and more inclined spicules, including nearly horizontal jets. Some of the spicules appear to be bent and show internal structure such as splitting into parallel jets. The maximum projected height above the limb is $\approx 8\,250$ km, while the mean height at this wavelength is $\approx$3700 km. The image has been rotated to show a horizontal limb in the presentation. Figure 5.8: Image representation of the mean measured spicular profiles from image 5.6. the x-axis is the height above limb, while the y-axis is the wavelength around H$\alpha$ line center (black horizontal line). Horizontal cuts at $\lambda-\lambda_{0}=[0,\pm 0.5,\pm 1]$Å are shown in Fig. 5.9. Figure 5.9: Average over $11\farcs 2$ of H$\alpha$ intensity profiles inside and outside the limb, for several line positions, observed near the solar north pole. _Dotted line_ : broadband intensity at 6300 Å, the inflection point defines the position of the solar limb; _green thick line_ : intensity at H$\alpha$ line center, which is nearly constant till $\approx$5 above the limb and then decreases outwards; _dashed lines_ : intensities at $-0.5$ Å (blue) and $+0.5$ Å (red) off line center, with height of spicular visibility decreasing at $\approx 3\farcs 7$; _solid lines_ : intensities at $\pm 1$Å off line center. It is seen that the H$\alpha$ line turns from absorption inside the limb into an emission line (line intensities higher than the continuum intensity) above the limb. ## Chapter 6 Conclusions and outlook We have studied the dynamics of the solar chromosphere, both on the disc and above the limb, using two spectral regions (H$\alpha$ in visible light and the infrared He i 10830 Å multiplet). By means of real-time correction and different post-processing techniques we have reduced the image degradation induced by the Earth’s atmosphere achieving resolutions in H$\alpha$ up to $0\farcs 5$ and better. This Chapter summarize the main conclusions of this work. ##### Observations and analysis The basic results from the observations taken from the disc are: * • Data taken in the combination with the “Göttingen”-Fabry Perot Interferometer (G-FPI), adaptive optics and speckle interferometry have high quality. We have obtained a time sequence in H$\alpha$ of 55 min from the active region AR 10875 at heliocentric angle $\vartheta\approx 36\degr$. The time cadence is 22 seconds, and its field of view $77\arcsec\times 94\arcsec$. For each interpolated time step we can retrieve 23 filtergrams along the H$\alpha$ spectral line with 45 mÅ FWHM and spatial resolution better than $0\farcs 5$ . Simultaneous broadband images at 6300 Å were also obtained, with spatial resolution of $0\farcs 25$, close to the telescopic diffraction limit. * • We have observed the dynamics of a small surge in detail: It showed repetitive occurrence with a rate of some 10 min. The surge developed from initial small active fibrils to a straight, thin structure of approximately 15 Mm length, then retreated back to its mouth to reappear again two times. The gas velocities reach approximately 100 km s-1. The rebound shock model by Sterling and Hollweg (1989) seems to be a viable explanation. * • The region was very active during the observations. We studied two small- scale, synchronous, possibly related flashes, or mini-flares. The simultaneity is within seconds, while their total evolution time was $\sim 45$s. The brightenings were separated by $\sim 14$ Mm. The used scanning parameters of the G-FPI were slow for this fast evolution, yet we could follow it with a temporal resolution of 2 s by analysing filtergrams taken consecutively at different wavelengths across the H$\alpha$ line. One of the two flashes showed an apparent proper motion with a speed up to 200 km s-1, while developing a high emission in H$\alpha$, above the continuum intensity. * • For the observations of waves we restricted our study to two areas exhibiting long fibrils. Yet the results likely represent the typical behavior of chromospheric magnetoacoustic waves within this active region. By means of high-pass frequency filtering, we observe waves running parallel to the fibrils, thus presumably also parallel to the magnetic field. They were mostly solitary waves, although sometimes repetitive wave trains could be seen with periods of 100–180 s. Most pulses start with velocities on the order of 12–14 km s-1 and get accelerated to reach phase speeds of approximately 30 km s-1. Furthermore, we observe that the slow waves have strong transversal (LOS) velocity components with $\sim$3 km s-1, i.e. are not purely longitudinal, and that the fast waves consist of short (1–2), thin ($\sim$05) blobs and apparently move along sinuous lines. Further, we have analyzed observations of spicules inside the disc and above the limb with the G-FPI data. Given the properties for this kind of observations we could not use the speckle interferometry method to reduce the atmospheric distortions. Instead we have used the blind deconvolution approach, in particular the version developed at the Swedish Institute for Solar Physics for multiple simultaneous objects with multiple frames per object (van Noort et al., 2005). The observations and analysis yielded the following main results: * • It is possible to successfully use multi-object multi-frame blind deconvolution methods with the G-FPI to reduce atmospheric distortions. This is specially important for on-limb observations, where the current speckle interferometry method is not applicable. * • We have observed spicules in H$\alpha$ at both solar polar caps. Compared with the solar north pole, we find much stronger spicular emission at the south pole that could be related to the presence of a coronal hole. The maximum projected height reaches 8250 km, while we see inclinations of the spicules up to 70 form the local vertical. We can resolve the detailed structure of the spicules as well as the presence of kinks or bends in some cases. The width of a single spicule ranges from 1 000 km down to the resolution limit of around 250 km. * • Using the retrieved spectral profile we can observe that spicules outside the limb continue as dark fibrils inside the disc, as shown in Fig. 5.6. This answers a long standing question, e.g. cited by Grossmann-Doerth and Schmidt (1992). Thus, we have used two different post-processing approaches to reduce the image degradation with the H$\alpha$ spectral line data. Since both methods are based on different approaches, we have reduced the same observational data with both techniques and compared the results. These are the main conclusions: * • The agreement of the images from both approaches is high. The achieved resolution comes close to the diffraction limit in both cases. Even though both methods split the image into isoplanatic subfields for individual reconstruction, there is no difference of subfield re-composition when comparing the results. * • In general, the biggest advantage of speckle interferometry over blind deconvolution is the small computational time required. A complete restoration of one full H$\alpha$ scan like the ones used in this work is roughly $\sim 400$ times faster with speckle interferometry than with blind deconvolution methods. * • The main advantage of blind deconvolution methods is their versatility. It can be applied with only few frames, with one or few simultaneous objects, or both at the same time. This method is highly advisable when aiming for e.g., fast evolving targets, or limb observations. The perfect sub-alignment of simultaneous objects avoids spurious signals on deduced quantities, like magnetograms. * • For the broadband channel we find that the speckle interferometry gives images with high contrast. Only when forcing the blind deconvolution method to reconstruct up to high ($100$) Karhunen-Loeve modes, the results are similar. We note that the speed of the reconstruction is proportional to the number of modes. * • The narrow-band images are clearly better reconstructed with the blind deconvolution, even with only 17 Karhunen-Loeve modes. The noise treatment gives smoother images with details at smaller scales (of the order of $\approx 0\farcs 3$). Further, we have obtained and analysed spectroscopic measurements in the infrared. We have centered our studies here on the intensity profiles of the He i 10830 Å multiplet above the limb. * • Recent work, like e.g. by Trujillo Bueno et al. (2005); Centeno (2006), has demonstrated the importance of the intensity ratio between the blue and red component of this triplet as tracer of the coronal irradiance. In this work we present novel observations showing the variation of this parameter with distance to the limb with a resolution of $0\farcs 35$ up to 7 above the solar visible limb (See Fig. 5.4). ##### Interpretation of observations For the interpretation of the observed data we have used several models and previous theoretical results to compare with the presented data. The main results from this analysis are: * • From the intensity profiles of the H$\alpha$ spectral line inside the disc we can infer many physical parameters. We have applied the lambdameter method as a fast and easy way to retrieve qualitative velocity maps. Also we have used Beckers’s 1964 cloud model. Our simple non-LTE inversion code provides the possibility to infer many physical parameters, e.g. hydrogen and electron density, mass density, temperature, …The results are in agreement with the data given in the current literature. * • From the linearization of the MHD problem, we discuss the interpretation of the observed waves as magnetoacoustic waves. We assume estimates with reasonable magnetic field strengths in the chromosphere of the active region of 30–100 Gauss and reasonable mass densities in the fibrils of 2$\times 10^{-13}$ g cm-3. The observed wave speeds are much lower than the expected Alfvén velocities. We conclude from these findings that a linear theory of wave propagation in straight magnetic flux tubes is not sufficient. * • From the infrared observations we have calculated the ratio of amplitudes in the two main components of the He i 10830 Å multiplet. Centeno (2006) has modelled synthetic limb observations according to the current theories of formation of this triplet and chromospheric models. The agreement is only qualitative. The failure to reproduce the observed profiles is very likely due to the density stratification not being adequate for spicule modelling used and to the limited vertical extension of the atmospheric models. Modelling of the intensity ratio should account for the fact that the solar chromosphere is inhomogeneous on small scales and that the spicules are small-scale intrusions of chromospheric matter into the hot corona. Future models of the solar chromosphere should be constrained by the observational evidences presented within this work. ##### Outlook The solar chromosphere represents a lively and exciting field of research. The wealth of structures, its dynamics and the wide range of evolution timescales are the consequence of the peculiar properties of this atmospheric layer. Current instruments like the ones used here, are able to observe and study in great detail new phenomena, that test current models and, as a consequence, helps our understanding of the solar atmosphere. This thesis aimed at contributing to the understanding. Yet, work to extend this research is already in progress. Here we shortly describe some of this work and give an outlook to further steps to be undertaken next. * • The blind convolution method provides a practical way to study the spicules in H$\alpha$ near and above the limb. Data from a short time sequence, taken under very good seeing conditions, are currently under reconstruction with phase diversity methods. The analysis will shed light onto the dynamic phenomena in spicules. * • We have learned that the sequential scanning, with the G-FPI, with cadence of 22 s is not fast enough in some cases. For future observations, we can design scanning modes of 2–3 second resolution taking images at fewer wavelength positions in a spectral line, like H$\alpha$. * • New infrared data of spicules near the solar poles and the equator, below coronal holes or in coronal active regions will help us to understand the detailed formation process of the He i 10830 Å lines. * • Full Stokes spectropolarimetric data of the He i 10830 Å multiplet are available from an earlier observing campaign. Scans above the limb were performed under very good seeing conditions. We can therefore extend our present analysis and study the polarization. We aim to investigate the Hanle effect as suggested by Trujillo Bueno et al. (2005) and make use of available inversion techniques like e.g. from Lagg et al. (2004). * • The new Gregor telescope (Balthasar et al., 2007) will host the G-FPI from the coming year on. The combination of this new facility with other instruments like Hinode (Kosugi et al., 2007), will provide new exciting resources for further research. ## References * Al et al. (2004) Al, N., Bendlin, C., Hirzberger, J., Kneer, F., Trujillo Bueno, J., 2004, Dynamics of an enhanced network region observed in H$\alpha$, A&A, 418, 1131–1139 * Alissandrakis et al. (1990) Alissandrakis, C. E., Tsiropoula, G., Mein, P., 1990, Physical parameters of solar H-alpha absorption features derived with the cloud model, A&A, 230, 200–212 * Andretta and Jones (1997) Andretta, V., Jones, H. P., 1997, On the Role of the Solar Corona and Transition Region in the Excitation of the Spectrum of Neutral Helium, ApJ, 489, 375–394 * Athay (1976) Athay, R. G., 1976, The solar chromosphere and corona: Quiet Sun, Reidel, Dordrecht, XII * Avrett (1995) Avrett, E. H., 1995, Two-component modeling of the solar IR CO lines, in Infrared Tools for Solar Astrophysics: What’s Next?, 15th NSO Sac Peak Workshop, (Eds.) J. Kuhn, M. Penn, pp. 303–311 * Avrett et al. (1994) Avrett, E. H., Fontenla, J. M., Loeser, R., 1994, Formation of the solar 10830 A line, in Infrared Solar Physics, IAU Symp. No. 154, (Eds.) D. Rabin, J. Jefferies, C. Lindsey, pp. 35–47 * Balthasar et al. (2007) Balthasar, H., von der Lühe, O., Kneer, F., Staude, J., Volkmer, R., Berkefeld, T., Caligari, P., Collados, M., Halbgewachs, C., Heidecke, F., Hofmann, A., Klvaňa, M., Nicklas, H., Popow, E., Puschmann, K., Schmidt, W., Sobotka, M., Soltau, D., Strassmeier, K., Wittmann, A., 2007, GREGOR: the New German Solar Telescope, in The Physics of Chromospheric Plasmas, (Eds.) P. Heinzel, I. Dorotovič, R. J. Rutten, vol. 368 of Astronomical Society of the Pacific Conference Series, pp. 605–610 * Beck et al. (2005) Beck, C., Schlichenmaier, R., Collados, M., Bellot Rubio, L., Kentischer, T., 2005, A polarization model for the German Vacuum Tower Telescope from in situ and laboratory measurements, A&A, 443, 1047–1053 * Beckers (1964) Beckers, J. M., 1964, A study of the fine structures in the solar chromosphere, Ph.D. thesis, University of Utrecht * Beckers (1968) Beckers, J. M., 1968, Solar Spicules (Invited Review Paper), Sol. Phys., 3, 367–433 * Beckers (1972) Beckers, J. M., 1972, Solar Spicules, ARA&A, 10, 73–100 * Beckers and Tallant (1969) Beckers, J. M., Tallant, P. E., 1969, Chromospheric inhomogeneities in sunspot umbrae, Sol. Phys., 7, 351–365 * Bello González and Kneer (2008) Bello González, N., Kneer, F., 2008, Narrow-band full Stokes polarimetry of small structures on the Sun with speckle methods, A&A, in press * Bendlin (1993) Bendlin, C., 1993, Hochauflösende zweidimensionale Spektroskopie der solaren Granulation mit einem Fabry-Perot-Interferometer, Ph.D. thesis, Göttingen * Bendlin and Volkmer (1995) Bendlin, C., Volkmer, R., 1995, The two-dimensional spectrometer in the German Vacuum Tower Telescope/Tenerife. From observations to results., A&AS, 112, 371–382 * Bendlin et al. (1992) Bendlin, C., Volkmer, R., Kneer, F., 1992, A new instrument for high resolution, two-dimensional solar spectroscopy, A&A, 257, 817–823 * Berkefeld (2007) Berkefeld, T., 2007, Solar adaptive optics, in Modern solar facilities - advanced solar science, (Eds.) F. Kneer, K. G. Puschmann, A. D. Wittmann, pp. 107–114, Universitätsverlag Göttingen * Born and Wolf (1999) Born, M., Wolf, E., 1999, Principles of Optics, Cambridge University Press, 7th edition * Bray and Loughhead (1974) Bray, R. J., Loughhead, R. E., 1974, The Solar Chromosphere, Chapman and Hall, London * Carlsson and Hansteen (2005) Carlsson, M., Hansteen, V., 2005, Chromospheric waves, in International Scientific Conference on Chromospheric and Coronal Magnetic Fields (ESA SP-596) on CD-Rom, (Eds.) D. Innes, A. Lagg, S. Solanki, D. Danesy * Centeno (2006) Centeno, R., 2006, Investigación de la propagación de ondas en la atmósfera solar mediante espectropolarimetría en Helio I 10830 Å, Ph.D. thesis, Universidad de La Laguna * Centeno et al. (2007) Centeno, R., Trujillo Bueno, J., Uitenbroek, H., Collados, M., 2007, The influence of coronal EUV irradiance on the emission in the He I 10830 A and D3 multiplets, ArXiv e-prints, 712, 0712.2203 * Chandrasekhar (1960) Chandrasekhar, S., 1960, Radiative Transfer, Dover Publications * Choudhuri et al. (1993) Choudhuri, A. R., Auffret, H., Priest, E. R., 1993, Implications of rapid footpoint motions of photospheric flux tubes for coronal heating, Sol. Phys., 143, 49–68 * Collados et al. (2007) Collados, M., Lagg, A., Díaz García, J. J., Hernández Suárez, E., López López, R., Páez Mañá, E., Solanki, S. K., 2007, Tenerife Infrared Polarimeter II, in The Physics of Chromospheric Plasmas, (Eds.) P. Heinzel, I. Dorotovič, R. J. Rutten, vol. 368 of PASP Conf. Ser., pp. 611–616 * de Boer (1996) de Boer, C. R., 1996, Noise filtering in solar speckle masking reconstructions, A&AS, 120, 195–199 * De Moortel et al. (2002) De Moortel, I., Ireland, J., Hood, A. W., Walsh, R. W., 2002, The detection of 3& 5 min period oscillations in coronal loops, A&A, 387, 13–16 * De Pontieu et al. (2004) De Pontieu, B., Erdélyi, R., James, S. P., 2004, Solar chromospheric spicules from the leakage of photospheric oscillations and flows, Nature, 430, 536–539 * De Pontieu et al. (2007) De Pontieu, B., McIntosh, S. W., Hansteen, V. H., Carlsson, M., Schrijver, C. J., Tarbell, T. D., Title, A. M., Shine, R. A., Suematsu, Y., Tsuneta, S., Katsukawa, Y., Ichimoto, K., Shimizu, T., Nagata, S., 2007, A Tale Of Two Spicules: The Impact of Spicules on the Magnetic Chromosphere, ArXiv e-prints, 710, 0710.2934 * Defouw (1976) Defouw, R. J., 1976, Wave propagation along a magnetic tube, ApJ, 209, 266–269 * Ferraro and Plumpton (1966) Ferraro, V. C. A., Plumpton, A., 1966, An Introduction to Magneto-Fluid Mechanics, Oxford University Press, Oxford * Fontenla et al. (1991) Fontenla, J. M., Avrett, E. H., Loeser, R., 1991, Energy balance in the solar transition region. II - effects of pressure and energy input on hydrostatic models, ApJ, 377, 712–725 * Fried (1965) Fried, D. L., 1965, Statistics of a Geometric Representation of Wavefront Distortion, J. Opt. Soc. Am. (1917-1983), 55, 1427–1435 * Georgakilas et al. (1990) Georgakilas, A. A., Alissandrakis, C. E., Zachariadis, T. G., 1990, Mass motions associated with H-alpha active region arch structures, Sol. Phys., 129, 277–293 * Giovanelli (1975) Giovanelli, R., 1975, Wave systems in the chromosphere, Sol. Phys., 44, 299–314 * Grossmann-Doerth and Schmidt (1992) Grossmann-Doerth, U., Schmidt, W., 1992, Chromospheric fine structure revisited, A&A, 264, 236–242 * Hansteen et al. (2006) Hansteen, V. H., De Pontieu, B., Rouppe van der Voort, L., van Noort, M., Carlsson, M., 2006, Dynamic fibrils are driven by magnetoacoustic shocks, ApJ, 647, 73–76 * Heinzel et al. (2007) Heinzel, P., Dorotovic, I., Rutten, R. J. (Eds.), 2007, The Physics of Chromospheric Plasmas, 368, ASP Conference Series * Hildner (1974) Hildner, E., 1974, The formation of solar quiescent prominences by condensation, Sol. Phys., 35, 123–136 * Innes et al. (2005) Innes, D. E., Lagg, A., Solanki, S. A., Danesy, D. (Eds.), 2005, International Scientific Conference on Chromospheric and Coronal Magnetic Fields, vol. 596 of ESA Special Publication (CD-ROM) * Janssen (2003) Janssen, K., 2003, Structure and dynamics of small scale magnetic fields in the solar atmosphere. Results of high resolution polarimetry and image reconstruction, Ph.D. thesis, Universitäts-Sternwarte Göttingen * Keller and von der Lühe (1992) Keller, C. U., von der Lühe, O., 1992, Solar speckle polarimetry, A&A, 261, 321–328 * Kippenhahn and Möllenhoff (1975) Kippenhahn, R., Möllenhoff, C., 1975, Elementare Plasmaphysik, Bibliograpisches Institut, Zürich * Korff (1973) Korff, D., 1973, Analysis of a Method for Obtaining Near Diffraction Limited Information in the Presence of Atmospheric Turbulence, J. Opt. Soc. Am. (1917-1983), 63, 971–980 * Koschinsky et al. (2001) Koschinsky, M., Kneer, F., Hirzberger, J., 2001, Speckle spectro-polarimetry of solar magnetic structures, A&A, 365, 588–597 * Kosugi et al. (2007) Kosugi, T., Matsuzaki, K., Sakao, T., Shimizu, T., Sone, Y., Tachikawa, S., Hashimoto, T., Minesugi, K., Ohnishi, A., Yamada, T., Tsuneta, S., Hara, H., Ichimoto, K., Suematsu, Y., Shimojo, M., Watanabe, T., Shimada, S., Davis, J. M., Hill, L. D., Owens, J. K., Title, A. M., Culhane, J. L., Harra, L. K., Doschek, G. A., Golub, L., 2007, The Hinode (Solar-B) Mission: An Overview, Sol. Phys., 243, 3–17 * Kukhianidze et al. (2006) Kukhianidze, V., Zaqarashvili, T. V., Khutsishvili, E., 2006, Observation of kink waves in solar spicules, A&A, 449, L35–L38, arXiv:astro-ph/0602534 * Labeyrie (1970) Labeyrie, A., 1970, Attainment of Diffraction Limited Resolution in Large Telescopes by Fourier Analysing Speckle Patterns in Star Images, A&A, 6, 85–87 * Lagg (2007) Lagg, A., 2007, Recent advances in measuring chromospheric magnetic fields in the He I 10830 Å line, Advances in Space Research, 39, 1734–1740 * Lagg et al. (2004) Lagg, A., Woch, J., Krupp, N., Solanki, S. K., 2004, Retrieval of the full magnetic vector with the He I multiplet at 1083 nm. Maps of an emerging flux region, A&A, 414, 1109–1120 * Löfdahl (2002) Löfdahl, M. G., 2002, Image Reconstruction from Incomplete Data II, in SPIE, (Eds.) P. J. Bones, M. A. Fiddy, R. P. Millane, vol. 4792, pp. 146–155 * Löfdahl et al. (2007) Löfdahl, M. G., van Noort, M. J., Denker, C., 2007, Solar image restoration, in Modern solar facilities - advanced solar science, Göttingen September 27-29, 2006, (Eds.) F. Kneer, K. G. Puschmann, A. D. Wittmann, p. 119, Universitätsverlag Göttingen * Martínez Pillet et al. (1999) Martínez Pillet, V., Collados, M., Sánchez Almeida, J., González, V., Cruz-López, A., Manescau, A., Joven, E., Paez, E., Díaz, J., Feeney, O., Sánchez, V., Scharmer, G., Soltau, D., 1999, LPSP & TIP: Full Stokes Polarimeters for the Canary Islands Observatories, in High Resolution Solar Physics: Theory, Observations, and Techniques, (Eds.) T. R. Rimmele, K. S. Balasubramaniam, R. R. Radick, vol. 183 of PASP Conf. Ser., pp. 264–272 * Mitalas and Sills (1992) Mitalas, R., Sills, K. R., 1992, On the photon diffusion time scale for the sun, ApJ, 401, 759–760 * Musielak et al. (2007) Musielak, Z. E., Routh, S., Hammer, R., 2007, Cutoff-free Propagation of Torsional Alfvén Waves along Thin Magnetic Flux Tubes, ApJ, 659, 650–654 * Neckel (1999) Neckel, H., 1999, Announcement spectral atlas of solar absolute disk-averaged and disk-center intensity from 3290 $\mathrm{\AA}$ to 12510 $\mathrm{\AA}$ (Brault and Neckel, 1987) now available from Hamburg observatory anonymous FTP site, Sol. Phys., 184, 421–422 * Noll (1976) Noll, R. J., 1976, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am. (1917-1983), 66, 207–211 * Paxman et al. (1996) Paxman, R. G., Seldin, J. H., Löfdahl, M. G., Scharmer, G. B., Keller, C. U., 1996, Evaluation of Phase-Diversity Techniques for Solar-Image Restoration, ApJ, 466, 1087–1099 * Priest (1984) Priest, E. R., 1984, Solar Magnetohydrodynamics, Reidel, Dortrecht * Puschmann and Sailer (2006) Puschmann, K. G., Sailer, M., 2006, Speckle reconstruction of photometric data observed with adaptive optics, A&A, 454, 1011–1019 * Puschmann et al. (2006) Puschmann, K. G., Kneer, F., Seelemann, T., Wittmann, A. D., 2006, The new Göttingen Fabry-Perot spectrometer for two-dimensional observations of the Sun, A&A, 451, 1151–1158 * Robbrecht et al. (2001) Robbrecht, E., Verwichte, E., Berghmans, D., Hochedez, J. F., Poedts, S., Nakariakov, V. M., 2001, Slow magnetoacoustic waves in coronal loops: EIT and TRACE, A&A, 370, 591–601 * Roddier (1990) Roddier, N. A., 1990, Atmospheric wavefront simulation and Zernike polynomials, in Amplitude and Intensity Spatial Interferometry, (Ed.) J. B. Breckinridge, vol. 1237 of SPIE, pp. 668–679 * Rouppe van der Voort et al. (2007) Rouppe van der Voort, L. H. M., De Pontieu, B., Hansteen, V. H., Carlsson, M., van Noort, M., 2007, Magnetoacoustic shocks as a driver of quiet-sun mottles, ApJ, 660, 169–172 * Saha (2002) Saha, S. K., 2002, Modern optical astronomy: technology and impact of interferometry, Reviews of Modern Physics, 74, 551–600, arXiv:astro-ph/0202115 * Sánchez-Andrade Nuño et al. (2005) Sánchez-Andrade Nuño, B., Puschmann, K. G., Sánchez Cuberes, M., Blanco Rodríguez, J., Kneer, F., 2005, Chromospheric Dynamics of a Solar Active Region, in The Dynamic Sun: Challenges for Theory and Observations, vol. 600 of ESA Special Publication * Sánchez-Andrade Nuño et al. (2007) Sánchez-Andrade Nuño, B., Bello González, N., Blanco Rodríguez, J., Kneer, F., Puschmann, K. G., 2007, Fast events and waves in solar H$\alpha$ structures at high resolution, A&A, submitted * Sánchez-Andrade Nuño et al. (2007a) Sánchez-Andrade Nuño, B., Centeno, R., Puschmann, K. G., Trujillo Bueno, J., Blanco Rodríguez, J., Kneer, F., 2007a, Spicule emission profiles observed in He I 10830 Å, A&A, 472, 51–54, arXiv:0707.4421 * Sánchez-Andrade Nuño et al. (2007b) Sánchez-Andrade Nuño, B., Puschmann, K. G., Kneer, F., 2007b, Observations of a flaring active region in H$\alpha$, in Modern solar facilities - advanced solar science, Proceedings of a Workshop held at Göttingen, (Eds.) F. Kneer, K. G. Puschmann, A. D. Wittmann, pp. 273–276 * Secchi (1877) Secchi, P., 1877, Le Soleil, 2 * Soltau (1985) Soltau, D., 1985, The German 60-cm Vacuum Tower Telescope and Its Post-Focus Facilities, in Solar Physics and Interplanetary Travelling Phenomena, (Eds.) C. de Jager, B. Chen, pp. 1191–1198 * Spruit (1981) Spruit, H. C., 1981, Motion of magnetic flux tubes in the solar convection zone and chromosphere, A&A, 98, 155–160 * Spruit (1982) Spruit, H. C., 1982, Propagation speeds and acoustic damping of waves in magnetic flux tubes, Sol. Phys., 75, 3–17 * Sterling (2000) Sterling, A. C., 2000, Solar Spicules: A Review of Recent Models and Targets for Future Observations - (Invited Review), Sol. Phys., 196, 79–111 * Sterling and Hollweg (1989) Sterling, A. C., Hollweg, J. V., 1989, A rebound shock mechanism for solar fibrils, ApJ, 343, 985–993 * Stix (2002) Stix, M., 2002, The Sun. An Introduction, Springer, Berlin. Second Edition * Tandberg-Hanssen (1977) Tandberg-Hanssen, E., 1977, Prominences, in Illustrated Glossary for Solar and Solar-Terrestrial Physics, (Eds.) A. Bruzek, C. J. Durrant, vol. 69 of Astrophysics and Space Science Library, pp. 97–111 * Tomczyk et al. (2007) Tomczyk, S., McIntosh, S. W., Keil, S. L., Judge, P. G., Schad, T., Seeley, D. H., Edmondson, J., 2007, Alfvén Waves in the Solar Corona, Science, 317, 1192–1196 * Tothova et al. (2007) Tothova, D., Innes, D. E., Solanki, S. K., 2007, Wavelet-based method for coronal loop oscillations analysis, in Modern solar facilities - advanced solar science, Proceedings of a Workshop held at Göttingen, (Eds.) F. Kneer, K. G. Puschmann, A. D. Wittmann, pp. 265–268 * Tripathi et al. (2007) Tripathi, D., Solanki, S. K., Mason, H. E., Webb, D. F., 2007, A bright coronal downflow seen in multi-wavelength observations: evidence of a bifurcating flux-rope?, A&A, 472, 633–642 * Trujillo Bueno et al. (2005) Trujillo Bueno, J., Merenda, L., Centeno, R., Collados, M., Landi Degl’Innocenti, E., 2005, The Hanle and Zeeman Effects in Solar Spicules: A Novel Diagnostic Window on Chromospheric Magnetism, ApJ, 619, 191–194 * Tsiropoula (2000) Tsiropoula, G., 2000, Physical parameters and flows along chromospheric penumbral fibrils, A&A, 357, 735–742 * Tsiropoula and Schmieder (1997) Tsiropoula, G., Schmieder, B., 1997, Determination of physical parameters in dark mottles., A&A, 324, 1183–1189 * Tsiropoula and Tziotziou (2004) Tsiropoula, G., Tziotziou, K., 2004, The role of chromospheric mottles in the mass balance and heating of the solar atmosphere, A&A, 424, 279–288 * Tsiropoula et al. (1993) Tsiropoula, G., Alissandrakis, C. E., Schmieder, B., 1993, The fine structure of a chromospheric rosette, A&A, 271, 574–586 * Tziotziou (2007) Tziotziou, K., 2007, Chromospheric Cloud-Model Inversion Techniques, in The Physics of Chromospheric Plasmas, (Eds.) P. Heinzel, I. Dorotovič, R. J. Rutten, vol. 368 of PASP Conf. Ser., pp. 217–237 * Tziotziou et al. (2003) Tziotziou, K., Tsiropoula, G., Mein, P., 2003, On the nature of the chromospheric fine structure. I. Dynamics of dark mottles and grains, A&A, 402, 361–372 * Tziotziou et al. (2004) Tziotziou, K., Tsiropoula, G., Mein, P., 2004, On the nature of the chromospheric fine structure. II. Intensity and velocity oscillations of dark mottles and grains, A&A, 423, 1133–1146 * van Noort et al. (2005) van Noort, M., Rouppe van der Voort, L., Löfdahl, M. G., 2005, Solar Image Restoration By Use Of Multi-frame Blind De-convolution With Multiple Objects And Phase Diversity, Sol. Phys., 228, 191–215 * van Noort and Rouppe van der Voort (2006) van Noort, M. J., Rouppe van der Voort, L. H. M., 2006, High-Resolution Observations of Fast Events in the Solar Chromosphere, ApJ, 648, 67–70 * Volkmer et al. (1995) Volkmer, R., Kneer, F., Bendlin, C., 1995, Short-period waves in small-scale magnetic flux tubes on the Sun., A&A, 304, 1–4 * von der Lühe (1984) von der Lühe, O., 1984, Estimating Fried’s parameter from a time series of an arbitrary resolved object imaged through atmospheric turbulence, J. Opt. Soc. Am., 1, 510–519 * von der Lühe et al. (2003) von der Lühe, O., Soltau, D., Berkefeld, T., Schelenz, T., 2003, KAOS: Adaptive optics system for the Vacuum Tower Telescope at Teide Observatory, in Innovative Telescopes and Instrumentation for Solar Astrophysics., (Eds.) S. L. Keil, S. V. Avakyan, vol. 4853 of SPIE Conf., pp. 187–193 * von Uexküll et al. (1983) von Uexküll, M., Kneer, F., Mattig, W., 1983, The chromosphere above sunspot umbrae. IV - Frequency analysis of umbral oscillations, A&A, 123, 263–270 * Weigelt and Wirnitzer (1983) Weigelt, G., Wirnitzer, B., 1983, Image reconstruction by the speckle-masking method, Optics Letters, 8, 389–391 * Weigelt (1977) Weigelt, G. P., 1977, Modified astronomical speckle interferometry ’speckle masking’, Optics Communications, 21, 55–59 * Wentzel (1979) Wentzel, D. G., 1979, Hydromagnetic surface waves on cylindrical fluxtubes, A&A, 76, 20–23 * Wikipedia (Stokes parameters) Wikipedia, t. f. e., Stokes parameters, (http://en.wikipedia.org/wiki/Stokes_parameters) * Wikipedia (Sun) Wikipedia, t. f. e., Sun, (http://en.wikipedia.org/wiki/sun) * Wilhelm (2000) Wilhelm, K., 2000, Solar spicules and macrospicules observed by SUMER, A&A, 360, 351–362 * Wittmann (1969) Wittmann, A., 1969, Some properties of umbral flashes, Sol. Phys., 7, 366–369 * Yi et al. (1992) Yi, Z., Darvann, T., Molowny-Horas, R., 1992, Software for Solar Image Processing - Proceedings from lest Mini Workshop, Tech. Rep. 56, Lest foundation, Institute of Theoretical Astrophysics, University of Oslo ## Publications 1. _Refereed papers_ 2. 1. B. Sánchez-Andrade Nuño, R. Centeno, K. G. Puschmann, J. Trujillo Bueno, J. Blanco Rodríguez, and F. Kneer. Spicule emission profiles observed in He i 10830 Å. A&A, 472:L51–L54, Sept. 2007. 3. 2. B. Sánchez-Andrade Nuño, N. Bello González, J. Blanco Rodríguez, F. Kneer, and K. G. Puschmann. Fast events and waves in an active region of the Sun observed in H$\alpha$ with high spatial resolution. A&A, submitted, Dec. 2007. 4. 3. J. Blanco Rodríguez, O. V. Okunev, K. G. Puschmann, F. Kneer, and B. Sánchez- Andrade Nuño. On the properties of faculae at the poles of the Sun. A&A, 474:251–259, Oct. 2007. 5. _Conference contributions_ 6. 4. B. Sánchez-Andrade Nuño, R. Centeno, K. G. Puschmann, J. Trujillo Bueno, and F. Kneer. Off-limb spectroscopy of the He I 10830 Å multiplet: observations vs. modelling. In F. Kneer, K. G. Puschmann, and A. D. Wittmann, editors, Modern solar facilities - advanced solar science, pages 177–180, 2007. 7. 5. B. Sánchez-Andrade Nuño, K. G. Puschmann, and F. Kneer. Observations of a flaring active region in H$\alpha$. In F. Kneer, K. G. Puschmann, and A. D. Wittmann, editors, Modern solar facilities - advanced solar science, pages 273–276, 2007. 8. 6. B. Sánchez-Andrade Nuño, K. G. Puschmann, M. Sánchez Cuberes, J. Blanco Rodríguez, and F. Kneer. Analysis of a Wide Chromospheric Active Region. In D. E. Innes, A. Lagg, S. A. Solanki, and D. Danesy, editors, Chromospheric and Coronal Magnetic Fields, volume 596 of ESA Special Publication, Nov. 2005. 9. 7. B. Sánchez-Andrade Nuño, K. G. Puschmann, M. Sánchez Cuberes, J. Blanco Rodríguez, and F. Kneer. Chromospheric Dynamics of a Solar Active Region. In The Dynamic Sun: Challenges for Theory and Observations, volume 600 of ESA Special Publication, Dec. 2005. 10. 8. B. Sánchez-Andrade Nuño. Study case: Solar Science Communication. In L. Lindberg Christensen, and M. Zoulias, editors, Communicating Astronomy with the Public 2007. An IAU/Nat. Obs. of Athens/Eugenides Foundation Conference, Oct. 2007. 11. 9. J. Blanco Rodríguez, B. Sánchez-Andrade Nuño, K. G. Puschmann, and F. Kneer. Study of Polar Faculae. In The Dynamic Sun: Challenges for Theory and Observations, volume 600 of ESA Special Publication, Dec. 2005. 12. 10. J. Blanco Rodríguez, B. Sánchez-Andrade Nuño, K. G. Puschmann, and F. Kneer. Study of Polar Faculae. In D. E. Innes, A. Lagg, S. A. Solanki, and D. Danesy, editors, Chromospheric and Coronal Magnetic Fields, volume 596 of ESA Special Publication, Nov. 2005. 13. 11. F. Kneer, K. G. Puschmann, J. Blanco Rodríguez, B. Sánchez-Andrade Nuño, and A. D. Wittmann. Magnetic Structures on the Sun: Observations with the New ”GÖTTINGEN” Two-Dimensional Spectrometer on Tenerife. In D. E. Innes, A. Lagg, and S. A. Solanki, editors, Chromospheric and Coronal Magnetic Fields, volume 596 of ESA Special Publication, Nov. 2005. 14. 12. L. Valdivielso Casas, N. Bello González, K. G. Puschmann, B. Sánchez-Andrade Nuño, and F. Kneer. Analysis of Polarimetric Sunspot Data from Tesos/vtt/tenerife. In D. E. Innes, A. Lagg, S. A. Solanki, and D. Danesy, editors, Chromospheric and Coronal Magnetic Fields, volume 596 of ESA Special Publication, Nov. 2005. 15. 13. C. Denker, A.P. Verdoni, F. Wöger, A. Tritschler, T.R. Rimmele, F. Kneer, K. Reardon, B. Sánchez-Andrade Nuño, I. Domínguez Cerdeña and K.G. Puschmann Speckle Interferometry of Solar Adaptive Optics Imagery DFG-NSF Astrophysics Research Conference “Advanced Photonics in Application to Astrophysical Problems”, June 2007 ## Acknowledgements I would like to acknowledge here the contributions from many people to the successful completion of this work. I want to express my gratitude to all those who helped me, from my supervisor this three last years to my teachers at my first university who encouraged me to start this adventure in astrophysics. During all this time I have been also surrounded and supported by many people to whom I want to thank within these lines. They all made this experience of doing a PhD, professionally and personally, one of the best times of my life. My supervisor of this research work was Prof. Dr. Franz Kneer, or better _franz_. He was the first person I saw when I came to the train station back three years ago (along with Julian). He was the guide to all my work, the observations, the interpretation, …basically, my formation as solar physicist. The amount of knowledge I learnt from him cannot be measured by any means, and for that I am professionally extremely grateful. In a personal way he was also very kind and helped me always with good advices whenever I needed it. He understood me perfectly when I had problems and also encouraged me to travel as much as I wanted (which was not a little). Working with him this time made the whole experience of PhD leaps better. Ich kann nicht diese Arbeit ohne Ihn zu vorstellen. Danke Franz. Above all, I am and will be always grateful to my reference in life, my parents Conchita Nuño López and Julio Sánchez-Andrade Fernández, and my sister Deva Sánchez Nuño. They always supported me, although a bit far in physical distance during this last years. When I was 5 years old I was going home with my mother from the playground when I saw a poster announcing a conference about starts. I asked my mother to read it loud for me, and to her surprise I had to explained her: “Don’t you yet know that I want to be a _researcher of stars_?”. The day after she gave me a children book about stars and explained me that if I wanted to be so I should know that they are called _astronomers_. And that’s how it all began. My work was supported by two institutions: the _Max Planck Institute für Sonnensystemforschung_ (MPS) granted me the fellowship and the _Institut für Astrophysik Göttingen_ (IAG) provided me the facilities to work with Franz at Göttingen and support for the observations at Tenerife. Also, being part of the _International Max Planck Research School on Physical Processes in the Solar System and Beyond at the Universities of Braunschweig and Göttingen_ (IMPRS) I could attend seminars, retreats and courses on various astrophysical topics. I am very grateful to these institutions for providing such a broad curricula. I would like also to thank the coordinator of the MPS and IMPRS, Dr. Dieter Schmitt. The Vacuum Tower Telescope used for the observations is operated by the Kiepenheuer-Institut für Sonnenphysik, Freiburg, at the Spanish Observatorio del Teide of the Instituto de Astrofísica de Canarias. Here at the solar group of IAG we had a really stimulating environment with long discussions: Franz, Klaus, Markus, Nazaret, Julian and the various Erasmus people that went by (like Luisa, Manu, …). We worked in the beginning at the beautiful historical _Sternwarte_ and then at the Faculty of Physics. I would like to thank specially Klaus for all the help and scientific discussions during all the time he was here. Many other professional colleagues contributed to this work. Their input was extremely helpful. A short list with few names would include: (from the MPS) Andreas Lagg, Vasili Zakarov, (from the IAG) the solar group, Axel Wittman, Volker Bothmer, (from the IAC) Basilio Ruiz Cobos, Manolo Collados, Javier Trujillo, Rebecca Centeno. Thanks to Nurol Al and H. Schleicher for their computing codes. For the Blind Deconvolution section I got much help from Jaime and Mats Löfdahl at the Institute for Solar Physics, Stockholm. From Asturias to here there is long way, specially passing through Canary Island. Without the advices and guidance from many people I would have been lost. In these sense I specially thank Cristina, Juanjo, Basilio and Fernando. Throughout all these years I had the luck to be surrounded by the best companion of friends. I would like to mention lastly some few friends that helped me to keep my feet on the ground. Here in Göttingen Thorsten, Gonzaga, Klaus, Miguel, Cristiano, Vladi, Cathi, Niko, Lorne, Diego, Iria, Teresa, Nora, Carlos, Olga, Julian, Nazaret, Benoit, Markus, Manu and Luisa. Thorsten in particular was not only a good friend but also my flat-mate, climbing-mate, snowboard-mate and of course party-mate. My new flat-mate, Richard, had to stand me this last months with my thesis mood, though. Thanks a lot!. Mark, Lucas, Clementina, Pedro, Emre or Martin, all the guys from Lindau, always warmly welcomed his lost member of the family emigrated to the civilization. During my short intense period in Tenerife I made good friends: Onti, Borja, Bendinat, Adriana, Manu, Miguel. Some time later I met Rebecca, with whom I also worked these years. Whenever a crazy trip came up I had AEGEE people coming along. They would join me to any place in Europe. With them I had some of my best times: Luis, David, Neila, Saioa, Javiero, Marti, Marta and recently of course Carol. All those friends with crazy names: Annia, Annamary, Bir, Konstantina, Marios or Lutzn. I am sure I’ll meet you again, somewhere. Since I left Asturias, I have always missed the green and astounding landscapes, but also my old friends: Roberto, Raul, Estela, Flaci, Nacho, Cova, Lisa, David and of course my big and warm family. Por ultimo me gustaría terminar esta sección, y con ello el fin de este trabajo, con la memoria de tres personas a quienes siempre echaré de menos. En mi primer año en Alemania tuve que despedir a mi abuela materna y el segundo a mi abuela paterna. Aunque sea ley de vida no deja de ser doloroso. Este último año, a mi primo Abraham, a quien tuve la suerte de ver una última vez. Con ellos he aprendido la lección más importante: _Disfruten de la vida._ Göttingen, January 2008 ## Lebenslauf Name: \| Bruno Sánchez-Andrade Nuño ---\|--- Geburtsdatum: \| 6\. Mai 1981 Geburstort: \| Oviedo, Asturias, Spanien Familienstand: \| Ledig Eltern: \| Julio Miguel Sánchez-Andrade Fernández \| Concepción Nuño López Staatsangehörigkeit: \| Spanisch Schulbildung: \| September 1987 - Juni 1995 \| Grundschule am _Colegio Público “Río Piles”_ in Gijón, Asturias \| September 1995 - Juni 1999 \| Weiter führende Schule am _I. B. “Río Piles”_ in Gijón, Asturias Studium: \| September 1999 - September 2003 \| Physikalische Fakultät der Universität Oviedo, Asturias \| September 2003 - September 2004 \| Physikalische Fakultät der Universität La Laguna, Teneriffa (Fachrichtung Astrophysik) Promotion: \| Januar 2005 - Januar 2008 \| Promotion an der Universitäts-Sternwarte Göttingen (seit Juni 2005 Institut für Astrophysik Göttingen, IAG) \| Januar 2005 - Januar 2008 \| Stipendium des Max-Planck-Instituts für Sonnensystemforschung	arxiv-papers	2009-02-18T15:51:57	2024-09-04T02:49:00.683696	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Bruno S\\'anchez-Andrade Nu\\~no", "submitter": "Bruno S\\'anchez-Andrade Nu\\~no", "url": "https://arxiv.org/abs/0902.3174" }
0902.3229	# Analysis of the Decay $e^{+}e^{-}\rightarrow\text{ invisible }+H(\rightarrow\mu\mu)$ at a Collision Energy of 500 GeV Jan Strube and Marcel Stanitzki Rutherford Appleton Laboratory - PPD Didcot Oxfordshire OX110QX - UK ###### Abstract The analysis of $e^{+}e^{-}\rightarrow\text{ invisible }+H(\rightarrow\mu\mu)$ at a next generation linear collider presents an opportunity to study the coupling of the Yukawa couplings of the second generation in a clean environment. We give an overview over the experimental challenges of this analysis at a collision energy of 500 GeV and present an outlook to the results of the analysis at a collision energy of 250 GeV. ## 1 Motivation The environment at the ILC is primed for analyses of precision measurements of Higgs couplings. The analysis of the decay $e^{+}e^{-}\rightarrow\text{ invisible }+H\rightarrow\mu\mu$ is an ideal opportunity to study the Yukawa couplings of the second generation in a clean environment and complement the recoil method measurement of this coupling by taking advantage of the large branching fraction of the decay of the Z boson to neutrinos. In addition to the physics interest, the decay of $H\rightarrow\mu\mu$ in an otherwise (almost) empty detector permits precision studies of the tracking performance and is used for detector benchmarking. ## 2 Software In absence of fully reconstructed events this analysis was carried out on events generated by Pythia 6.4[2] and using the fast detector simulation in the org.lcsim framework. Events were simulated with the sid01 version of the SiD detector concept[3]. For the classification of events we used the TMVA[4] libraries that provide a variety of multi-variate classifiers. ## 3 Presentation of Samples (a) t channel (b) s channel Figure 1: Feynman diagrams of the leading-order signal processes Events in the signal sample are computed with either of the two diagrams in Figure 1. The fraction of events in the signal sample containing a Z boson is 16.2%, the other 83.8% contain neutrinos from the vector boson fusion diagram111The relative fraction of the two contributing diagrams to the observed final state are obtained from PYTHIA 6.4. A mixture of different decays containing a pair of muons in the final state make up the background sample. The different samples and their cross-sections at an energy of 500 GeV in the center-of-momentum system are listed in Table 1. Process \| cross-section (pb) \| # events $(10^{5}/2000fb^{-1}$) ---\|---\|--- 4 fermion \| $4.1\times 10^{2}$ \| 8.3 $WW\rightarrow\nu\mu\nu\mu$ \| $3.1\times 10^{2}$ \| 6.3 $ZZ\rightarrow\nu\nu\mu\mu$ \| $2.7\times 10^{-3}$ \| $5.4\times 10^{-5}$ $Z\rightarrow\mu\mu$ \| $1.1\times 10^{3}$ \| 21.4 $Z\rightarrow\tau\tau$ \| $1.1\times 10^{3}$ \| 21.0 invisible + $H\rightarrow\mu\mu$ \| $2.4\times 10^{-3}$ \| $48\times 10^{-5}$ Table 1: Considered processes and their cross-sections as obtained from PYTHIA 6.4 ## 4 Event and Candidate Selection Events were selected by requiring exactly two muons. We assume a muon identification efficiency of 100%, leading to this cut being 98.7% efficient on signal, while rejecting 67.0 % of the considered background events. ### 4.1 Signal Candidates After pre-selecting events by requiring a pair of identified muons, signal candidates are selected by applying a loose cut on the invariant mass of the muon pair. In order to avoid introducing systematic dependencies, the cut on the invariant mass is very wide around the nominal mass of the Higgs boson of $120\pm 0.5$ GeV. Additional cuts on the visible energy in the event, the oblateness, and the acoplanarity result in a sample consisting of 25 signal events and 8891 background events. The cuts are listed in Table 2. Cut \| Signal \| Background ---\|---\|--- \| efficiency \| efficiency 100 GeV $<$ di-muon mass $<$ 140 GeV \| 95.4% \| 4.1% 130 GeV $<$ visible energy $<$ 260 GeV \| 92.2% \| 44.8% 0 $<$ acoplanarity $<$ 0.5 \| 77.7% \| 62.3% oblateness $>$ 0 \| 75.9% \| 33.8% Table 2: Efficiencies of the signal selection cuts ## 5 Signal Extraction Since the other variables available to us do not exhibit clear distinctive features between signal and background, a square cut would reduce the signal efficiency to unacceptable levels. With the help of multivariate classifiers, we take advantage of the full statistical information available to us. Splitting the data sample into a training and a validation set, and providing the classifier with a number of variables in the training set results then in the optimal case in maximally separated signal and background classes. In this analysis, we found that boosted decision trees (also called “random forest”) [5] exhibit the best performance of the available classifiers and indeed outperform the more commonly used neural nets. With the following set of input variables, we achieve a classification of signal events with a statistical significance of 1.85 sigma. Figure 2 shows a stack of the distributions of the di-muon invariant mass for each of the samples after cuts. Figure 2: Stack of the distributions of the di-muon invariant mass after cuts for each of the samples * • opening angle of the muon pair * • missing momentum * • $\cos(\theta)$ for each muon * • energy for each muon * • transverse momentum for each muon * • whether the muons traversed barrel or endcap * • ratio of the muon energies * • ratio of $p_{T}$ of the muons ## 6 Summary and Outlook We have presented the status of the analysis of the decay $e^{+}e^{-}\rightarrow\text{ invisible }+H(\rightarrow\mu\mu)$ in the framework of a fast simulation of the sid01 detector. We expect to improve upon the achieved signal significance of 1.85 sigma by including a larger sample and by adding a maximum likelihood fit for the signal extraction. This study was carried out at a collision energy of 500 GeV. Events mediated by the s-channel diagram 1b are easier to separate from background, because the missing mass exhibits a peak at the nominal Z mass. The fraction of these events in the signal sample is 16.2% at 500 GeV. For the LOI benchmarking effort, the analysis will be repeated at a collision energy of 250 GeV, reducing the relative fraction of t-channel events in the signal sample to 16.0%. ## 7 Acknowledgments The authors would like to expresses thanks to the org.lcsim development team for producing a stable simulation and reconstruction framework for future linear colliders. ## References * [1] Presentation: `http://ilcagenda.linearcollider.org/contributionDisplay.py?contribId=397&sessionId=16&confId=2628` * [2] Torbjorn Sjostrand, Stephen Mrenna, Peter Skands, JHEP 0605:026 (2006). * [3] The SiD detector outline document http://hep.uchicago.edu/ oreglia/siddod.pdf * [4] Andreas Hocker et al., PoS ACAT 040 (2007) * [5] Leo Breiman, Machine Learning 45 (1), 5-32 (2001)	arxiv-papers	2009-02-18T20:04:17	2024-09-04T02:49:00.700447	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jan Strube and Marcel Stanitzki", "submitter": "Jan Strube", "url": "https://arxiv.org/abs/0902.3229" }
0902.3437	# Effective mass suppression in a ferromagnetic two-dimensional electron liquid Reza Asgari School of Physics, Institute for Research in Fundamental Sciences, (IPM) 19395-5531 Tehran, Iran T. Gokmen Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA B. Tanatar Department of Physics, Bilkent University, Bilkent, Ankara 06800, Turkey Medini Padmanabhan Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA M. Shayegan Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA ###### Abstract We present numerical calculations of the electron effective mass in an interacting, ferromagnetic, two-dimensional electron system. We consider quantum interaction effects associated with the charge-density fluctuation induced many-body vertex corrections. Our theory, which is free of adjustable parameters, reveals that the effective mass is suppressed (relative to its band value) in the strong coupling limit, in good agreement with recent experimental results. ###### pacs: 71.10.Ca, 73.20.Mf ## I Introduction Two-dimensional electron systems (2DESs) realized at semiconductor interfaces are of continuing interest ando ; gv_book from both basic physics and technological points of view. As a function of the interaction strength, which is characterized by the ratio $r_{s}$ of the Coulomb energy to Fermi energy, many novel correlated ground states have been predicted such as a paramagnetic liquid ($r_{s}<26$), ferromagnetic liquid ($26<r_{s}<35$) and Wigner crystal ($r_{s}>35$) AttaccalitePRL02 . In the paramagnetic liquid phase, interaction typically leads to an enhancement of effective mass ($m^{}$) and spin susceptibility (${\chi}^{}$ ${\propto}$ $g^{}m^{}$), where $g^{}$ is the Landé $g^{}$-factor. Effective mass is an important concept in Landau’s Fermi liquid theory since it provides a direct measure of the many-body interactions in the electron system as characterized by increasing $r_{s}$. The effective mass $m^{}$ renormalized by interactions has been experimentally studied SmithPRL72 ; PanPRB99 ; ShashkinPRL03 ; TanPRL05 ; PadmanabhanPRL08 for various paramagnetic 2DESs as a function of $r_{s}$. In the highly interacting, dilute, paramagnetic regime ($3<r_{s}<26$), $m^{}$ is typically significantly enhanced compared to its band value, $m_{b}$, and tends to increase with increasing $r_{s}$ SmithPRL72 ; PanPRB99 ; ShashkinPRL03 ; TanPRL05 ; PadmanabhanPRL08 ; KwonPRB94 ; AsgariSSC04 ; AsgariPRB05 ; AsgariPRB06 ; GangadharaiahPRL05 ; ZhangPRL05 . A question of particular interest is the dependence of $m^{}$ on the 2D electrons’ spin and valley degrees of freedom as these affect the exchange interaction. Recent measurements of $m^{}$ for 2D electrons confined to AlAs quantum wells revealed that, when the 2DES is fully valley- and spin-polarized, $m^{}$ is suppressed down to values near or even slightly below $m_{b}$ PadmanabhanPRL08 ; GokmenPRL08 ; GokmenUNP09 . Note that in these experiments, $r_{s}<22$ so that the 2DES is in the paramagnetic regime, but a strong magnetic field is applied in order to fully spin-polarize the electrons. Here we present theoretical calculations indicating that the $m^{}$ suppression is caused by the absence (freezing out) of the spin fluctuations. The results of our $m^{}$ calculations are indeed in semi-quantitative agreement with the measurements. Previous theoretical calculations of the effective mass are mostly performed within the framework of Landau’s Fermi liquid theory whose key ingredient is the quasiparticle (QP) concept and its interactions. This entails the calculation of effective electron-electron interactions which enter the many- body formalism allowing the calculation of effective mass. A number of works considered different variants of the leading order in the screened interaction for the self-energy em ; yarlagadda_1994_2 ; em_bohm ; em_dassarma ; DasPRB04 ; zhang ; AsgariSSC04 ; AsgariPRB05 ; AsgariPRB06 from which density, spin- polarization, and temperature dependence of effective mass are obtained. In these calculations the on-shell approximation em_bohm ; em_dassarma ; DasPRB04 yields a diverging effective mass but the full solution of the Dyson equation yields only a mild enhancement. AsgariSSC04 ; AsgariPRB05 ; AsgariPRB06 Almost all these works considered a paramagnetic 2DES as past experiments concentrated on the effective mass enhancement in partially spin polarized 2D systems with $r_{s}<26$. ## II Theory We consider a ferromagnetic 2DES as a model for a system of electronic carriers with band mass $m_{b}$ in a semiconductor heterostructure with dielectric constant $\kappa$. The bare electron-electron interaction is given by $v_{\bf q}=2\pi e^{2}/(\kappa q)$. At zero temperature there is only one relevant parameter for the homogeneous, ferromagnetic 2DES, the usual Wigner- Seitz density parameter $r_{s}=(\pi na^{2}_{B})^{-1/2}$ in which $a_{B}=\hbar^{2}\kappa/(m_{b}e^{2})$ is the Bohr radius in the medium of interest. Figure 1: (color online). Many-body effective mass as a function of $r_{s}$ for $0\leq r_{s}\leq 22$ for a ferromagnetic 2DES. The QP self-energy with momentum $\bf k$ and frequency $\omega$ in a fully polarized electron system can be written as $\Sigma^{\uparrow}({\bf k},\omega)=-\int\frac{d^{2}{\bf q}}{i(2\pi)^{2}}v_{\bf q}\int_{-\infty}^{\infty}\frac{d\Omega}{2\pi}\frac{1}{\varepsilon({\bf q},\Omega)}\,\left[\frac{1-n_{\rm F}(\xi^{\uparrow}_{\bf k})}{\omega+\Omega-\xi^{\uparrow}_{{\bf k}+{\bf q}}/\hbar+i\eta}+\frac{n_{\rm F}(\xi^{\uparrow}_{\bf k})}{\omega+\Omega-\xi^{\uparrow}_{{\bf k}+{\bf q}}/\hbar-i\eta}\right]\,.$ (1) Here $\xi^{\uparrow}_{\bf k}=\varepsilon_{\bf k}-\varepsilon_{F}$ where $\varepsilon_{\bf k}=\hbar^{2}{\bf k}^{2}/(2m_{b})$ is the single-particle energy with $\varepsilon_{F}=\hbar^{2}{k^{\uparrow}_{F}}^{2}/(2m_{b})$ and $k^{\uparrow}_{F}=(4\pi n_{\rm\scriptscriptstyle 2D})^{1/2}=2/(r_{s}a_{B})$, respectively, being the Fermi energy and wave vector; $n_{\rm F}(k)$ is the Fermi function. In Eq. (1), $\varepsilon({\bf q},\omega)$ is the dynamical screening function for which we use the form appropriate for a ferromagnetic 2DES derived from Kukkonen-Overhauser effective interaction. kukkonen The many-body exchange and correlation (XC) effects are introduced through the local-field factors (LFF) $G_{\sigma,\sigma^{\prime}}(q,\omega)$ ($\sigma$ and $\sigma^{\prime}$ are spin indices) take the Pauli-Coulomb hole around a charged particle into account. The dynamical screening function reads $\frac{1}{\varepsilon({\bf q},\omega)}=1+v_{\bf q}\,\left[1-G^{+}_{\uparrow}({\bf q},\omega)\right]^{2}\,\chi_{\rm\scriptstyle C}({\bf q},\omega)~{},$ (2) where $G^{+}_{\uparrow}$ is the LFF associated with charge-fluctuations. This expression is similar to the Kukkonen and Overhauser interaction kukkonen where the spin-fluctuation term is dropped. A similar expression has also been reported in Refs. [ng, ; tanaka, ]. In Eq. (2) $\chi_{\rm\scriptstyle C}({\bf q},\omega)$ represents the density-density response function, which in turn is determined by the local-field factor $G^{+}_{\uparrow}({\bf q},\omega)$ via the relation $\chi_{\rm\scriptstyle C}({\bf q},\omega)=\frac{\chi^{0}_{\uparrow}({\bf q},\omega)}{1-v_{\bf q}[1-G^{+}_{\uparrow}({\bf q},\omega)]\chi^{0}_{\uparrow}({\bf q},\omega)}\,,$ (3) in which $\chi^{0}_{\uparrow}(q,\omega)$ is the density response function of the spin-polarized electrons. The expression for the noninteracting density response function on the imaginary frequency axis is obtained for use in Eq. (3) as $\chi^{0}_{\uparrow}({\bf q},i\Omega)=\frac{m_{b}^{2}}{2\pi\hbar^{2}q^{2}}\left(\sqrt{2}\sqrt{a_{\uparrow}+\sqrt{a_{\uparrow}^{2}+\left(\frac{q^{2}\Omega}{\hbar m_{b}}\right)^{2}}}-\frac{q^{2}}{m_{b}}\right)\,,$ (4) where we have defined $a_{\uparrow}=q^{4}/4m_{b}^{2}-q^{2}{k^{\uparrow}_{F}}^{2}/m_{b}^{2}-\Omega^{2}/\hbar^{2}$. It is evident that setting $G^{+}_{\uparrow}({\bf q},\omega)=0$, we recover the standard random phase approximation (RPA). In what follows, we shall make the common approximation of neglecting the frequency dependence of $G^{+}_{\uparrow}$. Quite generally, once the QP retarded self-energy is known, the QP excitation energy $\delta{\mathcal{E}}^{\uparrow}_{\rm QP}({\bf k})$, which is the QP energy measured from the chemical potential $\mu^{\uparrow}$ of the interacting ferromagnetic 2DES, can be calculated by solving self-consistently the Dyson equation $\delta{\mathcal{E}}^{\uparrow}_{\rm QP}({\bf k})=\xi^{\uparrow}_{\bf k}+\left.\Re e\Sigma^{R}_{\rm\scriptstyle ret}({\bf k},\omega)\right\|_{\omega=\,\delta{\mathcal{E}}^{\uparrow}_{\rm QP}({\bf k})/\hbar}\,.$ (5) Alternatively, the QP excitation energy can also be calculated from $\delta{\mathcal{E}}^{\uparrow}_{\rm QP}({\bf k})=\xi^{\uparrow}_{\bf k}+\left.\Re e\Sigma^{R}_{ret}({\bf k},\omega)\right\|_{\omega=\xi^{\uparrow}_{\bf k}/\hbar}\,.$ (6) This is called the on-shell approximation (OSA) and it is argued rice to be a better approach than solving the full Dyson equation since noninteracting Green function is used in Eq. (1). Here $\Re e\Sigma^{R}_{ret}({\bf k},\omega)$ is defined as $\Re e\Sigma^{\uparrow}_{ret}({\bf k},\omega)-\Sigma^{\uparrow}_{ret}({\bf k^{\uparrow}_{F}},0)$. The effective mass $m^{}_{\uparrow}(k)$ is now calculated from $\frac{1}{m^{}_{\uparrow}(k)}=\frac{1}{\hbar^{2}k}\frac{d\delta{\mathcal{E}}^{\uparrow}_{\rm QP}(k)}{dk}\,,$ (7) where for $\delta{\mathcal{E}}^{\uparrow}_{\rm QP}$ we have at our disposal the Dyson and OSA approaches. Evaluating $m^{}_{\uparrow}(k)$ at $k=k^{\uparrow}_{F}$, one gets the QP effective mass at the Fermi contour. Clearly from Eqs. (2) and (3) LFF is the basic quantity for an evaluation of the QP properties. We have used the parameterized forms of LFFs $G^{+}(q,\zeta)$ and $G^{-}(q,\zeta)$ (and in particular $G^{+}_{\uparrow}(q)=G^{+}(q,\zeta=1)$ where $\zeta$ is the spin polarization) of Moreno and Marinescu. moreno ## III Results and discussion We now present our numerical results, which are based on the LFF $G^{+}_{\uparrow}(q)$ as input. In Fig. 1 we show our numerical results of the QP effective mass both in OSA and Dyson approximations. The QP effective mass suppression is substantially smaller in the Dyson equation calculation than in the OSA; the reason is that a significant cancellation occurs between the numerator and the denominator in the effective mass expression in the Dyson approach. To clarify the effect of charge-density fluctuation we have also shown the RPA results which do not take the strong many-body fluctuations into account. Note that the LFF takes into account multiple scattering events to infinite order as compared to the RPA where these effects are neglected. In the limit of small $\zeta$ and $r_{s}\rightarrow 0$, the effective mass can be analytically shown to be $m^{}_{\uparrow}/m_{b}=1+(1-\zeta/2.0)r_{s}\ln r_{s}/(\sqrt{2}\pi)$ which our numerical calculations faithfully reproduce. Figure 2: (color online). Many-body effective mass as a function of $r_{s}$ for $0\leq r_{s}\leq 22$ for the ferromagnetic 2DES in comparison to experiments in Ref. [PadmanabhanPRL08, ; GokmenUNP09, ]. Different symbols denote different samples; triangles: A, squares: B, circles: C, and diamonds: D. In Fig. 2 we compare our effective mass calculations with the experimental results PadmanabhanPRL08 ; GokmenPRL08 ; GokmenUNP09 . The measurements were made on 2DESs confined to modulation-doped AlAs quantum wells (QWs) of width 4.5, 11, 12, and 15 nm (samples A, B, C, and D). These samples were grown on GaAs substrates using molecular beam epitaxy. In bulk AlAs, electrons occupy three degenerate ellipsoidal conduction band valleys at the X-points of the Brillouin zone with longitudinal and transverse effective masses $m_{l}$=1.05 and $m_{t}$=0.205 (in units of the free electron mass). Thanks to the slightly larger lattice constant of AlAs compared to GaAs, the AlAs QW layer is under bi-axial compressive strain. Because of this compression, the 2DES in the wider QW samples (B, C, and D) occupy two in-plane valleys with their major axes lying in the plane ShayeganPSS06 . In our measurements on these samples, we applied uni-axial, in-plane strain to break the symmetry between these two valleys so that only one in-plane valley, with an $anisotropic$ Fermi contour and band effective mass of $m_{b}=\sqrt{m_{l}m_{t}}=0.46$ is occupied ShayeganPSS06 . In sample A, however, thanks to its very small QW width, the confinement energy of the out-of-plane valley is lower (because of its larger mass along the growth direction), so that the electrons occupy this valley and therefore have an $isotropic$ Fermi contour and band effective mass is $m_{b}=m_{t}=0.205$ ShayeganPSS06 . The effective masses were deduced from the temperature dependence of the Shubnikov-de Haas oscillations, the details of which are given in Refs. PadmanabhanPRL08 ; GokmenPRL08 ; GokmenUNP09 . We emphasize that the data shown here (Fig. 2) were taken on single-valley 2DESs which were subjected to sufficiently large magnetic fields to fully spin polarize the electrons . It appears in Fig. 2 that the OSA accounts overall for the observed reduction of $m^{}_{\uparrow}$ below the band value reasonably well. The agreement is particularly good for the wider samples (B, C, and D) which have $r_{s}>7$. The $m^{}$ data for the narrowest sample (A), however, fall above the theoretical predictions. We do not know the reason for this discrepancy. However, we point out that, besides the difference in the shapes of the Fermi contour, there is another difference between sample A and the other three samples. Because of the very narrow width of sample A’s quantum well and the prevalence of interface roughness scattering VakiliAPL06 , the mobility of the electrons in this sample is much lower (about a factor of 6) than in other samples for comparable $r_{s}$. It is possible that the higher disorder in sample A is responsible for $m^{}$ being larger; this conjecture is indeed consistent with the results of calculations AsgariSSC04 which predict a larger $m^{}$ for more disordered samples. From Figs. 1 and 2 we draw two main conclusions. (i) The RPA and present results are rather similar in the weak coupling limit ($r_{s}<1$). (ii) In the strong coupling regime ($r_{s}>3$), however, our theoretical calculations which incorporate the proper many-body effects exhibit a mass suppression, similar to the experimental data, while the RPA results show a mass enhancement and are far from the experimental data. We emphasize that the effective mass at the Fermi contour is significantly suppressed in the fully polarized case because of the absence of spin-fluctuation contribution. This suppression suggests that the anti-symmetric Landau parameter $F^{a}_{1}<0$ and thus higher angular momentum Landau parameters may be negligible in a fully spin-polarized 2DES. Figure 3: (color online). Many-body on-shell effective mass as a function of $k/k_{F}$ at $r_{s}=5$ for 2DES with the combined effect of charge fluctuations in comparison to paramagnetic 2DES. Figure 4: (color online). Renormalization constant $Z_{\uparrow}$ as a function of $r_{s}$ for $0<r_{s}<22$ for a ferromagnetic 2DES. To gain further insight to the density dependence of $m^{\ast}$, we have calculated the on-shell effective mass as a function of particle momentum $k$ using Eq. (7) evaluated at $\omega(k)=\xi^{\uparrow}_{\bf k}/\hbar$ and $r_{s}=5$. More specifically, we use $\frac{m_{b}}{m^{}_{{\uparrow}}(k)}=1+\frac{m_{b}}{\hbar^{2}k}\frac{d}{dk}\Re e\Sigma^{\uparrow}_{\rm ret}(k,\xi^{\uparrow}_{k}),$ (8) for a ferromagnetic case. The results for both paramagnetic and ferromagnetic cases are shown in Fig. 3. $m^{}(k)$ for a paramagnetic 2DES by using $G^{+}(q,\zeta=0)$ and $G^{-}(q,\zeta=0)$ has a sharp peak around $k\approx k_{\rm F}$ where $k_{\rm F}=2/(r_{s}a_{B})$ and a resonance like divergent behavior around $k\approx 2k_{\rm F}$. The peak around $k_{\rm F}$ is associated with spin fluctuations and the divergent behavior around $2k_{\rm F}$ is related to density fluctuations. ng ; zhang In particular, the latter divergence has been extensively studied by Zhang et al. zhang within the RPA. It is related to the dispersion instability and coincides with the plasmon emission. $m^{}_{\uparrow}(k)$ for the ferromagnetic 2DES, on the other hand, clearly shows the disappearance of the peak associated with spin fluctuations. Thus, $m^{\ast}_{\uparrow}(k)$ is very weakly momentum dependent for $k<k^{\uparrow}_{F}$, since there is a substantial cancelation between the residue and the exchange plus line self-energy contribution in this regime which make the real part of the retarded self-energy approximately linear with respect to $k$. AsgariPRB05 The divergence associated with charge fluctuations is still present, showing a negative peak around $k=2k_{\rm F}$. $m^{\ast}_{\uparrow}(k)$ calculated within the RPA reproduces quantitatively the divergent behavior associated with charge fluctuations but shows some structure for $k\leq k^{\uparrow}_{\rm F}$, therefore failing to account for the absence of spin fluctuations. Our density-dependent effective mass results (Figs. 1 and 2) are consistent with $m^{\ast}(k)$ calculations which we have checked for a range of $r_{s}$ values. We have also calculated the renormalization factor $Z_{\uparrow}(r_{s})$ which is equal to the discontinuity in the momentum distribution at $k_{F}$ and defined by $Z^{-1}_{\uparrow}=1-\hbar^{-1}\left.\partial_{\omega}\Re e\Sigma^{\uparrow}_{\rm ret}(k,\omega)\right\|_{k=k^{\uparrow}_{F},\omega=0}$. The effect of charge-fluctuations is to make the $Z_{\uparrow}$ values larger at large $r_{s}$ compared to the case when they are not included as shown in Fig. 4. This means that charge-density fluctuations tend to stabilize the system, whereas the RPA works in the opposite direction.AsgariPRB05 In the present case including the LFF helps preserve the Fermi liquid picture in the low density regime. We have performed our numerical calculations for strictly 2DES. As indicated above, experimental samples have a finite thickness in the range of $5-15$ nm. Our theoretical model may be extended to include the finite quantum well width effects in the following manner. Choosing, say, an infinite square well model with width $L$ will modify the bare Coulomb interaction, $v_{\bf q}\rightarrow v_{\bf q}F(qL)$, in which $F(x)$ is a form factor.gold For consistency, one should also calculate the local-field factor $G^{+}_{\uparrow}(q)$ using the same model for the finite width effects. This would provide a better comparison with experiments. In our case, the local-field factor we use was constructedmoreno by the quantum Monte Carlo data for a strictly 2DES and it is not straightforward to incorporate the finite with effects within such an approach. Previous calculationsAsgariPRB06 of the effective mass for a paramagnetic 2DES suggest that the effect of a finite thickness is to suppress $m^{}$. Therefore, we surmise that a similar qualitative effect would occur for the ferromagnetic 2DES. On the other hand, the finite temperature and disorder effects have a tendency to enhance the effective massDasPRB04 ; AsgariSSC04 which may lead to a cancellation. These issues require a more systematic study. ## IV Summary In conclusion, our theoretical calculations incorporating the proper Pauli- Coulomb hole and multi-scattering processes show that in an interacting, fully spin-polarized 2DES the absence of spin fluctuations reduces the effective mass below its band value, in agreement with experimental data. Our results also demonstrate the inadequacy of RPA to account for the observed effective mass suppression. ###### Acknowledgements. R. A. thanks M. Polini for helpful discussions. The work at Princeton University was supported by the NSF. B. T. is supported by TUBITAK (No. 108T743) and TUBA. ## References (1) T. Ando, A.B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437-672 (1982) . * (2) G.F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, England, 2005) . * (3) C. Attaccalite, S. Moroni, P. Gori-Giorgi, and G. B. Bachelet, Phys. Rev. Lett. 88, 256601 (2002); B. Tanatar and D.M. Ceperley, Phys. Rev. B 39, 5005 (1989) . * (4) J.L. Smith and P.J. Stiles, Phys. Rev. Lett. 29, 102 (1972). * (5) W. Pan, D.C. Tsui, and B.L. Draper, Phys. Rev. B 59, 10208 (1999). * (6) A.A. Shashkin, S.V. Kravchenko, V.T. Dolgopolov, and T.M. Klapwijk, Phys. Rev. B 66, 073303 (2002). * (7) Y.-W. Tan, J. Zhu, H.L. Stormer, L.N. Pfeiffer, K.W. Baldwin, and K.W. West Phys. Rev. Lett. 94, 016405 (2005) . * (8) M. Padmanabhan, T. Gokmen, N.C. Bishop, and M. Shayegan, Phys. Rev. Lett 101, 026402 (2008) . * (9) Y. Kwon, D.M. Ceperley, and R.M. Martin, Phys. Rev. B 50, 1684 (1994); M. Holzmann, B. Bernu, V. Olevano, R.M. Martin, and D.M. Ceperley, Phys. Rev. B 79, 041308(R) (2009). * (10) R. Asgari, B. Davoudi, and B. Tanatar, Solid State Commun. 130, 13 (2004). * (11) R. Asgari, B. Davoudi, M. Polini, G.F. Giuliani, M.P. Tosi, and G. Vignale, Phys. Rev. B 71, 045323 (2005) . * (12) R. Asgari and B. Tanatar, Phys. Rev. B 74, 075301 (2006) . * (13) S. Gangadharaiah and D.L. Maslov, Phys. Rev. Lett. 95, 186801 (2005). * (14) Y. Zhang and S. Das Sarma, Phys. Rev. Lett. 95, 256603 (2005). * (15) T. Gokmen, M. Padmanabhan, and M. Shayegan, Phys. Rev. Lett 101, 146405 (2008) . * (16) T. Gokmen, M. Padhamadnan, K. Vakili, E. Tutuc, and M. Shayegan, Phys. Rev. B 79, 195311 (2009). * (17) I.K. Marmorkos and S. Das Sarma, Phys. Rev. B 44, R3451 (1991); H.-J. Schulze, P. Schuck, and N. Van Giai, Phys. Rev. B 61, 8026 (2000) . * (18) S. Yarlagadda and G.F. Giuliani, Phys. Rev. B 49, 7887 (1994); 61, 12556 (2000); C.S. Ting, T.K. Lee, and J.J. Quinn, Phys. Rev. Lett. 34, 870 (1975). * (19) H. M. Böhm and K. Schörkhuber, J. Phys.: Condens. Matter 12, 2007 (2000) . * (20) Y. Zhang and S. Das Sarma, Phys. Rev. B 71, 045322 (2005) . * (21) S. Das Sarma, Victor M. Galitski and Ying Zhang, Phys. Rev. B 69, 125334 (2004) . * (22) Y. Zhang, V.M. Yakovenko, and S. Das Sarma, Phys. Rev. B 71, 115105 (2005) . * (23) C.A. Kukkonen and A.W. Overhauser, Phys. Rev. B 20, 550 (1979) . * (24) T.-K. Ng and K.S. Singwi, Phys. Rev. B 34, 7743 (1986) . * (25) S. Tanaka and S. Ichimaru, Phys. Rev. B 39, 1036 (1989) . * (26) T.M. Rice, Ann. Phys. (N.Y.) 31, 100 (1965) . * (27) J. Moreno and D.C. Marinescu, Phys. Rev. B 68, 195210 (2003) . * (28) M. Shayegan, E.P. De Poortere, O. Gunawan, Y.P. Shkolnikov, E. Tutuc, K. Vakili, Phys. Stat. Sol. (b) 243, 3629 (2006). * (29) K. Vakili, Y.P. Shkolnikov, E. Tutuc, E.P. De Poortere, M. Padmanabhan, and M. Shayegan, Appl. Phys. Lett. 89, 172118 (2006) . * (30) A. Gold, Phys. Rev. B 35, 723 (1987).	arxiv-papers	2009-02-19T19:08:23	2024-09-04T02:49:00.706583	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Reza Asgari, T. Gokmen, B. Tanatar, Medini Padmanabhan, M. Shayegan", "submitter": "Reza Asgari", "url": "https://arxiv.org/abs/0902.3437" }
0902.3443	# Leptonic Decay Constants of $D_{s}$ and $B_{s}$ Mesons at Finite Temperature Elşen Veli Veliev , Gülşah Kaya * Physics Department, Kocaeli University, Umuttepe Yerleşkesi 41380 Izmit, Turkey * e-mail: elsen@kocaeli.edu.tr ** e-mail: gulsahbozkir@kocaeli.edu.tr ###### Abstract In the present work, $D_{s}$ and $B_{s}$ meson parameters are investigated in the framework of thermal QCD sum rules. The temperature dependences of the mass and the leptonic decay constants are investigated by using Borel transform sum rules and Hilbert moment sum rules. To increase sensitivity, the vacuum contributions are subtracted from thermal expressions and the temperature dependences of the leptonic decay constants and meson masses are studied. ## 1 Introduction In order to explain the heavy ion collision results, some information about hadrons parameters at finite temperature and density is required. Some of the characteristic parameters at finite temperature and density are the masses and leptonic decay constants of hadrons. The investigation of these parameters requires non-perturbative approaches. One of these non-perturbative methods is the QCD sum rules [1], formulated by Shifman, Vainshtein and Zakharov. The extension of the QCD sum rules method to finite temperatures has been made by Bochkarev and Shaposhnikov [2]. This extension is based on two basic assumptions that the OPE and notion of quark-hadron duality remain valid, but the vacuum condensates are replaced by their thermal expectation values. The thermal QCD sum rules method has been extensively used for studying thermal properties of both light and heavy hadrons as a reliable and well-establish method [3]-[8]. The investigation of heavy meson decay constants at zero temperature has been widely discussed in the literature [9]. The knowledge of these constants is needed in order to predict numerous heavy flavor electroweak transitions and to determine Standard Model parameters from the experimental data. Also leptonic decay constants play essential role in the analysis of CKM matrix, CP violation and the mixings $\overline{B_{d}}B_{d}$, $\overline{B_{s}}B_{s}$ . The first determination of these constants were made twenty years ago [10]-[12] and due to further theoretical and experimental progress, this problem was reconsidered taking into account the running quark masses and perturbative three-loop $\alpha_{s}^{2}$ corrections to the correlation function [13], [14]. At finite temperatures the nonperturbative nature of QCD vacuum induces temperature dependences of the leptonic decay constants and masses. Recently, first attempts have been made in order to calculate the leptonic decay constants of heavy mesons at finite temperature in the framework of thermal QCD sum rules [15]. In the present paper, we investigate the temperature behavior of the masses and leptonic decay constants of $D_{s}$ and $B_{s}$ mesons using QCD sum rules. Taking into account perturbative two-loop order $\alpha_{s}$ corrections to the correlation function and nonperturbative corrections up to the dimension six condensates [16] we investigated the temperature dependences of masses and leptonic decay constants using Borel transform sum rules and Hilbert moment sum rules. For increased sensitivity, we subtract the vacuum contributions from thermal expressions and study the temperature dependences of the leptonic decay constants and meson masses. ## 2 Pseudoscalar thermal correlator at finite temperature We start with pseudoscalar two-point thermal correlator $\psi_{5}(q^{2})=i\int d^{4}xe^{iq\cdot x}\langle T(J(x)J^{+}(0))\rangle,\\\ $ (1) where $J(x)=(m_{Q}+m_{s}):\bar{s}(x)i\gamma_{5}Q(x):$ is the heavy-light quark current and has the quantum numbers of the $D_{s}$ and $B_{s}$ mesons, $m_{Q}$ and $m_{s}$ are heavy and strange quark masses respectively. We shall not neglect $s$ quark mass throughout this work. Thermal average of any operator _O_ is defined in the following way $\langle O\rangle=Tre^{-\beta H}O/Tre^{-\beta H},\\\ $ (2) where $H$ is the QCD Hamiltonian, $\beta=1/T$ stands for the inverse of the temperature $T$ and traces are over any complete set of states. Up to a subtraction polynomial, which depends on the large $q^{2}$ behavior, $\psi_{5}(q^{2})$ satisfies the following dispersion relation [1],[9] $\psi_{5}(Q^{2})=\int ds\frac{\rho(s)}{s+Q^{2}}+subtractions,\\\ $ (3) where $Q^{2}=-q^{2}$ is Euclidean momentum, $\rho(s)=\frac{1}{\pi}Im\psi_{5}(s)$ is spectral density and in perturbation theory at zero temperature in the leading order has the following form [16]: $\rho(s)=\frac{3(m_{s}+m_{Q})^{2}}{8\pi^{2}s}v(s)q^{2}(s)\Big{[}1+\frac{4\alpha_{s}}{3\pi}f(x)\Big{]},\\\ $ (4) where $x={m_{Q}^{2}}/{s}$, $\alpha_{s}=\alpha_{s}(m_{Q}^{2})$ and $q(s)=s-(m_{Q}-m_{s})^{2},~{}~{}v(s)=(1-4m_{s}m_{Q}/q(s))^{1/2},\\\ $ (5) $f(x)=\frac{9}{4}+2Li_{2}(x)+\ln{x}\ln{(1-x)}-\frac{3}{2}\ln{\Big{(}\frac{1}{x}-1\Big{)}}-\ln{(1-x)}+x\ln{\Big{(}\frac{1}{x}-1\Big{)}}-\frac{x}{1-x}\ln{x}.\\\ $ (6) The subtraction terms are removed by using the Borel transformation or Hilbert moment methods. Therefore we will omit these terms. The thermal propagator contains on-shell soft quarks which do not exist in the confined phase. Therefore, in obtaining the OPE of the thermal correlator (1), vacuum propagators must be used [4]. The non-perturbative contributions at zero temperature to the correlator has the following form $\displaystyle\psi_{5,np}(Q^{2})=-m_{Q}\lambda\langle 0\|\bar{s}s\|0\rangle\Big{[}1+\frac{1}{2}\varepsilon(3-\lambda)-\lambda\varepsilon^{2}(1-\lambda)-\frac{1}{2}\varepsilon^{3}(1+\lambda-4\lambda^{2}+2\lambda^{3})\Big{]}$ $\displaystyle+\frac{1}{12\pi}\lambda\langle 0\|\alpha_{s}G^{2}\|0\rangle\Big{[}1+3\varepsilon\Big{(}1-\frac{8}{3}\lambda+2\lambda^{2}-2\lambda(1-\lambda)\ln{(\varepsilon\lambda)}\Big{)}\Big{]}$ $\displaystyle-\frac{M_{0}^{2}}{2m_{Q}}\langle 0\|\bar{s}s\|0\rangle\lambda^{2}(1-\lambda)(1+\varepsilon(2-\lambda))-\frac{8\pi\rho}{27m_{Q}^{2}}\alpha_{s}\langle 0\|\bar{s}s\|0\rangle^{2}\lambda^{2}(2-\lambda-\lambda^{2}),$ (7) where $\lambda=m_{Q}^{2}/(Q^{2}+m_{Q}^{2})$ and $\varepsilon=m_{s}/m_{Q}$. Also, for the mixed condensate the parameterization: $g\langle 0\|\bar{q}\sigma_{\mu\nu}\frac{\lambda_{a}}{2}G_{a}^{\mu\nu}q\|0\rangle=M_{0}^{2}\langle 0\|\bar{q}q\|0\rangle\\\ $ (8) is used. It is assumed, that the expansion (7) also remains valid at finite temperatures, but the vacuum condensates must be replaced by their thermal expectation values [2]. For the light quark condensate at finite temperature we use the results of [17], [18] obtained in chiral perturbation theory. Temperature dependence of quark condensate in a good approximation can be written as $\langle\bar{q}q\rangle=\langle 0\|\bar{q}q\|0\rangle\Big{[}1-0.4\Big{(}\frac{T}{T_{c}}\Big{)}^{4}-0.6\Big{(}\frac{T}{T_{c}}\Big{)}^{8}\Big{]},\\\ $ (9) where $T_{c}=160~{}MeV$ is the critical temperature. The low temperature expansion of the gluon condensate is proportional to the trace of the energy momentum tensor [19] and can be approximated by [15] $\langle\alpha_{s}G^{2}\rangle=\langle 0\|\alpha_{s}G^{2}\|0\rangle\Big{[}1-\Big{(}\frac{T}{T_{c}}\Big{)}^{8}\Big{]}.\\\ $ (10) The value of the QCD scale $\Lambda$ is extracted from the value of $\alpha_{s}(M_{Z})=0.1176$ [20]. Equating OPE and hadron representations of the correlation function and using quark-hadron duality the sum rules is obtained as $\frac{f_{H}^{2}m_{H}^{4}}{Q^{2}+m_{H}^{2}}=\int^{s_{0}}_{(m_{Q}+m_{s})^{2}}ds\frac{\rho(s)}{s+Q^{2}}+\psi_{5,np}(Q^{2}),\\\ $ (11) where $f_{H}$ is the leptonic decay constant and is defined by the matrix element of the axial-vector current between the corresponding meson and the vacuum as: $\langle 0\|\bar{s}\gamma_{\mu}\gamma_{5}Q\|H(q)\rangle=if_{H}q_{\mu},\\\ $ (12) where $Q=c,b$ and $H=D_{s},B_{s}$ in the same normalization as $f_{\pi}=130.56~{}MeV$. In thermal field theories the parameters $m_{H}$ and $f_{H}$ must be replaced by their temperature dependent values. The continuum threshold $s_{0}$ also depends on temperature; to a very good approximation it scales universally as the quark condensate [15] $s_{0}(T)=s_{0}\frac{\langle\bar{q}q\rangle}{\langle 0\|\bar{q}q\|0\rangle}\Big{[}1-\frac{(m_{Q}+m_{s})^{2}}{s_{0}}\Big{]}+(m_{Q}+m_{s})^{2},\\\ $ (13) where in the right hand side $s_{0}$ is hadronic treshold at zero temperature: $s_{0}\equiv s_{0}(0)$. Analysis shows that thermal non-perturbative correlator is basically driven by the quark condensates. ## 3 Numerical analysis of masses and leptonic decay constants In this section we present our results for the temperature dependence of $D_{s}$ and $B_{s}$ meson masses and leptonic decay constants. Performing Borel transformation with respect to $Q_{0}^{2}$ on both sides of equation (11) and differentiating with respect to $1/M^{2}$, we obtain: $m_{H}^{2}(T)=\frac{f_{H}^{2}m_{H}^{6}exp(-m_{H}^{2}/M^{2})+\overline{B}(T)}{f_{H}^{2}m_{H}^{4}exp(-m_{H}^{2}/M^{2})+\overline{A}(T)},\\\ $ (14) $f_{H}^{2}(T)=\frac{1}{m_{H}^{4}(T)}\Big{[}\overline{A}(T)+f_{H}^{2}m_{H}^{4}exp\Big{(}-\frac{m_{H}^{2}}{M^{2}}\Big{)}\Big{]}exp\Big{[}\frac{m_{H}^{2}(T)}{M^{2}}\Big{]},\\\ $ (15) where the bar on the operators means subtractions of their vacuum expectation values from thermal expectation values; for example $\overline{\psi_{5,np}}(M^{2},T)=\psi_{5,np}(M^{2},T)-\psi_{5,np}(M^{2},T=0)$. Here $\overline{A}(T)=\int^{s_{0}(T)}_{s_{0}}ds\rho(s)exp\Big{(}-\frac{s}{M^{2}}\Big{)}+\overline{\psi_{5,np}}(M^{2},T),\\\ $ (16) $\displaystyle\overline{\psi_{5,np}}(M^{2},T)=-m_{Q}^{3}\overline{\langle 0\|\overline{s}s\|0\rangle}e^{-\beta}\Big{[}1+\frac{3}{2}\varepsilon-\frac{1}{2}\varepsilon\beta-\beta\varepsilon^{2}\Big{(}1-\frac{1}{2}\beta\Big{)}-\frac{1}{2}\varepsilon^{3}\Big{(}1+\beta-2\beta^{2}+\frac{1}{3}\beta^{3}\Big{)}\Big{]}$ $\displaystyle+\frac{1}{12\pi}\overline{\langle 0\|\alpha_{s}G^{2}\|0\rangle}m_{Q}^{2}e^{-\beta}\Big{[}1+3\varepsilon\Big{(}1-\frac{8}{3}\beta+\beta^{2}-2\beta(\ln(\beta\varepsilon)+\gamma-1)+\beta^{2}\Big{(}\ln(\beta\varepsilon)+\gamma-\frac{3}{2}\Big{)}\Big{)}\Big{]}$ $\displaystyle-\frac{1}{2}M_{0}^{2}m_{Q}\beta\overline{\langle 0\|\overline{s}s\|0\rangle}e^{-\beta}\Big{[}1-\frac{1}{2}\beta+2\varepsilon\Big{(}1-\frac{3}{4}\beta\Big{(}1-\frac{1}{9}\beta\Big{)}\Big{)}\Big{]}-\frac{4}{81}\pi\rho\alpha_{s}\overline{\langle 0\|\bar{s}s\|0\rangle^{2}}\beta e^{-\beta}$ $\displaystyle\times(12-3\beta-\beta^{2}),$ (17) where $\gamma$ is the Euler constant, $\beta=m_{Q}^{2}/M^{2}$ and $\overline{B}(T)=-m_{Q}^{2}\frac{d\overline{A}(T)}{d\beta}$. To investigate the meson parameters at finite temperature we also use Hilbert moments methods, which eliminate the subtraction terms. Calculating Hilbert moments at $Q^{2}=-q^{2}=0$ and using first two moments we obtain $m_{H}^{2}(T)=\frac{F(T)-\int^{s_{0}}_{s_{0}(T)}ds\rho(s)s^{-3}+f_{H}^{2}/m_{H}^{2}}{G(T)-\int^{s_{0}}_{s_{0}(T)}ds\rho(s)s^{-4}+f_{H}^{2}/m_{H}^{4}},\\\ $ (18) $f_{H}^{2}(T)=m_{H}^{2}(T)\big{[}F(T)-\int^{s_{0}}_{s_{0}(T)}ds\rho(s)s^{-3}+f_{H}^{2}/m_{H}^{2}\big{]},\\\ $ (19) where $F(T)$ and $G(T)$ functions are expressed by thermal expectation values of condensates $\displaystyle F(T)=-\frac{1}{m_{Q}^{3}}\overline{\langle 0\|\bar{s}s\|0\rangle}(1+3\varepsilon^{2})+\frac{1}{12\pi m_{Q}^{4}}\overline{\langle 0\|\alpha_{s}G^{2}\|0\rangle}[1+\varepsilon(21+18\ln\varepsilon)]$ $\displaystyle+\frac{1}{2m_{Q}^{5}}M_{0}^{2}\overline{\langle 0\|\overline{s}s\|0\rangle}(3+2\varepsilon)+\frac{80}{27m_{Q}^{6}}\pi\rho\alpha_{s}\overline{\langle 0\|\bar{s}s\|0\rangle^{2}},$ (20) $\displaystyle G(T)=-\frac{1}{m_{Q}^{5}}\overline{\langle 0\|\overline{s}s\|0\rangle}\Big{(}1-\frac{1}{2}\varepsilon+6\varepsilon^{2}-\frac{5}{2}\varepsilon^{3}\Big{)}+\frac{1}{12\pi m_{Q}^{6}}\overline{\langle 0\|\alpha_{s}G^{2}\|0\rangle}[1+\varepsilon(52+36\ln\varepsilon)]$ $\displaystyle+\frac{1}{m_{Q}^{7}}M_{0}^{2}\overline{\langle 0\|\overline{s}s\|0\rangle}(3+\varepsilon)+\frac{176}{27m_{Q}^{8}}\pi\rho\alpha_{s}\overline{\langle 0\|\bar{s}s\|0\rangle^{2}}.$ (21) Table 1: QCD input parameters used in the analysis. Parameters \| References ---\|--- $m_{D_{s}}=1968$ MeV \| [20] $m_{B_{s}}=5366$ MeV \| [20] $m_{s}=120$ MeV \| [20] $m_{c}=1.47$ GeV \| [13, 20] $m_{b}=4.4$ GeV \| [13, 20] $f_{D_{s}}=235$ MeV \| [13, 20] $f_{B_{s}}=240$ MeV \| [13, 20] $\rho=4$ \| [15, 16] $\langle 0\|\overline{q}q\|0\rangle=-0.014~{}$GeV3 \| [1] $\langle 0\|\frac{1}{\pi}\alpha_{s}G^{2}\|0\rangle=0.012~{}$GeV4 \| [1] $\alpha_{s}\langle 0\|\overline{q}q\|0\rangle^{2}=5.8\times 10^{-4}~{}$GeV6 \| [13] $M_{0}^{2}=0.8~{}$GeV2 \| [13] $\langle 0\|\overline{s}s\|0\rangle=0.8\langle 0\|\overline{q}q\|0\rangle$ \| [13] For the numerical evolution of the above sum rule, the values of the QCD parameters used are shown in Table 1. The criterion we adopt here is to fix $s_{0}$ in such a way as to reproduce the zero temperature values of meson masses and leptonic decay constants. For $D_{s}$ meson $s_{0}$ is $6~{}GeV^{2}$ and $8~{}GeV^{2}$, for $B_{s}$ meson $s_{0}$ is $34~{}GeV^{2}$ and $35~{}GeV^{2}$ in Borel and Hilbert moment sum rules methods, respectively. The temperature dependences of the $D_{s}$ and $B_{s}$ meson masses and leptonic decay constants obtained using the Borel and Hilbert moment methods are shown in Fig. 1 and Fig. 2, respectively. The results for leptonic decay constants are shown in Fig. 3 and Fig. 4. As seen in figures, $f_{D_{s}}$ and $f_{B_{s}}$ decrease with increasing temperature and vanish approximately at critical temperature $T_{c}=160~{}MeV$. This may be interpreted as a signal for deconfinement and agrees with light and heavy- light mesons investigations [15], [21]. Numerical analysis shows that the temperature dependence of $f_{D_{s}}$ is independent of $M^{2}$, when $M^{2}$ changes between $3~{}GeV^{2}$ and $4~{}GeV^{2}$ and $f_{B_{s}}$ is independent of the Borel parameter, when $M^{2}$ changes between $16~{}GeV^{2}$ and $24~{}GeV^{2}$. Obtained results can be used for the interpretation of heavy ion collision experiments. It is also essential to compare these results with other model calculations. We believe these studies to be of great importance for understanding phenomenological and theoretical aspects of thermal QCD. ## 4 Acknowledgement The authors much pleasure to thank T. M. Aliev and A. Özpineci for useful discussions. This work is supported by the Scientific and Technological Research Council of Turkey (TUBITAK), research project no.105T131. ## References * [1] M. A. Shifman, A. I. Vainstein and V. I. Zakharov, Nucl. Phys. B147, 385 (1979); M. A. Shifman, A. I. Vainstein and V. I. Zakharov, Nucl. Phys. B147, 448 (1979). * [2] A. I. Bochkarev and M. E. Shaposhnikov, Nucl. Phys. B268, 220 (1986). * [3] E. V. Shuryak, Rev. Mod. Phys. 65, 1 (1993). * [4] T. Hatsuda, Y. Koike, S. H. Lee, Nucl. Phys. B394, 221 (1993). * [5] T. Hatsuda, Y. Koike, S. H. Lee, Phys. Rev. D47, 1225 (1993). * [6] V. L. Eletsky and B. L. Ioffe, Phys. Rev. Lett. 78, 1010 (1997). * [7] S. Mallik, Phys. Lett. B416, 373 (1998). * [8] S. Mallik and K. Mukherjee, Phys. Rev. D58, 096011 (1998). * [9] P. Colangelo, A. Khodjamirian, In: At the Frontier of Particle Physics, vol.3, ed. M. Shifman,World Scientific, Singapore, 1495 (2001). * [10] T. M. Aliev and V. L. Eletsky, Sov.J. Nucl. Phys. 38, 936 (1983). * [11] C. A. Dominguez and N. Paver, Phys. Lett. B197, 423 (1987). * [12] L. J. Reinders, Phys. Rev. D38, 947 (1988). * [13] S. Narison, Phys. Lett. B520, 115 (2001); S. Narison, Phys. Lett. B 605, 319 (2005). * [14] M. Jamin and B. O. Lange, Phys. Rev. D65, 056005 (2002). * [15] C. A. Dominguez, M. Loewe, J.C. Rojas, JHEP 08, 040 (2008). * [16] C. A. Dominguez and N. Paver, Phys. Lett. B318, 629 (1993). * [17] J. Gasser and H. Leutwyler, Phys. Lett. B184, 83 (1987). * [18] P. Gerber and H. Leutwyler, Nucl. Phys. B321, 387 (1989). * [19] D. E. Miller, Acta Phys. Pol. B28, 2937 (1997), D.E. Miller, arXiv: hep-ph/0008031. * [20] PDG 2008, C. Amsler, et al., Phys. Lett B667, 1 (2008). * [21] E. V. Veliev, T. M. Aliev, J. Phys. G: Nucl. Part. Phys. 35, 125002 (2008). Figure 1: Temperature dependence of $D_{s}$ meson mass in Hilbert and Borel sum rules methods. Here Borel parameter is $M^{2}=3~{}GeV^{2}$, hadronic threshold $s_{0}=6~{}GeV^{2}$ for Borel and $s_{0}=8~{}GeV^{2}$ for Hilbert moment sum rules methods. Figure 2: Temperature dependence of $B_{s}$ meson mass in Hilbert and Borel sum rules methods. Here Borel parameter is $M^{2}=20~{}GeV^{2}$, hadronic threshold $s_{0}=34~{}GeV^{2}$ for Borel and $s_{0}=35~{}GeV^{2}$ for Hilbert moment sum rules methods. Figure 3: Temperature dependence of $f_{D_{s}}$ in Hilbert and Borel sum rules methods. Here Borel parameter is $M^{2}=3~{}GeV^{2}$, hadronic threshold $s_{0}=6~{}GeV^{2}$ for Borel and $s_{0}=8~{}GeV^{2}$ for Hilbert moment sum rules methods. Figure 4: Temperature dependence of $f_{B_{s}}$ in Hilbert and Borel sum rules methods. Here Borel parameter is $M^{2}=20~{}GeV^{2}$, hadronic threshold $s_{0}=34~{}GeV^{2}$ for Borel and $s_{0}=35~{}GeV^{2}$ for Hilbert moment sum rules methods.	arxiv-papers	2009-02-19T19:51:46	2024-09-04T02:49:00.711182	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Elsen Veli Veliev, Gulsah Kaya", "submitter": "Elsen Veli Veliev", "url": "https://arxiv.org/abs/0902.3443" }
0902.3485	# Pricing strategies for viral marketing on Social Networks David Arthur 111Department of Computer Science, Stanford University Rajeev Motwani ††footnotemark: Aneesh Sharma222Institute for Computational and Mathematical Engineering, Stanford University Ying Xu333Work done when author was a student at the Department of Computer Science, Stanford University {darthur,rajeev,aneeshs,xuying}@cs.stanford.edu ###### Abstract We study the use of viral marketing strategies on social networks to maximize revenue from the sale of a single product. We propose a model in which the decision of a buyer to buy the product is influenced by friends that own the product and the price at which the product is offered. The influence model we analyze is quite general, naturally extending both the Linear Threshold model and the Independent Cascade model, while also incorporating price information. We consider sales proceeding in a cascading manner through the network, i.e. a buyer is offered the product via recommendations from its neighbors who own the product. In this setting, the seller influences events by offering a cashback to recommenders and by setting prices (via coupons or discounts) for each buyer in the social network. Finding a seller strategy which maximizes the expected revenue in this setting turns out to be NP-hard. However, we propose a seller strategy that generates revenue guaranteed to be within a constant factor of the optimal strategy in a wide variety of models. The strategy is based on an influence-and-exploit idea, and it consists of finding the right trade-off at each time step between: generating revenue from the current user versus offering the product for free and using the influence generated from this sale later in the process. We also show how local search can be used to improve the performance of this technique in practice. ## 1 Introduction Social networks such as Facebook, Orkut and MySpace are free to join, and they attract vast numbers of users. Maintaining these websites for such a large group of users requires substantial investment from the host companies. To help recoup these investments, these companies often turn to monetizing the information that their users provide for free on these websites. This information includes both detailed profiles of users and also the network of social connections between the users. Not surprisingly, there is a widespread belief that this information could be a gold mine for targeted advertising and other online businesses. Nonetheless, much of this potential still remains untapped today. Facebook, for example, was valued at $15 billion by Microsoft in 2007 [13], but its estimated revenue in 2008 was only $300 million [17]. With so many users and so much data, higher profits seem like they should be possible. Facebook’s Beacon advertising system does attempt to provide targeted advertisements but it has only obtained limited success due to privacy concerns [16]. This raises the question of how companies can better monetize the already public data on social networks without requiring extra information and thereby compromising privacy. In particular, most large-scale monetization technologies currently used on social networks are modeled on the sponsored search paradigm of contextual advertising and do not effectively leverage the networked nature of the data. Recently, however, people have begun to consider a different monetization approach that is based on selling products through the spread of influence. Often, users can be convinced to purchase a product if many of their friends are already using it, even if these same users would be hard to convince through direct advertising. This is often a result of personal recommendations – a friend’s opinion can carry far more weight than an impersonal advertisement. In some cases, however, adoption among friends is important for even more practical reasons. For example, instant messenger users and cell phone users will want a product that allows them to talk easily and cheaply with their friends. Usually, this encourages them to adopt the same instant messenger program and the same cell phone carrier that their friends have. We refer the reader to previous work and the references therein for further explanations behind the motivation of the influence model [6, 4]. In fact, many sellers already do try to utilize influence-and-exploit strategies that are based on these tendencies. In the advertising world, this has recently led to the adoption of viral marketing, where a seller attempts to artificially create word-of-mouth advertising among potential customers [8, 9, 14]. A more powerful but riskier technique has been in use much longer: the seller gives out free samples or coupons to a limited set of people, hoping to convince these people to try out the product and then recommend it to their friends. Without any extra data, however, this forces sellers to make some very difficult decisions. Who do they give the free samples to? How many free samples do they need to give out? What incentives can they afford to give to recommenders without jeopardizing the overall profit too much? In this paper, we are interested in finding systematic answers to these questions. In general terms, we can model the spread of a product as a process on a social network. Each node represents a single person, and each edge represents a friendship. Initially, one or more nodes is “active”, meaning that person already has the product. This could either be a large set of nodes representing an established customer base, or it could be just one node – the seller – whose neighbors consist of people who independently trust the seller, or who are otherwise likely to be interested in early adoption. At this point, the seller can encourage the spread of influences in two ways. First of all, it can offer cashback rewards to individuals who recommend the product to their friends. This is often seen in practice with “referral bonuses” – each buyer can optionally name the person who referred them, and this person then receives a cash reward. This gives existing buyers an incentive to recommend the product to their friends. Secondly, a seller can offer discounts to specific people in order to encourage them to buy the product, above and beyond any recommendations they receive. It is important to choose a good discount from the beginning here. If the price is not acceptable when a prospective buyer first receives recommendations, they might not bother to reconsider even if the price is lowered later. After receiving discount offers and some set of recommendations, it is up to the prospective buyers to decide whether to actually go through with a purchase. In general, they will do so with some probability that is influenced by the discount and by the set of recommendations they have received. The form of this probability is a parameter of the model and it is determined by external factors, for instance, the quality of the product and various exogenous market conditions. While it is impossible for a seller to calculate the form of these probability exactly, they can estimate it from empirical observations, and use that estimate to inform their policies. One could interpret the probabilities according to a number of different models that have been proposed in the literature (for instance, the Independent Cascade and Linear Threshold models), and hence it is desirable for the seller to be able to come up with a strategy that is applicable to a wide variety of models. Now let us suppose that a seller has access to data from a social network such as Facebook, Orkut, or MySpace. Using this, the seller can estimate what the real, true, underlying friendship structure is, and while this estimate will not be perfect, it is getting better over time, and any information is better than none. With this information in hand, a seller can model the spread of influence quite accurately, and the formerly inscrutable problems of who to offer discounts to, and at what price, become algorithmic questions that one can legitimately hope to solve. For example, if a seller knows the structure of the network, she can locate individuals that are particularly well connected and do everything possible to ensure they adopt the product and exert their considerable influence. In this paper, we are interested in the algorithmic side of this question: Given the network structure and a model of the purchase probabilities, how should the seller decide to offer discounts and cashback rewards? ### 1.1 Our contributions We investigate seller strategies that address the above questions in the context of expected revenue maximization. We will focus much of our attention on non-adaptive strategies for the seller: the seller chooses and commits to a discount coupon and cashback offer for each potential buyer before the cascade starts. If a recommendation is given to this node at any time, the price offered will be the one that the seller committed to initially, irrespective of the current state of the cascade. A wider class of strategies that one could consider are adaptive strategies, which do not have this restriction. For example, in an adaptive strategy, the seller could choose to observe the outcome of the (random) cascading process up until the last minute before making very well informed pricing decisions for each node. One might imagine that this additional flexibility could allow for potentially large improvements over non-adaptive strategies. Unfortunately, there is a price to be paid, in that good adaptive strategies are likely to be very complicated, and thus difficult and expensive to implement. The ratio of the revenue generated from the optimal adaptive strategy to the revenue generated from the optimal non-adaptive strategy is termed the “adaptivity gap”. Our main theoretical contribution is a very efficient non-adaptive strategy whose expected revenue is within a constant factor of the optimal revenue from an adaptive strategy. This guarantee holds for a wide variety of probability functions, including natural extensions of both the Linear Threshold and Independent Cascade models444More precisely, the strategy achieves a constant- factor approximation for any fixed model, independent of the social network. If one changes the model, the approximation factor does vary, as made precise in Section 3.. Note that a surprising consequence of this result is that the adaptivity gap is constant, so one can make the case that not much is lost by restricting our attention to non-adaptive policies. We also show that the problem of finding an optimal non-adaptive strategy is NP-hard, which means an efficient approximation algorithm is the best theoretical result that one could hope for. Intuitively, the seller strategy we propose is based on an influence-and- exploit idea, and it consists of categorizing each potential buyer as either an influencer or a revenue source. The influencers are offered the product for free and the revenue sources are offered the product at a pre-determined price, chosen based on the exact probability model. Briefly, the categorization is done by finding a spanning tree of the social network with as many leaves as possible, and then marking the leaves as revenue sources and the internal nodes as influencers. We can find such a tree in near-linear time [7, 10]. Cashback amounts are chosen to be a fixed fraction of the total revenue expected from this process. The full details are presented in section 3. In practice, we propose using this approach to find a strategy that has good global properties, and then using local search to improve it further. This kind of combination has been effective in the past, for example on the k-means problem [1]. Indeed, experiments (see section 4) show that combining local search with the above influence-and-exploit strategy is more effective than using either approach on its own. ### 1.2 Related work The problem of social contagion or spread of influence was first formulated by the sociological community, and introduced to the computer science community by Domingos and Richardson [2]. An influential paper by Kempe, Kleinberg and Tardos [5] solved the target set selection problem posed by [2] and sparked interest in this area from a theoretical perspective (see [6]). This work has mostly been limited to the influence maximization paradigm, where influence has been taken to be a proxy for the revenue generated through a sale. Although similar to our work in spirit, there is no notion of price in this model, and therefore, our central problem of setting prices to encourage influence spread requires a more complicated model. A recent work by Hartline, Mirrokni and Sundararajan [4] is similar in flavor to our work, and also considers extending social contagion ideas with pricing information, but the model they examine differs from our model in a several aspects. The main difference is that they assume that the seller is allowed to approach arbitrary nodes in the network at any time and offer their product at a price chosen by the seller, while in our model the cascade of recommendations determines the timing of an offer and this cannot be directly manipulated. In essence, the model proposed in [4] is akin to advertising the product to arbitrary nodes, bypassing the network structure to encourage a desired set of early adopters. Our model restricts such direct advertising as it is likely to be much less effective than a direct recommendation from a friend, especially when the recommender has an incentive to convince the potential buyer to purchase the product (for instance, the recommender might personalize the recommendation, increasing its effectiveness). Despite the different models, the algorithms proposed by us and [4] are similar in spirit and are based on an influence-and-exploit strategy. This work has also been inspired by a direction mentioned by Kleinberg [6], and is our interpretation of the informal problem posed there. Finally, we point out that the idea of cashbacks has been implemented in practice, and new retailers are also embracing the idea [8, 9, 14]. We note that some of the systems being implemented by retailers are quite close to the model that we propose, and hence this problem is relevant in practice. ## 2 The Formal Model Let us start by formalizing the setting stated above. We represent the social network as an undirected graph $G(V,E)$, and denote the initial set of adopters by $S^{0}\subseteq V$. We also denote the active set at time $t$ by $S^{t-1}$ (we call a node active if it has purchased the product and inactive otherwise). Given this setting, the recommendations cascade through the network as follows: at each time step $t\geq 1$, the nodes that became active at time $t-1$ (i.e. $S^{0}$ for $t=1$, and $u\in S^{t-1}\setminus S^{t-2}$ for $t\geq 2$) send recommendations to their currently inactive friends in the network: $N^{t-1}=\\{v\in V\setminus S^{t-1}\|(u,v)\in E,u\in S^{t-1}\setminus S^{t-2}\\}$. Each such node $v\in N^{t-1}$ is also given a price $c_{v,t}\in\mathbb{R}$ at which it can purchase the product. This price is chosen by the seller to either be full price or some discounted fraction thereof. The node $v$ must then decide whether to purchase the product or not (we discuss this aspect in the next section). If $v$ does accept the offer, a fixed cashback $r>0$ is given to a recommender $u\in S^{t-1}$ (note that we are fixing the cashback to be a positive constant for all the nodes as the nodes are assumed to be non-strategic and any positive cashback provides incentive for them to provide recommendations). If there are multiple recommenders, the buyer must choose only one of them to receive the cashback; this is a system that is quite standard in practice. In this way, offers are made to all nodes $v\in N^{t-1}$ through the recommendations at time $t$ and these nodes make a decision at the end of this time period. The set of active nodes is then updated and the same process is repeated until the process quiesces, which it must do in finite time since any step with no purchases ends the process. In the model described above, the only degree of freedom that the seller has is in choosing the prices and the cashback amounts. It wants to do this in a way that maximizes its own expected revenue (the expectation is over randomness in the buyer strategies). Since the seller may not have any control over the seed set, we are looking for a strategy that can maximize the expected revenue starting from any seed set on any graph. In most online scenarios, producing extra copies of the product has negligible cost, so maximizing expected revenue will also maximize expected profit. Now we can formally state the problem of finding a revenue maximizing strategy as follows: ###### Problem 1. Given a connected undirected graph $G(V,E)$, a seed set $S^{0}$, a fixed cashback amount $r$, and a model M for determining when nodes will purchase a product, find a strategy that maximizes the expected revenue from the cascading process described above. We are particularly interested in non-adaptive policies, which correspond to choosing a price for each node in advance, making the price independent of the time of the recommendation and the state of the cascade at the time of the offer. Our goal will be threefold: (1) to show that this problem is NP-hard even for simple models M, (2) to construct a constant-factor approximation algorithm for a wide variety of models, and (3) to show that restricting to non-adaptive policies results in at most a constant factor loss of profit. To simplify the exposition, we will assume the cashback $r=0$ for now. At the end of Section 4, we will show how the results can be generalized to work for positive $r$, which should be sufficient incentive for buyers to pass on recommendations. ### 2.1 Buyer decisions In this section, we discuss how to model the probability that a node will actually buy the product given a set of recommendations and a price. We use a very general model in this work that naturally extends the most popular traditional models proposed in the influence maximization literature, including both Independent Cascade and Linear Threshold. Consider an abstract model M for determining the probability that a node will buy a product given a price and what recommendations it has received. We allow M to take on virtually any form, imposing only the following conditions: 1. 1. The seller has full information about M. This is a standard assumption, and it can be approximated in practice by running experiments and observing people’s behavior. 2. 2. A node will never pay more than full price for the product (we assume this full price is 1 without loss of generality). Without an assumption like this, the seller could potentially achieve unbounded revenue on a single network, which makes the problem degenerate. 3. 3. A node will always accept the product and recommend it to friends if it receives a recommendation with price 0 (i.e. if a friend offers the product for free). Since nodes are given positive cash rewards for making recommendations, this condition is true for any rational buyer. 4. 4. If the social network is a single line graph with $S^{0}$ being the two endpoints, the maximum expected revenue is at most a constant $L$. Intuitively, this states that each prospective buyer on a social network should have some chance of rejecting the product (unless it’s given to them for free), and therefore the maximum revenue on a line is bounded by a geometric series, and is therefore constant. 5. 5. There exist constants $f$, $c$, $q$ so that if more than fraction $f$ of a given node’s neighbors recommend the product to the node at cost $c$, the node will purchase the product with probability $q$. This rules out extreme inertia, for example the case where no buyer will consider purchasing a product unless almost all of its neighbors have already done so. The fourth and fifth conditions here are used to parametrize how complicated the model is, and our final approximation bound will be in terms of this model “complexity”, which is defined to be $\frac{L}{(1-f)cq}$. While it may not be obvious that all these conditions are met in general, we will show that they are for both the Independent Cascade and Linear Threshold models, and indeed, the arguments there extend naturally to many other cases as well. In the traditional Independent Cascade model, there is a fixed probability $p$ that a node will purchase a product each time it is recommended to them. These decisions are made independently for each recommendation, but each node will buy the product at most once. To generalize this to multiple prices, it is natural to make $p$ a function $[0,1]\rightarrow[0,1]$ where $p(x)$ represents the probability that a node will buy the product at price $x$. For technical reasons, however, it is convenient to work with the inverse of $p$, which we call $C$.555It is sometimes useful to consider functions $p(\cdot)$ that are not one-to-one. These functions have no formal inverse, but in this case, $c$ can still be formally defined as $C(x)=\max_{y}\|p(y)\|\geq x$. Our general conditions on the model reduce to setting $C(0)=1$ and $C(1)=0$ in this case. To ensure bounded complexity, we also impose a minor smoothness condition. ###### Definition 1. Fix a cost function $C:[0,1]\rightarrow[0,1]$ with $C(0)=1,C(1)=0$ and with $C$ differentiable at 0 and 1. We define the Independent Cascade Model ICMc as follows: Every time a node receives a recommendation at price $C(x)$, it buys the product with probability $x$ and does nothing otherwise. If a node receives multiple recommendations, it performs this check independently for each recommendation but it never purchases the product more than once. ###### Lemma 1. Fix a cost function $C$. Then: 1. 1. ICMC has bounded (model) complexity. 2. 2. If $C$ has maximum slope $m$ (i.e. $\|C(x)-C(y)\|\leq m\|x-y\|$ for all $x,y$), then $ICM_{C}$ has $O(m^{2})$ complexity. 3. 3. If $C$ is a step function with $n$ regularly spaced steps (i.e. $C(x)=C(y)$ if $\lfloor\frac{x}{n}\rfloor=\lfloor\frac{y}{n}\rfloor$), then ICMC has $O(n^{2})$ complexity. ###### Proof. We show that the complexity of ICMC can be bounded in terms of the maximum slope of $C$ near 0 and 1. Recall that if $C$ is differentiable at 0, then, by definition, there exists $\epsilon>0$ so that $\frac{\|C(x)-C(0)\|}{x}\leq\|C^{\prime}(0)\|+1$ for $x<\epsilon$. A similar argument can be made for $x=1$, and thus we can say formally that there exist $m$ and $\epsilon>0$ such that: $\displaystyle C(x)\geq 1-mx$ $\displaystyle\textrm{ for }x\leq\epsilon,\textrm{and}$ $\displaystyle C(x)\leq m(1-x)$ $\displaystyle\textrm{ for }x\geq 1-\epsilon.$ In this case, we will show that ICMC has complexity at most $8\max(\frac{1}{\epsilon},m)^{2}$, proving part 1. Note that parts 2 and 3 of the lemma will also follow immediately. We begin by analyzing $L_{n}$, the maximum expected revenue that can be achieved on a path of length $n$ if one of the endpoints is a seed. Note that $L\leq 2\max_{n}L_{n}$ since selling a product on a line graph with two seeds can be thought of as two independent sales, each with one seed, that are cut short if the sales ever meet. Now we have: $\displaystyle L_{n}=\max_{x}x(C(x)+L_{n-1}).$ This is because offering the product at cost $C(x)$ will lead to a purchase with probability $x$, and in that case, we get $C(x)$ revenue immediately and $L_{n-1}$ expected revenue in the future. Since $L_{n}$ is obviously increasing in $n$, this can be simplified further: $\displaystyle L_{n}\leq\max_{x}x(C(x)+L_{n})$ $\displaystyle\implies$ $\displaystyle L\leq 2L_{n}\leq\max_{0<x<1}\frac{2x\cdot C(x)}{1-x}$ For $x\geq 1-\epsilon$, we have $\frac{2x\cdot C(x)}{1-x}\leq\frac{2x\cdot m(1-x)}{1-x}\leq 2m$, and for $x<1-\epsilon$, we have $\frac{2x\cdot C(x)}{1-x}\leq\frac{2}{\epsilon}$. Either way, $L\leq 2\max(\frac{1}{\epsilon},m)$. It remains to choose $f,c$ and $q$ as per the first complexity condition. We use $f=0$, $q=\min(\epsilon,\frac{1}{2m})$ and $c=C(q)\geq\frac{1}{2}$. Indeed, if a node has more than 0 active neighbors, it will accept a recommendation at cost $C(q)$ with probability $q$. Thus ICMc has complexity at most $\frac{L}{(1-f)cq}\leq 8\max(\frac{1}{\epsilon},m)^{2}$, as required. ∎ In the traditional Linear Threshold model, there are fixed influences $b_{v,w}$ on each directed edge $(v,w)$ in the network. Each node independently chooses a threshold $\theta$ uniformly at random from $[0,1]$, and then purchases the product if and when the total influence on it from nodes that have recommended the product exceeds $\theta$. To generalize this to multiple prices, it is natural to make $b_{v,w}$ a function $[0,1]\rightarrow[0,1]$ where $b_{v,w}(x)$ indicates the influence $v$ exerts on $w$ as a result of recommending the product at price $x$. To simplify the exposition, we will focus on the case where a node is equally influenced by all its neighbors. (This is not strictly necessary but removing this assumptions requires rephrasing the definition of $f$ to be a weighted fraction of a node’s neighbors.) Finally, we assume for all $v,w$ that $b_{v,w}(0)=1$ to satisfy the second general condition for models. ###### Definition 2. Fix a max influence function $B:(0,1]\rightarrow[0,1]$, not uniformly 0. We define the Linear Threshold Model LTMB as follows: Every node independently chose a threshold $\theta$ uniformly at random from $[0,1]$. A node will buy the product at price $x>0$ only if $B(x)\geq\frac{\alpha}{\theta}$ where $\alpha$ denotes the fraction of the node’s neighbors that have recommended the product. A node will always accept a recommendation if the product is offered for free. ###### Lemma 2. Fix a max influence function $B$ and let $K=\max_{x}x\cdot B(x)$. Then LTMB has complexity $O(\frac{1}{K})$. We omit the proof since it is similar to that of Lemma 1. In fact, it is simpler since, on a line graph, a node either gets the product for free or it has probability at most $\frac{1}{2}$ of buying the product and passing on a recommendation. ## 3 Approximating the Optimal Revenue In this section, we present our main theoretical contribution: a non-adaptive seller strategy that achieves expected revenue within a constant factor of the revenue from the optimal adaptive strategy. We show the problem of finding the exact optimal strategy is NP-hard (see section 8.1 in the appendix), so this kind of result is the best we can hope for. Note that our approximation guarantee is against the strongest possible optimum, which is perhaps surprising: it is unclear a priori whether such a strategy should even exist. The strategy we propose is based on computing a maximum-leaf spanning tree (MAXLEAF) of the underlying social network graph, i.e., computing a spanning tree of the graph with the maximum number of leaf nodes. The MAXLEAF problem is known to be NP-Hard, and it is in fact also MAX SNP-Complete, but there are several constant-factor approximation algorithms known for the problem [3, 7, 10, 15]. In particular, one of these is nearly linear-time [10], making it practical to apply on large online social network graphs. The seller strategy we attain through this is an influence-and-exploit strategy that offers the product to all of the interior nodes of the spanning tree for free, and charges a fixed price from the leaves. Note that this strategy works for all the buyer decision models discussed above, including multi-price generalizations of both Independent Cascade and Linear Threshold. We consider the setting of Problem 1, where we are given an undirected social network graph $G(V,E)$, a seed set $S^{0}\subseteq V$ and a buyer decision model M. Throughout this section, we will let $L$, $f$, $c$ and $q$ denote the quantities that parametrize the model complexity, as described in Section 2.1. To simplify the exposition, we will assume for now that the seed set is a singleton node (i.e., $\|S^{0}\|=1$). If this is not the case, the seed nodes can be merged into a single node, and we can make much the same argument in that case. We will ignore cashbacks for now, and return to address them at the end of the section. The exact algorithm we will use is stated below: * • Use the MAXLEAF algorithm [10] to compute an approximate max-leaf spanning tree $T$ for $G$ that is rooted at $S_{0}$. * • Offer the product to each internal node of $T$ for free. * • For each leaf of $T$ (excluding $S_{0}$), independently flip a biased coin. With probability $\frac{1+f}{2}$, offer the product to the node for free. With probability $\frac{1-f}{2}$, offer the product to the node at cost $c$. We henceforth refer to this strategy as STRATEGYMAXLEAF. Our analysis will revolve around what we term as “good” vertices, defined formally as follows: ###### Definition 3. Given a graph $G(V,E)$, we define the good vertices to be the vertices with degree at least 3 and their neighbors. On the one hand, we show that if $G$ has $g$ good vertices, then the MAXLEAF algorithm will find a spanning tree with $\Omega(g)$ leaves. We then show that each leaf of this tree leads to $\Omega(1)$ revenue, implying STRATEGYMAXLEAF gives $\Omega(g)$ revenue overall. Conversely, we can decompose $G$ into at most $g$ line-graphs joining high-degree vertices, and the total revenue from these is bounded by $gL=O(g)$ for all policies, which gives the constant- factor approximation we need. We begin by bounding the number of leaves in a max-leaf spanning tree. For dense graphs, we can rely on the following fact [7, 10]: ###### Fact 1. The max-leaf spanning tree of a graph with minimum degree at least 3 has at least $n/4+2$ leaves [7, 10]. In general graphs, we cannot apply this result directly. However, we can make any graph have minimum degree 3 by replacing degree-1 vertices with small, complete graphs and by contracting along edges to remove degree-2 vertices. We can then apply Fact 1 to analyze this auxiliary graph, which leads to the following result: ###### Lemma 3. Suppose a connected graph $G$ has $n_{3}$ vertices with degree at least $3$. Then $G$ has a spanning tree with at least $\frac{n_{3}}{8}+1$ leaves. ###### Proof. Let $n_{1}$ and $n_{2}$ denote the number of vertices of degree 1 and 2 respectively, and let $M$ denote the number of leaves in a max-leaf spanning tree of $G$. If $n_{1}=n_{2}=0$, the result follows from Fact 1. Now, suppose $n_{2}=0$ but $n_{1}>0$. Clearly, every spanning tree has at least $n_{1}$ leaves, so the result is obvious if $n_{1}\geq\frac{n_{3}}{8}+1$. Otherwise, we replace each degree-1 vertex with a copy of $K_{4}$ (the complete graph on 4 vertices), one of whose vertices connects back to the rest of the graph. Let $G^{\prime}$ denote the resulting graph. Then $G^{\prime}$ has $4n_{1}+n_{3}$ vertices, and they are all at least degree 3, so $G^{\prime}$ has a spanning tree $T^{\prime}$ with at least $n_{1}+\frac{n_{3}}{4}+2$ leaves. We can transform this into a spanning tree $T$ on $G$ by contracting each copy of $K_{4}$ down to a single point. Each contraction could transform up to 3 leaves into a single leaf, but it will not affect other leaves. Since there are exactly $n_{1}$ contractions that need to be done altogether, $T$ has at least $n_{1}+\frac{n_{3}}{4}+2-2n_{1}\geq\frac{n_{3}}{8}+1$ leaves, as required. We now prove the result holds in general by induction on $n_{2}$. We have already shown the base case $(n_{2}=0)$. For the inductive step, we will define an auxiliary graph $G^{\prime}$ with $n_{2}^{\prime},n_{3}^{\prime}$ and $M^{\prime}$ defined as for $G$. We will then show $n_{2}^{\prime}<n_{2},n_{3}^{\prime}\geq n_{3}$, and for every spanning tree $T^{\prime}$ on $G^{\prime}$, there is a spanning tree $T$ on $G$ with at least as many leaves. This implies $M^{\prime}\leq M$, and using the inductive hypothesis, it follows that $M\geq M^{\prime}\geq\frac{n_{3}^{\prime}}{8}+1\geq\frac{n_{3}}{8}+1$, which will complete the proof. Towards that end, suppose $v$ is a degree-2 vertex in $G$, and let its neighbors be $u$ and $w$. If $u$ and $w$ are not adjacent, we let $G^{\prime}$ be the graph attained by contracting along the edge $(u,v)$. Then $n_{2}^{\prime}=n_{2}-1$ and $n_{3}^{\prime}=n_{3}$. Any spanning tree $T^{\prime}$ on $G^{\prime}$ can be extended back to a spanning tree $T$ on $G$ by uncontracting the edge $(u,v)$ and adding it to $T$. This does not decrease the number of leaves in the tree, so we are done. Next, suppose instead that $u$ and $w$ are adjacent. We cannot contract $(u,v)$ here since it will create a duplicate edge in $G^{\prime}$. However, a different construction can be used. If the entire graph is just these 3 vertices, the lemma is trivial. Otherwise, let $G^{\prime}$ be the graph attained by adding a degree-1 vertex $x$ adjacent to $v$. Then $n_{2}^{\prime}=n_{2}-1$ and $n_{3}^{\prime}=n_{3}+1$. Now consider a spanning tree $T^{\prime}$ of $G^{\prime}$. We can transform this into a spanning tree $T$ on $G$ by removing the edge $(v,x)$ that must be in $T^{\prime}$. This removes the leaf $x$ but if $v$ has degree 2 in $T^{\prime}$, it makes $v$ a leaf. In this case, $T$ and $T^{\prime}$ have the same number of leaves, so we are done. Otherwise, $(u,v)$ and $(v,w)$ are also in $T^{\prime}$, and since $G$ was assumed to have more than 3 vertices, $u$ and $w$ cannot both be leaves in $T^{\prime}$. Assume without loss of generality that $u$ is not a leaf. We then further modify $T$ by replacing $(v,w)$ with $(u,w)$. Now, $v$ is a leaf in $T$ and the only vertex whose degree has changed is $u$, which is not a leaf in either $T$ or $T^{\prime}$. Therefore, $T$ and $T^{\prime}$ again have the same number of leaves, and we are once again done. The result now follows from induction, as discussed above. ∎ We must further extend this to be in terms of the number of good vertices $g$, rather than being in terms of $n_{3}$: ###### Lemma 4. Given an undirected graph $G$ with $g$ good vertices, the MAXLEAF algorithm [10] will construct a spanning tree with $\max(\frac{g}{50}+0.5,2)$ leaves. ###### Proof. If $g=0$, the result is trivial. Otherwise, let $n_{3}$ denote the number of vertices in $G$ with degree at least 3, and let $M$ denote the number of leaves in a max-leaf spanning tree of $G$. By Lemma 3, we know $M\geq\frac{n_{3}}{8}+1$. Now consider constructing a spanning tree as follows: * 1. Let $A$ denote the set of vertices in $G$ with degree at least 3. * 2. Set $T$ to be a minimal subtree of $G$ that connects all vertices in $A$. * 3. Add all remaining vertices in $G$ to $T$ one at a time. If a vertex $v$ could be connected to $T$ in multiple ways, connect it to a vertex in $A$ if possible. To analyze this, note that $G-A$ can be decomposed into a collection of “primitive” paths. Given a primitive path $P$, let $g_{P}$ denote the number of good vertices on $P$ and let $l_{P}$ denote the number of leaves $T$ has on $P$. In Step 2 of the algorithm above, exactly $n_{3}-1$ of these paths are added to $T$. For each such path $P$, we have $g_{P}\leq 2$ and $l_{P}=0$. On the remaining paths, we have $g_{P}=l_{P}$. Therefore, the total number of leaves on $T$ is at least $\displaystyle\sum_{P}l_{P}=(g-n_{3})+\sum_{P}(l_{P}-g_{P})$ $\displaystyle\geq$ $\displaystyle(g-n_{3})-2(n_{3}-1).$ Thus, $\displaystyle M$ $\displaystyle\geq$ $\displaystyle\max\left(\frac{n_{3}}{8}+1,g-3n_{3}+1\right)$ $\displaystyle\geq$ $\displaystyle\frac{24}{25}\cdot\left(\frac{n_{3}}{8}+1\right)+\frac{1}{25}\cdot(g-3n_{3}+1)=\frac{g}{25}+1$ The result now follows from the fact that the MAXLEAF algorithm gives a 2-approximation for the max-leaf spanning tree, and that every non-degenerate tree has at least two leaves. ∎ We can now use this to prove a guarantee on the performance of STRATEGYMAXLEAF in terms of the number of good vertices on an arbitrary graph: ###### Lemma 5. Given a social network $G$ with $g$ good vertices, STRATEGYMAXLEAF guarantees an expected revenue of $\Omega((1-f)cq\cdot g)$. ###### Proof. Let $T$ denote the spanning tree found by the MAXLEAF algorithm. Let $U$ denote the set of interior nodes of $T$, and let $V$ denote the leaves of $T$ (excluding $S_{0}$). Since we assumed $\|S_{0}\|=1$, Lemma 4 guarantees $\|V\|\geq\max(\frac{g}{50}-0.5,1)=\Omega(g)$. Note every vertex can be reached from $S_{0}$ by passing through nodes in $U$, each of which is offered the product for free. These nodes are guaranteed to accept the product, and therefore, they will collectively pass on at least one recommendation to each vertex. Now consider the expected revenue from a vertex $v\in V$. Let $M$ be the random variable giving the fraction of $v$’s neighbors in $V$ that were not offered the product for free. We know $E[M]=\frac{1-f}{2}$, so with probability $\frac{1}{2}$, we have $M\leq 1-f$. In this case, $v$ is guaranteed to receive recommendations from a fraction $f$ of its neighbors in $V$, as well as all of its neighbors in $U\cup S_{0}$ (of which there is at least 1). If we charge $v$ a total of $c$ for the product, it will then purchase the product with probability at least $q$, by the original definitions of $f$, $c$ and $q$. Furthermore, independent of $v$’s neighbors, we will ask this price from $v$ with probability $\frac{1-f}{2}$. Therefore, our expected revenue from $v$ is at least $\frac{1}{2}\cdot q\cdot\frac{1-f}{2}\cdot c$. The result now follows from linearity of expectation. ∎ Now that we have computed the expected revenue from STRATEGYMAXLEAF, we need to characterize the optimal revenue to bound the approximation ratio. This bound is given by the following lemma. ###### Lemma 6. The maximum expected revenue achievable by any strategy (adaptive or not) on a social network $G$ with $g$ good vertices is $O(L\cdot g)$. ###### Proof. Let $A$ denote the set of vertices in $G$ with degree at least 3, and let $n_{3}=\|A\|$. Clearly, no strategy can achieve more than $n_{3}$ revenue directly from the nodes in $A$. As observed in the proof of Lemma 4, however, $G-A$ can be decomposed into a collection of primitive paths. Since each primitive path contains at least one unique good vertex with degree less than 3, there is at most $g-n_{3}$ such paths. Even if each endpoint of a path is guaranteed to recommend the product, the total revenue from the path is at most $L$. Therefore, the total revenue from any strategy on such a graph is at most $n_{3}+(g-n_{3})L=O(L\cdot g)$. ∎ Now, we can combine the above lemmas to state the main theorem of the paper, which states that STRATEGYMAXLEAF provides a constant factor approximation guarantee for the revenue. ###### Theorem 1. Let $K$ denote the complexity of our buyer decision model M. Then, the expected revenue generated by STRATEGYMAXLEAF on an arbitrary social network is $O(K)$-competitive with the expected revenue generated by the optimal (adaptive or not) strategy. ###### Proof. This follows immediately from Lemmas 5 and 6, as well as the fact that $K=\frac{L}{(1-f)cq}$. ∎ As a corollary, we get the fact that the adaptivity gap is also constant: ###### Corollary 1. Let $K$ denote the complexity of our buyer decision model M. Then the adaptivity gap is $O(K)$. Now we briefly address the issue of cashbacks that were ignored in this exposition. We set the cashback $r$ to be a small fraction of our expected revenue from each individual $r_{0}$, i.e. $r=z\cdot r_{0}$, where $z<1$. Then, our total profit will be $n\cdot r_{0}\cdot(1-z)$. Adding this cashback decreases our total profit by a constant factor that depends on $z$, but otherwise the argument now carries through as before, and nodes now have a positive incentive to pass on recommendations. In light of Corollary 1, one might ask whether the adaptivity gap is not just 1. In other words, is there any benefit at all to be gained from using non- adaptive strategies? In fact, there is. For example, consider a social network consisting of 4 nodes $\\{v_{1},v_{2},v_{3},v_{4}\\}$ in a cycle, with $v_{3}$ connected to two other isolated vertices. Suppose furthermore that a node will accept a recommendation with probability 0.5 unless the price is 0, in which case the node will accept it with probability 1. On this network, with seed set $S^{0}=\\{v_{1}\\}$, the optimal adaptive strategy is to always demand full price unless exactly one of $v_{2}$ and $v_{4}$ purchases the product initially, in which case $v_{3}$ should be offered the product for free. This beats the optimal non-adaptive strategy by a factor of 1.0625. ## 4 Local Search In this section, we discuss how an arbitrary seller strategy can be tweaked by the use of a local search algorithm. Taken on its own, this technique can sometimes be problematic since it can take a long time to converge to a good strategy. However, it performs very well when applied to an already good strategy, such as STRATEGYMAXLEAF. This approach of combining theoretically sound results with local search to generate strong techniques in practice is similar in spirit to the recent k-means++ algorithm [1]. Intuitively, the local search strategy for pricing on social networks works as follows: * • Choose an arbitrary seller strategy $S$ and an arbitrary node $v$ to edit. * • Choose a set of prices $\\{p_{1},p_{2},\ldots,p_{k}\\}$ to consider. * • For each price $p_{i}$, empirically estimate the expected revenue $r_{i}$ that is achieved by using the price $p_{i}$ for node $v$. * • If any revenue $r_{i}$ beats the current expected revenue (also estimated empirically) by some threshold $\epsilon$, then change $S$ to use the price $p_{i}$ for node $v$. * • Repeat the preceding steps for different nodes until there are no more improvements. Henceforth, we call this the LOCALSEARCH algorithm for improving seller strategies. To empirically estimate the revenue from a seller strategy, we can always just simulate the entire process. We know who has the product initially, we know what price each node will be offered, and we know the probability each node will purchase the product at that price after any number of recommendations. Simulating this process a number of times and taking the average revenue, we can arrive at a fair approximation at how good a strategy is in practice. In fact, we can prove that performing local search on any input policy will ensure that the seller gets at least as much revenue as the original policy with high probability. The proof of this fact holds for any simulatable input policy, and proceeds by induction on the evolution tree of the process. The proof is somewhat technical, so we will skip it, and instead focus on the empirical question of the advantage provided by local search. In light of the fact that local search can only improve the revenue (and never hurt it), it seems that one should always implement local search for any policy. There is a important technical detail that complicates this, however. Suppose we wish to evaluate strategies $S_{1}$ and $S_{2}$, differing only on one node $v$. If we independently run simulations for each strategy, it could take thousands of trials (or more!) before the systematic change to one node becomes visible over the noise resulting from random choices made by the other nodes. It is impractical to perform these many simulations on a large network every time we want to change the strategy for a single node. Fortunately, it is possible to circumvent this problem using an observation first noted in [5]. Let us consider the Linear Threshold model LTMB. In this case, all randomness occurs before the process begins when each node chooses a threshold that encodes how resistant it is to buying the product. Once these thresholds have been fixed, the entire sales process is deterministic. We can now change the strategy slightly and maintain the same thresholds to isolate exactly what effect this strategy change had. Any model, including Independent Cascade, can be rephrased in terms of thresholds, making this technique possible. The LOCALSEARCH algorithm relies heavily on this observation. While comparing strategies, we choose several threshold lists, and simulate each strategy against the same threshold lists. If these lists are not representative, we might still make a mistake drawing conclusions from this, but we will not lose a universally good signal or a universally bad signal under the weight of random noise. With this implementation, empirical tests (see the next section) show the LOCALSEARCH algorithm does do its job: given enough time, it will improve virtually any strategy enough to be competitive. It is not a perfect solution, however. First of all, it can still make small mistakes while doing the random estimates, possibly causing a strategy to become worse over time666Note that if we choose $\epsilon$ and the number of trials carefully, we can make this possibility vanishingly small (this is also the intuition behind the local search guarantee, as we had mentioned earlier. In practice, however, it is usually better to run fewer trials and accept the possibility of regressing slightly.. Secondly, it is possible to end up with a sub-optimal strategy that simply cannot be improved by any local changes. Finally, the LOCALSEARCH algorithm can often take many steps to improve a bad strategy, making it occasionally too slow to be useful in practice. Nonetheless, these drawbacks really only becomes a serious problem if one begins with a bad strategy. If one begins with a relatively good strategy – for example STRATEGYMAXLEAF – the LOCALSEARCH algorithm performs well, and is almost always worth doing in practice. We justify this claim in the next section. ### 4.1 Experimental Results In this section, we provide experimental evidence for the efficacy of the LOCALSEARCH algorithm in improving the revenue guarantee. Note that in these experiments, we need to assume a benchmark strategy as finding the optimal strategy is NP-hard (see section 8.1). We pick a very simple strategy RANDOMPRICING, which picks a random price independently for each node. The results demonstrate that even this naive strategy can be coupled with the LOCALSEARCH algorithm to do well in practice. (a) Random preferential attachment graph (b) Youtube subgraph Figure 1: The variation in revenue generated by RANDOMPRICING and STRATEGYMAXLEAF with the iterations of the LOCALSEARCH algorithm. The data is averaged over $10$ runs of a $1000$ node random preferential attachment graph 1(a) or a $10000$ node subgraph of YouTube 1(b), starting with a random seed each time. We simulate the cascading process on two kind of graphs. The first graph we study is a randomly generated graph, based on the preferential attachment model that is a popular model for representing social networks [12]. We generate a $1000$ node preferential attachment graph at random, and simulate the cascading process by picking a random node as the seed in the network. The probability model we examine is a step function (see the second example given in Lemma 1) of probabilities. We note that the function is necessarily arbitrary. The result of one particular parameter settings are shown in figure 1(a), which plots average revenue obtained by the two pricing strategies: RANDOMPRICING and STRATEGYMAXLEAF. Each point on the figure is obtained by average revenue over 10 runs on the same graph but with a different (random) seed. The horizontal axis indicates the number of LOCALSEARCH iterations that were done on the graph, where each iteration consisted of simulating the process 50 times, and choosing the best value over the runs. It is clear from the graph that STRATEGYMAXLEAF does quite well even without the addition of LOCALSEARCH, although the addition of LOCALSEARCH does increase the revenue. On the other hand, the RANDOMPRICING strategy performs poorly on its own, but its revenue increases steadily with the iterations of the LOCALSEARCH algorithm. We note that the difference between the revenue from the two policies does vary (as expected) with the probability model, and the difference between the revenue is not as large in all the different runs. But the difference does persist across the runs, especially when the strategies are run without the local search improvement. We also conduct a similar simulation with a real-world network, namely the links between users of the video-sharing site YouTube.777The network can be freely downloaded; see [11] for details. The YouTube network has millions of nodes, and we only study a subset of $10,000$ nodes of the network. We simulate the random process as earlier, and the results are shown in figure 1(b). Again, we note that STRATEGYMAXLEAF does very well on its own, easily beating the revenue of RANDOMPRICING. The RANDOMPRICING strategy does improve a lot with LOCALSEARCH, but it fails to equalize the revenue of STRATEGYMAXLEAF. The large size of the YouTube graph and the expensive nature of the LOCALSEARCH algorithm restrict the size of the experiments we can conduct with the graph, but the results from the above does experiments do offer some insights. In particular, STRATEGYMAXLEAF succeeds in extracting a good portion of the revenue from the graph, if we consider the revenue obtained from STRATEGYMAXLEAF combined with LOCALSEARCH based improvements to be the benchmark. Further, LOCALSEARCH can improve the revenue from any strategy by a substantial margin, though it may not be able to attain enough revenue when starting with a sub-optimal strategy such as RANDOMPRICING. Finally, we observe that the combination of STRATEGYMAXLEAF and LOCALSEARCH generates the best revenue among our strategies, and it is an open question as to whether this is the optimal adaptive strategy. ## 5 Conclusions In this work, we discussed pricing strategies for sellers distributing a product over social networks through viral marketing. We show that computing the optimal (one that maximizes expected revenue) non-adaptive strategy for a seller is NP-Hard. In a positive result, we show that there exists a non- adaptive strategy for the seller which generates expected revenue that is within a constant factor of the expected revenue generated by the optimal adaptive strategy. This strategy is based on an influence-and-exploit policy which computes a max-leaf spanning tree of the graph, and offers the product to the interior nodes of the spanning tree for free, later on exploiting this influence by extracting its profit from the leaf nodes of the tree. The approximation guarantee of the strategy holds for fairly general conditions on the probability function. ## 6 Open Questions The added dimension of pricing to influence maximization models poses a host of interesting questions, many of which are open. An obvious direction in which this work could be extended is to think about influence models stronger than the model examined here. It is also unclear whether the assumptions on the function $C(\cdot)$ are the minimal set that is required, and it would be interesting to remove the assumption that there exists a price at which the probability of acceptance is 1. A different direction of research would be to consider the game-theoretic issues involved in a practical system. Namely, in the model presented here, we think of each buyer as just sending the recommendations to all its friends and ignore the issue of any “cost” involved in doing so, thereby assuming all the nodes to be non-strategic. It would be very interesting to model a system where the nodes were allowed to behave strategically, trying to maximize their payoff, and characterize the optimal seller strategy (especially w.r.t. the cashback) in such a setting. ## 7 Acknowledgments Supported in part by NSF Grant ITR-0331640, TRUST (NSF award number CCF-0424422), and grants from Cisco, Google, KAUST, Lightspeed, and Microsoft. The third author is grateful to Mukund Sundararajan and Jason Hartline for useful discussions. ## References * [1] David Arthur and Sergei Vassilvitskii. k-means++: the advantages of careful seeding. In SODA ’07: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, pages 1027–1035, Philadelphia, PA, USA, 2007\. Society for Industrial and Applied Mathematics. * [2] P. Domingos and M. Richardson. Mining the network value of customers. Proceedings of the seventh ACM SIGKDD international conference on Knowledge discovery and data mining, pages 57–66, 2001. * [3] M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. WH Freeman & Co. New York, NY, USA, 1979. * [4] J. Hartline, V. Mirrokni, and M. Sundararajan. Optimal Marketing Strategies over Social Networks. Proceedings of the 17th international conference on World Wide Web, 2008. * [5] D. Kempe, J. Kleinberg, and É. Tardos. Maximizing the spread of influence through a social network. Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 137–146, 2003. * [6] J. Kleinberg. Cascading Behavior in Networks: Algorithmic and Economic Issues. In N. Nisan, T. Roughgarden, E. Tardos, and V.V. Vazirani, editors, Algorithmic Game Theory. Cambridge University Press New York, NY, USA, 2007. * [7] D.J. Kleitman and D.B. West. Spanning Trees with Many Leaves. SIAM Journal on Discrete Mathematics, 4:99, 1991. * [8] J. Leskovec, A. Singh, and J. Kleinberg. Patterns of influence in a recommendation network. Pacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD), 2006. * [9] Jure Leskovec, Lada A. Adamic, and Bernardo A. Huberman. The dynamics of viral marketing. ACM Trans. Web, 1(1):5, 2007. * [10] H.I. Lu and R. Ravi. Approximating Maximum Leaf Spanning Trees in Almost Linear Time. Journal of Algorithms, 29(1):132–141, 1998. * [11] A. Mislove, M. Marcon, K.P. Gummadi, P. Druschel, and B. Bhattacharjee. Measurement and analysis of online social networks. In Proceedings of the 7th ACM SIGCOMM conference on Internet measurement, pages 29–42. ACM New York, NY, USA, 2007. * [12] MEJ Newman, DJ Watts, and SH Strogatz. Random graph models of social networks, 2002. * [13] BBC News. Facebook valued at $15 billion. http://news.bbc.co.uk/2/hi/business/7061042.stm, 2007. * [14] Erick Schonfeld. Amiando makes tickets go viral and widgetizes event management. http://www.techcrunch.com/2008/07/17/amiando-makes-tickets-go-viral-and%-widgetizes-event-management-200-discount-for-techcrunch-readers/, 2008. * [15] R. Solis-Oba. 2-Approximation Algorithm for finding a Spanning Tree with Maximum Number of leaves. Proceedings of the Sixth European Symposium on Algorithms, pages 441–452, 1998. * [16] Wikipedia. The beacon advertisement system. http://en.wikipedia.org/wiki/Facebook_Beacon, 2008. * [17] Wikipedia. Facebook revenue in 2008. http://en.wikipedia.org/wiki/Facebook, 2008. ## 8 Appendix ### 8.1 Hardness of finding the optimal strategy In this section, we show that Problem 1 is NP-hard even for a very simple buyer model M by a reduction from vertex cover with bounded degree (see [3] for the hardness of bounded-degree vertex cover). Letting $d$ denote the degree bound, and letting $p=\frac{1}{4d}$, we will use an Independent Cascade Model ICMC with: $\displaystyle C(x)=\left\\{\begin{array}[]{ll}1&\textrm{ if }x<p,\\\ 0&\textrm{ if }x\geq p\end{array}\right.$ Intuitively, the seller has to partition the nodes into “free” nodes and “full-price” nodes. In the former case, nodes are offered the product for free, and they accept it with probability 1 as soon as they receive a recommendation. In the latter case, nodes are offered the product for price 1, and they accept each recommendation with probability $p$. (Note that the seller is allowed to use other prices between $0$ and $1$ but a price of $1$ is always better.) We are going to use a special family of graphs illustrated in Figure 2. The graph consists of four layers: * • A singleton node $s$, which we will use as the only initially active node (i.e., $S^{0}=\\{s\\}$); * • $s$ links to a set of $n$ nodes, denoted by $V_{1}$; * • Nodes in $V_{1}$ also link to another set of nodes, denoted by $V_{2}$. Each node in $V_{1}$ will be adjacent to $d$ nodes in $V_{2}$, and each node in $V_{2}$ will be adjacent to $2$ nodes in $V_{1}$ (so $\|V_{2}\|=dn/2$); * • Each node $v\in V_{2}$ also links to $k=20d$ new nodes, denoted by $W_{v}$; these $k$ nodes do not link to any other nodes. The union of all $W_{v}$’s is denoted by $V_{3}$. Figure 2: Reducing Bounded-Degree Vertex Cover to Optimal Network Pricing We first sketch the idea of the hardness proof. The connection between $V_{1}$ and $V_{2}$ will be decided by the vertex cover instance: given a vertex cover instance $G^{\prime}(V,E)$ with bounded degree $d$, we construct a graph $G$ as above where $V_{1}=V$ and $V_{2}=E$, adding an edge between $V_{1}$ and $V_{2}$ if the corresponding vertex is incident to the corresponding edge in $G^{\prime}$. The key lemma is that, in the optimal pricing strategy for $G$, the subset of nodes in $V_{1}$ that are given the product for free is the minimum set that covers $V_{2}$ (i.e., a minimum vertex cover of $G^{\prime}$). To formalize this, first note that, in an optimal strategy, all nodes in $V_{3}$ should be full-price. Giving the product to them for free gets 0 immediate revenue, and offers no long-term benefit since nodes in $V_{3}$ cannot recommend the product to anyone else. If the nodes are full-price, on the other hand, there is at least a chance at some revenue. On the other hand, we show the optimal strategy must also ensure each vertex in $V_{2}$ eventually becomes active with probability 1. ###### Lemma 7. In an optimal strategy, every node $v\in V_{2}$ is free, and can be reached from $s$ by passing through free nodes. ###### Proof. Suppose, by way of contradiction, that the optimal strategy has a node $v\in V_{2}$ that does not satisfy these conditions. Let $u_{1}$ and $u_{2}$ be the two neighbors of $v$ in $V_{1}$, and let $q$ denote the probability that $v$ eventually becomes active. We first claim that $q<2dp$. Indeed, if $v$ is full-price, then even if $u_{1}$ and $u_{2}$ become active, the probability that $v$ becomes active is $1-(1-p)^{2}<2p$. Otherwise, $u_{1}$ and $u_{2}$ are both full-price. Since $u_{1}$ and $u_{2}$ connect to at most $2d$ edges other than $v$, the probability that one of them becomes active before $v$ is at most $1-(1-p)^{2d}<2dp$. Thus, $q<2dp$. It follows that the total revenue that this strategy can achieve from $u_{1}$, $v$, and $W_{v}$ is $2+kqp<2+2kdp^{2}=4.5$. Conversely, if we make $u_{1}$ and $v$ free, we can achieve $kp=5$ revenue from the same buyers. Furthermore, doing this cannot possibly lose revenue elsewhere, which contradicts the assumption that our original strategy was optimal. ∎ It follows that, in an optimal strategy, all of $V_{3}$ is full-price, all of $V_{2}$ is free, and every node in $V_{2}$ is adjacent to a free node in $V_{1}$. It remains only to determine $C$, the nodes in $V_{1}$, that an optimal strategy should make free. At this point, it should be intuitively clear that $C$ should correspond to a minimum vertex-cover of $V_{2}$. We now formalize this as follows: ###### Lemma 8. Let $C$ denote the set of free nodes in $V_{1}$, as chosen by an optimal strategy. Then $C$ corresponds to a minimum vertex cover of $G^{\prime}$. ###### Proof. As noted above, every node in $V_{2}$ must be adjacent to a node in $C$, which implies $C$ does indeed correspond to a vertex cover in $G^{\prime}$. Now we know an optimal strategy makes every node in $V_{2}$ free, and every node in $V_{3}$ full-price. Once we know $C$, the strategy is determined completely. Let $x_{C}$ denote the expected revenue obtained by this strategy. Since all nodes in $V_{2}$ are free and are activated with probability $1$, we know the strategy achieves 0 revenue from $V_{2}$ and $p\|V_{3}\|$ expected revenue from $V_{3}$. Among nodes in $V_{1}$, the strategy achieves 0 revenue for free nodes, and exactly $1-(1-p)^{d+1}$ expected revenue for each full-price node. This is because each full-price node is adjacent to exactly $d+1$ other nodes, and each of these nodes is activated with probability 1. Therefore, $x_{C}=(\|V_{1}\|-\|C\|)(1-(1-p)^{d+1})+p\|V_{3}\|$, which is clearly minimized when $C$ is a minimum-vertex cover. ∎ Therefore, optimal pricing, even in this limited scenario, can be used to calculate the minimum-vertex cover of any bounded-degree graph, from which NP- hardness follows. ###### Theorem 2. Two Coupon Optimal Strategy Problem is NP-Hard.	arxiv-papers	2009-02-20T00:06:42	2024-09-04T02:49:00.716289	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "David Arthur, Rajeev Motwani, Aneesh Sharma, Ying Xu", "submitter": "Aneesh Sharma", "url": "https://arxiv.org/abs/0902.3485" }
0902.3501	# Measurement of the Branching Fractions for $J/\psi$ $\rightarrow$ $p\bar{p}\eta$ and $p\bar{p}\eta^{\prime}$ M. Ablikim1, J. Z. Bai1, Y. Bai1, Y. Ban11, X. Cai1, H. F. Chen16, H. S. Chen1, H. X. Chen1, J. C. Chen1, Jin Chen1, X. D. Chen5, Y. B. Chen1, Y. P. Chu1, Y. S. Dai18, Z. Y. Deng1, S. X. Du1a, J. Fang1, C. D. Fu1, C. S. Gao1, Y. N. Gao14, S. D. Gu1, Y. T. Gu4, Y. N. Guo1, Z. J. Guo15b, F. A. Harris15, K. L. He1, M. He12, Y. K. Heng1, H. M. Hu1, T. Hu1, G. S. Huang1c, X. T. Huang12, Y. P. Huang1, X. B. Ji1, X. S. Jiang1, J. B. Jiao12, D. P. Jin1, S. Jin1, G. Li1, H. B. Li1, J. Li1, L. Li1, R. Y. Li1, W. D. Li1, W. G. Li1, X. L. Li1, X. N. Li1, X. Q. Li10, Y. F. Liang13, B. J. Liu1d, C. X. Liu1, Fang Liu1, Feng Liu6, H. M. Liu1, J. P. Liu17, H. B. Liu4e, J. Liu1, Q. Liu15, R. G. Liu1, S. Liu8, Z. A. Liu1, F. Lu1, G. R. Lu5, J. G. Lu1, C. L. Luo9, F. C. Ma8, H. L. Ma2, Q. M. Ma1, M. Q. A. Malik1, Z. P. Mao1, X. H. Mo1, J. Nie1, S. L. Olsen15, R. G. Ping1, N. D. Qi1, J. F. Qiu1, G. Rong1, X. D. Ruan4, L. Y. Shan1, L. Shang1, C. P. Shen15, X. Y. Shen1, H. Y. Sheng1, H. S. Sun1, S. S. Sun1, Y. Z. Sun1, Z. J. Sun1, X. Tang1, J. P. Tian14, G. L. Tong1, G. S. Varner15, X. Wan1, L. Wang1, L. L. Wang1, L. S. Wang1, P. Wang1, P. L. Wang1, Y. F. Wang1, Z. Wang1, Z. Y. Wang1, C. L. Wei1, D. H. Wei3, N. Wu1, X. M. Xia1, G. F. Xu1, X. P. Xu6, Y. Xu10, M. L. Yan16, H. X. Yang1, M. Yang1, Y. X. Yang3, M. H. Ye2, Y. X. Ye16, C. X. Yu10, C. Z. Yuan1, Y. Yuan1, Y. Zeng7, B. X. Zhang1, B. Y. Zhang1, C. C. Zhang1, D. H. Zhang1, H. Q. Zhang1, H. Y. Zhang1, J. W. Zhang1, J. Y. Zhang1, X. Y. Zhang12, Y. Y. Zhang13, Z. X. Zhang11, Z. P. Zhang16, D. X. Zhao1, J. W. Zhao1, M. G. Zhao1, P. P. Zhao1, Z. G. Zhao16, B. Zheng1, H. Q. Zheng11, J. P. Zheng1, Z. P. Zheng1, B. Zhong9 L. Zhou1, K. J. Zhu1, Q. M. Zhu1, X. W. Zhu1, Y. S. Zhu1, Z. A. Zhu1, Z. L. Zhu3, B. A. Zhuang1, B. S. Zou1 (BES Collaboration) 1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China 2 China Center for Advanced Science and Technology(CCAST), Beijing 100080, People’s Republic of China 3 Guangxi Normal University, Guilin 541004, People’s Republic of China 4 Guangxi University, Nanning 530004, People’s Republic of China 5 Henan Normal University, Xinxiang 453002, People’s Republic of China 6 Huazhong Normal University, Wuhan 430079, People’s Republic of China 7 Hunan University, Changsha 410082, People’s Republic of China 8 Liaoning University, Shenyang 110036, People’s Republic of China 9 Nanjing Normal University, Nanjing 210097, People’s Republic of China 10 Nankai University, Tianjin 300071, People’s Republic of China 11 Peking University, Beijing 100871, People’s Republic of China 12 Shandong University, Jinan 250100, People’s Republic of China 13 Sichuan University, Chengdu 610064, People’s Republic of China 14 Tsinghua University, Beijing 100084, People’s Republic of China 15 University of Hawaii, Honolulu, HI 96822, USA 16 University of Science and Technology of China, Hefei 230026, People’s Republic of China 17 Wuhan University, Wuhan 430072, People’s Republic of China 18 Zhejiang University, Hangzhou 310028, People’s Republic of China a Current address: Zhengzhou University, Zhengzhou 450001, People’s Republic of China b Current address: Johns Hopkins University, Baltimore, MD 21218, USA c Current address: University of Oklahoma, Norman, Oklahoma 73019, USA d Current address: University of Hong Kong, Pok Fu Lam Road, Hong Kong e Current address: Graduate University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China ###### Abstract Using 58$\times 10^{6}$ $J/\psi$ events collected with the Beijing Spectrometer (BESII) at the Beijing Electron Positron Collider (BEPC), the branching fractions of $J/\psi$ to $p\bar{p}\eta$ and $p\bar{p}\eta^{\prime}$ are determined. The ratio $\frac{\Gamma(J/\psi\rightarrow p\bar{p}\eta)}{\Gamma(J/\psi\rightarrow p\bar{p})}$ obtained by this analysis agrees with expectations based on soft-pion theorem calculations. ## I Introduction The $J/\psi$ meson has hadronic, electromagnetic, and radiative decays to light hadrons, and a radiative transition to the $\eta_{c}$. In Ref. PCX , direct hadronic, electromagnetic and radiative decays are estimated to account for 69.2$\%$, 13.4$\%$, and 4.3$\%$, respectively, of all $J/\psi$ decays. However, individual exclusive $J/\psi$ decays are more difficult to analyze quantitatively in QCD. To date, two-body decay modes such as $J/\psi\rightarrow B_{8}\bar{B_{8}}$ or $P_{9}V_{9}$, where $B_{8}$, $P_{9}$ and $V_{9}$ refer to baryon octet, pseudoscalar nonet, and vector nonet particle, respectively, have been studied with some success using an effective model, and other similar methods SPIT . Studies of three-body decays of $J/\psi$ are a natural extension of studies of two-body decays. Since most $J/\psi$ decays proceed via two-body intermediate states, including wide resonances, it is hard to experimentally extract the non-resonant three-body contribution JPDC . Specific models based on proton and $N^{}$ pole diagrams have been introduced to deal with these problems SPIT . In the calculation, the soft-pion theorem LABW has been applied to the decay $J/\psi\rightarrow p\bar{p}\pi^{0}$ successfully. This method has also been used for $J/\psi\rightarrow p\bar{p}\eta$ and $J/\psi\rightarrow p\bar{p}\eta^{\prime}$ decays SPIT . This paper reports measurement of the branching fractions for $p\bar{p}\eta$ and $p\bar{p}\eta^{\prime}$, and tests of the soft-pion theorem for $J/\psi\rightarrow p\bar{p}\eta$, which states SPIT : $\displaystyle\frac{\Gamma(J/\psi\rightarrow p\bar{p}\eta)}{\Gamma(J/\psi\rightarrow p\bar{p})}\simeq 0.64\pm 0.52.$ ## II The BES detector and Monte Carlo simulation BESII is a conventional solenoidal magnet detector that is described in detail in Refs. JZB . A 12-layer vertex chamber (VC) surrounding the beam pipe provides trigger and track information. A forty-layer main drift chamber (MDC), located radially outside the VC, provides trajectory and energy loss ($dE/dx$) information for tracks over $85\%$ of the total solid angle. The momentum resolution is $\sigma_{p}/p=0.017\sqrt{1+p^{2}}$ ($p$ in $\hbox{\rm GeV}/c$), and the $dE/dx$ resolution for hadron tracks is $\sim 8\%$. An array of 48 scintillation counters surrounding the MDC measures the time-of-flight (TOF) of tracks with a resolution of $\sim 200$ ps for hadrons. Radially outside the TOF system is a 12 radiation length, lead-gas barrel shower counter (BSC). This measures the energies of electrons and photons over $\sim 80\%$ of the total solid angle with an energy resolution of $\sigma_{E}/E=22\%/\sqrt{E}$ ($E$ in GeV). Outside of the solenoidal coil, which provides a 0.4 Tesla magnetic field over the tracking volume, is an iron flux return that is instrumented with three double layers of counters that identify muons of momentum greater than 0.5 GeV/$c$. In the analysis, a GEANT3-based Monte Carlo (MC) simulation program (SIMBES) GEANT with detailed consideration of detector performance is used. The consistency between data and MC has been validated using many high purity physics channels NIM . In this analysis, the detection efficiency for each decay mode is determined by a MC simulation that takes into account the angular distributions. For $J/\psi\rightarrow p\bar{p}\eta$, the angle ($\theta$) between the directions of $e^{+}$ and $p$ in the laboratory frame is generated according to $1+\alpha\cdot\cos^{2}\theta$ distribution, where $\alpha$ is obtained by fitting the data from $J/\psi\rightarrow p\bar{p}\eta$. A uniform phase space distribution is used for $J/\psi$ decaying into $p\bar{p}\eta^{\prime}$. ## III General Selection Criteria Candidate events are required to satisfy the following common selection criteria: ### III.1 Charged track selection Each charged track must: (1) have a good helix fit in order to ensure a correct error matrix in the kinematic fit; (2) originate from the interaction region, $\sqrt{V^{2}_{x}+V^{2}_{y}}<2$ cm and $\|V_{z}\|<20$ cm, where $V_{x}$, $V_{y}$, and $V_{z}$ are the $x$, $y$ and $z$ coordinates of the point of closest approach of the track to the beam axis; (3) have a transverse momentum greater than 70 MeV/$c$; and (4) have $\|\cos\theta\|\leq 0.80$, where $\theta$ is the polar angle of the track. ### III.2 Photon selection A neutral cluster in the BSC is assumed to be a photon candidate if the following requirements are satisfied: (1) the energy deposited in the BSC is greater than 0.05 GeV; (2) energy is deposited in more than one layer; (3) the angle between the direction of photon emission and the direction of shower development is less than $30^{\circ}$; and (4) the angle between the photon and the nearest charged track is greater than $5^{\circ}$ (if the charged track is an antiproton, the angle is required to be great than $25^{\circ}$). ### III.3 Particle Identification (PID) For each charged track in an event, $\chi^{2}_{PID}(i)$ is determined using both $dE/dx$ and TOF information: $\chi^{2}_{PID}(i)=\chi^{2}_{dE/dx}(i)+\chi^{2}_{TOF}(i)$, where $i$ corresponds to the particle hypothesis. A charged track is identified as a pion if $\chi^{2}_{PID}$ for the $\pi$ hypothesis is less than those for the kaon and proton hypotheses. For $p$ or $\bar{p}$ identification, the same method is used. In this analysis, all charged tracks are required to be positively identified. ## IV Analysis of $J/\psi\rightarrow p\bar{p}\eta$ The decay modes for the $J/\psi\rightarrow p\bar{p}\eta$ measurement are $\eta\rightarrow\gamma\gamma$ and $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$. The use of different decay modes allows us to cross check our measurements, as well as to obtain higher statistical precision. Figure 1: The two-photon invariant mass distribution for $J/\psi\rightarrow p\bar{p}\gamma\gamma$ candidate events. Data are represented by rectangles; the error bars are too small to be seen. The curves are the results of the fit described in the text. The shaded part is background from MC simulation. ### IV.1 $\eta\rightarrow\gamma\gamma$ Events with two charged tracks and two photons are selected. A four-constraint (4C) kinematic fit is performed to the hypothesis $J/\psi\rightarrow p\bar{p}\gamma\gamma$. For events with more than two photons, all combinations are tried, and the combination with the smallest $\chi^{2}$ is retained. $\chi^{2}_{\gamma\gamma p\bar{p}}$ is required to be less than 20. The $\gamma\gamma$ invariant mass $(m_{\gamma\gamma})$ distribution for selected events is shown in Fig. 1. A peak around the $\eta$ mass is evident. The curves in the figure indicate the best fit to the signal and background. The shaded part is the background estimated from a MC simulation of inclusive $J/\psi$ events LUND . The main background comes from $J/\psi\rightarrow p\bar{p}\pi^{0}\pi^{0}$ and $\Sigma^{+}\bar{\Sigma}^{-}$. By fitting the $\eta$ signal with a MC-simulated signal histogram plus a third order polynomial background function, the number of $\eta$ signal events is determined to be $(12220\pm 149)$. For the signal MC simulation, the events are generated with a proton angle distribution of $1+\alpha\cos^{2}\theta$, where $\alpha$ is taken to be -0.6185 in order to describe the data. In the decay, intermediate resonances, N(1440), N(1535), N(1650), and N(1800) and antiparticles, with fractional contribution of $(8\pm 4)$%, $(56\pm 15)$%, $(24^{+5}_{-15})$%, and $(12\pm 7)$% LHBD , respectively, are included. The resulting detection efficiency for $J/\psi\rightarrow p\bar{p}\eta$ $(\eta\rightarrow\gamma\gamma)$ is determined to be $28.70$%. The $p\bar{p}\eta$ branching fraction, calculated using $\displaystyle B(J/\psi\rightarrow p\bar{p}\eta)=\frac{N_{obs}}{\epsilon\cdot N_{J/\psi}\cdot B(\eta\rightarrow 2\gamma)\cdot f_{1}},$ is $(1.93\pm 0.02)\times 10^{-3}$, where the error is statistical only. Here $N_{obs}$ represents the number of observed events, $\epsilon$ is the detection efficiency for $J/\psi\rightarrow p\bar{p}\eta(\eta\to 2\gamma)$, $f_{1}=0.9739$ is the efficiency correction factor (see Section VI), and $N_{J/\psi}$ is the total number of $J/\psi$ events. Figure 2: The $\pi^{+}\pi^{-}\pi^{0}$ invariant mass distribution for $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}\pi^{0}$ candidate events. The curves are results of the fit described in the text. The shaded part is background from MC simulation. ### IV.2 $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$ Similar to the above analysis, events with four charged tracks and two photons are selected. A 4C kinematic fit is performed to the $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}\gamma\gamma$ hypothesis, and the $\chi^{2}_{\gamma\gamma p\bar{p}\pi^{+}\pi^{-}}$ value is required to be less than 20. In order to suppress multi-photon backgrounds, the number of photons is required to be two. The invariant mass of the $\gamma\gamma$ is required to be between 0.095 and 0.175 GeV/$c^{2}$. The $\pi^{+}\pi^{-}\pi^{0}$ invariant mass $(m_{\pi^{+}\pi^{-}\pi^{0}})$ distribution is shown in Fig. 2, where a peak at the $\eta$ mass is observed. The curves in the figure are the results of a fit to the signal and background. The shaded part is background estimated from MC simulation of inclusive $J/\psi$ decay events LUND . Here the main background comes from $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}\pi^{0}$ and $p\bar{p}\pi^{+}\pi^{-}\gamma$ decays. By fitting the distribution with a MC simulated signal histogram plus a third order polynomial background function, $(954\pm 45)$ signal events are obtained. Similar to the $\eta\rightarrow 2\gamma$ decay, contributions from the baryon excited states N(1440), N(1535), N(1650), and N(1800), as well as their anti-particles LHBD , are considered. The detection efficiency of $J/\psi\rightarrow p\bar{p}\eta$ $(\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$, $\pi^{0}\rightarrow\gamma\gamma)$ is determined to be $4.20$%. The branching fraction is determined from the calculation $\displaystyle B(J/\psi\rightarrow p\bar{p}\eta)$ $\displaystyle=$ $\displaystyle\frac{N_{obs}}{\epsilon\cdot N_{J/\psi}\cdot B(\eta\rightarrow\pi^{+}\pi^{-}\pi^{0})}$ $\displaystyle\cdot$ $\displaystyle\frac{1}{B(\pi^{0}\rightarrow\gamma\gamma)\cdot f_{2}},$ where $f_{2}=0.9582$ is a correction factor for the efficiency that is described below in Section VI. We determine a branching fraction for $J/\psi\rightarrow p\bar{p}\eta$ of $(1.83\pm 0.09)\times 10^{-3}$, where the error is statistical only. Figure 3: The $\pi^{+}\pi^{-}\eta$ invariant mass distribution for $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}\eta$ candidate events. The curves are results of the fit described in the text. The shaded part is background from MC simulation. ## V Analysis of $J/\psi\rightarrow p\bar{p}\eta^{\prime}$ There are three main decay modes of the $\eta^{\prime}$: $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$, $\eta^{\prime}\rightarrow\gamma\rho^{0}$ and $\eta^{\prime}\rightarrow\pi^{0}\pi^{0}\eta$. Here the first two decay modes are used. ### V.1 $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$, $\eta\rightarrow\gamma\gamma$ In the search for $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ decays, events with four charged tracks and two photons are selected. A five-constraint (5C) kinematic fit is performed to the hypothesis of $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}\gamma\gamma$, in which the $2\gamma$ invariant mass is constrained to equal the $\eta$ mass, and the $\chi^{2}_{\gamma\gamma p\bar{p}\pi^{+}\pi^{-}}$ value is required to be less than 20. The $\pi^{+}\pi^{-}\eta$ invariant mass $(m_{\pi^{+}\pi^{-}\eta})$ distribution for events that survive the selection criteria is shown in Fig. 3. A clear $\eta^{\prime}$ signal is observed. The curves in the figure are the best fit to the signal and background. The shaded part is background estimated from MC simulation of inclusive $J/\psi$ decay events LUND . The main background comes from $J/\psi\rightarrow\Delta^{+}\bar{\Delta}^{-}\eta$, and $\Delta^{0}\bar{\Delta}^{0}\eta$ decays. By fitting the distribution with a MC simulated signal histogram plus a third order polynomial background function, a signal yield of $(65\pm 12)$ events is observed. According to a MC simulation, in which the events are generated with uniform phase space, the detection efficiency of $J/\psi\rightarrow p\bar{p}\eta^{\prime}$ $(\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$, $\eta\rightarrow\gamma\gamma)$ is $3.38$%. The effect of intermediate resonances is considered as a source of systematic error. Using $\displaystyle B(J/\psi\rightarrow p\bar{p}\eta^{\prime})$ $\displaystyle=$ $\displaystyle\frac{N_{obs}}{\epsilon\cdot N_{J/\psi}\cdot B(\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta)}$ $\displaystyle\cdot$ $\displaystyle\frac{1}{B(\eta\rightarrow\gamma\gamma)\cdot f_{3}}$ with $f_{3}=0.8228$ being the efficiency correction factor (see Section VI), we determine the branching fraction for $J/\psi\rightarrow p\bar{p}\eta^{\prime}$ to be $(2.31\pm 0.43)\times 10^{-4}$, where the error is statistical only. Figure 4: The $\gamma\pi^{+}\pi^{-}$ invariant mass distribution for $J/\psi\rightarrow p\bar{p}\gamma\pi^{+}\pi^{-}$ candidate events. The curves are results of the fit described in the text. The shaded part is background from MC simulation. ### V.2 $\eta^{\prime}\rightarrow\gamma\rho^{0},\rho^{0}\rightarrow\pi^{+}\pi^{-}$ In order to select $\eta^{\prime}\rightarrow\gamma\rho^{0}$, a 4C kinematic fit is performed under the hypothesis of $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}\gamma$. The $\chi^{2}_{\gamma p\bar{p}\pi^{+}\pi^{-}}$ value is required to be less than 20. To ensure the events are from $\gamma\rho^{0}$, a $\|m_{\pi^{+}\pi^{-}}-m_{\rho}\|<0.20$ GeV/$c^{2}$ requirement is imposed, where $m_{\pi^{+}\pi^{-}}$ is the $\pi^{+}\pi^{-}$ invariant mass, and $m_{\rho}$ is the $\rho$ mass. In order to exclude the background from $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}$, it is required that the invariant mass of the four charged tracks is less than 3.02 GeV/$c^{2}$. The $\gamma\rho^{0}$ invariant mass $(m_{\gamma\rho^{0}})$ distribution for selected events, where a clear $\eta^{\prime}$ signal is observed, is shown in Fig. 4. The curves in the figure are the best fit to the signal and background. The shaded part is the background estimated from MC simulation of inclusive $J/\psi$ decay events LUND . The main background comes from $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}\gamma$, ${\Delta}^{++}\bar{\Delta}^{--}\pi^{0}$ and $p\bar{p}\pi^{+}\pi^{-}\pi^{0}$ decays. By fitting the $m_{\gamma\pi^{+}\pi^{-}}$ distribution with a MC simulated signal shape and a third order polynomial background function, we determine the number of $\eta^{\prime}$ signal events to be $(200\pm 29)$. The detection efficiency for $J/\psi\rightarrow p\bar{p}\eta^{\prime}$ $(\eta^{\prime}\rightarrow\gamma\rho^{0})$ is determined to be $7.48$%, assuming phase space production, where the $\pi^{+}\pi^{-}$ mass distribution is generated according to measurements from$J/\psi\rightarrow\phi\eta^{\prime},\eta^{\prime}\rightarrow\gamma\pi^{+}\pi^{-}$ FPRD71 . Using $\displaystyle B(J/\psi\rightarrow p\bar{p}\eta^{\prime})$ $\displaystyle=$ $\displaystyle\frac{N_{obs}}{\epsilon\cdot N_{J/\psi}\cdot B(\eta^{\prime}\rightarrow\gamma\rho^{0})\cdot f_{4}}$ with the $f_{4}$ correction factor of 0.8522 (see Section VI). The resulting branching fraction for $J/\psi\rightarrow p\bar{p}\eta^{\prime}$ is $(1.85\pm 0.27)\times 10^{-4}$, where the error is statistical only. Table 1: Numbers used in the calculations of branching fractions and results. Decay mode \| $N_{obs}$ \| $\epsilon(\%)$ \| Branching fraction ---\|---\|---\|--- $J/\psi\rightarrow p\bar{p}\eta,\eta\rightarrow\gamma\gamma$ \| $12220\pm 149$ \| $28.70$ \| $B(J/\psi\rightarrow p\bar{p}\eta)=(1.93\pm 0.02\pm 0.18)\times 10^{-3}$ $J/\psi\rightarrow p\bar{p}\eta,\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$ \| $954\pm 45$ \| $4.20$ \| $B(J/\psi\rightarrow p\bar{p}\eta)=(1.83\pm 0.09\pm 0.24)\times 10^{-3}$ $J/\psi\rightarrow p\bar{p}\eta^{\prime},\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ \| $65\pm 12$ \| $3.38$ \| $B(J/\psi\rightarrow p\bar{p}\eta^{\prime})=(2.31\pm 0.43\pm 0.34)\times 10^{-4}$ $J/\psi\rightarrow p\bar{p}\eta^{\prime},\eta^{\prime}\rightarrow\gamma\rho^{0}$ \| $200\pm 29$ \| $7.48$ \| $B(J/\psi\rightarrow p\bar{p}\eta^{\prime})=(1.85\pm 0.27\pm 0.31)\times 10^{-4}$ ## VI Systematic errors In our analysis, the systematic errors on the branching fractions come from the uncertainties in the MDC tracking, photon efficiency, particle identification, photon identification, kinematic fit, background shapes, hadronic interaction model, intermediate decay branching fraction, the $\pi^{0}$ and $\rho$ selection requirements, intermediate resonance states, and the total number of $J/\psi$ events. The errors from the different sources are listed in Table 2. The MDC tracking efficiency has been measured using $J/\psi\rightarrow\rho\pi$, $\Lambda\bar{\Lambda}$, and $\psi(2S)\rightarrow\pi^{+}\pi^{-}J/\psi$, $J/\psi$ to $\mu^{+}\mu^{-}$. The MC simulation agrees with data within 1 to 2$\%$ for each charged track NIM . Thus $4\%$ is regarded as the systematic error for the two charged-track mode, and $8\%$ for the four charged-track final states. The photon detection efficiency has been studied using a sample of $J/\psi\rightarrow\rho\pi$ NIM decays; the difference between data and MC simulation is about $2\%$ for each photon. In this analysis, $2\%$ is included in the systematic error for one-photon modes and $4\%$ for two-photon modes. The charged pion PID efficiency has been studied using $J/\psi\rightarrow\rho\pi$ decays NIM . The PID efficiency from data is in good agreement with that from MC simulation with an average difference that is less than $1\%$ for each charged pion. Here $2\%$ is taken as the systematic error for identifying two pions. The proton PID efficiencies have been studied using $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}$ decays. The main difference between data and MC simulation occurs for tracks with momentum less than 0.35 GeV/$c$. We determine a weighting factor for identifying a proton or anti-proton as a function of momentum from studies of the $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}$ channel. After considering the weight of each particle in an event, the difference between data and MC is determined to be $\frac{\epsilon_{DT}}{\epsilon_{MC}}=0.9739\pm 0.0078$ for $\eta\rightarrow 2\gamma$, $0.9582\pm 0.0199$ for $\eta\rightarrow 3\pi$, $0.8228\pm 0.0211$ for $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$, and $0.8522\pm 0.0140$ for $\eta^{\prime}\rightarrow\gamma\rho^{0}$. We take $f_{1}=0.9739$, $f_{2}=0.9582$, $f_{3}=0.8228$, and $f_{4}=0.8522$ as efficiency correction factors for the corresponding decay channel, and $0.8\%$, $2.1\%$, $2.6\%$, and $1.6\%$ are taken as the errors associated with identifying protons and anti-protons, respectively. The PID systematic errors for the four decay modes are listed in Table 2. For the systematic error of photon ID, which arises mainly from the simulation of fake photons, $p\bar{p}$ and $J/\psi\to p\bar{p}\pi^{+}\pi^{-}$ data samples were selected and $10^{5}$ simulated $p\bar{p}$ and $J/\psi\to p\bar{p}\pi^{+}\pi^{-}$ events were generated, with real and fake photons. The decay mode $J/\psi\to p\bar{p}$ is used for the photon ID systematic error of $J/\psi\to p\bar{p}\eta$ $(\eta\to 2\gamma)$, and the decay mode $J/\psi\to p\bar{p}\pi^{+}\pi^{-}$ for $J/\psi\to p\bar{p}\eta$ $(\eta\to 3\pi)$ and $J/\psi\to p\bar{p}\eta^{\prime}$. From the decay mode $J/\psi\to p\bar{p}$, the fake photon differences between data and MC is about $2.0\%$, while for the decay mode $J/\psi\to p\bar{p}\pi^{+}\pi^{-}$, the difference is $1.6\%$. Here $2.0\%$ is taken as the systematic error associated with photon ID for the decay mode determined to be $J/\psi\to p\bar{p}\eta$ $(\eta\to\gamma\gamma)$, and $1.6\%$ for the decay modes $J/\psi\to p\bar{p}\eta$ $(\eta\to 3\pi)$ and $J/\psi\to p\bar{p}\eta^{\prime}$. In Ref. PRD7 , the uncertainty of the 4C kinematic fit is $4\%$, which we include here in the systematic error. The uncertainty of the 5C kinematic fit is $4.1\%$ in Ref. RHOP . Here we conservatively take $5\%$ as the systematic error from the 5C kinematic fit for the decay mode $\eta^{\prime}\to\pi^{+}\pi^{-}\eta$. The systematic errors of the background uncertainty is obtained by changing the range of the fit and varying the order of the polynomial background. The errors range from 0.8$\%$ to 7.3$\%$ in all decay modes (see Table 2 for detail). There are two models, FLUKA and GCALOR, used for simulating hadronic interactions; the different models lead to different detection efficiencies. The difference between them is regarded as a systematic error. For the decay $J/\psi\to p\bar{p}\eta$ $(\eta\rightarrow 2\gamma)$, the difference is very small and negligible. For the other decay modes, it is about 1.4$\%$ for $J/\psi\rightarrow p\bar{p}\eta$ $(\eta\to\pi^{+}\pi^{-}\pi^{0})$, $J/\psi\rightarrow p\bar{p}\eta^{\prime}$ $(\eta^{\prime}\to\pi^{+}\pi^{-}\eta)$, and 5.2$\%$ for $J/\psi\rightarrow p\bar{p}\eta^{\prime}$ $(\eta^{\prime}\to\gamma\rho^{0})$. The branching fractions for the decays $\pi^{0}\rightarrow 2\gamma$, $\eta\rightarrow 2\gamma$, $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$, $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$, and $\eta^{\prime}\rightarrow\gamma\rho$ are taken from the PDG PDG . The errors on these branching fractions are systematic errors in our measurements. For the $\eta\to 3\pi$ mode, the $\pi^{0}$ mass is required to satisfy $\|M_{\gamma\gamma}-M_{\pi^{0}}\|<0.04$ GeV/$c^{2}$. To study the systematic error associated with this requirement, $\pi^{0}$ samples are selected and simulated using $J/\psi\to\rho\pi$, and the data and MC efficiencies in the 3$\sigma$ signal region are compared with using the requirement or not, the difference is about $1\%$. Here it is taken as the systematic error caused by the $\pi^{0}$ requirement. For the $\eta^{\prime}\to\gamma\rho$ mode, we require that $\|M_{\pi^{+}\pi^{-}}-M_{\rho}\|<0.20$ GeV/$c^{2}$. According to Ref. FANGP , the uncertainty associated with this requirement is $5.9\%$. Here we take this as the systematic error for the $\rho$ mass requirement. In the signal MC simulation, we assume the presence of N(1440), N(1535), N(1650), and N(1800) in the $p\bar{p}\eta$ channel. If some of these resonances are not included, the efficiency of this channel changes. These differences are taken as the systematic error associated with possible intermediate states. The total systematic error associated with this is taken as the sum added in quadrature. For the decay modes with an $\eta^{\prime}$, we take the difference in efficiency determined assuming the decay proceeds via an intermediate $N(2090)$ resonance compared with phase space generation as the systematic error (see Table 2 for detail). The uncertainty of the total number of $J/\psi$ events is $4.7\%$ FSS . Combining all errors in quadrature gives total systematic errors of $9.3\%$ for $\eta\rightarrow\gamma\gamma$, $12.9\%$ for $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$, $14.8\%$ for $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$, and $16.6\%$ for $\eta^{\prime}\rightarrow\gamma\rho$. Table 2: Summary of systematic errors; “-” means no contribution. Sources \| \| Relative error ($\%$) \| \| ---\|---\|---\|---\|--- Decay modes \| $\eta\rightarrow 2\gamma$ \| $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$ \| $\eta^{\prime}\rightarrow\pi^{+}\pi^{-}\eta$ \| $\eta^{\prime}\rightarrow\gamma\rho^{0}$ MDC tracking \| 4 \| 8 \| 8 \| 8 Photon detection efficiency \| 4 \| 4 \| 4 \| 2 Particle ID \| $\sim$1 \| 4.1 \| 4.6 \| 3.6 Photon ID \| 2.0 \| 1.6 \| 1.6 \| 1.6 Kinematic fit \| 4.0 \| 4.0 \| 5.0 \| 4.0 Background uncertainty \| $\sim$1 \| 3.1 \| 7.3 \| 5.8 Hadronic Interaction Model \| $\sim$0 \| 1.4 \| 1.4 \| 5.2 Intermediate decay Br. Fr. \| $\sim$ 1 \| 1.2 \| 3.1 \| 3.1 $\pi^{0}$ selection \| - \| $\sim$1 \| - \| $\rho$ selection \| - \| - \| - \| 5.9 Intermediate resonances \| 3.0 \| 4.0 \| 2.0 \| 7.1 Number of $J/\psi$ events \| 4.7 \| 4.7 \| 4.7 \| 4.7 Total systematic error \| 9.3 \| 12.9 \| 14.8 \| 16.6 ## VII Results Table 1 shows the branching fractions of the two channels into their different decay modes; the first error is statistical and the second is systematic. The results for the different decay modes in the same channel are consistent within errors and are combined after taking out the common systematic errors (8.37 % for the $\eta$ mode and 10.8% for the $\eta^{{}^{\prime}}$ mode): $\displaystyle Br(J/\psi\rightarrow p\bar{p}\eta)=(1.91\pm 0.17)\times 10^{-3},$ $\displaystyle Br(J/\psi\rightarrow p\bar{p}\eta^{\prime})=(2.00\pm 0.36)\times 10^{-4}.$ In comparison with previous measurements of $J/\psi\to p\bar{p}\eta$ and $J/\psi\to p\bar{p}\eta^{\prime}$, the present results are of higher precision. Using the result of $J/\psi\to p\bar{p}\eta$ from this analysis and that of $J/\psi\to p\bar{p}$ in Ref. LXLM , we determine: $\displaystyle\frac{\Gamma(J/\psi\rightarrow p\bar{p}\eta)}{\Gamma(J/\psi\rightarrow p\bar{p})}=0.85\pm 0.08.$ This is consistent with the calculation based on the soft-pion theorem, and indicates that the contribution of $N^{}$\- pole diagram is dominant for the $J/\psi\to p\bar{p}\eta$ mode. ## VIII Acknowledgments The BES collaboration thanks the staff of BEPC and computing center for their hard efforts. This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 10491300, 10225524, 10225525, 10425523, 10625524, 10521003, 10821063, 10825524, the Chinese Academy of Sciences under contract No. KJ 95T-03, the 100 Talents Program of CAS under Contract Nos. U-11, U-24, U-25, and the Knowledge Innovation Project of CAS under Contract Nos. U-602, U-34 (IHEP), the National Natural Science Foundation of China under Contract No. 10225522 (Tsinghua University), and the Department of Energy under Contract No. DE-FG02-04ER41291 (U. Hawaii). ## References * (1) P. Wang, C. Z. Yuan, X. H. Mo, Phys. Rev. D 70 (2004) 114014. * (2) Rahul Sinha and Susumu Okubo, Phys. Rev. D 30 (1984) 2333. * (3) L. Köpke and N. Wermes, $J/\psi$ Decays, CERN, CH-1211 Geneva 23 Switzerland. * (4) L. Adler and R. F. Dashen, Current Algebra and Application to Particle Physics (Benjamin, New York, 1968); B. W. Lee Chiral Dynamics (Gordon and Breach, New York, 1972). * (5) J. Z. Bai et al., Nucl. Instrum. Meth. A 458 (2001) 627. * (6) CERN Application Software Group, GEANT Detector Description and simulation Tool, CERN Program Library Writeup W5013, Geneva (1994). * (7) M. Ablikim et al., Nucl. Instrum. Meth. A 552 (2005) 344. * (8) J. C. Chen et al., Phys. Rev. D 62 (2000) 034003. * (9) J. Z. Bai et al., Phys. Lett. B 510 (2001) 75. * (10) M.Ablikim et al., Phys. Rev. D 71 (2005) 032003. * (11) M.Ablikim et al., Phys. Rev. D 74 (2006) 012004. * (12) J. Z. Bai et al., Phys. Rev. D 70 (2004) 012005. * (13) Particle Data Group, C. Amsler et al., Phys. Lett. B 667 (2008) issue 1-5. * (14) M.Ablikim et al., Phys. Rev. Lett. 95 (2005) 262001. * (15) S. S. Fang et al., High Energy Phys. Nucl. Phys. 27 (2003) 277 (in Chinese). * (16) J. Z. Bai et al., Phys. Lett. B 591 (2004) 42.	arxiv-papers	2009-02-20T02:16:54	2024-09-04T02:49:00.722809	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "BES collaboration", "submitter": "Bingxin Zhang", "url": "https://arxiv.org/abs/0902.3501" }
0902.3614	size [PostScript=dvips,height=2em,width=3.0em,tight,midshaft,balance,dpi=80,heads=vee,labelstyle=,] ++$\parallel$$\parallel$====red—-¿ redexistsdashdash¿ redpara–++-¿ equal===== antired¡—- Syntactic Confluence Criteria for Positive/Negative-Conditional Term Rewriting Systems Claus-Peter Wirth SEKI Report SR–95–09 Searchable Online Edition July 23, 1995 Revised March 6, 1996 (“noetherian” replaced with “terminating”. Example 14.8 added.) Revised January and October 2005 Universität Kaiserslautern Fachbereich Informatik D-67663 Kaiserslautern Abstract: We study the combination of the following already known ideas for showing confluence of unconditional or conditional term rewriting systems into practically more useful confluence criteria for conditional systems: Our syntactic separation into constructor and non-constructor symbols, Huet’s introduction and Toyama’s generalization of parallel closedness for non- terminating unconditional systems, the use of shallow confluence for proving confluence of terminating and non-terminating conditional systems, the idea that certain kinds of limited confluence can be assumed for checking the fulfilledness or infeasibility of the conditions of conditional critical pairs, and the idea that (when termination is given) only prime superpositions have to be considered and certain normalization restrictions can be applied for the substitutions fulfilling the conditions of conditional critical pairs. Besides combining and improving already known methods, we present the following new ideas and results: We strengthen the criterion for overlay joinable terminating systems, and, by using the expressiveness of our syntactic separation into constructor and non-constructor symbols, we are able to present criteria for level confluence that are not criteria for shallow confluence actually and also able to weaken the severe requirement of normality (stiffened with left-linearity) in the criteria for shallow confluence of terminating and non-terminating conditional systems to the easily satisfied requirement of quasi-normality. Finally, the whole paper also gives a practically useful overview of the syntactic means for showing confluence of conditional term rewriting systems. This research was supported by the Deutsche Forschungsgemeinschaft, SFB 314 (D4-Projekt) ###### Contents 1. 1 Introduction and Overview 2. 2 Positive/Negative-Conditional Rule Systems 3. 3 Confluence 4. 4 Critical Peaks 5. 5 Basic Forms of Joinability of Critical Peaks 6. 6 Basic Forms of Shallow and Level Joinability 7. 7 Sophisticated Forms of Shallow Joinability 8. 8 Sophisticated Forms of Level Joinability 9. 9 Quasi Overlay Joinability 10. 10 Some Unconditional Examples 11. 11 Normality 12. 12 Counterexamples for Closed Systems 13. 13 Criteria for Confluence 14. 14 Criteria for Confluence of Terminating Systems 15. 15 Criteria for Confluence of the Constructor Sub-System 16. A Further Lemmas for Section 13 17. B Further Lemmas for Section 14 18. C $\omega$-Coarse Level Joinability 19. D The Proofs ## 1 Introduction and Overview While111Please do try not to read the footnotes for a first reading! powerful confluence criteria for conditional term rewriting systems222For an introduction to the subject cf. Avenhau & Madlener (1989) or Klop (1992). are in great demand and while there are interesting new results for unconditional systems333Cf. Oostrom (1994a) and Oostrom (1994b). Note that the lemmas 5.1 and 5.3 of Oostrom (1994b) do not apply for conditional systems because they are not subsumed by the notion of “patterm rewriting systems” of Oostrom (1994b)., hardly any new results on confluence of conditional term rewriting systems (besides some on modularity444Cf. Middeldorp (1993), Gramlich (1994). and on the treatment of extra-variables in conditions555Cf. Avenhau & Loría- Sáenz (1994) for the case of decreasing systems and Suzuki &al. (1995) for the case of orthogonal systems.) have been published since Dershowitz &al. (1988), Toyama (1988), and Bergstra & Klop (1986), and not even a common generalization (as given by our theorems 13.6 and 15.1) of the main confluence theorems of the latter two papers (i.e. something like confluence of parallel closed conditional systems) has to our knowledge been published. We guess that this is due to the following problems: 1. 1. A proper treatment is very tedious and technically most complicated, especially in the case of non-terminating reduction relations.666The technique we apply for proving our confluence criteria for non-terminating reduction relations is in essence to show strong confluence of relations whose reflexive & transitive closures are equal to that of the reduction relation. In Bergstra & Klop (1986) another technique is used. Instead of an actual presentation of the proof there is only a pointer to Klop (1980). It would be worthwhile to reformulate this proof in modern notions (including path orderings) and notations. While we did not do this, we just try to describe here the abstract global idea of this proof: The field of the reduction relation is changed from terms to terms with licenses in such a way that the projection to terms just yields the original reduction relation again. The transformed reduction relation becomes terminating since it consumes and inherits licenses in a wellfounded manner; thus its confluence is implied by its local confluence that is to be shown. Finally, each diverging peak of the original reduction relation is a projection of a diverging peak in the transformed reduction relation when one only provides enough licenses. We did not apply this global proof idea since (while we were able to generalize it for allowing parallel closed critical pairs as in the corollary on page 815 in Huet (1980)) we were not able to generalize it for proving Corollary 3.2 of Toyama (1988) (which generalizes this corollary of Huet (1980)). 2. 2. There is a big gap between the known criteria and those criteria that are supposed to be true, even for unconditional systems.777Cf. e.g. Problem 13 of Dershowitz &al. (1991). 3. 3. The usual framework for conditional term rewriting systems does not allow us to model some simple and straightforward applications naturally in such a way that the resulting reduction relation is known to be confluent, unless some sophisticated semantic or termination knowledge is postulated a priori. 4. 4. For conditional rule systems there is another big gap between the known criteria and those criteria that are required for practical purposes. This results from the difficulty to capture (with effective means) the infinite number of substitutions that must be tested for fulfilling the conditions of critical pairs. While we are not able to contribute too much regarding the first two problems, we are able to present some progress with the latter two. Our positive/negative-conditional rule systems including a syntactic separation between constructor and non-constructor symbols as presented in Wirth & Gramlich (1994a) offer more expressive power than the standard positive conditional rule systems and therefore allow us to model more applications naturally in such a way that their confluence is given by the new confluence criteria presented in this paper. Using the separation into constructor and non-constructor rules (generated by the syntactic separation into constructor and non-constructor function symbols) it is possible to divide the problem of showing confluence of the whole rule system into three smaller sub-problems, namely confluence of the constructor rules, confluence of the non-constructor rules, and their commutation. The important advantage of this modularization is not only the division into smaller problems, but is due to the possibility to tackle the sub-problems with different confluence criteria. E.g., when confluence of the constructor rules is not trivial then its confluence often can only be shown by sophisticated semantical considerations or by criteria that are applicable to terminating systems only. For the whole rule system, however, neither semantic confluence criteria nor confluence criteria requiring termination of the reduction relation are practically feasible in general. This is because, on the one hand, an effective application of semantic confluence criteria requires that the specification given by the whole rule system has actually been modeled before in some formalism. On the other hand, termination of the whole rule system may not be given or difficult to be shown without some confluence assumptions.888When termination is assumed, there are approaches to prove confluence automatically, cf. Becker (1993) and Becker (1994). Fortunately, without requiring termination of the whole rule system the syntactic confluence criteria999Cf. our theorems 13.3, 13.4, and 15.3. presented in this paper guarantee confluence of the non-constructor rules of a class of rule systems that is sufficient for practical specification. This class of rule systems generalizes the function specification style used in the framework of classic inductive theorem proving101010Cf. Walther (1994). Note that we can even keep the notation style similar to this function specification style, cf. Wirth & Lunde (1994). by allowing of partial functions resulting from incomplete specification as well as from non-termination. Together with the notions of inductive validity presented in Wirth & Gramlich (1994b) this extends the area of semantically clearly understood inductive specification considerably. Regarding the last problem of the above problem list (occuring in case of conditional rule systems), by carefully including the invariants of the proofs for the confluence criteria into the conditions of the joinability tests for the conditional critical pairs we allow of more reasoning on those substitutions that fulfill the condition of a critical pair. E.g. consider the following example: ###### Example 1.1 Let R: $\begin{array}[t]{lllll}{{\mathsf{f}}{(}{{{\mathsf{s}}{(}{{{\mathsf{s}}{(}{x}{)}}}{)}}}{)}}&{=}&{{\mathsf{s}}{(}{{{\mathsf{0}}}}{)}}&{\longleftarrow}&{{\mathsf{f}}{(}{x}{)}}{=}{{\mathsf{0}}}\\\ {{\mathsf{f}}{(}{{{\mathsf{s}}{(}{{{\mathsf{s}}{(}{x}{)}}}{)}}}{)}}&{=}&{{\mathsf{0}}}&{\longleftarrow}&{{\mathsf{f}}{(}{x}{)}}{=}{{\mathsf{s}}{(}{{{\mathsf{0}}}}{)}}\\\ {{\mathsf{f}}{(}{{{\mathsf{s}}{(}{{{\mathsf{0}}}}{)}}}{)}}&{=}&{{\mathsf{s}}{(}{{{\mathsf{0}}}}{)}}\\\ {{\mathsf{f}}{(}{{{\mathsf{0}}}}{)}}&{=}&{{\mathsf{0}}}\\\ \end{array}$ Assume ${\mathsf{0}}$ and ${\mathsf{s}}{(}{{{\mathsf{0}}}}{)}$ to be irreducible. The experts may notice that the part of R we are given in this example is rather well-behaved: It is left-linear and normal; it may be decreasing; and the only critical pair is an overlay. Now, for showing the critical pair between the first two rules to be joinable, one has to show that it is impossible that both conditions hold simultaneously for a substitution $\\{x\\!\mapsto\\!t\\}$. One could argue the following way: If both conditions were fulfilled, then ${\mathsf{f}}{(}{t}{)}$ would reduce to ${\mathsf{0}}$ as well as to ${\mathsf{s}}{(}{{{\mathsf{0}}}}{)}$, which contradicts confluence below ${\mathsf{f}}{(}{t}{)}$. But, as our aim is to establish confluence, it is not all clear that we are allowed to assume confluence for the joinability test here. None of the theorems in Dershowitz &al. (1988) or Bergstra & Klop (1986) provides us with such a confluence assumption, even if their proofs could do so with little additional effort. For practical purposes, however, it is important that the joinability test allows us to assume a sufficient kind of confluence for the condition terms. Therefore, all our joinability notions provide us with sufficient assumptions that allow us to easily establish the infeasibility of the condition of a critical pair, without knowing the proofs for the confluence criteria by heart. This applies for example, when two rules with same left-hand side are meant to express a case distinction that is established by the condition of the one containing a condition literal “$p{=}{{\mathsf{true}}}$” or “$u{=}v$” and the condition of the other containing the condition literal “$p{=}{{\mathsf{false}}}$” or “$u{\not=}v$”.111111In Definition 4.4 of Avenhau & Loría-Sáenz (1994) the critical pair resulting from such two rules is called “infeasible” (in the case with “$p{=}{{\mathsf{true}}}$” and “$p{=}{{\mathsf{false}}}$”). We will call it “complementary” instead (in both cases), cf. Theorem 13.3. For terminating reduction relations we carefully investigate whether the joinability test can be restricted by certain irreducibility requirements, e.g. whether the substitutions which must be tested for fulfilling the conditions of critical pairs can be required to be normalized, cf. 14, esp. Example 14.3. The restrictions on the infinite number of substitutions for which the condition of a critical pair must be tested for fulfilledness may be a great help in practice. However, they do not solve the principle problem that the number of substitutions is still infinite. Another important point is that we weaken the severe restriction imposed on terminating systems by Theorem 2 of Dershowitz &al. (1988) and on non- terminating systems by Theorem 3.5 of Bergstra & Klop (1986), namely normality, which in our framework can be considerably weakened to the so- called quasi-normality, cf. our theorems 13.6 and 14.5. Moreover, besides these two criteria for shallow confluence, we present to our knowledge the first criteria for level confluence that are not criteria for shallow confluence actually121212as is the case with Suzuki &al. (1995)., cf. our theorems 13.9 and 14.6. Finally, we considerably improve the notion of “quasi overlay joinability” of Wirth & Gramlich (1994a), generalizing the notion of “overlay joinability” of Dershowitz &al. (1988). This results in a stronger criterion with a simpler proof, cf. 9 and Theorem 14.7. Since our main interest is in positive/negative-conditional rule systems with two kinds of variables and two kinds of function symbols as presented in Wirth &al. (1993) and Wirth & Gramlich (1994a), the whole paper is based on this framework. We know that this is problematic because the paper may also be of interest for readers interested in positive conditional rule systems with one kind of variables and function symbols only: With the exception of our generalization of normality to quasi-normality and our criteria for level confluence, our results also have interesting implications for this special case (which is subsumed by our approach). Nevertheless we prefer our more expressive framework for this presentation because it provides us with more power for most of our confluence criteria which is lost when restricting them to the standard framework. Therefore in the following section we are going to repeat those results of Wirth & Gramlich (1994a) which are essential for this paper. Those readers who are only interested in the implications of this paper for standard positive conditional rule systems with one kind of variables and function symbols should try to read only the theorems presented or pointed at in 15, which have been supplied with independent proofs for allowing a direct understanding. The contents of the other sections are explained by their titles. For a first reading sections 7 and 8 should only be skimmed and its definitions looked up by need. Due to their enormous length, most of the proofs have been put into D. We conclude this section with a list on where in this paper to find generalizations of known theorems: Parallel Closed + Left-Linear + Unconditional: The corollary on page 815 in Huet (1980) as well as Corollary 3.2 in Toyama (1988) are generalized by our theorems 13.6(I), 13.6(III), 13.6(IV), 13.9(I), 13.9(III), 13.9(IV), and 15.1(I). No Critical Pairs + Left-Linear + Normal: Theorem 3.5 in Bergstra & Klop (1986) as well as Theorem 1 in Dershowitz &al. (1988) are generalized by our theorems 13.3, 13.4, 13.6, 15.1, and 15.3. Strongly Joinable + Strong Variable Restriction: Lemma 3.2 of Huet (1980) as well as the translation of Theorem 5.2 in Avenhau & Becker (1994) into our framework is generalized by our theorems 13.6(II) and 13.9(II). Shallow Joinable + Left-Linear + Normal + Terminating: Theorem 2 in Dershowitz &al. (1988) is generalized by our theorems 14.5 and 15.4. Overlay Joinable + Terminating: Theorem 4 in Dershowitz &al. (1988) as well as Theorem 6.3 in Wirth & Gramlich (1994a) are generalized by our theorem 14.7. Joinable + Variable Restriction + Terminating: Theorem 7.18 in Wirth & Gramlich (1994a) is generalized by our theorem 14.4. Joinable + Decreasing: Theorem 3.3 in Kaplan (1987), Theorem 4.2 in Kaplan (1988), Theorem 3 in Dershowitz &al. (1988), as well as Theorem 7.17 in Wirth & Gramlich (1994a) are generalized by our theorems 14.2 and 14.4. ## 2 Positive/Negative-Conditional Rule Systems We use ‘$\uplus$’ for the union of disjoint classes and ‘id’ for the identity function. ‘${\bf N}$’ denotes the set of natural numbers and we define ${{\bf N}}_{+}:={{\\{\ }n{\,\in\,}{{\bf N}}}~{}{\|}\penalty-9\,\ {0{\,\not=\,}n{\ \\}}}.$ For classes $A,B$ we define: ${{\rm dom}({A})}:={{\\{\ }\\!a}~{}{\|}\penalty-9\,\ {\exists b{.}\penalty-1\,\,(a,b){\,\in\,}A\\!{\ \\}}};$ ${{\rm ran}({A})}:={{\\{\ }\\!b}~{}{\|}\penalty-9\,\ {\exists a{.}\penalty-1\,\,(a,b){\,\in\,}A\\!{\ \\}}};$ $B[A]:={{\\{\ }\\!b}~{}{\|}\penalty-9\,\ {\exists a{\,\in\,}A{.}\penalty-1\,\,(a,b){\,\in\,}B\\!{\ \\}}}.$ This use of “[…]” should not be confused with our habit of stating two definitions, lemmas, or theorems (and their proofs &c.) in one, where the parts between ‘[’ and ‘]’ are optional and are meant to be all included or all omitted. Furthermore, we use ‘$\emptyset$’ to denote the empty set as well as the empty function or empty word. ### 2.1 Terms and Substitutions Since our approach is based on the consequent syntactic distinction of constructors, we have to be quite explicit about terms and substitutions. We will consider terms of fixed arity over many-sorted signatures. A signature ${\rm sig}=({{\mathbb{F}}},{{\mathbb{S}}},{\alpha})$ consists of an enumerable set of function symbols ${\mathbb{F}}$, a finite set of sorts ${\mathbb{S}}$ (disjoint from ${\mathbb{F}}$), and a computable arity-function ${{{{\alpha}}:{{{{{\mathbb{F}}}}\rightarrow{{{\mathbb{S}}}^{+}}}}}}.$ For $f\in{{\mathbb{F}}}{.}\penalty-1\,\,$ ${\alpha}(f)$ is the list of argument sorts augmented by the sort of the result of $f$; to ease reading we will sometimes insert a ‘$\rightarrow$’ between a nonempty list of argument sorts and the result sort. A constructor sub-signature of the signature $({{\mathbb{F}}},{{\mathbb{S}}},{\alpha})$ is a signature ${\rm cons}=({{\mathbb{C}}\,},{{\mathbb{S}}},{{{}_{{{\mathbb{C}}\,}}{\upharpoonleft}{\alpha}}})$ such that the set ${\mathbb{C}}\,$ is a decidable subset of ${\mathbb{F}}$. ${\mathbb{C}}\,$ is called the set of constructor symbols; the complement ${{\mathbb{N}}}={{\mathbb{F}}}\setminus{{\mathbb{C}}\,}$ is called the set of non-constructor symbols. ###### Example 2.1 (Signature with Constructor Sub-Signature) $\begin{array}[t]{rclcl}{{\mathbb{C}}}&=&\lx@intercol\\{{{\mathsf{0}}},{\mathsf{s}},{{\mathsf{false}}},{{\mathsf{true}}},{{\mathsf{nil}}},{\mathsf{cons}}\\}\hfil\lx@intercol\\\ {{\mathbb{N}}}&=&\lx@intercol\\{{\mathsf{-}},{{\mathsf{mbp}}}\\}\hfil\lx@intercol\\\ {{\mathbb{S}}}&=&\lx@intercol\\{{\mathsf{nat}},{\mathsf{bool}},{\mathsf{list}}\\}\hfil\lx@intercol\\\ \\\ {\alpha}({{\mathsf{0}}})&=&&&{\mathsf{nat}}\\\ {\alpha}({\mathsf{s}})&=&{\mathsf{nat}}&{\ {\rightarrow}}&{\mathsf{nat}}\\\ \end{array}\hfill\begin{array}[t]{\|@{~~~}rclcl@{~~~~}}{}~{}~{}\lx@intercol\hfil{\alpha}({{\mathsf{false}}})&=&&&{\mathsf{bool}}\hfil~{}~{}~{}~{}\\\ {}~{}~{}\lx@intercol\hfil{\alpha}({{\mathsf{true}}})&=&&&{\mathsf{bool}}\hfil~{}~{}~{}~{}\\\ {}~{}~{}\lx@intercol\hfil{\alpha}({{\mathsf{nil}}})&=&&&{\mathsf{list}}\hfil~{}~{}~{}~{}\\\ {}~{}~{}\lx@intercol\hfil{\alpha}({\mathsf{cons}})&=&{\mathsf{nat}}\ {\mathsf{list}}&{\ {\rightarrow}}&{\mathsf{list}}\hfil~{}~{}~{}~{}\\\ {}~{}~{}\lx@intercol\hfil{\alpha}({\mathsf{-}})&=&{\mathsf{nat}}\ {\mathsf{nat}}&{\ {\rightarrow}}&{\mathsf{nat}}\hfil~{}~{}~{}~{}\\\ {}~{}~{}\lx@intercol\hfil{\alpha}({{\mathsf{mbp}}})&=&{\mathsf{nat}}\ {\mathsf{list}}&{\ {\rightarrow}}&{\mathsf{bool}}\hfil~{}~{}~{}~{}\\\ \end{array}$ To reduce declaration effort, in all examples (unless stated otherwise) in this and the following sections we will have only one sort; ‘${\mathsf{a}}$’, ‘${\mathsf{b}}$’, ‘${\mathsf{c}}$’, ‘${\mathsf{d}}$’, ‘${\mathsf{e}}$’, and ‘${\mathsf{0}}$’ will always be constants; ‘$\mathsf{s}$’, ‘$\mathsf{p}$’, ‘$\mathsf{f}$’, ‘$\mathsf{g}$’, and ‘$\mathsf{h}$’ will always denote functions with one argument; ‘$\mathsf{+}$’ and ‘$\mathsf{-}$’ take two arguments in infix notation; ‘$W$’, ‘$X$’, ‘$Y$’, ‘$Z$’ are variables from ${{\rm V}}\\!_{{\rm SIG}}$ (cf. below). A variable-system for a signature $({{\mathbb{F}}},{{\mathbb{S}}},{\alpha})$ is an ${\mathbb{S}}$-sorted family of decidable sets of variable symbols which are mutually disjoint and disjoint from ${\mathbb{F}}$. By abuse of notation we will use the symbol ‘$X$’ for an ${\mathbb{S}}$-sorted family to denote not only the family $X=(X_{s})_{s\in{{\mathbb{S}}}}$ itself, but also the union of its ranges: $\bigcup_{s\in{{\mathbb{S}}}}X_{s}.$ As the basis for our terms throughout the whole paper we assume two fixed disjoint variable-systems ${{{\rm V}}\\!_{{\rm SIG}}}$ of general variables and ${{{\rm V}}\\!_{{\mathcal{C}}}}$ of constructor variables such that ${{\rm V}}\\!_{{{\rm SIG}},{s}}$ as well as ${{\rm V}}\\!_{{{\mathcal{C}}},{s}}$ contain infinitely many elements for each $s\in{{\mathbb{S}}}$. ${{\mathcal{T}}({{\rm sig},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ denotes the ${\mathbb{S}}$-sorted family of all well-sorted (variable-mixed) terms over ‘sig/${{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}$’, while ${{\mathcal{GT}}({{\rm sig}})}$ denotes the ${\mathbb{S}}$-sorted family of all well-sorted ground terms over ‘sig’. Similarly, ${{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ denotes the ${\mathbb{S}}$-sorted family of all (variable-mixed) constructor terms, ${{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ denotes the ${\mathbb{S}}$-sorted family of all pure constructor terms, while ${{\mathcal{GT}}({{\rm cons}})}$ denotes the ${\mathbb{S}}$-sorted family of all constructor ground terms. To avoid problems with empty sorts, we assume ${\mathcal{GT}}({{\rm cons}})$ to have nonempty ranges only. We define ${{\rm V}}:=({{{\rm V}}\\!_{{\varsigma},{s}}})_{(\varsigma,s)\in{{\\{{\rm SIG},{\mathcal{C}}\\}}{\times}{{\mathbb{S}}}}}$ and call it a variable-system for a signature $({{\mathbb{F}}},{{\mathbb{S}}},{\alpha})$ with constructor sub-signature. We use ${{\mathcal{V}}}({A})$ to denote the ${\\{{\rm SIG},{\mathcal{C}}\\}}{\times}{{\mathbb{S}}}$-sorted family of variables occurring in a structure $A$ (e.g. a term or a set or list of terms). Let ${{\rm X}}\subseteq{{\rm V}}$ be a variable-system. We define ${{\mathcal{T}}({{{\rm X}}})}=({{\mathcal{T}}({{{\rm X}}})}_{\varsigma,s})_{(\varsigma,s)\in{{\\{{\rm SIG},{\mathcal{C}}\\}}{\times}{{\mathbb{S}}}}}$ by ($s\in{{\mathbb{S}}}$): ${{\mathcal{T}}({{{\rm X}}})}_{{\rm SIG},s}:={{\mathcal{T}}({{\rm sig},{{\rm X}}})}_{s}$ and ${{\mathcal{T}}({{{\rm X}}})}_{{\mathcal{C}},s}:={{\mathcal{T}}({{\rm cons},{{{\rm X}}_{{\mathcal{C}}}}})}_{s}.$ To avoid confusion: Note that ${{\mathcal{T}}({{{\rm X}}})}_{{\mathcal{C}},s}\subseteq{{\mathcal{T}}({{{\rm X}}})}_{{\rm SIG},s}$ for $s\in{{\mathbb{S}}}$, whereas ${{{\rm V}}\\!_{{{\mathcal{C}}},{s}}}\cap{{{\rm V}}\\!_{{{\rm SIG}},{s}}}=\emptyset.$ Furthermore we write $\mathcal{GT}$ for ${\mathcal{T}}({\emptyset})$ as well as $\mathcal{T}$ for ${\mathcal{T}}({{{\rm V}}})$. Our custom of reusing the symbol of a family for the union of its ranges now allows us to write ${\mathcal{T}}$ as a shorthand for ${{\mathcal{T}}({{\rm sig},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ For a term $t\in{\mathcal{T}}$ we denote by ${{\mathcal{POS}}}({t})$ the set of its positions (which are lists of positive natural numbers), by $t/p$ the subterm of $t$ at position $p$, and by $t\penalty-1{[\,p\leftarrow t^{\prime}\,]}$ the result of replacing $t/p$ with $t^{\prime}$ at position $p$ in $t$. We write ${{p}\,{\parallel}\,{q}}$ to express that neither p is a prefix of q, nor q a prefix of p. For $\mathchar 261\relax\subseteq{{{\mathcal{POS}}}({t})}$ with $\forall p,q{\,\in\,}\mathchar 261\relax{.}\penalty-1\,\,(p{\,=\,}\penalty-1q\ {\vee}\penalty-2\ {{p}\,{\parallel}\,{q}})$ we denote by $t\penalty-1{{[\,p\leftarrow t_{p}^{\prime}\ \|\ p{\,\in\,}\mathchar 261\relax\,]}}$ the result of replacing, for each $p\in\mathchar 261\relax$, the subterm at position $p$ in the term $t$ with the term $t_{p}^{\prime}$. $t$ is linear if $\forall p,q{\,\in\,}{{{\mathcal{POS}}}({t})}{.}\penalty-1\,\,(\mbox{$t/p\\!=\\!t/q\\!\in\\!{{\rm V}}$}\ {\Rightarrow}\penalty-2\ p\\!=\\!q)\ .$ The set of substitutions from ${\rm V}$ to a ${\\{{\rm SIG},{\mathcal{C}}\\}}{\times}{{\mathbb{S}}}$-sorted family of sets $T=(T_{\varsigma,s})_{(\varsigma,s)\in{{\\{{\rm SIG},{\mathcal{C}}\\}}{\times}{{\mathbb{S}}}}}$ is defined to be $\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}\rule{0.0pt}{8.43889pt}{{{\mathcal{SUB}}}({{{\rm V}}},{T})}:={{\\{\ }{{{\sigma}:{{{{{\rm V}}}\rightarrow{T}}}}}}~{}{\|}\penalty-9\,\ {\forall(\varsigma,s){\,\in\,}{{\\{{\rm SIG},{\mathcal{C}}\\}}{\times}{{\mathbb{S}}}}{.}\penalty-1\,\,\forall x{\,\in\,}{{{\rm V}}\\!_{{\varsigma},{s}}}{.}\penalty-1\,\,\sigma(x){\,\in\,}T_{\varsigma,s}{\ \\}}}.$ Note that $\forall\sigma{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{\mathcal{T}}})}{.}\penalty-1\,\,\forall(\varsigma,s){\,\in\,}{{\\{{\rm SIG},{\mathcal{C}}\\}}{\times}{{\mathbb{S}}}}{.}\penalty-1\,\,\forall t{\,\in\,}{\mathcal{T}\\!_{\varsigma,s}}{.}\penalty-1\,\,\ t\sigma{\,\in\,}{\mathcal{T}\\!_{\varsigma,s}}.$ Let $E$ be a finite set of equations and ${\rm X}$ a finite subset of ${\rm V}$. A substitution $\sigma$ $\in{{{\mathcal{SUB}}}({{{\rm V}}},{{\mathcal{T}}})}$ is called a unifier for $E$ if $E\sigma\subseteq{\rm id}.$ Such a unifier is called most general on ${\rm X}$ if for each unifier $\mu$ for $E$ there is some $\tau\in{{{\mathcal{SUB}}}({{{\rm V}}},{{\mathcal{T}}})}$ such that ${{{}_{{{\rm X}}}{\upharpoonleft}{(\sigma\tau)}}}={{{}_{{{\rm X}}}{\upharpoonleft}\mu}}.$ If $E$ has a unifier, then it also has a most general unifier131313For this most general unifier $\sigma$ we could, as usual, even require $\sigma\sigma=\sigma$ but not ${{{\mathcal{V}}}({\sigma[{{{\mathcal{V}}}({E})}]})}\ \subseteq\ {{{\mathcal{V}}}({E})}.$ on ${\rm X}$, denoted by ${\rm mgu}({E},{{{\rm X}}})$. ### 2.2 Relations Let ${{\rm X}}{\subseteq}{{\rm V}}$. Let ${\rm T}\subseteq{\mathcal{T}}$. A relation $R$ on $\mathcal{T}$ is called: > sort-invariant if $\forall(t,t^{\prime}){\,\in\,}R{.}\penalty-1\,\,\exists > s{\,\in\,}{{\mathbb{S}}}{.}\penalty-1\,\,t,t^{\prime}{\,\in\,}{\mathcal{T}\\!_{{\rm > SIG},s}}$ > > ${\rm X}$-stable (w.r.t. substitution) if > $\forall(t_{0},\ldots,t_{n-1}){\,\in\,}R{.}\penalty-1\,\,\forall\sigma{\,\in\,}{{{\mathcal{SUB}}}({{{\rm > V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ > $(t_{0}\sigma,\ldots,t_{n-1}\sigma)\in R$ > > $\rm T$-monotonic if > $\forall(t^{\prime},t^{\prime\prime}){\,\in\,}R{.}\penalty-1\,\,\forall > t{\,\in\,}{\mathcal{T}}{.}\penalty-1\,\,\forall > p{\,\in\,}{{{\mathcal{POS}}}({t})}{.}\penalty-1\,\,$ > ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&\exists > s{\,\in\,}{{\mathbb{S}}}{.}\penalty-1\,\,t/p,t^{\prime},t^{\prime\prime}{\,\in\,}{\mathcal{T}\\!_{{\rm > SIG},s}}\\\ {\wedge}&{t\penalty-1{[\,p\leftarrow t^{\prime}\,]}}\in{\rm > T}\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ > {{\left({{\begin{array}[]{ll}&({t\penalty-1{[\,p\leftarrow > t^{\prime}\,]}},{t\penalty-1{[\,p\leftarrow t^{\prime\prime}\,]}})\in R\\\ > {\wedge}&{t\penalty-1{[\,p\leftarrow t^{\prime\prime}\,]}}\in{\rm T}\\\ > \end{array}}}\right)}}\end{array}\right)}$ The subterm ordering $\lhd_{{}_{\rm ST}}$ on $\mathcal{T}$ is the ${\rm V}$-stable and wellfounded ordering defined by: $t{\trianglelefteq_{{}_{\rm ST}}}t^{\prime}$ if $\exists p{\,\in\,}{{{\mathcal{POS}}}({t^{\prime}})}{.}\penalty-1\,\,t{\,=\,}\penalty-1t^{\prime}/p$. A termination-pair over sig/${\rm V}$ is a pair $(>,\rhd)$ of ${\rm V}$-stable, wellfounded orderings on $\mathcal{T}$ such that $>$ is $\mathcal{T}$-monotonic, ${>}\subseteq{\rhd},$ and ${{\rhd_{{}_{\rm ST}}}}\subseteq{\rhd}.$ Cf. Wirth & Gramlich (1994a) for further theoretical aspects of termination-pairs, and Geser (1994) for interesting practical examples. For further details on orderings cf. Dershowitz (1987). The reflexive, symmetric, transitive, and reflexive & transitive closure of a relation $\longrightarrow$ will be denoted by ${\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}$, $\longleftrightarrow$, ${\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}$, and ${\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}$, resp..141414Note that this is actually an abuse of notation since $A^{+}$ now denotes the transitive closure of $A$ as well as the set of nonempty words over $A$ and since $A^{\ast}$ now denotes the reflexive & transitive closure of $A$ as well as the set of words over $A$. In our former papers we prefered to denote different things different but now we have found back to this standard abuse of notion for the sake of convenient readability, because the reader will easily find out what is meant with any application with the exception of those in the proof of Lemma B.7. Two terms $v$, $w$ are called joinable w.r.t. $\longrightarrow$ if $v{\downarrow}w$, i.e. if $v{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}w$. They are strongly joinable w.r.t. $\longrightarrow$ if $v{\downdownarrows}w$, i.e. if $v{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}w{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v$. $\longrightarrow$ is called terminating below $u$ if there is no ${{s}:{{{{{\bf N}}}\rightarrow{{{\rm dom}({{\longrightarrow}})}}}}}$ such that $u{\,=\,}\penalty-1s_{0}\ {\wedge}\penalty-2\ \forall i{\,\in\,}{{\bf N}}{.}\penalty-1\,\,s_{i}{\longrightarrow}s_{i+1}.$ ### 2.3 The Reduction Relation In the definition below we restrict our constructor rules to contain no non- constructor function symbols, to be extra-variable free, and to contain no negative literals. This is important for our approach (cf. Lemma 2.10, Lemma 2.11, and Lemma 2.12) and should always be kept in mind when reading the following sections. ###### Definition 2.2 (Syntax of CRS) ${{{\mathcal{COND}}{\mathcal{LIT}}}}({{\rm sig},{{\rm V}}})$ is the set of condition literals over the following predicate symbols on terms from ${\mathcal{T}}({{\rm sig},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})$: ‘$=$’, ‘$\not=$’ (binary, symmetric, sort- invariant), and ‘Def’ (singulary). The terms151515To avoid misunderstanding: For a condition list, say “ $s{=}t,\ u{\not=}v,\ \mbox{{\rm Def}}\,w$ ”, we mean the top level terms $s,t,u,v,w\in{{\mathcal{T}}({{\rm sig},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ but neither their proper subterms nor the literals “$s{=}t$”, “$u{\not=}v$”, “$\mbox{{\rm Def}}\,w$” themselves. of a list $C$ of condition literals are called condition terms and their set is denoted by ${{\mathcal{TERMS}}}({C})$. A (positive/negative-) conditional rule system (CRS) R over sig/cons/${\rm V}$ is a finite subset of the set of rules over sig/cons/${\rm V}$, which is defined by $\left\\{\left.\ {\left(\begin{array}[c]{l}(l,r),\ C\end{array}\right)}\ \right\|\right.$ $\left.{{\left({{\begin{array}[]{ll}&\exists s{\,\in\,}{{\mathbb{S}}}{.}\penalty-1\,\,l,r{\,\in\,}{{\mathcal{T}}({{\rm sig},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}_{s}\\\ {\wedge}&C\in({{{{\mathcal{COND}}{\mathcal{LIT}}}}({{\rm sig},{{\rm V}}})})^{\ast}\\\ {\wedge}&{\left(\begin{array}[c]{l}l\in{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\ \ {\Rightarrow\penalty-2}\\\ {{\left({{\begin{array}[]{ll}&\\{r\\}\cup{{{\mathcal{TERMS}}}({C})}\ {\ {\subseteq}\ }\ {{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&{{{\mathcal{V}}}({\\{r\\}\cup{{{\mathcal{TERMS}}}({C})}})}\ {\ {\subseteq}\ }\ {{{\mathcal{V}}}({l})}\\\ {\wedge}&\forall L\mbox{ in }C{.}\penalty-1\,\,\forall u,v{.}\penalty-1\,\,\ L\not=(u{\not=}v)\\\ \end{array}}}\right)}}\end{array}\right)}\\\ \end{array}}}\right)}}\right\\}.$ A rule $((l,r),\emptyset)$ with an empty condition will be written $l{=}r$. Note that $l{=}r$ differs from $r{=}l$ whenever the equation is used as a reduction rule. A rule $((l,r),C)$ with condition $C$ will be written $l{=}r{\longleftarrow}C$. We call $l$ the left-hand side and $r$ the right- hand side of the rule $l{=}r{\longleftarrow}C$. A rule is said to be left- linear (or else right-linear) if its left-hand (or else right-hand) side is a linear term. A rule $l{=}r{\longleftarrow}C$ is said to be extra-variable free if ${{{\mathcal{V}}}({\\{r\\}\cup{{{\mathcal{TERMS}}}({C})}})}\ {\ {\subseteq}\ }\ {{{\mathcal{V}}}({l})}.$ The whole CRS R is said to have one of these properties if each of its rules has it. A rule $l{=}r{\longleftarrow}C$ is called a constructor rule if its left-hand side is a constructor term, i.e. $l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ In the following example we define the subtraction operation ‘$\mathsf{-}$’ partially (due to a non-complete defining case distinction), whereas we define a member-predicate ‘${\mathsf{mbp}}$’ totally on the constructor ground terms. ###### Example 2.3 (continuing Example 2.1) Let $x,y\in{{{\rm V}}\\!_{{{\mathcal{C}}},{{\mathsf{nat}}}}}$ and $l\in{{{\rm V}}\\!_{{{\mathcal{C}}},{{\mathsf{list}}}}}$. ${\rm R}_{\,\rm\ref{exb}}$: $\begin{array}[t]{l@{\ \ }c@{\ \ }lcl}{{x}\,{\mathsf{-}}\,{{{\mathsf{0}}}}}&{=}&x\\\ {{{{\mathsf{s}}{(}{x}{)}}}\,{\mathsf{-}}\,{{{\mathsf{s}}{(}{y}{)}}}}&{=}&{{x}\,{\mathsf{-}}\,{y}}\\\ \end{array}~{}~{}~{}~{}\begin{array}[t]{\|l@{\ \ }c@{\ \ }lcl}{{{\mathsf{mbp}}}{(}{x}{,\,}{{{\mathsf{nil}}}}{)}}&{=}&{{\mathsf{false}}}\\\ {{{\mathsf{mbp}}}{(}{x}{,\,}{{{\mathsf{cons}}{(}{y}{,\,}{l}{)}}}{)}}&{=}&{{\mathsf{true}}}&{\>{\longleftarrow}}&x{=}y\\\ {{{\mathsf{mbp}}}{(}{x}{,\,}{{{\mathsf{cons}}{(}{y}{,\,}{l}{)}}}{)}}&{=}&{{{\mathsf{mbp}}}{(}{x}{,\,}{l}{)}}&{\>{\longleftarrow}}&x{\not=}y\\\ \end{array}$ ###### Definition 2.4 (Fulfilledness) A list $D\in{{{{\mathcal{COND}}{\mathcal{LIT}}}}({{\rm sig},{{\rm X}}})}^{\ast}$ of condition literals is said to be fulfilled w.r.t. some relation $\longrightarrow$ if $\forall u,v\in{\mathcal{T}}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\begin{array}[]{l @{\ \ \mbox{\bf(}\ \mbox{(}(} c @{)\mbox{ in }D\mbox{)}\ \ \Rightarrow\ \ } r l }&u{=}v&&u{\downarrow}v\ \ \mbox{\bf)}\\\ \wedge&{{\rm Def}\>}u&\exists\hat{u}\in{{\mathcal{GT}}({{\rm cons}})}{.\penalty-1}&u{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{u}\ \mbox{\bf)}\\\ \wedge&u{\not=}v&\exists\hat{u},\hat{v}\in{{\mathcal{GT}}({{\rm cons}})}{.\penalty-1}&u{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{u}{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}\hat{v}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v\ \mbox{\bf)}\\\ \end{array}\end{array}\right)}.$ To avoid a non-monotonic behaviour of our negative conditions, we define our reduction relation ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ via a double closure: First we define ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ by using the constructor rules only. Then we define ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\omega}}$ via a second closure including all rules. ###### Definition 2.5 (${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$) Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}$. Let $\prec$ denote the ordering on the ordinal numbers. For $\beta\preceq\omega{+}\omega$ and $p\in{{\bf N}}_{+}^{\ast}$ the reduction relations ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\beta}}$ and ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\beta,p}}$ on ${\mathcal{T}}({{\rm sig},{{\rm X}}})$ are inductively defined as follows: For $s,t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$: $s{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\beta}}}t$ if $\exists p{\,\in\,}{{{\mathcal{POS}}}({s})}{.}\penalty-1\,\,s{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\beta,p}}}t.$ For $p\in{{\bf N}}_{+}^{\ast}$: ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},0,p}}}:=\emptyset.$ For $i\in{{\bf N}}$; $s,t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$: $s{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},i+1,p}}}t$ if $\ \exists\left\langle\mbox{$\begin{array}[]{l}{((l,r),C)}{\,\in\,}{\rm R}\\\ \sigma{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}\end{array}$}\right\rangle{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&l\in{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&s/p=l\sigma\\\ {\wedge}&t={s\penalty-1{[\,p\leftarrow r\sigma\,]}}\\\ {\wedge}&C\sigma\mbox{ is fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},i}}}\\\ \end{array}}}\right)}}.$ ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega,p}}}:=\bigcup_{{}_{i\in{{\bf N}}}}{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},i,p}}}.$ For $i\in{{\bf N}}$; $s,t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$: $s{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+i+1,p}}}t$ if $s{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega,p}}}t\ \ {\vee}\penalty-2\ \ \exists\left\langle\mbox{$\begin{array}[]{l}{((l,r),C)}{\,\in\,}{\rm R}\\\ \sigma{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}\end{array}$}\right\rangle{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&s/p=l\sigma\\\ {\wedge}&t={s\penalty-1{[\,p\leftarrow r\sigma\,]}}\\\ {\wedge}&C\sigma\mbox{ is fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}},\omega+i}}}\\\ \end{array}}}\right)}}.$ ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\omega,p}}}:=\bigcup_{{}_{i\in{{\bf N}}}}{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}},\omega+i,p}}};$ ${{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}:={{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\omega}}}$ . We will drop “${\rm R},{{\rm X}}$” in ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ and ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\beta}}$ &c. when referring to some fixed ${\rm R},{{\rm X}}$. ###### Corollary 2.6 ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is the minimum (w.r.t. set-inclusion) of all relations $\rightsquigarrow$ on $\mathcal{T}$ satisfying for all $s,t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$: $s\rightsquigarrow t$ if $\exists\left\langle\mbox{$\begin{array}[]{l}p{\,\in\,}{{{\mathcal{POS}}}({s})}\\\ {((l,r),C)}{\,\in\,}{\rm R}\\\ \sigma{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}\\\ \end{array}$}\right\rangle{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&s/p\\!=\\!l\sigma\\\ {\wedge}&t\\!=\\!{s\penalty-1{[\,p\leftarrow r\sigma\,]}}\\\ {\wedge}&C\sigma\mbox{ is fulfilled w.r.t.\/ }\rightsquigarrow\\\ \end{array}}}\right)}}.$ ###### Lemma 2.7 Let $S_{{}_{{\rm R},{{\rm X}}}}$ be the set of all relations $\rightsquigarrow$ on $\mathcal{T}$ satisfying 1. 1. $(\,\,\rightsquigarrow\ \cap\ ({{\mathcal{GT}}({{\rm cons}})}{\times}{\mathcal{T}})\,)\ \subseteq\ {{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ as well as 2. 2. for all $s,t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$: $s\rightsquigarrow t$ if $\exists\left\langle\mbox{$\begin{array}[]{l}p{\,\in\,}{{{\mathcal{POS}}}({s})}\\\ {((l,r),C)}{\,\in\,}{\rm R}\\\ \sigma{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}\\\ \end{array}$}\right\rangle{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&s/p\\!=\\!l\sigma\\\ {\wedge}&t\\!=\\!{s\penalty-1{[\,p\leftarrow r\sigma\,]}}\\\ {\wedge}&C\sigma\mbox{ is fulfilled w.r.t.\/ }\rightsquigarrow\\\ \end{array}}}\right)}}.$ Now ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is the minimum (w.r.t. set-inclusion) in $S_{{}_{{\rm R},{{\rm X}}}}$, and $S_{{}_{{\rm R},{{\rm X}}}}$ is closed under nonempty intersection. ###### Corollary 2.8 (Monotonicity of $\longrightarrow$ w.r.t. Replacement) ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}},\beta}}$ (for $\beta\preceq\omega{+}\omega$) and ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ are ${\mathcal{T}}({{\rm sig},{{\rm X}}})$-monotonic as well as ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}{[{\rm T}]}$-monotonic for each ${\rm T}\subseteq{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$. ###### Corollary 2.9 (Stability of $\longrightarrow$) ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}},\beta}}$ (for $\beta\preceq\omega{+}\omega$), ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$, and their respective fulfilledness-predicates are ${\rm X}$-stable. ###### Lemma 2.10 For ${{\rm X}}\subseteq{\rm Y}\subseteq{{\rm V}}$: $\forall n{\,\in\,}{{\bf N}}{.}\penalty-1\,\,\forall s{\,\in\,}{{\mathcal{T}}({{\rm cons},{{\rm X}}})}{.}\penalty-1\,\,\forall t{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\\!\\!s{\stackrel{{\scriptstyle n}}{{{\longrightarrow}}}_{{}_{\\!{\rm R},{\rm Y}}}}t\ \ {\Rightarrow}\penalty-2\ \ (s{\stackrel{{\scriptstyle n}}{{{\longrightarrow}}}_{{}_{\\!{\rm R},{\rm Y},\omega}}}t\in{{\mathcal{T}}({{\rm cons},{{\rm X}}})})\\!\\!\end{array}\right)}\\!\\!$ ###### Lemma 2.11 $\downarrow\cap\ (\,{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\times{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\,)\ \ \subseteq\ \ {\downarrow_{{}_{\omega}}}$ ###### Lemma 2.12 (Monotonicity of ${\longrightarrow}_{{}_{\\!\beta}}$ and of Fulfilledness w.r.t. ${\longrightarrow}_{{}_{\\!\beta}}$ in $\beta$) For $\beta\preceq\gamma\preceq\omega\\!+\\!\omega$: ${{\longrightarrow}_{{}_{\\!\beta}}}\ \subseteq\ {{\longrightarrow}_{{}_{\\!\gamma}}}\ \subseteq\ {\longrightarrow}$ ; and if $C$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\beta}}$ and $\omega\preceq\beta\ \vee\ \forall u,v{.}\penalty-1\,\,((u{\not=}v)\mbox{ is not in }C)$ , then $C$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\gamma}}$ and w.r.t. $\longrightarrow$. Note that monotonicity of fulfilledness is not given in general for $\beta{\,\prec\,}\omega$ and a negative literal which may become invalid during the growth of the reduction relation on constructor terms. For the proofs cf. Wirth & Gramlich (1994a). ### 2.4 The Parallel Reduction Relation The following relation is essential for sophisticated joinability notions as well as for most of our proofs: ###### Definition 2.13 (Parallel Reduction) For $\beta\preceq\omega{+}\omega$ we define the parallel reduction relation ${\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\beta}$ on ${\mathcal{T}}({{\rm sig},{{\rm X}}})$: $s{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\beta}}t$ if $\exists\,\mathchar 261\relax\subseteq{{{\mathcal{POS}}}({s})}{.}\penalty-1\,\,s{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\beta,\mathchar 261\relax}}t,$ where $s{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\beta,\mathchar 261\relax}}t$ if ${{\left({{\begin{array}[]{ll}&\forall p,q{\,\in\,}\mathchar 261\relax{.}\penalty-1\,\,{\left(\begin{array}[c]{l}p{\,=\,}\penalty-1q\ \ {\vee}\penalty-2\ \ {{p}\,{\parallel}\,{q}}\end{array}\right)}\\\ {\wedge}&t{\,=\,}\penalty-1{s\penalty-1{{[\,p\leftarrow t/p\ \|\ p{\,\in\,}\mathchar 261\relax\,]}}}\\\ {\wedge}&\forall p{\,\in\,}\mathchar 261\relax{.}\penalty-1\,\,s/p{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\beta}}}t/p\\\ \end{array}}}\right)}}.$ ###### Corollary 2.14 $\forall\beta{\,\preceq\,}\omega{+}\omega{.}\penalty-1\,\,{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\beta}}}\subseteq{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\beta}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\beta}}}.$ ## 3 Confluence The following notions and lemmas have become folklore, cf. e.g. Klop (1980) or Huet (1980) for more information. ###### Definition 3.1 (Commutation and Confluence) Two relations ${\longrightarrow}_{{}_{\\!0}}$ and ${\longrightarrow}_{{}_{\\!1}}$ are commuting if $\forall s,t_{0},t_{1}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}s{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}t_{1}\ \ {\Rightarrow}\penalty-2\ \ \ t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}\end{array}\right)}.$ ${\longrightarrow}_{{}_{\\!0}}$ and ${\longrightarrow}_{{}_{\\!1}}$ are locally commuting if $\forall s,t_{0},t_{1}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}t_{0}{{\longleftarrow}_{{}_{\\!0}}}s{{\longrightarrow}_{{}_{\\!1}}}t_{1}\ \ {\Rightarrow}\penalty-2\ \ \ t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}\end{array}\right)}.$ ${\longrightarrow}_{{}_{\\!1}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!0}}$ if $\forall s,t_{0},t_{1}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}t_{0}{{\longleftarrow}_{{}_{\\!0}}}s{{\longrightarrow}_{{}_{\\!1}}}t_{1}\ \ {\Rightarrow}\penalty-2\ \ t_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}\end{array}\right)}.$ A single relation $\longrightarrow$ is called [ locally] confluent if $\longrightarrow$ and $\longrightarrow$ are [locally] commuting. It is called strongly confluent if $\longrightarrow$ strongly commutes over $\longrightarrow$. It is called confluent below $u$ if $\forall v,w{.}\penalty-1\,\,{(\ v{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}u{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w\ \ {\Rightarrow}\penalty-2\ \ v{\downarrow}w\ )}.$ ###### Lemma 3.2 (Generalized Newman Lemma) If ${\longrightarrow}_{{}_{\\!0}}$ and ${\longrightarrow}_{{}_{\\!1}}$ are commuting, then they are locally commuting, too. Furthermore, if ${{\longrightarrow}_{{}_{\\!0}}}\cup{{\longrightarrow}_{{}_{\\!1}}}$ is terminating or if ${\longrightarrow}_{{}_{\\!0}}$ or ${\longrightarrow}_{{}_{\\!1}}$ is transitive, then also the converse is true, i.e. ${\longrightarrow}_{{}_{\\!0}}$ and ${\longrightarrow}_{{}_{\\!1}}$ are commuting iff they are locally commuting. ###### Lemma 3.3 The following three properties are logically equivalent: 1. 1. ${\longrightarrow}_{{}_{\\!1}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!0}}$. 2. 2. ${\longrightarrow}_{{}_{\\!1}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}}_{{}_{\\!0}}$. 3. 3. ${\longrightarrow}_{{}_{\\!1}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0}}$. Moreover, each of them implies that ${\longrightarrow}_{{}_{\\!0}}$ and ${\longrightarrow}_{{}_{\\!1}}$ are commuting. ###### Lemma 3.4 (Church-Rosser) Assume that $\longrightarrow$ is confluent. Now: ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftrightarrow}}}}}\subseteq{\downarrow}.$ Besides strong confluence there are two other important versions of strengthened confluence for conditional rule systems. They are based on the depth of the reduction steps, i.e. on the $\beta$ of ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\beta}}$. Therefore they actually are properties of ${\rm R},{{\rm X}}$ instead of ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$, unless one considers ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ to be the family $({{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\beta}}})_{\beta\preceq\omega+\omega}.$ These two strengthened versions of confluence are shallow confluence and level confluence. Their generalizations to our generalized framework here are called $0$-shallow confluence for the closure w.r.t. our constructor rules, as well as $\omega$-shallow confluence and $\omega$-level confluence for our second closure. Shallow and level confluence are interesting: On the on hand, they provide us with stronger induction hypotheses for the proofs of our confluence criteria. On the other hand, the stronger confluence properties may be essential for certain kinds of reasoning with the specification of a rule system; for level joinability cf. Middeldorp & Hamoen (1994). Before we define our notions of shallow and level confluence we present some operations on ordinal numbers: ###### Definition 3.5 ($+_{\\!\\!{}_{0}}$, $+_{\\!\\!{}_{\omega}}$, $\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$) Let $\alpha\in\\{0,\omega\\}$. Let ‘$+$’ be the addition of ordinal numbers. Define ‘$+_{\\!\\!{}_{0}}$’, ‘$+_{\\!\\!{}_{\omega}}$’, and ‘$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$’ for $n_{0},n_{1}\prec\omega$: $\begin{array}[t]{lll}0{{+_{\\!\\!{}_{\alpha}}}}n_{1}&:=&n_{1}\\\ n_{0}{{+_{\\!\\!{}_{\alpha}}}}0&:=&n_{0}\\\ (n_{0}{+}1){{+_{\\!\\!{}_{\alpha}}}}(n_{1}{+}1)&:=&\alpha+n_{0}{+}1+n_{1}{+}1\\\ \\\ (n_{0}{+}n_{1}){\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}n_{1}&:=&n_{0}\\\ n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}(n_{0}{+}n_{1})&:=&0\\\ \\\ \end{array}$ Note that the subscript of the operator ‘$+_{\\!\\!{}_{\omega}}$’ is chosen to remind that it adds an extra $\omega$ to the left if both arguments are different from $0$. Moreover, note that ${{{}_{{{\bf N}}\times{{\bf N}}}{\upharpoonleft}{+_{\\!\\!{}_{0}}}}}{\,=\,}\penalty-1{{{}_{{{\bf N}}\times{{\bf N}}}{\upharpoonleft}{+}}}.$ ‘$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$’ is sometimes called monus. Since we want to use shallow and level confluence also for terminating reduction relations we have to parameterize them w.r.t. wellfounded orderings. Let ‘$\succ$’ as before be the wellordering of the ordinal numbers. Let ‘$\rhd$’ be some wellfounded ordering on $\mathcal{T}$. We denote the lexicographic combination of $\succ$ and $\rhd$ by ‘$\,\,{\succ\\!\\!\\!\rhd}\,\,$’, its reverse by ‘$\,\,{\prec\\!\\!\lhd}\,\,$’, and the reflexive closure of the latter by ‘$\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,$’. ###### Definition 3.6 ( $0$-Shallow Confluent / $\omega$-Shallow Confluent ) Let $\alpha\in\\{0,\omega\\}$. Let $\beta\preceq\omega{+}\omega.$ Let $s\in{\mathcal{T}}$. ${\rm R},{{\rm X}}$ is said to be $\alpha$-shallow confluent up to $\beta$ and $s$ in $\lhd$ if $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,\forall u,v,w{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}n_{0}{{+_{\\!\\!{}_{\alpha}}}}n_{1},\ u\end{array}\right)}{\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}{\left(\begin{array}[c]{l}\beta,\ s\end{array}\right)}\\\ {\wedge}&v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}w\\\ \end{array}}}\right)}}\\\ \ {\Rightarrow}\penalty-2\ \ v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}w\end{array}\right)}.$ ${\rm R},{{\rm X}}$ is said to be $\alpha$-shallow confluent up to $\beta$ if ${\rm R},{{\rm X}}$ is $\alpha$-shallow confluent up to $\beta$ and161616Note that reference to a special $\lhd$ becomes irrelevant here $s$ for all $s\in{\mathcal{T}}$. ${\rm R},{{\rm X}}$ is said to be $\alpha$-shallow confluent if ${\rm R},{{\rm X}}$ is $\alpha$-shallow confluent up to $\omega{+}\alpha$. ###### Definition 3.7 ($\omega$-Level Confluent) Let $\beta\preceq\omega$. Let $s\in{\mathcal{T}}$. ${\rm R},{{\rm X}}$ is said to be $\omega$-level confluent up to $\beta$ and $s$ in $\lhd$ if $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,\forall u,v,w{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}\max\\{n_{0},n_{1}\\},\ u\end{array}\right)}{\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}{\left(\begin{array}[c]{l}\beta,\ s\end{array}\right)}\\\ {\wedge}&v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}w\\\ \end{array}}}\right)}}\\\ \ {\Rightarrow}\penalty-2\ \ v{\downarrow_{{}_{{{\rm R},{{\rm X}}},\omega+\max\\{n_{0},n_{1}\\}}}}w\end{array}\right)}.$ ${\rm R},{{\rm X}}$ is said to be $\omega$-level confluent up to $\beta$ if ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $\beta$ and${}^{\ref{footnote lhd does not matter}}$ $s$ for all $s\in{\mathcal{T}}$. ${\rm R},{{\rm X}}$ is said to be $\omega$-level confluent if ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $\omega$. Note that $\omega$-level and $\omega$-shallow confluence specialize to the standard definitions of level and shallow confluence, resp., for the case that all symbols are considered to be non-constructor symbols (where $n$ becomes the standard depth of ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}$); and that $0$-shallow confluence specializes to the standard definition of shallow confluence for the case that all symbols are considered to be constructor symbols. ###### Corollary 3.8 ( $\omega$-Shallow Confluent $\Rightarrow$ $\omega$-Level Confluent $\Rightarrow$ Confluent ) If ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent, then ${\rm R},{{\rm X}}$ is $\omega$-level confluent. If ${\rm R},{{\rm X}}$ is $\omega$-level confluent, then ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is confluent. ###### Corollary 3.9 ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $0$ iff ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $0$ iff ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent. ## 4 Critical Peaks Critical peaks describe those possible sources of non-confluence that directly arise from the syntax of the given rule system. While the so-called variable overlaps can hardly be approached via syntactic means, the critical peaks describe the non-variable overlaps resulting from an instantiated left-hand side being subterm of an instantiated left-hand side at a non-variable position. Our critical peaks capture more information than the standard critical pairs: Besides the pair, they contain the peak term and its overlap position. Furthermore, each element of the pair is augmented with the condition that must be fulfilled for enabling the reduction step down from the peak term, and with a bit indicating whether the rule applied was a non- constructor rule or not. ###### Definition 4.1 (Critical Peak) If the left-hand side of a rule $l_{0}{=}r_{0}{\longleftarrow}C_{0}$ and the subterm at non-variable (i.e. $l_{1}/p\notin{{\rm V}}$) position $p\in{{{\mathcal{POS}}}({l_{1}})}$ of the left-hand side of a rule $l_{1}{=}r_{1}{\longleftarrow}C_{1}$ (assuming ${{{\mathcal{V}}}({{l_{0}{=}r_{0}{\longleftarrow}C_{0}}})}\cap{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}=\emptyset$ w.l.o.g.171717To achieve this, let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{l_{0}{=}r_{0}{\longleftarrow}C_{0}}})}]\cap{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}=\emptyset$ and then replace $l_{0}{=}r_{0}{\longleftarrow}C_{0}$ with $({l_{0}{=}r_{0}{\longleftarrow}C_{0}})\xi.$ ) are unifiable by $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}\sigma={{\rm mgu}({\ \\{(l_{0},l_{1}/p)\\}},{\ {{{\mathcal{V}}}({l_{0}{=}r_{0}{\longleftarrow}C_{0},l_{1}{=}r_{1}{\longleftarrow}C_{1}})}})},$ if (for $i{\,\prec\,}2$) $\ \ \mathchar 259\relax_{i}=\left\\{\mbox{$\begin{array}[]{ll}0&\mbox{ if }l_{i}\in{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ 1&\mbox{ otherwise}\\\ \end{array}$}\right\\},$ and if the resulting critical pair is non-trivial (i.e. ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\,]}}\sigma\not=r_{1}\sigma$), then $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}{\left(\begin{array}[c]{l}({l_{1}\penalty-1{[\,p\leftarrow r_{0}\,]}},\ C_{0},\ \mathchar 259\relax_{0}),\ \ (r_{1},\ C_{1},\ \mathchar 259\relax_{1}),\ \ l_{1},\ \ \sigma,\ \ p\end{array}\right)}$ is a (non-trivial) critical peak (of the form $(\mathchar 259\relax_{0},\mathchar 259\relax_{1})$) consisting of the conditional critical pair, its peak term $l_{1}$, the most general unifier $\sigma$, and the overlap position $p$. For convenience we usually identify this critical peak with its instantiated version $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}{\left(\begin{array}[c]{l}({l_{1}\penalty-1{[\,p\leftarrow r_{0}\,]}}\sigma,\ C_{0}\sigma,\ \mathchar 259\relax_{0}),\ \ (r_{1}\sigma,\ C_{1}\sigma,\ \mathchar 259\relax_{1}),\ \ l_{1}\sigma,\ \ p\end{array}\right)}$ which should not lead to confusion because the tuple is shorter. The set of all critical peaks of a CRS R is denoted by ${\rm CP}({\rm R})$. ###### Example 4.2 (continuing Example 2.3) ${\rm CP}({\rm R}_{\,\rm\ref{exb}})$ contains two critical peaks, namely (in the instantiated version) $\left(\begin{array}[c]{l}({{\mathsf{true}}},(x{=}y),1),\ ({{{\mathsf{mbp}}}{(}{x}{,\,}{l}{)}},(x{\not=}y),1),\ {{{\mathsf{mbp}}}{(}{x}{,\,}{{{\mathsf{cons}}{(}{y}{,\,}{l}{)}}}{)}},\ \emptyset\end{array}\right)$ and $\left(\begin{array}[c]{l}({{{\mathsf{mbp}}}{(}{x}{,\,}{l}{)}},(x{\not=}y),1),\ ({{\mathsf{true}}},(x{=}y),1),\ {{{\mathsf{mbp}}}{(}{x}{,\,}{{{\mathsf{cons}}{(}{y}{,\,}{l}{)}}}{)}},\ \emptyset\end{array}\right)$ which we would (partially) display as Note that we omit the position at the arrow to the right because it is always $\emptyset$. Furthermore, note that the two critical peaks are different although they look similar. Namely, the one is the symmetric overlay (cf. below) of the other. ## 5 Basic Forms of Joinability of Critical Peaks A critical peak $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}{((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)}$ is joinable w.r.t. ${\rm R},{{\rm X}}$ (for ${{\rm X}}{\subseteq}{{\rm V}}$) if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}((D_{0}D_{1})\sigma\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}})\ {\Rightarrow}\penalty-2\ t_{0}\sigma\varphi{\downarrow_{{}_{{\rm R},{{\rm X}}}}}t_{1}\sigma\varphi\end{array}\right)}.$ It is an overlay if $p{\,=\,}\penalty-1\emptyset$. It is a non-overlay if $p{\,\not=\,}\emptyset$. It is overlay joinable w.r.t. ${\rm R},{{\rm X}}$ if it is joinable w.r.t. ${\rm R},{{\rm X}}$ and is an overlay. In the following two definitions ‘${\mathsf{true}}$’ and ‘${\mathsf{false}}$’ denote two arbitrary irreducible ground terms. Their special names have only been chosen to make clear the intuition behind. The above critical peak is complementary w.r.t. ${\rm R},{{\rm X}}$ if ${{\left({{\begin{array}[]{ll}&\exists u,v{\,\in\,}{\mathcal{T}}{.}\penalty-1\,\,\exists i{\,\prec\,}2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&(u{=}v)\mbox{ occurs in }D_{i}\sigma\\\ {\wedge}&(u{\not=}v)\mbox{ occurs in }D_{1-i}\sigma\\\ \end{array}}}\right)}}\\\ {\vee}&\exists p{\,\in\,}{\mathcal{T}}{.}\penalty-1\,\,\exists{{\mathsf{true}}},{{\mathsf{false}}}{\,\in\,}{\mathcal{GT}}{\setminus}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}{.}\penalty-1\,\,\exists i{\,\prec\,}2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&(p{=}{{\mathsf{true}}})\mbox{ occurs in }D_{i}\sigma\\\ {\wedge}&(p{=}{{\mathsf{false}}})\mbox{ occurs in }D_{1-i}\sigma\\\ {\wedge}&{{\mathsf{true}}}{\,\not=\,}{{\mathsf{false}}}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}.$ It is weakly complementary w.r.t. ${\rm R},{{\rm X}}$ if ${{\left({{\begin{array}[]{ll}&\exists u,v{\,\in\,}{\mathcal{T}}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}(u{=}v)\mbox{ and }\\\ (u{\not=}v)\mbox{ occur in }(D_{0}D_{1})\sigma\end{array}\right)}\\\ {\vee}&\exists p{\,\in\,}{\mathcal{T}}{.}\penalty-1\,\,\exists{{\mathsf{true}}},{{\mathsf{false}}}{\,\in\,}{\mathcal{GT}}{\setminus}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&(p{=}{{\mathsf{true}}})\mbox{ and }\\\ &(p{=}{{\mathsf{false}}})\mbox{ occur in }(D_{0}D_{1})\sigma\\\ {\wedge}&{{\mathsf{true}}}{\,\not=\,}{{\mathsf{false}}}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}.$ It is strongly joinable w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}((D_{0}D_{1})\sigma\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}})\ {\Rightarrow}\penalty-2\ t_{0}\sigma\varphi{{\downdownarrows}_{{}_{{\rm R},{{\rm X}}}}}t_{1}\sigma\varphi\end{array}\right)}.$ In the following definition ‘$A$’ is an arbitrary function from positions to sets of terms. The above critical peak is $\rhd$-weakly joinable w.r.t. ${\rm R},{{\rm X}}$ [besides $A$] if $\ \forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}\\!\\!{{\left({{\begin{array}[]{ll}&(D_{0}D_{1})\sigma\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\\\ {\wedge}&\forall u{.}\penalty-1\,\,{\left(\begin{array}[c]{l}u\lhd\hat{t}\sigma\varphi\ \ {\Rightarrow}\penalty-2\ \ {{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\mbox{ is confluent below }u\end{array}\right)}\\\ {\wedge}&\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\\\ {\wedge}&{\left(\begin{array}[c]{l}p{\,\not=\,}\emptyset\ \ {\Rightarrow}\penalty-2\ \ \forall x{\,\in\,}{{{\mathcal{V}}}({\hat{t}})}{.}\penalty-1\,\,x\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\end{array}\right)}\\\ \lx@intercol\left[\begin{array}[]{@{\wedge\ \ \ }l@{\ \ \ \ }}\hat{t}\sigma\varphi{\,\not\in\,}A(p)\end{array}\right]\hfil\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ t_{0}\sigma\varphi{\downarrow_{{}_{{\rm R},{{\rm X}}}}}t_{1}\sigma\varphi\\\ \end{array}\right)}.$ Note that $\rhd$-weak joinability adds several useful features to the single condition of joinability, forming a conjunctive condition list. The first new feature allows to assume confluence below all terms that are strictly smaller than the peak term. The following features allow us to assume some irreducibilities for the joinability test, where the optional one is an interface that is to be specified by the confluence criteria using it, cf. our theorems 14.2 and 14.4. For a demonstration of the usefulness of these additional features cf. Example 14.3. ###### Lemma 5.1 (Joinability of Critical Peaks is Necessary for Confluence) If ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is confluent, then all critical peaks in ${\rm CP}({\rm R})$ are joinable w.r.t. ${\rm R},{{\rm X}}$. ## 6 Basic Forms of Shallow and Level Joinability Just like confluence and strong confluence, also level and shallow confluence have their corresponding joinability notion. Sorry to say, they are pretty complicated, however. ###### Definition 6.1 ( $0$-Shallow Joinable / $\omega$-Shallow Joinable ) Let $\alpha\in\\{0,\omega\\}$. Let $\beta\preceq\omega{+}\alpha$. Let $s\in{\mathcal{T}}$. A critical peak $((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)$ is $\alpha$-shallow joinable up to $\beta$ and $s$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$] if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n_{0},n_{1}\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1},\ \hat{t}\sigma\varphi\end{array}\right)}{\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}{\left(\begin{array}[c]{l}\beta,\ s\end{array}\right)}\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{i}{\,=\,}\penalty-10{\,\prec\,}n_{i}\end{array}\right)}\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-1\omega\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{i}{\,\preceq\,}n_{i}\end{array}\right)}\\\ {\wedge}&D_{i}\sigma\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall{\left(\begin{array}[c]{l}\delta,\ s^{\prime}\end{array}\right)}{\prec\\!\\!\lhd}{\left(\begin{array}[c]{l}n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1},\ \hat{t}\sigma\varphi\end{array}\right)}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\begin{array}[]{@{}l@{}}{{\rm R},{{\rm X}}}\mbox{ is $\alpha$-shallow confluent}\\\ \mbox{up to }\delta\mbox{ and }s^{\prime}\mbox{ in }\lhd\\\ \end{array}\end{array}\right)}\\\ {\wedge}&\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+\min\\{n_{0},n_{1}\\}}}}})}\\\ {\wedge}&{\left(\begin{array}[c]{l}p{\,\not=\,}\emptyset\ \ {\Rightarrow}\penalty-2\ \ \forall x{\,\in\,}{{{\mathcal{V}}}({\hat{t}})}{.}\penalty-1\,\,x\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+\min\\{n_{0},n_{1}\\}}}}})}\end{array}\right)}\\\ \lx@intercol\left[\begin{array}[]{@{\wedge\ \ \ }l@{\ \ \ \ }}\hat{t}\sigma\varphi{\,\not\in\,}A(p,\min\\{n_{0},n_{1}\\})\end{array}\right]\hfil\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{\left(\begin{array}[c]{l}t_{0}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}t_{1}\sigma\varphi\end{array}\right)}\\\ \end{array}}}\right)}}.$ It is called $\alpha$-shallow joinable up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$] if it is $\alpha$-shallow joinable up to $\beta$ and $s$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$] for all $s\in{\mathcal{T}}$. It is called $\alpha$-shallow joinable w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$] if it is $\alpha$-shallow joinable up to $\omega{+}\alpha$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$]. When $\lhd$ is not specified, we tacitly assume it to be $\lhd_{{}_{\rm ST}}$. ###### Definition 6.2 ($\omega$-Level Joinable) Let $\beta\preceq\omega$. Let $s\in{\mathcal{T}}$. A critical peak $((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)$ is $\omega$-level joinable up to $\beta$ and $s$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$] if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n_{0},n_{1}\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}\max\\{n_{0},n_{1}\\},\ \hat{t}\sigma\varphi\end{array}\right)}{\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}{\left(\begin{array}[c]{l}\beta,\ s\end{array}\right)}\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\mathchar 259\relax_{i}\preceq n_{i}\\\ {\wedge}&D_{i}\sigma\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall{\left(\begin{array}[c]{l}\delta,\ s^{\prime}\end{array}\right)}{\prec\\!\\!\lhd}{\left(\begin{array}[c]{l}\max\\{n_{0},n_{1}\\},\ \hat{t}\sigma\varphi\end{array}\right)}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\begin{array}[]{@{}l@{}}{{\rm R},{{\rm X}}}\mbox{ is $\omega$-level confluent}\\\ \mbox{up to }\delta\mbox{ and }s^{\prime}\mbox{ in }\lhd\end{array}\end{array}\right)}\\\ {\wedge}&\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\max\\{n_{0},n_{1}\\}}}}})}\\\ {\wedge}&{\left(\begin{array}[c]{l}p{\,\not=\,}\emptyset\ \ {\Rightarrow}\penalty-2\ \ \forall x{\,\in\,}{{{\mathcal{V}}}({\hat{t}})}{.}\penalty-1\,\,x\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\max\\{n_{0},n_{1}\\}}}}})}\end{array}\right)}\\\ \lx@intercol\left[\begin{array}[]{@{\wedge\ \ \ }l@{\ \ \ \ }}\hat{t}\sigma\varphi{\,\not\in\,}A(p,\max\\{n_{0},n_{1}\\})\end{array}\right]\hfil\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{\left(\begin{array}[c]{l}t_{0}\sigma\varphi{\downarrow_{{}_{{{\rm R},{{\rm X}}},\omega+\max\\{n_{0},n_{1}\\}}}}t_{1}\sigma\varphi\end{array}\right)}\\\ \end{array}}}\right)}}.$ It is called $\omega$-level joinable up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$] if it is $\omega$-level joinable up to $\beta$ and $s$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$] for all $s\in{\mathcal{T}}$. It is called $\omega$-level joinable w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$] if it is $\omega$-level joinable up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$]. When $\lhd$ is not specified, we tacitly assume it to be $\lhd_{{}_{\rm ST}}$. Please notice the generic structure of these and the following definitions that makes them actually less complicated than they look like. While the conclusions of their implications should be clear, the elements of their conjunctive condition lists have the following purposes: The first just parameterizes the notion in $\beta$ and $s$. The second requires the appropriate fulfilledness of the conditions of the critical peak, where $\mathchar 259\relax_{i}{\,\preceq\,}n_{i}$ allows us to assume $1{\,\preceq\,}n_{i}$ when the term $t_{i}$ is generated by a non-constructor rule which is important since otherwise the conclusion is very unlikely to be fulfilled, cf. also below. The third allows us to assume a certain confluence property which can be applied when checking the fulfilledness of the conditions. E.g., this condition sometimes implies that the fulfilledness assumptions of the second element for “$i{\,=\,}\penalty-10$” and “$i{\,=\,}\penalty-11$” are contradictory. An example for this are the critical peaks of Example 4.2 which are both $\omega$-level and $\omega$-shallow confluent since the condition list can never be fulfilled. But how do we know that? Suppose that $(x{=}y)\varphi$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}$ and that $(x{\not=}y)\varphi$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}$. Then there are $\hat{u},\hat{v}\in{{\mathcal{GT}}({{\rm cons}})}$ such that $x\varphi{\downarrow_{{}_{{{\rm R},{{\rm X}}},\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}y\varphi$ and $x\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\hat{u}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{{\rm R},{{\rm X}}},\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\hat{v}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}y\varphi.$ By $x,y{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}$ we get $x\varphi,y\varphi{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and thus by Lemma 2.10 we get $x\varphi{\downarrow_{{}_{{{\rm R},{{\rm X}}},\omega}}}y\varphi$ and $x\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\hat{u}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{{\rm R},{{\rm X}}},\omega}}}\hat{v}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}y\varphi.$ This contradicts confluence of ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ and then by Corollary 3.9 it also contradicts $\omega$-level and $\omega$-shallow confluence up to $0$. However, we are allowed to assume this since we know $0\prec\max\\{n_{0},n_{1}\\}$ and $0\prec n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}$ due to $\mathchar 259\relax_{0}{\,=\,}\penalty-1\mathchar 259\relax_{1}{\,=\,}\penalty-11$ (and $\mathchar 259\relax_{i}{\,\preceq\,}n_{i}$). A more general argumentation of this kind proves theorems 13.3, 13.4, and 15.3, which are confluence criteria for rule systems with complementary critical peaks. Finally, the following items in the conjunctive condition lists allow us to assume some irreducibilities similar to those for $\rhd$-weak joinability but less powerful. ###### Lemma 6.3 ($\alpha$-Shallow Joinability is Necessary for $\alpha$-Shallow Confluence) Let $\alpha\in\\{0,\omega\\}$. If ${\rm R},{{\rm X}}$ is $\alpha$-shallow confluent [up to $\beta$ [and $s$ in $\lhd$]], then all critical peaks in ${\rm CP}({\rm R})$ are $\alpha$-shallow joinable [up to $\beta$ [and $s$]] w.r.t. ${\rm R},{{\rm X}}$ [[and $\lhd$]]. ###### Lemma 6.4 ($\omega$-Level Joinability is Necessary for $\omega$-Level Confluence) If ${\rm R},{{\rm X}}$ is $\omega$-level confluent [up to $\beta$ [and $s$ in $\lhd$]], then all critical peaks in ${\rm CP}({\rm R})$ are $\omega$-level joinable [up to $\beta$ [and $s$]] w.r.t. ${\rm R},{{\rm X}}$ [[and $\lhd$]]. ## 7 Sophisticated Forms of Shallow Joinability For a first reading this section should only be skimmed and its definitions looked up by need. At least 12 should be read before.181818We put this section here because we do not want to scatter our later discussion with a big definition section and because we do not want to use the (for a first reading not essential) joinability labels in the boxes of the examples in the following sections before defining them. The $\omega$-shallow joinability notions of this section are only necessary for understanding the sophisticated Theorem 13.6 and its interrelation with the examples in the following sections, but not for the important practical consequence of this theorem, namely Theorem 13.3, which is easy to understand and sufficient for many practical applications. The $0$-shallow joinability notions are needed for Theorem 15.1 only. The following notion will be applied for non-overlays of the forms $(1,0)$ and $(1,1)$ for “$\alpha{\,=\,}\penalty-1\omega$” and of the form $(0,0)$ for “$\alpha{\,=\,}\penalty-10$”: ###### Definition 7.1 ( $0$-Shallow Parallel Closed / $\omega$-Shallow Parallel Closed ) Let $\alpha\in\\{0,\omega\\}$. Let $\beta\preceq\omega{+}\alpha$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\alpha$-shallow parallel closed up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n_{0},n_{1}\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&0{\,\prec\,}n_{0}{\,\succeq\,}n_{1}\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{i}{\,=\,}\penalty-10{\,\prec\,}n_{i}\end{array}\right)}\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-1\omega\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{i}{\,\preceq\,}n_{i}\end{array}\right)}\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\alpha$-shallow confluent up to }\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}n_{1}{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ t_{0}\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\alpha}}t_{1}\varphi\end{array}\right)}\\\ {\wedge}&t_{0}\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\alpha+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha}}}t_{1}\varphi\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}.$ It is called $\alpha$-shallow parallel closed w.r.t. ${\rm R},{{\rm X}}$ if it is $\alpha$-shallow parallel closed up to $\omega{+}\alpha$ w.r.t. ${\rm R},{{\rm X}}$. The following notion will be applied for critical peaks of the forms $(0,1)$ and $(1,1)$ for “$\alpha{\,=\,}\penalty-1\omega$” and of the form $(0,0)$ for “$\alpha{\,=\,}\penalty-10$”: ###### Definition 7.2 ( $0$-Shallow / $\omega$-Shallow [Noisy] Parallel Joinable) Let $\alpha\in\\{0,\omega\\}$. Let $\beta\preceq\omega{+}\alpha$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\alpha$-shallow [noisy] parallel joinable up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n_{0},n_{1}\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}{\,\succ\,}0\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{i}{\,=\,}\penalty-10{\,\prec\,}n_{i}\end{array}\right)}\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-1\omega\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{i}{\,\preceq\,}n_{i}\end{array}\right)}\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\alpha$-shallow confluent up to }\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&t_{0}\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\alpha+n_{1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}t_{1}\varphi\\\ \end{array}}}\right)}}.$ It is called $\alpha$-shallow [noisy] parallel joinable w.r.t. ${\rm R},{{\rm X}}$ if it is $\alpha$-shallow [noisy] parallel joinable up to $\omega{+}\alpha$ w.r.t. ${\rm R},{{\rm X}}$. Note that $\alpha$-shallow parallel closedness specializes to the standard definition of parallel closedness of Huet (1980) for the case that all symbols are considered to be non-constructor symbols in case of $\alpha{\,=\,}\penalty-1\omega$ (or else constructor symbols in case of $\alpha{\,=\,}\penalty-10$) and the rule system is unconditional (since then ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha}}}{\,=\,}\penalty-1\emptyset$ and ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+1}}}{\,=\,}\penalty-1{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}$). Similarly, $\alpha$-shallow parallel joinability specializes for these cases to the joinability required for overlays in Toyama (1988). Moreover, note that the notions whose names end with “closed” are always restricted to “$0{\,\prec\,}n_{0}{\,\succeq\,}n_{1}$”, whereas those whose names end with “joinable” are always restricted to “$n_{0}{\,\preceq\,}n_{1}{\,\succ\,}0$”. Finally, note that some notions have “noisy” variants which are weaker since they allow some “noise”, i.e. some reduction on a smaller depth than the preceding reduction step.191919The name for the notion was inspired by Oostrom (1994a). The following notion will be applied for non-overlays of the forms $(1,0)$ and $(1,1)$ for “$\alpha{\,=\,}\penalty-1\omega$” and of the form $(0,0)$ for “$\alpha{\,=\,}\penalty-10$”: ###### Definition 7.3 ( $0$-Shallow / $\omega$-Shallow [Noisy] Anti-Closed ) Let $\alpha\in\\{0,\omega\\}$. Let $\beta\preceq\omega{+}\alpha$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\alpha$-shallow [noisy] anti-closed up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n_{0},n_{1}\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&0{\,\prec\,}n_{0}{\,\succeq\,}n_{1}\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{i}{\,=\,}\penalty-10{\,\prec\,}n_{i}\end{array}\right)}\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-1\omega\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{i}{\,\preceq\,}n_{i}\end{array}\right)}\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\alpha$-shallow confluent up to }\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}n_{1}{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha[+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}t_{1}\varphi\end{array}\right)}\\\ {\wedge}&t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha{[+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha}}}t_{1}\varphi\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}.$ It is called $\alpha$-shallow [noisy] anti-closed w.r.t. ${\rm R},{{\rm X}}$ if it is $\alpha$-shallow [noisy] anti-closed up to $\omega{+}\alpha$ w.r.t. ${\rm R},{{\rm X}}$. The following notion will be applied for critical peaks of the form $(0,1)$ and $(1,1)$ for “$\alpha{\,=\,}\penalty-1\omega$” and of the form $(0,0)$ for “$\alpha{\,=\,}\penalty-10$”: ###### Definition 7.4 ( $0$-Shallow / $\omega$-Shallow [Noisy] Strongly Joinable ) Let $\alpha\in\\{0,\omega\\}$. Let $\beta\preceq\omega{+}\alpha$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\alpha$-shallow [noisy] strongly joinable up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n_{0},n_{1}\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}{\,\succ\,}0\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{i}{\,=\,}\penalty-10{\,\prec\,}n_{i}\end{array}\right)}\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-1\omega\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{i}{\,\preceq\,}n_{i}\end{array}\right)}\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\alpha$-shallow confluent up to }\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}n_{0}{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha}}}t_{1}\varphi\end{array}\right)}\\\ {\wedge}&t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha}}}\circ{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}t_{1}\varphi\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}.$ It is called $\alpha$-shallow [noisy] strongly joinable w.r.t. ${\rm R},{{\rm X}}$ if it is $\alpha$-shallow [noisy] strongly joinable up to $\omega{+}\alpha$ w.r.t. ${\rm R},{{\rm X}}$. The following notion will be applied for non-overlays of the forms $(1,0)$ and $(1,1)$: ###### Definition 7.5 ($\omega$-Shallow Closed) Let $\beta\preceq\omega{+}\omega$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\omega$-shallow closed up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n_{0},n_{1}\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&0{\,\prec\,}n_{0}{\,\succeq\,}n_{1}\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\mathchar 259\relax_{i}\preceq n_{i}\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}n_{1}{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}t_{1}\varphi\end{array}\right)}\\\ {\wedge}&t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}t_{1}\varphi\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}.$ It is called $\omega$-shallow closed w.r.t. ${\rm R},{{\rm X}}$ if it is $\omega$-shallow closed up to $\omega{+}\omega$ w.r.t. ${\rm R},{{\rm X}}$. The following notion will be applied for critical peaks of the forms $(0,1)$ and $(1,1)$: ###### Definition 7.6 ($\omega$-Shallow [Noisy] Weak Parallel Joinable) Let $\beta\preceq\omega{+}\omega$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\omega$-shallow [noisy] weak parallel joinable up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n_{0},n_{1}\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}{\,\succ\,}0\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\mathchar 259\relax_{i}\preceq n_{i}\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n_{1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}t_{1}\varphi\\\ \end{array}}}\right)}}.$ It is called $\omega$-shallow [noisy] weak parallel joinable w.r.t. ${\rm R},{{\rm X}}$ if it is $\omega$-shallow [noisy] weak parallel joinable up to $\omega{+}\omega$ w.r.t. ${\rm R},{{\rm X}}$. The following are corollaries of Corollary 2.14: ###### Corollary 7.7 Let $\alpha\in\\{0,\omega\\}$. Now w.r.t. ${\rm R},{{\rm X}}$ the following holds: If a critical peak is $\omega$-shallow [noisy] parallel joinable up to $\beta\preceq\omega{+}\omega$, then it is $\omega$-shallow [noisy] weak parallel joinable up to $\beta$. If a critical peak is $\omega$-shallow [noisy] strongly joinable up to $\beta\preceq\omega{+}\omega$, then it is $\omega$-shallow [noisy] weak parallel joinable up to $\beta$. If a critical peak is $\alpha$-shallow [noisy] strongly joinable up to $\beta\preceq\omega$, then it is $\alpha$-shallow [noisy] parallel joinable up to $\beta$. ###### Corollary 7.8 Let $\alpha\in\\{0,\omega\\}$. Let $\beta\preceq\omega{+}\alpha$. Now w.r.t. ${\rm R},{{\rm X}}$ the following holds: If a critical peak is $\alpha$-shallow parallel closed or (for $\alpha{\,=\,}\penalty-1\omega$) $\alpha$-shallow closed up to $\beta$, then it is $\alpha$-shallow [noisy] anti-closed up to $\beta$. Overview over sophisticated forms of $\omega$-Shallow … of $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ Generally assumed condition for $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$; $n_{0},n_{1}\prec\omega$: ${\left({{\begin{array}[]{ll}&\mbox{``Property~{}1''}\ \ {\wedge}\penalty-2\ \ n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\mathchar 259\relax_{i}\preceq n_{i}\ \ {\wedge}\penalty-2\ \ D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\end{array}\right)}\\\ {\wedge}&\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\delta\\\ \end{array}}}\right)}$ Required conclusion (P := Parallel; C := Closed; N := Noisy; J := Joinable; W := Weak; A := Anti-; S := Strongly): Property 1 := … $0\prec n_{0}\succeq n_{1}$ $n_{0}\preceq n_{1}\succ 0$ In case of … $n_{1}=0$ $n_{1}\succ 0$ ## 8 Sophisticated Forms of Level Joinability For a first reading this section should only be skimmed and its definitions looked up by need. At least 7 should be read before. This section is only necessary for understanding the sophisticated Theorem 13.9 and its interrelation with the examples in the following sections, but not for the easy to understand consequence of this theorem, namely Theorem 13.4. Having completed our special notions for shallow confluence, we now present some for level confluence. The following notion will be applied for non-overlays of the form $(1,1)$: ###### Definition 8.1 ($\omega$-Level Parallel Closed) Let $\beta\preceq\omega$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\omega$-level parallel closed up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&0{\,\prec\,}n\\\ {\wedge}&n\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\mathchar 259\relax_{i}\preceq n\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-level confluent up to }\delta\\\ {\wedge}&{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\omega\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&t_{0}\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}t_{1}\varphi\\\ \end{array}}}\right)}}.$ It is called $\omega$-level parallel closed w.r.t. ${\rm R},{{\rm X}}$ if it is $\omega$-level parallel closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$. The following notion will be applied for critical peaks of the form $(1,1)$: ###### Definition 8.2 ($\omega$-Level Parallel Joinable) Let $\beta\preceq\omega$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\omega$-level parallel joinable up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n{\,\succ\,}0\\\ {\wedge}&n\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\mathchar 259\relax_{i}\preceq n\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-level confluent up to }\delta\\\ {\wedge}&{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\omega\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&t_{0}\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}t_{1}\varphi\\\ \end{array}}}\right)}}.$ It is called $\omega$-level parallel joinable w.r.t. ${\rm R},{{\rm X}}$ if it is $\omega$-level parallel joinable up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$. The following notion will be applied for non-overlays of the form $(1,1)$: ###### Definition 8.3 ($\omega$-Level Anti-Closed) Let $\beta\preceq\omega$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\omega$-level anti-closed up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&0{\,\prec\,}n\\\ {\wedge}&n\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\mathchar 259\relax_{i}\preceq n\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-level confluent up to }\delta\\\ {\wedge}&{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\omega\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}t_{1}\varphi\\\ \end{array}}}\right)}}.$ It is called $\omega$-level anti-closed w.r.t. ${\rm R},{{\rm X}}$ if it is $\omega$-level anti-closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$. The following notion will be applied for critical peaks of the form $(1,1)$: ###### Definition 8.4 ($\omega$-Level Strongly Joinable) Let $\beta\preceq\omega$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\omega$-level strongly joinable up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n{\,\succ\,}0\\\ {\wedge}&n\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\mathchar 259\relax_{i}\preceq n\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-level confluent up to }\delta\\\ {\wedge}&{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\omega\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}t_{1}\varphi\\\ \end{array}}}\right)}}.$ It is called $\omega$-level strongly joinable w.r.t. ${\rm R},{{\rm X}}$ if it is $\omega$-level strongly joinable up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$. The following notion will be applied for non-overlays of the form $(1,1)$: ###### Definition 8.5 ($\omega$-Level Closed) Let $\beta\preceq\omega$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\omega$-level closed up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&0{\,\prec\,}n\\\ {\wedge}&n\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\mathchar 259\relax_{i}\preceq n\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-level confluent up to }\delta\\\ {\wedge}&{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\omega\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}t_{1}\varphi\\\ \end{array}}}\right)}}.$ It is called $\omega$-level closed w.r.t. ${\rm R},{{\rm X}}$ if it is $\omega$-level closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$. The following notion will be applied for critical peaks of the form $(1,1)$: ###### Definition 8.6 ($\omega$-Level Weak Parallel Joinable) Let $\beta\preceq\omega$. A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\omega$-level weak parallel joinable up to $\beta$ w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall n\prec\omega{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n{\,\succ\,}0\\\ {\wedge}&n\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\mathchar 259\relax_{i}\preceq n\\\ {\wedge}&D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\\\ \end{array}}}\right)}}\\\ {\wedge}&\forall\delta{\,\prec\,}n{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-level confluent up to }\delta\\\ {\wedge}&{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\omega\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}t_{1}\varphi\\\ \end{array}}}\right)}}.$ It is called $\omega$-level weak parallel joinable w.r.t. ${\rm R},{{\rm X}}$ if it is $\omega$-level weak parallel joinable up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$. Overview over sophisticated forms of $\omega$-Level … of $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ Generally assumed condition for $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$; $n\prec\omega$: ${\left({{\begin{array}[]{ll}&0{\,\prec\,}n\ \ {\wedge}\penalty-2\ \ n\preceq\beta\\\ {\wedge}&\forall i\prec 2{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\mathchar 259\relax_{i}\preceq n\ \ {\wedge}\penalty-2\ \ D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\end{array}\right)}\\\ {\wedge}&\forall\delta{\,\prec\,}n{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-level confluent up to }\delta\\\ {\wedge}&{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\omega\\\ \end{array}}}\right)}$ Required conclusion (P := Parallel; C := Closed; J := Joinable; W := Weak; A := Anti-; S := Strongly): ## 9 Quasi Overlay Joinability According to Theorem 4 of Dershowitz &al. (1988), a terminating positive conditional rule system is confluent if it is overlay joinable. The remainder of this section is only relevant for Theorem 14.7 and even this can be applied without knowing about $\rhd$-quasi overlay joinability when one just knows: ###### Lemma 9.1 ( Overlay Joinable $\Rightarrow$ $\rhd$-Quasi Overlay Joinable ) W.r.t. ${\rm R},{{\rm X}}$ the following holds for each critical peak: If it is overlay joinable, then it is $\rhd$-quasi overlay joinable. In Wirth & Gramlich (1994a) we introduced the following definition: A critical peak $((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)$ is quasi overlay joinable w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}(D_{0}D_{1})\sigma\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\end{array}\right)}\\\ {\Rightarrow}&{{\left({{\begin{array}[]{ll}&t_{1}\sigma\varphi{\,=\,}\penalty-1{t_{0}\sigma\varphi\penalty-1{[\,p\leftarrow t_{1}\sigma\varphi/p\,]}}\\\ {\wedge}&(t_{0}/p)\sigma\varphi\,\,{\downarrow_{{}_{{\rm R},{{\rm X}}}}}\,\,t_{1}\sigma\varphi/p\,\,\,{{({{\longleftarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\cup{\lhd_{{}_{\rm ST}}})}^{\scriptscriptstyle+}}\,\,\,(\hat{t}/p)\sigma\varphi\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}.$ This notion of quasi overlay joinability, however, has turned out to produce a wondrous effect in case that for some critical peak, w.l.o.g. say $(({l_{1}\penalty-1{[\,p\leftarrow r_{0}\,]}},C_{0},\mathchar 259\relax_{0}),\ (r_{1},C_{1},\mathchar 259\relax_{1}),\ l_{1},\ \sigma,\ p\ )$ generated by two rules $((l_{0},r_{0}),C_{0})$, $((l_{1},r_{1}),C_{1})$ (with w.l.o.g. no variables in common) due to $\sigma{\,=\,}\penalty-1{{\rm mgu}({\\{(l_{0},l_{1}/p)\\}},{{\rm Y}})}$ for ${\rm Y}:={{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})},{((l_{1},r_{1}),C_{1})}})},$ and for some $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $(C_{0}\,C_{1})\sigma\varphi$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$, there is some $p^{\prime}\in{{{\mathcal{POS}}}({l_{1}})}{\setminus}\\{p\\}$ with $l_{1}/p^{\prime}{\,\not\in\,}{{\rm V}}$ and $l_{0}\sigma\varphi{\,=\,}\penalty-1(l_{1}/p^{\prime})\sigma\varphi;$ i.e. the left-hand side of the rule $((l_{0},r_{0}),C_{0})$ occurs a second time in the instantiated peak term (or superposition term) at a non-variable position $p^{\prime}$. In this case due to $l_{0}\sigma\varphi{\,=\,}\penalty-1(l_{1}/p^{\prime})\sigma\varphi$ there are $\sigma^{\prime}{\,=\,}\penalty-1{{\rm mgu}({\\{(l_{0},l_{1}/p^{\prime})\\}},{{\rm Y}})}$ and $\varphi^{\prime}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma^{\prime}\varphi^{\prime})}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}$ and then (unless ${l_{1}\penalty-1{[\,p^{\prime}\leftarrow r_{0}\,]}}\sigma^{\prime}{\,=\,}\penalty-1r_{1}\sigma^{\prime}$) we get another critical peak $(({l_{1}\penalty-1{[\,p^{\prime}\leftarrow r_{0}\,]}},C_{0},\mathchar 259\relax_{0}),\ (r_{1},C_{1},\mathchar 259\relax_{1}),\ l_{1},\ \sigma^{\prime},\ p^{\prime}\ ).$ Now (since $(C_{0}\,C_{1})\sigma^{\prime}\varphi^{\prime}=(C_{0}\,C_{1})\sigma\varphi$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$), if both critical peaks are quasi overlay joinable, then we get by the first conclusion in the above definition: $\begin{array}[t]{lll@{}l@{}l@{}ll}r_{1}\sigma\varphi&{\,=\penalty-1}&l_{1}&{[\,p\leftarrow r_{0}\,]}&\sigma\varphi&{[\,p\leftarrow r_{1}\sigma\varphi/p\,]}&;\\\ r_{1}\sigma^{\prime}\varphi^{\prime}&{\,=\penalty-1}&l_{1}&{[\,p^{\prime}\leftarrow r_{0}\,]}&\sigma^{\prime}\varphi^{\prime}&{[\,p^{\prime}\leftarrow r_{1}\sigma^{\prime}\varphi^{\prime}/p^{\prime}\,]}&\\\ \end{array}$ (unless ${l_{1}\penalty-1{[\,p^{\prime}\leftarrow r_{0}\,]}}\sigma^{\prime}\varphi^{\prime}{\,=\,}\penalty-1r_{1}\sigma^{\prime}\varphi^{\prime}$). Simplified, this means: $\begin{array}[t]{llll}r_{1}\sigma\varphi&{\,=\penalty-1}&l_{1}\sigma\varphi{[\,p\leftarrow r_{1}\sigma\varphi/p\,]}&;\\\ r_{1}\sigma\varphi&{\,=\penalty-1}&l_{1}\sigma\varphi{[\,p^{\prime}\leftarrow r_{1}\sigma\varphi/p^{\prime}\,]}&\\\ \end{array}$ (unless $r_{1}\sigma\varphi{\,=\,}\penalty-1{l_{1}\sigma\varphi\penalty-1{[\,p^{\prime}\leftarrow r_{0}\sigma\varphi\,]}}$). Thus, in any case, we get ${l_{1}\sigma\varphi\penalty-1{[\,p\leftarrow\ldots\,]}}{\,=\,}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1{l_{1}\sigma\varphi\penalty-1{[\,p^{\prime}\leftarrow\ldots\,]}}.$ Since (due to $p{\,\not=\,}p^{\prime}$ and $(l_{1}/p^{\prime})\sigma\varphi{\,=\,}\penalty-1l_{0}\sigma\varphi{\,=\,}\penalty-1(l_{1}/p)\sigma\varphi$) we have ${{p^{\prime}}\,{\parallel}\,{p}},$ this has the wondrous result $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}l_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\sigma\varphi.$ (!) Using the second conclusion of the quasi overlay joinability we get $l_{1}\sigma\varphi/p{\,=\,}\penalty-1r_{1}\sigma\varphi/p\;{{{{({\longleftarrow}\cup{\lhd_{{}_{\rm ST}}})}}}^{\scriptscriptstyle+}}\;(l_{1}/p)\sigma\varphi$ which implies $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}l_{0}\sigma\varphi\,\,{{{{({\longleftarrow}\cup{\lhd_{{}_{\rm ST}}})}}}^{\scriptscriptstyle+}}\,\,l_{0}\sigma\varphi.$ (!!) Since both results (!) and (!!) are absurd for a property which is only to be used for a noethe-rian reduction relation ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$, we now generalize our notion of quasi overlay joinability. ###### Definition 9.2 ($\rhd$-Quasi Overlay Joinability) A critical peak $((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)$ is $\rhd$-quasi overlay joinable w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,\forall\mathchar 257\relax{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&(D_{0}D_{1})\sigma\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\\\ {\wedge}&\mathchar 257\relax{\,=\,}\penalty-1{{\\{\ }p^{\prime}{\,\in\,}{{{\mathcal{POS}}}({\hat{t}})}{\setminus}\\{p\\}}~{}{\|}\penalty-9\,\ {\hat{t}/p^{\prime}{\,\not\in\,}{{\rm V}}\ {\wedge}\penalty-2\ (\hat{t}/p^{\prime})\sigma\varphi{\,=\,}\penalty-1(\hat{t}/p)\sigma\varphi{\ \\}}}\\\ {\wedge}&\forall w\,{{({{\longleftarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}{\cup}\,\lhd)}^{\scriptscriptstyle+}}\,\,(\hat{t}/p)\sigma\varphi{.}\penalty-1\,\,\ {{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\mbox{ is confluent below }w\\\ {\wedge}&\forall p^{\prime\prime}{\,\in\,}{{{\mathcal{POS}}}({(\hat{t}/p)\sigma\varphi})}{\setminus}\\{\emptyset\\}{.}\penalty-1\,\,(\hat{t}/p)\sigma\varphi/p^{\prime\prime}{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&\exists\bar{n}{\,\in\,}{{\bf N}}{.}\penalty-1\,\,\exists\bar{p}{.}\penalty-1\,\,\exists\bar{u}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{{\bar{p}}:{{{\\{0,\ldots,\bar{n}{-}1\\}}\rightarrow{{{\bf N}}^{\ast}}}}}}\\\ {\wedge}&{{{\bar{u}}:{{{\\{0,\ldots,\bar{n}\\}}\rightarrow{{\mathcal{T}}}}}}}\\\ {\wedge}&{t_{0}\penalty-1{{[\,p^{\prime}\leftarrow t_{0}/p\ \|\ p^{\prime}{\,\in\,}\mathchar 257\relax\,]}}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}\bar{u}_{\bar{n}}\\\ {\wedge}&\forall i{\,\prec\,}\bar{n}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\bar{u}_{i+1}{\,=\,}\penalty-1{\bar{u}_{i}\penalty-1{[\,\bar{p}_{i}\leftarrow\bar{u}_{i+1}/\bar{p}_{i}\,]}}\\\ {\wedge}&\bar{u}_{i+1}/\bar{p}_{i}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}\bar{u}_{i}/\bar{p}_{i}\;{{({{\longleftarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}{\cup}\,\lhd)}^{\scriptscriptstyle+}}\;(\hat{t}/p)\sigma\varphi\\\ \end{array}}}\right)}}\\\ {\wedge}&\bar{u}_{0}{\,=\,}\penalty-1t_{1}\sigma\varphi\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}.$ For $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ and $\mathchar 257\relax\subseteq{{{\mathcal{POS}}}({\hat{t}})}{\setminus}\\{\emptyset\\}$ with $(D_{0}D_{1})\sigma\varphi$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ and $\forall p^{\prime}{\,\in\,}\mathchar 257\relax{.}\penalty-1\,\,{(\ \hat{t}/p^{\prime}{\,\not\in\,}{{\rm V}}\ \ {\wedge}\penalty-2\ \ (\hat{t}/p^{\prime})\sigma\varphi{\,=\,}\penalty-1(\hat{t}/p)\sigma\varphi\ )}$ the critical peak, the further reduction of its left part, and the required joinability after this reduction can be depicted as follows:202020It should be noted that the fact that the parallel reduction can be restricted not only to non-variable positions of $\hat{t}$ but also to the same identical redex $(\hat{t}/p)\sigma\varphi$ (and the necessity of the analogous restriction in the proof) was especially brought to our attention by Bernhard Gramlich (cf. Gramlich (1995b)) who already had similar but less general ideas on the weakening of overlay joinability. It is rather easy to see that $\rhd_{{}_{\rm ST}}$-quasi overlay joinability of a critical peak generalizes the old notion of quasi overlay joinability: In case that $\mathchar 257\relax{\,=\,}\penalty-1\emptyset:$ For quasi overlay joinability of $((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)$, i.e. for $t_{1}\sigma\varphi{\,=\,}\penalty-1{t_{0}\sigma\varphi\penalty-1{[\,p\leftarrow t_{1}\sigma\varphi/p\,]}};$ $(t_{0}/p)\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}w{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t_{1}\sigma\varphi/p\,{{({{\longleftarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\cup{\lhd_{{}_{\rm ST}}})}^{\scriptscriptstyle+}}\,(\hat{t}/p)\sigma\varphi;$ we simply choose $\bar{n}:=1;$ $\bar{u}_{0}:=t_{1}\sigma\varphi;$ $\bar{u}_{1}:={\bar{u}_{0}\penalty-1{[\,p\leftarrow w\,]}};$ and get $t_{0}\sigma\varphi{\,=\,}\penalty-1{t_{1}\sigma\varphi\penalty-1{[\,p\leftarrow t_{0}\sigma\varphi/p\,]}}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}\penalty-1{t_{1}\sigma\varphi\penalty-1{[\,p\leftarrow w\,]}}{\,=\,}\penalty-1\bar{u}_{1}$ and $\bar{u}_{1}/p{\,=\,}\penalty-1w{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t_{1}\sigma\varphi/p{\,=\,}\penalty-1\bar{u}_{0}/p{\,=\,}\penalty-1t_{1}\sigma\varphi/p\,{{({{\longleftarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\cup{\lhd_{{}_{\rm ST}}})}^{\scriptscriptstyle+}}\,(\hat{t}/p)\sigma\varphi.$ In case that $\mathchar 257\relax{\,\not=\,}\emptyset:$ $\rhd_{{}_{\rm ST}}$-quasi overlay joinability of some critical peak, w.l.o.g. say $(({l_{1}\penalty-1{[\,p\leftarrow r_{0}\,]}},C_{0},\mathchar 259\relax_{0}),\ (r_{1},C_{1},\mathchar 259\relax_{1}),\ l_{1},\ \sigma,\ p\ )$ generated by two rules $((l_{0},r_{0}),C_{0})$, $((l_{1},r_{1}),C_{1})$ (with w.l.o.g. no variables in common) due to $\sigma{\,=\,}\penalty-1{{\rm mgu}({\\{(l_{0},l_{1}/p)\\}},{{\rm Y}})}$ for ${\rm Y}:={{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})},{((l_{1},r_{1}),C_{1})}})},$ generalizes quasi overlay joinability of the critical peaks resulting from overlapping $((l_{0},r_{0}),C_{0})$ into $((l_{1},r_{1}),C_{1})$. While we are not going to discuss the (then obvious) general case in detail here, the case of $\mathchar 257\relax{\,=\,}\penalty-1\\{p^{\prime}\\}$ was just discussed before the definition above and we complete this discussion now as follows: Defining $\hat{t}:=l_{1};$ $t_{0}:={l_{1}\penalty-1{[\,p\leftarrow r_{0}\,]}};$ $t_{1}:=r_{1};$ $\bar{n}:=2;$ $\bar{u}_{0}:=t_{1}\sigma\varphi;$ $\bar{u}_{1}:={\bar{u}_{0}\penalty-1{[\,p\leftarrow r_{0}\sigma\varphi\,]}};$ $\bar{u}_{2}:={\bar{u}_{1}\penalty-1{[\,p^{\prime}\leftarrow r_{0}\sigma\varphi\,]}};$ due to (!) we have ${t_{0}\penalty-1{{[\,p^{\prime}\leftarrow t_{0}/p\ \|\ p^{\prime}{\,\in\,}\mathchar 257\relax\,]}}}\sigma\varphi{\,=\,}\penalty-1{{l_{1}\penalty-1{[\,p\leftarrow r_{0}\,]}}\penalty-1{[\,p^{\prime}\leftarrow r_{0}\,]}}\sigma\varphi{\,=\,}\penalty-1{{r_{1}\sigma\varphi\penalty-1{[\,p\leftarrow r_{0}\sigma\varphi\,]}}\penalty-1{[\,p^{\prime}\leftarrow r_{0}\sigma\varphi\,]}}{\,=\,}\penalty-1\bar{u}_{2}$ and due to (!!) we have $\bar{u}_{2}/p^{\prime}{\,=\,}\penalty-1\bar{u}_{1}/p{\,=\,}\penalty-1r_{0}\sigma\varphi{{\longleftarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}l_{0}\sigma\varphi\,{{{{({\longleftarrow}\cup{\lhd_{{}_{\rm ST}}})}}}^{\scriptscriptstyle+}}\,l_{0}\sigma\varphi{\,=\,}\penalty-1(\hat{t}/p)\sigma\varphi$ where by (!) $l_{0}\sigma\varphi{\,=\,}\penalty-1(l_{1}/p)\sigma\varphi{\,=\,}\penalty-1l_{1}\sigma\varphi/p{\,=\,}\penalty-1t_{1}\sigma\varphi/p{\,=\,}\penalty-1\bar{u}_{0}/p$ and $l_{0}\sigma\varphi{\,=\,}\penalty-1(l_{1}/p)\sigma\varphi{\,=\,}\penalty-1(l_{1}/p^{\prime})\sigma\varphi{\,=\,}\penalty-1l_{1}\sigma\varphi/p^{\prime}{\,=\,}\penalty-1t_{1}\sigma\varphi/p^{\prime}{\,=\,}\penalty-1\bar{u}_{0}/p^{\prime}{\,=\,}\penalty-1\bar{u}_{1}/p^{\prime}.$ In the case of an arbitrary $\mathchar 257\relax{\,\not=\,}\emptyset,$ quasi overlay joinability of any two of the critical peaks involved implies that the diagram from above then looks the following way (where $\bar{n}:={\,\|{\\{p\\}{\cup}\mathchar 257\relax}\|\,}$): That the wondrous results of quasi overlay joinability in the above reported case can be overcome with the new notion of $\rhd$-quasi overlay joinability can be seen from the following example: ###### Example 9.3 $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{\mathsf{f}},{\mathsf{g}},{{\mathsf{a}}},{{\mathsf{c}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{+}}\\}\\\ {\rm R}_{\,\rm\ref{ex overlay problems overcome}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}ll}{{{\mathsf{f}}{(}{X}{)}}}&=&{{{\mathsf{g}}{(}{X}{)}}}\\\ {{\mathsf{a}}}&=&{{\mathsf{c}}}\\\ {{{{\mathsf{f}}{(}{X}{)}}}\,{\mathsf{+}}\,{{{\mathsf{f}}{(}{X}{)}}}}&=&{{\mathsf{a}}}\\\ {{{{\mathsf{g}}{(}{X}{)}}}\,{\mathsf{+}}\,{{{\mathsf{g}}{(}{X}{)}}}}&=&{{\mathsf{c}}}&{\longleftarrow}\ {{{{\mathsf{f}}{(}{X}{)}}}\,{\mathsf{+}}\,{{{\mathsf{f}}{(}{X}{)}}}}\;{=}\;{{\mathsf{c}}}\end{array}$}\end{array}$ Now the unconditional version of ${\rm R}_{\,\rm\ref{ex overlay problems overcome}}$ is compatible with the lexicographic path ordering $\rhd$ resulting from the following precedence on function symbols (in decreasing order): $\mathsf{f}$, $\mathsf{g}$, ${\mathsf{a}}$, ${\mathsf{c}}$. The critical peak $(({{{{\mathsf{g}}{(}{X}{)}}}\,{\mathsf{+}}\,{{{\mathsf{f}}{(}{X}{)}}}},\emptyset,0),\ ({{\mathsf{a}}},\emptyset,1),\ {{{{\mathsf{f}}{(}{X}{)}}}\,{\mathsf{+}}\,{{{\mathsf{f}}{(}{X}{)}}}},\ \emptyset,\ 1\ )$ cannot be quasi overlay joinable because ${{\mathsf{a}}}/1$ is undefined. It is, however, $\rhd$-quasi overlay joinable: That the $\mathchar 257\relax$ in the notion of ${\rhd_{{}_{\rm ST}}}$-quasi overlay joinability cannot be restricted to be empty can be seen from Example 12.2. ## 10 Some Unconditional Examples Our main goal in this and the following sections is to find confluence criteria that do not depend on termination arguments but on the structure of the joinability of critical peaks only. Finally in 14 we will investigate how termination can strengthen our criteria. Up to then, however, we are not going to use termination arguments. Instead, we are looking for confluence criteria of the form “If all critical peaks of a … (e.g. normal, left-linear, &c.) rule system are joinable according to the pattern … (e.g. shallow joinable, parallel closed, &c.) then the reduction relation is confluent.” First we want to make clear that this approach has its limits. We do this by giving some examples. To distinguish confluent from non-confluent examples the rule systems of the latter ones are displayed in a box at the right margin while in a connected box to the left we list the example’s crucial properties, concerning joinability structure of their critical peaks, variable occurrence, condition properties, &c.. The reader should not try to understand the sophisticated joinability labels in the boxes at a first reading. This is not necessary for understanding the examples. The sophisticated joinability labels are only needed for 13. In this section we start with some unconditional examples. The first one shows that left-linearity is essential212121Since this counterexample for confluence is unconditional it must be non-terminating of course. For conditional systems, however, left-linearity is essential also for terminating systems for joinability of critical peaks to imply confluence, cf. the transformation described in 11 applied to Example 11.3 as described in 11.: ###### Example 10.1 (Huet (1980)) No Critical Peaks Not Left-Linear Unconditional Not Terminating $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{0}}},{\mathsf{s}},{{\mathsf{c}}},{{\mathsf{d}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{+}}\\}\\\ \end{array}$ $\begin{array}[t]{\|lll}{\rm R}_{\,\rm\ref{ex not left linear}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l}{{\mathsf{0}}}&=&{{\mathsf{s}}{(}{{{\mathsf{0}}}}{)}}\\\ {{X}\,{\mathsf{+}}\,{X}}&=&{{\mathsf{c}}}\\\ {{X}\,{\mathsf{+}}\,{{{\mathsf{s}}{(}{X}{)}}}}&=&{{\mathsf{d}}}\end{array}$}\end{array}$ There are no critical peaks. Nevertheless, ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex not left linear}},\emptyset}}$ is not confluent: ###### Example 10.2 $\,\omega$-Level [[Weak] Parallel] Joinable $\,\omega$-Level Strongly Joinable Not $\omega$-Shallow [[Noisy] Parallel] Joinable up to $\omega$ Ground Unconditional Not Terminating $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{a}}},{{\mathsf{b}}},{{\mathsf{c}}},{{\mathsf{d}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{f}}\\}\\\ {\rm R}_{\,\rm\ref{ex a}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l}{{\mathsf{a}}}&=&{{\mathsf{c}}}\\\ {{\mathsf{b}}}&=&{{\mathsf{d}}}\\\ {{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}}&=&{{\mathsf{f}}{(}{{{\mathsf{b}}}}{)}}\\\ {{\mathsf{f}}{(}{{{\mathsf{b}}}}{)}}&=&{{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}}\end{array}$}\end{array}$ The critical peaks are all of the form $(0,1)$ and can be closed as follows: However, ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex a}},\emptyset}}$ is not confluent: ###### Example 10.3 $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{0}}},{\mathsf{s}},{\mathsf{p}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{+}}\\}\\\ \end{array}$ $\begin{array}[t]{\|lll}{\rm R}_{\,\rm\ref{ex b}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l}{{\mathsf{s}}{(}{{{\mathsf{p}}{(}{X}{)}}}{)}}&=&X\\\ {{\mathsf{p}}{(}{{{\mathsf{s}}{(}{X}{)}}}{)}}&=&X\\\ {{{{\mathsf{0}}}}\,{\mathsf{+}}\,{Y}}&=&Y\\\ {{{{\mathsf{s}}{(}{X}{)}}}\,{\mathsf{+}}\,{Y}}&=&{{\mathsf{s}}{(}{{{X}\,{\mathsf{+}}\,{Y}}}{)}}\\\ {{{{\mathsf{p}}{(}{X}{)}}}\,{\mathsf{+}}\,{Y}}&=&{{\mathsf{p}}{(}{{{X}\,{\mathsf{+}}\,{Y}}}{)}}\end{array}$}\end{array}$ The critical peaks are all of the form $(0,1)$ and can be closed as follows: Since the reduction relation is terminating, we have confluence here. However, note that the structure of the joinability of the critical peaks is identical to that of Example 10.2 (with the exception of the positions). Thus, argumentation on the joinability structure of critical peaks must fail to infer confluence for this example (at least if we do not take positions into account). The following example results from Example 10.2 just by changing ‘${\mathsf{a}}$’ and ‘${\mathsf{b}}$’ into non-constructors. While Example 10.2 was able to discourage generalizations of Theorem 13.9, by the slight change the following example is able to discourage generalizations of Theorem 13.6 regarding the required $\omega$-shallow parallel closedness (for part (I) of Theorem 13.6), $\omega$-shallow noisy anti-closedness (for part (II)), or $\omega$-shallow closedness (for parts (III) and (IV)) of the non-overlays of the form $(1,1)$. ###### Example 10.4 $\,\omega$-Shallow [[Noisy] [Weak] Parallel] Joinable $\,\omega$-Shallow [Noisy] Strongly Joinable Non-Overlay is Not $\omega$-Shallow [Parallel] Closed Non-Overlay is Not $\omega$-Shallow [Noisy] Anti-Closed Ground Unconditional Not Terminating $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{c}}},{{\mathsf{d}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{{\mathsf{a}}},{{\mathsf{b}}},{\mathsf{f}}\\}\\\ {\rm R}_{\,\rm\ref{ex a modified}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l}{{\mathsf{a}}}&=&{{\mathsf{c}}}\\\ {{\mathsf{b}}}&=&{{\mathsf{d}}}\\\ {{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}}&=&{{\mathsf{f}}{(}{{{\mathsf{b}}}}{)}}\\\ {{\mathsf{f}}{(}{{{\mathsf{b}}}}{)}}&=&{{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}}\end{array}$}\end{array}$ The critical peaks are all of the form $(1,1)$ now and can be closed as follows: However, ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex a modified}},\emptyset}}$ is not confluent: ###### Example 10.5 $\\!\rm{\rhd_{{}_{\rm ST}}}$-Quasi Overlay Joinable $\,\omega$-Shallow [[Noisy] Weak Parallel] Joinable Not $\omega$-Shallow [Noisy] Parallel Joinable up to $\omega$ Not $\omega$-Shallow [Noisy] Strongly Joinable up to $\omega$ Ground Unconditional Not Terminating $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{a}}},{{\mathsf{b}}},{{\mathsf{c}}},{{\mathsf{d}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{f}}\\}\\\ {\rm R}_{\,\rm\ref{ex c}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l}{{\mathsf{a}}}&=&{{\mathsf{b}}}\\\ {{\mathsf{b}}}&=&{{\mathsf{a}}}\\\ {{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}}&=&{{\mathsf{c}}}\\\ {{\mathsf{f}}{(}{{{\mathsf{b}}}}{)}}&=&{{\mathsf{d}}}\end{array}$}\end{array}$ The critical peaks are all of the form $(0,1)$ and can be closed as follows: However, ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex c}},\emptyset}}$ is not confluent: ###### Example 10.6 $\begin{array}[t]{lll\|}{{\mathbb{C}}}&:=&\\{{{\mathsf{0}}},{\mathsf{s}},{\mathsf{p}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{+}}\\}\\\ \end{array}$ $\begin{array}[t]{lll}{\rm R}_{\,\rm\ref{ex d}}&:&{\rm R}_{\,\rm\ref{ex b}}\mbox{~{}~{}}+\mbox{~{}~{}}\mbox{$\begin{array}[t]{r@{\,}l@{\,}l}X&=&{{\mathsf{s}}{(}{{{\mathsf{p}}{(}{X}{)}}}{)}}\\\ X&=&{{\mathsf{p}}{(}{{{\mathsf{s}}{(}{X}{)}}}{)}}\end{array}$}\end{array}$ Note that we have added two rules to the system from Example 10.3: The critical peaks of the form $(0,1)$ of Example 10.3 still exist but can now be closed in different way; e.g., the first one can be closed as follows: Since ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex b}},\emptyset}}$ is confluent and ${{\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex b}},\emptyset}}}\subseteq{{\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex d}},\emptyset}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftrightarrow}}}}}_{{}_{\\!{\rm R}_{\,\rm\ref{ex b}},\emptyset}}},$ ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex d}},\emptyset}}$ is confluent, too (cf. Lemma 3.4). However, note that the structure of the joinability of the critical peaks is identical to that of Example 10.5. Thus, argumentation on the joinability structure of critical peaks must fail to infer confluence for this example. According to Lemma 3.2 of Huet (1980), unconditional left- and right-linear rule systems with strongly joinable critical peaks are [strongly] confluent. That the severe restriction of right-linearity is essential here can be seen from the following example: ###### Example 10.7 (Jean-Jacque Lévy as cited in Huet (1980)) $\omega$-Level [Parallel] Joinable [$\omega$-Level] [Strongly] Joinable Not $\omega$-Shallow [Noisy] Parallel Joinable up to $\omega$ Not $\omega$-Shallow [Noisy] Strongly Joinable up to $\omega$ Left-Linear Right-Linear Constructor Rules Not Right-Linear Unconditional Not Terminating --- $\mid$ [$\omega$-Shallow] Joinable Not $\omega$-Shallow [Noisy] Parallel Joinable up to $\omega$ Not $\omega$-Shallow [Noisy] Strongly Joinable up to $\omega$ Left-Linear Right-Linear Constructor Rules Not Right-Linear Unconditional Not Terminating $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{a}}},{{\mathsf{b}}},{{\mathsf{c}}},{{\mathsf{d}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{+}},{\mathsf{-}}\\}\\\ {\rm R}_{\,\rm\ref{ex levy a}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l}{{\mathsf{a}}}&=&{{\mathsf{c}}}\\\ {{\mathsf{b}}}&=&{{\mathsf{d}}}\\\ {{{{\mathsf{a}}}}\,{\mathsf{+}}\,{{{\mathsf{a}}}}}&=&{{{{\mathsf{b}}}}\,{\mathsf{-}}\,{{{\mathsf{b}}}}}\\\ {{{{\mathsf{c}}}}\,{\mathsf{+}}\,{X}}&=&{{X}\,{\mathsf{+}}\,{X}}\\\ {{X}\,{\mathsf{+}}\,{{{\mathsf{c}}}}}&=&{{X}\,{\mathsf{+}}\,{X}}\\\ {{{{\mathsf{b}}}}\,{\mathsf{-}}\,{{{\mathsf{b}}}}}&=&{{{{\mathsf{a}}}}\,{\mathsf{+}}\,{{{\mathsf{a}}}}}\\\ {{{{\mathsf{d}}}}\,{\mathsf{-}}\,{X}}&=&{{X}\,{\mathsf{-}}\,{X}}\\\ {{X}\,{\mathsf{-}}\,{{{\mathsf{d}}}}}&=&{{X}\,{\mathsf{-}}\,{X}}\end{array}$}\end{array}$ There are only four critical peaks and they are all of the form $(0,1)$. Using the symmetry of $\mathsf{+}$ in its arguments as well the symmetry of ${\mathsf{a}}$, ${\mathsf{c}}$, $\mathsf{+}$ with ${\mathsf{b}}$, ${\mathsf{d}}$, $\mathsf{-}$, all other critical peaks are symmetric to the following one, which can be closed in the following two different ways: Nevertheless, ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex levy a}},\emptyset}}$ is not confluent: We now use the same ${\rm R}_{\,\rm\ref{ex levy a}}$ to show that even another structure of joinability is insufficient for confluence. We do this by changing the separation into constructors and non-constructors: $\omega$-Shallow [[Noisy] [Weak] Parallel] Joinable Non-Overlay is Not $\omega$-Shallow [Parallel] Closed [$\omega$-Shallow] Strongly Joinable $\omega$-Shallow Anti-Closed Left-Linear [Constructor Rules] Not Right-Linear Unconditional Not Terminating --- $\mid$ $\omega$-Shallow [[Noisy] Weak Parallel] Joinable Non-Overlay is Not $\omega$-Shallow [Parallel] Closed $\omega$-Shallow Strongly Joinable $\omega$-Shallow Anti-Closed Left-Linear [Constructor Rules] Not Right-Linear Unconditional Not Terminating $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{c}}},{{\mathsf{d}}},{\mathsf{+}},{\mathsf{-}}\\}\\\ {{\mathbb{N}}}&:=&\\{{{\mathsf{a}}},{{\mathsf{b}}}\\}\\\ {\rm R}_{\,\rm\ref{ex levy a}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l}{{{{\mathsf{c}}}}\,{\mathsf{+}}\,{X}}&=&{{X}\,{\mathsf{+}}\,{X}}\\\ {{X}\,{\mathsf{+}}\,{{{\mathsf{c}}}}}&=&{{X}\,{\mathsf{+}}\,{X}}\\\ {{{{\mathsf{d}}}}\,{\mathsf{-}}\,{X}}&=&{{X}\,{\mathsf{-}}\,{X}}\\\ {{X}\,{\mathsf{-}}\,{{{\mathsf{d}}}}}&=&{{X}\,{\mathsf{-}}\,{X}}\\\ {{\mathsf{a}}}&=&{{\mathsf{c}}}\\\ {{\mathsf{b}}}&=&{{\mathsf{d}}}\\\ {{{{\mathsf{a}}}}\,{\mathsf{+}}\,{{{\mathsf{a}}}}}&=&{{{{\mathsf{b}}}}\,{\mathsf{-}}\,{{{\mathsf{b}}}}}\\\ {{{{\mathsf{b}}}}\,{\mathsf{-}}\,{{{\mathsf{b}}}}}&=&{{{{\mathsf{a}}}}\,{\mathsf{+}}\,{{{\mathsf{a}}}}}\end{array}$}\end{array}$ Note that the rule system is not changed, but only reordered to have the constructor rules precede the non-constructor rules. The rewrite relation ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex levy a}},\emptyset}}$ is not changed by this constructor re-declaration. (Note $X{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}$.) The critical peaks only have changed their form from $(0,1)$ to $(1,1)$ and are still all symmetric to the following one that closes in the two following ways: Finally, the divergence looks the following way now (Please note that now ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex levy a}},\emptyset,\omega}}$ and ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex levy a}},\emptyset}}$ are commuting, which was not the case before.): The following example is a slight variation of Example 10.7 which is interesting w.r.t. Example 10.9. ###### Example 10.8 $\omega$-Shallow [[Noisy] Parallel] Joinable Non-Overlay is Not $\omega$-Shallow [Parallel] Closed $\omega$-Shallow Strongly Joinable $\omega$-Shallow Anti-Closed Left-Linear [Constructor Rules] Not Right-Linear Unconditional Not Terminating --- $\mid$ $\omega$-Shallow [[Noisy] Weak Parallel] Joinable Non-Overlay is Not $\omega$-Shallow [Parallel] Closed $\omega$-Shallow Strongly Joinable $\omega$-Shallow Anti-Closed Left-Linear [Constructor Rules] Not Right-Linear Unconditional Not Terminating $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{c}}},{{\mathsf{d}}},{\mathsf{+}},{\mathsf{-}},{\mathsf{f}},{\mathsf{g}}\\}\\\ {{\mathbb{N}}}&:=&\\{{{\mathsf{a}}},{{\mathsf{b}}}\\}\\\ {\rm R}_{\,\rm\ref{ex for asso}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l}{{{{\mathsf{c}}}}\,{\mathsf{+}}\,{X}}&=&{{X}\,{\mathsf{+}}\,{{{\mathsf{f}}{(}{X}{)}}}}\\\ {{X}\,{\mathsf{+}}\,{{{\mathsf{c}}}}}&=&{{X}\,{\mathsf{+}}\,{{{\mathsf{f}}{(}{X}{)}}}}\\\ {{\mathsf{f}}{(}{X}{)}}&=&X\\\ {{{{\mathsf{d}}}}\,{\mathsf{-}}\,{X}}&=&{{X}\,{\mathsf{-}}\,{{{\mathsf{g}}{(}{X}{)}}}}\\\ {{X}\,{\mathsf{-}}\,{{{\mathsf{d}}}}}&=&{{X}\,{\mathsf{-}}\,{{{\mathsf{g}}{(}{X}{)}}}}\\\ {{\mathsf{g}}{(}{X}{)}}&=&X\\\ {{\mathsf{a}}}&=&{{\mathsf{c}}}\\\ {{\mathsf{b}}}&=&{{\mathsf{d}}}\\\ {{{{\mathsf{a}}}}\,{\mathsf{+}}\,{{{\mathsf{a}}}}}&=&{{{{\mathsf{b}}}}\,{\mathsf{-}}\,{{{\mathsf{b}}}}}\\\ {{{{\mathsf{b}}}}\,{\mathsf{-}}\,{{{\mathsf{b}}}}}&=&{{{{\mathsf{a}}}}\,{\mathsf{+}}\,{{{\mathsf{a}}}}}\end{array}$}\end{array}$ There are only four critical peaks and they are all of the form $(1,1)$. Using the symmetry of $\mathsf{+}$ in its relevant arguments as well the symmetry of ${\mathsf{a}}$, ${\mathsf{c}}$, $\mathsf{+}$, $\mathsf{f}$ with ${\mathsf{b}}$, ${\mathsf{d}}$, $\mathsf{-}$, $\mathsf{g}$, all other critical peaks are symmetric to the following one, which can be closed in the following two different ways: Finally, the divergence looks the following way now: ###### Example 10.9 $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{0}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{+}}\\}\\\ {\rm R}_{\,\rm\ref{ex asso}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l}{{{({{X}\,{\mathsf{+}}\,{Y}})}}\,{\mathsf{+}}\,{Z}}&=&{{X}\,{\mathsf{+}}\,{{({{Y}\,{\mathsf{+}}\,{Z}})}}}\end{array}$}\end{array}$ There is only one critical peak. It is of the form $(1,1)$ and can be closed as follows: However, note that the structure of the joinability of the critical peak is weaker than the first alternative of Example 10.8. Thus, argumentation on the joinability structure of critical peaks must fail to infer confluence for this example. ## 11 Normality When we now start to consider conditional besides unconditional rule systems, the first to notice is that we have to impose some normality restriction, as can be seen from Example 11.2 below. A rule system is called normal if for all equations “$u_{0}{=}u_{1}$” in the condition lists of the rules, at least one of $u_{0}$, $u_{1}$ is an irreducible ground term. Normality is no serious restriction unless left-linearity is required, too. This is because each non-normal system can be transformed into a normal but then not left-linear system without changing the reduction relation on the old sorts: One just adds for each old sort $s$ a new constructor function symbol ${\mathsf{eq}}_{s}$ with arity $s\,s{\ {\rightarrow}\ }s_{\rm new}$ (where $s_{\rm new}$ is a new sort) and a new constructor constant symbol $\bot$ of the sort $s_{\rm new}$. Then in each condition of each rule one transforms each equation of the form “$u{=}v$” with $u,v{\,\in\,}{{\mathcal{T}}({{\rm sig},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}_{s}$ into “${{{\mathsf{eq}}_{s}}{(}{u}{,\,}{v}{)}}{=}\bot$” and adds for each old sort $s$ the rule ${{{\mathsf{eq}}_{s}}{(}{X_{s}}{,\,}{X_{s}}{)}}{\,=\,}\penalty-1\bot$ (where $X_{s}{\,\in\,}{{{\rm V}}\\!_{{{\rm SIG}},{s}}}$). Furthermore one adds the condition “${{{\mathsf{eq}}_{s}}{(}{{{\mathsf{a}}}}{,\,}{{{\mathsf{a}}}}{)}}{=}\bot$” to each unconditional rule for some arbitrary constant ${\mathsf{a}}$ of an arbitrary old sort $s$. The only change this transformation brings for the old sorts is that exactly those reductions which were possible with ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},n}}$ (for $n\prec\omega$) become exactly those reductions which are possible with ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},n+1}}$ after the transformation. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}$, however, is not changed by the transformation. E.g. for the rule system of Example 11.3 the transformation yields a $\omega$-shallow [parallel] joinable, terminating system that is normal now but not left-linear anymore. Now we return to the question whether joinability implies confluence. While Lemma 5.1 states the converse, actually little is known about the other direction unless the rule system is decreasing. Theorems 1 (which is taken from Bergstra & Klop (1986)) and 2 of Dershowitz &al. (1988) state that left- linear and normal rule systems are confluent if they have no critical pairs or are both shallow joinable and terminating. That normality is essential to imply confluence of systems with no critical pairs can be seen from Example 11.2. That normality is also essential to imply confluence of shallow joinable and terminating systems can be seen from Example 11.3. That left-linearity too is essential in both cases follows from the transformation described above. In our framework, normality can be generalized and weakened to quasi- normality, which is a major result of this paper. ###### Definition 11.1 (Quasi-Normal) Let $\alpha\in\\{0,\omega\\}$. A rule $l{=}r{\longleftarrow}C$ is said to be $\alpha$-quasi-normal w.r.t. ${\rm R},{{\rm X}}$ if $\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}\end{array}\right)}\\\ {\Rightarrow}&\forall(u_{0}{=}u_{1})\mbox{ in }C{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&\alpha{\,=\,}\penalty-1\omega\\\ {\wedge}&{{{\mathcal{V}}}({u_{0},u_{1}})}\subseteq{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\\\ {\vee}&{{{\mathcal{V}}}({u_{0},u_{1}})}\subseteq\emptyset\\\ {\vee}&\exists i{\prec}2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&u_{i}\tau{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}})}\\\ {\vee}&{{\left({{\begin{array}[]{ll}&\alpha{\,=\,}\penalty-1\omega\\\ {\wedge}&({{\rm Def}\>}u_{i}\tau)\mbox{ occurs in }C\tau\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}$. ${\rm R},{{\rm X}}$ is said to be $\omega$-quasi-normal if all rules in R are $\omega$-quasi-normal w.r.t. ${\rm R},{{\rm X}}$. ${\rm R},{{\rm X}}$ is said to be $0$-quasi-normal if all constructor rules in R are $0$-quasi-normal w.r.t. ${\rm R},{{\rm X}}$. Since the case of “$\alpha{\,=\,}\penalty-1\omega$” is more important than the case of “$\alpha{\,=\,}\penalty-10$”, we use “quasi-normal” as an abbreviation for “$\omega$-quasi-normal”. First note that we have added a condition that may reduce the instantiations of a rule we have to consider. While this may be useless in practice most of the time, it may allow of further theoretical treatment. Also the fact that we have given up the requirement that the irreducible term has to be ground may be of minor importance: In practice this usually allows only for constructor variables or variables of sorts having only irreducible terms. Important, however, is the fact that equations containing only constructor variables are not restricted by quasi-normality anymore. E.g., the rule system of Example 2.3 is quasi-normal but not normal. Besides this, it is important that quasi-normality also allows to make any system quasi-normal simply by replacing any equation “$u{=}v$” in a condition with “$u{=}v{,\ \ }{{\rm Def}\>}v$”. Furthermore, note that no restrictions are imposed on Def\- and $\not=$-literals. ###### Example 11.2 (Bergstra & Klop (1986)) No Critical Peaks Left- & Right-Linear Not [Quasi-] Normal Not Terminating $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{d}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{{\mathsf{b}}},{\mathsf{g}}\\}\\\ {\rm R}_{\,\rm\ref{ex bergstra klop}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l@{}l}{{\mathsf{b}}}&=&{{\mathsf{g}}{(}{{{\mathsf{b}}}}{)}}\\\ {{\mathsf{g}}{(}{X}{)}}&=&{{\mathsf{d}}}&{\>{\longleftarrow}\>\>}{{\mathsf{g}}{(}{X}{)}}{\,=\,}\penalty-1X\end{array}$}\end{array}$ There are no critical peaks. Nevertheless, ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex bergstra klop}},\emptyset}}$ is not confluent: The following example shows that normality is also required for terminating systems. Note that this was already shown by Example C of Dershowitz &al. (1988) which, however, is more complicated because it has there additional critical peaks. ###### Example 11.3 $\omega$-Shallow [[Noisy] Parallel] Joinable $\omega$-Shallow [Noisy] Strongly Joinable Non-Overlay is Neither $\omega$-Shallow [Parallel] Closed Nor $\omega$-Shallow [Noisy] Anti-Closed Not [$\rhd_{{}_{\rm ST}}$-Quasi] Overlay Joinable Left- & Right-Linear Not [Quasi-] Normal Terminating $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{c}}},{{\mathsf{d}}},{{\mathsf{e}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{{\mathsf{a}}},{{\mathsf{b}}},{\mathsf{f}},{\mathsf{g}},{\mathsf{h}}\\}\\\ {\rm R}_{\,\rm\ref{ex cpw not normal}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l@{}l}{{\mathsf{a}}}&=&{{\mathsf{c}}}\\\ {{\mathsf{b}}}&=&{{\mathsf{d}}}\\\ {{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}}&=&{{\mathsf{g}}{(}{{{\mathsf{b}}}}{)}}\\\ {{\mathsf{f}}{(}{{{\mathsf{c}}}}{)}}&=&{{\mathsf{h}}{(}{{{\mathsf{c}}}}{)}}\\\ {{\mathsf{g}}{(}{{{\mathsf{d}}}}{)}}&=&{{\mathsf{h}}{(}{{{\mathsf{a}}}}{)}}\\\ {{\mathsf{g}}{(}{X}{)}}&=&{{\mathsf{e}}}&{\>{\longleftarrow}\>\>}X{=}{{\mathsf{b}}}\\\ {{\mathsf{h}}{(}{X}{)}}&=&{{\mathsf{e}}}&{\>{\longleftarrow}\>\>}{{\mathsf{f}}{(}{X}{)}}{=}{{\mathsf{e}}}\end{array}$}\end{array}$ There are three critical peaks and they are all of the form $(1,1)$. Since the third is the symmetric overlay of the second, we do not depict it. The first and the second are joinable as follows: Nevertheless, ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex cpw not normal}},\emptyset}}$ is not confluent: Note that the overlay would lose its shallow joinability if we made the system normal (or else quasi-normal) by writing the condition of the one but last rule in the form “$X{=}{{\mathsf{d}}}$” (or else in the form “$X{=}{{\mathsf{b}}}{,\ \ }{{\rm Def}\>}{{\mathsf{b}}}$” and declaring ${\mathsf{b}}$ to be a constructor), since then we would have ${{\mathsf{g}}{(}{{{\mathsf{d}}}}{)}}{{\longrightarrow}_{{}_{\\!\omega+1}}}{{\mathsf{e}}}.$ Similarly, the overlay would lose its shallow joinability if we made the system quasi-normal by writing the condition of the one but last rule in the form “$X{=}{{\mathsf{b}}}{,\ \ }{{\rm Def}\>}{{\mathsf{b}}}$” or by substituting $X$ with a variable from ${{\rm V}}\\!_{{\mathcal{C}}}$, since then we would have ${{\mathsf{h}}{(}{{{\mathsf{a}}}}{)}}{{\longrightarrow}_{{}_{\\!\omega+3}}}{{\mathsf{e}}}$ only (since ${{\mathsf{g}}{(}{{{\mathsf{b}}}}{)}}{\,\,\,\not\\!\\!\\!\\!{\longrightarrow}_{{}_{\\!\omega+1}}}{{\mathsf{e}}}$). ## 12 Counterexamples for Closed Systems From the examples of the previous sections we can draw the following conclusions: 1. 1. For being able to apply syntactic confluence criteria to non-terminating conditional rule systems, some kind of [quasi-] normality must be required. 2. 2. Syntactic confluence criteria based solely on the joinability structure of the critical peaks must fail on some rather simple and common joinability structures. Therefore, it is now the time to have a look at the two most simple non- trivial joinability structures under the requirement of normality. These two most simple joinability structures of critical peaks are closedness and anti-closedness, cf. below. Regarding the names of notions below, “parallel closed” is taken from Huet (1980), “closed” and “anti-closed” have been derived from “parallel closed” in an obvious manner, and “parallel joinable” was the simplest name222222The only obvious wrong intuitions it could rise are either meaningless (since the transitive closures of reduction and parallel reduction are always identical) or an unnecessary sharpening of our notion. we found for the last important variant. Closed: Anti-Closed: Parallel Closed: Parallel Joinable: It may seem to be surprising that the question whether anti-closedness of critical peaks implies confluence for left-linear, non-right-linear, unconditional systems was listed as Problem 13 in the list of open problems of Dershowitz &al. (1991) and still seems to be open. For the question whether closedness of critical peaks, a positive answer follows from the corollary on page 815 in Huet (1980) which says that a left- linear and unconditional system is confluent if all its critical pairs are parallel closed. The condition of parallel closedness was weakened in Corollary 3.2 of Toyama (1988) for the overlays which are required to be only parallel joinable instead of parallel closed. For conditional systems, however, neither closedness nor anti-closedness implies confluence. And this situation does not change when we additionally require the rule systems to be terminating and normal, as can be seen from the following examples: ###### Example 12.1 (Aart Middeldorp, modified by Bernhard Gramlich) Anti-Closed Strongly Joinable Not $\omega$-Level Joinable Not $\omega$-Shallow Joinable Not [$\rhd_{{}_{\rm ST}}$-Quasi] Overlay Joinable Left-Linear & Right-Linear [Quasi-] Normal Terminating Not Decreasing $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{a}}},{{\mathsf{c}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{f}},{\mathsf{g}}\\}\\\ {\rm R}_{{\,\rm\ref{ex gramlich}}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l@{}l}{{\mathsf{a}}}&=&{{\mathsf{c}}}\\\ {{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}}&=&{{\mathsf{g}}{(}{{{\mathsf{a}}}}{)}}\\\ {{\mathsf{g}}{(}{X}{)}}&=&{{\mathsf{f}}{(}{{{\mathsf{c}}}}{)}}&{\>{\longleftarrow}\>\>}{{\mathsf{f}}{(}{X}{)}}{\,=\,}\penalty-1{{\mathsf{g}}{(}{{{\mathsf{c}}}}{)}}\end{array}$}\end{array}$ There is only the following critical peak and is of the form $(0,1)$: Nevertheless, ${\longrightarrow}_{{}_{\\!{\rm R}_{{\,\rm\ref{ex gramlich}}},\emptyset}}$ is not confluent: Since all critical peaks are joinable, ${\rm R}_{{\,\rm\ref{ex gramlich}}}$ is necessarily non-decreasing and not compatible with a termination- pair.232323Cf. Definition 14.1 and Theorem 14.2 Nevertheless, it is obviously terminating, since $\\{X\mapsto{{\mathsf{a}}}\\}$ is the only solution for the condition of the last equation. Furthermore, ${\rm R}_{\,\rm\ref{ex gramlich}}$ is left-linear, right-linear, and normal242424even if some authors would not call it “normal” since the left-hand side of the last rule matches the right-hand side of the equation of its condition. Thus (since it is not confluent), it can be neither overlay joinable nor $\omega$-shallow joinable.252525Cf. theorems 14.7 and 14.5 It is, however, not $\omega$-level joinable and we did not find a $\omega$-level anti-closed but non-confluent system, though we spent some time searching for such an example. ###### Example 12.2 [$\omega$-Level [Parallel]] Closed $\omega$-Level Anti-Closed [$\omega$-Level] [Strongly] Joinable $\omega$-Level [Weak] Parallel Joinable Not $\omega$-Shallow Joinable Not [$\rhd_{{}_{\rm ST}}$-Quasi] Overlay Joinable Left-Linear & Right-Linear Conditions contain General variables [Quasi-] Normal Terminating Not Decreasing $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{c}}},{{\mathsf{d}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{{\mathsf{a}}},{{\mathsf{b}}},{\mathsf{+}}\\}\\\ {\rm R}_{{\,\rm\ref{ex toll}}}&:&\mbox{$\begin{array}[t]{r@{\,}l@{\,}l@{}l}{{\mathsf{a}}}&=&{{\mathsf{c}}}&{\>{\longleftarrow}\>\>}{{\mathsf{b}}}{\,=\,}\penalty-1{{\mathsf{d}}}\\\ {{\mathsf{b}}}&=&{{\mathsf{d}}}\\\ {{{{\mathsf{a}}}}\,{\mathsf{+}}\,{{{\mathsf{a}}}}}&=&{{\mathsf{d}}}\\\ {{{{\mathsf{c}}}}\,{\mathsf{+}}\,{X}}&=&{{\mathsf{d}}}&{\>{\longleftarrow}\>\>}{{X}\,{\mathsf{+}}\,{X}}{\,=\,}\penalty-1{{\mathsf{d}}}\\\ {{X}\,{\mathsf{+}}\,{{{\mathsf{c}}}}}&=&{{\mathsf{d}}}&{\>{\longleftarrow}\>\>}{{X}\,{\mathsf{+}}\,{X}}{\,=\,}\penalty-1{{\mathsf{d}}}\end{array}$}\end{array}$ There are only two critical peaks and they are of the form $(1,1)$. Using the symmetry of $\mathsf{+}$ in its arguments, the other critical peak is symmetric to the following one. Nevertheless, ${\longrightarrow}_{{}_{\\!{\rm R}_{{\,\rm\ref{ex toll}}},\emptyset}}$ is not confluent: Since all critical peaks are joinable, our system is necessarily non- decreasing, cf. Theorem 14.2. Nevertheless, it is obviously terminating, left- linear, right-linear, and normal. Thus (since it is not confluent), it can be neither overlay joinable nor $\omega$-shallow joinable, cf. theorems 14.7 and 14.5. Due to the given forms of $\omega$-level joinability, the occurrence of general variables in the conditions is essential for this example, cf. theorems 13.9 and 14.6. ## 13 Criteria for Confluence Most of the theorems we present in this and the following section assume the constructor sub-system ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ to be confluent and then suggest how to find out that the whole system ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is confluent, too. How to find out that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent will be discussed in 15. In this section we present confluence criteria that do not rely on termination. They are, of course, also applicable to terminating systems, which might be very attractive if one does not know how to show termination or if the correctness of the technique for proving termination requires confluence. Before we state our main theorems it is convenient to introduce some further syntactic restriction. By disallowing non-constructor variables in conditions of constructor equations we disentangle the fulfilledness of conditions of constructor equations from the influence of non-constructor rules. ###### Definition 13.1 (Conservative Constructors) R is said to have conservative constructors if $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\ \ {\Rightarrow}\penalty-2\ \ {{{\mathcal{V}}}({C})}\subseteq{{{\rm V}}\\!_{{\mathcal{C}}}}\end{array}\right)}.$ Let us consider a rule system with conservative constructors. Together with our global restrictions on constructor rules (cf. Definition 2.2) this means that the condition terms of constructor rules are pure constructor terms. This has the advantage that (contrary to the general case) the condition terms of constructor rules still are constructor terms after they have been instantiated with some substitution. By Lemma 2.10 this means that the reducibility with constructor rules does not depend on the new possibilities which could be added by the non-constructor rules later on, i.e. that the constructor rules are conservative w.r.t. their decision not to reduce a given term because non-constructor rules cannot generate additional solutions for their conditions.262626Since “conservative constructors” is actually a property not of the constructors (i.e. constructor function symbols) but of the constructor rules, the notion should actually be called “conservative constructor rules”. But the commonplace notion of “free constructors” is just the same. The condition of conservative constructors is very natural and not very restrictive. (Note that even now constructor rules may have general variables in their left- and right-hand sides.) That conservative constructors make the construction of confluence criteria much easier can be seen from the following lemma which can treat a special case of possible divergence, namely a sub-case of the “variable overlap case”. In this case it is important that a reduction with a certain rule can still be done after the instantiating substitution has been reduced. ###### Lemma 13.2 Let $\mu,\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$. Let ${((l,r),C)}\in{\rm R}$ with $l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Assume that ${\left({{\begin{array}[]{ll}&{\rm R}\mbox{ has conservative constructors}\\\ {\vee}&{{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ {\vee}&{{{\mathcal{TERMS}}}({C\mu})}{\subseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ \end{array}}}\right)}$. Assume ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ to be confluent. Now, if $C\mu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ and $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}x\nu,$ then $C\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ and $l\nu{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}r\nu.$ While the conditions of our main theorems of this section, Theorem 13.6 and Theorem 13.9, are rather complicated and difficult to check, they are always satisfied for a certain class of rule systems captured by Theorem 13.3 (being a consequence of Theorem 13.6) and Theorem 13.4 (being a consequence of Theorem 13.9) below. This class consists of left-linear rule systems with conservative constructors that achieve quasi-normality just by requiring the presence of a Def-literal for each equation not containing an irreducible ground term in a condition of a rule, and satisfy the joinability requirements due to the critical peaks being complementary, i.e. having complementary literals in their condition lists, cf. 5. Furthermore, rule systems of this class are quite useful in practice. It generalizes the function specification style that is usually required in the framework of classic inductive theorem proving (cf. e.g. Walther (1994)) by allowing for partial functions resulting from non-complete defining case distinctions as well as resulting from non-termination. ###### Theorem 13.3 (Syntactic Confluence Criterion) Let R be a left-linear CRS over sig/cons/${\rm V}$ with conservative constructors. Assume $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall(u_{0}{=}u_{1})\mbox{ in }C{.}\penalty-1\,\,\exists i{\,\prec\,}2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&({{\rm Def}\>}u_{i})\mbox{ occurs in }C\\\ {\vee}&u_{i}\in{\mathcal{GT}}{\setminus}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\\\ \end{array}}}\right)}}.$ Assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent. Now: If each critical peak in ${\rm CP}({\rm R})$ of the form $(0,1)$, $(1,0)$, or $(1,1)$ is complementary, then ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is confluent. ###### Theorem 13.4 (Syntactic Confluence Criterion) Let R be a left-linear CRS over sig/cons/${\rm V}$ with $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,{{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Assume $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall(u_{0}{=}u_{1})\mbox{ in }C{.}\penalty-1\,\,\exists i{\,\prec\,}2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&({{\rm Def}\>}u_{i})\mbox{ occurs in }C\\\ {\vee}&u_{i}\in{\mathcal{GT}}{\setminus}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\\\ \end{array}}}\right)}}.$ Assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent. Now: If each critical peak in ${\rm CP}({\rm R})$ of the form $(0,1)$ or $(1,0)$ is complementary and each critical peak in ${\rm CP}({\rm R})$ of the form $(1,1)$ is weakly complementary, then ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is confluent. Note that both theorems are applicable272727The careful reader may have noticed that the last two rules of ${\rm R}_{{\,\rm\ref{exb}}}$ actually are lacking the required Def-literals. For practical specification, however, this Def-literal can be omitted here because it is tautological for ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ if ${{\rm X}}{\subseteq}{{{\rm V}}\\!_{{\rm SIG}}}$ . Note that in practice of specification one is only interested in ${\longrightarrow}_{{}_{\\!{\rm R},\emptyset}}$ and ${\longrightarrow}_{{}_{\\!{\rm R},{{{\rm V}}\\!_{{\rm SIG}}}}}$ cf. Wirth & Gramlich (1994a) and Wirth & Gramlich (1994b). (This, however, does not mean that we do not need formulas containing ${{\rm V}}\\!_{{\mathcal{C}}}$ for inductive theorem proving.) to the rule system of Example 2.3 where the subtraction on natural numbers is defined via a non-complete syntactic case distinction that does not yield critical peaks at all and where the member- predicate is defined by a syntactic case distinction followed (for the case of a nonempty list) by a semantic case distinction via condition literals which yields only critical peaks with complementary equations. To illustrate the possibility of partiality due to non-termination as well as the possibility of critical peaks with complementary predicate literals, here is another toy example to which we can apply Theorem 13.3 (but not Theorem 13.4). ###### Example 13.5 (continuing Example 2.3) $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{0}}},{\mathsf{s}},{{\mathsf{true}}},{{\mathsf{false}}},{{\mathsf{nil}}},{\mathsf{cons}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{-}},{{\mathsf{mbp}}},{\mathsf{while}}\\}\\\ {{\mathbb{S}}}&:=&\\{{\mathsf{nat}},{\mathsf{bool}},{\mathsf{list}}\\}\\\ \end{array}$ $\begin{array}[t]{lll}{\rm R}_{\,\rm\ref{ex while}}&:&{\rm R}_{\,\rm\ref{exb}}\\\ &&\vdots\\\ &&\mbox{$\begin{array}[t]{@{}r@{\,}l@{\,}ll}{{\mathsf{while}}{(}{X}{,\,}{Y}{)}}&=&Y&{\longleftarrow}\ X{=}{{\mathsf{false}}}\\\ {{\mathsf{while}}{(}{X}{,\,}{Y}{)}}&=&{{\mathsf{while}}{(}{\ldots}{,\,}{\ldots}{)}}&{\longleftarrow}\ X{=}{{\mathsf{true}}},\ \ldots\end{array}$}\\\ &&\vdots\end{array}$ We have added two rules to the system from Example 2.3 for a function ‘$\mathsf{while}$’ with arity “$\ {\mathsf{bool}}\;{\mathsf{nat}}{\ {\rightarrow}\ }{\mathsf{nat}}\ $” where $X$ is meant to be a variable from ${{\rm V}}\\!_{{{\rm SIG}},{{\mathsf{bool}}}}$ and $Y$ from ${{\rm V}}\\!_{{{\rm SIG}},{{\mathsf{nat}}}}$. The two resulting critical peaks are of the form $(1,1)$ and complementary. Furthermore, we assume that there are no rules with ${\mathsf{true}}$, ${\mathsf{false}}$, or a variable of the sort $\mathsf{bool}$ as left-hand sides, such that we have ${{\mathsf{true}}},{{\mathsf{false}}}\in{\mathcal{GT}}{\setminus}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex while}},{{\rm X}}}}}})}.$ The main part of the following theorem is part (I). Parts (III) and (IV) only weaken the required $\omega$-shallow noisy parallel joinability for critical peaks of the form $(1,1)$ to $\omega$-shallow noisy weak parallel joinability but have to pay a considerable price for it. It would be of practical importance (cf. Example 10.6) to achieve this weakening for critical peaks of the form $(0,1)$, but this is not possible, cf. Example 10.5. Furthermore, the difference between (III) and (IV) is marginal since non-overlays of the form $(1,0)$ are pathological282828A critical peak of the form $(1,0)$ requires a non-constructor rule whose left-hand side has a constructor function symbol as top symbol, and also requires a constructor rule with a general variable in its left-hand side. anyway. (II) is rather interesting for the cases where it is possible to restrict the right-hand sides to be linear w.r.t. general variables; this severe restriction is necessary, however; cf. the second version of Example 10.7 or cf. Example 10.8. Besides these examples, also Example 10.4 may be able to discourage the search for a further generalization of the theorem. Finally note that the ‘$i$’ and ‘$j$’ in the theorem range over $\\{0,1\\}$. ###### Theorem 13.6 (Syntactic Criterion for $\omega$-Shallow Confluence) Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume R to have conservative constructors, ${\rm R},{{\rm X}}$ to be quasi- normal, and the following weak kind of left-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent. 1. (I) Now if each critical peak in ${\rm CP}({\rm R})$ of the form $(i,1)$ is $\omega$-shallow noisy parallel joinable up to $\omega{+}i{}\omega$ w.r.t. ${\rm R},{{\rm X}}$, and each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,j)$ is $\omega$-shallow parallel closed up to $\omega{+}j{}\omega$ w.r.t. ${\rm R},{{\rm X}}$, then ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent. 2. (II) If we have the following kind of right-linearity w.r.t. general variables $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({r})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}r/p{\,=\,}\penalty-1x{\,=\,}\penalty-1r/q\ \ {\Rightarrow}\penalty-2\ \ p{\,=\,}\penalty-1q\end{array}\right)},$ and if each critical peak in ${\rm CP}({\rm R})$ of the form $(i,1)$ is $\omega$-shallow noisy strongly joinable up to $\omega{+}i{}\omega$ w.r.t. ${\rm R},{{\rm X}}$, and each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,j)$ is $\omega$-shallow noisy anti-closed up to $\omega{+}j{}\omega$ w.r.t. ${\rm R},{{\rm X}}$, then ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent. Now additionally assume the following very weak kind of right-linearity of constructor rules: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({r})}{.}\penalty-1\,\,\\!\\!{\left(\begin{array}[c]{l}\\!\\!{{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\!\\!\\\ {\wedge}&r/p{\,=\,}\penalty-1x{\,=\,}\penalty-1r/q\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ p{\,=\,}\penalty-1q\end{array}\right)}.$ Furthermore, additionally assume that each critical peak in ${\rm CP}({\rm R})$ of the form $(0,1)$ is $\omega$-shallow noisy strongly joinable up to $\omega$, that each critical peak in ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-shallow noisy weak parallel joinable w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-shallow closed w.r.t. ${\rm R},{{\rm X}}$. 1. (III) Now if each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,0)$ is $\omega$-shallow parallel closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, then ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent. Now additionally assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is strongly confluent. 1. (IV) Now if each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,0)$ is $\omega$-shallow closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, then ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent. If we consider all symbols to be non-constructor symbols, then each of the parts (I), (III), and (IV) of Theorem 13.6 is strong enough to imply Theorem 1 of Dershowitz &al. (1988) (which is taken from Bergstra & Klop (1986)). If we, moreover, restrict to unconditional rule systems, then Theorem 13.6(I) specializes to Corollary 3.2 of Toyama (1988) (which is stronger than the more restrictive corollary on page 815 in Huet (1980) which says that a left-linear and unconditional system is confluent if all its critical pairs are parallel closed). Moreover, Theorem 13.6(II) is a generalization of Theorem 5.2 of Avenhau & Becker (1994) translated into our framework. The proof of Theorem 13.6 is similar to that of Corollary 3.2 of Toyama (1988) for unconditional systems, but with a global induction loop on the depth of reduction for using the shallow joinability to get along with the conditions of the rules, and this whole proof twice due to our separation into constructors and non-constructors, and this again for each part of the theorem. Since it is very long, tedious, and uninteresting we have put most its lemmas into A and the proofs into D. The only lemmas we consider to be interesting are those which make clear why it is possible to generalize from normal to quasi-normal rule systems. The problematic case is always the variable-overlap case since it is not covered by critical peaks. The hard step in this case is to show that an equation “$u_{0}{=}u_{1}$” which had been joinable when instantiated with substitution $\mu$ is still joinable after the instantiations for its variables have been reduced, yielding a new substitution $\nu$. Thus one has to show that for two natural numbers $n_{0}$ and $n_{1}$ with $u_{0}\mu{\downarrow_{{}_{{{\rm R},{{\rm X}}},\omega+n_{1}}}}u_{1}\mu$ and $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}x\nu$ we always have $u_{0}\nu{\downarrow_{{}_{{{\rm R},{{\rm X}}},\omega+n_{1}}}}u_{1}\nu$ . This means that the fulfilledness of the instantiated equation “$u_{0}{=}u_{1}$” is not changed by the reduction of its instantiating substitution. For showing this we may use the global induction hypothesis implying that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}$. The reader may verify that we do not seem to have a chance for being successful here unless we require some kind of normality. Lemma 13.7(4) depicts the situation we are in (matching its $s_{i}$ to $u_{i}\mu$ and its $s_{i}^{\prime}$ to $u_{i}\nu$) and shows that irreducibility of $u_{1}\nu$ (roughly speaking i.e. normality) is just as helpful as some literal “${{\rm Def}\>}u_{1}\mu$” in the condition list (i.e. an alternative allowed by quasi- normality) (because the latter implies the existence of some $t_{1}\in{{\mathcal{GT}}({{\rm cons}})}$ with $u_{1}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}$ ). Finally, Lemma 13.8 states that the other alternative given by quasi-normality (i.e. that the equation contains no non-constructor variables) is no problem either, and that Def\- and $\not=$-literals do not make any problems and therefore need not at all be restricted by normality requirements. Since we consider the proofs of the following two lemmas to be interesting, we did not put them into the appendix but included them here. The form of presentation is very general. This enables the proof to present the idea of quasi-normality in its essential form and also enables more than a dozen of applications of Lemma 13.8 in the proofs of the theorems in this and the following sections. When reading the lemmas please note that the optional parts are only necessary for reusing the lemmas in the proofs of the theorems of the following sections where termination arguments will be included into the confluence criteria. Moreover for a first reading only the second cases of their initial disjunctive assumptions should be considered. The others are uninteresting special cases. ###### Lemma 13.7 [Let $\rhd$ be a wellfounded ordering.] Let $n_{0},n_{1}\prec\omega$. Let $\alpha\in\\{0,\omega\\}$. Assume that $\forall i{\,\prec\,}2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&s_{i}{\,=\,}\penalty-1s_{i}^{\prime}\\\ {\vee}&\mbox{${\rm R},{{\rm X}}$\ is $\alpha$-shallow confluent up to }n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}\mbox{ [and }s_{i}\mbox{ in }\lhd\mbox{]}\\\ \end{array}}}\right)}}.$ Now: 1. 1. $n_{0}{\,\preceq\,}n_{1}$ and $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{0}$ implies $s_{0}^{\prime}{\downarrow_{{}_{{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{0}.$ 2. 2. $n_{0}{\,\preceq\,}n_{1}$ and $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}s_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{1}^{\prime}$ implies $s_{0}^{\prime}{\downarrow_{{}_{{{\rm R},{{\rm X}}},\alpha+n_{1}}}}s_{1}^{\prime}.$ 3. 3. $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{2}{\,\in\,}{{\mathcal{GT}}({{\rm cons}})}$ implies $\exists t_{3}{\,\in\,}{{\mathcal{GT}}({{\rm cons}})}{.}\penalty-1\,\,s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{3}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{2}.$ 4. 4. $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}s_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{1}^{\prime}$ together with either $s_{1}{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}})}$ or ${{\left({{\begin{array}[]{ll}&\alpha{\,=\,}\penalty-1\omega\\\ {\wedge}&s_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}{\,\in\,}{{\mathcal{GT}}({{\rm cons}})}\\\ {\wedge}&\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\delta\mbox{ [and }s_{1}\mbox{ in }{\lhd}\mbox{]}\\\ \end{array}}}\right)}}$ implies $s_{0}^{\prime}{\downarrow_{{}_{{{\rm R},{{\rm X}}},\alpha+n_{1}}}}s_{1}^{\prime}.$ Proof of Lemma 13.7 1: Consider the peak $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{0}.$ If $s_{0}{\,=\,}\penalty-1s_{0}^{\prime},$ then we are finished due to $s_{0}^{\prime}{\,=\,}\penalty-1s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{0}.$ Otherwise: We have assumed that ${\rm R},{{\rm X}}$ is $\alpha$-shallow confluent up to $n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}$ [and $s_{0}$ in $\lhd$]. Thus we get $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}t_{0}$ and then due to $n_{0}{\,\preceq\,}n_{1}$ and Lemma 2.12 we get $s_{0}^{\prime}{\downarrow_{{}_{{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{0}$ . 2: By (1) we get $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{0}$ for some $t_{1}$. Finally, consider the peak $t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}s_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{1}^{\prime}.$ By (1) again we get $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{1}{\downarrow_{{}_{{{\rm R},{{\rm X}}},\alpha+n_{1}}}}s_{1}^{\prime}$ as desired. 3: Consider the peak $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{2}.$ If $s_{0}{\,=\,}\penalty-1s_{0}^{\prime},$ then we are finished due to $s_{0}^{\prime}{\,=\,}\penalty-1s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{2}.$ Otherwise: By $\alpha$-shallow confluence up to $n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}$ [and $s_{0}$ in $\lhd$] we get $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{3}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}t_{2}$ for some $t_{3}$. By $t_{2}{\,\in\,}{{\mathcal{GT}}({{\rm cons}})}$ and Lemma 2.10 we get ${{\mathcal{GT}}({{\rm cons}})}{\,\ni\,}t_{3}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}t_{2}.$ Thus we have $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{3}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{2}$ as desired. 4: $s_{1}{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}})}$: If $s_{0}{\,=\,}\penalty-1s_{0}^{\prime},$ then we are finished due to $s_{0}^{\prime}{\,=\,}\penalty-1s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{0}{\,=\,}\penalty-1s_{1}{\,=\,}\penalty-1s_{1}^{\prime}.$ Otherwise: Consider the peak $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{0}.$ By $\alpha$-shallow confluence up to $n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}$ [and $s_{0}$ in $\lhd$] we get $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}t_{0}$ for some $t_{2}$. Since $s_{1}{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}})}$ this finishes the proof in this case due to $t_{2}{\,=\,}\penalty-1t_{0}{\,=\,}\penalty-1s_{1}{\,=\,}\penalty-1s_{1}^{\prime}.$ $s_{1}{\,\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}})}$: Then we have $\alpha{\,=\,}\penalty-1\omega,$ $s_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}{\,\in\,}{{\mathcal{GT}}({{\rm cons}})},$ and $\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\delta\mbox{ [and }s_{1}\mbox{ in }{\lhd}\mbox{]},$ cf. the diagram below. Consider the peak $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}s_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}.$ We may assume $n_{1}{\,\prec\,}n_{0}$ because in case of $n_{0}{\,\preceq\,}n_{1}$ the proof is finished due to (2). Then we have $n_{1}{+_{\\!\\!{}_{\omega}}}n_{1}\prec n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}.$ Thus by $\omega$-shallow confluence up to $n_{1}{+_{\\!\\!{}_{\omega}}}n_{1}$ [and $s_{1}$ in $\lhd$] we get $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}$ for some $t_{2}$. By $t_{1}{\,\in\,}{{\mathcal{GT}}({{\rm cons}})}$ and Lemma 2.10 we get ${{\mathcal{GT}}({{\rm cons}})}{\,\ni\,}t_{2}.$ Consider the peak $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}s_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{2}.$ Due to $t_{2}{\,\in\,}{{\mathcal{GT}}({{\rm cons}})}$ and (3) there is some $t_{3}\in{{\mathcal{GT}}({{\rm cons}})}$ with $s_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{3}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{2}.$ By (3) again, the peak $t_{3}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}s_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}s_{1}^{\prime}$ implies $t_{3}{\downarrow_{{}_{{{\rm R},{{\rm X}}},\omega+n_{1}}}}s_{1}^{\prime}$ as desired. Q.e.d. (Lemma 13.7) ###### Lemma 13.8 [Let $\rhd$ be a wellfounded ordering.] Let $\alpha\in\\{0,\omega\\}$. Let $n_{0},n_{1}\prec\omega$. Let $\mu,\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$. Let ${((l,r),C)}\in{\rm R}$ with $\alpha{\,=\,}\penalty-10\ {\Rightarrow}\penalty-2\ l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Assume that $n_{0}{\,\preceq\,}n_{1}$ or that $((l,r),C)$ is $\alpha$-quasi- normal w.r.t. ${\rm R},{{\rm X}}$. Assume that $\forall L\mbox{ in }C{.}\penalty-1\,\,\forall u{\,\in\,}{{{\mathcal{TERMS}}}({L})}{.}\penalty-1\,\,$ ${\left({{\begin{array}[]{ll}&u\mu{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}})}\\\ {\vee}&{{\rm R},{{\rm X}}}\mbox{ is $\alpha$-shallow confluent up to\/ }n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}\mbox{ [and }u\mu\mbox{ in }\lhd\mbox{]}\\\ {\vee}&{{\left({{\begin{array}[]{ll}&\forall x{\,\in\,}{{{\mathcal{V}}}({u})}{.}\penalty-1\,\,x\mu{\,=\,}\penalty-1x\nu\\\ {\wedge}&{{\left({{\begin{array}[]{ll}&\alpha{\,=\,}\penalty-10\\\ {\vee}&\forall v{.}\penalty-1\,\,L{\,\not\in\,}\\{(u{=}v),(v{=}u)\\}\\\ {\vee}&\forall x{\,\in\,}{{{\mathcal{V}}}({L})}{.}\penalty-1\,\,x\mu{\,=\,}\penalty-1x\nu\\\ {\vee}&\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}{.\penalty-1}\\\ &{{\rm R},{{\rm X}}}\mbox{ is $\alpha$-shallow confluent up to\/ }\delta\mbox{ [and }u\mu\mbox{ in }\lhd\mbox{]}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}$. Now, if $C\mu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}$ and $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}x\nu,$ then $C\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}$ and $l\nu{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}+1}}}r\nu.$ Proof of Lemma 13.8 Since $\alpha{\,=\,}\penalty-10\ {\Rightarrow}\penalty-2\ l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ it suffices to show that for each literal $L$ in $C$: $L\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}$. Note that we already know that $L\mu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}$. In case of $u\mu{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}})}$ we get $u\mu{\,=\,}\penalty-1u\nu$ due to $u\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}u\nu.$ In case of $\forall x{\,\in\,}{{{\mathcal{V}}}({u})}{.}\penalty-1\,\,x\mu{\,=\,}\penalty-1x\nu$ we get $u\mu{\,=\,}\penalty-1u\nu$ again. Thus we may assume $\forall u{\,\in\,}{{{\mathcal{TERMS}}}({L})}{.}\penalty-1\,\,{(\ u\mu{\,=\,}\penalty-1u\nu\ \ {\vee}\penalty-2\ \ {{\rm R},{{\rm X}}}\mbox{ is $\alpha$-shallow confluent up to }n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}\mbox{ [and }u\mu\mbox{ in }\lhd\mbox{]}\ )}.$ $L=(s_{0}{=}s_{1})$: We have $s_{0}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{0}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}\penalty-1t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}\penalty-1s_{1}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{1}\nu$ for some $t_{0}.$ In case of $n_{0}{\,\preceq\,}n_{1}$ we get the desired $s_{0}\nu{\downarrow_{{}_{{{\rm R},{{\rm X}}},\alpha+n_{1}}}}s_{1}\nu$ by Lemma 13.7(2). Otherwise, by assumption of the lemma, $((l,r),C)$ must be $\alpha$-quasi-normal. Since $C\mu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}$, according to the definition of $\alpha$-quasi-normality and the disjunctive assumption of the lemma we have two distinguish several cases here. First we treat the case in which $\exists i{\,\prec\,}2{.}\penalty-1\,\,s_{i}\mu{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}})}.$ W.l.o.g. say $s_{1}\mu{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}})}.$ By Lemma 13.7(4) we get the desired $s_{0}\nu{\downarrow_{{}_{{{\rm R},{{\rm X}}},\alpha+n_{1}}}}s_{1}\nu.$ Second, in case of $\forall x{\,\in\,}{{{\mathcal{V}}}({L})}{.}\penalty-1\,\,x\mu{\,=\,}\penalty-1x\nu$ we know that $L\nu{\,=\,}\penalty-1L\mu$ which is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}$. Note that now we may assume that $\alpha{\,=\,}\penalty-1\omega$ because the second case includes the only case left for $0$-quasi-normality, namely ${{{\mathcal{V}}}({s_{0},s_{1}})}{\subseteq}\emptyset.$ Third, in case of ${{{\mathcal{V}}}({s_{0},s_{1}})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}$ we have for all $x\in{{{\mathcal{V}}}({s_{0},s_{1}})}$: $x\mu{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})};$ and then $x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}x\nu$ by Lemma 2.10. This means $s_{i}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}s_{i}\nu.$ By Lemma 13.7(2) (matching its $n_{0}$ to $0$) due to $0{+_{\\!\\!{}_{\omega}}}n_{1}\preceq n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}$ we get the desired $s_{0}\nu{\downarrow_{{}_{{{\rm R},{{\rm X}}},\omega+n_{1}}}}s_{1}\nu.$ Finally we come to the fourth case where w.l.o.g. $({{\rm Def}\>}s_{1}\mu)$ occurs in $C\mu$. Then there is some $t_{1}\in{{\mathcal{GT}}({{\rm cons}})}$ with $s_{1}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}.$ Since we may assume that we are not in any of the previous cases, the disjunctive assumption of the lemma now states that $\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\delta\mbox{ [and $u\mu$ in $\lhd$]}.$ By Lemma 13.7(4) we get the desired $s_{0}\nu{\downarrow_{{}_{{{\rm R},{{\rm X}}},\omega+n_{1}}}}s_{1}\nu.$ $L=({{\rm Def}\>}s)$: We know the existence of $t\in{{\mathcal{GT}}({{\rm cons}})}$ with $s\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t.$ By Lemma 13.7(3) there is some $t^{\prime}\in{{\mathcal{GT}}({{\rm cons}})}$ with $s\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t.$ $L=(s_{0}{\not=}s_{1})$: There exist some $t_{0},t_{1}\in{{\mathcal{GT}}({{\rm cons}})}$ with $\forall i{\,\prec\,}2{.}\penalty-1\,\,s_{i}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}s_{i}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{i}$ and $t_{0}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{1}.$ Just like above we get $t_{0}^{\prime},\ t_{1}^{\prime}\in{{\mathcal{GT}}({{\rm cons}})}$ with $\forall i{\,\prec\,}2{.}\penalty-1\,\,s_{i}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}\penalty-1t_{i}^{\prime}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{i}.$ Finally $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{0}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{{\rm R},{{\rm X}}},\alpha+n_{1}}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}^{\prime}$ implies $t_{0}^{\prime}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}^{\prime}$ since we have $\alpha{\,=\,}\penalty-1\omega$ due to $l{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ in this case of a negative literal. Q.e.d. (Lemma 13.8) We do not have to discuss the following theorem in detail here, because it is very similar to Theorem 13.6, but weakens the required $\omega$-shallow joinabilities to $\omega$-level joinabilities wherever possible. Note that from Example 10.2 we can conclude that the $\omega$-shallow joinabilities required for critical peaks of the form $(0,1)$ cannot be weakened to $\omega$-level joinabilities in any of the four parts of the theorem.292929Note that with the exception of part (II) of the theorem we could also use the first version of Example 10.7 for this conclusion. However, the price we have to pay for weakening shallow to level joinability is to extend our requirement that the conditions contain constructor variables only, from constructor rules (“conservative constructors”) to all rules! That this restriction is necessary indeed can be seen from Example 12.2. On the other hand, this restriction gives quasi-normality for free. We prefer to discuss and apply Theorem 13.6 wherever possible because contrary to Theorem 13.9 it has interesting implications for the standard framework without the separation into constructor and non-constructor symbols where “only constructor variables in conditions” means “no variables in conditions” which again can (in general not effectively) be reduced to “no conditions” by removing the fulfilled conditions and the rules with non-fulfilled conditions. The main part of the following theorem is part (I). Parts (III) and (IV) only weaken the required $\omega$-level parallel joinability for critical peaks of the form $(1,1)$ to $\omega$-level weak parallel joinability but have to pay a considerable price for it. Furthermore, the difference between (III) and (IV) is marginal since non-overlays of the form $(1,0)$ are pathological anyway. (II) is rather interesting for the cases where it is possible to restrict the right-hand sides to be linear w.r.t. general variables; this severe restriction is necessary, however; cf. the second version of Example 10.7 or cf. Example 10.8. ###### Theorem 13.9 (Syntactic Criterion for $\omega$-Level Confluence) Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume the following important restriction on variables in conditions to hold: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,{{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Moreover, assume the following weak kind of left-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent. 1. (I) Now if each critical peak in ${\rm CP}({\rm R})$ of the form $(0,1)$ is $\omega$-shallow parallel joinable up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,0)$ is $\omega$-shallow parallel closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, each critical peak in ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level parallel joinable w.r.t. ${\rm R},{{\rm X}}$, and each non- overlay in ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level parallel closed w.r.t. ${\rm R},{{\rm X}}$, then ${\rm R},{{\rm X}}$ is $\omega$-level confluent. 2. (II) If we have the following kind of right-linearity w.r.t. general variables $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({r})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}r/p{\,=\,}\penalty-1x{\,=\,}\penalty-1r/q\ \ {\Rightarrow}\penalty-2\ \ p{\,=\,}\penalty-1q\end{array}\right)},$ and if each critical peak in ${\rm CP}({\rm R})$ of the form $(0,1)$ is $\omega$-shallow strongly joinable up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,0)$ is $\omega$-shallow anti-closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, each critical peak in ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level strongly joinable w.r.t. ${\rm R},{{\rm X}}$, and each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level anti-closed w.r.t. ${\rm R},{{\rm X}}$, then ${\rm R},{{\rm X}}$ is $\omega$-level confluent. Now additionally assume the following very weak kind of right-linearity of constructor rules: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({r})}{.}\penalty-1\,\,\\!\\!{\left(\begin{array}[c]{l}\\!\\!{{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\!\\!\\\ {\wedge}&r/p{\,=\,}\penalty-1x{\,=\,}\penalty-1r/q\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ p{\,=\,}\penalty-1q\end{array}\right)}.$ Furthermore, additionally assume that each critical peak in ${\rm CP}({\rm R})$ of the form $(0,1)$ is $\omega$-shallow strongly joinable up to $\omega$, that each critical peak in ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level weak parallel joinable w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level closed w.r.t. ${\rm R},{{\rm X}}$. 1. (III) Now if each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,0)$ is $\omega$-shallow parallel closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, then ${\rm R},{{\rm X}}$ is $\omega$-level confluent. Now additionally assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is strongly confluent. 1. (IV) Now if each non-overlay in ${\rm CP}({\rm R})$ of the form $(1,0)$ is $\omega$-shallow closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, then ${\rm R},{{\rm X}}$ is $\omega$-level confluent. ## 14 Criteria for Confluence of Terminating Systems In this section we examine how we can relax our joinability requirements when we additionally require termination for our reduction relation. Note that in confluence criteria whose proof is by induction on an extension of the reduction relation the joinability requirement can be weakened to a sub- connectedness requirement, cf. Kuchlin (1985). We here, however, present the simpler versions only, where the connectedness is required to have the form of a single “valley”. Due to its fundamental importance, we first repeat Theorem 7.17 of Wirth & Gramlich (1994a) here, which generalizes Theorem 3 of Dershowitz &al. (1988) by weakening decreasingness to compatibility with a termination-pair (defined in 2.2) as well as joinability to $\rhd$-weak joinability (defined in 5) which provides us with some confluence assumption when checking the fulfilledness of the condition of a critical peak. ###### Definition 14.1 (Compatibility with a Termination-Pair) A rule ${((l,r),C)}$ is is ${\rm R},{{\rm X}}$-compatible with a termination- pair $(>,\rhd)$ over sig/${\rm V}$ if $\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}C\tau\mbox{ fulfilled w.r.t.\ ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$}\ \ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l\tau>r\tau\\\ {\wedge}&\forall u{\,\in\,}{{{\mathcal{TERMS}}}({C})}{.}\penalty-1\,\,l\tau\rhd u\tau\\\ \end{array}}}\right)}}\end{array}\right)}.$303030We could require the weaker $\forall u{\,\in\,}{{{\mathcal{TERMS}}}({C})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&u\tau{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\\\ {\vee}&l\tau\rhd u\tau\\\ \end{array}}}\right)}}$ instead of $\forall u{\,\in\,}{{{\mathcal{TERMS}}}({C})}{.}\penalty-1\,\,l\tau\rhd u\tau$ here. Theorem 14.2 would still be true since its proof need not be modified. We did not do this because we did not see an interesting application that would justify the change of the notion already introduced in Wirth & Gramlich (1993), Wirth &al. (1993), and Wirth & Gramlich (1994a). A CRS R over sig/cons/${\rm V}$ is ${\rm X}$-compatible with a termination- pair $(>,\rhd)$ over sig/${\rm V}$ if $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,{((l,r),C)}\mbox{ is ${\rm R},{{\rm X}}$-compatible with }(>,\rhd).$ ###### Theorem 14.2 (Syntactic Test for Confluence) Let R be a CRS over sig/cons/${\rm V}$ and ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume that R is ${\rm X}$-compatible with a termination-pair $(>,\rhd)$ over sig/${\rm V}$. [For each $t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ assume $\lll_{t}$ to be a wellfounded ordering on ${{\mathcal{POS}}}({t})$. Define ($p{\,\in\,}{{\bf N}}_{+}^{\ast}$) $A(p):={{\\{\ }t{\,\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\omega,q}}}})}}~{}{\|}\penalty-9\,\ {\emptyset{\,\not=\,}q\lll_{t}p{\ \\}}}.$ ] The following two are logically equivalent: 1. 1. Each critical peak in ${\rm CP}({\rm R})$ is $\rhd$-weakly joinable w.r.t. ${\rm R},{{\rm X}}$ [besides $A$]. 2. 2. ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is confluent. Due to a weakening of the notion of $\rhd$-weak joinability, Theorem 14.2 actually differs from Theorem 7.17 of Wirth & Gramlich (1994a) in that it provides several irreducibility assumptions intended to restrict the number of substitutions $\varphi$ for which for a critical peak ${\left(\begin{array}[c]{l}{{({l_{1}\penalty-1{[\,p\leftarrow r_{0}\,]}},C_{0},\ldots)}},\ {{(r_{1},C_{1},\ldots)}},\ l_{1},\ \sigma,\ p\end{array}\right)}$ resulting from two rules $l_{0}{=}r_{0}{\longleftarrow}C_{0}$ and $l_{1}{=}r_{1}{\longleftarrow}C_{1}$ (with no variables in common) we have to show ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\,]}}\sigma\varphi{\downarrow_{{}_{{\rm R},{{\rm X}}}}}r_{1}\sigma\varphi$ in case of $(C_{0}C_{1})\sigma\varphi$ being fulfilled. This means that Theorem 14.2 provides further means to tackle problem 4 of our 1. The first assumption allowed is that the substitution $\varphi$ itself is normalized: $\forall x\in{{\rm V}}{.}\penalty-1\,\,x\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}.$ The second allows to assume that for non-overlays (i.e. for $p{\,\not=\,}\emptyset$) even $\sigma\varphi$ is normalized on all variables occurring in the left-hand side $l_{1}$. Moreover, by weakening “$\rhd$-weak joinability” to “$\rhd$-weak joinability besides $A$” with $A$ defined as in the theorem via some family $\ggg$ = $(\ggg_{t})_{t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}}$ of arbitrary wellfounded orderings $\ggg_{t}$ on ${{\mathcal{POS}}}({t})$, we have added a new feature which allows to assume the instantiated peak term (or superposition term) $l_{1}\sigma\varphi$ to be irreducible at all nonempty positions which are $\lll_{l_{1}\sigma\varphi}$-smaller than the overlap position $p$. Generally, beyond our first two assumptions, we may use $\lll$ to further reduce the number of instantiations for which the joinability test must succeed in the following way: If we can choose $\lll_{l_{1}\sigma\varphi}$ such that ${\left(\begin{array}[c]{l}p{\,=\,}\penalty-1\emptyset\ \ {\Rightarrow}\penalty-2\ \ \forall x{\,\in\,}{{{\mathcal{V}}}({l_{1}})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\sigma{\,\not=\,}x\\\ {\Rightarrow}&\exists q{\,\in\,}{{{\mathcal{POS}}}({l_{1}})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&l_{1}/q{\,=\,}\penalty-1x\\\ {\wedge}&\forall q^{\prime}{\,\in\,}{{{\mathcal{POS}}}({x\sigma\varphi})}{.}\penalty-1\,\,qq^{\prime}\lll_{l_{1}\sigma\varphi}p\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}\end{array}\right)}$ as well as $\forall x{\,\in\,}{{{\mathcal{V}}}({l_{0}})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\sigma{\,\not=\,}x\\\ {\Rightarrow}&\exists q{\,\in\,}{{{\mathcal{POS}}}({l_{0}})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&l_{0}/q{\,=\,}\penalty-1x\\\ {\wedge}&\forall q^{\prime}{\,\in\,}{{{\mathcal{POS}}}({x\sigma\varphi})}{.}\penalty-1\,\,pqq^{\prime}\lll_{l_{1}\sigma\varphi}p\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}},$ then we may assume $\sigma\varphi$ to be normalized: $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}.$ This can be a considerable help for showing that $(C_{0}C_{1})\sigma\varphi$ is not fulfilled when we have a certain knowledge on the normal forms of the terms of the sorts of the variables occurring in $C_{0}C_{1}$. E.g., when we define the depth of a term $t\in{\mathcal{T}}$ by $0pt{t}:=\max{{\\{\ }{\,\|{p^{\prime}}\|\,}}~{}{\|}\penalty-9\,\ {p^{\prime}{\,\in\,}{{{\mathcal{POS}}}({t})}{\ \\}}}$ and then define ($p,q{\,\in\,}{{{\mathcal{POS}}}({t})}$) $q\lll_{t}p$ if $0ptt-{\,\|{q}\|\,}\prec 0ptt-{\,\|{p}\|\,},$ then we can forget about all critical peaks which are called “composite” in 2.3 of Kapur &al. (1988) — and even some more, namely all those whose peak term is reducible at some position that is longer than the overlap position of the critical peak. Kapur &al. (1988) already states in Corollary 5 that (unless $l_{0}{\,\in\,}{{\rm V}}$, which some authors generally disallow) the irreducibility of these positions implies the irreducibility of all terms introduced by the unifying substitution $\sigma$; more precisely, the joinability test may assume: $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}x\sigma{\,\not=\,}x\ \ {\Rightarrow}\penalty-2\ \ x\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\end{array}\right)},$ which, by our first irreducibility assumption can be simplified to $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}.$ If we, however, revert $\lll$ by defining $q\lll_{t}p$ if ${\,\|{q}\|\,}\prec{\,\|{p}\|\,},$ then we can forget about all critical peaks which are called “composite” in 4.1 of Kapur &al. (1988) — and even some more, namely all those whose peak term is reducible at some nonempty position that is shorter than the overlap position of the critical peak. The power of the combination of the two weakenings of the joinability requirement, i.e. the confluence and the irreducibility assumptions, is demonstrated by the following simple but non-trivial example whose predicate ‘$\mathsf{nonnegp}$’ checks whether an integer number is non-negative: ###### Example 14.3 $\begin{array}[t]{lll}{{\mathbb{C}}}&:=&\\{{{\mathsf{0}}},{\mathsf{s}},{\mathsf{p}},{{\mathsf{true}}},{{\mathsf{false}}}\\}\\\ {{\mathbb{N}}}&:=&\\{{\mathsf{nonnegp}}\\}\\\ {\rm R}_{\,\rm\ref{example integers}}&:&\begin{array}[t]{llll}{{\mathsf{s}}{(}{{{\mathsf{p}}{(}{y}{)}}}{)}}&=&y\\\ {{\mathsf{p}}{(}{{{\mathsf{s}}{(}{y}{)}}}{)}}&=&y\\\ {{\mathsf{nonnegp}}{(}{{{\mathsf{0}}}}{)}}&=&{{\mathsf{true}}}\\\ {{\mathsf{nonnegp}}{(}{{{\mathsf{s}}{(}{x}{)}}}{)}}&=&{{\mathsf{true}}}&{\>{\longleftarrow}\>\>}{{\mathsf{nonnegp}}{(}{x}{)}}={{\mathsf{true}}}\\\ {{\mathsf{nonnegp}}{(}{{{\mathsf{p}}{(}{{{\mathsf{0}}}}{)}}}{)}}&=&{{\mathsf{false}}}\\\ {{\mathsf{nonnegp}}{(}{{{\mathsf{p}}{(}{x}{)}}}{)}}&=&{{\mathsf{false}}}&{\>{\longleftarrow}\>\>}{{\mathsf{nonnegp}}{(}{x}{)}}={{\mathsf{false}}}\\\ \end{array}\end{array}$ Let ${\mathsf{0}}$, $\mathsf{s}$, $\mathsf{p}$ be constructor symbols of the sort $\mathsf{int}$ and ${\mathsf{true}}$, ${\mathsf{false}}$ constructor symbols of the sort $\mathsf{bool}$. Let $\mathsf{nonnegp}$ be a non- constructor predicate with arity “$\ {\mathsf{int}}{\ {\rightarrow}\ }{\mathsf{bool}}\ $”. Let $x$, $y$ be constructor variables of the sort $\mathsf{int}$. Obviously, ${\rm R}_{\,\rm\ref{example integers}},{{\rm V}}$ is ${\rm V}$-compatible with the termination-pair $({\rhd},{\rhd})$ where $\rhd$ is the lexicographic path ordering generated by $\mathsf{nonnegp}$ being bigger than ${\mathsf{true}}$ and ${\mathsf{false}}$. There are only the following two critical peaks which are both of the form $(0,1)$: where $\sigma:=\\{x\mapsto{{\mathsf{p}}{(}{y}{)}}\\}$ and $\sigma^{\prime}:=\\{x\mapsto{{\mathsf{s}}{(}{y}{)}}\\}$. Their respective condition lists are the following two lists containing each one literal only: Now the following is easy to show: The irreducible constructor terms of the sort $\mathsf{int}$ are exactly the terms of the form ${\mathsf{s}}^{n}{(}{z}{)}$ or ${\mathsf{p}}^{n+1}{(}{z}{)}$ with $n{\,\in\,}{{\bf N}}$ and $z{\,\in\,}{{{\rm V}}\\!_{{{\mathcal{C}}},{{\mathsf{int}}}}}{\cup}\\{{{\mathsf{0}}}\\}$. The irreducible constructor terms of the sort $\mathsf{bool}$ are ${{{\rm V}}\\!_{{{\mathcal{C}}},{{\mathsf{bool}}}}}{\cup}\\{{{\mathsf{true}}},{{\mathsf{false}}}\\}$ . Furthermore, by induction on $n{\,\in\,}{{\bf N}}$ one easily shows ${{\mathsf{nonnegp}}{(}{{{\mathsf{s}}^{n}{(}{{{\mathsf{0}}}}{)}}}{)}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R}_{\,\rm\ref{example integers}},\emptyset}}}{{\mathsf{true}}}$ and ${{\mathsf{nonnegp}}{(}{{{\mathsf{p}}^{n+1}{(}{{{\mathsf{0}}}}{)}}}{)}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R}_{\,\rm\ref{example integers}},\emptyset}}}{{\mathsf{false}}}.$ Finally by induction on $n{\,\in\,}{{\bf N}}$ one easily shows that ${{\mathsf{nonnegp}}{(}{t}{)}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R}_{\,\rm\ref{example integers}},{{\rm V}},\omega+n}}}{{\mathsf{true}}}\ \ {\vee}\penalty-2\ \ {{\mathsf{nonnegp}}{(}{t}{)}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R}_{\,\rm\ref{example integers}},{{\rm V}},\omega+n}}}{{\mathsf{false}}}$ implies ${{{\mathcal{V}}}({t})}{\,=\,}\penalty-1\emptyset,$ which we only need to show confluence besides ground confluence. Define $\lll$ via ($p,q{\,\in\,}{{{\mathcal{POS}}}({t})}$): $q\lll_{t}p$ if $0ptt-{\,\|{q}\|\,}\prec 0ptt-{\,\|{p}\|\,}.$ Now the new combined weakening of joinability to $\rhd$-weak joinability w.r.t. ${\rm R}_{\,\rm\ref{example integers}},{{\rm V}}$ besides $A$ (with $A$ defined as in the theorem) allows us to show joinability of the above critical peaks very easily. Since the second critical peak can be treated analogous to the first, we explain how to treat the first only: By the new additional feature for assuming irreducibility, our weakened joinability allows to assume that $x\sigma\varphi$ is irreducible for the first critical peak, which can be seen in two different ways: First, since the critical peak is a non-overlay and $x$ occurs in the peak term ${\mathsf{nonnegp}}{(}{{{\mathsf{s}}{(}{x}{)}}}{)}$. Second, since the overlap position is $1,$ ${{\mathsf{nonnegp}}{(}{{{\mathsf{s}}{(}{x}{)}}}{)}}/1\ 1=x$ and $\forall q^{\prime}{\,\in\,}{{{\mathcal{POS}}}({x\sigma\varphi})}{.}\penalty-1\,\,\ 1\ 1\ q^{\prime}\lll_{{{\mathsf{nonnegp}}{(}{{{\mathsf{s}}{(}{x}{)}}}{)}}\sigma\varphi}1.$ Furthermore, we are allowed to assume that the condition of the critical peak is fulfilled, i.e. that ${{\mathsf{nonnegp}}{(}{x}{)}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R}_{\,\rm\ref{example integers}},{{\rm V}}}}}{{\mathsf{true}}}.$ Together with the irreducibility of $x\sigma\varphi{\,=\,}\penalty-1{{\mathsf{p}}{(}{y}{)}}\varphi$ this implies that $y\varphi$ is of the form ${\mathsf{p}}^{n}{(}{{{\mathsf{0}}}}{)}$. This again implies ${{\mathsf{nonnegp}}{(}{x}{)}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R}_{\,\rm\ref{example integers}},{{\rm V}}}}}{{\mathsf{false}}}.$ But since we may assume confluence below the condition term ${{\mathsf{nonnegp}}{(}{x}{)}}\sigma\varphi$ we get ${{\mathsf{true}}}{\downarrow_{{}_{{\rm R}_{\,\rm\ref{example integers}},{{\rm V}}}}}{{\mathsf{false}}},$ which is impossible. Thus the properties that weak joinability allows us to assume for the joinability test are inconsistent and the critical pair need not be joined at all. All in all, Theorem 14.2 implies confluence of ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{example integers}},{{\rm V}}}}$ without solving the task of showing that for each arbitrary (not necessarily normalized) substitution $\varphi$ either ${{\mathsf{nonnegp}}{(}{{{\mathsf{p}}{(}{y}{)}}}{)}}\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R}_{\,\rm\ref{example integers}},{{\rm V}}}}}{{\mathsf{true}}}$ does not hold or ${{\mathsf{nonnegp}}{(}{y}{)}}\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R}_{\,\rm\ref{example integers}},{{\rm V}}}}}{{\mathsf{true}}}$ holds, which is more difficult to show than our simple properties above. The following theorem is a generalization of Theorem 7.18 in Wirth & Gramlich (1994a). In comparison with Theorem 14.2 it offers for each condition term $u$ of a rule $l{=}r{\longleftarrow}C$ the possibility to replace the requirement $l\tau\rhd u\tau$ (roughly speaking i.e. decreasingness) with ${{{\mathcal{V}}}({u})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}$ (i.e. the absence of general variables). The basic idea of its proof is first to show $\omega$-shallow confluence up to $\omega$ (i.e. commutation of ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ and ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$) with the usual argumentation on quasi-normality, left-linearity, termination and $\omega$-shallow joinability (cf. Theorem 14.5), and then to use decreasingness argumentation for the confluence of ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$. ###### Theorem 14.4 (Syntactic Test for Confluence) Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume the following very weak kind of left-linearity of constructor rules w.r.t. general variables: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ p{\,=\,}\penalty-1q\end{array}\right)}.$ Furthermore, assume that constructor rules are quasi-normal w.r.t. ${\rm R},{{\rm X}}$: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ {((l,r),C)}\mbox{ is quasi-normal w.r.t.\ ${\rm R},{{\rm X}}$}\end{array}\right)}.$ Moreover, assume the following compatibility property for a termination-pair $(>,\rhd)$ over sig/${\rm V}$: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}}}}}\ \ {\Rightarrow}\penalty-2\ {{\left({{\begin{array}[]{ll}&l\tau>r\tau\\\ {\wedge}&\forall u\in{{{\mathcal{TERMS}}}({C})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&l\tau\rhd u\tau\\\ {\vee}&u\tau{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\\\ {\vee}&{{{\mathcal{V}}}({u})}\subseteq{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\\!\\!\\\ \end{array}}}\right)}}\\!\\!\end{array}\right)}.$ Assume ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ to be confluent. Assume that each critical peak ${((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)}\in{\rm CP}({\rm R})$ (with $(\mathchar 259\relax_{0},\mathchar 259\relax_{1}){\,\not=\,}(1,1)$ and $(\ (\mathchar 259\relax_{0},\mathchar 259\relax_{1}){\,\not=\,}(0,0)\ \ {\vee}\penalty-2\ \ {{{\mathcal{TERMS}}}({D_{0}\sigma\,D_{1}\sigma})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}\ )$) is $\omega$-shallow joinable up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$. [For each $t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ assume $\lll_{t}$ to be a wellfounded ordering on ${{\mathcal{POS}}}({t})$. Define ($p{\,\in\,}{{\bf N}}_{+}^{\ast}$) $A(p):={{\\{\ }t{\,\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\omega,q}}}})}}~{}{\|}\penalty-9\,\ {\emptyset{\,\not=\,}q\lll_{t}p{\ \\}}}\ {\cup}\ {{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}})}.$ ] Now the following two are logically equivalent: 1. 1. Each critical peak ${((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)}\in{\rm CP}({\rm R})$ (with $\forall k{\,\prec\,}2{.}\penalty-1\,\,{(\ \mathchar 259\relax_{k}{\,=\,}\penalty-11\ \ {\vee}\penalty-2\ \ {{{\mathcal{TERMS}}}({D_{k}\sigma})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}\ )}$) is $\rhd$-weakly joinable w.r.t. ${\rm R},{{\rm X}}$ [besides $A$]. 2. 2. ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is confluent. The following theorem generalizes Theorem 2 in Dershowitz &al. (1988) by weakening normality to quasi-normality. ###### Theorem 14.5 (Syntactic Test for $\omega$-Shallow Confluence) Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume the following weak kind of left-linearity w.r.t. general variables: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\ \ {\Rightarrow}\penalty-2\ \ p{\,=\,}\penalty-1q\end{array}\right)}.$ Furthermore, assume ${\rm R},{{\rm X}}$ to be quasi-normal. Let $({>},{\rhd})$ be a termination-pair over sig/${\rm V}$ such that the following compatibility property for constructor rules holds (which is always satisfied when R has conservative constructors): $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l\in{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ \forall u{\,\in\,}{{{\mathcal{TERMS}}}({C})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&l\tau\rhd u\tau\\\ {\vee}&u\tau{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\\\ {\vee}&{{{\mathcal{V}}}({u})}\subseteq{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume that the system is terminating: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{\left(\begin{array}[c]{l}C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}}}}}\end{array}\right)}\ \ {\Rightarrow}\penalty-2\ \ l\tau>r\tau\end{array}\right)}.$ [For each $t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ assume $\lll_{t}$ to be a wellfounded ordering on ${{\mathcal{POS}}}({t})$. Define ($p{\,\in\,}{{\bf N}}_{+}^{\ast}$, $n{\,\prec\,}\omega$) $A(p,n):={{\\{\ }t{\,\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n,q}}}})}}~{}{\|}\penalty-9\,\ {\emptyset{\,\not=\,}q\lll_{t}p{\ \\}}}.$ ] Now the following two are logically equivalent: 1. 1. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent and each critical peak ${((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)}\in{\rm CP}({\rm R})$ (with $(\ (\mathchar 259\relax_{0},\mathchar 259\relax_{1}){\,\not=\,}(0,0)\ \ {\vee}\penalty-2\ \ {{{\mathcal{TERMS}}}({D_{0}\sigma\,D_{1}\sigma})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}\ )$) is $\omega$-shallow joinable w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$]. 2. 2. ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent. The following theorem weakens the $\omega$-shallow joinability requirement to that of $\omega$-level joinability, but disallows general variables in conditions of rules. That this restriction is necessary indeed can be seen from Example 12.2. ###### Theorem 14.6 (Syntactic Test for $\omega$-Level Confluence) Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,{{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Let $({>},{\rhd})$ be a termination-pair over sig/${\rm V}$. Assume that the system is terminating: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{\left(\begin{array}[c]{l}C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\end{array}\right)}\ {\Rightarrow}\penalty-2\ \ l\tau>r\tau\end{array}\right)}.$ [For each $t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ assume $\lll_{t}$ to be a wellfounded ordering on ${{\mathcal{POS}}}({t})$. Define ($p{\,\in\,}{{\bf N}}_{+}^{\ast}$, $n{\,\prec\,}\omega$) $A(p,n):={{\\{\ }t{\,\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n,q}}}})}}~{}{\|}\penalty-9\,\ {\emptyset{\,\not=\,}q\lll_{t}p{\ \\}}}.$ ] Now the following two are logically equivalent: 1. 1. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent and each critical peak in ${\rm CP}({\rm R})$ of the forms $(0,1)$, $(1,0)$, or $(1,1)$ is $\omega$-level joinable w.r.t. ${\rm R},{{\rm X}}$ and $\rhd$ [besides $A$]. 2. 2. ${\rm R},{{\rm X}}$ is $\omega$-level confluent. The following theorem generalizes Theorem 4 in Dershowitz &al. (1988) and Theorem 6.3 in Wirth & Gramlich (1994a) by weakening overlay joinability to $\rhd$-quasi overlay joinability. For a discussion of the notion of $\rhd$-quasi overlay joinability cf. 9. The proof is discussed above the key lemma B.8. ###### Theorem 14.7 (Syntactic Confluence Criterion) Let R be a CRS over sig/cons/${\rm V}$ and ${{\rm X}}{\subseteq}{{\rm V}}$. Assume either that ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is terminating313131Actually innermost termination is enough here when we require overlay joinability instead of $\rhd$-quasi overlay joinability, cf. Gramlich (1995a). and ${\rhd}{\,=\,}\penalty-1{\rhd_{{}_{\rm ST}}}$ or that ${{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\subseteq{\rhd},$ ${\rhd_{{}_{\rm ST}}}\subseteq{\rhd},$ and $\rhd$ is a wellfounded ordering on $\mathcal{T}$. Now, if all critical peaks in ${\rm CP}({\rm R})$ are $\rhd$-quasi overlay joinable w.r.t. ${\rm R},{{\rm X}}$, then ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is confluent. ###### Example 14.8 Let ${{\rm X}}{\subseteq}{{\rm V}}$. The following system is neither decreasing, nor left-linear, nor overlay joinable; but it is terminating and $\rhd_{{}_{\rm ST}}$-quasi overlay joinable w.r.t. ${\rm R}_{\,\rm\ref{ex quasi over}},{{\rm X}}$. Thus Theorem 14.7 is the only one that implies confluence of ${\longrightarrow}_{{}_{\\!{\rm R}_{\,\rm\ref{ex quasi over}},{{\rm X}}}}$. Note that Theorem 14.4 becomes applicable when we replace the non-constructor variable in ($\mathsf{p}$1) with a constructor variable. Moreover, if we additionally do the same with ($\mathsf{p}$2), then Theorem 14.6 becomes applicable, too. Even though it is irrelevant for Theorem 14.7, let $X,Y\in{{{\rm V}}\\!_{{\rm SIG}}}$, ${{\mathsf{0}}},{\mathsf{s}},{{\mathsf{a}}},{{\mathsf{true}}},{{\mathsf{false}}}\in{{\mathbb{C}}\,}$, and ${\mathsf{less}},{\mathsf{p}},{\mathsf{f}},{\mathsf{g}}\in{{\mathbb{F}}}$. Note that ${{\mathsf{0}}},{\mathsf{s}},{{\mathsf{a}}},{\mathsf{less}}$ model the ordinal number $\omega{+}1$. ${\rm R}_{\,\rm\ref{ex quasi over}}$: $\begin{array}[t]{@{}l@{\mbox{~~~~~~~~~}}l@{\ =\ }ll}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\ \mbox{(\/$\mathsf{s}$1)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{s}}{(}{{{\mathsf{a}}}}{)}}&{{\mathsf{a}}}&\\\ \mbox{(\/$\mathsf{less}$1)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{less}}{(}{{{\mathsf{s}}{(}{X}{)}}}{,\,}{{{\mathsf{s}}{(}{Y}{)}}}{)}}&{{\mathsf{less}}{(}{X}{,\,}{Y}{)}}&\\\ \mbox{(\/$\mathsf{less}$2)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{less}}{(}{X}{,\,}{X}{)}}&{{\mathsf{false}}}\\\ \mbox{(\/$\mathsf{less}$3)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{less}}{(}{{{\mathsf{0}}}}{,\,}{{{\mathsf{s}}{(}{Y}{)}}}{)}}&{{\mathsf{true}}}\\\ \mbox{(\/$\mathsf{less}$4)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{less}}{(}{X}{,\,}{{{\mathsf{0}}}}{)}}&{{\mathsf{false}}}\\\ \mbox{(\/$\mathsf{less}$5)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{less}}{(}{{{\mathsf{0}}}}{,\,}{{{\mathsf{a}}}}{)}}&{{\mathsf{true}}}\\\ \mbox{(\/$\mathsf{less}$6)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{less}}{(}{{{\mathsf{a}}}}{,\,}{{{\mathsf{s}}{(}{Y}{)}}}{)}}&{{\mathsf{less}}{(}{{{\mathsf{a}}}}{,\,}{Y}{)}}&\\\ \mbox{(\/$\mathsf{less}$7)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{less}}{(}{{{\mathsf{s}}{(}{X}{)}}}{,\,}{{{\mathsf{a}}}}{)}}&{{\mathsf{less}}{(}{X}{,\,}{{{\mathsf{a}}}}{)}}&\\\ \mbox{(\/$\mathsf{p}$1)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{p}}{(}{X}{)}}&{{\mathsf{true}}}&{\>{\longleftarrow}\>\>}{{\mathsf{p}}{(}{{{\mathsf{s}}{(}{X}{)}}}{)}}{=}{{\mathsf{true}}}\\\ \mbox{(\/$\mathsf{p}$2)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{p}}{(}{X}{)}}&{{\mathsf{true}}}&{\>{\longleftarrow}\>\>}{{\mathsf{less}}{(}{{{\mathsf{f}}{(}{X}{)}}}{,\,}{{{\mathsf{g}}{(}{X}{)}}}{)}}{=}{{\mathsf{true}}}\\\ \mbox{(\/$\mathsf{f}$$i$)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{f}}{(}{X}{)}}&\ldots&\\\ \mbox{(\/$\mathsf{g}$$i$)}\hfil\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}&{{\mathsf{g}}{(}{X}{)}}&\ldots&\\\ \end{array}}}}}}}}}}}}}}$ The critical peaks are the following: From ($\mathsf{s}$1) into ($\mathsf{less}$1) we get: From ($\mathsf{s}$1) into ($\mathsf{less}$3) we get: The criticial peaks resulting from ($\mathsf{s}$1) into ($\mathsf{less}$6) and ($\mathsf{less}$7) are trivial. From ($\mathsf{less}$1) into ($\mathsf{less}$2) we get: From ($\mathsf{less}$2) into ($\mathsf{less}$1) we get: The criticial peaks resulting from ($\mathsf{less}$2) into ($\mathsf{less}$4), ($\mathsf{less}$4) into ($\mathsf{less}$2), ($\mathsf{p}$1) into ($\mathsf{p}$2), and ($\mathsf{p}$2) into ($\mathsf{p}$1) are trivial. ## 15 Criteria for Confluence of the Constructor Sub-System Define the constructor sub-system of a rule system R to be ${\rm R}_{\mathcal{C}}:={{\\{\ }{((l,r),C)}{\,\in\,}{\rm R}}~{}{\|}\penalty-9\,\ {l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}{\ \\}}},$ i.e. the system of the constructor rules of R. In this section we discuss the problem how to find out that ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}={{\longrightarrow}_{{}_{\\!{\rm R}_{\mathcal{C}},{{\rm X}},\omega}}}$ is confluent. Note that this is a necessary ingredient for achieving confluence via any of the theorems 13.3, 13.4, 13.6, 13.9, 14.4, 14.5, and 14.6. The easiest way to achieve confluence of ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is to have no constructor rules at all, i.e. ${\rm R}_{\mathcal{C}}{\,=\,}\penalty-1\emptyset.$ While it is rather restrictive, this case of free constructors is very important in practice since a lot of data structures can be specified this way. Moreover, it is economic to restrict to this case because non-free constructors make a lot of trouble when working with the specification, e.g., most techniques for proving inductive validity get into tremendous trouble with non-free constructors — if they are able to handle them at all. The second case where confluence of ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is immediate is when for each rule $l{=}r{\longleftarrow}C$ in ${\rm R}_{\mathcal{C}}$ also $r{=}l{\longleftarrow}C$ is an instance of a rule of R, and then also of ${\rm R}_{\mathcal{C}}$ due to the restriction on the constructor rule $l{=}r{\longleftarrow}C$ given by Definition 2.2. An example for this is the commutativity rule which is equal to a renamed version of the reverse of itself. In this case it may be worthwhile to consider reduction modulo a constructor congruence as described in Avenhau & Becker (1992) and Avenhau & Becker (1994). A third way to achieve confluence of ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is to use semantic confluence criteria in the style of Plaisted (1985), cf. also Theorem 6.5 in Wirth & Gramlich (1994a). While this semantic argumentation is very powerful when one has sufficient knowledge about the constructor domain, it is, however, not at all obvious how to formalize or even automate such semantic considerations. Above that, these semantic confluence criteria are based on the existence of normal forms and therefore require termination of the constructor sub-system (at least in some weak form). Termination of the constructor sub-system, of course, does not mean termination of the whole rule system. We may, e.g., apply Theorem 14.2 to infer confluence of a terminating constructor sub-system containing the associativity rule of Example 10.8 (whose confluence can hardly be inferred without termination) and then infer the confluence of the whole non- terminating rule system by some of the theorems of 13. This case where a terminating constructor sub-system is part of a non-terminating rule system seems to be important in practice since confluence of non-free constructors often can hardly be inferred without termination whereas termination is usually not needed for then inferring confluence of the whole system because the non-constructor rules can be chosen in such a way that their critical peaks are complementary, cf. Theorem 13.3. Moreover note that the reverse case, i.e. that of a non-terminating constructor sub-system of a terminating rule system, is impossible in our framework but not in the abovementioned one of Avenhau & Becker (1992) and Avenhau & Becker (1994) where the notion of reduction is different, namely reduction via ${\rm R}{\setminus}{\rm R}_{\mathcal{C}}$ modulo ${\rm R}_{\mathcal{C}}$. In the rest of this section we will present syntactic criteria for confluence of ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. First note that the theorems 14.2 and 14.7 can directly be applied to infer confluence of ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ simply by instantiating the ‘R’ of these theorems with ${\rm R}_{\mathcal{C}}$. The other theorems we will present in the following are nothing but informal corollaries of other theorems of the sections 13 and 14. To apply the latter theorems to our special case here, it is not sufficient only to throw away the non-constructor rules, but we also have to transform the constructor function symbols of the constructor rules into non-constructor function symbols. For consistency we then also have to rename their constructor variables with general variables. Then the constructor sub-system of the transformed system is empty and therefore trivially confluent, such that these theorems can be applied. If the constructor rules contain general variables or Def-literals, then, however, this transformation brings us beyond the two layered framework presented in this paper: As we translate constructor variables (level $0$) into general variables (level $1$), then, for consistency, since ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is a relation on the terms of the whole signature, we also have to translate general variables (level $1$) into some kind of variables of level $2$, and non-constructor function symbols (level $1$) into some kind of function symbols of level $2$. Symbols of level $2$, however, are not present in the framework presented in this paper. Moreover we have to translate our Def-literals (which test for reducibility to a ground term of level $0$) into predicate literals that test for reducibility to a ground term of level $1$, which are also not present in our framework. While it would be possible and beautiful to present our confluence criteria of the sections 13 and 14 in a framework with a special signature and variable-system for the level of each natural number, we have decided not to do so for the following reasons: First, it would make the paper even more technically and conceptually difficult as it is. Second, the infinitely layered framework may be of little importance (since its only useful application so far is this section). Third, the step of level $0$ we want to treat here may in principle allow of more powerful criteria than an arbitrary level $i$ and therefore it does not seem to be a good idea to achieve its confluence criteria as corollaries of the theorems for an arbitrary level. Fourth, by proving the theorems of this section separately, we provide the reader interested only in the standard positive conditional rule systems without constructor sub-signature and constructor sub-system with a direct approach to this special case. This can clearly be seen when one translates a system of the standard positive conditional framework into our framework by simply saying that all its symbols are constructor symbols. For all the following theorems let ${\rm R}_{\mathcal{C}}$ be the constructor sub-system of a CRS R over sig/cons/${\rm V}$ as defined above, and let ${{\rm X}}{\subseteq}{{\rm V}}.$ Note that the critical peaks in ${\rm CP}({\rm R}_{\mathcal{C}})$ are exactly the critical peaks of the form $(0,0)$ in ${\rm CP}({\rm R})$. The following is the analogue of parts (I) and (II) of Theorem 13.6. Note that we do not present the analogues of parts (III) and (IV) because they are subsumed323232This is because the notion of $0$-shallow [noisy] weak parallel joinability (when defined analogous to the notion of $\omega$-shallow [noisy] weak parallel joinability) is identical to the notion of $0$-shallow [noisy] parallel joinability. by the analogue of part (I). ###### Theorem 15.1 (Syntactic Criterion for $0$-Shallow Confluence) Assume ${{\rm R},{{\rm X}}}$ to be $0$-quasi-normal and ${\rm R}_{\mathcal{C}}$ to be left-linear. 1. (I) Now if each critical peak in ${\rm CP}({\rm R})$ of the form $(0,0)$ is $0$-shallow noisy parallel joinable w.r.t. ${\rm R},{{\rm X}}$, and each non- overlay in ${\rm CP}({\rm R})$ of the form $(0,0)$ is $0$-shallow parallel closed w.r.t. ${\rm R},{{\rm X}}$, then ${\rm R},{{\rm X}}$ is $0$-shallow confluent. 2. (IIa) If ${\rm R}_{\mathcal{C}}$ is right-linear and if each critical peak in ${\rm CP}({\rm R})$ of the form $(0,0)$ is $0$-shallow noisy strongly joinable w.r.t. ${\rm R},{{\rm X}}$, and each non-overlay in ${\rm CP}({\rm R})$ of the form $(0,0)$ is $0$-shallow noisy anti-closed w.r.t. ${\rm R},{{\rm X}}$, then ${\rm R},{{\rm X}}$ is $0$-shallow confluent. 3. (IIb) If ${\rm R}_{\mathcal{C}}$ is right-linear and if each critical peak in ${\rm CP}({\rm R})$ of the form $(0,0)$ is $0$-shallow strongly joinable w.r.t. ${\rm R},{{\rm X}}$, and each non-overlay in ${\rm CP}({\rm R})$ of the form $(0,0)$ is $0$-shallow anti-closed w.r.t. ${\rm R},{{\rm X}}$, then ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is strongly confluent. ###### Corollary 15.2 If ${\rm R},{{\rm X}}$ is $0$-shallow confluent, then ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent. We omit the analogue of Theorem 13.9 here because it requires that the conditions of the constructor rules do not contain any variables. In this case ${\rm R}_{\mathcal{C}}$ can (in general not effectively) be transformed into an unconditional system with identical reduction relation (with possibly different depths) to which we can then apply Theorem 15.1 instead. The following is the analogue of Theorem 13.3. ###### Theorem 15.3 (Syntactic Confluence Criterion) If ${\rm R}_{\mathcal{C}}$ is left-linear and normal and all critical peaks of ${\rm R}_{\mathcal{C}}$ are complementary, then ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent. The analogue of theorems 14.2 and 14.4 is just Theorem 14.2 with ‘R’ instantiated with ${\rm R}_{\mathcal{C}}$. The following is the analogue of Theorem 14.5. ###### Theorem 15.4 (Syntactic Test for $0$-Shallow Confluence) Let $({>},{\rhd})$ be a termination-pair over sig/${\rm V}$. Assume ${{\rm R},{{\rm X}}}$ to be $0$-quasi-normal and ${\rm R}_{\mathcal{C}}$ to be left-linear. Furthermore, assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is terminating: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ l\tau>r\tau\end{array}\right)}.$ [For each $t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ assume $\lll_{t}$ to be a wellfounded ordering on ${{\mathcal{POS}}}({t})$. Define ($p{\,\in\,}{{\bf N}}_{+}^{\ast}$, $n{\,\prec\,}\omega$) $A(p,n):={{\\{\ }t{\,\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},n,q}}}})}}~{}{\|}\penalty-9\,\ {\emptyset{\,\not=\,}q\lll_{t}p{\ \\}}}.$ ] Now the following two are logically equivalent: 1. 1. Each critical peak in ${\rm CP}({\rm R})$ of the form $(0,0)$ is $0$-shallow joinable w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$]. 2. 2. ${\rm R},{{\rm X}}$ is $0$-shallow confluent. We omit the analogue of Theorem 14.6 here because it requires that the conditions of the constructor rules do not contain any variables. In this case ${\rm R}_{\mathcal{C}}$ can be transformed into an unconditional system with identical reduction relation to which we can then apply Theorem 14.2 with ‘R’ instantiated with ${\rm R}_{\mathcal{C}}$. The analogue of Theorem 14.7 is just Theorem 14.7 with ‘R’ instantiated with ${\rm R}_{\mathcal{C}}$. ## References 1 * 2 * 3 * 4 * 5 Jurgen Avenhau, Klau Becker (1992). Conditional Rewriting modulo a Built-in Algebra. SEKI-Report SR–92–11, FB Informatik, Univ. Kaiserlautern(SFB). 6 * 7 Jurgen Avenhau, Klau Becker (1994). Operational Specifications with Built-Ins. 11 th STACS 1994, LNCS 775, pp. 263–274, Springer. 8 * 9 Jurgen Avenhau, Carlo A. Loría-Sáenz (1994). On conditional rewrite systems with extra variables and deterministic logic programs. 5 th LPAR 1994, LNAI 822, pp. 215–229, Springer. 10 * 11 Jurgen Avenhau, Klau Madlener (1989). Term Rewriting and Equational Reasoning. In: R. B. Banerji (eds.). Formal Techniques in Artificial Intelligence. Academic Press (Elsevier). 12 * 13 Klau Becker (1993). Proving Ground Confluence and Inductive Validity in Constructor Based Equational Specifications. TAPSOFT 1993, LNCS 668, pp. 46–60, Springer. * 14 * 15 Klau Becker (1994). Rewrite Operationalization of Clausal Specifications with Predefined Structures. PhD thesis, Fachbereich Informatik, Universitat Kaiserlautern. 16 * 17 Jan A. Bergstra, Jan Willem Klop (1986). Conditional Rewrite Rules: Confluence and Termination. J. Computer and System Sci. 32, pp. 323–362, Academic Press (Elsevier). * 18 * 19 Nachum Dershowitz (1987). Termination of Rewriting. J. Symbolic Computation (1987) 3, pp. 69–116, Academic Press (Elsevier). * 20 * 21 Nachum Dershowitz, Mitsuhiro Okada, G. Sivakumar (1988). Confluence of Conditional Rewrite Systems. 1 st CTRS 1987, LNCS 308, pp. 31–44, Springer. * 22 * 23 Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop (1991). Open Problems in Rewriting. 4 th RTA 1991, LNCS 488, pp. 445–456, Springer. * 24 * 25 Alfons Geser (1994). An Improved General Path Order. MIP-9407, Universität Passau. Accepted by J. Applicable Algebra in Engineering, Communication and Computing (AAECC), 1995\. * 26 * 27 Bernhard Gramlich (1994). On Modularity of Termination and Confluence Properties of Conditional Rewrite Systems. 4 th Algebraic and Logic Programming 1994, LNCS 850, pp. 186–203, Springer. * 28 * 29 Bernhard Gramlich (1995a). On Termination and Confluence of Conditional Rewrite Systems. 4 th CTRS 1994, LNCS 968, pp. ?–?, Springer. * 30 * 31 Bernhard Gramlich (1995b). On Weakening Overlay Joinability. Personal Communication, June 1995. * 32 * 33 Gérard Huet (1980). Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems. J. ACM 27 (4), pp. 797–821, ACM Press. * 34 * 35 * 36 Stéphane Kaplan (1987). Simplifying Conditional Term Rewriting Systems: Unification, Termination and Confluence. J. Symbolic Computation (1987) 4, pp. 295–334, Academic Press (Elsevier). * 37 * 38 Stéphane Kaplan (1988). Positive/Negative Conditional Rewriting. 1 st CTRS 1987, LNCS 308, pp. 129–143, Springer. * 39 * 40 Deepak Kapur, David R. Musser, Paliath Narendran (1988). Only Prime Superpositions Need be Considered in the Knuth-Bendix Completion Procedure. J. Symbolic Computation (1988) 6, pp. 19–36, Academic Press (Elsevier). * 41 * 42 Jan Willem Klop (1980). Combinatory Reduction Systems. Mathematical Centre Tracts 127, Mathematisch Centrum, Amsterdam. * 43 * 44 Jan Willem Klop (1992). Term Rewriting Systems. In: S. Abramsky, Dov M. Gabbay, T. S. E. Maibaum (eds.). Handbook of Logic in Computer Science, Vol. 2. Clarendon Press. * 45 * 46 Wolfgang Kuchlin (1985). A Confluence Criterion Based on the Generalized Newman Lemma. EUROCAL ’85, LNCS 204, pp. 390–399, Springer. 47 * 48 Aart Middeldorp (1993). Modular Properties of Conditional Term Rewriting Systems. Information and Computation 104, pp. 110–158, Academic Press (Elsevier). * 49 * 50 Aart Middeldorp, Erik Hamoen (1994). Completeness Results for Basic Narrowing. J. Applicable Algebra in Engineering, Communication and Computing (AAECC) 5, pp. 313–253, Springer. * 51 * 52 Vincent van Oostrom (1994a). Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit te Amsterdam. * 53 * 54 Vincent van Oostrom (1994b). Developing Developments. ISRL–94–4, Information Science and Research Laboratory, Nippon Telegraph and Telephone Corporation. * 55 * 56 David A. Plaisted (1985). Semantic Confluence Tests and Completion Methods. Information and Control 65, pp. 182–215. * 57 * 58 Taro Suzuki, Aart Middeldorp, Tetsuo Ida (1995). Level-Confluence of Conditional Rewrite Systems with Extra Variables in Right-Hand Sides. 6 th RTA 1995, LNCS 914, pp. 179–193, Springer. * 59 * 60 Yoshihito Toyama (1988). Commutativity of Term Rewriting Systems. In: K. Fuchi, L. Kott (eds.). Programming of Future Generation Computers II. Elsevier. Also in: Toyama (1990). * 61 * 62 Yoshihito Toyama (1990). Term Rewriting Systems and the Church-Rosser Property. PhD thesis, Tohoku University / Nippon Telegraph and Telephone Corporation. * 63 * 64 Christoph Walther (1994). Mathematical Induction. In: Handbook of Logic in Artificial Intelligence and Logic Programming. Vol. 2, Clarendon Press. * 65 * 66 Clau-Peter Wirth, Bernhard Gramlich (1993). A Constructor-Based Approach for Positive/Negative-Conditional Equational Specifications. 3 rd CTRS 1992, LNCS 656, pp. 198–212, Springer. Revised and extended version is Wirth & Gramlich (1994a). * 67 * 68 * 69 Clau-Peter Wirth, Bernhard Gramlich (1994a). A Constructor-Based Approach for Positive/Negative-Conditional Equational Specifications. J. Symbolic Computation (1994) 17, pp. 51–90, Academic Press (Elsevier). * 70 * 71 Clau-Peter Wirth, Bernhard Gramlich (1994b). On Notions of Inductive Validity for First-Order Equational Clauses. 12 th CADE 1994, LNAI 814, pp. 162–176, Springer. * 72 * 73 Clau-Peter Wirth, Rudiger Lunde (1994). Writing Positive/Negative-Conditional Equations Conveniently. SEKI-Working-Paper SWP–94–04, FB Informatik, Univ. Kaiserlautern(SFB). 74 * 75 Clau-Peter Wirth, Bernhard Gramlich, Ulrich Kuhler, Horst Prote (1993). Constructor-Based Inductive Validity in Positive/Negative-Conditional Equational Specifications. SEKI-Report SR–93–05, FB Informatik, Univ. Kaiserlautern(SFB). Revised and extended version of first part is Wirth & Gramlich (1994a), revised version of second part is Wirth & Gramlich (1994b). 76 * 77 Acknowledgements: I would like to thank Bernhard Gramlich for many fruitful discussions and J*urgen Avenhau, Roland Fettig, Klau Madlener, Birgit Reinert, and Andrea Sattler-Klein for some useful hints. I also would like to thank Thomas Deiß for providing me with a TeX-version with huge semantic stack size and Paul Taylor for his support with his diagram typesetting TeX-package. ## Appendix A Further Lemmas for Section 13 ###### Lemma A.1 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume R to have conservative constructors, ${\rm R},{{\rm X}}$ to be quasi- normal, and the following weak kind of left-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent, that each critical peak from ${\rm CP}({\rm R})$ of the form $(0,1)$ is $\omega$-shallow [noisy] parallel joinable up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay from ${\rm CP}({\rm R})$ of the form $(1,0)$ is $\omega$-shallow parallel closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$. Now for each $n\prec\omega$: ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. A fortiori ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$. ###### Lemma A.2 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume R to have conservative constructors, ${\rm R},{{\rm X}}$ to be quasi- normal, and the following very weak kind of left-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\vee}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume that for each $n\prec\omega$: ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. Moreover, assume that each critical peak from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-shallow noisy parallel joinable w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-shallow parallel closed w.r.t. ${\rm R},{{\rm X}}$. Now for all $n_{0}\preceq n_{1}\prec\omega$: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}$. A fortiori ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent. ###### Lemma A.3 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume R to have conservative constructors, ${\rm R},{{\rm X}}$ to be quasi- normal, and the following very weak kind of left-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\end{array}\right)}.$ Furthermore, assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is strongly confluent, that each critical peak from ${\rm CP}({\rm R})$ of the form $(0,1)$ is $\omega$-shallow [noisy] weak parallel joinable up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay from ${\rm CP}({\rm R})$ of the form $(1,0)$ is $\omega$-shallow closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$. Now for each $n\prec\omega$: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. A fortiori ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$. ###### Lemma A.4 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume R to have conservative constructors, ${\rm R},{{\rm X}}$ to be quasi- normal, and the following weak kinds of left- and right-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}\\\ {\wedge}&\forall p,q{\,\in\,}{{{\mathcal{POS}}}({r})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&r/p{\,=\,}\penalty-1x{\,=\,}\penalty-1r/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\end{array}\right)}\\\ \end{array}}}\right)}}.$ Furthermore, assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent, that each critical peak from ${\rm CP}({\rm R})$ of the form $(0,1)$ is $\omega$-shallow [noisy] strongly joinable up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay from ${\rm CP}({\rm R})$ of the form $(1,0)$ is $\omega$-shallow [noisy] anti-closed up to $\omega$ w.r.t. ${\rm R},{{\rm X}}$. Now for each $n\prec\omega$: ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. A fortiori ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$. ###### Lemma A.5 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume R to have conservative constructors, ${\rm R},{{\rm X}}$ to be quasi- normal, and the following very weak kind of left-linearity $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\vee}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume that for each $n\prec\omega$: ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\ \subseteq\ {{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}.$ Moreover, assume that that each critical peak from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-shallow noisy weak parallel joinable w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-shallow closed w.r.t. ${\rm R},{{\rm X}}$. Now for all $n_{0}\preceq n_{1}\prec\omega$: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}$. A fortiori ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent. ###### Lemma A.6 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume R to have conservative constructors, ${\rm R},{{\rm X}}$ to be quasi- normal, and the following very weak kind of left- and right-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\vee}&r/p{\,=\,}\penalty-1x{\,=\,}\penalty-1r/q\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&p{\,=\,}\penalty-1q\\\ {\vee}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\vee}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume that for each $n\prec\omega$: ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. Moreover, assume that each critical peak from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-shallow noisy strongly joinable w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-shallow noisy anti-closed w.r.t. ${\rm R},{{\rm X}}$. Now for all $n_{0}\preceq n_{1}\prec\omega$: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}$. A fortiori ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent. ###### Lemma A.7 Let $n_{0},n_{1}\prec\omega$. Let $\mu,\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$. Let ${((l,r),C)}\in{\rm R}$. Assume that $n_{0}{\,\preceq\,}n_{1}$ or that ${{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Assume that ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n_{1}$. Now, if $C\mu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}$ and $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}x\nu,$ then $C\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}$ and $l\nu{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}+1}}}r\nu.$ ###### Lemma A.8 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,{{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}$ and the following very weak kind of left-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\vee}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume333333Contrary to analogous lemma for shallow joinability (i.e. Lemma A.2), this strong commutation assumption is not really essential for this lemma if we are confident with the result that ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ (instead of ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$) strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}$ (which directly allows to get rid of the application of the strong commutation assumption in the proof of Claim 2). Then it is sufficient to assume that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$ (which means that Claim 0 of the proof holds directly), that ${{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega}}\circ{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}\ {\ {\subseteq}\ }\ {{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}$ (which replaces the application of the strong commutation assumption in the proof of Claim 5), and that the non-overlays of the form $(1,1)$ satisfy instead of $\omega$-level parallel closedness (which allows to replace the application of the strong commutation assumption at the end of “The critical peak case”). that for each $n\prec\omega$: ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. Moreover, assume that each critical peak from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level parallel joinable w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level parallel closed w.r.t. ${\rm R},{{\rm X}}$. Now for all $n\prec\omega$: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}$. A fortiori ${\rm R},{{\rm X}}$ is $\omega$-level confluent. ###### Lemma A.9 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,{{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}},$ and the following very weak kind of left-linearity $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\vee}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume that for each $n\prec\omega$: ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\ \subseteq\ {{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}.$ Moreover, assume that that each critical peak from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level weak parallel joinable w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level closed w.r.t. ${\rm R},{{\rm X}}$. Now for all $n\prec\omega$: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}$. A fortiori ${\rm R},{{\rm X}}$ is $\omega$-level confluent. ###### Lemma A.10 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,{{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}$ and the following very weak kind of left- and right- linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\vee}&r/p{\,=\,}\penalty-1x{\,=\,}\penalty-1r/q\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&p{\,=\,}\penalty-1q\\\ {\vee}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\vee}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume343434Contrary to analogous lemma for shallow joinability (i.e. Lemma A.6), this strong commutation assumption is not really essential for this lemma if we are confident with the result that ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ (instead of ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$) strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}$ (which directly allows to get rid of the application of the strong commutation assumption in the proof of Claim 2). Then it is sufficient to assume that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$ (which means that Claim 0 of the proof holds directly), that ${{\longleftarrow}_{{}_{\\!\omega}}}\circ{{\longrightarrow}_{{}_{\\!\omega+n}}}\ {\ {\subseteq}\ }\ {{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}$ (which replaces the application of the strong commutation assumption in the proof of Claim 5), that the critical peaks of the form $(1,1)$ satisfy instead of $\omega$-level strong joinability (which allows to complete “The second critical peak case” for the new induction hypothesis), that the non- overlays of the form $(1,1)$ satisfy instead of $\omega$-level anti- closedness (which allows to complete “The critical peak case” for the new induction hypothesis). that for each $n\prec\omega$: ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. Moreover, assume that each critical peak from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level strongly joinable w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-level anti-closed w.r.t. ${\rm R},{{\rm X}}$. Now for all $n\prec\omega$: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}$. A fortiori ${\rm R},{{\rm X}}$ is $\omega$-level confluent. ## Appendix B Further Lemmas for Section 14 ###### Lemma B.1 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}$. Let $\alpha\in\\{0,\omega\\}$. Let $(>,\rhd)$ be a termination-pair over sig/${\rm V}$. If $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\end{array}\right)}\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ l\tau>r\tau\end{array}\right)},$ then ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}\subseteq{\rhd}.$ ###### Lemma B.2 Let $\mu,\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$. Let ${((l,r),C)}\in{\rm R}$. Let $({>},{\rhd})$ be a termination-pair over sig/${\rm V}$ such that: $\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}}}}}\ \ {\Rightarrow}\penalty-2\ {{\left({{\begin{array}[]{ll}&l\tau>r\tau\\\ {\wedge}&\forall u\in{{{\mathcal{TERMS}}}({C})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&l\tau\rhd u\tau\\\ {\vee}&u\tau{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\\\ \lx@intercol\left[\begin{array}[]{@{\vee\ \ }l@{\ }}{{{\mathcal{V}}}({u})}\subseteq{{{\rm V}}\\!_{{\mathcal{C}}}}\end{array}\right]\hfil\\\ \end{array}}}\right)}}\\!\\!\\\ \end{array}}}\right)}}\\!\\!\end{array}\right)}.$ Assume that $\forall u{\lhd}l\mu{.}\penalty-1\,\,{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\mbox{ is confluent below }u.$ [Assume that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}\subseteq{\downarrow_{{}_{{\rm R},{{\rm X}}}}}.$ ] Now, if $C\mu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}}}}$ and $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}}}}}x\nu,$ then $C\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}}}}$ and $l\nu{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}}}}}r\nu.$ ###### Lemma B.3 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Let $({>},{\rhd})$ be a termination-pair over sig/${\rm V}$ such that: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}}}}}\ \ {\Rightarrow}\penalty-2\ {{\left({{\begin{array}[]{ll}&l\tau>r\tau\\\ {\wedge}&\forall u\in{{{\mathcal{TERMS}}}({C})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&l\tau\rhd u\tau\\\ {\vee}&u\tau{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\\\ \lx@intercol\left[\begin{array}[]{@{\vee\ \ }l@{\ }}{{{\mathcal{V}}}({u})}\subseteq{{{\rm V}}\\!_{{\mathcal{C}}}}\end{array}\right]\hfil\\\ \end{array}}}\right)}}\\!\\!\\\ \end{array}}}\right)}}\\!\\!\end{array}\right)}.$ For each $t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ assume $\lll_{t}$ to be a wellfounded ordering on ${{\mathcal{POS}}}({t})$. Define ($p{\,\in\,}{{\bf N}}_{+}^{\ast}$) $A(p):={{\\{\ }t{\,\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\omega,q}}}})}}~{}{\|}\penalty-9\,\ {\emptyset{\,\not=\,}q\lll_{t}p{\ \\}}}\ {[\ {\cup}\ {{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}})}]}\ .$ [Assume that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}\subseteq{\downarrow_{{}_{{\rm R},{{\rm X}}}}}.$ ] Assume that each critical peak ${((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)}\in{\rm CP}({\rm R})$ [with $\forall k{\,\prec\,}2{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\mathchar 259\relax_{k}{\,=\,}\penalty-11\ \ {\vee}\penalty-2\ \ {{{\mathcal{TERMS}}}({D_{k}\sigma})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}\end{array}\right)}$ ] is $\rhd$-weakly joinable w.r.t. ${\rm R},{{\rm X}}$ besides $A$. Now: ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is confluent. ###### Lemma B.4 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}$. Let $\beta\preceq\omega$. Let $\hat{s}{\,\in\,}{\mathcal{T}}$. Assume the following very weak kind of left-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ p{\,=\,}\penalty-1q\end{array}\right)}.$ Furthermore, assume the following compatibility property for a termination- pair $(>,\rhd)$ over sig/${\rm V}$: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&l\in{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{{\left({{\begin{array}[]{ll}&{((l,r),C)}\mbox{ is quasi-normal w.r.t.\ ${\rm R},{{\rm X}}$}\\\ {\wedge}&\forall u\in{{{\mathcal{TERMS}}}({C})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&l\tau\rhd u\tau\\\ {\vee}&u\tau{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}\\\ {\vee}&{{{\mathcal{V}}}({u})}\subseteq{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}$ and $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{\left(\begin{array}[c]{l}C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}}}}}\end{array}\right)}\ \ {\Rightarrow}\penalty-2\ \ l\tau>r\tau\end{array}\right)}.$ [For each $t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ assume $\lll_{t}$ to be a wellfounded ordering on ${{\mathcal{POS}}}({t})$. Define ($p{\,\in\,}{{\bf N}}_{+}^{\ast}$, $n{\,\prec\,}\omega$) $A(p,n):={{\\{\ }t{\,\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n,q}}}})}}~{}{\|}\penalty-9\,\ {\emptyset{\,\not=\,}q\lll_{t}p{\ \\}}}.$ ] Assume ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ to be confluent. Assume that each critical peak ${((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)}\in{\rm CP}({\rm R})$ with $(\mathchar 259\relax_{0},\mathchar 259\relax_{1}){\,\not=\,}(1,1)$ and $\left(\begin{array}[c]{l}(\mathchar 259\relax_{0},\mathchar 259\relax_{1}){\,\not=\,}(0,0)\ \ {\vee}\penalty-2\ \ {{{\mathcal{TERMS}}}({D_{0}\sigma\,D_{1}\sigma})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}\end{array}\right)$ is $\omega$-shallow joinable up to $\beta$ and $\hat{s}$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$]. Now: ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\beta$ and $\hat{s}$ in $\lhd$. ###### Lemma B.5 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Let $\alpha\in\\{0,\omega\\}$. Let $\beta\preceq\omega{+}\alpha$. Let $\hat{s}{\,\in\,}{\mathcal{T}}.$ Assume the following weak kind of left-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\end{array}\right)}\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-1\omega\ \ {\Rightarrow}\penalty-2\ \ x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}\end{array}\right)}\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ p{\,=\,}\penalty-1q\end{array}\right)}.$ Furthermore, assume ${\rm R},{{\rm X}}$ to be $\alpha$-quasi-normal. Let $({>},{\rhd})$ be a termination-pair over sig/${\rm V}$ such that the following compatibility property holds: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+\alpha}}}\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\end{array}\right)}\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ l\tau>r\tau\end{array}\right)}$ [For each $t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ assume $\lll_{t}$ to be a wellfounded ordering on ${{\mathcal{POS}}}({t})$. Define ($p{\,\in\,}{{\bf N}}_{+}^{\ast}$, $n{\,\prec\,}\omega$) $A(p,n):={{\\{\ }t{\,\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n,q}}}})}}~{}{\|}\penalty-9\,\ {\emptyset{\,\not=\,}q\lll_{t}p{\ \\}}}.$ ] Assume ${\rm R},{{\rm X}}$ to be $\alpha$-shallow confluent up to $\alpha$. Assume that each critical peak ${((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ \sigma,\ p)}\in{\rm CP}({\rm R})$ with $\forall k{\,\prec\,}2{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ \mathchar 259\relax_{k}{\,=\,}\penalty-10\end{array}\right)}\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-1\omega\ \ {\Rightarrow}\penalty-2\ \ {\left(\begin{array}[c]{l}\mathchar 259\relax_{k}{\,=\,}\penalty-11\ \ {\vee}\penalty-2\ \ {{{\mathcal{TERMS}}}({D_{k}\sigma})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}\end{array}\right)}\end{array}\right)}\\\ \end{array}}}\right)}}$ is $\alpha$-shallow joinable up to $\beta$ and $\hat{s}$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$]. Now: ${\rm R},{{\rm X}}$ is $\alpha$-shallow confluent up to $\beta$ and $\hat{s}$ in $\lhd$. ###### Lemma B.6 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Let $\beta\preceq\omega$. Let $\hat{s}{\,\in\,}{\mathcal{T}}.$ Assume $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,{{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Let $({>},{\rhd})$ be a termination-pair over sig/${\rm V}$ such that the following compatibility property holds: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall\tau{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{\left(\begin{array}[c]{l}C\tau\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\end{array}\right)}\ {\Rightarrow}\penalty-2\ \ l\tau>r\tau\end{array}\right)}.$ [For each $t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ assume $\lll_{t}$ to be a wellfounded ordering on ${{\mathcal{POS}}}({t})$. Define ($p{\,\in\,}{{\bf N}}_{+}^{\ast}$, $n{\,\prec\,}\omega$) $A(p,n):={{\\{\ }t{\,\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n,q}}}})}}~{}{\|}\penalty-9\,\ {\emptyset{\,\not=\,}q\lll_{t}p{\ \\}}}.$ ] Assume ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ to be confluent. Assume that each critical peak in ${\rm CP}({\rm R})$ of the forms $(0,1)$, $(1,0)$, or $(1,1)$ is $\omega$-level joinable up to $\beta$ and $\hat{s}$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$]. Now: ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $\beta$ and $\hat{s}$ in $\lhd$. The following lemma generalizes Lemma 7.6 of Wirth & Gramlich (1994a) by requiring $\rightrightarrows$ to be terminating only below a restricted set of terms T: ###### Lemma B.7 Let ${\rm T}\subseteq{\mathcal{T}}.$ Let ${\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}$ denote the set of subterms of T. Let $\rightrightarrows$ be a sort-invariant (This can always be achieved by identifying all sorts.) and T-monotonic relation on $\mathcal{T}$. Define ${\rhd}\ {\ {\ {:=}\ }\ }\ {{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{(\rightrightarrows\cup{\rhd_{{}_{\rm ST}}})}^{\scriptscriptstyle+}}.$ Now: 1. 1. $\begin{array}[t]{@{}r@{\nottight{\nottight{\nottight{\nottight=}}}}cl}{{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\ {\ {\ {=}\ }\ }\ }\ &{{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\circ}\ {{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}&;\\\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\ {\ {\ {=}\ }\ }\ }\ &{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\circ}\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}&.\\\ \end{array}$ 2. 2. ${{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}\ {\circ}\ {\rhd_{{}_{\rm ST}}}\ {\circ}\ {\rightrightarrows}\ {\ {\ {\subseteq}\ }\ }\ {{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}\ {\circ}\ {\rightrightarrows}\ {\circ}\ {{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}\ {\circ}\ {\rhd_{{}_{\rm ST}}}.$ Moreover, for ${\rm T}{\,=\,}\penalty-1{\mathcal{T}}$: $\ \ {\rhd_{{}_{\rm ST}}}\circ\rightrightarrows\ \ \subseteq\ \ \rightrightarrows\circ{\rhd_{{}_{\rm ST}}}$. 3. 3. $\begin{array}[t]{@{}r@{}c@{}ll}\rhd&\ {\ {\ {\subseteq}}}&{{\trianglelefteq_{{}_{\rm ST}}}}\ {\circ}\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{(\rightrightarrows\cup{\rhd_{{}_{\rm ST}}})}^{\scriptscriptstyle+}}&;\\\ \rhd&\ {\ {\ {=}}}&{{{\left(\begin{array}[c]{l}{{({{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ \rightrightarrows)}}\ {\cup}\ {{({{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}})}}\end{array}\right)}}^{\scriptscriptstyle+}}\ {\circ}\ {{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}&;\\\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{(\rightrightarrows\cup{\rhd_{{}_{\rm ST}}})}^{\scriptscriptstyle+}}&\ {\ {\ {=}}}&{\left(\begin{array}[c]{l}{{{{}_{\rm T}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}}\end{array}\right)}\ {\ {\cup}\ }\ {\left(\begin{array}[c]{l}{{{(\ {{{{}_{\rm T}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ )}}^{\scriptscriptstyle+}}\ {\circ}\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\trianglerighteq_{{}_{\rm ST}}}}\end{array}\right)}&.\\\ \end{array}$ Moreover, for ${\rm T}{\,=\,}\penalty-1{\mathcal{T}}$: $\ {{\rhd}\ {\ {\ {=}\ }\ }\ {\rhd_{{}_{\rm ST}}}\ {\cup}\ ({{\rightrightarrows}^{\scriptscriptstyle+}}\circ{\trianglerighteq_{{}_{\rm ST}}})}\ .$ 4. 4. If $\rightrightarrows$ is terminating (below all $t\in{\rm T}$) [and $\rightrightarrows$ and T are ${\rm X}$-stable], then $\rhd$ is a wellfounded [and ${\rm X}$-stable] ordering on ${\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}$ (which does not need to be sort-invariant or T-monotonic). 5. 5. (4) does not hold in general if one of the two conditions “$\rightrightarrows$ sort-invariant” or “$\rightrightarrows$ T-monotonic” is removed. Moreover, (4) does not hold in general for ${{{(\rightrightarrows\cup{\rhd_{{}_{\rm ST}}})}}}^{\scriptscriptstyle+}$ instead of $\rhd$. The proof of the following lemma and its far more restrictive predecessors has an interesting history. After its first occurrence in Dershowitz &al. (1988) for overlay joinable positive conditional systems, in our proof for quasi overlay joinable positive/negative-conditional systems in Wirth & Gramlich (1994a) we changed the third component of the induction ordering from ${{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}$ to $\succ$, the ordering of the ordinals. This change was done because it allowed us to check for generalizations more easily but did not result in a stronger criterion at first. Later, however, this change of the induction ordering turned out to be essential for Theorem 21 of Gramlich (1995a) saying that an innermost terminating overlay joinable positive conditional rule system is terminating and confluent: Due to the mutual dependency of the termination and the confluence proof, when proving confluence it was not possible to assume global termination but local termination only. And it was especially impossible to assume termination for that part of ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ which was necessary for the third component of the induction ordering. The following lemma (just like Theorem 7 of Gramlich (1995a)) requires local termination instead of global termination, which is not really necessary for proving Theorem 14.7 but again allows us to check for future generalizations more easily. Moreover, note that the form of the proof has been considerably improved compared to any previous publication: Claim 0 of the proof does not only provide us with the new irreducibility assumptions we have included into the notion of $\rhd$-quasi overlay joinability but also subsumes the whole second case of the global case distinction of the proof (as presented in Dershowitz (1987) as well as presented in Wirth & Gramlich (1994a)). As a consequence, in the whole new proof now the second and the third component of the induction ordering are used only once. ###### Lemma B.8 (Syntactic Confluence Criterion) Let R be a CRS over sig/cons/${\rm V}$ and ${{\rm X}}{\subseteq}{{\rm V}}$. Let $\hat{s}\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$. Define ${\rm T}:={{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}{[\\{\hat{s}\\}]}.$ Assume either that ${{{}_{{\rm T}}{\upharpoonleft}{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}}}$ is terminating and ${\rhd}={\rhd_{{}_{\rm ST}}}$ or that ${{{}_{\trianglerighteq{[{\rm T}]}}{\upharpoonleft}{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}}}{\ {\subseteq}\ }{\rhd},$ ${\rhd_{{}_{\rm ST}}}{\ {\subseteq}\ }{\rhd},$ and $\rhd$ is a wellfounded ordering on $\mathcal{T}$. Now, if all critical peaks in ${\rm CP}({\rm R})$ are $\rhd$-quasi overlay joinable w.r.t. ${\rm R},{{\rm X}}$, then ${{{}_{\trianglerighteq{[{\rm T}]}}{\upharpoonleft}{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}}}$ is confluent. ## Appendix C $\omega$-Coarse Level Joinability Using the following notions for $\omega$-coarse level joinability one can work out a whole analogue of Theorem 13.9. We did not do so because this analogue does not allow of a corollary theorem analogous to Theorem 13.4 because the information on confluence provided by the joinability notion for testing the conditions of critical peaks is to poor for practically applicable reasoning. To those who are interested in this notion, however, we present here the analogues of Definition 8.1, Definition 8.2, Lemma A.7, and Lemma A.8, for which we also have included the proofs. ###### Definition C.1 ($\omega$-Coarse Level Parallel Closed) A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\omega$-coarse level parallel closed w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&\forall i\prec 2{.}\penalty-1\,\,D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}}}}}\\\ {\wedge}&{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\mbox{ and }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\mbox{ are commuting}\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&t_{0}\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}t_{1}\varphi\\\ \end{array}}}\right)}}.$ ###### Definition C.2 ($\omega$-Coarse Level Parallel Joinable) A critical peak $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ is $\omega$-coarse level parallel joinable w.r.t. ${\rm R},{{\rm X}}$ if $\forall\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&\forall i\prec 2{.}\penalty-1\,\,D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}}}}}\\\ {\wedge}&{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}\mbox{ and }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\mbox{ are commuting}\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&t_{0}\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}}}}}t_{1}\varphi\\\ \end{array}}}\right)}}.$ ###### Lemma C.3 Let R be a CRS over sig/cons/${\rm V}$. Let ${{\rm X}}{\subseteq}{{\rm V}}.$ Assume $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,{{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}$ and the following very weak kind of left-linearity: $\forall{((l,r),C)}{\,\in\,}{\rm R}{.}\penalty-1\,\,\forall p,q{\,\in\,}{{{\mathcal{POS}}}({l})}{.}\penalty-1\,\,\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&l/p{\,=\,}\penalty-1x{\,=\,}\penalty-1l/q\\\ {\wedge}&p{\,\not=\,}q\\\ \end{array}}}\right)}}\ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\vee}&x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Furthermore, assume that ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ is confluent and that ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. Moreover, assume that each critical peak from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-coarse level parallel joinable w.r.t. ${\rm R},{{\rm X}}$, and that each non-overlay from ${\rm CP}({\rm R})$ of the form $(1,1)$ is $\omega$-coarse level parallel closed w.r.t. ${\rm R},{{\rm X}}$. Now: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}}}}$. A fortiori ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ is confluent. ###### Lemma C.4 Let $\mu,\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$. Let ${((l,r),C)}\in{\rm R}$. Assume that ${{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Assume ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\subseteq{\downarrow_{{}_{{\rm R},{{\rm X}}}}}.$ Now, if $C\mu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ and $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}x\nu,$ then $C\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$ and $l\nu{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}r\nu.$ ## Appendix D The Proofs Proof of Lemma 3.2 Assume ${\longrightarrow}_{{}_{\\!0}}$ and ${\longrightarrow}_{{}_{\\!1}}$ to be locally commuting. For the first claim we assume that ${{\longrightarrow}_{{}_{\\!0}}}\cup{{\longrightarrow}_{{}_{\\!1}}}$ is terminating. We show commutation by induction over the wellfounded ordering ${{{{\longrightarrow}_{{}_{\\!0}}}\cup{{\longrightarrow}_{{}_{\\!1}}}}^{\scriptscriptstyle+}}.$ Suppose $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}s{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}t_{1}^{\prime}.$ We have to show $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}^{\prime}.$ In case there is some $i\prec 2$ with $t_{i}^{\prime}{\,=\,}\penalty-1s$ the proof is finished due to $t_{i}^{\prime}{\,=\,}\penalty-1s{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1-i}}}t_{1-i}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!i}}}t_{1-i}^{\prime}.$ Otherwise $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{0}{{\longleftarrow}_{{}_{\\!0}}}s{{\longrightarrow}_{{}_{\\!1}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}t_{1}^{\prime}$ for some $t_{0}$, $t_{1}$ (cf. diagram below). By local commutation there is some $s^{\prime}$ with $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}s^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}.$ Due to $s\;{{{{\longrightarrow}_{{}_{\\!0}}}\cup{{\longrightarrow}_{{}_{\\!1}}}}^{\scriptscriptstyle+}}\;t_{0},$ by induction hypothesis we get some $s^{\prime\prime}$ with $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}s^{\prime\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}s^{\prime}.$ Due to $s\;{{{{\longrightarrow}_{{}_{\\!0}}}\cup{{\longrightarrow}_{{}_{\\!1}}}}^{\scriptscriptstyle+}}\;t_{1},$ by induction hypothesis we get $s^{\prime\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}^{\prime}.$ For the second claim we now assume that ${\longrightarrow}_{{}_{\\!0}}$ or ${\longrightarrow}_{{}_{\\!1}}$ is transitive. W.l.o.g. (due to symmetry in $0$ and $1$) say ${\longrightarrow}_{{}_{\\!0}}$ is transitive. It is sufficient to show $\forall n{\,\in\,}{{\bf N}}{.}\penalty-1\,\,\forall s,t_{0},t_{1}{.}\penalty-1\,\,(t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}s{\stackrel{{\scriptstyle n}}{{{\longrightarrow}}}_{{}_{\\!1}}}t_{1}\ \ {\Rightarrow}\penalty-2\ \ t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}).$ $n{\,=\,}\penalty-10$: $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}s{\,=\,}\penalty-1t_{1}.$ $n\ {\Rightarrow}\penalty-2\ (n{+}1)$: Assume $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}s{\stackrel{{\scriptstyle n}}{{{\longrightarrow}}}_{{}_{\\!1}}}t^{\prime}{{\longrightarrow}_{{}_{\\!1}}}t_{1}$ (cf. diagram below). By induction hypothesis there is some $w$ with $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}w{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t^{\prime}.$ In case of $w{\,=\,}\penalty-1t^{\prime}$ the proof is finished by $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}w{\,=\,}\penalty-1t^{\prime}{{\longrightarrow}_{{}_{\\!1}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}.$ Otherwise, since ${\longrightarrow}_{{}_{\\!0}}$ is transitive, we have $w{{\longleftarrow}_{{}_{\\!0}}}t^{\prime}{{\longrightarrow}_{{}_{\\!1}}}t_{1}.$ By the local commutation of ${\longrightarrow}_{{}_{\\!0}}$ and ${\longrightarrow}_{{}_{\\!1}}$ this implies $w{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{0}.$ Proof of Lemma 3.3 That (3) (or else (2)) implies (1) is trivial. For (1) implying (2) and (3) it is sufficient to show under the assumption of (1) that $\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}\forall n{\,\in\,}{{\bf N}}{.}\penalty-1\,\,\forall s,t_{0},t_{1}{.}\penalty-1\,\,(t_{0}{\stackrel{{\scriptstyle n}}{{{\longleftarrow}}}_{{}_{\\!0}}}s{{\longrightarrow}_{{}_{\\!1}}}t_{1}\ \ {\Rightarrow}\penalty-2\ \ t_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}).$ $n{\,=\,}\penalty-10$: $t_{0}{\,=\,}\penalty-1s{{\longrightarrow}_{{}_{\\!1}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}.$ $n\ {\Rightarrow}\penalty-2\ (n{+}1)$: Suppose $t_{0}{{\longleftarrow}_{{}_{\\!0}}}t^{\prime}{\stackrel{{\scriptstyle n}}{{{\longleftarrow}}}_{{}_{\\!0}}}s{{\longrightarrow}_{{}_{\\!1}}}t_{1}$ (cf. diagram below). By induction hypothesis there is some $w$ with $t^{\prime}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}w{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}.$ In case of $t^{\prime}{\,=\,}\penalty-1w$ the proof is finished due to $t_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}t_{0}{{\longleftarrow}_{{}_{\\!0}}}t^{\prime}{\,=\,}\penalty-1w{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}t_{1}.$ Otherwise we have $t_{0}{{\longleftarrow}_{{}_{\\!0}}}t^{\prime}{{\longrightarrow}_{{}_{\\!1}}}w$ and get by the assumed strong commutation $t_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}w.$ For proving the final implication of the lemma, we may assume that ${\longrightarrow}_{{}_{\\!1}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}}_{{}_{\\!0}}$. A fortiori ${{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}}_{{}_{\\!0}}$ and ${\longrightarrow}_{{}_{\\!1}}$ are locally commuting. By Lemma 3.2 they are commuting. Therefore ${\longrightarrow}_{{}_{\\!0}}$ and ${\longrightarrow}_{{}_{\\!1}}$ are commuting, too. Proof of Lemma 3.4 It is trivial to show $\forall n{\,\in\,}{{\bf N}}{.}\penalty-1\,\,{\stackrel{{\scriptstyle n}}{{{\longleftrightarrow}_{{}_{\\!}}}}}\subseteq{\downarrow}$ by induction on $n$. Proof of Lemma 5.1 Just like the proof of Lemma 6.3 when the depth considerations are omitted. Proof of Lemma 6.3 For ${\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ }{\,\in\,}{\rm CP}({\rm R})$ there are two rules $l_{0}{=}r_{0}{\longleftarrow}C_{0}$ and $l_{1}{=}r_{1}{\longleftarrow}C_{1}$ in R (assuming ${{{\mathcal{V}}}({{l_{0}{=}r_{0}{\longleftarrow}C_{0}}})}\cap{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}=\emptyset$ w.l.o.g.) and $\sigma\in{{{\mathcal{SUB}}}({{{\rm V}}},{{\mathcal{T}}})}$ with $l_{0}\sigma=l_{1}\sigma/p;$ $\ (t_{0},\ D_{0},\ t_{1},\ D_{1},\ \hat{t})=({l_{1}\penalty-1{[\,p\leftarrow r_{0}\,]}},\ C_{0},\ r_{1},\ C_{1},\ l_{1})\sigma$ and $\mathchar 259\relax_{i}=\left\\{\mbox{$\begin{array}[]{ll}0&\mbox{ if }l_{i}\in{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ 1&\mbox{ otherwise}\\\ \end{array}$}\right\\}.$ Let $\varphi{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})};$ $n_{0},n_{1}\prec\omega;$ and assume [$(n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1},\ \hat{t}\varphi){\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}(\beta,\ s)\ $ and] for all $i\prec 2$: ${(\ \alpha{\,=\,}\penalty-10\ {\Rightarrow}\penalty-2\ \mathchar 259\relax_{i}{\,=\,}\penalty-10{\,\prec\,}n_{i}\ )};$ ${(\ \alpha{\,=\,}\penalty-1\omega\ {\Rightarrow}\penalty-2\ \mathchar 259\relax_{i}{\,\preceq\,}n_{i}\ )};$ $D_{i}\varphi$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}$; i.e. $C_{i}\sigma\varphi$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}$. In case of $n_{i}{\,=\,}\penalty-10$ we have $\mathchar 259\relax_{i}{\,=\,}\penalty-10$ and $\alpha{\,=\,}\penalty-1\omega$ and therefore by Corollary 2.6 $l_{i}\sigma\varphi{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{i}}}}r_{i}\sigma\varphi.$ In case of $n_{i}{\,\succ\,}0$ we have $n_{i}{\,=\,}\penalty-1(n_{i}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){+}1$ and therefore $l_{i}\sigma\varphi{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{i}}}}r_{i}\sigma\varphi$ again due to $\alpha{\,=\,}\penalty-10\ {\Rightarrow}\penalty-2\ \mathchar 259\relax_{i}{\,=\,}\penalty-10.$ Then $t_{0}\varphi={l_{1}\sigma\varphi\penalty-1{[\,p\leftarrow r_{0}\sigma\varphi\,]}}{{\longleftarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}l_{1}\sigma\varphi{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}r_{1}\sigma\varphi=t_{1}\varphi.$ By $\alpha$-shallow confluence [up to $\beta$ [and $s$ in $\lhd$]] we have $t_{0}\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\alpha+n_{0}}}}t_{1}\varphi$ . Proof of Lemma 6.4 The proof is analogous to the proof of Lemma 6.3. Proof of Lemma 9.1 In case of $(\hat{t}/p^{\prime})\sigma\varphi{\,=\,}\penalty-1(\hat{t}/\emptyset)\sigma\varphi$ we get $p^{\prime}{\,=\,}\penalty-1\emptyset.$ Thus $\mathchar 257\relax\subseteq{{{\mathcal{POS}}}({\hat{t}})}{\setminus}\\{\emptyset\\}$ together with $\forall p^{\prime}{\,\in\,}\mathchar 257\relax{.}\penalty-1\,\,(\hat{t}/p^{\prime})\sigma\varphi{\,=\,}\penalty-1(\hat{t}/\emptyset)\sigma\varphi$ implies $\mathchar 257\relax{\,=\,}\penalty-1\emptyset.$ If there is some $\bar{u}_{1}$ with $t_{0}\sigma\mu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\bar{u}_{1}\penalty-1{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}t_{1}\sigma\mu;$ define $\bar{n}:=1$; $\bar{u}_{0}:=t_{1}\sigma\mu$; $\bar{p}_{0}:=\emptyset$; and note that $t_{1}\sigma\varphi{\longleftarrow}\hat{t}\sigma\varphi$ when $D_{1}\sigma\varphi$ is fulfilled. Proof of Lemma 13.2 If R has conservative constructors we get ${{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}}$ (since $l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$). If ${{{\mathcal{V}}}({C})}\subseteq{{{\rm V}}\\!_{{\mathcal{C}}}},$ then ${{{\mathcal{TERMS}}}({C\mu})}{\subseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ (since $l{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$). Thus we can always assume ${{{\mathcal{TERMS}}}({C\mu})}{\subseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Then we have $\forall x{\,\in\,}{{{\mathcal{V}}}({C})}{.}\penalty-1\,\,x\mu{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ and thus $\forall x{\,\in\,}{{{\mathcal{V}}}({C})}{.}\penalty-1\,\,x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}x\nu$ by Lemma 2.10. Moreover $C\mu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ by Lemma 2.10. By confluence of ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$ and Lemma 2.10 $C\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. By Corollary 2.6 we finally get $l\nu{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}r\nu.$ Proof of Theorem 13.3 and Theorem 13.4 Due to Corollary 3.8, it suffices to show that the conditions of Theorem 13.6(I) or else (in case of Theorem 13.4) Theorem 13.9(I) are satisfied. The only non-trivial part are the joinability requirements for the critical pairs. We just have to show that the conjunctive condition lists of the joinability notions are never satisfied. Assume $\ ((t_{0},D_{0},\mathchar 259\relax_{0}),\ (t_{1},D_{1},\mathchar 259\relax_{1}),\ \hat{t},\ p)\ $ to be a critical peak. We first treat the critical peaks of the form $(0,1)$ or $(1,0)$, and, in case of Theorem 13.3, also of the form $(1,1)$. For these we have to show $\omega$-shallow parallel joinability or else $\omega$-shallow parallel closedness. Thus, assume $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ and $n_{0},n_{1}\prec\omega$ such that $\forall i{\,\prec\,}2{.}\penalty-1\,\,{(\ D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\ )}$ and $\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{.}\penalty-1\,\,{(\ {{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\delta\ )}.$ By the assumed complementarity there must be complementary equation literals in $D_{0}$ and $D_{1}$. Due to our symmetry in $0$ and $1$ so far, we may w.l.o.g. assume that $(u{=}v)$ occurs in $D_{0}$ and $(u{\not=}v)$ occurs in $D_{1}$ or else that $(p{=}{{\mathsf{true}}})$ occurs in $D_{0}$ and $(p{=}{{\mathsf{false}}})$ occurs in $D_{1}$. We treat the first case first. Then there are $\hat{u},\hat{v}\in{{\mathcal{GT}}({{\rm cons}})}$ with $\hat{u}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}u\varphi{\downarrow_{{}_{\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}v\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\hat{v}$ and $\hat{u}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{\omega}}}\hat{v}.$ In case of $n_{0},n_{1}{\,\preceq\,}1$ this contradicts the required confluence of ${\longrightarrow}_{{}_{\\!\omega}}$, cf. Lemma 3.4. Otherwise, in case of $n_{0}{\,\succeq\,}1$ we have $(n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){+_{\\!\\!{}_{\omega}}}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)\prec n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}$ and thus by our above assumption ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $(n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){+_{\\!\\!{}_{\omega}}}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)$. Due to the assumption of the theorem at least one of $u\varphi$, $v\varphi$, w.l.o.g. say $v\varphi$, must be either irreducible or have a $v^{\prime}\in{{\mathcal{GT}}({{\rm cons}})}$ with $v\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}v^{\prime}.$ Now Lemma 13.7(4) implies $\hat{u}{\downarrow_{{}_{\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\hat{v},$ and then Lemma 2.11 implies the contradicting $\hat{u}{\downarrow_{{}_{\omega}}}\hat{v}.$ Now we treat the case that that $(p{=}{{\mathsf{true}}})$ occurs in $D_{0}$ and $(p{=}{{\mathsf{false}}})$ occurs in $D_{1}$. Due to the definition of complementarity, ${\mathsf{true}}$ and ${\mathsf{false}}$ are distinct irreducible ground terms. Thus we have $p\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}{{\mathsf{true}}}$ and $p\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}{{\mathsf{false}}}.$ In case of $n_{0},n_{1}{\,\preceq\,}1$ this contradicts the required confluence of ${\longrightarrow}_{{}_{\\!\omega}}$. Otherwise, in case of $n_{0}{\,\succeq\,}1$ we have $(n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){+_{\\!\\!{}_{\omega}}}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)\prec n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}$ and thus by our above assumption ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $(n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){+_{\\!\\!{}_{\omega}}}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)$. This again implies the contradicting ${{\mathsf{true}}}\downarrow{{\mathsf{false}}}.$ Finally we treat the critical peaks of the form $(1,1)$ in case of Theorem 13.4. For these we have to show $\omega$-level parallel joinability or else $\omega$-level parallel closedness. Thus, assume $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ and $n\prec\omega$ with $0{\,\prec\,}n$ such that $\forall i{\,\prec\,}2{.}\penalty-1\,\,{(\ D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\ )}$ and $\forall\delta{\,\prec\,}n{.}\penalty-1\,\,{(\ {{\rm R},{{\rm X}}}\mbox{ is $\omega$-level confluent up to }\delta\ )}.$ Due to $0{\,\prec\,}n$ we have $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1{\,\prec\,}n$ and thus ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$. By the assumed weak complementarity there must be complementary equation literals in $D_{0}D_{1}$. First we treat the case that $(u{=}v)$ and $(u{\not=}v)$ occur in $D_{0}D_{1}$. Then there are $\hat{u},\hat{v}\in{{\mathcal{GT}}({{\rm cons}})}$ and $v^{\prime}\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ with $\hat{u}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}u\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}v\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\hat{v}$ and $\hat{u}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{\omega}}}\hat{v}.$ Now, by $\omega$-level confluence up to $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$, there is some $u^{\prime}$ with $\hat{u}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}v^{\prime}$ and then by $\omega$-level confluence up to $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$ again $u^{\prime}{\downarrow_{{}_{\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\hat{v},$ and then Lemma 2.11 implies the contradicting $\hat{u}{\downarrow_{{}_{\omega}}}\hat{v}.$ Now we treat the case that that $(p{=}{{\mathsf{true}}})$ and $(p{=}{{\mathsf{false}}})$ occur in $D_{0}D_{1}$. Due to the definition of weak complementarity, ${\mathsf{true}}$ and ${\mathsf{false}}$ are distinct irreducible ground terms. Thus we have ${{\mathsf{true}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}p\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}{{\mathsf{false}}}.$ By $\omega$-level confluence up to $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$ this again implies the contradicting ${{\mathsf{true}}}\downarrow{{\mathsf{false}}}.$ Q.e.d. (Theorem 13.3 and Theorem 13.4) Proof of Theorem 13.6 (I) follows from the lemmas A.1 and A.2. (II) follows from the lemmas A.4 and A.6. (III) follows from the lemmas A.1, A.4, and A.5, since for critical peaks of the form $(0,1)$ $\omega$-shallow noisy strong joinability up to $\omega$ implies $\omega$-shallow noisy parallel joinability up to $\omega$ (cf. Corollary 7.7) and for non-overlays of the form $(1,0)$ $\omega$-shallow parallel closedness up to $\omega$ implies $\omega$-shallow noisy anti- closedness up to $\omega$ (cf. Corollary 7.8). (IV) follows from the lemmas A.3, A.4, and A.5, since for critical peaks of the form $(0,1)$ $\omega$-shallow noisy strong joinability up to $\omega$ implies $\omega$-shallow noisy weak parallel joinability up to $\omega$ (cf. Corollary 7.7) and for critical peaks of the form $(1,0)$ $\omega$-shallow closedness up to $\omega$ implies $\omega$-shallow anti-closedness up to $\omega$ (cf. Corollary 7.8). Proof of Theorem 13.9 (I) follows from the lemmas A.1 and A.8. (II) follows from the lemmas A.4 and A.10 (III) follows from the lemmas A.1, A.4, and A.9, since for critical peaks of the form $(0,1)$ $\omega$-shallow strong joinability up to $\omega$ implies $\omega$-shallow parallel joinability up to $\omega$ (cf. Corollary 7.7) and for non-overlays of the form $(1,0)$ $\omega$-shallow parallel closedness up to $\omega$ implies $\omega$-shallow anti-closedness up to $\omega$ (cf. Corollary 7.8). (IV) follows from the lemmas A.3, A.4, and A.9, since for critical peaks of the form $(0,1)$ $\omega$-shallow strong joinability up to $\omega$ implies $\omega$-shallow weak parallel joinability up to $\omega$ (cf. Corollary 7.7) and for critical peaks of the form $(1,0)$ $\omega$-shallow closedness up to $\omega$ implies $\omega$-shallow anti-closedness up to $\omega$ (cf. Corollary 7.8). Proof of Theorem 14.2 1 $\Rightarrow$ 2: By Lemma B.3. 2 $\Rightarrow$ 1: By Lemma 5.1. Proof of Theorem 14.4 1 $\Rightarrow$ 2: Directly by the lemmas B.4 and B.3. 2 $\Rightarrow$ 1: By Lemma 5.1. Proof of Theorem 14.5 1 $\Rightarrow$ 2: Directly by the lemmas B.4 and B.5. 2 $\Rightarrow$ 1: By Corollary 3.9 and Lemma 6.3. Proof of Theorem 14.6 1 $\Rightarrow$ 2: Directly by Lemma B.6. 2 $\Rightarrow$ 1: By Corollary 3.9 and Lemma 6.4. Proof of Theorem 14.7 Directly by Lemma B.8. Proof of Theorem 15.1(I) Claim 1: If ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}$, then ${\longrightarrow}_{{}_{\\!n_{1}}}$ and ${\longrightarrow}_{{}_{\\!n_{0}}}$ are commuting. Proof of Claim 1: ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}$ are commuting by Lemma 3.3. Since by Corollary 2.14 and Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!n_{1}}}}\subseteq{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}},$ now ${\longrightarrow}_{{}_{\\!n_{1}}}$ and ${\longrightarrow}_{{}_{\\!n_{0}}}$ are commuting, too. Q.e.d. (Claim 1) For $n_{0}\preceq n_{1}\prec\omega$ we are going to show by induction on $n_{0}{+}n_{1}$ the following property: $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle n_{0}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{1}.$ Claim 2: Let $\delta\prec\omega$. If $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}\\\ {\wedge}&n_{0}{+}n_{1}{\,\preceq\,}\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&\forall w_{0},w_{1},u{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle n_{0}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}w_{1}\\\ {\Rightarrow}&w_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{1}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}},$ then $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}\\\ {\wedge}&n_{0}{+}n_{1}{\,\preceq\,}\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}\mbox{ strongly commutes over }{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}}\\\ \end{array}}}\right)}},$ and ${\rm R},{{\rm X}}$ is $0$-shallow confluent up to $\delta$. Proof of Claim 2: By induction on $\delta$ in $\,\prec\,$. First we show the strong commutation. Assume $n_{0}\preceq n_{1}\prec\omega$ with $n_{0}{+}n_{1}{\,\preceq\,}\delta$. By Lemma 3.3 it suffices to show that ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!n_{0}}}$. Assume $w_{0}{{\longleftarrow}_{{}_{\\!n_{0}}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}w_{2}$ (cf. diagram below). By the above property there is some $w_{1}^{\prime}$ with $w_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{1}.$ Next we show that we can close the peak $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}w_{2}$ according to $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}w_{2}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{2}$ for some $w_{2}^{\prime}$. In case of $n_{1}{\,=\,}\penalty-10$ this is possible due $w_{1}{\,=\,}\penalty-1w_{2}.$ Otherwise we have $n_{0}{+}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){\,\prec\,}n_{0}{+}n_{1}{\,\preceq\,}\delta$ and due to our induction hypothesis (saying that ${\rm R},{{\rm X}}$ is $0$-shallow confluent up to all $\delta^{\prime}\prec\delta$) this is possible again. Finally we show $0$-shallow confluence up to $\delta$. Assume $n_{0}{+}n_{1}{\,\preceq\,}\delta$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}w_{1}.$ Due to symmetry in $n_{0}$ and $n_{1}$ we may assume $n_{0}{\,\preceq\,}n_{1}.$ Above we have shown that ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}$. By Claim 1 we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{1}$ as desired. Q.e.d. (Claim 2) Note that for $n_{0}{\,=\,}\penalty-10$ our property follows from ${{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle 0}}\subseteq{\rm id}.$ The benefit of Claim 2 is twofold: First, it says that our theorem is valid if the above property holds for all $n_{0}\preceq n_{1}\prec\omega$. Second, it strengthens the property when used as induction hypothesis. Thus (writing $n_{i}{+}1$ instead of $n_{i}$ since we may assume $0{\,\prec\,}n_{0}{\,\preceq\,}n_{1}$) it now suffices to show for $n_{0}\preceq n_{1}\prec\omega$ that $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle n_{0}+1,\mathchar 261\relax_{0}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1,\mathchar 261\relax_{1}}}w_{1}$ together with our induction hypotheses that $\rule{0.0pt}{8.43889pt}\forall\delta{\,\prec\,}(n_{0}{+}1){+}(n_{1}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $0$-shallow confluent up to }\delta$ and (due to $n_{0}{\,\preceq\,}n_{1}{+}1$ and $n_{0}{+}(n_{1}{+}1){\,\prec\,}(n_{0}{+}1){+}(n_{1}{+}1)$) ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}$ implies $w_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}+1}}}w_{1}.\ \ $ Note that for the availability of our second induction hypothesis it is important that we have imposed the restriction “$n_{0}{\,\preceq\,}n_{1}$” in opposition to the restriction “$n_{0}{\,\succeq\,}n_{1}$”. In the latter case the availability of our second induction hypothesis would require $n_{0}{+}1{\,\succeq\,}n_{1}{+}1\ {\Rightarrow}\penalty-2\ n_{0}{\,\succeq\,}n_{1}{+}1$ which is not true for $n_{0}{\,=\,}\penalty-1n_{1}.$ The additional hypothesis ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}+1}}$ of the latter restriction is useless for our proof. W.l.o.g. let the positions of $\mathchar 261\relax_{i}$ be maximal in the sense that for any $p\in\mathchar 261\relax_{i}$ and $\mathchar 260\relax\subseteq{{{\mathcal{POS}}}({u})}{\cap}(p{{\bf N}}^{+})$ we do not have $u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{i}+1,(\mathchar 261\relax_{i}\setminus\\{p\\})\cup\mathchar 260\relax}}w_{i}$ anymore. Then for each $i\prec 2$ and $p\in\mathchar 261\relax_{i}$ there are ${((l_{i,p},r_{i,p}),C_{i,p})}\in{\rm R}$ and $\mu_{i,p}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $l_{i,p}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ $u/p{\,=\,}\penalty-1l_{i,p}\mu_{i,p},$ $r_{i,p}\mu_{i,p}{\,=\,}\penalty-1w_{i}/p,$ $C_{i,p}\mu_{i,p}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!n_{i}}}$. Finally, for each $i\prec 2$: $w_{i}{\,=\,}\penalty-1{u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}\,]}}}.$ Define the set of inner overlapping positions by $\displaystyle\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1}):=\bigcup_{i\prec 2}{{\\{\ }p{\,\in\,}\mathchar 261\relax_{1-i}}~{}{\|}\penalty-9\,\ {\exists q{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}},$ and the length of a term by $\lambda({{f}{(}{t_{0}}{,\,}\ldots{,\,}{t_{m-1}}{)}}):=1+\sum_{j\prec m}\lambda(t_{j}).$ Now we start a second level of induction on $\displaystyle\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})$ in $\,\prec\,$. Define the set of top positions by $\displaystyle\mathchar 258\relax:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{0}{\cup}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {\neg\exists q{\,\in\,}\mathchar 261\relax_{0}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{+}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}}.$ Since the prefix ordering is wellfounded we have $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,\exists q{\,\in\,}\mathchar 258\relax{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}.$ Then $\forall i{\,\prec\,}2{.}\penalty-1\,\,w_{i}{\,=\,}\penalty-1{w_{i}\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{{u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}\,]}}}\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}.$ Thus, it now suffices to show for all $q\in\mathchar 258\relax$ $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}+1}}}w_{1}/q$ because then we have $w_{0}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}+1}}}{u\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1w_{1}.$ Therefore we are left with the following two cases for $q\in\mathchar 258\relax$: $q{\,\not\in\,}\mathchar 261\relax_{1}$: Then $q{\,\in\,}\mathchar 261\relax_{0}.$ Define $\mathchar 261\relax_{1}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{1}{\ \\}}}$. We have two cases: “The variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}{.}\penalty-1\,\,l_{0,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{0,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}{\ \\}}}.$ Claim 7: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\mu_{0,q}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}x\nu\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\nu{\,=\,}\penalty-1{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\\\ \end{array}}}\right)}}.$ Proof of Claim 7: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{0,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{0,q}&{\,=\penalty-1}&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}\\\ &&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{0,q}$ is not linear in $x$, which contradicts the left-linearity assumption of the theorem. Q.e.d. (Claim 7) Claim 8: $l_{0,q}\nu{\,=\,}\penalty-1w_{1}/q.$ Proof of Claim 8: By Claim 7 we get $w_{1}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{0,q}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0,q}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}r_{0,q}\nu.$ Proof of Claim 9: Since $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q},$ this follows directly from Claim 7. Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $l_{0,q}\nu{{\longrightarrow}_{{}_{\\!n_{0}+1}}}r_{0,q}\nu,$ which again follows from Lemma 13.8 since $((l_{0,q},r_{0,q}),C_{0,q})$ is $0$-quasi- normal w.r.t. ${\rm R},{{\rm X}}$ (due to $l_{0,q}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ and the assumption of our theorem), since ${\rm R},{{\rm X}}$ is $0$-shallow confluent up to $(n_{1}{+}1){+}n_{0}$ (by our induction hypothesis), and since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}+1}}}x\nu$ by Claim 7 and Corollary 2.14. Q.e.d. (“The variable overlap (if any) case”) “The critical peak case”: There is some $p\in\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}$ with $l_{0,q}/p{\,\not\in\,}{{\rm V}}$: Claim 10: $p{\,\not=\,}\emptyset.$ Proof of Claim 10: If $p{\,=\,}\penalty-1\emptyset,$ then $\emptyset{\,\in\,}\mathchar 261\relax_{1}^{\prime},$ then $q{\,\in\,}\mathchar 261\relax_{1},$ which contradicts our global case assumption. Q.e.d. (Claim 10) Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}\\\ x\xi^{-1}\mu_{1,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1,qp}\xi\varrho{\,=\,}\penalty-1l_{1,qp}\xi\xi^{-1}\mu_{1,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p{\,=\,}\penalty-1l_{0,q}\varrho/p{\,=\,}\penalty-1(l_{0,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1,qp}\xi},{l_{0,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{0,q}\mu_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}$. We get $\begin{array}[]{l@{}l}u^{\prime}{\,=\penalty-1}&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}\\\ &{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}\,]}}}{\,=\,}\penalty-1w_{1}/q.\end{array}$ If ${l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1r_{0,q}\sigma\varphi{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ Otherwise we have $(\,({l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}},C_{1,qp}\xi,0),\penalty-1\,(r_{0,q},C_{0,q},0),\penalty-1\,l_{0,q},\,\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R});$ $p{\,\not=\,}\emptyset$ (due to Claim 10); $C_{1,qp}\xi\sigma\varphi=C_{1,qp}\mu_{1,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!n_{1}}}$; $C_{0,q}\sigma\varphi=C_{0,q}\mu_{0,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!n_{0}}}$. Since $\forall\delta{\,\prec\,}(n_{1}{+}1){+}(n_{0}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $0$-shallow confluent up to }\delta$ (by our induction hypothesis) due to our assumed $0$-shallow parallel closedness (matching the definition’s $n_{0}$ to our $n_{1}{+}1$ and its $n_{1}$ to our $n_{0}{+}1$) we have $u^{\prime}{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi\penalty-1{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{0}+1}}\penalty-1v_{1}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}}r_{0,q}\sigma\varphi{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1w_{0}/q$ for some $v_{1}$. We then have $v_{1}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle n_{0}+1,\mathchar 261\relax^{\prime\prime}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q$ for some $\mathchar 261\relax^{\prime\prime}$. By $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 266\relax(\mathchar 261\relax^{\prime\prime},\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\})}\lambda(u^{\prime}/p^{\prime\prime})\ \ \preceq\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}\lambda(u^{\prime}/p^{\prime\prime})\ \ =\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}\lambda(u/qp^{\prime\prime})\ \ \prec\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}}\lambda(u/qp^{\prime\prime})\ \ =$ $\displaystyle\sum_{p^{\prime}\in q\mathchar 261\relax_{1}^{\prime}}\lambda(u/p^{\prime})\ \ =\sum_{p^{\prime}\in\mathchar 266\relax(\\{q\\},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})\ \ \preceq\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime}),$ due to our second induction level we get some $v_{1}^{\prime}$ with $v_{1}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}+1}}}w_{1}/q.$ Finally by our induction hypothesis that ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}$ the peak at $v_{1}$ can be closed according to $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}v_{1}^{\prime}.$ Q.e.d. (“The critical peak case”) Q.e.d. (“$q{\,\not\in\,}\mathchar 261\relax_{1}$”) $q{\,\in\,}\mathchar 261\relax_{1}$: Define $\mathchar 261\relax_{0}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{0}{\ \\}}}$. We have two cases: “The second variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{0}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{1,q}})}{.}\penalty-1\,\,l_{1,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{1,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}{\ \\}}}.$ Claim 11: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle n_{0}+1}}x\mu_{1,q}\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu\\\ \end{array}}}\right)}}.$ Proof of Claim 11: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{1,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{1,q}&{\,=\penalty-1}&{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{0}+1}}\\\ &&{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{1,q}$ is not linear in $x$, which contradicts the left-linearity assumption of the theorem. Q.e.d. (Claim 11) Claim 12: $w_{0}/q{\,=\,}\penalty-1l_{1,q}\nu.$ Proof of Claim 12: By Claim 11 we get $w_{0}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{1,q}\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1,q}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1,q}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle n_{0}+1}}w_{1}/q.$ Proof of Claim 13: Since $r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q,$ this follows directly from Claim 11. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1,q}\nu{{\longrightarrow}_{{}_{\\!n_{1}+1}}}r_{1,q}\nu,$ which again follows from Claim 11, Corollary 2.14, Lemma 13.8 (matching its $n_{0}$ to our $n_{0}{+}1$ and its $n_{1}$ to our $n_{1}$), and our induction hypothesis that ${\rm R},{{\rm X}}$ is $0$-shallow confluent up to $(n_{0}{+}1){+}n_{1}.$ Q.e.d. (“The second variable overlap (if any) case”) “The second critical peak case”: There is some $p\in\mathchar 261\relax_{0}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{1,q}})}$ with $l_{1,q}/p{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0,qp},r_{0,qp}),C_{0,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0,qp},r_{0,qp}),C_{0,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}\\\ x\xi^{-1}\mu_{0,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0,qp}\xi\varrho{\,=\,}\penalty-1l_{0,qp}\xi\xi^{-1}\mu_{0,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p{\,=\,}\penalty-1l_{1,q}\varrho/p{\,=\,}\penalty-1(l_{1,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0,qp}\xi},{l_{1,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{1,q}\mu_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\mu_{0,qp}\,]}}$. We get $\begin{array}[]{l@{}l@{}l}w_{0}/q&{\,=\penalty-1}&{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{0,qp^{\prime}}\mu_{0,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{0}^{\prime}\,]}}}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle n_{0}+1,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}\\\ &&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{0,qp^{\prime}}\mu_{0,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{0}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{0,qp}\mu_{0,qp}\,]}}{\,=\,}\penalty-1u^{\prime}.\end{array}$ If ${l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1,q}\sigma,$ then the proof is finished due to $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle n_{0}+1,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}u^{\prime}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Otherwise we have $(\,({l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}},C_{0,qp}\xi,0),\penalty-1\,(r_{1,q},C_{1,q},0),\penalty-1\,l_{1,q},\,\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R});$ $C_{0,qp}\xi\sigma\varphi=C_{0,qp}\mu_{0,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!n_{0}}}$; $C_{1,q}\sigma\varphi=C_{1,q}\mu_{1,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!n_{1}}}$. Since $\forall\delta{\,\prec\,}(n_{0}{+}1){+}(n_{1}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $0$-shallow confluent up to }\delta$ (by our induction hypothesis) due to our assumed $0$-shallow noisy parallel joinability (matching the definition’s $n_{0}$ to our $n_{0}{+}1$ and its $n_{1}$ to our $n_{1}{+}1$ ) we have $u^{\prime}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}\penalty-1v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}\penalty-1v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}+1}}}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q$ for some $v_{1}$, $v_{2}$. We then have $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle n_{0}+1,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1,\mathchar 261\relax^{\prime\prime}}}v_{1}$ for some $\mathchar 261\relax^{\prime\prime}$. Since $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\},\mathchar 261\relax^{\prime\prime})}\lambda(u^{\prime}/p^{\prime\prime})\ \ \preceq\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}\lambda(u^{\prime}/p^{\prime\prime})\ \ =\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}\lambda(u/qp^{\prime\prime})\ \ \prec\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}}\lambda(u/qp^{\prime\prime})\ \ =\sum_{p^{\prime}\in q\mathchar 261\relax_{0}^{\prime}}\lambda(u/p^{\prime})\ \ =\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\\{q\\})}\lambda(u/p^{\prime})\ \ \preceq\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})$ due to our second induction level we get some $v_{1}^{\prime}$ with $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}+1}}}v_{1}.$ Finally the peak at $v_{1}$ can be closed according to $v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}+1}}}v_{2}$ by our induction hypothesis saying that ${\rm R},{{\rm X}}$ is $0$-shallow confluent up to $(n_{0}{+}1){+}n_{1}$. Q.e.d. (“The second critical peak case”) Q.e.d. (Theorem 15.1(I)) Proof of Theorem 15.1(II) The parts in the following proof which are only for Theorem 15.1(IIa) are in optional brackets. Claim 1: If ${{\longrightarrow}_{{}_{\\!n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}$, then ${\longrightarrow}_{{}_{\\!n_{1}}}$ and ${\longrightarrow}_{{}_{\\!n_{0}}}$ are commuting. Proof of Claim 1: ${{\longrightarrow}_{{}_{\\!n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}$ are commuting by Lemma 3.3. Since by Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!n_{1}}}}\subseteq{{\longrightarrow}_{{}_{\\!n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}},$ now ${\longrightarrow}_{{}_{\\!n_{1}}}$ and ${\longrightarrow}_{{}_{\\!n_{0}}}$ are commuting, too. Q.e.d. (Claim 1) For $n_{0}\preceq n_{1}\prec\omega$ we are going to show by induction on $n_{0}{+}n_{1}$ the following property: $w_{0}{{\longleftarrow}_{{}_{\\!n_{0}}}}u{{\longrightarrow}_{{}_{\\!n_{1}}}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{1}.$ Claim 2: Let $\delta\prec\omega$. If $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}\\\ {\wedge}&n_{0}{+}n_{1}{\,\preceq\,}\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&\forall w_{0},w_{1},u{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&w_{0}{{\longleftarrow}_{{}_{\\!n_{0}}}}u{{\longrightarrow}_{{}_{\\!n_{1}}}}w_{1}\\\ {\Rightarrow}&w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{1}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}},$ then $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}\\\ {\wedge}&n_{0}{+}n_{1}{\,\preceq\,}\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{{\longrightarrow}_{{}_{\\!n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}\mbox{ strongly commutes over }{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}}\\\ \end{array}}}\right)}},$ and ${\rm R},{{\rm X}}$ is $0$-shallow confluent up to $\delta$. Proof of Claim 2: By induction on $\delta$ in $\,\prec\,$. First we show the strong commutation. Assume $n_{0}\preceq n_{1}\prec\omega$ with $n_{0}{+}n_{1}{\,\preceq\,}\delta$. By Lemma 3.3 it suffices to show that ${{\longrightarrow}_{{}_{\\!n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!n_{0}}}$. Assume $w_{0}{{\longleftarrow}_{{}_{\\!n_{0}}}}u{{\longrightarrow}_{{}_{\\!n_{1}}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}w_{2}$ (cf. diagram below). By the above property there is some $w_{1}^{\prime}$ with $w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{1}.$ Next we show that we can close the peak $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}w_{2}$ according to $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}w_{2}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{2}$ for some $w_{2}^{\prime}$. In case of $n_{1}{\,=\,}\penalty-10$ this is possible due to $w_{1}{\,=\,}\penalty-1w_{2}.$ Otherwise we have $n_{0}{+}(0{[{+}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}){\,\prec\,}n_{0}{+}n_{1}{\,\preceq\,}\delta$ and due to our induction hypothesis (saying that ${\rm R},{{\rm X}}$ is $0$-shallow confluent up to all $\delta^{\prime}\prec\delta$) this is possible again. Finally we show $0$-shallow confluence up to $\delta$. Assume $n_{0}{+}n_{1}{\,\preceq\,}\delta$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}w_{1}.$ Due to symmetry in $n_{0}$ and $n_{1}$ we may assume $n_{0}{\,\preceq\,}n_{1}.$ Above we have shown that ${{\longrightarrow}_{{}_{\\!n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}}}$. By Claim 1 we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}}}}w_{1}$ as desired. Q.e.d. (Claim 2) Note that for $n_{0}{\,=\,}\penalty-10$ our property follows from ${{\longleftarrow}_{{}_{\\!n_{0}}}}\subseteq{\rm id}.$ The benefit of Claim 2 is twofold: First, it says that our theorem is valid if the above property holds for all $n_{0}\preceq n_{1}\prec\omega$. For part (IIb) this is because then by Lemma 3.3 ${\longrightarrow}_{{}_{\\!n_{1}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!n_{0}}}$ for all $n_{0}\preceq n_{1}\prec\omega$, i.e. ${\longrightarrow}_{{}_{\\!\omega}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!n_{0}}}$, i.e. ${\longrightarrow}_{{}_{\\!\omega}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!\omega}}$, i.e. ${\longrightarrow}_{{}_{\\!\omega}}$ is strongly confluent. Second, it strengthens the property when used as induction hypothesis. Thus (writing $n_{i}{+}1$ instead of $n_{i}$ since we may assume $0{\,\prec\,}n_{0}{\,\preceq\,}n_{1}$) it now suffices to show for $n_{0}\preceq n_{1}\prec\omega$ that $w_{0}{{\longleftarrow}_{{}_{\\!n_{0}+1,\bar{p}_{0}}}}u{{\longrightarrow}_{{}_{\\!n_{1}+1,\bar{p}_{1}}}}w_{1}$ together with our induction hypotheses that $\rule{0.0pt}{8.43889pt}\forall\delta{\,\prec\,}(n_{0}{+}1){+}(n_{1}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $0$-shallow confluent up to }\delta$ implies $w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}+1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+n_{1}]}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}+1}}}w_{1}.$ Now for each $i\prec 2$ there are ${((l_{i},r_{i}),C_{i})}\in{\rm R}$ and $\mu_{i}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/\bar{p}_{i}{\,=\,}\penalty-1l_{i}\mu_{i},$ $w_{i}{\,=\,}\penalty-1{u\penalty-1{[\,\bar{p}_{i}\leftarrow r_{i}\mu_{i}\,]}},$ $C_{i}\mu_{i}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!n_{i}}}$, and $l_{i}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ In case of ${{\bar{p}_{0}}\,{\parallel}\,{\bar{p}_{1}}}$ we have $w_{i}/\bar{p}_{1-i}{\,=\,}\penalty-1{u\penalty-1{[\,\bar{p}_{i}\leftarrow r_{i}\mu_{i}\,]}}/\bar{p}_{1-i}{\,=\,}\penalty-1u/\bar{p}_{1-i}{\,=\,}\penalty-1l_{1-i}\mu_{1-i}$ and therefore $w_{i}{{\longrightarrow}_{{}_{\\!n_{i}+1}}}{u\penalty-1{{[\,\bar{p}_{k}\leftarrow r_{k}\mu_{k}\ \|\ k{\,\prec\,}2\,]}}},$ i.e. our proof is finished. Thus, according to whether $\bar{p}_{0}$ is a prefix of $\bar{p}_{1}$ or vice versa, we have the following two cases left: There is some $\bar{p}_{1}^{\prime}$ with $\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1\bar{p}_{1}$ and $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ : We have two cases: “The variable overlap case”: There are $x\in{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{0}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{1}^{\prime}$: Claim 6: We have $x\mu_{0}/p^{\prime\prime}{\,=\,}\penalty-1l_{1}\mu_{1}.$ Proof of Claim 6: We have $x\mu_{0}/p^{\prime\prime}{\,=\,}\penalty-1l_{0}\mu_{0}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{0}p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{1}{\,=\,}\penalty-1l_{1}\mu_{1}.$ Q.e.d. (Claim 6) Claim 7: We can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by $x\nu{\,=\,}\penalty-1{x\mu_{0}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{0}.$ Then we have $x\mu_{0}{{\longrightarrow}_{{}_{\\!n_{1}+1}}}x\nu.$ Proof of Claim 7: This follows directly from Claim 6. Q.e.d. (Claim 7) Claim 8: $l_{0}\nu{\,=\,}\penalty-1w_{1}/\bar{p}_{0}.$ Proof of Claim 8: By the left-linearity assumption of our theorem we may assume ${{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}=\\{p^{\prime}\\}.$ Thus, by Claim 7 we get $w_{1}/\bar{p}_{0}{\,=\,}\penalty-1{u/\bar{p}_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow x\mu_{0}\,]}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow{x\mu_{0}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}\,]}}{\,=\,}\penalty-1\\\ {l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/\bar{p}_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}+1}}}r_{0}\nu.$ Proof of Claim 9: By the right-linearity assumption of our theorem we may assume ${\,\|{{{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}}\|\,}{\,\preceq\,}1.$ Thus by Claim 7 we get: $w_{0}/\bar{p}_{0}{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}+1}}}\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{\,=\,}\penalty-1\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{\,=\,}\penalty-1r_{0}\nu.$ Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $l_{0}\nu{{\longrightarrow}_{{}_{\\!n_{0}+1}}}r_{0}\nu,$ which again follows from Lemma 13.8 (matching its $n_{0}$ to our $n_{1}{+}1$ and its $n_{1}$ to our $n_{0}$) since ${\rm R},{{\rm X}}$ is $0$-quasi-normal and $0$-shallow confluent up to $(n_{1}{+}1){+}n_{0}$ by our induction hypothesis, and since $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\mu_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}+1}}}y\nu$ by Claim 7. Q.e.d. (“The variable overlap case”) “The critical peak case”: $\bar{p}_{1}^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{0}})}\ {\wedge}\penalty-2\ l_{0}/\bar{p}_{1}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}]\cap{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}]\cup{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}\\\ x\xi^{-1}\mu_{1}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1}\xi\varrho{\,=\,}\penalty-1l_{1}\xi\xi^{-1}\mu_{1}{\,=\,}\penalty-1u/\bar{p}_{1}{\,=\,}\penalty-1u/\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1l_{0}\mu_{0}/\bar{p}_{1}^{\prime}{\,=\,}\penalty-1l_{0}\varrho/\bar{p}_{1}^{\prime}{\,=\,}\penalty-1(l_{0}/\bar{p}_{1}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1}\xi},{l_{0}/\bar{p}_{1}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0}\sigma,$ then the proof is finished due to $w_{0}/\bar{p}_{0}{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1r_{0}\sigma\varphi{\,=\,}\penalty-1{l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1{l_{0}\mu_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1w_{1}/\bar{p}_{0}.$ Otherwise we have $(\,({l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}},C_{1}\xi,0),\penalty-1\,(r_{0},C_{0},0),\penalty-1\,l_{0},\penalty-1\,\sigma,\penalty-1\,\bar{p}_{1}^{\prime}\,)\in{\rm CP}({\rm R});$ $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ (due the global case assumption); $C_{1}\xi\sigma\varphi=C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!n_{1}}}$; $C_{0}\sigma\varphi=C_{0}\mu_{0}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!n_{0}}}$. Since $\forall\delta{\,\prec\,}(n_{1}{+}1){+}(n_{0}{+}1){.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $0$-shallow confluent up to }\delta$ (by our induction hypothesis), due to our assumed $0$-shallow [noisy] anti-closedness (matching the definition’s $n_{0}$ to our $n_{1}{+}1$ and its $n_{1}$ to $n_{0}{+}1$) we have $w_{1}/\bar{p}_{0}{\,=\,}\penalty-1{l_{0}\mu_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1{l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}+1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0{[+n_{1}]}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!n_{1}+1}}}r_{0}\sigma\varphi{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1w_{0}/\bar{p}_{0}.$ Q.e.d. (“The critical peak case”) Q.e.d. (“There is some $\bar{p}_{1}^{\prime}$ with $\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1\bar{p}_{1}$ and $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ ”) There is some $\bar{p}_{0}^{\prime}$ with $\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1\bar{p}_{0}$ : We have two cases: “The second variable overlap case”: There are $x{\,\in\,}{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{1}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{0}^{\prime}$: Claim 11a: We have $x\mu_{1}/p^{\prime\prime}{\,=\,}\penalty-1l_{0}\mu_{0}.$ Proof of Claim 11a: We have $x\mu_{1}/p^{\prime\prime}{\,=\,}\penalty-1l_{1}\mu_{1}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{1}p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1l_{0}\mu_{0}.$ Q.e.d. (Claim 11a) Claim 11b: We can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by $x\nu{\,=\,}\penalty-1{x\mu_{1}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{1}.$ Then we have $x\mu_{1}{{\longrightarrow}_{{}_{\\!n_{0}+1}}}x\nu.$ Proof of Claim 11b: This follows directly from Claim 11a. Q.e.d. (Claim 11b) Claim 12: $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1l_{1}\nu.$ Proof of Claim 12: By the left-linearity assumption of our theorem we may assume ${{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}=\\{p^{\prime}\\}.$ Thus, by Claim 11b we get $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{u/\bar{p}_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow x\mu_{1}\,]}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow{x\mu_{1}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}\,]}}{\,=\,}\penalty-1\\\ {l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle n_{0}+1}}w_{1}/\bar{p}_{1}.$ Proof of Claim 13: Since $r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1},$ this follows directly from Claim 11b. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1}\nu{{\longrightarrow}_{{}_{\\!n_{1}+1}}}r_{1}\nu,$ which again follows from Claim 11b, Lemma 13.8 (matching its $n_{0}$ to our $n_{0}{+}1$ and its $n_{1}$ to our $n_{1}$), and our induction hypothesis that ${\rm R},{{\rm X}}$ is $0$-shallow confluent up to $(n_{0}{+}1){+}n_{1}.$ Q.e.d. (“The second variable overlap case”) “The second critical peak case”: $\bar{p}_{0}^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{1}})}\ {\wedge}\penalty-2\ l_{1}/\bar{p}_{0}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}]\cap{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}]\cup{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}\\\ x\xi^{-1}\mu_{0}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0}\xi\varrho{\,=\,}\penalty-1l_{0}\xi\xi^{-1}\mu_{0}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1u/\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1}\mu_{1}/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1}\varrho/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1(l_{1}/\bar{p}_{0}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0}\xi},{l_{1}/\bar{p}_{0}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1}\sigma,$ then the proof is finished due to $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1}.$ Otherwise we have $(\,({l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}},C_{0}\xi,0),\penalty-1\,(r_{1},C_{1},0),\penalty-1\,l_{1},\penalty-1\,\sigma,\penalty-1\,\bar{p}_{0}^{\prime}\,)\in{\rm CP}({\rm R});$ $C_{0}\xi\sigma\varphi=C_{0}\mu_{0}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!n_{0}}}$; $C_{1}\sigma\varphi=C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!n_{1}}}$. Since $\forall\delta{\,\prec\,}(n_{0}{+}1){+}(n_{1}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $0$-shallow confluent up to }\delta$ (by our induction hypothesis) due to our assumed $0$-shallow [noisy] strong joinability (matching the definition’s $n_{0}$ to our $n_{0}{+}1$ and its $n_{1}$ to our $n_{1}{+}1$) we have $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}+1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!0{[+n_{1}]}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!n_{0}+1}}}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1}.$ Q.e.d. (“The second critical peak case”) Q.e.d. (Theorem 15.1(II)) Proof of Theorem 15.3 Due to Corollary 15.2 it suffices to show that the conditions of Theorem 15.1 are satisfied. Since ${\rm R}_{\mathcal{C}}$ is normal, ${\rm R},{{\rm X}}$ is $0$-quasi-normal. Thus we only have to show that the conjunctive condition lists of the $0$-shallow joinability notions are never satisfied for critical peaks of the form $(0,0)$. Thus, assume $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ and $n_{0},n_{1}\prec\omega$ such that $\forall i{\,\prec\,}2{.}\penalty-1\,\,{(\ D_{i}\varphi\mbox{ fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}\ )}$ and $\forall\delta{\,\prec\,}n_{0}{+}n_{1}{.}\penalty-1\,\,{(\ {{\rm R},{{\rm X}}}\mbox{ is $0$-shallow confluent up to }\delta\ )}.$ By the assumed complementarity there must be complementary equation literals in $D_{0}$ and $D_{1}$. Due to our symmetry in $0$ and $1$ so far, we may w.l.o.g. assume that $(u{=}v)$ occurs in $D_{0}$ and $(u{\not=}v)$ occurs in $D_{1}$ or else that $(p{=}{{\mathsf{true}}})$ occurs in $D_{0}$ and $(p{=}{{\mathsf{false}}})$ occurs in $D_{1}$. Since negative conditions are not allowed for constructor rules we must be in the latter case here. Due to the definition of complementarity, ${\mathsf{true}}$ and ${\mathsf{false}}$ are distinct irreducible ground terms. Thus we have $p\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}{{\mathsf{true}}}$ and $p\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1}}}{{\mathsf{false}}}.$ In case of $n_{0},n_{1}{\,\preceq\,}1$ this implies the contradicting ${{\mathsf{true}}}{\,=\,}\penalty-1p\varphi{\,=\,}\penalty-1{{\mathsf{false}}}.$ Otherwise, in case of $n_{0}{\,\succeq\,}1$ we have $(n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){+}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)\prec n_{0}{+}n_{1}$ and thus by our above assumption ${\rm R},{{\rm X}}$ is $0$-shallow confluent up to $(n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){+}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)$. This implies the contradicting ${{\mathsf{true}}}\downarrow{{\mathsf{false}}}.$ Q.e.d. (Theorem 15.3) Proof of Theorem 15.4 1 $\Rightarrow$ 2: Directly by Lemma B.5. 2 $\Rightarrow$ 1: Directly by Lemma 6.3. Proof of Lemma A.1 For $n\prec\omega$ we are going to show by induction on $n$ the following property: $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}.$ Claim 1: If the above property holds for a fixed $n\prec\omega$, and $\forall k{\,\prec\,}n{.}\penalty-1\,\,({{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }k),$ then ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}$. Proof of Claim 1: By Lemma 3.3 it suffices to show that ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!\omega}}$. Assume $w_{0}{{\longleftarrow}_{{}_{\\!\omega}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}w^{\prime}$ (cf. diagram below). By the above property there is some $v^{\prime}$ with $w_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}.$ We only have to show that we can close the peak $v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}w^{\prime}$ according to $v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w^{\prime}.$ [In case of $n{\,=\,}\penalty-10:$ ] This is possible due to confluence of ${\longrightarrow}_{{}_{\\!\omega}}$. [Otherwise we have $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1{\,\prec\,}n$ and due to the assumed $\omega$-shallow confluence up to $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$ this is possible again.] Q.e.d. (Claim 1) Claim 2: If the above property holds for a fixed $n\prec\omega$, and $\forall k{\,\prec\,}n{.}\penalty-1\,\,({{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }k),$ then ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ are commuting. Proof of Claim 2: ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}$ are commuting by Lemma 3.3 and Claim 1. Since by Corollary 2.14 and Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!\omega+n}}}\subseteq{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}},$ now ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ are commuting, too. Q.e.d. (Claim 2) Claim 3: If the above property holds for all $n\preceq m$ for some $m\prec\omega$, then ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $m$. Proof of Claim 3: By induction on $m$ in $\,\prec\,$. Assume $i{+_{\\!\\!{}_{\omega}}}n{\,\preceq\,}m$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+i}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}.$ By definition of ‘$+_{\\!\\!{}_{\omega}}$’ and $i{+_{\\!\\!{}_{\omega}}}n{\,\prec\,}\omega$ w.l.o.g. we have $i{\,=\,}\penalty-10$ and $n{\,\preceq\,}m.$ By Claim 2 and our induction hypothesis we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}$ as desired. Q.e.d. (Claim 3) Note that our property for is trivial for $n{\,=\,}\penalty-10$ since then by Corollary 2.14 we have ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}={{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ is confluent. The benefit of claims 1 and 3 is twofold: First, they say that our lemma is valid if the above property holds for all $n\prec\omega$. Second, they strengthen the property when used as induction hypothesis. Thus (writing $n{+}1$ instead of $n$ since we may assume $0{\,\prec\,}n$) it now suffices to show for $n\prec\omega$ that $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega,\mathchar 261\relax_{0}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}}}w_{1}$ together with our induction hypothesis that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n$ implies $w_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}.$ W.l.o.g. let the positions of $\mathchar 261\relax_{0}$ (and $\mathchar 261\relax_{1}$) be maximal in the sense that for any $p\in\mathchar 261\relax_{0}$ (or else $p\in\mathchar 261\relax_{1}$) and $\mathchar 260\relax\subseteq{{{\mathcal{POS}}}({u})}{\cap}(p{{\bf N}}^{+})$ we do not have $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega,(\mathchar 261\relax_{0}\setminus\\{p\\})\cup\mathchar 260\relax}}u$ (or else $\ u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,(\mathchar 261\relax_{1}\setminus\\{p\\})\cup\mathchar 260\relax}}w_{1}$) anymore. Then for each $i\prec 2$ and $p\in\mathchar 261\relax_{i}$ there are ${((l_{i,p},r_{i,p}),C_{i,p})}\in{\rm R}$ and $\mu_{i,p}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/p{\,=\,}\penalty-1l_{i,p}\mu_{i,p},$ $r_{i,p}\mu_{i,p}{\,=\,}\penalty-1w_{i}/p.$ Moreover, for each $p\in\mathchar 261\relax_{0}$: $l_{0,p}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ and $\ C_{0,p}\mu_{0,p}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$. Similarly, for each $p\in\mathchar 261\relax_{1}$: $\ C_{1,p}\mu_{1,p}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Finally, for each $i\prec 2$: $w_{i}{\,=\,}\penalty-1{u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}\,]}}}.$ Claim 5: We may assume $\forall p{\,\in\,}\mathchar 261\relax_{1}{.}\penalty-1\,\,l_{1,p}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 5: Define $\mathchar 260\relax:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {l_{1,p}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}{\ \\}}}$ and $u^{\prime}:={u\penalty-1{{[\,p\leftarrow r_{1,p}\mu_{1,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{1}{\setminus}\mathchar 260\relax\,]}}}$. If we have succeeded with our proof under the assumption of Claim 5, then we have shown $w_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}u^{\prime}$ for some $v^{\prime}$ (cf. diagram below). By Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{1,p}$) we get $\forall p{\,\in\,}\mathchar 260\relax{.}\penalty-1\,\,l_{1,p}\mu_{1,p}{{\longrightarrow}_{{}_{\\!\omega}}}r_{1,p}\mu_{1,p}.$ Thus from $v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{1}$ we get $v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}$ by confluence of ${\longrightarrow}_{{}_{\\!\omega}}$. Q.e.d. (Claim 5) Define the set of inner overlapping positions by $\displaystyle\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1}):=\bigcup_{i\prec 2}{{\\{\ }p{\,\in\,}\mathchar 261\relax_{1-i}}~{}{\|}\penalty-9\,\ {\exists q{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}},$ and the length of a term by $\lambda({{f}{(}{t_{0}}{,\,}\ldots{,\,}{t_{m-1}}{)}}):=1+\sum_{j\prec m}\lambda(t_{j}).$ Now we start a second level of induction on $\displaystyle\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})$ in $\,\prec\,$. Define the set of top positions by $\displaystyle\mathchar 258\relax:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{0}{\cup}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {\neg\exists q{\,\in\,}\mathchar 261\relax_{0}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{+}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}}.$ Since the prefix ordering is wellfounded we have $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,\exists q{\,\in\,}\mathchar 258\relax{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}.$ Then $\forall i{\,\prec\,}2{.}\penalty-1\,\,w_{i}{\,=\,}\penalty-1{w_{i}\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{{u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}\,]}}}\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}.$ Thus, it now suffices to show for all $q\in\mathchar 258\relax$ $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}/q$ because then we have $w_{0}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}{u\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1w_{1}.$ Therefore we are left with the following two cases for $q\in\mathchar 258\relax$: $q{\,\not\in\,}\mathchar 261\relax_{1}$: Then $q{\,\in\,}\mathchar 261\relax_{0}.$ Define $\mathchar 261\relax_{1}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{1}{\ \\}}}$. We have two cases: “The variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}{.}\penalty-1\,\,l_{0,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{0,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}{\ \\}}}.$ Claim 7: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\mu_{0,q}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}x\nu\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\\\ \end{array}}}\right)}}.$ Proof of Claim 7: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{0,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{0,q}&{\,=\penalty-1}&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}\\\ &&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{0,q}$ is not linear in $x$. By the conditions of our lemma, this implies $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Therefore $x\mu_{0,q}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}.$ Together with $\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}$ this implies $\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\in{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ by Lemma 2.10. By confluence of ${\longrightarrow}_{{}_{\\!\omega}}$ and Lemma 2.10 again, there is some $t\in{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ with $\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}t.$ Therefore we can define $x\nu:=t$ in this case. This is appropriate since by $\exists p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\penalty-1{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}x\nu$ we have $x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}x\nu.$ Q.e.d. (Claim 7) Claim 8: $l_{0,q}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}/q.$ Proof of Claim 8: By Claim 7 we get $w_{1}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{0,q}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0,q}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}r_{0,q}\nu.$ Proof of Claim 9: Since $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q},$ this follows directly from Claim 7. Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $r_{0,q}\nu{{\longleftarrow}_{{}_{\\!\omega}}}l_{0,q}\nu,$ which again follows from Lemma 13.2 since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}x\nu$ by Claim 7 and Corollary 2.14. Q.e.d. (“The variable overlap (if any) case”) “The critical peak case”: There is some $p\in\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}$ with $l_{0,q}/p{\,\not\in\,}{{\rm V}}$: Claim 10: $p{\,\not=\,}\emptyset.$ Proof of Claim 10: If $p{\,=\,}\penalty-1\emptyset,$ then $\emptyset{\,\in\,}\mathchar 261\relax_{1}^{\prime},$ then $q{\,\in\,}\mathchar 261\relax_{1},$ which contradicts our global case assumption. Q.e.d. (Claim 10) Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}\\\ x\xi^{-1}\mu_{1,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1,qp}\xi\varrho{\,=\,}\penalty-1l_{1,qp}\xi\xi^{-1}\mu_{1,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p{\,=\,}\penalty-1l_{0,q}\varrho/p{\,=\,}\penalty-1(l_{0,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1,qp}\xi},{l_{0,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{0,q}\mu_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}$. We get $\begin{array}[]{l@{}l}u^{\prime}{\,=\penalty-1}&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}\\\ &{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}\,]}}}{\,=\,}\penalty-1w_{1}/q.\end{array}$ If ${l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1r_{0,q}\sigma\varphi{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ Otherwise we have $(\,({l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma,C_{1,qp}\xi\sigma,1),\penalty-1\,(r_{0,q}\sigma,C_{0,q}\sigma,0),\penalty-1\,l_{0,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $p{\,\not=\,}\emptyset$ (due to Claim 10); $C_{1,qp}\xi\sigma\varphi=C_{1,qp}\mu_{1,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $C_{0,q}\sigma\varphi=C_{0,q}\mu_{0,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$. Since ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n$ (by our induction hypothesis), due to our assumed $\omega$-shallow parallel closedness up to $\omega$ (matching the definition’s $n_{0}$ to our $n{+}1$ and its $n_{1}$ to $0$) we have $u^{\prime}{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega}}r_{0,q}\sigma\varphi{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1w_{0}/q.$ We then have $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega,\mathchar 261\relax^{\prime\prime}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q$ for some $\mathchar 261\relax^{\prime\prime}$. We can finish the proof in this case due to our second induction level since $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 266\relax(\mathchar 261\relax^{\prime\prime},\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\})}\lambda(u^{\prime}/p^{\prime\prime})\ \ \preceq\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}\lambda(u^{\prime}/p^{\prime\prime})\ \ =\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}\lambda(u/qp^{\prime\prime})$ $\displaystyle\ \ \prec\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}}\lambda(u/qp^{\prime\prime})\ \ =\sum_{p^{\prime}\in q\mathchar 261\relax_{1}^{\prime}}\lambda(u/p^{\prime})\ \ =\sum_{p^{\prime}\in\mathchar 266\relax(\\{q\\},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})\ \ \preceq\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime}).$ Q.e.d. (“The critical peak case”) Q.e.d. (“$q{\,\not\in\,}\mathchar 261\relax_{1}$”) $q{\,\in\,}\mathchar 261\relax_{1}$: Define $\mathchar 261\relax_{0}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{0}{\ \\}}}$. We have two cases: “The second variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{0}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{1,q}})}{.}\penalty-1\,\,l_{1,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{1,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}{\ \\}}}.$ Claim 11: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega}}x\mu_{1,q}\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu\\\ \end{array}}}\right)}}.$ Proof of Claim 11: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{1,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{1,q}&{\,=\penalty-1}&{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega}}\\\ &&{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{1,q}$ is not linear in $x$. By the conditions of our lemma, this contradicts Claim 5. Q.e.d. (Claim 11) Claim 12: $w_{0}/q{\,=\,}\penalty-1l_{1,q}\nu.$ Proof of Claim 12: By Claim 11 we get $w_{0}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{1,q}\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1,q}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1,q}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega}}w_{1}/q.$ Proof of Claim 13: Since $r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q,$ this follows directly from Claim 11. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{1,q}\nu,$ which again follows from Claim 11, Lemma 13.8 (matching its $n_{0}$ to $0$ and its $n_{1}$ to our $n$) and our induction hypothesis that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n$. Q.e.d. (“The second variable overlap (if any) case”) “The second critical peak case”: There is some $p\in\mathchar 261\relax_{0}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{1,q}})}$ with $l_{1,q}/p{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0,qp},r_{0,qp}),C_{0,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0,qp},r_{0,qp}),C_{0,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}\\\ x\xi^{-1}\mu_{0,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0,qp}\xi\varrho{\,=\,}\penalty-1l_{0,qp}\xi\xi^{-1}\mu_{0,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p{\,=\,}\penalty-1l_{1,q}\varrho/p{\,=\,}\penalty-1(l_{1,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0,qp}\xi},{l_{1,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{1,q}\mu_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\mu_{0,qp}\,]}}.$ We get $\begin{array}[]{l@{}l@{}l}w_{0}/q&{\,=\penalty-1}&{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{0,qp^{\prime}}\mu_{0,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{0}^{\prime}\,]}}}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}\\\ &&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{0,qp^{\prime}}\mu_{0,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{0}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{0,qp}\mu_{0,qp}\,]}}{\,=\,}\penalty-1u^{\prime}.\end{array}$ If ${l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1,q}\sigma,$ then the proof is finished due to $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}u^{\prime}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Otherwise we have $(\,({l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma,C_{0,qp}\xi\sigma,0),\penalty-1\,(r_{1,q}\sigma,C_{1,q}\sigma,1),\penalty-1\,l_{1,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $C_{0,qp}\xi\sigma\varphi=C_{0,qp}\mu_{0,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$; $C_{1,q}\sigma\varphi=C_{1,q}\mu_{1,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Since ${\rm R},{{\rm X}}$ $\omega$-shallow confluent up to $n$ (by our induction hypothesis), due to our assumed $\omega$-shallow [noisy] parallel joinability up to $\omega$ (matching the definition’s $n_{0}$ to $0$ and its $n_{1}$ to our $n{+}1$) we have $u^{\prime}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q$ for some $v_{1}$, $v_{2}$. We then have $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax^{\prime\prime}}}v_{1}$ for some $\mathchar 261\relax^{\prime\prime}$. Since $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\},\mathchar 261\relax^{\prime\prime})}\lambda(u^{\prime}/p^{\prime\prime})\ \ \preceq\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}\lambda(u^{\prime}/p^{\prime\prime})\ \ =\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}\lambda(u/qp^{\prime\prime})\ \ \prec\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}}\lambda(u/qp^{\prime\prime})\ \ =\sum_{p^{\prime}\in q\mathchar 261\relax_{0}^{\prime}}\lambda(u/p^{\prime})\ \ =\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\\{q\\})}\lambda(u/p^{\prime})\ \ \preceq\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})$ due to our second induction level we get some $v_{1}^{\prime}$ with $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{1}.$ From the peak $v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}v_{2}$ we finally get $v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{2}$ by $\omega$-shallow confluence up to $0[+n]$. Q.e.d. (“The second critical peak case”) Q.e.d. (Lemma A.1) Proof of Lemma A.2 Claim 0: ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$. Proof of Claim 0: Directly by the assumed strong commutation, cf. the proofs of the claims 2 and 3 of the proof of Lemma A.1. Q.e.d. (Claim 0) Claim 1: If ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$, then ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$ and ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$ are commuting. Proof of Claim 1: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$ are commuting by Lemma 3.3. Since by Corollary 2.14 and Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}},$ now ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$ and ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$ are commuting, too. Q.e.d. (Claim 1) For $n_{0}\preceq n_{1}\prec\omega$ we are going to show by induction on $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}$ the following property: $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}.$ Claim 2: Let $\delta\prec\omega{+}\omega$. If $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&\forall w_{0},w_{1},u{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}w_{1}\\\ {\Rightarrow}&w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}},$ then $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\mbox{ strongly commutes over }{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}}\\\ \end{array}}}\right)}},$ and ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\delta$. Proof of Claim 2: By induction on $\delta$ in $\,\prec\,$. First we show the strong commutation. Assume $n_{0}\preceq n_{1}\prec\omega$ with $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta$. By Lemma 3.3 it suffices to show that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$. Assume $u^{\prime\prime}{{\longleftarrow}_{{}_{\\!\omega+n_{0}}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}$ (cf. diagram below). By the strong commutation assumed for our lemma and Corollary 2.14, there are $w_{0}$ and $w_{0}^{\prime}$ with $u^{\prime\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}}}u.$ By the above property there are some $w_{3}$, $w_{1}^{\prime}$ with $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}.$ Next we show that we can close the peak $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}$ according to $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{2}$ for some $w_{2}^{\prime}$. In case of $n_{1}{\,=\,}\penalty-10$ this is possible due to the $\omega$-shallow confluence up to $\omega$ given by Claim 0. Otherwise we have $n_{0}{+_{\\!\\!{}_{\omega}}}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta$ and due to our induction hypothesis (saying that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to all $\delta^{\prime}\prec\delta$) this is possible again. By Claim 0 again, we can close the peak $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ according to $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{3}$ for some $w_{3}^{\prime}$. To close the whole diagram, we only have to show that we can close the peak $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{3}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}^{\prime}$ according to $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}^{\prime}.$ In case of $n_{0}{\,=\,}\penalty-10$ this is possible due to the strong commutation assumed for our lemma. Otherwise we have $n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1{\,\prec\,}n_{0}{\,\preceq\,}n_{1}$ and $(n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){+_{\\!\\!{}_{\omega}}}n_{1}{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta,$ and then due to our induction hypothesis this is possible again. Finally we show $\omega$-shallow confluence up to $\delta$. Assume $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}w_{1}.$ Due to symmetry in $n_{0}$ and $n_{1}$ we may assume $n_{0}{\,\preceq\,}n_{1}.$ Above we have shown that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$. By Claim 1 we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}$ as desired. Q.e.d. (Claim 2) Note that for $n_{0}{\,=\,}\penalty-10$ our property follows from ${{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}$ (by Corollary 2.14) and the assumption of our lemma that for each $n_{1}\prec\omega$: ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$. The benefit of Claim 2 is twofold: First, it says that our lemma is valid if the above property holds for all $n_{0}\preceq n_{1}\prec\omega$. Second, it strengthens the property when used as induction hypothesis. Thus (writing $n_{i}{+}1$ instead of $n_{i}$ since we may assume $0{\,\prec\,}n_{0}{\,\preceq\,}n_{1}$) it now suffices to show for $n_{0}\preceq n_{1}\prec\omega$ that $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}+1,\mathchar 261\relax_{0}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1,\mathchar 261\relax_{1}}}w_{1}$ together with our induction hypotheses that $\rule{0.0pt}{8.43889pt}\forall\delta{\,\prec\,}(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}(n_{1}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $\omega$-shallow confluent up to }\delta$ and (due to $n_{0}{\,\preceq\,}n_{1}{+}1$ and $n_{0}{+_{\\!\\!{}_{\omega}}}(n_{1}{+}1){\,\prec\,}(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}(n_{1}{+}1)$) ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$ implies $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}.$ Note that for the availability of our second induction hypothesis it is important that we have imposed the restriction “$n_{0}{\,\preceq\,}n_{1}$” in opposition to the restriction “$n_{0}{\,\succeq\,}n_{1}$”. In the latter case the availability of our second induction hypothesis would require $n_{0}{+}1{\,\succeq\,}n_{1}{+}1\ {\Rightarrow}\penalty-2\ n_{0}{\,\succeq\,}n_{1}{+}1$ which is not true for $n_{0}{\,=\,}\penalty-1n_{1}.$ The additional hypothesis ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}$ of the latter restriction is useless for our proof. W.l.o.g. let the positions of $\mathchar 261\relax_{i}$ be maximal in the sense that for any $p\in\mathchar 261\relax_{i}$ and $\mathchar 260\relax\subseteq{{{\mathcal{POS}}}({u})}{\cap}(p{{\bf N}}^{+})$ we do not have $u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{i}+1,(\mathchar 261\relax_{i}\setminus\\{p\\})\cup\mathchar 260\relax}}w_{i}$ anymore. Then for each $i\prec 2$ and $p\in\mathchar 261\relax_{i}$ there are ${((l_{i,p},r_{i,p}),C_{i,p})}\in{\rm R}$ and $\mu_{i,p}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/p{\,=\,}\penalty-1l_{i,p}\mu_{i,p},$ $r_{i,p}\mu_{i,p}{\,=\,}\penalty-1w_{i}/p,$ $C_{i,p}\mu_{i,p}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{i}}}$. Finally, for each $i\prec 2$: $w_{i}{\,=\,}\penalty-1{u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}\,]}}}.$ Claim 5: We may assume $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,l_{i,p}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 5: Define $\mathchar 260\relax_{i}:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{i}}~{}{\|}\penalty-9\,\ {l_{i,p}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}{\ \\}}}$ and $u_{i}^{\prime}:={u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}{\setminus}\mathchar 260\relax_{i}\,]}}}$. If we have succeeded with our proof under the assumption of Claim 5, then we have shown $u_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}u_{1}^{\prime}$ for some $v_{0}$, $v_{1}$ (cf. diagram below). By Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{i,p}$) we get $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 260\relax_{i}{.}\penalty-1\,\,l_{i,p}\mu_{i,p}{{\longrightarrow}_{{}_{\\!\omega}}}r_{i,p}\mu_{i,p}$ and therefore $\forall i{\,\prec\,}2{.}\penalty-1\,\,u_{i}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{i}.$ Thus from $v_{1}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}\penalty-1u_{1}^{\prime}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\penalty-1w_{1}$ we get $v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}$ for some $v_{2}$ by $\omega$-shallow confluence up to $\omega$ (cf. Claim 0). For the same reason we can close the peak $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}u_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{0}$ according to $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{0}$ for some $v_{0}^{\prime}$. By the assumption of our lemma that ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}$, from $v_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}v_{2}$ we can finally conclude $v_{0}^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{2}.$ Q.e.d. (Claim 5) Define the set of inner overlapping positions by $\displaystyle\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1}):=\bigcup_{i\prec 2}{{\\{\ }p{\,\in\,}\mathchar 261\relax_{1-i}}~{}{\|}\penalty-9\,\ {\exists q{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}},$ and the length of a term by $\lambda({{f}{(}{t_{0}}{,\,}\ldots{,\,}{t_{m-1}}{)}}):=1+\sum_{j\prec m}\lambda(t_{j}).$ Now we start a second level of induction on $\displaystyle\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})$ in $\,\prec\,$. Define the set of top positions by $\displaystyle\mathchar 258\relax:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{0}{\cup}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {\neg\exists q{\,\in\,}\mathchar 261\relax_{0}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{+}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}}.$ Since the prefix ordering is wellfounded we have $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,\exists q{\,\in\,}\mathchar 258\relax{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}.$ Then $\forall i{\,\prec\,}2{.}\penalty-1\,\,w_{i}{\,=\,}\penalty-1{w_{i}\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{{u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}\,]}}}\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}.$ Thus, it now suffices to show for all $q\in\mathchar 258\relax$ $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}w_{0}/q{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}/q$ because then we have $w_{0}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}{u\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1w_{1}.$ Therefore we are left with the following two cases for $q\in\mathchar 258\relax$: $q{\,\not\in\,}\mathchar 261\relax_{1}$: Then $q{\,\in\,}\mathchar 261\relax_{0}.$ Define $\mathchar 261\relax_{1}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{1}{\ \\}}}$. We have two cases: “The variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}{.}\penalty-1\,\,l_{0,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{0,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}{\ \\}}}.$ Claim 7: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\mu_{0,q}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}x\nu\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\nu{\,=\,}\penalty-1{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\\\ \end{array}}}\right)}}.$ Proof of Claim 7: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{0,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{0,q}&{\,=\penalty-1}&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}\\\ &&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{0,q}$ is not linear in $x$. By the conditions of our lemma and Claim 5 this implies $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Since there is some $(p^{\prime},p^{\prime\prime})\in\mathchar 256\relax(x)$ with $x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ this implies $l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then $l_{1,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which contradicts Claim 5. Q.e.d. (Claim 7) Claim 8: $l_{0,q}\nu{\,=\,}\penalty-1w_{1}/q.$ Proof of Claim 8: By Claim 7 we get $w_{1}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{0,q}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0,q}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}r_{0,q}\nu.$ Proof of Claim 9: Since $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q},$ this follows directly from Claim 7. Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $l_{0,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n_{0}+1}}}r_{0,q}\nu,$ which again follows from Lemma 13.8 since ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $(n_{1}{+}1){+_{\\!\\!{}_{\omega}}}n_{0}$ by our induction hypothesis and since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}+1}}}x\nu$ by Claim 7 and Corollary 2.14. Q.e.d. (“The variable overlap (if any) case”) “The critical peak case”: There is some $p\in\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}$ with $l_{0,q}/p{\,\not\in\,}{{\rm V}}$: Claim 10: $p{\,\not=\,}\emptyset.$ Proof of Claim 10: If $p{\,=\,}\penalty-1\emptyset,$ then $\emptyset{\,\in\,}\mathchar 261\relax_{1}^{\prime},$ then $q{\,\in\,}\mathchar 261\relax_{1},$ which contradicts our global case assumption. Q.e.d. (Claim 10) Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}\\\ x\xi^{-1}\mu_{1,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1,qp}\xi\varrho{\,=\,}\penalty-1l_{1,qp}\xi\xi^{-1}\mu_{1,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p{\,=\,}\penalty-1l_{0,q}\varrho/p{\,=\,}\penalty-1(l_{0,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1,qp}\xi},{l_{0,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{0,q}\mu_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}$. We get $\begin{array}[]{l@{}l}u^{\prime}{\,=\penalty-1}&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}\\\ &{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}\,]}}}{\,=\,}\penalty-1w_{1}/q.\end{array}$ If ${l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1r_{0,q}\sigma\varphi{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ Otherwise we have $(\,({l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma,C_{1,qp}\xi\sigma,1),\penalty-1\,(r_{0,q}\sigma,C_{0,q}\sigma,1),\penalty-1\,l_{0,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $p{\,\not=\,}\emptyset$ (due to Claim 10); $C_{1,qp}\xi\sigma\varphi=C_{1,qp}\mu_{1,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$; $C_{0,q}\sigma\varphi=C_{0,q}\mu_{0,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$. Since $\forall\delta{\,\prec\,}(n_{1}{+}1){+_{\\!\\!{}_{\omega}}}(n_{0}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $\omega$-shallow confluent up to }\delta$ (by our induction hypothesis) due to our assumed $\omega$-shallow parallel closedness (matching the definition’s $n_{0}$ to our $n_{1}{+}1$ and its $n_{1}$ to our $n_{0}{+}1$) we have $u^{\prime}{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi\penalty-1{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{0}+1}}\penalty-1v_{1}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}}v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}r_{0,q}\sigma\varphi{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1w_{0}/q$ for some $v_{1}$, $v_{2}$. We then have $v_{1}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}+1,\mathchar 261\relax^{\prime\prime}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q$ for some $\mathchar 261\relax^{\prime\prime}$. By $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 266\relax(\mathchar 261\relax^{\prime\prime},\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\})}\lambda(u^{\prime}/p^{\prime\prime})\ \ \preceq\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}\lambda(u^{\prime}/p^{\prime\prime})\ \ =\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}\lambda(u/qp^{\prime\prime})\ \ \prec\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}}\lambda(u/qp^{\prime\prime})\ \ =$ $\displaystyle\sum_{p^{\prime}\in q\mathchar 261\relax_{1}^{\prime}}\lambda(u/p^{\prime})\ \ =\sum_{p^{\prime}\in\mathchar 266\relax(\\{q\\},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})\ \ \preceq\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime}),$ due to our second induction level we get some $v_{1}^{\prime}$ with $v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}/q.$ Finally by our induction hypothesis that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$ the peak at $v_{1}$ can be closed according to $v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}v_{1}^{\prime}.$ Q.e.d. (“The critical peak case”) Q.e.d. (“$q{\,\not\in\,}\mathchar 261\relax_{1}$”) $q{\,\in\,}\mathchar 261\relax_{1}$: Define $\mathchar 261\relax_{0}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{0}{\ \\}}}$. We have two cases: “The second variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{0}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{1,q}})}{.}\penalty-1\,\,l_{1,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{1,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}{\ \\}}}.$ Claim 11: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}+1}}x\mu_{1,q}\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu\\\ \end{array}}}\right)}}.$ Proof of Claim 11: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{1,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{1,q}&{\,=\penalty-1}&{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{0}+1}}\\\ &&{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{1,q}$ is not linear in $x$. By the conditions of our lemma and Claim 5 this implies $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Since there is some $(p^{\prime},p^{\prime\prime})\in\mathchar 256\relax(x)$ with $x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}$ this implies $l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then $l_{0,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which contradicts Claim 5. Q.e.d. (Claim 11) Claim 12: $w_{0}/q{\,=\,}\penalty-1l_{1,q}\nu.$ Proof of Claim 12: By Claim 11 we get $w_{0}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{1,q}\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1,q}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1,q}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}+1}}w_{1}/q.$ Proof of Claim 13: Since $r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q,$ this follows directly from Claim 11. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n_{1}+1}}}r_{1,q}\nu,$ which again follows from Claim 11, Corollary 2.14, Lemma 13.8 (matching its $n_{0}$ to our $n_{0}{+}1$ and its $n_{1}$ to our $n_{1}$), and our induction hypothesis that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}n_{1}.$ Q.e.d. (“The second variable overlap (if any) case”) “The second critical peak case”: There is some $p\in\mathchar 261\relax_{0}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{1,q}})}$ with $l_{1,q}/p{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0,qp},r_{0,qp}),C_{0,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0,qp},r_{0,qp}),C_{0,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}\\\ x\xi^{-1}\mu_{0,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0,qp}\xi\varrho{\,=\,}\penalty-1l_{0,qp}\xi\xi^{-1}\mu_{0,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p{\,=\,}\penalty-1l_{1,q}\varrho/p{\,=\,}\penalty-1(l_{1,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0,qp}\xi},{l_{1,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{1,q}\mu_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\mu_{0,qp}\,]}}$. We get $\begin{array}[]{l@{}l@{}l}w_{0}/q&{\,=\penalty-1}&{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{0,qp^{\prime}}\mu_{0,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{0}^{\prime}\,]}}}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}+1,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}\\\ &&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{0,qp^{\prime}}\mu_{0,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{0}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{0,qp}\mu_{0,qp}\,]}}{\,=\,}\penalty-1u^{\prime}.\end{array}$ If ${l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1,q}\sigma,$ then the proof is finished due to $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}+1,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}u^{\prime}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Otherwise we have $(\,({l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma,C_{0,qp}\xi\sigma,1),\penalty-1\,(r_{1,q}\sigma,C_{1,q}\sigma,1),\penalty-1\,l_{1,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $C_{0,qp}\xi\sigma\varphi=C_{0,qp}\mu_{0,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$; $C_{1,q}\sigma\varphi=C_{1,q}\mu_{1,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$. Since $\forall\delta{\,\prec\,}(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}(n_{1}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $\omega$-shallow confluent up to }\delta$ (by our induction hypothesis) due to our assumed $\omega$-shallow noisy parallel joinability (matching the definition’s $n_{0}$ to our $n_{0}{+}1$ and its $n_{1}$ to our $n_{1}{+}1$ ) we have $u^{\prime}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}\penalty-1v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\penalty-1v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q$ for some $v_{1}$, $v_{2}$. We then have $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}+1,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1,\mathchar 261\relax^{\prime\prime}}}v_{1}$ for some $\mathchar 261\relax^{\prime\prime}$. Since $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\},\mathchar 261\relax^{\prime\prime})}\lambda(u^{\prime}/p^{\prime\prime})\ \ \preceq\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}\lambda(u^{\prime}/p^{\prime\prime})\ \ =\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}\lambda(u/qp^{\prime\prime})\ \ \prec\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}}\lambda(u/qp^{\prime\prime})\ \ =\sum_{p^{\prime}\in q\mathchar 261\relax_{0}^{\prime}}\lambda(u/p^{\prime})\ \ =\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\\{q\\})}\lambda(u/p^{\prime})\ \ \preceq\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})$ due to our second induction level we get some $v_{1}^{\prime}$ with $w_{0}/q{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}v_{1}.$ Finally the peak at $v_{1}$ can be closed according to $v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}v_{2}$ by our induction hypothesis saying that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}n_{1}$. Q.e.d. (“The second critical peak case”) Q.e.d. (Lemma A.2) Proof of Lemma A.3 For $n\prec\omega$ we are going to show by induction on $n$ the following property: $w_{0}{{\longleftarrow}_{{}_{\\!\omega}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}.$ Claim 1: If the above property holds for a fixed $n\prec\omega$, and $\forall k{\,\prec\,}n{.}\penalty-1\,\,({{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }k),$ then ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}$. Proof of Claim 1: By Lemma 3.3 it suffices to show that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!\omega}}$. Assume $u^{\prime\prime}{{\longleftarrow}_{{}_{\\!\omega}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}w_{2}$ (cf. diagram below). By the strong confluence of ${\longrightarrow}_{{}_{\\!\omega}}$ assumed for our lemma we can close the peak $u^{\prime\prime}{{\longleftarrow}_{{}_{\\!\omega}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}u$ according to $u^{\prime\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}u$ for some $w_{0}$. By the above property there is some $w_{1}^{\prime}$ with $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}.$ We only have to show that we can close the peak $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}w_{2}$ according to $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{2}.$ [In case of $n{\,=\,}\penalty-10:$ ] This is possible due to confluence of ${\longrightarrow}_{{}_{\\!\omega}}$. [Otherwise we have $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1{\,\prec\,}n$ and due to the assumed $\omega$-shallow confluence up to $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$ this is possible again.] Q.e.d. (Claim 1) Claim 2: If the above property holds for a fixed $n\prec\omega$, and $\forall k{\,\prec\,}n{.}\penalty-1\,\,({{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }k),$ then ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ are commuting. Proof of Claim 2: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}$ are commuting by Lemma 3.3 and Claim 1. Since by Corollary 2.14 and Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!\omega+n}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}},$ now ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ are commuting, too. Q.e.d. (Claim 2) Claim 3: If the above property holds for all $n\preceq m$ for some $m\prec\omega$, then ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $m$. Proof of Claim 3: By induction on $m$ in $\,\prec\,$. Assume $i{+_{\\!\\!{}_{\omega}}}n{\,\preceq\,}m$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+i}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}.$ By definition of ‘$+_{\\!\\!{}_{\omega}}$’ and $i{+_{\\!\\!{}_{\omega}}}n{\,\prec\,}\omega$ w.l.o.g. we have $i{\,=\,}\penalty-10$ and $n{\,\preceq\,}m.$ By Claim 2 and our induction hypothesis we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}$ as desired. Q.e.d. (Claim 3) Note that our property for is trivial for $n{\,=\,}\penalty-10$ since then by Corollary 2.14 we have ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}={{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ is confluent. The benefit of claims 1 and 3 is twofold: First, they say that our lemma is valid if the above property holds for all $n\prec\omega$. Second, they strengthen the property when used as induction hypothesis. Thus (writing $n{+}1$ instead of $n$ since we may assume $0{\,\prec\,}n$) it now suffices to show for $n\prec\omega$ that $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega,\bar{p}_{0}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}}}w_{1}$ together with our induction hypothesis that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n$ implies $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}.$ There are ${((l_{0,\bar{p}_{0}},r_{0,\bar{p}_{0}}),C_{0,\bar{p}_{0}})}\in{\rm R}$ and $\mu_{0,\bar{p}_{0}}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ such that $l_{0,\bar{p}_{0}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ $u/\bar{p}_{0}{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}},$ $\ C_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$, and $w_{0}{\,=\,}\penalty-1{u\penalty-1{[\,\bar{p}_{0}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}.$ W.l.o.g. let the positions of $\mathchar 261\relax_{1}$ be maximal in the sense that for any $p\in\mathchar 261\relax_{1}$ and $\mathchar 260\relax\subseteq{{{\mathcal{POS}}}({u})}{\cap}(p{{\bf N}}^{+})$ we do not have $\ u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,(\mathchar 261\relax_{1}\setminus\\{p\\})\cup\mathchar 260\relax}}w_{1}$ anymore. Then for each $p\in\mathchar 261\relax_{1}$ there are ${((l_{1,p},r_{1,p}),C_{1,p})}\in{\rm R}$ and $\mu_{1,p}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ such that $u/p{\,=\,}\penalty-1l_{1,p}\mu_{1,p},$ $r_{1,p}\mu_{1,p}{\,=\,}\penalty-1w_{1}/p$ , $\ C_{1,p}\mu_{1,p}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$, and $w_{1}{\,=\,}\penalty-1{u\penalty-1{{[\,p\leftarrow r_{1,p}\mu_{1,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{1}\,]}}}.$ Claim 5: We may assume $\forall p{\,\in\,}\mathchar 261\relax_{1}{.}\penalty-1\,\,l_{1,p}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 5: Define $\mathchar 260\relax:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {l_{1,p}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}{\ \\}}}$ and $u^{\prime}:={u\penalty-1{{[\,p\leftarrow r_{1,p}\mu_{1,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{1}{\setminus}\mathchar 260\relax\,]}}}$. If we have succeeded with our proof under the assumption of Claim 5, then we have shown $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}u^{\prime}$ for some $v^{\prime}$ (cf. diagram below). By Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{1,p}$) we get $\forall p{\,\in\,}\mathchar 260\relax{.}\penalty-1\,\,l_{1,p}\mu_{1,p}{{\longrightarrow}_{{}_{\\!\omega}}}r_{1,p}\mu_{1,p}.$ Thus from $v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{1}$ we get $v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}$ by confluence of ${\longrightarrow}_{{}_{\\!\omega}}$. Q.e.d. (Claim 5) Now we start a second level of induction on ${\,\|{\mathchar 261\relax_{1}}\|\,}$ in $\,\prec\,$. Define the set of top positions by $\displaystyle\mathchar 258\relax:={{\\{\ }p{\,\in\,}\\{\bar{p}_{0}\\}{\cup}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {\neg\exists q{\,\in\,}\\{\bar{p}_{0}\\}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{+}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}}.$ Since the prefix ordering is wellfounded we have $\forall p{\,\in\,}\\{\bar{p}_{0}\\}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q{\,\in\,}\mathchar 258\relax{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}.$ It now suffices to show for all $q\in\mathchar 258\relax$ $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}w_{0}/q{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}/q$ because then we have $w_{0}{\,=\,}\penalty-1{w_{0}\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{{u\penalty-1{[\,\bar{p}_{0}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}{u\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1\\\ {{u\penalty-1{{[\,p\leftarrow r_{1,p}\mu_{1,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{1}\,]}}}\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{w_{1}\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1w_{1}.$ Therefore we are left with the following two cases for $q\in\mathchar 258\relax$: $q{\,\not\in\,}\mathchar 261\relax_{1}$: Then $q{\,=\,}\penalty-1\bar{p}_{0}.$ Define $\mathchar 261\relax_{1}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{1}{\ \\}}}$. We have two cases: “The variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}{.}\penalty-1\,\,l_{0,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{0,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}{\ \\}}}.$ Claim 7: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\mu_{0,q}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}x\nu\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\\\ \end{array}}}\right)}}.$ Proof of Claim 7: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{0,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{0,q}&{\,=\penalty-1}&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}\\\ &&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{0,q}{\,=\,}\penalty-1l_{0,\bar{p}_{0}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ is not linear in $x$. By the conditions of our lemma, this implies $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Therefore $x\mu_{0,q}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}.$ Together with $\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}$ this implies $\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\in{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ by Lemma 2.10. By confluence of ${\longrightarrow}_{{}_{\\!\omega}}$ and Lemma 2.10 again, there is some $t\in{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ with $\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}t.$ Therefore we can define $x\nu:=t$ in this case. This is appropriate since by $\exists p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\penalty-1{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}x\nu$ we have $x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}x\nu.$ Q.e.d. (Claim 7) Claim 8: $l_{0,q}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}/q.$ Proof of Claim 8: By Claim 7 we get $w_{1}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{0,q}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0,q}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}r_{0,q}\nu.$ Proof of Claim 9: Since $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q},$ this follows from Claim 7. Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $r_{0,q}\nu{{\longleftarrow}_{{}_{\\!\omega}}}l_{0,q}\nu,$ which again follows from Lemma 13.2 since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}x\nu$ by Claim 7 and Corollary 2.14. Q.e.d. (“The variable overlap (if any) case”) “The critical peak case”: There is some $p\in\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}$ with $l_{0,q}/p{\,\not\in\,}{{\rm V}}$: Claim 10: $p{\,\not=\,}\emptyset.$ Proof of Claim 10: If $p{\,=\,}\penalty-1\emptyset,$ then $\emptyset{\,\in\,}\mathchar 261\relax_{1}^{\prime},$ then $q{\,\in\,}\mathchar 261\relax_{1},$ which contradicts our global case assumption. Q.e.d. (Claim 10) Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}\\\ x\xi^{-1}\mu_{1,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1,qp}\xi\varrho{\,=\,}\penalty-1l_{1,qp}\xi\xi^{-1}\mu_{1,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p{\,=\,}\penalty-1l_{0,q}\varrho/p{\,=\,}\penalty-1(l_{0,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1,qp}\xi},{l_{0,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{0,q}\mu_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}$. We get $\begin{array}[]{l@{}l}u^{\prime}{\,=\penalty-1}&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}\\\ &{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}\,]}}}{\,=\,}\penalty-1w_{1}/q.\end{array}$ If ${l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1r_{0,q}\sigma\varphi{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ Otherwise we have $(\,({l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma,C_{1,qp}\xi\sigma,1),\penalty-1\,(r_{0,q}\sigma,C_{0,q}\sigma,0),\penalty-1\,l_{0,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $p{\,\not=\,}\emptyset$ (due to Claim 10); $C_{1,qp}\xi\sigma\varphi=C_{1,qp}\mu_{1,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $C_{0,q}\sigma\varphi=C_{0,q}\mu_{0,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$. Since ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n$ (by our induction hypothesis), due to our assumed $\omega$-shallow closedness up to $\omega$ (matching the definition’s $n_{0}$ to our $n{+}1$ and its $n_{1}$ to $0$) we have $u^{\prime}{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}r_{0,q}\sigma\varphi{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1w_{0}/q$ for some $v$. We then have $v{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ We can finish the proof in this case due to our second induction level since ${\,\|{\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}}\|\,}\prec{\,\|{\mathchar 261\relax_{1}^{\prime}}\|\,}\preceq{\,\|{\mathchar 261\relax_{1}}\|\,}.$ Q.e.d. (“The critical peak case”) Q.e.d. (“$q{\,\not\in\,}\mathchar 261\relax_{1}$”) $q{\,\in\,}\mathchar 261\relax_{1}$: If there is no $\bar{p}_{0}^{\prime}$ with $q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1\bar{p}_{0},$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1u/q{{\longrightarrow}_{{}_{\\!\omega+n+1}}}w_{1}/q.$ Otherwise, we can define $\bar{p}_{0}^{\prime}$ by $q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1\bar{p}_{0}.$ We have two cases: “The second variable overlap case”: There are $x\in{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{1,q}/p^{\prime}{\,=\,}\penalty-1x$ and $p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{0}^{\prime}$ : Claim 11: For $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ defined by $x\nu{\,=\,}\penalty-1{x\mu_{1,q}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{1,q}$ we get $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\mu_{1,q}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega}}y\nu.$ Proof of Claim 11: Due to $x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}$ we have $x\mu_{1,q}{\,=\,}\penalty-1{x\mu_{1,q}\penalty-1{[\,p^{\prime\prime}\leftarrow l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{{\longrightarrow}_{{}_{\\!\omega}}}{x\mu_{1,q}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1x\nu.$ Q.e.d. (Claim 11) Claim 12: $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega}}l_{1,q}\nu.$ Proof of Claim 12: By Claim 11 we get $w_{0}/q{\,=\,}\penalty-1{u/q\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1\\\ {{l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1,q}\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1\\\ {{{l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1,q}\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ x{\,\not=\,}y\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\mu_{1,q}\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime\prime\prime}{\,\not=\,}p^{\prime}\,]}}}}\\\ {[\,p^{\prime}\leftarrow{x\mu_{1,q}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}\,]}{\,=\,}\penalty-1\\\ {{{l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ x{\,\not=\,}y\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\mu_{1,q}\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime\prime\prime}{\,\not=\,}p^{\prime}\,]}}}}{[\,p^{\prime}\leftarrow x\nu\,]}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega}}\\\ {{{l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ x{\,\not=\,}y\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\nu\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime\prime\prime}{\,\not=\,}p^{\prime}\,]}}}}{[\,p^{\prime}\leftarrow x\nu\,]}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1,q}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1,q}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega}}w_{1}/q.$ Proof of Claim 13: Since $r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q,$ this follows directly from Claim 11. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{1,q}\nu,$ which again follows from Claim 11, Lemma 13.8 (matching its $n_{0}$ to $0$ and its $n_{1}$ to our $n$) and our induction hypothesis that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n$. Q.e.d. (“The second variable overlap case”) “The second critical peak case”: $\bar{p}_{0}^{\prime}\in{{{\mathcal{POS}}}({l_{1,q}})}$ with $l_{1,q}/\bar{p}_{0}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0,\bar{p}_{0}},r_{0,\bar{p}_{0}}),C_{0,\bar{p}_{0}})}})}]\cap{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0,\bar{p}_{0}},r_{0,\bar{p}_{0}}),C_{0,\bar{p}_{0}})}})}]\cup{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}\\\ x\xi^{-1}\mu_{0,\bar{p}_{0}}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0,\bar{p}_{0}}\xi\varrho{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\xi\xi^{-1}\mu_{0,\bar{p}_{0}}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1,q}\varrho/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1(l_{1,q}/\bar{p}_{0}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0,\bar{p}_{0}}\xi},{l_{1,q}/\bar{p}_{0}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1{l_{1,q}\mu_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Otherwise we have $(\,({l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma,C_{0,\bar{p}_{0}}\xi\sigma,0),\penalty-1\,(r_{1,q}\sigma,C_{1,q}\sigma,1),\penalty-1\,l_{1,q}\sigma,\penalty-1\,\bar{p}_{0}^{\prime}\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $C_{0,\bar{p}_{0}}\xi\sigma\varphi=C_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$; $C_{1,q}\sigma\varphi=C_{1,q}\mu_{1,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Since ${\rm R},{{\rm X}}$ $\omega$-shallow confluent up to $n$ (by our induction hypothesis), due to our assumed $\omega$-shallow [noisy] weak parallel joinability up to $\omega$ (matching the definition’s $n_{0}$ to $0$ and its $n_{1}$ to our $n{+}1$) we have $w_{0}/q{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Q.e.d. (“The second critical peak case”) Q.e.d. (Lemma A.3) Proof of Lemma A.4 For $n\prec\omega$ we are going to show by induction on $n$ the following property: $w_{0}{{\longleftarrow}_{{}_{\\!\omega}}}u{{\longrightarrow}_{{}_{\\!\omega+n}}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}.$ Claim 1: If the above property holds for a fixed $n\prec\omega$, and $\forall k{\,\prec\,}n{.}\penalty-1\,\,({{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }k),$ then ${{\longrightarrow}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}$. Proof of Claim 1: By Lemma 3.3 it suffices to show that ${{\longrightarrow}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!\omega}}$. Assume $w_{0}{{\longleftarrow}_{{}_{\\!\omega}}}u{{\longrightarrow}_{{}_{\\!\omega+n}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}w^{\prime}$ (cf. diagram below). By the above property there is some $v^{\prime}$ with $w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}.$ We only have to show that we can close the peak $v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}w^{\prime}$ according to $v^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w^{\prime}.$ [In case of $n{\,=\,}\penalty-10:$ ] This is possible due to confluence of ${\longrightarrow}_{{}_{\\!\omega}}$. [Otherwise we have $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1{\,\prec\,}n$ and due to the assumed $\omega$-shallow confluence up to $n{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$ this is possible again.] Q.e.d. (Claim 1) Claim 2: If the above property holds for a fixed $n\prec\omega$, and $\forall k{\,\prec\,}n{.}\penalty-1\,\,({{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }k),$ then ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ are commuting. Proof of Claim 2: ${{\longrightarrow}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}$ are commuting by Lemma 3.3 and Claim 1. Since by Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!\omega+n}}}\subseteq{{\longrightarrow}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)]}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}},$ now ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ are commuting, too. Q.e.d. (Claim 2) Claim 3: If the above property holds for all $n\preceq m$ for some $m\prec\omega$, then ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $m$. Proof of Claim 3: By induction on $m$ in $\,\prec\,$. Assume $i{+_{\\!\\!{}_{\omega}}}n{\,\preceq\,}m$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+i}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}.$ By definition of ‘$+_{\\!\\!{}_{\omega}}$’ and $i{+_{\\!\\!{}_{\omega}}}n{\,\prec\,}\omega$ w.l.o.g. we have $i{\,=\,}\penalty-10$ and $n{\,\preceq\,}m.$ By Claim 2 and our induction hypothesis we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}$ as desired. Q.e.d. (Claim 3) Note that our property for is trivial for $n{\,=\,}\penalty-10$ since ${\longrightarrow}_{{}_{\\!\omega}}$ is confluent. The benefit of claims 1 and 3 is twofold: First, they say that our lemma is valid if the above property holds for all $n\prec\omega$. Second, they strengthen the property when used as induction hypothesis. Thus (writing $n{+}1$ instead of $n$ since we may assume $0{\,\prec\,}n$) it now suffices to show for $n\prec\omega$ that $w_{0}{{\longleftarrow}_{{}_{\\!\omega,\bar{p}_{0}}}}u{{\longrightarrow}_{{}_{\\!\omega+n+1,\bar{p}_{1}}}}w_{1}$ together with our induction hypothesis that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n$ implies $w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega[+n]}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{1}.$ Now for each $i\prec 2$ there are ${((l_{i},r_{i}),C_{i})}\in{\rm R}$ and $\mu_{i}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/\bar{p}_{i}{\,=\,}\penalty-1l_{i}\mu_{i},$ $w_{i}{\,=\,}\penalty-1{u\penalty-1{[\,\bar{p}_{i}\leftarrow r_{i}\mu_{i}\,]}},$ $l_{0}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ $\ C_{0}\mu_{0}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$, $C_{1}\mu_{1}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Claim 5: We may assume $l_{1}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 5: In case of $l_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ by Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{1}$) we get $l_{1}\mu_{1}{{\longrightarrow}_{{}_{\\!\omega}}}r_{1}\mu_{1}.$ Then the proof is finished by confluence of ${\longrightarrow}_{{}_{\\!\omega}}$. Q.e.d. (Claim 5) In case of ${{\bar{p}_{0}}\,{\parallel}\,{\bar{p}_{1}}}$ we have $w_{i}/\bar{p}_{1-i}{\,=\,}\penalty-1{u\penalty-1{[\,\bar{p}_{i}\leftarrow r_{i}\mu_{i}\,]}}/\bar{p}_{1-i}{\,=\,}\penalty-1u/\bar{p}_{1-i}{\,=\,}\penalty-1l_{1-i}\mu_{1-i}$ and therefore $w_{0}{{\longrightarrow}_{{}_{\\!\omega+n+1}}}{u\penalty-1{{[\,\bar{p}_{k}\leftarrow r_{k}\mu_{k}\ \|\ k{\,\prec\,}2\,]}}}{{\longleftarrow}_{{}_{\\!\omega}}}w_{1},$ i.e. our proof is finished. Thus, according to whether $\bar{p}_{0}$ is a prefix of $\bar{p}_{1}$ or vice versa, we have the following two cases left: There is some $\bar{p}_{1}^{\prime}$ with $\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1\bar{p}_{1}$ and $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ : We have two cases: “The variable overlap case”: There are $x\in{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{0}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{1}^{\prime}$: Claim 6: We have $x\mu_{0}/p^{\prime\prime}{\,=\,}\penalty-1l_{1}\mu_{1}$ and may assume $x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}.$ Proof of Claim 6: We have $x\mu_{0}/p^{\prime\prime}{\,=\,}\penalty-1l_{0}\mu_{0}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{0}p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{1}{\,=\,}\penalty-1l_{1}\mu_{1}.$ If $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}},$ then $x\mu_{0}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $x\mu_{0}/p^{\prime\prime}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $l_{1}\mu_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ and then $l_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which we may assume not to be the case by Claim 5. Q.e.d. (Claim 6) Claim 7: We can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by $x\nu{\,=\,}\penalty-1{x\mu_{0}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{0}.$ Then we have $x\mu_{0}{{\longrightarrow}_{{}_{\\!\omega+n+1}}}x\nu.$ Proof of Claim 7: This follows directly from Claim 6. Q.e.d. (Claim 7) Claim 8: $l_{0}\nu{\,=\,}\penalty-1w_{1}/\bar{p}_{0}.$ Proof of Claim 8: By the left-linearity assumption of our lemma and Claim 6 we may assume ${{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}=\\{p^{\prime}\\}.$ Thus, by Claim 7 we get $w_{1}/\bar{p}_{0}{\,=\,}\penalty-1{u/\bar{p}_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow x\mu_{0}\,]}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow{x\mu_{0}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}\,]}}{\,=\,}\penalty-1\\\ {l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/\bar{p}_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}r_{0}\nu.$ Proof of Claim 9: By the right-linearity assumption of our lemma and Claim 6 we may assume ${\,\|{{{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}}\|\,}{\,\preceq\,}1.$ Thus by Claim 7 we get: $w_{0}/\bar{p}_{0}{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{\,=\,}\penalty-1\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{\,=\,}\penalty-1r_{0}\nu.$ Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $r_{0,q}\nu{{\longleftarrow}_{{}_{\\!\omega}}}l_{0,q}\nu,$ which again follows from Lemma 13.2 since $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}y\nu$ by Claim 7. Q.e.d. (“The variable overlap case”) “The critical peak case”: $\bar{p}_{1}^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{0}})}\ {\wedge}\penalty-2\ l_{0}/\bar{p}_{1}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}]\cap{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}]\cup{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}\\\ x\xi^{-1}\mu_{1}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1}\xi\varrho{\,=\,}\penalty-1l_{1}\xi\xi^{-1}\mu_{1}{\,=\,}\penalty-1u/\bar{p}_{1}{\,=\,}\penalty-1u/\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1l_{0}\mu_{0}/\bar{p}_{1}^{\prime}{\,=\,}\penalty-1l_{0}\varrho/\bar{p}_{1}^{\prime}{\,=\,}\penalty-1(l_{0}/\bar{p}_{1}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1}\xi},{l_{0}/\bar{p}_{1}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0}\sigma,$ then the proof is finished due to $w_{0}/\bar{p}_{0}{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1r_{0}\sigma\varphi{\,=\,}\penalty-1{l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1{l_{0}\mu_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1w_{1}/\bar{p}_{0}.$ Otherwise we have $(\,({l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}},C_{1}\xi,1),\penalty-1\,(r_{0},C_{0},0),\penalty-1\,l_{0},\penalty-1\,\sigma,\penalty-1\,\bar{p}_{1}^{\prime}\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ (due the global case assumption); $C_{1}\xi\sigma\varphi=C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $C_{0}\sigma\varphi=C_{0}\mu_{0}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$. Since ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n$ (by our induction hypothesis), due to our assumed $\omega$-shallow [noisy] anti-closedness up to $\omega$ (matching the definition’s $n_{0}$ to our $n{+}1$ and its $n_{1}$ to $0$) we have $w_{1}/q{\,=\,}\penalty-1{l_{0}\mu_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1{l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega[+n]}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}r_{0}\sigma\varphi{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1w_{0}/q.$ Q.e.d. (“The critical peak case”) Q.e.d. (“There is some $\bar{p}_{1}^{\prime}$ with $\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1\bar{p}_{1}$ and $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ ”) There is some $\bar{p}_{0}^{\prime}$ with $\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1\bar{p}_{0}$ : We have two cases: “The second variable overlap case”: There are $x{\,\in\,}{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{1}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{0}^{\prime}$: We have $x\mu_{1}/p^{\prime\prime}{\,=\,}\penalty-1l_{1}\mu_{1}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{1}p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1l_{0}\mu_{0}.$ Claim 11: We can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by $x\nu{\,=\,}\penalty-1{x\mu_{1}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{1}.$ Then we have $x\mu_{1}{{\longrightarrow}_{{}_{\\!\omega}}}x\nu.$ Proof of Claim 11: This follows directly from the above equality and Lemma 2.10. Q.e.d. (Claim 11) Claim 12: $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1l_{1}\nu.$ Proof of Claim 12: By the left-linearity assumption of our lemma and Claim 5 we may assume ${{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}=\\{p^{\prime}\\}.$ Thus, by Claim 11 we get $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{u/\bar{p}_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow x\mu_{1}\,]}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow{x\mu_{1}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}\,]}}{\,=\,}\penalty-1\\\ {l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega}}w_{1}/\bar{p}_{1}.$ Proof of Claim 13: Since $r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1},$ this follows directly from Claim 11. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{1,q}\nu,$ which again follows from Claim 11, Lemma 13.8 (matching its $n_{0}$ to $0$ and its $n_{1}$ to our $n$) and our induction hypothesis that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n$. Q.e.d. (“The second variable overlap (if any) case”) “The second critical peak case”: $\bar{p}_{0}^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{1}})}\ {\wedge}\penalty-2\ l_{1}/\bar{p}_{0}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}]\cap{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}]\cup{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}\\\ x\xi^{-1}\mu_{0}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0}\xi\varrho{\,=\,}\penalty-1l_{0}\xi\xi^{-1}\mu_{0}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1u/\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1}\mu_{1}/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1}\varrho/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1(l_{1}/\bar{p}_{0}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0}\xi},{l_{1}/\bar{p}_{0}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1}\sigma,$ then the proof is finished due to $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1}.$ Otherwise we have $(\,({l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}},C_{0}\xi,0),\penalty-1\,(r_{1},C_{1},1),\penalty-1\,l_{1},\penalty-1\,\sigma,\penalty-1\,\bar{p}_{0}^{\prime}\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $C_{0}\xi\sigma\varphi=C_{0}\mu_{0}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$; $C_{1}\sigma\varphi=C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Since ${\rm R},{{\rm X}}$ $\omega$-shallow confluent up to $n$ (by our induction hypothesis), due to our assumed $\omega$-shallow [noisy] strong joinability up to $\omega$ (matching the definition’s $n_{0}$ to $0$ and its $n_{1}$ to our $n{+}1$) we have $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega{[+n]}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1}.$ Q.e.d. (“The second critical peak case”) Q.e.d. (Lemma A.4) Proof of Lemma A.5 Claim 0: ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$. Proof of Claim 0: Directly by the assumed strong commutation of ${{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+(n{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}$, cf. the proofs of the claims 2 and 3 of the proof of Lemma A.4. Q.e.d. (Claim 0) Claim 1: If ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$, then ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$ and ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$ are commuting. Proof of Claim 1: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$ are commuting by Lemma 3.3. Since by Corollary 2.14 and Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}},$ now ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$ and ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$ are commuting, too. Q.e.d. (Claim 1) For $n_{0}\preceq n_{1}\prec\omega$ we are going to show by induction on $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}$ the following property: $w_{0}{{\longleftarrow}_{{}_{\\!\omega+n_{0}}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}$ Claim 2: Let $\delta\prec\omega{+}\omega$. If $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&\forall w_{0},w_{1},u{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&w_{0}{{\longleftarrow}_{{}_{\\!\omega+n_{0}}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}w_{1}\\\ {\Rightarrow}&w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}},$ then $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\mbox{ strongly commutes over }{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}}\\\ \end{array}}}\right)}},$ and ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\delta$. Proof of Claim 2: By induction on $\delta$ in $\,\prec\,$. First we show the strong commutation. Assume $n_{0}\preceq n_{1}\prec\omega$ with $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta$. By Lemma 3.3 it suffices to show that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$. Assume $u^{\prime\prime}{{\longleftarrow}_{{}_{\\!\omega+n_{0}}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}$ (cf. diagram below). By the strong commutation assumption of our lemma there are $w_{0}$ and $w_{0}^{\prime}$ with $u^{\prime\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}u.$ By the above property there are some $w_{3}$, $w_{1}^{\prime}$ with $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}.$ Next we show that we can close the peak $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}$ according to $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{2}$ for some $w_{2}^{\prime}$. In case of $n_{1}{\,=\,}\penalty-10$ this is possible due to the $\omega$-shallow confluence up to $\omega$ given by Claim 0. Otherwise we have $n_{0}{+_{\\!\\!{}_{\omega}}}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta$ and due to our induction hypothesis (saying that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to all $\delta^{\prime}\prec\delta$) this is possible again. By Claim 0 again, we can close the peak $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ according to $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{3}$ for some $w_{3}^{\prime}$. To close the whole diagram, we only have to show that we can close the peak $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{3}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}^{\prime}$ according to $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}^{\prime}.$ In case of $n_{0}{\,=\,}\penalty-10$ this is possible since it is assumed for our lemma (below the strong commutation assumption). Otherwise we have $n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1{\,\prec\,}n_{0}{\,\preceq\,}n_{1}$ and $(n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){+_{\\!\\!{}_{\omega}}}n_{1}{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta,$ and then due to our induction hypothesis this is possible again. Finally we show $\omega$-shallow confluence up to $\delta$. Assume $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}w_{1}.$ Due to symmetry in $n_{0}$ and $n_{1}$ we may assume $n_{0}{\,\preceq\,}n_{1}.$ Above we have shown that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$. By Claim 1 we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}$ as desired. Q.e.d. (Claim 2) Note that for $n_{0}{\,=\,}\penalty-10$ our property follows from the assumption of our lemma (below the strong commutation assumption). The benefit of Claim 2 is twofold: First, it says that our lemma is valid if the above property holds for all $n_{0}\preceq n_{1}\prec\omega$. Second, it strengthens the property when used as induction hypothesis. Thus (writing $n_{i}{+}1$ instead of $n_{i}$ since we may assume $0{\,\prec\,}n_{0}{\,\preceq\,}n_{1}$) it now suffices to show for $n_{0}\preceq n_{1}\prec\omega$ that $w_{0}{{\longleftarrow}_{{}_{\\!\omega+n_{0}+1,\bar{p}_{0}}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1,\mathchar 261\relax_{1}}}w_{1}$ together with our induction hypotheses that $\rule{0.0pt}{8.43889pt}\forall\delta{\,\prec\,}(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}(n_{1}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $\omega$-shallow confluent up to }\delta$ and (due to $n_{0}{\,\preceq\,}n_{1}{+}1$ and $n_{0}{+_{\\!\\!{}_{\omega}}}(n_{1}{+}1){\,\prec\,}(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}(n_{1}{+}1)$) ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$ implies $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}.$ Note that for the availability of our second induction hypothesis it is important that we have imposed the restriction “$n_{0}{\,\preceq\,}n_{1}$” in opposition to the restriction “$n_{0}{\,\succeq\,}n_{1}$”. In the latter case the availability of our second induction hypothesis would require $n_{0}{+}1{\,\succeq\,}n_{1}{+}1\ {\Rightarrow}\penalty-2\ n_{0}{\,\succeq\,}n_{1}{+}1$ which is not true for $n_{0}{\,=\,}\penalty-1n_{1}.$ The additional hypothesis ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}$ of the latter restriction is useless for our proof. There are ${((l_{0,\bar{p}_{0}},r_{0,\bar{p}_{0}}),C_{0,\bar{p}_{0}})}\in{\rm R}$ and $\mu_{0,\bar{p}_{0}}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/p{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}},$ $C_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$, and $w_{0}{\,=\,}\penalty-1{u\penalty-1{[\,p\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}.$ W.l.o.g. let the positions of $\mathchar 261\relax_{1}$ be maximal in the sense that for any $p\in\mathchar 261\relax_{1}$ and $\mathchar 260\relax\subseteq{{{\mathcal{POS}}}({u})}{\cap}(p{{\bf N}}^{+})$ we do not have $u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1,(\mathchar 261\relax_{1}\setminus\\{p\\})\cup\mathchar 260\relax}}w_{1}$ anymore. Then for each $p\in\mathchar 261\relax_{1}$ there are ${((l_{1,p},r_{1,p}),C_{1,p})}\in{\rm R}$ and $\mu_{1,p}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/p{\,=\,}\penalty-1l_{1,p}\mu_{1,p},$ $r_{1,p}\mu_{1,p}{\,=\,}\penalty-1w_{1}/p,$ $C_{1,p}\mu_{1,p}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$. Finally, $w_{1}{\,=\,}\penalty-1{u\penalty-1{{[\,p\leftarrow r_{1,p}\mu_{1,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{1}\,]}}}.$ Claim 5: We may assume $l_{0,\bar{p}_{0}}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ and $\forall p{\,\in\,}\mathchar 261\relax_{1}{.}\penalty-1\,\,l_{1,p}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 5: In case of $l_{0,\bar{p}_{0}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ we get $w_{0}{{\longleftarrow}_{{}_{\\!\omega}}}u$ by Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{0,\bar{p}_{0}}$) and then our property follows from the assumption of our lemma (below the strong commutation assumption). For the second restriction define $\mathchar 260\relax_{1}:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {l_{1,p}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}{\ \\}}}$ and $u_{1}^{\prime}:={u\penalty-1{{[\,p\leftarrow r_{1,p}\mu_{1,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{1}{\setminus}\mathchar 260\relax_{1}\,]}}}$. If we have succeeded with our proof under the assumption of Claim 5, then we have shown $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}u_{1}^{\prime}$ for some $v_{1}$ (cf. diagram below). By Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{1,p}$) we get $\forall p{\,\in\,}\mathchar 260\relax_{1}{.}\penalty-1\,\,l_{1,p}\mu_{1,p}{{\longrightarrow}_{{}_{\\!\omega}}}r_{1,p}\mu_{1,p}$ and therefore $u_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{1}.$ Thus from $v_{1}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}\penalty-1u_{1}^{\prime}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\penalty-1w_{1}$ we get $v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}$ for some $v_{2}$ by $\omega$-shallow confluence up to $\omega$ (cf. Claim 0). Q.e.d. (Claim 5) Now we start a second level of induction on ${\,\|{\mathchar 261\relax_{1}}\|\,}$ in $\,\prec\,$. Define the set of top positions by $\displaystyle\mathchar 258\relax:={{\\{\ }p{\,\in\,}\\{\bar{p}_{0}\\}{\cup}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {\neg\exists q{\,\in\,}\\{\bar{p}_{0}\\}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{+}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}}.$ Since the prefix ordering is wellfounded we have $\forall p{\,\in\,}\\{\bar{p}_{0}\\}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q{\,\in\,}\mathchar 258\relax{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}.$ It now suffices to show for all $q\in\mathchar 258\relax$ $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}w_{0}/q{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}/q$ because then we have $w_{0}{\,=\,}\penalty-1{w_{0}\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{{u\penalty-1{[\,\bar{p}_{0}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}{u\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1\\\ {{u\penalty-1{{[\,p\leftarrow r_{1,p}\mu_{1,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{1}\,]}}}\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{w_{1}\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1w_{1}.$ Therefore we are left with the following two cases for $q\in\mathchar 258\relax$: $q{\,\not\in\,}\mathchar 261\relax_{1}$: Then $q{\,=\,}\penalty-1\bar{p}_{0}.$ Define $\mathchar 261\relax_{1}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{1}{\ \\}}}$. We have two cases: “The variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}{.}\penalty-1\,\,l_{0,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{0,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}{\ \\}}}.$ Claim 7: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\mu_{0,q}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}x\nu\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\nu{\,=\,}\penalty-1{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\\\ \end{array}}}\right)}}.$ Proof of Claim 7: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{0,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{0,q}&{\,=\penalty-1}&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}\\\ &&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{0,q}$ is not linear in $x$. By the conditions of our lemma and Claim 5 this implies $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Since there is some $(p^{\prime},p^{\prime\prime})\in\mathchar 256\relax(x)$ with $x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ this implies $l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then $l_{1,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which contradicts Claim 5. Q.e.d. (Claim 7) Claim 8: $l_{0,q}\nu{\,=\,}\penalty-1w_{1}/q.$ Proof of Claim 8: By Claim 7 we get $w_{1}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{0,q}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0,q}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}r_{0,q}\nu.$ Proof of Claim 9: Since $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q},$ this follows directly from Claim 7. Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $l_{0,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n_{0}+1}}}r_{0,q}\nu,$ which again follows from Lemma 13.8 since ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $(n_{1}{+}1){+_{\\!\\!{}_{\omega}}}n_{0}$ by our induction hypothesis and since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}+1}}}x\nu$ by Claim 7 and Corollary 2.14. Q.e.d. (“The variable overlap (if any) case”) “The critical peak case”: There is some $p\in\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}$ with $l_{0,q}/p{\,\not\in\,}{{\rm V}}$: Claim 10: $p{\,\not=\,}\emptyset.$ Proof of Claim 10: If $p{\,=\,}\penalty-1\emptyset,$ then $\emptyset{\,\in\,}\mathchar 261\relax_{1}^{\prime},$ then $q{\,\in\,}\mathchar 261\relax_{1},$ which contradicts our global case assumption. Q.e.d. (Claim 10) Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}\\\ x\xi^{-1}\mu_{1,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1,qp}\xi\varrho{\,=\,}\penalty-1l_{1,qp}\xi\xi^{-1}\mu_{1,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p{\,=\,}\penalty-1l_{0,q}\varrho/p{\,=\,}\penalty-1(l_{0,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1,qp}\xi},{l_{0,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{0,q}\mu_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}$. We get $\begin{array}[]{l@{}l}u^{\prime}{\,=\penalty-1}&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}\\\ &{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}\,]}}}{\,=\,}\penalty-1w_{1}/q.\end{array}$ If ${l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1r_{0,q}\sigma\varphi{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ Otherwise we have $(\,({l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma,C_{1,qp}\xi\sigma,1),\penalty-1\,(r_{0,q}\sigma,C_{0,q}\sigma,1),\penalty-1\,l_{0,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $p{\,\not=\,}\emptyset$ (due to Claim 10); $C_{1,qp}\xi\sigma\varphi=C_{1,qp}\mu_{1,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$; $C_{0,q}\sigma\varphi=C_{0,q}\mu_{0,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$. Since $\forall\delta{\,\prec\,}(n_{1}{+}1){+_{\\!\\!{}_{\omega}}}(n_{0}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $\omega$-shallow confluent up to }\delta$ (by our induction hypothesis) due to our assumed $\omega$-shallow closedness (matching the definition’s $n_{0}$ to our $n_{1}{+}1$ and its $n_{1}$ to our $n_{0}{+}1$) we have $u^{\prime}{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi\penalty-1{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}\penalty-1v_{1}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}}v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}r_{0,q}\sigma\varphi{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1w_{0}/q$ for some $v_{1}$, $v_{2}$. We then have $v_{1}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ By ${\,\|{\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}}\|\,}\prec{\,\|{\mathchar 261\relax_{1}^{\prime}}\|\,}\preceq{\,\|{\mathchar 261\relax_{1}}\|\,},$ due to our second induction level we get some $v_{1}^{\prime}$ with $v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}/q.$ Finally by our induction hypothesis that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$ the peak at $v_{1}$ can be closed according to $v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}v_{1}^{\prime}.$ Q.e.d. (“The critical peak case”) Q.e.d. (“$q{\,\not\in\,}\mathchar 261\relax_{1}$”) $q{\,\in\,}\mathchar 261\relax_{1}$: If there is no $\bar{p}_{0}^{\prime}$ with $q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1\bar{p}_{0},$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1u/q{\,=\,}\penalty-1l_{1,q}\mu_{1,q}{{\longrightarrow}_{{}_{\\!\omega+n_{1}+1}}}r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Otherwise, we can define $\bar{p}_{0}^{\prime}$ by $q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1\bar{p}_{0}.$ We have two cases: “The second variable overlap case”: There are $x{\,\in\,}{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{1,q}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{0}^{\prime}$: Claim 11a: We have $x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}$ and may assume $x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}.$ Proof of Claim 11a: We have $x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}.$ If $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}},$ then $x\mu_{1,q}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $x\mu_{1,q}/p^{\prime\prime}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ and then $l_{0,\bar{p}_{0}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which we may assume not to be the case by Claim 5. Q.e.d. (Claim 11a) Claim 11b: We can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by $x\nu{\,=\,}\penalty-1{x\mu_{1,q}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{1,q}.$ Then we have $x\mu_{1,q}{{\longrightarrow}_{{}_{\\!\omega+n_{0}+1}}}x\nu.$ Proof of Claim 11b: This follows directly from Claim 11a. Q.e.d. (Claim 11b) Claim 12: $w_{0}/q{\,=\,}\penalty-1l_{1,q}\nu.$ Proof of Claim 12: By the left-linearity assumption of our lemma, Claim 5, and Claim 11a we may assume ${{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}=\\{p^{\prime}\\}.$ Thus, by Claim 11b we get $w_{0}/q{\,=\,}\penalty-1{u/q\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1\\\ {{l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1,q}\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1\\\ {{{l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1,q}\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow x\mu_{1,q}\,]}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1\\\ {{l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow{x\mu_{1,q}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}\,]}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1,q}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1,q}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}+1}}w_{1}/q.$ Proof of Claim 13: Since $r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q,$ this follows directly from Claim 11b. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n_{1}+1}}}r_{1,q}\nu,$ which again follows from Claim 11b, Lemma 13.8 (matching its $n_{0}$ to our $n_{0}{+}1$ and its $n_{1}$ to our $n_{1}$), and our induction hypothesis that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}n_{1}.$ Q.e.d. (“The second variable overlap case”) “The second critical peak case”: $\bar{p}_{0}^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{1,q}})}\ {\wedge}\penalty-2\ l_{1,q}/\bar{p}_{0}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0,\bar{p}_{0}},r_{0,\bar{p}_{0}}),C_{0,\bar{p}_{0}})}})}]\cap{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0,\bar{p}_{0}},r_{0,\bar{p}_{0}}),C_{0,\bar{p}_{0}})}})}]\cup{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}\\\ x\xi^{-1}\mu_{0,\bar{p}_{0}}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0,\bar{p}_{0}}\xi\varrho{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\xi\xi^{-1}\mu_{0,\bar{p}_{0}}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1u/q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1,q}\varrho/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1(l_{1,q}/\bar{p}_{0}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0,\bar{p}_{0}}\xi},{l_{1,q}/\bar{p}_{0}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1{l_{1,q}\mu_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Otherwise we have $(\,({l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma,C_{0,\bar{p}_{0}}\xi\sigma,1),\penalty-1\,(r_{1,q}\sigma,C_{1,q}\sigma,1),\penalty-1\,l_{1,q}\sigma,\penalty-1\,\bar{p}_{0}^{\prime}\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $C_{0,\bar{p}_{0}}\xi\sigma\varphi=C_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$; $C_{1,q}\sigma\varphi=C_{1,q}\mu_{1,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$. Since $\forall\delta{\,\prec\,}(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}(n_{1}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $\omega$-shallow confluent up to }\delta$ (by our induction hypothesis) due to our assumed $\omega$-shallow noisy weak parallel joinability (matching the definition’s $n_{0}$ to our $n_{0}{+}1$ and its $n_{1}$ to our $n_{1}{+}1$) we have $w_{0}/q{\,=\,}\penalty-1{l_{1,q}\mu_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Q.e.d. (“The second critical peak case”) Q.e.d. (Lemma A.5) Proof of Lemma A.6 Claim 0: ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$. Proof of Claim 0: Directly by the assumed strong commutation, cf. the proofs of the claims 2 and 3 of the proof of Lemma A.1. Q.e.d. (Claim 0) Claim 1: If ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$, then ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$ and ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$ are commuting. Proof of Claim 1: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$ are commuting by Lemma 3.3. Since by Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}},$ now ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$ and ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$ are commuting, too. Q.e.d. (Claim 1) For $n_{0}\preceq n_{1}\prec\omega$ we are going to show by induction on $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}$ the following property: $w_{0}{{\longleftarrow}_{{}_{\\!\omega+n_{0}}}}u{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}.$ Claim 2: Let $\delta\prec\omega{+}\omega$. If $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&\forall w_{0},w_{1},u{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&w_{0}{{\longleftarrow}_{{}_{\\!\omega+n_{0}}}}u{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}w_{1}\\\ {\Rightarrow}&w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}},$ then $\forall n_{0},n_{1}{\,\prec\,}\omega{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&{{\left({{\begin{array}[]{ll}&n_{0}{\,\preceq\,}n_{1}\\\ {\wedge}&n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta\\\ \end{array}}}\right)}}\\\ {\Rightarrow}&{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\mbox{ strongly commutes over }{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}}\\\ \end{array}}}\right)}},$ and ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\delta$. Proof of Claim 2: By induction on $\delta$ in $\,\prec\,$. First we show the strong commutation. Assume $n_{0}\preceq n_{1}\prec\omega$ with $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta$. By Lemma 3.3 it suffices to show that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$. Assume $u^{\prime\prime}{{\longleftarrow}_{{}_{\\!\omega+n_{0}}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}u{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}$ (cf. diagram below). By the strong commutation assumed for our lemma, there are $w_{0}$ and $w_{0}^{\prime}$ with $u^{\prime\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}u.$ By the above property there are some $w_{3}$, $w_{1}^{\prime}$ with $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}.$ Next we show that we can close the peak $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}$ according to $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{2}$ for some $w_{2}^{\prime}$. In case of $n_{1}{\,=\,}\penalty-10$ this is possible due to the $\omega$-shallow confluence up to $\omega$ given by Claim 0. Otherwise we have $n_{0}{+_{\\!\\!{}_{\omega}}}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta$ and due to our induction hypothesis (saying that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to all $\delta^{\prime}\prec\delta$) this is possible again. By Claim 0 again, we can close the peak $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ according to $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{3}$ for some $w_{3}^{\prime}$. To close the whole diagram, we only have to show that we can close the peak $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{3}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}^{\prime}$ according to $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}w_{2}^{\prime}.$ In case of $n_{0}{\,=\,}\penalty-10$ this is possible due to the strong commutation assumed for our lemma. Otherwise we have $n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1{\,\prec\,}n_{0}{\,\preceq\,}n_{1}$ and $(n_{0}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){+_{\\!\\!{}_{\omega}}}n_{1}{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta,$ and then due to our induction hypothesis this is possible again. Finally we show $\omega$-shallow confluence up to $\delta$. Assume $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{\,\preceq\,}\delta$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}w_{1}.$ Due to symmetry in $n_{0}$ and $n_{1}$ we may assume $n_{0}{\,\preceq\,}n_{1}.$ Above we have shown that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}}}$. By Claim 1 we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}w_{1}$ as desired. Q.e.d. (Claim 2) Note that for $n_{0}{\,=\,}\penalty-10$ our property follows from the strong commutation assumption of our lemma. The benefit of Claim 2 is twofold: First, it says that our lemma is valid if the above property holds for all $n_{0}\preceq n_{1}\prec\omega$. Second, it strengthens the property when used as induction hypothesis. Thus (writing $n_{i}{+}1$ instead of $n_{i}$ since we may assume $0{\,\prec\,}n_{0}{\,\preceq\,}n_{1}$) it now suffices to show for $n_{0}\preceq n_{1}\prec\omega$ that $w_{0}{{\longleftarrow}_{{}_{\\!\omega+n_{0}+1,\bar{p}_{0}}}}u{{\longrightarrow}_{{}_{\\!\omega+n_{1}+1,\bar{p}_{1}}}}w_{1}$ together with our induction hypotheses that $\rule{0.0pt}{8.43889pt}\forall\delta{\,\prec\,}(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}(n_{1}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $\omega$-shallow confluent up to }\delta$ implies $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}+1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}.$ Now for each $i\prec 2$ there are ${((l_{i},r_{i}),C_{i})}\in{\rm R}$ and $\mu_{i}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/\bar{p}_{i}{\,=\,}\penalty-1l_{i}\mu_{i},$ $w_{i}{\,=\,}\penalty-1{u\penalty-1{[\,\bar{p}_{i}\leftarrow r_{i}\mu_{i}\,]}},$ and $C_{i}\mu_{i}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{i}}}$. Claim 5: We may assume $\forall i{\,\prec\,}2{.}\penalty-1\,\,l_{i}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 5: In case of $l_{i}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ we get $u{{\longrightarrow}_{{}_{\\!\omega}}}w_{i}$ by Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{i}$). In case of “$i{\,=\,}\penalty-10$” our property follows from the strong commutation assumption of our lemma. In case of “$i{\,=\,}\penalty-11$” our property follows from Claim 0. Q.e.d. (Claim 5) In case of ${{\bar{p}_{0}}\,{\parallel}\,{\bar{p}_{1}}}$ we have $w_{i}/\bar{p}_{1-i}{\,=\,}\penalty-1{u\penalty-1{[\,\bar{p}_{i}\leftarrow r_{i}\mu_{i}\,]}}/\bar{p}_{1-i}{\,=\,}\penalty-1u/\bar{p}_{1-i}{\,=\,}\penalty-1l_{1-i}\mu_{1-i}$ and therefore $w_{i}{{\longrightarrow}_{{}_{\\!\omega+n_{i}+1}}}{u\penalty-1{{[\,\bar{p}_{k}\leftarrow r_{k}\mu_{k}\ \|\ k{\,\prec\,}2\,]}}},$ i.e. our proof is finished. Thus, according to whether $\bar{p}_{0}$ is a prefix of $\bar{p}_{1}$ or vice versa, we have the following two cases left: There is some $\bar{p}_{1}^{\prime}$ with $\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1\bar{p}_{1}$ and $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ : We have two cases: “The variable overlap case”: There are $x\in{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{0}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{1}^{\prime}$: Claim 6: We have $x\mu_{0}/p^{\prime\prime}{\,=\,}\penalty-1l_{1}\mu_{1}$ and may assume $x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}.$ Proof of Claim 6: We have $x\mu_{0}/p^{\prime\prime}{\,=\,}\penalty-1l_{0}\mu_{0}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{0}p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{1}{\,=\,}\penalty-1l_{1}\mu_{1}.$ If $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}},$ then $x\mu_{0}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $x\mu_{0}/p^{\prime\prime}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $l_{1}\mu_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ and then $l_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which we may assume not to be the case by Claim 5. Q.e.d. (Claim 6) Claim 7: We can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by $x\nu{\,=\,}\penalty-1{x\mu_{0}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{0}.$ Then we have $x\mu_{0}{{\longrightarrow}_{{}_{\\!\omega+n_{1}+1}}}x\nu.$ Proof of Claim 7: This follows directly from Claim 6. Q.e.d. (Claim 7) Claim 8: $l_{0}\nu{\,=\,}\penalty-1w_{1}/\bar{p}_{0}.$ Proof of Claim 8: By the left-linearity assumption of our lemma and claims 5 and 6 we may assume ${{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}=\\{p^{\prime}\\}.$ Thus, by Claim 7 we get $w_{1}/\bar{p}_{0}{\,=\,}\penalty-1{u/\bar{p}_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow x\mu_{0}\,]}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow{x\mu_{0}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}\,]}}{\,=\,}\penalty-1\\\ {l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/\bar{p}_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}+1}}}r_{0}\nu.$ Proof of Claim 9: By the right-linearity assumption of our lemma and claims 5 and 6 we may assume ${\,\|{{{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}}\|\,}{\,\preceq\,}1.$ Thus by Claim 7 we get: $w_{0}/\bar{p}_{0}{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}+1}}}\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{\,=\,}\penalty-1\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{\,=\,}\penalty-1r_{0}\nu.$ Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $l_{0}\nu{{\longrightarrow}_{{}_{\\!\omega+n_{0}+1}}}r_{0}\nu,$ which again follows from Lemma 13.8 (matching its $n_{0}$ to our $n_{1}{+}1$ and its $n_{1}$ to our $n_{0}$) since ${\rm R},{{\rm X}}$ is quasi-normal and $\omega$-shallow confluent up to $(n_{1}{+}1){+_{\\!\\!{}_{\omega}}}n_{0}$ by our induction hypothesis, and since $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\mu_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}+1}}}y\nu$ by Claim 7. Q.e.d. (“The variable overlap case”) “The critical peak case”: $\bar{p}_{1}^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{0}})}\ {\wedge}\penalty-2\ l_{0}/\bar{p}_{1}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}]\cap{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}]\cup{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}\\\ x\xi^{-1}\mu_{1}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1}\xi\varrho{\,=\,}\penalty-1l_{1}\xi\xi^{-1}\mu_{1}{\,=\,}\penalty-1u/\bar{p}_{1}{\,=\,}\penalty-1u/\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1l_{0}\mu_{0}/\bar{p}_{1}^{\prime}{\,=\,}\penalty-1l_{0}\varrho/\bar{p}_{1}^{\prime}{\,=\,}\penalty-1(l_{0}/\bar{p}_{1}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1}\xi},{l_{0}/\bar{p}_{1}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0}\sigma,$ then the proof is finished due to $w_{0}/\bar{p}_{0}{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1r_{0}\sigma\varphi{\,=\,}\penalty-1{l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1{l_{0}\mu_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1w_{1}/\bar{p}_{0}.$ Otherwise we have $(\,({l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}},C_{1}\xi,1),\penalty-1\,(r_{0},C_{0},1),\penalty-1\,l_{0},\penalty-1\,\sigma,\penalty-1\,\bar{p}_{1}^{\prime}\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ (due the global case assumption); $C_{1}\xi\sigma\varphi=C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$; $C_{0}\sigma\varphi=C_{0}\mu_{0}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$. Since $\forall\delta{\,\prec\,}(n_{1}{+}1){+_{\\!\\!{}_{\omega}}}(n_{0}{+}1){.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\delta$ (by our induction hypothesis), due to our assumed $\omega$-shallow noisy anti- closedness (matching the definition’s $n_{0}$ to our $n_{1}{+}1$ and its $n_{1}$ to $n_{0}{+}1$) we have $w_{1}/\bar{p}_{0}{\,=\,}\penalty-1{l_{0}\mu_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1{l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{1}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{1}+1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}r_{0}\sigma\varphi{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1w_{0}/\bar{p}_{0}.$ Q.e.d. (“The critical peak case”) Q.e.d. (“There is some $\bar{p}_{1}^{\prime}$ with $\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1\bar{p}_{1}$ and $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ ”) There is some $\bar{p}_{0}^{\prime}$ with $\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1\bar{p}_{0}$ : We have two cases: “The second variable overlap case”: There are $x{\,\in\,}{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{1}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{0}^{\prime}$: Claim 11a: We have $x\mu_{1}/p^{\prime\prime}{\,=\,}\penalty-1l_{0}\mu_{0}$ and may assume $x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}.$ Proof of Claim 11a: We have $x\mu_{1}/p^{\prime\prime}{\,=\,}\penalty-1l_{1}\mu_{1}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{1}p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1l_{0}\mu_{0}.$ If $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}},$ then $x\mu_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $x\mu_{1}/p^{\prime\prime}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $l_{0}\mu_{0}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ and then $l_{0}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which we may assume not to be the case by Claim 5. Q.e.d. (Claim 11a) Claim 11b: We can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by $x\nu{\,=\,}\penalty-1{x\mu_{1}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{1}.$ Then we have $x\mu_{1}{{\longrightarrow}_{{}_{\\!\omega+n_{0}+1}}}x\nu.$ Proof of Claim 11b: This follows directly from Claim 11a. Q.e.d. (Claim 11b) Claim 12: $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1l_{1}\nu.$ Proof of Claim 12: By the left-linearity assumption of our lemma and claims 5 and 11a we may assume ${{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}=\\{p^{\prime}\\}.$ Thus, by Claim 11b we get $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{u/\bar{p}_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow x\mu_{1}\,]}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow{x\mu_{1}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}\,]}}{\,=\,}\penalty-1\\\ {l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n_{0}+1}}w_{1}/\bar{p}_{1}.$ Proof of Claim 13: Since $r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1},$ this follows directly from Claim 11b. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1}\nu{{\longrightarrow}_{{}_{\\!\omega+n_{1}+1}}}r_{1}\nu,$ which again follows from Claim 11b, Lemma 13.8 (matching its $n_{0}$ to our $n_{0}{+}1$ and its $n_{1}$ to our $n_{1}$), and our induction hypothesis that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}n_{1}.$ Q.e.d. (“The second variable overlap case”) “The second critical peak case”: $\bar{p}_{0}^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{1}})}\ {\wedge}\penalty-2\ l_{1}/\bar{p}_{0}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}]\cap{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}]\cup{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}\\\ x\xi^{-1}\mu_{0}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0}\xi\varrho{\,=\,}\penalty-1l_{0}\xi\xi^{-1}\mu_{0}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1u/\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1}\mu_{1}/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1}\varrho/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1(l_{1}/\bar{p}_{0}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0}\xi},{l_{1}/\bar{p}_{0}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1}\sigma,$ then the proof is finished due to $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1}.$ Otherwise we have $(\,({l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}},C_{0}\xi,1),\penalty-1\,(r_{1},C_{1},1),\penalty-1\,l_{1},\penalty-1\,\sigma,\penalty-1\,\bar{p}_{0}^{\prime}\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $C_{0}\xi\sigma\varphi=C_{0}\mu_{0}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{0}}}$; $C_{1}\sigma\varphi=C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$. Since $\forall\delta{\,\prec\,}(n_{0}{+}1){+_{\\!\\!{}_{\omega}}}(n_{1}{+}1){.}\penalty-1\,\,\mbox{${\rm R},{{\rm X}}$\ is $\omega$-shallow confluent up to }\delta$ (by our induction hypothesis) due to our assumed $\omega$-shallow noisy strong joinability (matching the definition’s $n_{0}$ to our $n_{0}{+}1$ and its $n_{1}$ to our $n_{1}{+}1$) we have $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}+1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1}.$ Q.e.d. (“The second critical peak case”) Q.e.d. (Lemma A.6) Proof of Lemma A.7 For each literal $L$ in $C$ we have to show that $L\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}$. Note that we already know that $L\mu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}$. If ${{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}},$ then for all $x$ in ${{\mathcal{V}}}({C})$ we have $x\mu{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then by Lemma 2.10 $x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+0}}}y\mu.$ Thus, by the disjunctive assumption of our lemma we may assume $n_{0}{\,\preceq\,}n_{1}.$ $L=(s_{0}{=}s_{1})$: We have $s_{0}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}s_{0}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}\penalty-1t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}\penalty-1s_{1}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}s_{1}\nu$ for some $t_{0}.$ By our $\omega$-level confluence up to $n_{1}$ and $n_{0}{\,\preceq\,}n_{1},$ we get some $v$ with $s_{0}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{0}$ and then (due to $v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}s_{1}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}s_{1}\nu)$ $v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}s_{1}\nu.$ $L=({{\rm Def}\>}s)$: We know the existence of $t\in{{\mathcal{GT}}({{\rm cons}})}$ with $s\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}s\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t.$ By our $\omega$-level confluence up to $n_{1}$ and $n_{0}{\,\preceq\,}n_{1},$ there is some $t^{\prime}$ with $s\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t.$ By Lemma 2.10 we get $t^{\prime}{\,\in\,}{{\mathcal{GT}}({{\rm cons}})}.$ $L=(s_{0}{\not=}s_{1})$: There exist some $t_{0},t_{1}\in{{\mathcal{GT}}({{\rm cons}})}$ with $\forall i{\,\prec\,}2{.}\penalty-1\,\,s_{i}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{0}}}}s_{i}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{i}$ and $t_{0}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}.$ Just like above we get $t_{0}^{\prime},\ t_{1}^{\prime}\in{{\mathcal{GT}}({{\rm cons}})}$ with $\forall i{\,\prec\,}2{.}\penalty-1\,\,s_{i}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}\penalty-1t_{i}^{\prime}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{i}.$ Finally $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{0}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}^{\prime}$ implies $t_{0}^{\prime}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{{\rm R},{{\rm X}}},\omega+n_{1}}}}t_{1}^{\prime}.$ Q.e.d. (Lemma A.7) Proof of Lemma A.8 Claim 0: ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$. Proof of Claim 0: Directly by the assumed strong commutation, cf. the proofs of the claims 2 and 3 of the proof of Lemma A.1. Q.e.d. (Claim 0) Claim 1: If ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}$, then ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega+n}}$ are commuting. Proof of Claim 1: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}$ are commuting by Lemma 3.3. Since by Corollary 2.14 and Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!\omega+n}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}},$ now ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega+n}}$ are commuting, too. Q.e.d. (Claim 1) For $n\prec\omega$ we are going to show by induction on $n$ the following property: $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}.$ Claim 2: Let $\delta\prec\omega$. If $\forall n{\,\preceq\,}\delta{.}\penalty-1\,\,\forall w_{0},w_{1},u{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}w_{1}\\\ {\Rightarrow}&w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}\\\ \end{array}}}\right)}},$ then $\forall n{\,\preceq\,}\delta{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\mbox{ strongly commutes over }{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}\end{array}\right)},$ and ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $\delta$. Proof of Claim 2: First we show the strong commutation. Assume $n{\,\preceq\,}\delta$. By Lemma 3.3 it suffices to show that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!\omega+n}}$. Assume $u^{\prime\prime}{{\longleftarrow}_{{}_{\\!\omega+n}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}$ (cf. diagram below). By the strong commutation assumed for our lemma and Corollary 2.14, there are $w_{0}$ and $w_{0}^{\prime}$ with $u^{\prime\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n}}u.$ By the above property there are some $w_{3}$, $w_{1}^{\prime}$ with $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}.$ By Claim 0 we can close the peak $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}$ according to $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{2}$ for some $w_{2}^{\prime}$. By Claim 0 again, we can close the peak $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ according to $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ for some $w_{3}^{\prime}$. To close the whole diagram, we only have to show that we can close the peak $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{3}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime}$ according to $w_{3}^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime},$ which is possible due to the strong commutation assumed for our lemma. Finally we show $\omega$-level confluence up to $\delta$. Assume $n_{0},n_{1}\prec\omega$ with ${{\rm max}\\{{n_{0}},{n_{1}}\\}}{\,\preceq\,}\delta$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}w_{1}.$ By Lemma 2.12 we get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}w_{1}.$ Since ${{\rm max}\\{{n_{0}},{n_{1}}\\}}{\,\preceq\,}\delta,$ above we have shown that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}$. By Claim 1 we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}w_{1}$ as desired. Q.e.d. (Claim 2) Note that for $n{\,=\,}\penalty-10$ our property follows from ${{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}$ (by Corollary 2.14) and Claim 0. The benefit of Claim 2 is twofold: First, it says that our lemma is valid if the above property holds for all $n\prec\omega$. Second, it strengthens the property when used as induction hypothesis. Thus (writing $n{+}1$ instead of $n$ since we may assume $0{\,\prec\,}n$) it now suffices to show for $n\prec\omega$ that $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{0}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}}}w_{1}$ together with our induction hypotheses that $\rule{0.0pt}{8.43889pt}\mbox{${\rm R},{{\rm X}}$\ is $\omega$-level confluent up to }n$ implies $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{1}+1}}}w_{1}.$ W.l.o.g. let the positions of $\mathchar 261\relax_{i}$ be maximal in the sense that for any $p\in\mathchar 261\relax_{i}$ and $\mathchar 260\relax\subseteq{{{\mathcal{POS}}}({u})}{\cap}(p{{\bf N}}^{+})$ we do not have $u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,(\mathchar 261\relax_{i}\setminus\\{p\\})\cup\mathchar 260\relax}}w_{i}$ anymore. Then for each $i\prec 2$ and $p\in\mathchar 261\relax_{i}$ there are ${((l_{i,p},r_{i,p}),C_{i,p})}\in{\rm R}$ and $\mu_{i,p}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/p{\,=\,}\penalty-1l_{i,p}\mu_{i,p},$ $r_{i,p}\mu_{i,p}{\,=\,}\penalty-1w_{i}/p,$ $C_{i,p}\mu_{i,p}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Finally, for each $i\prec 2$: $w_{i}{\,=\,}\penalty-1{u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}\,]}}}.$ Claim 5: We may assume $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,l_{i,p}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 5: Define $\mathchar 260\relax_{i}:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{i}}~{}{\|}\penalty-9\,\ {l_{i,p}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}{\ \\}}}$ and $u_{i}^{\prime}:={u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}{\setminus}\mathchar 260\relax_{i}\,]}}}$. If we have succeeded with our proof under the assumption of Claim 5, then we have shown $u_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}u_{1}^{\prime}$ for some $v_{0}$, $v_{1}$ (cf. diagram below). By Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{i,p}$) we get $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 260\relax_{i}{.}\penalty-1\,\,l_{i,p}\mu_{i,p}{{\longrightarrow}_{{}_{\\!\omega}}}r_{i,p}\mu_{i,p}$ and therefore $\forall i{\,\prec\,}2{.}\penalty-1\,\,u_{i}^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega,\mathchar 260\relax_{i}}}w_{i}.$ Thus from $v_{1}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}\penalty-1u_{1}^{\prime}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\penalty-1w_{1}$ we get $v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}$ for some $v_{2}$ by $\omega$-shallow confluence up to $\omega$ (cf. Claim 0). For the same reason we can close the peak $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}u_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{0}$ according to $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{0}$ for some $v_{0}^{\prime}$. By the assumption of our lemma that ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}$, from $v_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{2}$ we can finally conclude $v_{0}^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{2}.$ Q.e.d. (Claim 5) Define the set of inner overlapping positions by $\displaystyle\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1}):=\bigcup_{i\prec 2}{{\\{\ }p{\,\in\,}\mathchar 261\relax_{1-i}}~{}{\|}\penalty-9\,\ {\exists q{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}},$ and the length of a term by $\lambda({{f}{(}{t_{0}}{,\,}\ldots{,\,}{t_{m-1}}{)}}):=1+\sum_{j\prec m}\lambda(t_{j}).$ Now we start a second level of induction on $\displaystyle\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})$ in $\,\prec\,$. Define the set of top positions by $\displaystyle\mathchar 258\relax:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{0}{\cup}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {\neg\exists q{\,\in\,}\mathchar 261\relax_{0}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{+}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}}.$ Since the prefix ordering is wellfounded we have $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,\exists q{\,\in\,}\mathchar 258\relax{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}.$ Then $\forall i{\,\prec\,}2{.}\penalty-1\,\,w_{i}{\,=\,}\penalty-1{w_{i}\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{{u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}\,]}}}\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}.$ Thus, it now suffices to show for all $q\in\mathchar 258\relax$ $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}w_{0}/q{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{1}/q$ because then we have $w_{0}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}{u\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1w_{1}.$ Therefore we are left with the following two cases for $q\in\mathchar 258\relax$: $q{\,\not\in\,}\mathchar 261\relax_{1}$: Then $q{\,\in\,}\mathchar 261\relax_{0}.$ Define $\mathchar 261\relax_{1}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{1}{\ \\}}}$. We have two cases: “The variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}{.}\penalty-1\,\,l_{0,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{0,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}{\ \\}}}.$ Claim 7: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\mu_{0,q}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}x\nu\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\nu{\,=\,}\penalty-1{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\\\ \end{array}}}\right)}}.$ Proof of Claim 7: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{0,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{0,q}&{\,=\penalty-1}&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}\\\ &&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{0,q}$ is not linear in $x$. By the conditions of our lemma and Claim 5 this implies $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Since there is some $(p^{\prime},p^{\prime\prime})\in\mathchar 256\relax(x)$ with $x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ this implies $l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then $l_{1,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which contradicts Claim 5. Q.e.d. (Claim 7) Claim 8: $l_{0,q}\nu{\,=\,}\penalty-1w_{1}/q.$ Proof of Claim 8: By Claim 7 we get $w_{1}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{0,q}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0,q}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}r_{0,q}\nu.$ Proof of Claim 9: Since $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q},$ this follows directly from Claim 7. Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $l_{0,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{0,q}\nu,$ which again follows from Lemma A.7 (matching its $n_{0}$ to our $n{+}1$ and its $n_{1}$ to our $n$) since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ by our induction hypothesis and since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}x\nu$ by Claim 7 and Corollary 2.14. Q.e.d. (“The variable overlap (if any) case”) “The critical peak case”: There is some $p\in\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}$ with $l_{0,q}/p{\,\not\in\,}{{\rm V}}$: Claim 10: $p{\,\not=\,}\emptyset.$ Proof of Claim 10: If $p{\,=\,}\penalty-1\emptyset,$ then $\emptyset{\,\in\,}\mathchar 261\relax_{1}^{\prime},$ then $q{\,\in\,}\mathchar 261\relax_{1},$ which contradicts our global case assumption. Q.e.d. (Claim 10) Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}\\\ x\xi^{-1}\mu_{1,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1,qp}\xi\varrho{\,=\,}\penalty-1l_{1,qp}\xi\xi^{-1}\mu_{1,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p{\,=\,}\penalty-1l_{0,q}\varrho/p{\,=\,}\penalty-1(l_{0,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1,qp}\xi},{l_{0,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{0,q}\mu_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}$. We get $\begin{array}[]{l@{}l}u^{\prime}{\,=\penalty-1}&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}\\\ &{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}\,]}}}{\,=\,}\penalty-1w_{1}/q.\end{array}$ If ${l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1r_{0,q}\sigma\varphi{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ Otherwise we have $(\,({l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma,C_{1,qp}\xi\sigma,1),\penalty-1\,(r_{0,q}\sigma,C_{0,q}\sigma,1),\penalty-1\,l_{0,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $p{\,\not=\,}\emptyset$ (due to Claim 10); $C_{1,qp}\xi\sigma\varphi=C_{1,qp}\mu_{1,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $C_{0,q}\sigma\varphi=C_{0,q}\mu_{0,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ (by our induction hypothesis) and $\omega$-shallow confluent up to $\omega$ (by Claim 0) due to our assumed $\omega$-level parallel closedness (matching the definition’s $n$ to our $n{+}1$) we have $u^{\prime}{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi\penalty-1{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}\penalty-1v_{1}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}r_{0,q}\sigma\varphi{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1w_{0}/q$ for some $v_{1}$, $v_{2}$. We then have $v_{1}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax^{\prime\prime}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q$ for some $\mathchar 261\relax^{\prime\prime}$. By $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 266\relax(\mathchar 261\relax^{\prime\prime},\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\})}\lambda(u^{\prime}/p^{\prime\prime})\ \ \preceq\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}\lambda(u^{\prime}/p^{\prime\prime})\ \ =\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}\lambda(u/qp^{\prime\prime})\ \ \prec$ $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}}\lambda(u/qp^{\prime\prime})\ \ =\sum_{p^{\prime}\in q\mathchar 261\relax_{1}^{\prime}}\lambda(u/p^{\prime})\ \ =\sum_{p^{\prime}\in\mathchar 266\relax(\\{q\\},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})\ \ \preceq\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime}),$ due to our second induction level we get some $v_{1}^{\prime}$, $v_{3}$ with $v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{3}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{1}/q.$ By Claim 0 we can close the peak at $v_{1}$ according to $v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{4}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{3}$ for some $v_{4}$. Finally by the assumption of our lemma that ${{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle{{\rm R},{{\rm X}}},\omega+n_{1}+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}$, the peak at $v_{3}$ can be closed according to $v_{4}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{1}^{\prime}.$ Q.e.d. (“The critical peak case”) Q.e.d. (“$q{\,\not\in\,}\mathchar 261\relax_{1}$”) $q{\,\in\,}\mathchar 261\relax_{1}$: Define $\mathchar 261\relax_{0}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{0}{\ \\}}}$. We have two cases: “The second variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{0}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{1,q}})}{.}\penalty-1\,\,l_{1,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{1,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}{\ \\}}}.$ Claim 11: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n+1}}x\mu_{1,q}\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu\\\ \end{array}}}\right)}}.$ Proof of Claim 11: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{1,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{1,q}&{\,=\penalty-1}&{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}\\\ &&{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{1,q}$ is not linear in $x$. By the conditions of our lemma and Claim 5 this implies $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Since there is some $(p^{\prime},p^{\prime\prime})\in\mathchar 256\relax(x)$ with $x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}$ this implies $l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then $l_{0,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which contradicts Claim 5. Q.e.d. (Claim 11) Claim 12: $w_{0}/q{\,=\,}\penalty-1l_{1,q}\nu.$ Proof of Claim 12: By Claim 11 we get $w_{0}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{1,q}\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1,q}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1,q}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n+1}}w_{1}/q.$ Proof of Claim 13: Since $r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q,$ this follows directly from Claim 11. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{1,q}\nu,$ which again follows from Lemma A.7 (matching its $n_{0}$ to our $n{+}1$ and its $n_{1}$ to our $n$) since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ by our induction hypothesis and since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{1,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}x\nu$ by Claim 11 and Corollary 2.14. Q.e.d. (“The second variable overlap (if any) case”) “The second critical peak case”: There is some $p\in\mathchar 261\relax_{0}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{1,q}})}$ with $l_{1,q}/p{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0,qp},r_{0,qp}),C_{0,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0,qp},r_{0,qp}),C_{0,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}\\\ x\xi^{-1}\mu_{0,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0,qp}\xi\varrho{\,=\,}\penalty-1l_{0,qp}\xi\xi^{-1}\mu_{0,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p{\,=\,}\penalty-1l_{1,q}\varrho/p{\,=\,}\penalty-1(l_{1,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0,qp}\xi},{l_{1,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{1,q}\mu_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\mu_{0,qp}\,]}}$. We get $\begin{array}[]{l@{}l@{}l}w_{0}/q&{\,=\penalty-1}&{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{0,qp^{\prime}}\mu_{0,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{0}^{\prime}\,]}}}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}\\\ &&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{0,qp^{\prime}}\mu_{0,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{0}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{0,qp}\mu_{0,qp}\,]}}{\,=\,}\penalty-1u^{\prime}.\end{array}$ If ${l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1,q}\sigma,$ then the proof is finished due to $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}u^{\prime}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Otherwise we have $(\,({l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma,C_{0,qp}\xi\sigma,1),\penalty-1\,(r_{1,q}\sigma,C_{1,q}\sigma,1),\penalty-1\,l_{1,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $C_{0,qp}\xi\sigma\varphi=C_{0,qp}\mu_{0,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $C_{1,q}\sigma\varphi=C_{1,q}\mu_{1,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ (by our induction hypothesis) and $\omega$-shallow confluent up to $\omega$ (by Claim 0) due to our assumed $\omega$-level parallel joinability (matching the definition’s $n$ to our $n{+}1$) we have $u^{\prime}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma\varphi{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}\penalty-1v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\penalty-1v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q$ for some $v_{1}$, $v_{2}$. We then have $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax^{\prime\prime}}}v_{1}$ for some $\mathchar 261\relax^{\prime\prime}$. Since $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\},\mathchar 261\relax^{\prime\prime})}\lambda(u^{\prime}/p^{\prime\prime})\ \ \preceq\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}\lambda(u^{\prime}/p^{\prime\prime})\ \ =\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}\lambda(u/qp^{\prime\prime})\ \ \prec\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}}\lambda(u/qp^{\prime\prime})\ \ =\sum_{p^{\prime}\in q\mathchar 261\relax_{0}^{\prime}}\lambda(u/p^{\prime})\ \ =\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\\{q\\})}\lambda(u/p^{\prime})\ \ \preceq\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})$ due to our second induction level we get some $v_{1}^{\prime}$ with $w_{0}/q{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}v_{1}.$ Finally the peak at $v_{1}$ can be closed according to $v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}v_{2}$ by Claim 0. Q.e.d. (“The second critical peak case”) Q.e.d. (Lemma A.8) Proof of Lemma A.9 Claim 0: ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$. Proof of Claim 0: Directly by the assumed strong commutation, cf. the proofs of the claims 2 and 3 of the proof of Lemma A.1. Q.e.d. (Claim 0) Claim 1: If ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}$, then ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega+n}}$ are commuting. Proof of Claim 1: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}$ are commuting by Lemma 3.3. Since by Corollary 2.14 and Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!\omega+n}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}},$ now ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega+n}}$ are commuting, too. Q.e.d. (Claim 1) For $n\prec\omega$ we are going to show by induction on $n$ the following property: $w_{0}{{\longleftarrow}_{{}_{\\!\omega+n}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}.$ Claim 2: Let $\delta\prec\omega$. If $\forall n{\,\preceq\,}\delta{.}\penalty-1\,\,\forall w_{0},w_{1},u{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&w_{0}{{\longleftarrow}_{{}_{\\!\omega+n}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}w_{1}\\\ {\Rightarrow}&w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}\\\ \end{array}}}\right)}},$ then $\forall n{\,\preceq\,}\delta{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\mbox{ strongly commutes over }{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}\end{array}\right)},$ and ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $\delta$. Proof of Claim 2: First we show the strong commutation. Assume $n{\,\preceq\,}\delta$. By Lemma 3.3 it suffices to show that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!\omega+n}}$. Assume $u^{\prime\prime}{{\longleftarrow}_{{}_{\\!\omega+n}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}$ (cf. diagram below). By the strong commutation assumed for our lemma, there are $w_{0}$ and $w_{0}^{\prime}$ with $u^{\prime\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}u.$ By the above property there are some $w_{3}$, $w_{1}^{\prime}$ with $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}.$ By Claim 0 we can close the peak $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}$ according to $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{2}$ for some $w_{2}^{\prime}$. By Claim 0 again, we can close the peak $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ according to $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ for some $w_{3}^{\prime}$. To close the whole diagram, we only have to show that we can close the peak $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{3}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime}$ according to $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime},$ which is possible since it is assumed for our lemma (below the strong commutation assumption). Finally we show $\omega$-level confluence up to $\delta$. Assume $n_{0},n_{1}\prec\omega$ with ${{\rm max}\\{{n_{0}},{n_{1}}\\}}{\,\preceq\,}\delta$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}w_{1}.$ By Lemma 2.12 we get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}w_{1}.$ Since ${{\rm max}\\{{n_{0}},{n_{1}}\\}}{\,\preceq\,}\delta,$ above we have shown that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}$. By Claim 1 we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}w_{1}$ as desired. Q.e.d. (Claim 2) Note that for $n{\,=\,}\penalty-10$ our property follows from Corollary 2.14 and Claim 0. The benefit of Claim 2 is twofold: First, it says that our lemma is valid if the above property holds for all $n\prec\omega$. Second, it strengthens the property when used as induction hypothesis. Thus (writing $n{+}1$ instead of $n$ since we may assume $0{\,\prec\,}n$) it now suffices to show for $n\prec\omega$ that $w_{0}{{\longleftarrow}_{{}_{\\!\omega+n+1,\bar{p}_{0}}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}}}w_{1}$ together with our induction hypotheses that $\rule{0.0pt}{8.43889pt}\mbox{${\rm R},{{\rm X}}$\ is $\omega$-level confluent up to }n$ implies $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{1}.$ There are ${((l_{0,\bar{p}_{0}},r_{0,\bar{p}_{0}}),C_{0,\bar{p}_{0}})}\in{\rm R}$ and $\mu_{0,\bar{p}_{0}}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/p{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}},$ $C_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$, and $w_{0}{\,=\,}\penalty-1{u\penalty-1{[\,p\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}.$ W.l.o.g. let the positions of $\mathchar 261\relax_{1}$ be maximal in the sense that for any $p\in\mathchar 261\relax_{1}$ and $\mathchar 260\relax\subseteq{{{\mathcal{POS}}}({u})}{\cap}(p{{\bf N}}^{+})$ we do not have $u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,(\mathchar 261\relax_{1}\setminus\\{p\\})\cup\mathchar 260\relax}}w_{1}$ anymore. Then for each $p\in\mathchar 261\relax_{1}$ there are ${((l_{1,p},r_{1,p}),C_{1,p})}\in{\rm R}$ and $\mu_{1,p}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/p{\,=\,}\penalty-1l_{1,p}\mu_{1,p},$ $r_{1,p}\mu_{1,p}{\,=\,}\penalty-1w_{1}/p,$ $C_{1,p}\mu_{1,p}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Finally, $w_{1}{\,=\,}\penalty-1{u\penalty-1{{[\,p\leftarrow r_{1,p}\mu_{1,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{1}\,]}}}.$ Claim 5: We may assume $l_{0,\bar{p}_{0}}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ and $\forall p{\,\in\,}\mathchar 261\relax_{1}{.}\penalty-1\,\,l_{1,p}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 5: In case of $l_{0,\bar{p}_{0}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ we get $w_{0}{{\longleftarrow}_{{}_{\\!\omega}}}u$ by Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{0,\bar{p}_{0}}$) and then our property follows from the assumption of our lemma (below the strong commutation assumption). For the second restriction define $\mathchar 260\relax_{1}:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {l_{1,p}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}{\ \\}}}$ and $u_{1}^{\prime}:={u\penalty-1{{[\,p\leftarrow r_{1,p}\mu_{1,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{1}{\setminus}\mathchar 260\relax_{1}\,]}}}$. If we have succeeded with our proof under the assumption of Claim 5, then we have shown $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}u_{1}^{\prime}$ for some $v_{1}$ (cf. diagram below). By Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{1,p}$) we get $\forall p{\,\in\,}\mathchar 260\relax_{1}{.}\penalty-1\,\,l_{1,p}\mu_{1,p}{{\longrightarrow}_{{}_{\\!\omega}}}r_{1,p}\mu_{1,p}$ and therefore $u_{1}^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega,\mathchar 260\relax_{1}}}w_{1}.$ Thus from $v_{1}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}\penalty-1u_{1}^{\prime}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\penalty-1w_{1}$ we get $v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}+1}}}w_{1}$ for some $v_{2}$ by $\omega$-shallow confluence up to $\omega$ (cf. Claim 0). Q.e.d. (Claim 5) Now we start a second level of induction on ${\,\|{\mathchar 261\relax_{1}}\|\,}$ in $\,\prec\,$. Define the set of top positions by $\displaystyle\mathchar 258\relax:={{\\{\ }p{\,\in\,}\\{\bar{p}_{0}\\}{\cup}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {\neg\exists q{\,\in\,}\\{\bar{p}_{0}\\}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{+}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}}.$ Since the prefix ordering is wellfounded we have $\forall p{\,\in\,}\\{\bar{p}_{0}\\}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q{\,\in\,}\mathchar 258\relax{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}.$ It now suffices to show for all $q\in\mathchar 258\relax$ $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}w_{0}/q{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{1}/q$ because then we have $w_{0}{\,=\,}\penalty-1{w_{0}\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{{u\penalty-1{[\,\bar{p}_{0}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}{u\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1\\\ {{u\penalty-1{{[\,p\leftarrow r_{1,p}\mu_{1,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{1}\,]}}}\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{w_{1}\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1w_{1}.$ Therefore we are left with the following two cases for $q\in\mathchar 258\relax$: $q{\,\not\in\,}\mathchar 261\relax_{1}$: Then $q{\,=\,}\penalty-1\bar{p}_{0}.$ Define $\mathchar 261\relax_{1}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{1}{\ \\}}}$. We have two cases: “The variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}{.}\penalty-1\,\,l_{0,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{0,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}{\ \\}}}.$ Claim 7: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\mu_{0,q}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}x\nu\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\nu{\,=\,}\penalty-1{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\\\ \end{array}}}\right)}}.$ Proof of Claim 7: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{0,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{0,q}&{\,=\penalty-1}&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}\\\ &&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{0,q}$ is not linear in $x$. By the conditions of our lemma and Claim 5 this implies $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Since there is some $(p^{\prime},p^{\prime\prime})\in\mathchar 256\relax(x)$ with $x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ this implies $l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then $l_{1,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which contradicts Claim 5. Q.e.d. (Claim 7) Claim 8: $l_{0,q}\nu{\,=\,}\penalty-1w_{1}/q.$ Proof of Claim 8: By Claim 7 we get $w_{1}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{0,q}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0,q}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/q{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}r_{0,q}\nu.$ Proof of Claim 9: Since $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q},$ this follows directly from Claim 7. Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $l_{0,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{0,q}\nu,$ which again follows from Lemma A.7 (matching its $n_{0}$ to our $n{+}1$ and its $n_{1}$ to our $n$) since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ by our induction hypothesis and since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{0,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}x\nu$ by Claim 7 and Corollary 2.14. Q.e.d. (“The variable overlap (if any) case”) “The critical peak case”: There is some $p\in\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}$ with $l_{0,q}/p{\,\not\in\,}{{\rm V}}$: Claim 10: $p{\,\not=\,}\emptyset.$ Proof of Claim 10: If $p{\,=\,}\penalty-1\emptyset,$ then $\emptyset{\,\in\,}\mathchar 261\relax_{1}^{\prime},$ then $q{\,\in\,}\mathchar 261\relax_{1},$ which contradicts our global case assumption. Q.e.d. (Claim 10) Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}\\\ x\xi^{-1}\mu_{1,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1,qp}\xi\varrho{\,=\,}\penalty-1l_{1,qp}\xi\xi^{-1}\mu_{1,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p{\,=\,}\penalty-1l_{0,q}\varrho/p{\,=\,}\penalty-1(l_{0,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1,qp}\xi},{l_{0,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{0,q}\mu_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}$. We get $\begin{array}[]{l@{}l}u^{\prime}{\,=\penalty-1}&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}\\\ &{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}\,]}}}{\,=\,}\penalty-1w_{1}/q.\end{array}$ If ${l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1r_{0,q}\sigma\varphi{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ Otherwise we have $(\,({l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma,C_{1,qp}\xi\sigma,1),\penalty-1\,(r_{0,q}\sigma,C_{0,q}\sigma,1),\penalty-1\,l_{0,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $p{\,\not=\,}\emptyset$ (due to Claim 10); $C_{1,qp}\xi\sigma\varphi=C_{1,qp}\mu_{1,qp}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $C_{0,q}\sigma\varphi=C_{0,q}\mu_{0,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ (by our induction hypothesis) and $\omega$-shallow confluent up to $\omega$ (by Claim 0) due to our assumed $\omega$-level closedness (matching the definition’s $n$ to our $n{+}1$) we have $u^{\prime}{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi\penalty-1{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}\penalty-1v_{1}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}r_{0,q}\sigma\varphi{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1w_{0}/q$ for some $v_{1}$, $v_{2}$. We then have $v_{1}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ By ${\,\|{\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}}\|\,}\prec{\,\|{\mathchar 261\relax_{1}^{\prime}}\|\,}\preceq{\,\|{\mathchar 261\relax_{1}}\|\,},$ due to our second induction level we get some $v_{1}^{\prime}$ with $v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{1}/q.$ By Claim 0 we can close the peak at $v_{1}$ according to $v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{4}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{3}$ for some $v_{4}$. Finally by the assumption of our lemma (below the strong commutation assumption) the peak at $v_{3}$ can be closed according to $v_{4}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}v_{1}^{\prime}.$ Q.e.d. (“The critical peak case”) Q.e.d. (“$q{\,\not\in\,}\mathchar 261\relax_{1}$”) $q{\,\in\,}\mathchar 261\relax_{1}$: If there is no $\bar{p}_{0}^{\prime}$ with $q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1\bar{p}_{0},$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1u/q{\,=\,}\penalty-1l_{1,q}\mu_{1,q}{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Otherwise, we can define $\bar{p}_{0}^{\prime}$ by $q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1\bar{p}_{0}.$ We have two cases: “The second variable overlap case”: There are $x{\,\in\,}{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{1,q}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{0}^{\prime}$: Claim 11a: We have $x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}$ and may assume $x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}.$ Proof of Claim 11a: We have $x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}.$ If $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}},$ then $x\mu_{1,q}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $x\mu_{1,q}/p^{\prime\prime}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $l_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ and then $l_{0,\bar{p}_{0}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which we may assume not to be the case by Claim 5. Q.e.d. (Claim 11a) Claim 11b: We can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by $x\nu{\,=\,}\penalty-1{x\mu_{1,q}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{1,q}.$ Then we have $x\mu_{1,q}{{\longrightarrow}_{{}_{\\!\omega+n+1}}}x\nu.$ Proof of Claim 11b: This follows directly from Claim 11a. Q.e.d. (Claim 11b) Claim 12: $w_{0}/q{\,=\,}\penalty-1l_{1,q}\nu.$ Proof of Claim 12: By the left-linearity assumption of our lemma, Claim 5, and Claim 11a we may assume ${{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}=\\{p^{\prime}\\}.$ Thus, by Claim 11b we get $w_{0}/q{\,=\,}\penalty-1{u/q\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1\\\ {{l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1,q}\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1\\\ {{{l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1,q}\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow x\mu_{1,q}\,]}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1\\\ {{l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow{x\mu_{1,q}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}\,]}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1,q}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1,q}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1,q}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n+1}}w_{1}/q.$ Proof of Claim 13: Since $r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q,$ this follows directly from Claim 11b. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1,q}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{1,q}\nu,$ which again follows from Lemma A.7 (matching its $n_{0}$ to our $n{+}1$ and its $n_{1}$ to our $n$) since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ by our induction hypothesis and since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{1,q}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}x\nu$ by Claim 11b. Q.e.d. (“The second variable overlap case”) “The second critical peak case”: $\bar{p}_{0}^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{1,q}})}\ {\wedge}\penalty-2\ l_{1,q}/\bar{p}_{0}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0,\bar{p}_{0}},r_{0,\bar{p}_{0}}),C_{0,\bar{p}_{0}})}})}]\cap{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0,\bar{p}_{0}},r_{0,\bar{p}_{0}}),C_{0,\bar{p}_{0}})}})}]\cup{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}\\\ x\xi^{-1}\mu_{0,\bar{p}_{0}}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0,\bar{p}_{0}}\xi\varrho{\,=\,}\penalty-1l_{0,\bar{p}_{0}}\xi\xi^{-1}\mu_{0,\bar{p}_{0}}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1u/q\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1,q}\varrho/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1(l_{1,q}/\bar{p}_{0}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0,\bar{p}_{0}}\xi},{l_{1,q}/\bar{p}_{0}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1{l_{1,q}\mu_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Otherwise we have $(\,({l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma,C_{0,\bar{p}_{0}}\xi\sigma,1),\penalty-1\,(r_{1,q}\sigma,C_{1,q}\sigma,1),\penalty-1\,l_{1,q}\sigma,\penalty-1\,\bar{p}_{0}^{\prime}\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $C_{0,\bar{p}_{0}}\xi\sigma\varphi=C_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $C_{1,q}\sigma\varphi=C_{1,q}\mu_{1,q}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ (by our induction hypothesis) and $\omega$-shallow confluent up to $\omega$ (by Claim 0) due to our assumed $\omega$-level weak parallel joinability (matching the definition’s $n$ to our $n{+}1$) we have $w_{0}/q{\,=\,}\penalty-1{l_{1,q}\mu_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\mu_{0,\bar{p}_{0}}\,]}}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0,\bar{p}_{0}}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+n+1}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Q.e.d. (“The second critical peak case”) Q.e.d. (Lemma A.9) Proof of Lemma A.10 Claim 0: ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\omega$. Proof of Claim 0: Directly by the assumed strong commutation, cf. the proofs of the claims 2 and 3 of the proof of Lemma A.1. Q.e.d. (Claim 0) Claim 1: If ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}$, then ${\longrightarrow}_{{}_{\\!\omega+n}}$ is confluent. Proof of Claim 1: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}$ are commuting by Lemma 3.3. Since by Lemma 2.12 we have ${{\longrightarrow}_{{}_{\\!\omega+n}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}},$ now ${\longrightarrow}_{{}_{\\!\omega+n}}$ and ${\longrightarrow}_{{}_{\\!\omega+n}}$ are commuting, too. Q.e.d. (Claim 1) For $n\prec\omega$ we are going to show by induction on $n$ the following property: $w_{0}{{\longleftarrow}_{{}_{\\!\omega+n}}}u{{\longrightarrow}_{{}_{\\!\omega+n}}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}.$ Claim 2: Let $\delta\prec\omega$. If $\forall n{\,\preceq\,}\delta{.}\penalty-1\,\,\forall w_{0},w_{1},u{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&w_{0}{{\longleftarrow}_{{}_{\\!\omega+n}}}u{{\longrightarrow}_{{}_{\\!\omega+n}}}w_{1}\\\ {\Rightarrow}&w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}\\\ \end{array}}}\right)}}\end{array}\right)},$ then $\forall n{\,\preceq\,}\delta{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\mbox{ strongly commutes over }{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}\end{array}\right)},$ and ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $\delta$. Proof of Claim 2: First we show the strong commutation. Assume $n{\,\preceq\,}\delta$. By Lemma 3.3 it suffices to show that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${\longrightarrow}_{{}_{\\!\omega+n}}$. Assume $u^{\prime\prime}{{\longleftarrow}_{{}_{\\!\omega+n}}}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}u{{\longrightarrow}_{{}_{\\!\omega+n}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}$ (cf. diagram below). By the strong commutation assumed for our lemma, there are $w_{0}$ and $w_{0}^{\prime}$ with $u^{\prime\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}u.$ By the above property there are some $w_{3}$, $w_{1}^{\prime}$ with $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}.$ By Claim 0 we can close the peak $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}$ according to $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n}}}w_{2}$ for some $w_{2}^{\prime}$. By Claim 0 again, we can close the peak $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ according to $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ for some $w_{3}^{\prime}$. To close the whole diagram, we only have to show that we can close the peak $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{3}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime}$ according to $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime},$ which is possible due to the strong commutation assumed for our lemma or due to Claim 0. Finally we show $\omega$-level confluence up to $\delta$. Assume $n_{0},n_{1}\prec\omega$ with ${{\rm max}\\{{n_{0}},{n_{1}}\\}}{\,\preceq\,}\delta$ and $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n_{0}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n_{1}}}}w_{1}.$ By Lemma 2.12 we get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}w_{1}.$ Since ${{\rm max}\\{{n_{0}},{n_{1}}\\}}{\,\preceq\,}\delta,$ above we have shown that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{\longrightarrow}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}$. By Claim 1 we finally get $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+{{\rm max}\\{{n_{0}},{n_{1}}\\}}}}}w_{1}$ as desired. Q.e.d. (Claim 2) Note that for $n{\,=\,}\penalty-10$ our property follows from Claim 0. The benefit of Claim 2 is twofold: First, it says that our lemma is valid if the above property holds for all $n\prec\omega$. Second, it strengthens the property when used as induction hypothesis. Thus (writing $n{+}1$ instead of $n$ since we may assume $0{\,\prec\,}n$) it now suffices to show for $n\prec\omega$ that $w_{0}{{\longleftarrow}_{{}_{\\!\omega+n+1,\bar{p}_{0}}}}u{{\longrightarrow}_{{}_{\\!\omega+n+1,\bar{p}_{1}}}}w_{1}$ together with our induction hypotheses that $\rule{0.0pt}{8.43889pt}\mbox{${\rm R},{{\rm X}}$\ is $\omega$-level confluent up to }n$ implies $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{1}.$ Now for each $i\prec 2$ there are ${((l_{i},r_{i}),C_{i})}\in{\rm R}$ and $\mu_{i}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/\bar{p}_{i}{\,=\,}\penalty-1l_{i}\mu_{i},$ $w_{i}{\,=\,}\penalty-1{u\penalty-1{[\,\bar{p}_{i}\leftarrow r_{i}\mu_{i}\,]}},$ and $C_{i}\mu_{i}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Claim 5: We may assume $\forall i{\,\prec\,}2{.}\penalty-1\,\,l_{i}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 5: In case of $l_{i}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ we get $u{{\longrightarrow}_{{}_{\\!\omega}}}w_{i}$ by Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{i}$). In case of “$i{\,=\,}\penalty-10$” our property follows from the strong commutation assumption of our lemma. In case of “$i{\,=\,}\penalty-11$” our property follows from Claim 0. Q.e.d. (Claim 5) In case of ${{\bar{p}_{0}}\,{\parallel}\,{\bar{p}_{1}}}$ we have $w_{i}/\bar{p}_{1-i}{\,=\,}\penalty-1{u\penalty-1{[\,\bar{p}_{i}\leftarrow r_{i}\mu_{i}\,]}}/\bar{p}_{1-i}{\,=\,}\penalty-1u/\bar{p}_{1-i}{\,=\,}\penalty-1l_{1-i}\mu_{1-i}$ and therefore $w_{i}{{\longrightarrow}_{{}_{\\!\omega+n+1}}}{u\penalty-1{{[\,\bar{p}_{k}\leftarrow r_{k}\mu_{k}\ \|\ k{\,\prec\,}2\,]}}},$ i.e. our proof is finished. Thus, according to whether $\bar{p}_{0}$ is a prefix of $\bar{p}_{1}$ or vice versa, we have the following two cases left: There is some $\bar{p}_{1}^{\prime}$ with $\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1\bar{p}_{1}$ and $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ : We have two cases: “The variable overlap case”: There are $x\in{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{0}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{1}^{\prime}$: Claim 6: We have $x\mu_{0}/p^{\prime\prime}{\,=\,}\penalty-1l_{1}\mu_{1}$ and may assume $x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}.$ Proof of Claim 6: We have $x\mu_{0}/p^{\prime\prime}{\,=\,}\penalty-1l_{0}\mu_{0}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{0}p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{1}{\,=\,}\penalty-1l_{1}\mu_{1}.$ If $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}},$ then $x\mu_{0}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $x\mu_{0}/p^{\prime\prime}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $l_{1}\mu_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ and then $l_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which we may assume not to be the case by Claim 5. Q.e.d. (Claim 6) Claim 7: We can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by $x\nu{\,=\,}\penalty-1{x\mu_{0}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{0}.$ Then we have $x\mu_{0}{{\longrightarrow}_{{}_{\\!\omega+n+1}}}x\nu.$ Proof of Claim 7: This follows directly from Claim 6. Q.e.d. (Claim 7) Claim 8: $l_{0}\nu{\,=\,}\penalty-1w_{1}/\bar{p}_{0}.$ Proof of Claim 8: By the left-linearity assumption of our lemma and claims 5 and 6 we may assume ${{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}=\\{p^{\prime}\\}.$ Thus, by Claim 7 we get $w_{1}/\bar{p}_{0}{\,=\,}\penalty-1{u/\bar{p}_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow x\mu_{0}\,]}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1\\\ {{l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow{x\mu_{0}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{1}\mu_{1}\,]}}\,]}}{\,=\,}\penalty-1\\\ {l_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/\bar{p}_{0}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}r_{0}\nu.$ Proof of Claim 9: By the right-linearity assumption of our lemma and claims 5 and 6 we may assume ${\,\|{{{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}}\|\,}{\,\preceq\,}1.$ Thus by Claim 7 we get: $w_{0}/\bar{p}_{0}{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{0}\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{\,=\,}\penalty-1\\\ {{r_{0}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}\,]}}}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow x\nu\ \|\ r_{0}/p^{\prime\prime\prime}{\,=\,}\penalty-1x\,]}}}{\,=\,}\penalty-1r_{0}\nu.$ Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $l_{0}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{0}\nu,$ which again follows from Lemma A.7 since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ by our induction hypothesis and since $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\mu_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}y\nu$ by Claim 7. Q.e.d. (“The variable overlap case”) “The critical peak case”: $\bar{p}_{1}^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{0}})}\ {\wedge}\penalty-2\ l_{0}/\bar{p}_{1}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}]\cap{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}]\cup{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}\\\ x\xi^{-1}\mu_{1}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1}\xi\varrho{\,=\,}\penalty-1l_{1}\xi\xi^{-1}\mu_{1}{\,=\,}\penalty-1u/\bar{p}_{1}{\,=\,}\penalty-1u/\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1l_{0}\mu_{0}/\bar{p}_{1}^{\prime}{\,=\,}\penalty-1l_{0}\varrho/\bar{p}_{1}^{\prime}{\,=\,}\penalty-1(l_{0}/\bar{p}_{1}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1}\xi},{l_{0}/\bar{p}_{1}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0}\sigma,$ then the proof is finished due to $w_{0}/\bar{p}_{0}{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1r_{0}\sigma\varphi{\,=\,}\penalty-1{l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1{l_{0}\mu_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1w_{1}/\bar{p}_{0}.$ Otherwise we have $(\,({l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}},C_{1}\xi,1),\penalty-1\,(r_{0},C_{0},1),\penalty-1\,l_{0},\penalty-1\,\sigma,\penalty-1\,\bar{p}_{1}^{\prime}\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ (due the global case assumption); $C_{1}\xi\sigma\varphi=C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $C_{0}\sigma\varphi=C_{0}\mu_{0}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ (by our induction hypothesis) and $\omega$-shallow confluent up to $\omega$, due to our assumed $\omega$-level anti-closedness (matching the definition’s $n$ to our $n{+}1$) we have $w_{1}/\bar{p}_{0}{\,=\,}\penalty-1{l_{0}\mu_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\mu_{1}\,]}}{\,=\,}\penalty-1{l_{0}\penalty-1{[\,\bar{p}_{1}^{\prime}\leftarrow r_{1}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}r_{0}\sigma\varphi{\,=\,}\penalty-1r_{0}\mu_{0}{\,=\,}\penalty-1w_{0}/\bar{p}_{0}.$ Q.e.d. (“The critical peak case”) Q.e.d. (“There is some $\bar{p}_{1}^{\prime}$ with $\bar{p}_{0}\bar{p}_{1}^{\prime}{\,=\,}\penalty-1\bar{p}_{1}$ and $\bar{p}_{1}^{\prime}{\,\not=\,}\emptyset$ ”) There is some $\bar{p}_{0}^{\prime}$ with $\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1\bar{p}_{0}$ : We have two cases: “The second variable overlap case”: There are $x{\,\in\,}{{\rm V}}$ and $p^{\prime}$, $p^{\prime\prime}$ such that $l_{1}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1\bar{p}_{0}^{\prime}$: Claim 11a: We have $x\mu_{1}/p^{\prime\prime}{\,=\,}\penalty-1l_{0}\mu_{0}$ and may assume $x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}.$ Proof of Claim 11a: We have $x\mu_{1}/p^{\prime\prime}{\,=\,}\penalty-1l_{1}\mu_{1}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{1}p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1l_{0}\mu_{0}.$ If $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}},$ then $x\mu_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $x\mu_{1}/p^{\prime\prime}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ then $l_{0}\mu_{0}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ and then $l_{0}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which we may assume not to be the case by Claim 5. Q.e.d. (Claim 11a) Claim 11b: We can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by $x\nu{\,=\,}\penalty-1{x\mu_{1}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\nu{\,=\,}\penalty-1y\mu_{1}.$ Then we have $x\mu_{1}{{\longrightarrow}_{{}_{\\!\omega+n+1}}}x\nu.$ Proof of Claim 11b: This follows directly from Claim 11a. Q.e.d. (Claim 11b) Claim 12: $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1l_{1}\nu.$ Proof of Claim 12: By the left-linearity assumption of our lemma and claims 5 and 11a we may assume ${{\\{\ }p^{\prime\prime\prime}}~{}{\|}\penalty-9\,\ {l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1x{\ \\}}}=\\{p^{\prime}\\}.$ Thus, by Claim 11b we get $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{u/\bar{p}_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow x\mu_{1}\,]}}\penalty-1{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {{l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\ {\wedge}\penalty-2\ y{\,\not=\,}x\,]}}}\penalty-1{[\,p^{\prime}\leftarrow{x\mu_{1}\penalty-1{[\,p^{\prime\prime}\leftarrow r_{0}\mu_{0}\,]}}\,]}}{\,=\,}\penalty-1\\\ {l_{1}\penalty-1{{[\,p^{\prime\prime\prime}\leftarrow y\nu\ \|\ l_{1}/p^{\prime\prime\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1}\nu{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+n+1}}w_{1}/\bar{p}_{1}.$ Proof of Claim 13: Since $r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1},$ this follows directly from Claim 11b. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{1}\nu,$ which again follows from Claim 11b, Lemma A.7 (matching its $n_{0}$ to our $n{+}1$ and its $n_{1}$ to our $n$), and our induction hypothesis that ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$. Q.e.d. (“The second variable overlap case”) “The second critical peak case”: $\bar{p}_{0}^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{1}})}\ {\wedge}\penalty-2\ l_{1}/\bar{p}_{0}^{\prime}{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}]\cap{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0},r_{0}),C_{0})}})}]\cup{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1},r_{1}),C_{1})}})}\\\ x\xi^{-1}\mu_{0}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0}\xi\varrho{\,=\,}\penalty-1l_{0}\xi\xi^{-1}\mu_{0}{\,=\,}\penalty-1u/\bar{p}_{0}{\,=\,}\penalty-1u/\bar{p}_{1}\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1}\mu_{1}/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1l_{1}\varrho/\bar{p}_{0}^{\prime}{\,=\,}\penalty-1(l_{1}/\bar{p}_{0}^{\prime})\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0}\xi},{l_{1}/\bar{p}_{0}^{\prime})\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. If ${l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1}\sigma,$ then the proof is finished due to $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1}.$ Otherwise we have $(\,({l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}},C_{0}\xi,1),\penalty-1\,(r_{1},C_{1},1),\penalty-1\,l_{1},\penalty-1\,\sigma,\penalty-1\,\bar{p}_{0}^{\prime}\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $C_{0}\xi\sigma\varphi=C_{0}\mu_{0}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $C_{1}\sigma\varphi=C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Since ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ (by our induction hypothesis) and $\omega$-shallow confluent up to $\omega$, due to our assumed $\omega$-level strong joinability (matching the definition’s $n$ to our $n{+}1$) we have $w_{0}/\bar{p}_{1}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,\bar{p}_{0}^{\prime}\leftarrow r_{0}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\mu_{1}{\,=\,}\penalty-1w_{1}/\bar{p}_{1}.$ Q.e.d. (“The second critical peak case”) Q.e.d. (Lemma A.10) Proof of Lemma B.1 Due to $\mathcal{T}$-monotonicity of $>$ and ${>}\subseteq{\rhd},$ it is easy to show by induction over $\beta$ in $\prec$ that $\forall\beta{\,\preceq\,}\omega{+}\alpha{.}\penalty-1\,\,{{\longrightarrow}_{{}_{\\!{{\rm R},{{\rm X}}},\beta}}}\subseteq{\rhd}$ using Lemma 2.12. Proof of Lemma B.2 Claim 0: $\forall u{\,\in\,}{{{\mathcal{TERMS}}}({C})}{.}\penalty-1\,\,\forall\hat{u}{\,\in\,}{{\mathcal{T}}({{\rm sig},{{\rm X}}})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}u\mu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{u}\ \ {\Rightarrow}\penalty-2\ \ u\nu{\downarrow}\hat{u}\end{array}\right)}.$ Proof of Claim 1: We get the following cases: $l\mu\rhd u\mu$: $u\nu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}u\mu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{u}$ implies $u\nu{\downarrow}\hat{u}$ by the assumed confluence below $u\mu$. $u\mu{\,\not\in\,}{{\rm dom}({{{\longrightarrow}}})}$: $u\nu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}u\mu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{u}$ implies $u\nu{\,=\,}\penalty-1u\mu{\,=\,}\penalty-1\hat{u}.$ [${{{\mathcal{V}}}({u})}\subseteq{{{\rm V}}\\!_{{\mathcal{C}}}}$: By Lemma 2.10 we get $\forall x{\,\in\,}{{{\mathcal{V}}}({u})}{.}\penalty-1\,\,x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}x\nu.$ Thus from $u\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}u\mu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{u}$ due to the assumed ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}\subseteq{\downarrow}$ we get $u\nu{\downarrow}\hat{u}.$ ] Q.e.d. (Claim 0) By Lemma 2.7 it suffices to show that $C\nu$ is fulfilled. For each $L$ in $C$ we have to show that $L\nu$ is fulfilled. Note that we already know that $L\mu$ is fulfilled. $L=(u{=}v)$: There is some $\hat{u}$ with $u\mu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{u}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v\mu.$ By Claim 0 there is some $\hat{v}$ with $u\nu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{v}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\hat{u}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v\mu.$ Thus, by Claim 0 we get $u\nu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{v}{\downarrow}v\nu.$ $L=({{\rm Def}\>}u)$: We know the existence of $\hat{u}\in{{\mathcal{GT}}({{\rm cons}})}$ with $u\mu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{u}.$ By Claim 0 we get $u\nu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}u^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\hat{u}$ for some $u^{\prime}$. By Lemma 2.10 we get $u^{\prime}{\,\in\,}{{\mathcal{GT}}({{\rm cons}})}.$ $L=(u{\not=}v)$: We know the existence of $\hat{u},\hat{v}\in{{\mathcal{GT}}({{\rm cons}})}$ with $u\mu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{u}{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}\hat{v}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v\mu.$ Just like above we get $u^{\prime},v^{\prime}\in{{\mathcal{GT}}({{\rm cons}})}$ with $u\nu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}u^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\hat{u}$ and $\hat{v}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}v^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v\nu.$ Due to $\hat{u}{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}\hat{v}$ we finally get $u^{\prime}{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}v^{\prime}.$ Q.e.d. (Lemma B.2) Proof of Lemma B.3 First notice that the usual modularization of the proof for the unconditional analogue of the theorem (by showing first that local confluence is guaranteed except for the cases that are matched by critical peaks (the so-called “critical pair lemma”)) is not possible here because we need the confluence property to hold for the condition terms even for the cases that are not matched by critical peaks. Now to the proof: For all $s\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ we are going to prove confluence below $s$ by induction over $s$ in $\lhd$. Let $s$ be minimal in $\lhd$ such that ${\longrightarrow}$ is not confluent below $s$. Because of ${{\longrightarrow}}\subseteq\rhd$ (by Lemma B.1) and minimality of $s$, ${\longrightarrow}$ is not even locally confluent below $s$. Let $p,q\in{{{\mathcal{POS}}}({s})}$; $t_{0}{{\longleftarrow}_{{}_{\\!\omega+\omega,p}}}s{{\longrightarrow}_{{}_{\\!\omega+\omega,q}}}t_{1};$ $t_{0}{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}t_{1}.$ Now as one of $p,q$ must be a prefix of the other, w.l.o.g. say that $q$ is a prefix of $p$. As $s\unrhd s/q,$ by the minimality of $s$ we have $q{\,=\,}\penalty-1\emptyset.$ We start a second level of induction on $p$ in $\lll_{s}$. Thus assume that $p$ is minimal such that there are $p\in{{{\mathcal{POS}}}({s})}$ and $t_{0},t_{1}\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ with $t_{0}{{\longleftarrow}_{{}_{\\!\omega+\omega,p}}}s{{\longrightarrow}_{{}_{\\!\omega+\omega,\emptyset}}}t_{1}$ and $t_{0}{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}t_{1}.$ Now for $k<2$ there must be $((l_{k},r_{k}),C_{k})\in{\rm R}$; $\mu_{k}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$; with $C_{k}\mu_{k}$ fulfilled; $s{\,=\,}\penalty-1l_{1}\mu_{1};$ $s/p{\,=\,}\penalty-1l_{0}\mu_{0};$ $t_{0}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}};$ $t_{1}{\,=\,}\penalty-1r_{1}\mu_{1}.$ Moreover, for $k<2$ we define $\mathchar 259\relax_{k}:=\left\\{\mbox{$\begin{array}[]{ll}0&\mbox{ if }l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ 1&\mbox{ otherwise}\end{array}$}\right\\}.$ Claim 0: We may assume that $\forall q{\,\in\,}{{{\mathcal{POS}}}({s})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\emptyset{\,\not=\,}{q}\lll_{s}p\ \ {\Rightarrow}\penalty-2\ \ s{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega+\omega,q}}}})}\end{array}\right)}.$ Proof of Claim 0: Otherwise there must be some $q\in{{{\mathcal{POS}}}({s})}$; ${((l_{2},r_{2}),C_{2})}\in{\rm R}$; $\mu_{2}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$; with $C_{2}\mu_{2}$ fulfilled; $s/q{\,=\,}\penalty-1l_{2}\mu_{2};$ and $\emptyset{\,\not=\,}{q}\lll_{s}p.$ By our second induction level we get ${l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{1}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}r_{1}\mu_{1}$ for some $w_{1}$; cf. the diagram below. Next we are going to show that there is some $w_{0}$ with ${l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{0}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}.$ Note that (since ${{\longrightarrow}}\subseteq\rhd$ implies $s{\rhd}{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}$) this finishes the proof of Claim 0 since then $w_{0}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{1}$ by our first level of induction implies the contradictory $t_{0}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{0}{\downarrow}w_{1}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}t_{1}.$ In case of ${{p}\,{\parallel}\,{q}}$ we simply can choose $w_{0}:={{l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}.$ Otherwise, there must be some $\bar{p}$, $\hat{p}$, $\hat{q}$, with $p{\,=\,}\penalty-1\bar{p}\hat{p},$ $q{\,=\,}\penalty-1\bar{p}\hat{q},$ and ${{(\hat{p}{\,=\,}\penalty-1\emptyset\ {\vee}\penalty-2\ \hat{q}{\,=\,}\penalty-1\emptyset)}}.$ Now it suffices to show ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{0}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}$ for some $w_{0}^{\prime}$, because by ${\mathcal{T}}({{\rm sig},{{\rm X}}})$-monotonicity of ${\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}$ we then have ${l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{s\penalty-1{[\,\bar{p}\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{{s\penalty-1{[\,\bar{p}\leftarrow s/\bar{p}\,]}}\penalty-1{[\,\bar{p}\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{s\penalty-1{[\,\bar{p}\leftarrow{s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}\,]}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\\\ {s\penalty-1{[\,\bar{p}\leftarrow w_{0}^{\prime}\,]}}\\\ {{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}{s\penalty-1{[\,\bar{p}\leftarrow{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}\,]}}{\,=\,}\penalty-1{{s\penalty-1{[\,\bar{p}\leftarrow s/\bar{p}\,]}}\penalty-1{[\,\bar{p}\hat{q}\leftarrow r_{2}\mu_{2}\,]}}{\,=\,}\penalty-1{s\penalty-1{[\,\bar{p}\hat{q}\leftarrow r_{2}\mu_{2}\,]}}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}.$ Note that ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{\longleftarrow}_{{}_{\\!\omega+\omega,\hat{p}}}}s/\bar{p}{{\longrightarrow}_{{}_{\\!\omega+\omega,\hat{q}}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}.$ In case of $\bar{p}{\,\not=\,}\emptyset$ (since then ${\rhd_{{}_{\rm ST}}}\subseteq\rhd$ implies $s\rhd s/\bar{p}$) we get some $w_{0}^{\prime}$ with ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{0}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}$ by our first level of induction. Otherwise, in case of $\bar{p}{\,=\,}\penalty-1\emptyset,$ our disjunction from above means ${{(p{\,=\,}\penalty-1\emptyset\ {\vee}\penalty-2\ q{\,=\,}\penalty-1\emptyset)}}.$ Since we have $\emptyset{\,\not=\,}q$ by our initial assumption, we may assume $q{\,=\,}\penalty-1\hat{q}{\,\not=\,}\emptyset$ and $p{\,=\,}\penalty-1\hat{p}{\,=\,}\penalty-1\bar{p}{\,=\,}\penalty-1\emptyset.$ Then the above divergence reads ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{\longleftarrow}_{{}_{\\!\omega+\omega,\emptyset}}}s{{\longrightarrow}_{{}_{\\!\omega+\omega,q}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}$ and we get the required joinability by our second induction level due to $q\lll_{s}p.$ Q.e.d. (Claim 0) Claim 1: In case of ${{\longleftarrow}_{{}_{\\!\omega}}}\circ{{\longrightarrow}}\subseteq{\downarrow}$ we may assume $s{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega}}}})}.$ Proof of Claim 1: Assume ${{\longleftarrow}_{{}_{\\!\omega}}}\circ{{\longrightarrow}}\subseteq{\downarrow}.$ If there is a $t_{2}$ with $s{{\longrightarrow}_{{}_{\\!\omega}}}t_{2}$ then we get some $t_{0}^{\prime}$, $t_{1}^{\prime}$ with $t_{0}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}t_{0}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}t_{2}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}t_{1}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}t_{1}.$ Due ${{\longrightarrow}_{{}_{\\!\omega}}}\subseteq{{\longrightarrow}}\subseteq{\rhd}$ by our first level of induction we get the contradictory $t_{0}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}t_{0}^{\prime}\downarrow t_{1}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}t_{1}.$ Q.e.d. (Claim 1) Claim 2: In case of ${{\longleftarrow}_{{}_{\\!\omega}}}\circ{{\longrightarrow}}\subseteq{\downarrow}$ for each $k\prec 2$ we may assume: $l_{k}\mu_{k}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ and ${(\ l_{k}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\ {\vee}\penalty-2\ {{{\mathcal{TERMS}}}({C_{k}\mu_{k}})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\ )}.$ Proof of Claim 2: By Lemma 2.10 and $l_{k}\mu_{k}{\longrightarrow}r_{k}\mu_{k},$ $l_{k}\mu_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ implies ${l_{k}\mu_{k}}{{\longrightarrow}_{{}_{\\!\omega}}}{r_{k}\mu_{k}}$ which we may assume not to be the case by Claim 1. In case of $l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ and ${{{\mathcal{TERMS}}}({C_{k}\mu_{k}})}{\subseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ by Lemma 2.10 $C_{k}\mu_{k}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$ and then Corollary 2.6 implies ${l_{k}\mu_{k}}{{\longrightarrow}_{{}_{\\!\omega}}}{r_{k}\mu_{k}}$ again, which we may assume not to be the case by Claim 1. Q.e.d. (Claim 2) Now we have two cases: The variable overlap case: $p{\,=\,}\penalty-1q_{0}q_{1};\ l_{1}/q_{0}{\,=\,}\penalty-1x\in{{\rm V}}$ : We have $x\mu_{1}/q_{1}{\,=\,}\penalty-1l_{1}\mu_{1}/q_{0}q_{1}{\,=\,}\penalty-1s/p{\,=\,}\penalty-1l_{0}\mu_{0}.$ By Lemma 2.10 (in case of $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}$), we can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by ($y{\,\in\,}{{\rm V}}$): $y\nu:=\left\\{\begin{array}[]{ll}{x\mu_{1}\penalty-1{[\,q_{1}\leftarrow r_{0}\mu_{0}\,]}}&\mbox{if }y{\,=\,}\penalty-1x\\\ y\mu_{1}&\mbox{otherwise}\\\ \end{array}\right\\}$ and get $y\mu_{1}{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}y\nu$ for $y{\,\in\,}{{\rm V}}$. By Corollary 2.8: $t_{0}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,q_{0}q_{1}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{{l_{1}\penalty-1{[\,q_{0}\leftarrow x\nu\,]}}\penalty-1{{[\,q^{\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/q^{\prime}{\,=\,}\penalty-1y\in{{\rm V}}\ {\wedge}\penalty-2\ q^{\prime}{\,\not=\,}q_{0}\,]}}}\ {{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}$ ${l_{1}\penalty-1{{[\,q^{\prime}\leftarrow y\nu\ \|\ l_{1}/q^{\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1}\nu;$ $t_{1}{\,=\,}\penalty-1r_{1}\mu_{1}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}r_{1}\nu.$ It suffices to show $l_{1}\nu{\longrightarrow}r_{1}\nu,$ which follows from Lemma B.2 because of [${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\subseteq{\downarrow}$,] $l_{1}\mu_{1}{\,=\,}\penalty-1s$ and our first level of induction. Q.e.d. (The variable overlap case) The critical peak case: $p\in{{{\mathcal{POS}}}({l_{1}})};\ l_{1}/p\not\in{{\rm V}}$ : Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{l_{0}{=}r_{0}{\longleftarrow}C_{0}}})}]\cap{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}{\,=\,}\penalty-1\emptyset.$ Define ${\rm Y}:={{{\mathcal{V}}}({({l_{0}{=}r_{0}{\longleftarrow}C_{0}})\xi,{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}$ . Let $\varrho$ be given by $\ x\varrho{\,=\,}\penalty-1\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1}&\mbox{ if }x\in{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}\\\ x\xi^{-1}\mu_{0}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0}\xi\varrho{\,=\,}\penalty-1l_{0}\xi\xi^{-1}\mu_{0}{\,=\,}\penalty-1s/p{\,=\,}\penalty-1l_{1}\mu_{1}/p{\,=\,}\penalty-1l_{1}\varrho/p{\,=\,}\penalty-1(l_{1}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0}\xi},{l_{1}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Claim A: We may assume ${\left(\begin{array}[c]{l}p{\,=\,}\penalty-1\emptyset\ \ {\vee}\penalty-2\ \ \forall y{\,\in\,}{{{\mathcal{V}}}({l_{1}})}{.}\penalty-1\,\,y\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}}})}\end{array}\right)}.$ Proof of Claim A: Otherwise, when $p{\,\not=\,}\emptyset$ holds but $\forall y{\,\in\,}{{{\mathcal{V}}}({l_{1}})}{.}\penalty-1\,\,y\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}}})}$ is not the case, there are some $x\in{{{\mathcal{V}}}({l_{1}})},$ $\nu{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $x\sigma\varphi{\longrightarrow}x\nu$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\mu_{1}{\,=\,}\penalty-1y\nu.$ Due to $l_{1}\mu_{1}/p\lhd l_{1}\mu_{1}{\,=\,}\penalty-1s$ by our first level of induction from $r_{0}\xi\sigma\varphi{\longleftarrow}l_{0}\xi\sigma\varphi{\,=\,}\penalty-1l_{1}\sigma\varphi/p{\,=\,}\penalty-1l_{1}\mu_{1}/p{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}l_{1}\nu/p$ we know that there must be some $u$ with $r_{0}\xi\sigma\varphi{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}u{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}l_{1}\nu/p.$ Due to $l_{1}\mu_{1}{{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}}l_{1}\nu$ and ${{\longrightarrow}}\ {\subseteq}\ \rhd$ we get $l_{1}\nu\lhd l_{1}\mu_{1}{\,=\,}\penalty-1s.$ Thus, by our first level of induction, from ${l_{1}\nu\penalty-1{[\,p\leftarrow u\,]}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}l_{1}\nu{{\longrightarrow}}r_{1}\nu$ (which is due to Lemma B.2, [${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\subseteq{\downarrow}$,] $l_{1}\mu_{1}{\,=\,}\penalty-1s$ and our first level of induction) we get $t_{0}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\sigma\varphi\,]}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}{l_{1}\nu\penalty-1{[\,p\leftarrow r_{0}\xi\sigma\varphi\,]}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}{l_{1}\nu\penalty-1{[\,p\leftarrow u\,]}}\downarrow r_{1}\nu{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}r_{1}\mu_{1}{\,=\,}\penalty-1t_{1}.$ Q.e.d. (Claim A) If ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1}\sigma,$ then we are finished due to $t_{0}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1t_{1}.$ Otherwise $(({l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}},C_{0}\xi,\mathchar 259\relax_{0}),\ (r_{1},C_{1},\mathchar 259\relax_{1}),\ l_{1},\ \sigma,\ p\ )$ is a critical peak in ${\rm CP}({\rm R})$. Now $(C_{0}\xi\,C_{1})\sigma\varphi{\,=\,}\penalty-1C_{0}\mu_{0}\,C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}$. Due to $l_{1}\sigma\varphi{\,=\,}\penalty-1l_{1}\varrho{\,=\,}\penalty-1l_{1}\mu_{1}{\,=\,}\penalty-1s,$ by our first level of induction we get $\forall u\lhd l_{1}\sigma\varphi{.}\penalty-1\,\,{{({{\longrightarrow}}\mbox{ is confluent below }u)}}.$ [By Claim 1 we get $l_{1}\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega}}}})}.$] By Claim 0 we get $\forall q{\,\in\,}{{{\mathcal{POS}}}({l_{1}\sigma\varphi})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\emptyset{\,\not=\,}{q}\lll_{l_{1}\sigma\varphi}p\ \ {\Rightarrow}\penalty-2\ \ l_{1}\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega+\omega,q}}}})}\end{array}\right)}.$ This means $l_{1}\sigma\varphi{\,\not\in\,}A(p).$ [Define $D_{0}:=C_{0}\xi$ and $D_{1}:=C_{1}$. If $\mathchar 259\relax_{k}{\,=\,}\penalty-10$ for some $k\prec 2$, then $l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ which by Claim 2 implies ${{{\mathcal{TERMS}}}({D_{k}\sigma\varphi})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ and then ${{{\mathcal{TERMS}}}({D_{k}\sigma})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}.$ ] Thus, in case of $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}}})},$ by Claim A and the assumed $\rhd$-weak joinability w.r.t. ${\rm R},{{\rm X}}$ besides $A$ we get $t_{0}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi\downarrow r_{1}\sigma\varphi{\,=\,}\penalty-1t_{1}.$ Otherwise, when $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}}})}$ is not the case, by ${{\longrightarrow}}\subseteq{\rhd}$ and the Axiom of Choice there is some $\varphi^{\prime}{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\varphi{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}y\varphi^{\prime}{\,\not\in\,}{{\rm dom}({{{\longrightarrow}}})}.$ Then, of course, $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\xi\sigma\varphi{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}y\xi\sigma\varphi^{\prime}$ and $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\sigma\varphi{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}y\sigma\varphi^{\prime}.$ By Lemma B.2 (due to [${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\subseteq{\downarrow}$;] $l_{0}\xi\sigma\varphi,l_{1}\sigma\varphi{\trianglelefteq_{{}_{\rm ST}}}s;$ ${\trianglelefteq_{{}_{\rm ST}}}\subseteq{\trianglelefteq};$ and our first level of induction) we know that $C_{0}\xi\sigma\varphi^{\prime}$ and $C_{1}\sigma\varphi^{\prime}$ are fulfilled. Furthermore, we have ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi^{\prime}$ and $r_{1}\sigma\varphi^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}r_{1}\sigma\varphi.$ Therefore, in case of $l_{1}\sigma\varphi{\,=\,}\penalty-1l_{1}\sigma\varphi^{\prime}$ the proof succeeds like above with $\varphi^{\prime}$ instead of $\varphi$. Otherwise we have $l_{1}\sigma\varphi{{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}}l_{1}\sigma\varphi^{\prime}.$ Then due to ${{\longrightarrow}}\ {\subseteq}\ \rhd$ we get $s{\,=\,}\penalty-1l_{1}\sigma\varphi\rhd l_{1}\sigma\varphi^{\prime}.$ Therefore, by our first level of induction, from ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi^{\prime}{\longleftarrow}{l_{1}\penalty-1{[\,p\leftarrow l_{0}\xi\,]}}\sigma\varphi^{\prime}{\,=\,}\penalty-1l_{1}\sigma\varphi^{\prime}{{\longrightarrow}}r_{1}\sigma\varphi^{\prime}$ (which is due to [${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\subseteq{\downarrow}$;] $l_{0}\xi\sigma\varphi,l_{1}\sigma\varphi{\trianglelefteq_{{}_{\rm ST}}}s;$ ${\trianglelefteq_{{}_{\rm ST}}}\subseteq{\trianglelefteq};$ and our first level of induction) we conclude $t_{0}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi^{\prime}\downarrow r_{1}\sigma\varphi^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}r_{1}\sigma\varphi{\,=\,}\penalty-1t_{1}.$ Q.e.d. (The critical peak case) Q.e.d. (Lemma B.3) Proof of Lemma B.4 and Lemma B.5 Since the proofs of the two lemmas are very similar, we treat them together, indicating the differences where necessary and using ‘$\alpha$’ to denote $\omega$ in the proof of Lemma B.4. For $(\delta,s){\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}(\beta,\hat{s})$ we are going to show that ${\rm R},{{\rm X}}$ is $\alpha$-shallow confluent up to $\delta$ and $s$ in $\lhd$ by induction over $(\delta,s)$ in ${\,\,{\prec\\!\\!\lhd}\,\,}$. Suppose that for $n_{0},n_{1}\prec\omega$ we have $(n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1},s){\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}(\beta,\hat{s})$ and $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}s{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}t_{1}^{\prime}.$ We have to show $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}t_{1}^{\prime}.$ In case of $\exists i{\,\prec\,}2{.}\penalty-1\,\,t_{i}^{\prime}{\,=\,}\penalty-1s$ this is trivially true. Thus, for $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}t_{0}{{\longleftarrow}_{{}_{\\!\alpha+n_{0},p}}}s{{\longrightarrow}_{{}_{\\!\alpha+n_{1},q}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}t_{1}^{\prime}$ using the induction hypothesis that $\forall(\delta,w^{\prime}){\,\,{\prec\\!\\!\lhd}\,\,}(n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1},s){.}\penalty-1\,\,$ ${\rm R},{{\rm X}}$ is $\alpha$-shallow confluent up to $\delta$ and $w^{\prime}$ in $\lhd$ we have to show $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}t_{1}^{\prime}.$ Note that due to Lemma B.1 we have ${{\longrightarrow}_{{}_{\\!\omega+\alpha}}}\subseteq{\rhd}.$ Claim 0: Now it is sufficient to show $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}t_{1}$ for some $u$. Proof of Claim 0: Due to ${{\longrightarrow}_{{}_{\\!\omega+\alpha}}}\subseteq{\rhd}$ we have $s\rhd t_{0},t_{1}.$ Thus by our induction hypotheses $u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}t_{1}^{\prime}$ (cf. diagram below) implies the existence of some $v$ with $u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}t_{1}^{\prime}$ and then $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}v$ implies $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}v.$ Q.e.d. (Claim 0) In case of ${{p}\,{\parallel}\,{q}}$ we have $t_{0}/q={s\penalty-1{[\,p\leftarrow t_{0}/p\,]}}/q=s/q$ and $t_{1}/p={s\penalty-1{[\,q\leftarrow t_{1}/q\,]}}/p=s/p$ and therefore $t_{0}{{\longrightarrow}_{{}_{\\!\alpha+n_{1},q}}}{{s\penalty-1{[\,p\leftarrow t_{0}/p\,]}}\penalty-1{[\,q\leftarrow t_{1}/q\,]}}{{\longleftarrow}_{{}_{\\!\alpha+n_{0},p}}}t_{1},$ i.e. our proof is finished. Otherwise one of $p,q$ must be a prefix of the other, w.l.o.g. say that $q$ is a prefix of $p$. In case of $q{\,\not=\,}\emptyset$ due to ${\rhd_{{}_{\rm ST}}}\subseteq{\rhd}$ we get $s/q\lhd s$ and the proof finished by our induction hypothesis and ${\mathcal{T}}({{\rm sig},{{\rm X}}})$-monotonicity of ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{k}}}$. Thus we may assume $q{\,=\,}\penalty-1\emptyset.$ We start a second level of induction on $p$ in $\lll_{s}$. Thus we may assume the following induction hypothesis: $\forall q{\,\in\,}{{{\mathcal{POS}}}({s})}{.}\penalty-1\,\,\forall t_{0}^{\prime},t_{1}^{\prime}{.}\penalty-1\,\,\forall n_{0}^{\prime},n_{1}^{\prime}{.}\penalty-1\,\,$ ${\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&t_{0}^{\prime}{{\longleftarrow}_{{}_{\\!\alpha+n_{0}^{\prime},q}}}s{{\longrightarrow}_{{}_{\\!\alpha+n_{1}^{\prime},\emptyset}}}t_{1}^{\prime}\\\ {\wedge}&n_{0}^{\prime}{+_{\\!\\!{}_{\alpha}}}n_{1}^{\prime}\preceq n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}\\\ {\wedge}&q\lll_{s}p\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}^{\prime}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}^{\prime}}}}t_{1}^{\prime}\end{array}\right)}$ Now for $k\prec 2$ there must be $((l_{k},r_{k}),C_{k})\in{\rm R}$; $\mu_{k}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$; with $C_{k}\mu_{k}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\alpha+(n_{k}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}$; $s{\,=\,}\penalty-1l_{1}\mu_{1};$ $s/p{\,=\,}\penalty-1l_{0}\mu_{0};$ $t_{0}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}};$ $t_{1}{\,=\,}\penalty-1r_{1}\mu_{1};$ and $\mathchar 259\relax_{k}{\,\preceq\,}n_{k}$ and $\alpha{\,=\,}\penalty-10\ {\Rightarrow}\penalty-2\ {{\left({{\begin{array}[]{ll}&1{\,\preceq\,}n_{k}\\\ {\wedge}&\mathchar 259\relax_{k}{\,=\,}\penalty-10\\\ \end{array}}}\right)}}$ for $\mathchar 259\relax_{k}:=\left\\{\mbox{$\begin{array}[]{ll}0&\mbox{ if }l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ 1&\mbox{ otherwise}\end{array}$}\right\\}.$ Claim 1: We may assume that $\ \forall q{\,\in\,}{{{\mathcal{POS}}}({s})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\emptyset{\,\not=\,}{q}\lll_{s}p\ \ {\Rightarrow}\penalty-2\ \ s{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\},q}}}})}\end{array}\right)}.$ Proof of Claim 1: Otherwise there must be some $q\in{{{\mathcal{POS}}}({s})}$; ${((l_{2},r_{2}),C_{2})}\in{\rm R}$; $\mu_{2}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$; with $C_{2}\mu_{2}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\alpha+(\min\\{n_{0},n_{1}\\}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}$; $s/q{\,=\,}\penalty-1l_{2}\mu_{2};$ $\emptyset{\,\not=\,}{q}\lll_{s}p.$ By our second induction level we get ${l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}r_{1}\mu_{1}$ for some $w_{1}$; cf. the diagram below. Next we are going to show that there is some $w_{0}$ with ${l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}.$ Note that (since ${{\longrightarrow}_{{}_{\\!\omega+\alpha}}}\subseteq{\rhd}$ implies $s{\rhd}{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}$) this finishes the proof since then $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}w_{1}$ by our first level of induction implies $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}t_{1}.$ In case of ${{p}\,{\parallel}\,{q}}$ we simply can choose $w_{0}:={{l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}.$ Otherwise, there must be some $\bar{p}$, $\hat{p}$, $\hat{q}$, with $p{\,=\,}\penalty-1\bar{p}\hat{p},$ $q{\,=\,}\penalty-1\bar{p}\hat{q},$ and ${{(\hat{p}{\,=\,}\penalty-1\emptyset\ {\vee}\penalty-2\ \hat{q}{\,=\,}\penalty-1\emptyset)}}.$ Now it suffices to show ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}$ for some $w_{0}^{\prime}$, because by ${\mathcal{T}}({{\rm sig},{{\rm X}}})$-monotonicity of ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n^{\prime}}}$ we then have ${l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{s\penalty-1{[\,\bar{p}\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{{s\penalty-1{[\,\bar{p}\leftarrow s/\bar{p}\,]}}\penalty-1{[\,\bar{p}\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {s\penalty-1{[\,\bar{p}\leftarrow{s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}{s\penalty-1{[\,\bar{p}\leftarrow w_{0}^{\prime}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}{s\penalty-1{[\,\bar{p}\leftarrow{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}\,]}}{\,=\,}\penalty-1\\\ {{s\penalty-1{[\,\bar{p}\leftarrow s/\bar{p}\,]}}\penalty-1{[\,\bar{p}\hat{q}\leftarrow r_{2}\mu_{2}\,]}}{\,=\,}\penalty-1{s\penalty-1{[\,\bar{p}\hat{q}\leftarrow r_{2}\mu_{2}\,]}}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}.$ Note that $\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}{s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{\longleftarrow}_{{}_{\\!\alpha+n_{0},\hat{p}}}}s/\bar{p}{{\longrightarrow}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\},\hat{q}}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}.$ In case of $\bar{p}{\,\not=\,}\emptyset$ (since then ${\rhd_{{}_{\rm ST}}}\subseteq\rhd$ implies $s\rhd s/\bar{p}$) we get some $w_{0}^{\prime}$ with ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}$ by our first level of induction. Otherwise, in case of $\bar{p}{\,=\,}\penalty-1\emptyset,$ our disjunction from above means ${{(p{\,=\,}\penalty-1\emptyset\ {\vee}\penalty-2\ q{\,=\,}\penalty-1\emptyset)}}.$ Since we have $\emptyset{\,\not=\,}q$ by our initial assumption, we may assume $q{\,=\,}\penalty-1\hat{q}{\,\not=\,}\emptyset$ and $p{\,=\,}\penalty-1\hat{p}{\,=\,}\penalty-1\bar{p}{\,=\,}\penalty-1\emptyset.$ Then the above divergence reads ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{\longleftarrow}_{{}_{\\!\alpha+n_{0},\emptyset}}}s{{\longrightarrow}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\},q}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}$ and we get the required joinability by our second induction level due to $q\lll_{s}p.$ Q.e.d. (Claim 1) Claim 2 of the proof of Lemma B.4: We may assume that for some $i\prec 2$: $n_{i}{\,=\,}\penalty-10{\,\prec\,}n_{1-i};$ $l_{i}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})};$ $l_{1-i}\mu_{1-i}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})};$ and ${(\ l_{1-i}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\ {\vee}\penalty-2\ {{{\mathcal{TERMS}}}({C_{1-i}\mu_{1-i}})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\ )}.$ Proof of Claim 2 of the proof of Lemma B.4: If $\forall i{\,\prec\,}2{.}\penalty-1\,\,s{{\longrightarrow}_{{}_{\\!\omega}}}t_{1-i},$ then the whole proof is finished by confluence of ${\longrightarrow}_{{}_{\\!\omega}}$. Thus there is some $i\prec 2$ with $s{\,\,\,\not\\!\\!\\!\\!{\longrightarrow}_{{}_{\\!\omega}}}t_{1-i}.$ Then we get $0{\,\prec\,}n_{1-i}.$ The case of $0{\,\prec\,}n_{i}$ is empty, since then due to $\beta\preceq\omega\prec n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}$ the globally supposed ordering property $(n_{0}{+_{\\!\\!{}_{\omega}}}n_{1},s){\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}(\beta,\hat{s})$ cannot hold. Thus we get $n_{i}{\,=\,}\penalty-10{\,\prec\,}n_{1-i}.$ Due $\mathchar 259\relax_{i}{\,\preceq\,}n_{i}{\,=\,}\penalty-10$ we get $l_{i}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ By Lemma 2.10 and $l_{1-i}\mu_{1-i}{{\longrightarrow}_{{}_{\\!\omega+n_{1-i}}}}r_{1-i}\mu_{1-i},$ $l_{1-i}\mu_{1-i}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ would imply the contradictory ${l_{1-i}\mu_{1-i}}{{\longrightarrow}_{{}_{\\!\omega}}}{r_{1-i}\mu_{1-i}}.$ Finally, $l_{1-i}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ and ${{{\mathcal{TERMS}}}({C_{1-i}\mu_{1-i}})}{\subseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ by Lemma 2.10 would imply that $C_{1-i}\mu_{1-i}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$ and then Corollary 2.6 would imply the contradictory ${l_{1-i}\mu_{1-i}}{{\longrightarrow}_{{}_{\\!\omega}}}{r_{1-i}\mu_{1-i}}$ again. Q.e.d. (Claim 2 of the proof of Lemma B.4) Claim 2 of the proof of Lemma B.5: For each $k\prec 2$ we may assume: $0{\,\prec\,}n_{k};$ $\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})};$ and $\alpha{\,=\,}\penalty-1\omega\ \ {\Rightarrow}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&l_{k}\mu_{k}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\wedge}&{{\left({{\begin{array}[]{ll}&l_{k}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ {\vee}&{{{\mathcal{TERMS}}}({C_{k}\mu_{k}})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ \end{array}}}\right)}}\\\ \end{array}}}\right)}}.$ Proof of Claim 2 of the proof of Lemma B.5: In case of $\alpha{\,=\,}\penalty-10$ we have $0{\,\prec\,}n_{k}$ due to $1{\,\preceq\,}n_{k}$ and have $l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ due to $\mathchar 259\relax_{k}{\,=\,}\penalty-10.$ Now we treat the case of $\alpha{\,=\,}\penalty-1\omega:$ We may assume $\forall k{\,\prec\,}2{.}\penalty-1\,\,s{\,\,\,\not\\!\\!\\!\\!{\longrightarrow}_{{}_{\\!\omega}}}t_{k},$ since otherwise the whole proof is finished by $\omega$-shallow confluence up to $\omega$. Thus we have $0{\,\prec\,}n_{0},n_{1}.$ By Lemma 2.10 and $l_{k}\mu_{k}{{\longrightarrow}_{{}_{\\!\omega+n_{k}}}}r_{k}\mu_{k},$ $l_{k}\mu_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ would imply the contradictory ${l_{k}\mu_{k}}{{\longrightarrow}_{{}_{\\!\omega}}}{r_{k}\mu_{k}}.$ Finally, $l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ and ${{{\mathcal{TERMS}}}({C_{k}\mu_{k}})}{\subseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ by Lemma 2.10 would imply that $C_{k}\mu_{k}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$ and then Corollary 2.6 would imply the contradictory ${l_{k}\mu_{k}}{{\longrightarrow}_{{}_{\\!\omega}}}{r_{k}\mu_{k}}$ again. Q.e.d. (Claim 2 of the proof of Lemma B.5) Claim 3: For all $k{\,\prec\,}2$ we may assume: $\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\end{array}\right)$; $\left(\begin{array}[c]{l}\min\\{n_{0},n_{1}\\}{\,\preceq\,}(n_{k}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)\ \ {\vee}\penalty-2\ \ {((l_{k},r_{k}),C_{k})}\mbox{ is }\alpha\mbox{-quasi- normal w.r.t.\ }{{\rm R},{{\rm X}}}\end{array}\right)$; and ${\rm R},{{\rm X}}$ is $\alpha$-shallow confluent up to $\min\\{n_{0},n_{1}\\}{+_{\\!\\!{}_{\alpha}}}(n_{k}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)$. Proof of Claim 3 of the proof of Lemma B.4: The first property is trivial due to $\alpha{\,=\,}\penalty-1\omega.$ By Claim 2 we get $\min\\{n_{0},n_{1}\\}{\,=\,}\penalty-10{\,\preceq\,}(n_{k}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)$ as well as $\min\\{n_{0},n_{1}\\}{+_{\\!\\!{}_{\omega}}}(n_{k}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){\,=\,}\penalty-10{+_{\\!\\!{}_{\omega}}}(n_{k}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){\,=\,}\penalty-1(n_{k}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){\,\prec\,}\max\\{1,n_{k}\\}{\,\preceq\,}\max\\{n_{0},n_{1}\\}{\,=\,}\penalty-1n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}.$ Thus ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $\min\\{n_{0},n_{1}\\}{+_{\\!\\!{}_{\omega}}}(n_{k}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)$ by our first level of induction. Q.e.d. (Claim 3 of the proof of Lemma B.4) Proof of Claim 3 of the proof of Lemma B.5: The first property follows from Claim 2. Since ${\rm R},{{\rm X}}$ is $\alpha$-quasi-normal, $((l_{k},r_{k}),C_{k})$ is $\alpha$-quasi-normal w.r.t. ${\rm R},{{\rm X}}$. By Claim 2 we have $\min\\{n_{0},n_{1}\\}{+_{\\!\\!{}_{\alpha}}}(n_{k}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1){\,\prec\,}\min\\{n_{0},n_{1}\\}{+_{\\!\\!{}_{\alpha}}}n_{k}{\,\preceq\,}n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1}.$ Thus Claim 3 follows from our first level of induction. Q.e.d. (Claim 3 of the proof of Lemma B.5) Claim 4: For any $k\prec 2$ and $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$, if $C_{k}\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\alpha+(n_{k}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}$, then $l_{k}\nu{{\longrightarrow}_{{}_{\\!\alpha+n_{k}}}}r_{k}\nu.$ Proof of Claim 4 of the proof of Lemma B.4: By Claim 2 we have $0{\,\prec\,}n_{k}$ or $n_{k}{\,=\,}\penalty-10\ \ {\wedge}\penalty-2\ \ l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ In the first case Claim 4 is trivial due to $(n_{k}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)+1{\,=\,}\penalty-1n_{k}.$ In the second case $C_{k}\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$ and $l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Thus, by Corollary 2.6, we get $l_{k}\nu{{\longrightarrow}_{{}_{\\!\omega}}}r_{k}\nu,$ which completes the proof of Claim 4 due to $n_{k}{\,=\,}\penalty-10$ in this case. Q.e.d. (Claim 4 of the proof of Lemma B.4) Proof of Claim 4 of the proof of Lemma B.5: By Claim 2 we have $0{\,\prec\,}n_{k}$ and $\alpha{\,=\,}\penalty-10\ {\Rightarrow}\penalty-2\ l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Thus Claim 4 is trivial due to $(n_{k}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)+1{\,=\,}\penalty-1n_{k}.$ Q.e.d. (Claim 4 of the proof of Lemma B.5) Two cases: The variable-overlap case: There are $q_{0}^{\prime}$, $q_{1}^{\prime}$ such that $p{\,=\,}\penalty-1q_{0}^{\prime}q_{1}^{\prime};$ $l_{1}/q_{0}^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}:$ We have $x\mu_{1}/q_{1}^{\prime}{\,=\,}\penalty-1l_{1}\mu_{1}/q_{0}^{\prime}q_{1}^{\prime}{\,=\,}\penalty-1s/p{\,=\,}\penalty-1l_{0}\mu_{0}.$ Claim A of the proof of Lemma B.4: In case of “$i{\,=\,}\penalty-11$” for the ‘$i$’ of Claim 2 we may assume $x\in{{{\rm V}}\\!_{{\rm SIG}}}.$ Proof of Claim A of the proof of Lemma B.4: Otherwise we would have $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}$, which implies $x\mu_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then $l_{0}\mu_{0}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$. We may assume $l_{1-i}\mu_{1-i}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ for the $i$ of Claim 2. Q.e.d. (Claim A of the proof of Lemma B.4) Claim A of the proof of Lemma B.5: We may assume ${{\left({{\begin{array}[]{ll}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-10\ \ {\Rightarrow}\penalty-2\ \ l_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\end{array}\right)}\\\ {\wedge}&{\left(\begin{array}[c]{l}\alpha{\,=\,}\penalty-1\omega\ \ {\Rightarrow}\penalty-2\ \ x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}\end{array}\right)}\\\ \end{array}}}\right)}}.$ Proof of Claim A of the proof of Lemma B.5: The first statement follows from Claim 2. The second is show by contradiction: Suppose we would have $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}$, which implies $x\mu_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then $l_{0}\mu_{0}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$. By Claim 2 we can assume that this is not the case for $\alpha{\,=\,}\penalty-1\omega.$ Q.e.d. (Claim A of the proof of Lemma B.5) By Lemma 2.10 (in case of $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}$), we can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by ($y{\,\in\,}{{\rm V}}$): $y\nu:=\left\\{\begin{array}[]{ll}{x\mu_{1}\penalty-1{[\,q_{1}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}&\mbox{if }y=x\\\ y\mu_{1}&\mbox{otherwise}\\\ \end{array}\right\\}$ and get $y\mu_{1}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}y\nu$ for $y\in{{\rm V}}.$ By ${\mathcal{T}}({{\rm sig},{{\rm X}}})$-monotonicity of ${\longrightarrow}_{{}_{\\!\alpha+n_{0}}}$ we get $r_{1}\mu_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}r_{1}\nu$ and $\begin{array}[]{@{}l@{}}{l_{1}\mu_{1}\penalty-1{[\,q_{0}^{\prime}q_{1}^{\prime}\leftarrow r_{0}\mu_{0}\,]}}=\\\ {{l_{1}\penalty-1{[\,q_{0}^{\prime}\leftarrow x\nu\,]}}\penalty-1{{[\,q^{\prime\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/q^{\prime\prime}{\,=\,}\penalty-1y\in{{\rm V}}\ \wedge\ q^{\prime\prime}{\,\not=\,}q_{0}^{\prime}\,]}}}=\\\ {{{l_{1}\penalty-1{[\,q_{0}^{\prime}\leftarrow x\nu\,]}}\penalty-1{{[\,q^{\prime\prime}\leftarrow x\mu_{1}\ \|\ l_{1}/q^{\prime\prime}{=}x\ \wedge\ q^{\prime\prime}{\not=}q_{0}^{\prime}\,]}}}\penalty-1{{[\,q^{\prime\prime}\leftarrow y\nu\ \|\ x{\not=}l_{1}/q^{\prime\prime}{=}y{\,\in\,}{{\rm V}}\ \wedge\ q^{\prime\prime}{\,\not=\,}q_{0}^{\prime}\,]}}}\\\ {{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}\ \ {l_{1}\penalty-1{{[\,q^{\prime\prime}\leftarrow y\nu\ \|\ l_{1}/q^{\prime\prime}=y\in{{\rm V}}\,]}}}=l_{1}\nu.\end{array}$ Claim B: ${l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}l_{1}\nu.$ Proof of Claim B of the proof of Lemma B.4: By case distinction over the ‘$i$’ of Claim 2: “$i{\,=\,}\penalty-10$”: $n_{0}{\,=\,}\penalty-10{\,\prec\,}n_{1}$ implies ${{\longrightarrow}_{{}_{\\!\omega+n_{0}}}}\subseteq{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}$ by Lemma 2.12. “$i{\,=\,}\penalty-11$”: In this case we have $l_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ By Claim A we may assume $x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}$. Then $l_{1}$ is linear in $x$. Thus ${{\\{\ }q^{\prime\prime}}~{}{\|}\penalty-9\,\ {l_{1}/q^{\prime\prime}{=}x\ \wedge\ q^{\prime\prime}{\not=}q_{0}^{\prime}{\ \\}}}=\emptyset,$ which means that the above reduction takes $0$ steps, i.e. ${l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1l_{1}\nu.$ Q.e.d. (Claim B of the proof of Lemma B.4) Proof of Claim B of the proof of Lemma B.5: By Claim A and the assumption of our lemma we know that $l_{0}$ is linear in $x$. Thus ${{\\{\ }q^{\prime\prime}}~{}{\|}\penalty-9\,\ {l_{1}/q^{\prime\prime}{=}x\ \wedge\ q^{\prime\prime}{\not=}q_{0}^{\prime}{\ \\}}}=\emptyset,$ which means that the above reduction takes $0$ steps, i.e. ${l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}=l_{1}\nu.$ Q.e.d. (Claim B of the proof of Lemma B.5) Claim C: $l_{1}\nu{{\longrightarrow}_{{}_{\\!\alpha+n_{1}}}}r_{1}\nu.$ Proof of Claim C of the proof of Lemma B.4: By case distinction over the ‘$i$’ of Claim 2: “$i{\,=\,}\penalty-10$”: Due to $n_{0}{\,=\,}\penalty-10{\,\prec\,}n_{1}$ this follows directly from Lemma 13.8 (matching its $n_{0}$ to our $n_{0}{\,=\,}\penalty-10$ and its $n_{1}$ to our $n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$) (since $0{\,\preceq\,}n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$ and ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$ by our induction hypothesis). “$i{\,=\,}\penalty-11$”: In this case we have $n_{1}{\,=\,}\penalty-10$ and $l_{1}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Thus, since $C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega}}$, by assumption of the lemma we know that $((l_{1},r_{1}),C_{1})$ is quasi-normal w.r.t. ${\rm R},{{\rm X}}$ and that for all $u\in{{{\mathcal{TERMS}}}({C_{1}})}$ we have $l_{1}\mu_{1}\rhd\,u\mu_{1}$ or $u\mu_{1}{\,\not\in\,}{{\rm dom}({{{\longrightarrow}}})}$ or ${{{\mathcal{V}}}({u})}\subseteq{{{\rm V}}\\!_{{\mathcal{C}}}}.$ In the latter case, since we may assume $x{\,\in\,}{{{\rm V}}\\!_{{\rm SIG}}}$ by Claim A, we get $\forall y{\,\in\,}{{{\mathcal{V}}}({u})}{.}\penalty-1\,\,y\mu_{1}{\,=\,}\penalty-1y\nu$ and, moreover, $\forall\delta{\,\prec\,}n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}{.}\penalty-1\,\,{{\rm R},{{\rm X}}}\mbox{ is $\omega$-shallow confluent up to }\delta$ by our induction hypothesis. In the first case, due to $l_{1}\mu_{1}{\,=\,}\penalty-1s$ our induction hypothesis even implies that ${\rm R},{{\rm X}}$ is $\omega$-shallow confluent up to $n_{0}{+_{\\!\\!{}_{\omega}}}n_{1}$ and $u\mu_{1}$ in $\lhd$. Thus Lemma 13.8 (matching its $n_{0}$ to our $n_{0}$ and its $n_{1}$ to our $n_{1}$) implies that $C_{1}\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n_{1}}}$. Now since $n_{1}{\,=\,}\penalty-10,$ Corollary 2.6 implies $l_{1}\nu{{\longrightarrow}_{{}_{\\!\omega+n_{1}}}}r_{1}\nu.$ Q.e.d. (Claim C of the proof of Lemma B.4) Proof of Claim C of the proof of Lemma B.5: Directly Lemma 13.8 (matching its $n_{0}$ to our $n_{0}$ and its $n_{1}$ to our $n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$) (by Claim 2 and since ${\rm R},{{\rm X}}$ is $\alpha$-quasi-normal and $\alpha$-shallow confluent up to $n_{0}{+_{\\!\\!{}_{\alpha}}}(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)$ by our first level of induction due to $n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1\preceq n_{1}$ by Claim 2). Q.e.d. (Claim C of the proof of Lemma B.5) Q.e.d. (The variable-overlap case) The critical peak case: $p{\,\in\,}{{{\mathcal{POS}}}({l_{1}})};$ $l_{1}/p{\,\not\in\,}{{\rm V}}$: Let $\xi_{0}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi_{0}[{{{\mathcal{V}}}({{l_{0}{=}r_{0}{\longleftarrow}C_{0}}})}]\cap{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}=\emptyset.$ Define ${\rm Y}:={{{\mathcal{V}}}({({l_{0}{=}r_{0}{\longleftarrow}C_{0}})\xi_{0},{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}.$ Define $\xi_{1}:={{{}_{{{\rm V}}}{\upharpoonleft}{\rm id}}}$. Let $\varrho$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{{1}}&\mbox{ if }x\in{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}\\\ x\xi_{0}^{-1}\mu_{{0}}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{{0}}\xi_{0}\varrho{\,=\,}\penalty-1l_{0}\xi_{0}\xi_{0}^{-1}\mu_{0}{\,=\,}\penalty-1s/p{\,=\,}\penalty-1l_{1}\mu_{1}/p{\,=\,}\penalty-1l_{1}\varrho/p{\,=\,}\penalty-1(l_{1}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0}\xi_{0}},{l_{1}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}\\!=\\!{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}.$ Claim A: We may assume ${\left(\begin{array}[c]{l}p{\,=\,}\penalty-1\emptyset\ \ {\vee}\penalty-2\ \ \forall y{\,\in\,}{{{\mathcal{V}}}({l_{1}})}{.}\penalty-1\,\,y\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}})}\end{array}\right)}.$ Proof of Claim A: Otherwise, when $p{\,\not=\,}\emptyset$ holds but $\forall y{\,\in\,}{{{\mathcal{V}}}({l_{1}})}{.}\penalty-1\,\,y\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}})}$ is not the case, there are some $x\in{{{\mathcal{V}}}({l_{1}})},$ $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $x\mu_{1}{{\longrightarrow}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}x\nu$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\mu_{1}{\,=\,}\penalty-1y\nu.$ Due to $l_{1}\mu_{1}/p\lhd l_{1}\mu_{1}{\,=\,}\penalty-1s$ by our first level of induction from $r_{0}\xi_{0}\sigma\varphi{{\longleftarrow}_{{}_{\\!\alpha+n_{0}}}}l_{0}\xi_{0}\sigma\varphi{\,=\,}\penalty-1l_{1}\sigma\varphi/p{\,=\,}\penalty-1l_{1}\mu_{1}/p{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}l_{1}\nu/p$ we know that there must be some $u$ with $r_{0}\xi_{0}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}l_{1}\nu/p.$ Due to Claim 3, by Lemma 13.8 (matching its $n_{0}$ to our $\min\\{n_{0},n_{1}\\}$ and its $n_{1}$ to our $(n_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)$) $C_{1}\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\alpha+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}$. Then Claim 4 implies $l_{1}\nu{{\longrightarrow}_{{}_{\\!\alpha+n_{1}}}}r_{1}\nu.$ Due to $l_{1}\mu_{1}{{{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}l_{1}\nu$ and ${{\longrightarrow}_{{}_{\\!\omega+\alpha}}}\ {\subseteq}\ \rhd$ we get $l_{1}\nu\lhd l_{1}\mu_{1}{\,=\,}\penalty-1s.$ Thus, by our first level of induction, from ${l_{1}\nu\penalty-1{[\,p\leftarrow u\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}l_{1}\nu{{\longrightarrow}_{{}_{\\!\alpha+n_{1}}}}r_{1}\nu$ we get $t_{0}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\xi_{0}\sigma\varphi\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}{l_{1}\nu\penalty-1{[\,p\leftarrow r_{0}\xi_{0}\sigma\varphi\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}{l_{1}\nu\penalty-1{[\,p\leftarrow u\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}r_{1}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}r_{1}\mu_{1}{\,=\,}\penalty-1t_{1}.$ Q.e.d. (Claim A) If ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi_{0}\,]}}\sigma{\,=\,}\penalty-1r_{1}\sigma,$ then we are finished due to $t_{0}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi_{0}\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1t_{1}.$ Otherwise we have $(\,({l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi_{0}\,]}},C_{0}\xi_{0},\mathchar 259\relax_{0}),\,(r_{1},C_{1}\xi_{1},\mathchar 259\relax_{1}),\,l_{1},\,\sigma,\,p\,)\in{\rm CP}({\rm R})$ with the following additional structure: In the proof of Lemma B.4: By Claim 2 the critical peak cannot be of the form $(1,1)$. Moreover, if it is of the form $(0,0)$, then we have $\forall k{\,\prec\,}2{.}\penalty-1\,\,l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ which by Claim 2 for some $i\prec 2$ implies ${{{\mathcal{TERMS}}}({C_{1-i}\xi_{1-i}\sigma\varphi})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ and then ${{{\mathcal{TERMS}}}({C_{1-i}\xi_{1-i}\sigma})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})},$ i.e. ${{{\mathcal{TERMS}}}({C_{0}\xi_{0}\sigma\,C_{1}\xi_{1}\sigma})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}.$ In the proof of Lemma B.5: For all $k\prec 2$ we have: $\alpha{\,=\,}\penalty-10\ {\Rightarrow}\penalty-2\ \mathchar 259\relax_{k}{\,=\,}\penalty-10.$ If $\alpha{\,=\,}\penalty-1\omega$ and $\mathchar 259\relax_{k}{\,=\,}\penalty-10$ for some $k\prec 2$, then $l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ which by Claim 2 implies ${{{\mathcal{TERMS}}}({C_{k}\xi_{k}\sigma\varphi})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})},$ and then ${{{\mathcal{TERMS}}}({C_{k}\xi_{k}\sigma})}{\nsubseteq}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}.$ Now $C_{0}\xi_{0}\sigma\varphi=C_{0}\mu_{0}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\alpha+(n_{0}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}$; $C_{1}\xi_{1}\sigma\varphi=C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\alpha+(n_{1}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}$. Since $l_{1}\sigma\varphi{\,=\,}\penalty-1l_{1}\mu_{1}{\,=\,}\penalty-1s,$ by our induction hypothesis we have $\ \forall(\delta,s^{\prime}){\,\,{\prec\\!\\!\lhd}\,\,}(n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1},l_{1}\sigma\varphi){.}\penalty-1\,\,$ (${\rm R},{{\rm X}}$ is $\alpha$-shallow confluent up to $\delta$ and $s^{\prime}$ in $\lhd$). By Claim 1 we get $\forall q{\,\in\,}{{{\mathcal{POS}}}({l_{1}\sigma\varphi})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\emptyset{\,\not=\,}{q}\lll_{l_{1}\sigma\varphi}p\ \ {\Rightarrow}\penalty-2\ \ l_{1}\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\},q}}}})}\end{array}\right)}.$ This means $l_{1}\sigma\varphi{\,\not\in\,}A(p,\min\\{n_{0},n_{1}\\}).$ Furthermore, $(n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1},l_{1}\sigma\varphi)=(n_{0}{+_{\\!\\!{}_{\alpha}}}n_{1},s){\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}(\beta,\hat{s}).$ Therefore, in case of $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}})},$ by Claim A and by the assumed form of $\alpha$-shallow joinability up to $\beta$ and $\hat{s}$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$], we get $t_{0}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi_{0}\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}r_{1}\sigma\varphi{\,=\,}\penalty-1t_{1}.$ Otherwise, when $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}})}$ is not the case, by ${{\longrightarrow}_{{}_{\\!\omega+\alpha}}}\subseteq{\rhd}$ and the Axiom of Choice there is some $\varphi^{\prime}{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}y\varphi^{\prime}{\,\not\in\,}\linebreak{{\rm dom}({{{\longrightarrow}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}})}.$ Then, of course, $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\xi_{i}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}y\xi_{i}\sigma\varphi^{\prime}.$ Due to Claim 3, by Lemma 13.8 (matching its $n_{0}$ to our $\min\\{n_{0},n_{1}\\}$ and its $n_{1}$ to our $(n_{i}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)$) we know that $\forall i{\,\prec\,}2{.}\penalty-1\,\,C_{i}\xi_{i}\sigma\varphi^{\prime}\mbox{ is fulfilled w.r.t.\ }{{\longrightarrow}_{{}_{\\!\alpha+(n_{i}{\mathchoice{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.60275pt}{$\dot{\raisebox{0.60275pt}[0.0pt]{$-$}}$}}{\raisebox{-0.45209pt}{$\scriptstyle\dot{\raisebox{0.31645pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.30138pt}{$\scriptscriptstyle\dot{\raisebox{0.15068pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1)}}}.$ Then Claim 4 implies $\forall i{\,\prec\,}2{.}\penalty-1\,\,l_{i}\xi_{i}\sigma\varphi^{\prime}{{\longrightarrow}_{{}_{\\!\alpha+n_{i}}}}r_{i}\xi_{i}\sigma\varphi^{\prime}.$ Furthermore, we have ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi_{0}\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi_{0}\,]}}\sigma\varphi^{\prime}$ and $r_{1}\sigma\varphi^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+\min\\{n_{0},n_{1}\\}}}}r_{1}\sigma\varphi,$ cf. the diagram below. Therefore, in case of $l_{1}\sigma\varphi{\,=\,}\penalty-1l_{1}\sigma\varphi^{\prime}$ the proof succeeds like above with $\varphi^{\prime}$ instead of $\varphi$. Otherwise we have $l_{1}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+\alpha}}}l_{1}\sigma\varphi^{\prime}.$ Then due to ${{\longrightarrow}_{{}_{\\!\omega+\alpha}}}\ {\subseteq}\ \rhd$ we get $s{\,=\,}\penalty-1l_{1}\sigma\varphi\rhd l_{1}\sigma\varphi^{\prime}.$ Therefore, by our first level of induction, from ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi_{0}\,]}}\sigma\varphi^{\prime}{{\longleftarrow}_{{}_{\\!\alpha+n_{0},p}}}{l_{1}\penalty-1{[\,p\leftarrow l_{0}\xi_{0}\,]}}\sigma\varphi^{\prime}{\,=\,}\penalty-1l_{1}\sigma\varphi^{\prime}{{\longrightarrow}_{{}_{\\!\alpha+n_{1},\emptyset}}}r_{1}\sigma\varphi^{\prime}$ we conclude ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi_{0}\,]}}\sigma\varphi^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\alpha+n_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\alpha+n_{0}}}}r_{1}\sigma\varphi^{\prime}.$ Q.e.d. (The critical peak case) Q.e.d. (Lemma B.4 and Lemma B.5) Proof of Lemma B.6 For $(\delta,s){\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}(\beta,\hat{s})$ we are going to show that ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $\delta$ and $s$ in $\lhd$ by induction over $(\delta,s)$ in ${\,\,{\prec\\!\\!\lhd}\,\,}$. Suppose that for $\bar{n}_{0},\bar{n}_{1}\prec\omega$ we have $(\max\\{\bar{n}_{0},\bar{n}_{1}\\},s){\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}(\beta,\hat{s})$ and $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+\bar{n}_{0}}}}s{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}t_{1}^{\prime}.$ We have to show $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+\max\\{\bar{n}_{0},\bar{n}_{1}\\}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+\max\\{\bar{n}_{0},\bar{n}_{1}\\}}}}t_{1}^{\prime}.$ In case of $\exists i{\,\prec\,}2{.}\penalty-1\,\,t_{i}^{\prime}{\,=\,}\penalty-1s$ this is trivially true by Lemma 2.12. In case of $\bar{n}_{0}{\,=\,}\penalty-1\bar{n}_{1}{\,=\,}\penalty-10$ this is true by confluence of ${\longrightarrow}_{{}_{\\!\omega}}$. Using symmetry in $0$ and $1$, w.l.o.g. we may assume $\bar{n}_{0}{\,\preceq\,}\bar{n}_{1}.$ Thus, assuming $\bar{n}_{0}{\,\preceq\,}\bar{n}_{1}{\,\succ\,}0,$ for $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+\bar{n}_{0}}}}t_{0}{{\longleftarrow}_{{}_{\\!\omega+\bar{n}_{0}}}}s{{\longrightarrow}_{{}_{\\!\omega+\bar{n}_{1}}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}t_{1}^{\prime}$ using the induction hypothesis that $\forall(m,w^{\prime}){\,\,{\prec\\!\\!\lhd}\,\,}(\max\\{\bar{n}_{0},\bar{n}_{1}\\},s){.}\penalty-1\,\,$ ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $m$ and $w^{\prime}$ in $\lhd$ we have to show $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}t_{1}^{\prime}.$ Claim 0: Now it is sufficient to show $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}t_{1}$ for some $u$. Proof of Claim 0: By Lemma B.1 we have $s\rhd t_{0},t_{1}.$ Thus, due to353535Note that it is this change from $\bar{n}_{0}$ to $\bar{n}_{1}$ in $\max\\{\bar{n}_{0},\bar{n}_{1}\\}$ that makes a two level treatment similar to that for $\omega$-shallow confluence (i.e. considering $\bar{n}_{0}{+_{\\!\\!{}_{\omega}}}\bar{n}_{1}$ instead of $\bar{n}_{0}{+}\bar{n}_{1}$) impossible because then for $\bar{n}_{0}{\,=\,}\penalty-10{\,\prec\,}\bar{n}_{1}$ we would get $\max_{{}_{\omega}}\\{\bar{n}_{0},\bar{n}_{1}\\}{\,\prec\,}\omega{\,\preceq\,}\max_{{}_{\omega}}\\{\bar{n}_{1},\bar{n}_{1}\\}$ and thus would not be allowed to apply our induction hypothesis here. $(\max\\{\bar{n}_{1},\bar{n}_{1}\\},t_{1}){\,\,{\prec\\!\\!\lhd}\,\,}\penalty-1(\max\\{\bar{n}_{0},\bar{n}_{1}\\},s),$ by our induction hypotheses $u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}t_{1}^{\prime}$ (cf. diagram below) implies the existence of some $v$ with $u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}t_{1}^{\prime}$ and then $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+\bar{n}_{0}}}}t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}v$ implies $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+\bar{n}_{1}}}}v.$ Q.e.d. (Claim 0) Defining $n:=\bar{n}_{1}{\mathchoice{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-1.20552pt}{$\dot{\raisebox{1.20552pt}[0.0pt]{$-$}}$}}{\raisebox{-0.90417pt}{$\scriptstyle\dot{\raisebox{0.63292pt}[0.0pt]{$\scriptstyle-$}}$}}{\raisebox{-0.60275pt}{$\scriptscriptstyle\dot{\raisebox{0.30138pt}[0.0pt]{$\scriptscriptstyle-$}}$}}}1$ and using Lemma 2.12 we can now restate our proof task in the following symmetric way: For $n\prec\omega$, $t_{0}{{\longleftarrow}_{{}_{\\!\omega+n+1,p}}}s{{\longrightarrow}_{{}_{\\!\omega+n+1,q}}}t_{1}$ using the induction hypothesis that $\forall(m,w^{\prime}){\,\,{\prec\\!\\!\lhd}\,\,}(n{+}1,s){.}\penalty-1\,\,$ ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $m$ and $w^{\prime}$ in $\lhd$ we have to show $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}t_{1}.$ In case of ${p}\,{\parallel}\,{q}$ this is trivial. Otherwise one of $p,q$ must be a prefix of the other, w.l.o.g. say that $q$ is a prefix of $p$. In case of $q{\,\not=\,}\emptyset$ due to ${\rhd_{{}_{\rm ST}}}\subseteq{\rhd}$ we get $s/q\lhd s$ and the proof finished by our induction hypothesis and ${\mathcal{T}}({{\rm sig},{{\rm X}}})$-monotonicity of ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}$. Thus we may assume $q{\,=\,}\penalty-1\emptyset.$ We start a second level of induction on $p$ in $\lll_{s}$. Thus we may assume the following induction hypothesis: $\forall q{\,\in\,}{{{\mathcal{POS}}}({s})}{.}\penalty-1\,\,\forall t_{0}^{\prime},t_{1}^{\prime}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&q\lll_{s}p\\\ {\wedge}&t_{0}^{\prime}{{\longleftarrow}_{{}_{\\!\omega+n+1,q}}}s{{\longrightarrow}_{{}_{\\!\omega+n+1,\emptyset}}}t_{1}^{\prime}\\\ \end{array}}}\right)}}\ \ {\Rightarrow}\penalty-2\ \ t_{0}^{\prime}{\downarrow_{{}_{\omega+n+1}}}t_{1}^{\prime}\end{array}\right)}$ Now for $k<2$ there must be $((l_{k},r_{k}),C_{k})\in{\rm R}$; $\mu_{k}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$; with $C_{k}\mu_{k}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $s{\,=\,}\penalty-1l_{1}\mu_{1};$ $s/p{\,=\,}\penalty-1l_{0}\mu_{0};$ $t_{0}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}};$ $t_{1}{\,=\,}\penalty-1r_{1}\mu_{1}.$ Moreover, for $k<2$ we define $\mathchar 259\relax_{k}:=\left\\{\mbox{$\begin{array}[]{ll}0&\mbox{ if }l_{k}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}\\\ 1&\mbox{ otherwise}\end{array}$}\right\\}.$ Claim 1: We may assume that $\forall q{\,\in\,}{{{\mathcal{POS}}}({s})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\emptyset{\,\not=\,}{q}\lll_{s}p\ \ {\Rightarrow}\penalty-2\ \ s{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega+n+1,q}}}})}\end{array}\right)}.$ Proof of Claim 1: Otherwise there must be some $q\in{{{\mathcal{POS}}}({s})}$; ${((l_{2},r_{2}),C_{2})}\in{\rm R}$; $\mu_{2}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$; with $C_{2}\mu_{2}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$; $s/q{\,=\,}\penalty-1l_{2}\mu_{2};$ and $\emptyset{\,\not=\,}{q}\lll_{s}p.$ By our second induction level we get ${l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}r_{1}\mu_{1}$ for some $w_{1}$; cf. the diagram below. Next we are going to show that there is some $w_{0}$ with ${l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}.$ Note that (since ${{\longrightarrow}}\subseteq\rhd$ implies $s{\rhd}{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}$) this finishes the proof since then $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{1}$ by our first level of induction implies $t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{0}{\downarrow_{{}_{\omega+n+1}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}t_{1}.$ In case of ${{p}\,{\parallel}\,{q}}$ we simply can choose $w_{0}:={{l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}.$ Otherwise, there must be some $\bar{p}$, $\hat{p}$, $\hat{q}$, with $p{\,=\,}\penalty-1\bar{p}\hat{p},$ $q{\,=\,}\penalty-1\bar{p}\hat{q},$ and ${{(\hat{p}{\,=\,}\penalty-1\emptyset\ {\vee}\penalty-2\ \hat{q}{\,=\,}\penalty-1\emptyset)}}.$ Now it suffices to show ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}$ for some $w_{0}^{\prime}$, because by ${\mathcal{T}}({{\rm sig},{{\rm X}}})$-monotonicity of ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}$ we then have ${l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{s\penalty-1{[\,\bar{p}\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{{s\penalty-1{[\,\bar{p}\leftarrow s/\bar{p}\,]}}\penalty-1{[\,\bar{p}\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1\\\ {s\penalty-1{[\,\bar{p}\leftarrow{s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{s\penalty-1{[\,\bar{p}\leftarrow w_{0}^{\prime}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}{s\penalty-1{[\,\bar{p}\leftarrow{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}\,]}}{\,=\,}\penalty-1\\\ {{s\penalty-1{[\,\bar{p}\leftarrow s/\bar{p}\,]}}\penalty-1{[\,\bar{p}\hat{q}\leftarrow r_{2}\mu_{2}\,]}}{\,=\,}\penalty-1{s\penalty-1{[\,\bar{p}\hat{q}\leftarrow r_{2}\mu_{2}\,]}}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,q\leftarrow r_{2}\mu_{2}\,]}}.$ Note that ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{\longleftarrow}_{{}_{\\!\omega+n+1,\hat{p}}}}s/\bar{p}{{\longrightarrow}_{{}_{\\!\omega+n+1,\hat{q}}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}.$ In case of $\bar{p}{\,\not=\,}\emptyset$ (since then ${\rhd_{{}_{\rm ST}}}\subseteq\rhd$ implies $s\rhd s/\bar{p}$) we get some $w_{0}^{\prime}$ with ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}$ by our first level of induction. Otherwise, in case of $\bar{p}{\,=\,}\penalty-1\emptyset,$ our disjunction from above means ${{(p{\,=\,}\penalty-1\emptyset\ {\vee}\penalty-2\ q{\,=\,}\penalty-1\emptyset)}}.$ Since we have $\emptyset{\,\not=\,}q$ by our initial assumption, we may assume $q{\,=\,}\penalty-1\hat{q}{\,\not=\,}\emptyset$ and $p{\,=\,}\penalty-1\hat{p}{\,=\,}\penalty-1\bar{p}{\,=\,}\penalty-1\emptyset.$ Then the above divergence reads ${s/\bar{p}\penalty-1{[\,\hat{p}\leftarrow r_{0}\mu_{0}\,]}}{{\longleftarrow}_{{}_{\\!\omega+n+1,\emptyset}}}s{{\longrightarrow}_{{}_{\\!\omega+n+1,q}}}{s/\bar{p}\penalty-1{[\,\hat{q}\leftarrow r_{2}\mu_{2}\,]}}$ and we get the required joinability by our second induction level due to $q\lll_{s}p.$ Q.e.d. (Claim 1) Claim 2: We may assume: $\exists i{\,\prec\,}2{.}\penalty-1\,\,l_{i}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 2: Since $C_{i}\mu_{i}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$, by Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{1}$) $l_{i}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ implies ${l_{i}\mu_{i}}{{\longrightarrow}_{{}_{\\!\omega}}}{r_{i}\mu_{i}}$ and then $s{{\longrightarrow}_{{}_{\\!\omega}}}t_{i}.$ Thus, if the claim does not hold, we have $t_{0}{{\longleftarrow}_{{}_{\\!\omega}}}s{{\longrightarrow}_{{}_{\\!\omega}}}t_{1}$ and the proof is finished by confluence of ${\longrightarrow}_{{}_{\\!\omega}}$. Q.e.d. (Claim 2) Now we have two cases: The variable overlap case: $p{\,=\,}\penalty-1q_{0}q_{1};\ l_{1}/q_{0}{\,=\,}\penalty-1x\in{{\rm V}}$ : We have $x\mu_{1}/q_{1}{\,=\,}\penalty-1l_{1}\mu_{1}/q_{0}q_{1}{\,=\,}\penalty-1s/p{\,=\,}\penalty-1l_{0}\mu_{0}.$ By Lemma 2.10 (in case of $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}$), we can define $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ by ($y{\,\in\,}{{\rm V}}$): $y\nu:=\left\\{\begin{array}[]{ll}{x\mu_{1}\penalty-1{[\,q_{1}\leftarrow r_{0}\mu_{0}\,]}}&\mbox{if }y{\,=\,}\penalty-1x\\\ y\mu_{1}&\mbox{otherwise}\\\ \end{array}\right\\}$ and get $y\mu_{1}{{{\stackrel{{\scriptstyle\scriptscriptstyle=}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}y\nu$ for $y{\,\in\,}{{\rm V}}$. By Corollary 2.8: $t_{0}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,q_{0}q_{1}\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1{{l_{1}\penalty-1{[\,q_{0}\leftarrow x\nu\,]}}\penalty-1{{[\,q^{\prime}\leftarrow y\mu_{1}\ \|\ l_{1}/q^{\prime}{\,=\,}\penalty-1y\in{{\rm V}}\ {\wedge}\penalty-2\ q^{\prime}{\,\not=\,}q_{0}\,]}}}\ {{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}$ ${l_{1}\penalty-1{{[\,q^{\prime}\leftarrow y\nu\ \|\ l_{1}/q^{\prime}{\,=\,}\penalty-1y{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1}\nu;$ $t_{1}{\,=\,}\penalty-1r_{1}\mu_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}r_{1}\nu.$ It suffices to show $l_{1}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{1}\nu,$ which follows from our first level of induction saying that ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$ by Lemma A.7 (matching its $n_{0}$ to our $n{+}1$ and its $n_{1}$ to our $n$). Q.e.d. (The variable overlap case) The critical peak case: $p\in{{{\mathcal{POS}}}({l_{1}})};\ l_{1}/p\not\in{{\rm V}}$ : Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{l_{0}{=}r_{0}{\longleftarrow}C_{0}}})}]\cap{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}{\,=\,}\penalty-1\emptyset.$ Define ${\rm Y}:={{{\mathcal{V}}}({({l_{0}{=}r_{0}{\longleftarrow}C_{0}})\xi,{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}$ . Let $\varrho$ be given by $\ x\varrho{\,=\,}\penalty-1\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1}&\mbox{ if }x\in{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}\\\ x\xi^{-1}\mu_{0}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0}\xi\varrho{\,=\,}\penalty-1l_{0}\xi\xi^{-1}\mu_{0}{\,=\,}\penalty-1s/p{\,=\,}\penalty-1l_{1}\mu_{1}/p{\,=\,}\penalty-1l_{1}\varrho/p{\,=\,}\penalty-1(l_{1}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0}\xi},{l_{1}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Claim A: We may assume ${\left(\begin{array}[c]{l}p{\,=\,}\penalty-1\emptyset\ \ {\vee}\penalty-2\ \ \forall y{\,\in\,}{{{\mathcal{V}}}({l_{1}})}{.}\penalty-1\,\,y\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega+n+1}}}})}\end{array}\right)}.$ Proof of Claim A: Otherwise, when $p{\,\not=\,}\emptyset$ holds but $\forall y{\,\in\,}{{{\mathcal{V}}}({l_{1}})}{.}\penalty-1\,\,y\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega+n+1}}}})}$ is not the case, there are some $x\in{{{\mathcal{V}}}({l_{1}})},$ $\nu{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $x\sigma\varphi{{\longrightarrow}_{{}_{\\!\omega+n+1}}}x\nu$ and $\forall y{\,\in\,}{{\rm V}}{\setminus}\\{x\\}{.}\penalty-1\,\,y\mu_{1}{\,=\,}\penalty-1y\nu.$ Due to $l_{1}\mu_{1}/p\lhd l_{1}\mu_{1}{\,=\,}\penalty-1s$ by our first level of induction from $r_{0}\xi\sigma\varphi{{\longleftarrow}_{{}_{\\!\omega+n+1}}}l_{0}\xi\sigma\varphi{\,=\,}\penalty-1l_{1}\sigma\varphi/p{\,=\,}\penalty-1l_{1}\mu_{1}/p{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}l_{1}\nu/p$ we know that there must be some $u$ with $r_{0}\xi\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}u{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}l_{1}\nu/p.$ Due to $l_{1}\mu_{1}{{{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}l_{1}\nu$ and ${{\longrightarrow}}\ {\subseteq}\ \rhd$ we get $l_{1}\nu\lhd l_{1}\mu_{1}{\,=\,}\penalty-1s.$ Thus, by our first level of induction, from ${l_{1}\nu\penalty-1{[\,p\leftarrow u\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}l_{1}\nu{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{1}\nu$ (which is due to Lemma A.7 and our first level of induction saying that ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$) we get $t_{0}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\sigma\varphi\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{l_{1}\nu\penalty-1{[\,p\leftarrow r_{0}\xi\sigma\varphi\,]}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{l_{1}\nu\penalty-1{[\,p\leftarrow u\,]}}{\downarrow_{{}_{\omega+n+1}}}r_{1}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}r_{1}\mu_{1}{\,=\,}\penalty-1t_{1}.$ Q.e.d. (Claim A) If ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1}\sigma,$ then we are finished due to $t_{0}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1}\sigma\varphi{\,=\,}\penalty-1t_{1}.$ Otherwise $(({l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}},C_{0}\xi,\mathchar 259\relax_{0}),\ (r_{1},C_{1},\mathchar 259\relax_{1}),\ l_{1},\ \sigma,\ p\ )$ is a critical peak in ${\rm CP}({\rm R})$. Furthermore, due to Claim 2, this critical peak is not of the form $(0,0)$. Now $(C_{0}\xi\,C_{1})\sigma\varphi{\,=\,}\penalty-1C_{0}\mu_{0}\,C_{1}\mu_{1}$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Due to $l_{1}\sigma\varphi{\,=\,}\penalty-1l_{1}\varrho{\,=\,}\penalty-1l_{1}\mu_{1}{\,=\,}\penalty-1s,$ by our first level of induction we get $\ \forall(\delta,s^{\prime}){\,\,{\prec\\!\\!\lhd}\,\,}(n{+}1,l_{1}\sigma\varphi){.}\penalty-1\,\,$ (${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $\delta$ and $s^{\prime}$ in $\lhd$). By Claim 1 we get $\forall q{\,\in\,}{{{\mathcal{POS}}}({l_{1}\sigma\varphi})}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}\emptyset{\,\not=\,}{q}\lll_{l_{1}\sigma\varphi}p\ \ {\Rightarrow}\penalty-2\ \ l_{1}\sigma\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega+n+1,q}}}})}\end{array}\right)}.$ This means $l_{1}\sigma\varphi{\,\not\in\,}A(p,n{+}1).$ Furthermore, $(n{+}1,l_{1}\sigma\varphi){\,=\,}\penalty-1(\max\\{n_{0},n_{1}\\},s){\,\,\underline{{\>\\!\\!{\prec\\!\\!\lhd}\\!}}\,\,}(\beta,\hat{s}).$ Thus, in case of $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega+n+1}}}})},$ by Claim A and the assumed by $\omega$-level joinability up to $\beta$ and $\hat{s}$ w.r.t. ${\rm R},{{\rm X}}$ and $\lhd$ [besides $A$] (matching the definition’s $n_{0}$ and $n_{1}$ to our $n{+}1$) we get $t_{0}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi{\downarrow_{{}_{\omega+n+1}}}r_{1}\sigma\varphi{\,=\,}\penalty-1t_{1}.$ Otherwise, when $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\varphi{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega+n+1}}}})}$ is not the case, by ${{\longrightarrow}}\subseteq{\rhd}$ and the Axiom of Choice there is some $\varphi^{\prime}{\,\in\,}{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}y\varphi^{\prime}{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!\omega+n+1}}}})}.$ Then, of course, $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\xi\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}y\xi\sigma\varphi^{\prime}$ and $\forall y{\,\in\,}{{\rm V}}{.}\penalty-1\,\,y\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}y\sigma\varphi^{\prime}.$ By Lemma A.7 (due to our first level of induction saying that ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$) we know that $C_{0}\xi\sigma\varphi^{\prime}$ and $C_{1}\sigma\varphi^{\prime}$ are fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+n}}$. Furthermore, we have ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi^{\prime}$ and $r_{1}\sigma\varphi^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}r_{1}\sigma\varphi.$ Therefore, in case of $l_{1}\sigma\varphi{\,=\,}\penalty-1l_{1}\sigma\varphi^{\prime}$ the proof succeeds like above with $\varphi^{\prime}$ instead of $\varphi$. Otherwise we have $l_{1}\sigma\varphi{{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longrightarrow}}}}}l_{1}\sigma\varphi^{\prime}.$ Then due to ${{\longrightarrow}}\ {\subseteq}\ \rhd$ we get $s{\,=\,}\penalty-1l_{1}\sigma\varphi\rhd l_{1}\sigma\varphi^{\prime}.$ Therefore, by our first level of induction, from ${l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi^{\prime}{{\longleftarrow}_{{}_{\\!\omega+n+1}}}{l_{1}\penalty-1{[\,p\leftarrow l_{0}\xi\,]}}\sigma\varphi^{\prime}{\,=\,}\penalty-1l_{1}\sigma\varphi^{\prime}{{\longrightarrow}_{{}_{\\!\omega+n+1}}}r_{1}\sigma\varphi^{\prime}$ (which is due to Lemma A.7 and our first level of induction saying that ${\rm R},{{\rm X}}$ is $\omega$-level confluent up to $n$) we conclude $t_{0}{\,=\,}\penalty-1{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega+n+1}}}{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}\sigma\varphi^{\prime}{\downarrow_{{}_{\omega+n+1}}}r_{1}\sigma\varphi^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+n+1}}}r_{1}\sigma\varphi{\,=\,}\penalty-1t_{1}.$ Q.e.d. (The critical peak case) Q.e.d. (Lemma B.6) Proof of Lemma B.7 1.: Since the direction “$\supseteq$” is trivial we only have to show “$\subseteq$” and begin with the first equation. For $t^{\prime}\in{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}$ there are some $t\in{\rm T}$ and $p\in{{{\mathcal{POS}}}({t})}$ with $t/p{\,=\,}\penalty-1t^{\prime}.$ Now, in case of $t^{\prime}\rightrightarrows t^{\prime\prime}$ by sort-invariance and T-monotonicity of $\rightrightarrows$ we get $t{\,=\,}\penalty-1{t\penalty-1{[\,p\leftarrow t^{\prime}\,]}}\rightrightarrows{t\penalty-1{[\,p\leftarrow t^{\prime\prime}\,]}}{\,\in\,}{\rm T},$ which implies $t^{\prime\prime}{\,\in\,}{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}.$ Thus we have shown ${{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}{\ {\ {\subseteq}\ }\ }{{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\circ}\ {{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}.$ In case of $t^{\prime}{\,\in\,}{\rm T}$ we can choose $p{\,=\,}\penalty-1\emptyset$ and get $t^{\prime\prime}{\,\in\,}{\rm T},$ which proves ${{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}{\ {\ {\subseteq}\ }\ }{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\circ}\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}.$ 2.: For ${\rm T}\ni t{\rhd_{{}_{\rm ST}}}t^{\prime}\rightrightarrows t^{\prime\prime}$ there is a $p\in{{{\mathcal{POS}}}({t})}$; $p{\,\not=\,}\emptyset$ with $t^{\prime}{\,=\,}\penalty-1t/p.$ By sort- invariance and T-monotonicity of $\rightrightarrows$ we get $t={t\penalty-1{[\,p\leftarrow t^{\prime}\,]}}\rightrightarrows{t\penalty-1{[\,p\leftarrow t^{\prime\prime}\,]}}{\rhd_{{}_{\rm ST}}}t^{\prime\prime}$ and ${t\penalty-1{[\,p\leftarrow t^{\prime\prime}\,]}}{\,\in\,}{\rm T}.$ 3.: The subset relationship is simple: ${{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{(\rightrightarrows\cup{\rhd_{{}_{\rm ST}}})}^{\scriptscriptstyle+}}\ {\ {\ {\subseteq}\ }\ }\ {{\trianglelefteq_{{}_{\rm ST}}}}\ {\circ}\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\trianglerighteq_{{}_{\rm ST}}}}\ {\circ}\ {{(\rightrightarrows\cup{\rhd_{{}_{\rm ST}}})}^{\scriptscriptstyle+}}\ {\ {\ {\subseteq}\ }\ }\ {{\trianglelefteq_{{}_{\rm ST}}}}\ {\circ}\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{(\rightrightarrows\cup{\rhd_{{}_{\rm ST}}})}^{\scriptscriptstyle+}}.$ The first equality follows from (1) and $\ {{{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}}{\ {\ {\ {=}\ }\ }\ }{{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}}\ {\circ}\ {{{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}}}\ .$ For the second equality consider the following subset relationships as a word rewriting system over the alphabet $\\{{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}},{\rightrightarrows},{{\rhd_{{}_{\rm ST}}}}\\}$ (containing three letters): $\begin{array}[]{l@{$\nottight{\nottight{\nottight\subseteq}}$}l l}{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}}\ {\circ}\ {\rightrightarrows}\hfil$\ $\ {\ {\subseteq}\ }\ $\ &{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\circ}\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}}&;\\\ {{\rhd_{{}_{\rm ST}}}}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}}\hfil$\ $\ {\ {\subseteq}\ }\ $\ &{{\rhd_{{}_{\rm ST}}}}&;\\\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}}\hfil$\ $\ {\ {\subseteq}\ }\ $\ &{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\circ}\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}}&;\\\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\circ}\ {\rightrightarrows}\hfil$\ $\ {\ {\subseteq}\ }\ $\ &{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ {\circ}\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}&.\\\ \end{array}$ First note that the system is sound: The first rule was proved in (2). The second is transitivity of ${\rhd_{{}_{\rm ST}}}$. The third and fourth are implied by (1). Since the number of substrings from $\\{{\rightrightarrows},{{\rhd_{{}_{\rm ST}}}}\\}^{2}$ is decreased by 1 by each of the rules, the word rewriting system is terminating. Thus, since all normal forms from ${{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\\{{\rightrightarrows},{{\rhd_{{}_{\rm ST}}}}\\}^{+}$ are in $\\{{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}{{\rhd_{{}_{\rm ST}}}}\\}\cup\\{{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\,{\rightrightarrows}\\}^{+}[\\{{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}{{\rhd_{{}_{\rm ST}}}}\\}],$ we get ${{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{{(\ {\rightrightarrows}\cup{{\rhd_{{}_{\rm ST}}}}\ )}}^{\scriptscriptstyle+}}\ {\ {\subseteq}\ }\ {\left(\begin{array}[c]{l}{{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}}\end{array}\right)}\ {\cup}\ {\left(\begin{array}[c]{l}{{{(\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {\rightrightarrows}\ )}}^{\scriptscriptstyle+}}\ {\circ}\ {{{(\ {{{{}_{{\rm T}}{\upharpoonleft}{\rm id}}}}\ {\circ}\ {{\rhd_{{}_{\rm ST}}}}\ )}}^{\scriptscriptstyle=}}\end{array}\right)}.$ Using (1) again as well as ${{{\rhd_{{}_{\rm ST}}}}^{\scriptscriptstyle=}}\ {\subseteq}\ {\trianglerighteq_{{}_{\rm ST}}},$ this implies the one direction; the other direction as well as the special case are trivial. 4.: By the first equation of (3) we conclude ${\rhd}\ {\ {\subseteq}\ }\ {\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}\ {\times}\ {\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}$ as well as transitivity of $\rhd$. Suppose that $\rhd$ is not terminating. By the first equation of (3) there is some ${{r}:{{{{{\bf N}}}\rightarrow{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}}}}$ with $\forall i{\,\in\,}{{\bf N}}{.}\penalty-1\,\,{(\ r_{i}{\rightrightarrows}r_{i+1}\ {\vee}\penalty-2\ r_{i}{\rhd_{{}_{\rm ST}}}r_{i+1}\ )}.$ There is some $t_{0}\in{\rm T}$ and some $p_{0}\in{{{\mathcal{POS}}}({t_{0}})}$ with $t_{0}/p_{0}{\,=\,}\penalty-1r_{0}.$ Moreover, there is also some ${{p}:{{{{{\bf N}}_{+}}\rightarrow{{{\bf N}}^{\ast}}}}}$ such that $\forall i{\,\in\,}{{\bf N}}{.}\penalty-1\,\,{\left(\begin{array}[c]{l}{{\left({{\begin{array}[]{ll}&r_{i}\rightrightarrows r_{i+1}\\\ {\wedge}&p_{i+1}{\,=\,}\penalty-1\emptyset\\\ \end{array}}}\right)}}\ \ {\vee}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&r_{i}{\rhd_{{}_{\rm ST}}}r_{i+1}\\\ {\wedge}&r_{i}/p_{i+1}{\,=\,}\penalty-1r_{i+1}\\\ \end{array}}}\right)}}\end{array}\right)}.$ Define $(t_{n})_{n\in{{\bf N}}}$ inductively by $t_{n+1}:={t_{n}\penalty-1{[\,p_{0}\ldots p_{n+1}\leftarrow r_{n+1}\,]}}.$ Claim 2: For each $n\in{{\bf N}}$ we get ${{\left({{\begin{array}[]{ll}&t_{n},t_{n+1}{\,\in\,}{\rm T}\\\ {\wedge}&t_{n}/p_{0}\ldots p_{n}{\,=\,}\penalty-1r_{n}\\\ {\wedge}&t_{n+1}/p_{0}\ldots p_{n+1}{\,=\,}\penalty-1r_{n+1}\\\ {\wedge}&{\left(\begin{array}[c]{l}t_{n}{\rightrightarrows}t_{n+1}\ \ {\vee}\penalty-2\ \ {{\left({{\begin{array}[]{ll}&t_{n}{\,=\,}\penalty-1t_{n+1}\\\ {\wedge}&r_{n}{\rhd_{{}_{\rm ST}}}r_{n+1}\\\ \end{array}}}\right)}}\end{array}\right)}\\\ \end{array}}}\right)}}.$ Proof of Claim 2: We have $t_{n}{\,\in\,}{\rm T}$ and $t_{n}/p_{0}\ldots p_{n}{\,=\,}\penalty-1r_{n}$ in case of $n{\,=\,}\penalty-10$ by our choice above and otherwise inductively by Claim 2. In case of $r_{n}{\rightrightarrows}r_{n+1}\ {\wedge}\penalty-2\ p_{n+1}{\,=\,}\penalty-1\emptyset,$ since $\rightrightarrows$ is sort- invariant and T-monotonic, we thus get: $t_{n}{\,=\,}\penalty-1{t_{n}\penalty-1{[\,p_{0}\ldots p_{n}\leftarrow r_{n}\,]}}\rightrightarrows{t_{n}\penalty-1{[\,p_{0}\ldots p_{n}\leftarrow r_{n+1}\,]}}{\,=\,}\penalty-1{t_{n}\penalty-1{[\,p_{0}\ldots p_{n}p_{n+1}\leftarrow r_{n+1}\,]}}{\,=\,}\penalty-1t_{n+1}{\,\in\,}{\rm T}.$ Otherwise we have $r_{n}{\rhd_{{}_{\rm ST}}}r_{n+1}$ and $r_{n}/p_{n+1}{\,=\,}\penalty-1r_{n+1}$ and get: ${\rm T}{\,\ni\,}t_{n}{\,=\,}\penalty-1{t_{n}\penalty-1{[\,p_{0}\ldots p_{n}\leftarrow r_{n}\,]}}{\,=\,}\penalty-1{t_{n}\penalty-1{[\,p_{0}\ldots p_{n}\leftarrow{r_{n}\penalty-1{[\,p_{n+1}\leftarrow r_{n+1}\,]}}\,]}}{\,=\,}\penalty-1\\\ {{t_{n}\penalty-1{[\,p_{0}\ldots p_{n}\leftarrow r_{n}\,]}}\penalty-1{[\,p_{0}\ldots p_{n}p_{n+1}\leftarrow r_{n+1}\,]}}{\,=\,}\penalty-1{t_{n}\penalty-1{[\,p_{0}\ldots p_{n}p_{n+1}\leftarrow r_{n+1}\,]}}{\,=\,}\penalty-1t_{n+1}.$ In both cases we have $t_{n+1}/p_{0}\ldots p_{n+1}{\,=\,}\penalty-1{t_{n}\penalty-1{[\,p_{0}\ldots p_{n+1}\leftarrow r_{n+1}\,]}}/p_{0}\ldots p_{n+1}{\,=\,}\penalty-1r_{n+1}.$ Q.e.d. (Claim 2) Since $\rhd_{{}_{\rm ST}}$ is terminating, Claim 2 contradicts $\rightrightarrows$ being terminating (below all $t\in{\rm T}$). If $\rightrightarrows$ and T are ${\rm X}$-stable, additionally, then $\rhd$ is ${\rm X}$-stable too, because ${\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}$, ${{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}$, and $\rhd_{{}_{\rm ST}}$ are. Here is an example for $\rhd$ not sort-invariant and not T-monotonic: Let $A,B$ be two different sorts. Let ${\alpha}({{\mathsf{a}}})=A$ , ${\alpha}({\mathsf{f}})=A{\ {\rightarrow}\ }B$ , ${\alpha}({\mathsf{g}})=A{\ {\rightarrow}\ }A$ . Define $\rightrightarrows:=\emptyset$ and ${\rm T}:={\mathcal{T}}$. Then we have $\rhd\,={\rhd_{{}_{\rm ST}}}$ and therefrom: ${{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}}\rhd{{\mathsf{a}}}$ (hence not sort- invariant); and ${{\mathsf{g}}{(}{{{\mathsf{a}}}}{)}}\rhd{{\mathsf{a}}}$ but ${{\mathsf{f}}{(}{{{\mathsf{g}}{(}{{{\mathsf{a}}}}{)}}}{)}}\ntriangleright{{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}}$ (hence not T-monotonic). 5.: Take the signature from the example in the proof of (4). Define $\rightrightarrows\,:=\\{({{\mathsf{a}}},{{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}})\\}$ and ${\rm T}:={\mathcal{T}}$. Now $\rightrightarrows$ is a T-monotonic (indeed!), terminating relation on $\mathcal{T}$ that is not sort-invariant; whereas $\rhd$ is not irreflexive: ${{\mathsf{a}}}\ \rightrightarrows\ {{\mathsf{f}}{(}{{{\mathsf{a}}}}{)}}\ {\rhd_{{}_{\rm ST}}}\ {{\mathsf{a}}}$ . If one changes ${\alpha}({\mathsf{f}})$ to be ${\alpha}({\mathsf{f}})=A{\ {\rightarrow}\ }A$ , then $\rightrightarrows$ is a sort-invariant, terminating relation on $\mathcal{T}$ that is not T-monotonic but $\emptyset$-monotonic; whereas neither $\rhd$ nor ${{{(\rightrightarrows\cup{\rhd_{{}_{\rm ST}}})}}}^{\scriptscriptstyle+}$ (in contrast to ${{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[\emptyset]}}{\upharpoonleft}{\rm id}}}\circ{{{{(\rightrightarrows\cup{\rhd_{{}_{\rm ST}}})}}}^{\scriptscriptstyle+}}$) are irreflexive. Q.e.d. (Lemma B.7) Proof of Lemma B.8 For the proof of Claim 3 below, we enrich the signatures by a new sort $s_{\rm new}$ and new constructor symbols ${\mathsf{eq}}_{\bar{s}}$ for each old sort $\bar{s}\in{{\mathbb{S}}}$ with arity $\bar{s}\bar{s}{\ {\rightarrow}\ }s_{\rm new}$ and $\bot$ with arity $s_{\rm new}$. We take (in addition to R) the following set of new rules (with $X_{\bar{s}}\in{{{\rm V}}\\!_{{{\rm SIG}},{\bar{s}}}}$ for $\bar{s}\in{{\mathbb{S}}}$): ${\rm R}^{\prime}:={{\\{\ }{{{\mathsf{eq}}_{\bar{s}}}{(}{X_{\bar{s}}}{,\,}{X_{\bar{s}}}{)}}=\bot}~{}{\|}\penalty-9\,\ {\bar{s}\in{{\mathbb{S}}}{\ \\}}}.$ Since the sort restrictions do not allow ${\longrightarrow}_{{}_{\\!{\rm R}\cup{\rm R}^{\prime},{{\rm X}},\beta}}$ to make any use of terms of the sort $s_{\rm new}$ when rewriting terms of an “old” sort, we get $\forall\beta\preceq\omega{+}\omega{.}\penalty-1\,\,\ {{\longrightarrow}_{{}_{\\!{\rm R}\cup{\rm R}^{\prime},{{\rm X}},\beta}}}\cap({{\mathcal{T}}({{\rm sig},{{\rm X}}})}{\times}{\mathcal{T}})\ =\ {{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}},\beta/{\rm sig}/{\rm cons}}}}$ (the latter being defined over the non-enriched signatures). Thus, ${\rm T}:={{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}{[\\{\hat{s}\\}]},$ ${{{}_{{\rm T}}{\upharpoonleft}{\longrightarrow}}},$ and ${{{}_{\trianglerighteq{[{\rm T}]}}{\upharpoonleft}{\longrightarrow}}}$ do not change when we exchange the one ${\longrightarrow}$ with the other. We use ‘$\rhd_{{}_{\rm ST}}$’ to denote the subterm ordering over the enriched signature. For keeping the assumptions of our lemma valid for this subterm ordering (instead of the subterm ordering on the non-enriched signature) we have to extend $\rhd$ with ${{{\mathsf{eq}}_{\bar{s}}}{(}{t_{0}}{,\,}{t_{1}}{)}}\rhd t^{\prime}$ if $\exists i{\,\prec\,}2{.}\penalty-1\,\,t_{i}{\trianglerighteq_{{}_{\rm ST}}}t^{\prime}$ for some $\bar{s}\in{{\mathbb{S}}}$ and $t_{0},t_{1}\in{{\mathcal{T}}({{\rm sig},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}_{\bar{s}}$. This extension neither changes ${\trianglerighteq}{[{\rm T}]}$ nor ${{{}_{\trianglerighteq{[{\rm T}]}}{\upharpoonleft}{\longrightarrow}}}.$ Thus, since ${{{}_{\trianglerighteq{[{\rm T}]}}{\upharpoonleft}{\longrightarrow}}}$ is not changed by any of the extensions, it now suffices to show its confluence after the extensions. Since the sort restrictions do not allow a term of the sort $s_{\rm new}$ to be a proper subterm of any other term, it is obvious that after the extension of $\rhd$ we still may assume either that ${{{}_{{\rm T}}{\upharpoonleft}{{\longrightarrow}_{{}_{\\!{\rm R}\cup{\rm R}^{\prime},{{\rm X}}}}}}}$ is terminating and ${\rhd}={\rhd_{{}_{\rm ST}}}$ or that ${{{}_{\trianglerighteq{[{\rm T}]}}{\upharpoonleft}{{\longrightarrow}_{{}_{\\!{\rm R}\cup{\rm R}^{\prime},{{\rm X}}}}}}}{\ {\subseteq}\ }{\rhd},$ ${\rhd_{{}_{\rm ST}}}{\ {\subseteq}\ }{\rhd},$ and $\rhd$ is a wellfounded ordering on $\mathcal{T}$. Moreover, again due to the sort restrictions not allowing a term of the sort $s_{\rm new}$ to be a proper subterm of any other term, if $w\,{{({\longleftarrow}{\cup}\,\lhd)}^{\scriptscriptstyle+}}\,\,(\hat{t}/p)\sigma\varphi$ holds for the extended ${\longrightarrow}$ and $\rhd$ and if $\hat{t}$ is an old term, then this also holds for the non-extended ${\longrightarrow}$ and $\rhd$. Therefore, (as no new critical peaks occur) the critical peaks keep being $\rhd$-quasi overlay joinable. We define ${{\longrightarrow}_{{}_{\\!\beta}}}:={{\longrightarrow}_{{}_{\\!{\rm R}\cup{\rm R}^{\prime},{{\rm X}},\beta}}}$ for any ordinal $\beta$ with $\beta\prec\omega{+}\omega;$ and ${{\longrightarrow}}:={{\longrightarrow}_{{}_{\\!\omega+\omega}}}:={{\longrightarrow}_{{}_{\\!{\rm R}\cup{\rm R}^{\prime},{{\rm X}}}}}.$ Since ${\longrightarrow}$ is sort-invariant, T-monotonic (cf. Corollary 2.8), and terminating below all $t\in{\rm T}$, by Lemma B.7(4), ${\rhd}^{\prime}\ {\ {:=}\ }\ {{{}_{{\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}}{\upharpoonleft}{\rm id}}}\circ{{{{({{\longrightarrow}}\cup{\rhd_{{}_{\rm ST}}})}}}^{\scriptscriptstyle+}}$ is a wellfounded ordering on ${\trianglerighteq_{{}_{\rm ST}}}{[{\rm T}]}.$ In case of ${\rhd}{\,=\,}\penalty-1{\rhd_{{}_{\rm ST}}},$ we define ${>}:={\rhd}^{\prime}$ . Otherwise, in case that ${{{}_{\trianglerighteq{[{\rm T}]}}{\upharpoonleft}{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}}}{\ {\subseteq}\ }{\rhd},$ ${\rhd_{{}_{\rm ST}}}{\ {\subseteq}\ }{\rhd},$ and $\rhd$ is a wellfounded ordering, we define ${>}\ {:=}\ {\rhd}\cap({{\trianglerighteq}{[{\rm T}]}}\times{{\trianglerighteq}{[{\rm T}]}})$ . In any case, $>$ is a wellfounded ordering on ${\trianglerighteq}{[{\rm T}]}$ containing ${{{}_{\trianglerighteq{[{\rm T}]}}{\upharpoonleft}{\rm id}}}\circ{{{{({{\longrightarrow}}\cup{\rhd_{{}_{\rm ST}}}\cup{\rhd})}}}^{\scriptscriptstyle+}}.$ This means in particular that ${\trianglerighteq}{[{\rm T}]}$ is closed under ${\longrightarrow}$, $\rhd_{{}_{\rm ST}}$, and $\rhd$. We say that $P(v,u,s,t,\mathchar 261\relax)$ holds if for $v,u,t\in{{\mathcal{T}}({{\rm sig},{{\rm X}}})}$ and $s\in{\trianglerighteq}{[{\rm T}]}$ with $v{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}u;$ and $s{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}t;$ $\mathchar 261\relax\subseteq{{{\mathcal{POS}}}({u})}$ with $\forall p,q{\,\in\,}\mathchar 261\relax{.}\penalty-1\,\,{(\ p{\,\not=\,}q\ {\Rightarrow}\penalty-2\ {{p}\,{\parallel}\,{q}}\ )}$ and $\forall o{\,\in\,}\mathchar 261\relax{.}\penalty-1\,\,u/o{\,=\,}\penalty-1s;$ we have $v\downarrow{u\penalty-1{{[\,o\leftarrow t\ \|\ o{\,\in\,}\mathchar 261\relax\,]}}}.$ Now (by $\mathchar 261\relax:=\\{\emptyset\\}$) it suffices to show that $P(v,u,s,t,\mathchar 261\relax)$ holds for all appropriate $v,u,s,t,\mathchar 261\relax$. We will show this by terminating induction over the lexicographic combination of the following orderings: $\begin{array}[]{@{}lll}1.&>&\\\ 2.&\succ&\\\ 3.&\succ&\\\ \end{array}$ using the following measure on $(v,u,s,t,\mathchar 261\relax)$: $\begin{array}[]{@{}ll}1.&s\\\ 2.&\mbox{the smallest ordinal }\beta\preceq\omega{+}\omega\mbox{ for which }v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\beta}}}u\\\ 3.&\mbox{the smallest }n\in{{\bf N}}\mbox{ for which }v{\stackrel{{\scriptstyle n}}{{{\longleftarrow}}}_{{}_{\\!\beta}}}u\mbox{ for the }\beta\mbox{ of (2)}\hskip 59.50012pt\mbox{}\\\ \end{array}$ For the limit ordinals $0$, $\omega$, $\omega{+}\omega$ in the second position of the measure, the induction step is trivial ( ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!0}}}\subseteq{\rm id}$ ; ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}\subseteq\bigcup_{i\in{{\bf N}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!i}}}$ ; ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+\omega}}}\subseteq\bigcup_{i\in{{\bf N}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega+i}}}$ ). Thus, as we now suppose a smallest $(v,u,s,t,\mathchar 261\relax)$ with $P(v,u,s,t,\mathchar 261\relax)$ not holding for, the second position of the measure must be a non-limit ordinal $\beta{+}1.$ As $P(v,u,s,t,\mathchar 261\relax)$ holds trivially for $u=v$ or $s=t$ we have some $u^{\prime},s^{\prime}$ with $v{\stackrel{{\scriptstyle n}}{{{\longleftarrow}}}_{{}_{\\!\beta+1}}}u^{\prime}{{\longleftarrow}_{{}_{\\!\beta+1}}}u$ $(n{\,\in\,}{{\bf N}})$ (with $\forall m{\,\in\,}{{\bf N}}{.}\penalty-1\,\,(v{\stackrel{{\scriptstyle m}}{{{\longleftarrow}}}_{{}_{\\!\beta+1}}}u\ {\ {\Rightarrow}\penalty-2\ }\ m{\,\succ\,}n)$) and $s{\longrightarrow}s^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}t.$ Now for a contradiction it is sufficient to show Claim: There is some $z$ with $v{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}z{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}{u\penalty-1{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 261\relax\,]}}}.$ because then we have $z\downarrow{u\penalty-1{{[\,o\leftarrow t\ \|\ o\in\mathchar 261\relax\,]}}}$ by $P(z,{u\penalty-1{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 261\relax\,]}}},s^{\prime},t,\mathchar 261\relax)$, which is smaller than $(v,u,s,t,\mathchar 261\relax)$ in the first position of the measure by $s{\longrightarrow}s^{\prime}$. Claim 0: We may assume $\forall p^{\prime\prime}{\,\in\,}{{{\mathcal{POS}}}({s})}{\setminus}\\{\emptyset\\}{.}\penalty-1\,\,s/p^{\prime\prime}{\,\not\in\,}{{\rm dom}({{{\longrightarrow}}})}.$ Proof of Claim 0: Otherwise there are some $p^{\prime\prime}\in{{{\mathcal{POS}}}({s})}{\setminus}\\{\emptyset\\}$ and some $s^{\prime\prime}$ with $s/p^{\prime\prime}{\longrightarrow}s^{\prime\prime}.$ Then, by $P(s^{\prime},s,s/p^{\prime\prime},s^{\prime\prime},\\{p^{\prime\prime}\\})$, which is smaller in the first position of the measure by $s{\rhd_{{}_{\rm ST}}}s/p^{\prime\prime}$, we get $s^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}s^{\prime\prime\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}{s\penalty-1{[\,p^{\prime\prime}\leftarrow s^{\prime\prime}\,]}}$ for some $s^{\prime\prime\prime}$. Similarly, by $P(v,u,s/p^{\prime\prime},s^{\prime\prime},\mathchar 261\relax p^{\prime\prime})$ we get $v{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}v^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}{u\penalty-1{{[\,p\leftarrow s^{\prime\prime}\ \|\ p{\,\in\,}\mathchar 261\relax p^{\prime\prime}\,]}}}{\,=\,}\penalty-1{u\penalty-1{{[\,o\leftarrow{s\penalty-1{[\,p^{\prime\prime}\leftarrow s^{\prime\prime}\,]}}\ \|\ o{\,\in\,}\mathchar 261\relax\,]}}}$ for some $v^{\prime}$. Finally, by $P(v^{\prime},\ {u\penalty-1{{[\,o\leftarrow{s\penalty-1{[\,p^{\prime\prime}\leftarrow s^{\prime\prime}\,]}}\ \|\ o{\,\in\,}\mathchar 261\relax\,]}}},\ {s\penalty-1{[\,p^{\prime\prime}\leftarrow s^{\prime\prime}\,]}},\ s^{\prime\prime\prime},\ \mathchar 261\relax\ ),$ which is smaller in the first position of the measure by $s{{\longrightarrow}}{s\penalty-1{[\,p^{\prime\prime}\leftarrow s^{\prime\prime}\,]}},$ we get $v^{\prime}\downarrow{u\penalty-1{{[\,o\leftarrow s^{\prime\prime\prime}\ \|\ o{\,\in\,}\mathchar 261\relax\,]}}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}{u\penalty-1{{[\,o\leftarrow s^{\prime}\ \|\ o{\,\in\,}\mathchar 261\relax\,]}}}.$ Q.e.d. (Claim 0) By Claim 0 there are some ${((l_{0},r_{0}),C_{0})}\in{\rm R}\cup{\rm R}^{\prime}$; $\mu_{0}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$; with $s=l_{0}\mu_{0}$; $s^{\prime}=r_{0}\mu_{0}$; and $C_{0}\mu_{0}$ is fulfilled w.r.t. ${\longrightarrow}$. Furthermore, we have some $q\in{{{\mathcal{POS}}}({u})}$; ${((l_{1},r_{1}),C_{1})}\in{\rm R}\cup{\rm R}^{\prime}$; $\mu_{1}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$; with $u/q{\,=\,}\penalty-1l_{1}\mu_{1};$ $u^{\prime}{\,=\,}\penalty-1{u\penalty-1{[\,q\leftarrow r_{1}\mu_{1}\,]}};$ $C_{1}\mu_{1}$ fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\beta}}$; and if $C_{1}$ contains some inequality $(u{\not=}v)$ then $\omega{\,\preceq\,}\beta.$ By Claim 0 we may assume that $q$ is not strictly below any $p\in\mathchar 261\relax$, i.e. that there are no $p$, $p^{\prime}$ with $pp^{\prime}{\,=\,}\penalty-1q,$ $p^{\prime}{\,\not=\,}\emptyset,$ and $p{\,\in\,}\mathchar 261\relax.$ Define $\begin{array}[t]{lrll}\mathchar 260\relax&:=&\mathchar 261\relax\setminus(q{{\bf N}}^{\ast})&;\\\ \mathchar 261\relax^{\prime}&:=&{{\\{\ }p^{\prime}}~{}{\|}\penalty-9\,\ {\ qp^{\prime}{\,\in\,}\mathchar 261\relax\ \ {\wedge}\penalty-2\ \ {{(p^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{1}})}\ {\Rightarrow}\penalty-2\ l_{1}/p^{\prime}{\,\in\,}{{\rm V}})}}{\ \\}}}&;\\\ \mathchar 261\relax^{\prime\prime}&:=&{{\\{\ }p^{\prime}}~{}{\|}\penalty-9\,\ {qp^{\prime}{\,\in\,}\mathchar 261\relax{\setminus}(q\mathchar 261\relax^{\prime}){\ \\}}}&.\\\ \end{array}$ Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }p^{\prime\prime}}~{}{\|}\penalty-9\,\ {\exists p^{\prime}{.}\penalty-1\,\,(l_{1}/p^{\prime}=x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax^{\prime}){\ \\}}}.$ Since for $p^{\prime\prime}\in\mathchar 256\relax(x)$ we always have some $p^{\prime}$ with $l_{1}/p^{\prime}{\,=\,}\penalty-1x;$ $x\mu_{1}/p^{\prime\prime}{\,=\,}\penalty-1l_{1}\mu_{1}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1s;$ we have $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,\forall p^{\prime\prime}{\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{1}/p^{\prime\prime}{\,=\,}\penalty-1s.$ (#0) Since the proper subterm ordering is irreflexive we cannot have $s{\rhd_{{}_{\rm ST}}}s,$ and therefore get $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,\forall p^{\prime},p^{\prime\prime}{\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,{(\ p^{\prime}{\,=\,}\penalty-1p^{\prime\prime}\ {\vee}\penalty-2\ {{p^{\prime}}\,{\parallel}\,{p^{\prime\prime}}}\ )}.$ (#1) Due to (#0) and (#1) we can define $\mu_{1}^{\prime}$ by ($x{\,\in\,}{{\rm V}}$): $x\mu_{1}^{\prime}:={x\mu_{1}\penalty-1{{[\,p^{\prime\prime}\leftarrow s^{\prime}\ \|\ p^{\prime\prime}\in\mathchar 256\relax(x)\,]}}}.$ Define for $\bar{w}\in{\mathcal{T}}$: $\mathchar 258\relax_{\bar{w}}:={\\{\ }{p^{\prime}p^{\prime\prime}}~{}{\|}\penalty-9\,\ \exists x{.}\penalty-1\,\,{{(\bar{w}/p^{\prime}{\,=\,}\penalty-1x\ {\wedge}\penalty-2\ p^{\prime\prime}{\,\in\,}\mathchar 256\relax(x))}}{\ \\}}.$ By (#0) we get $\forall\bar{w}{\,\in\,}{\mathcal{T}}{.}\penalty-1\,\,\forall p^{\prime}{\,\in\,}\mathchar 258\relax_{\bar{w}}{.}\penalty-1\,\,\bar{w}\mu_{1}/p^{\prime}{\,=\,}\penalty-1s$ (#$\mathchar 258\relax$1) and by (#1) $\forall\bar{w}{\,\in\,}{\mathcal{T}}{.}\penalty-1\,\,\forall p^{\prime},p^{\prime\prime}{\,\in\,}\mathchar 258\relax_{\bar{w}}{.}\penalty-1\,\,{(\ p^{\prime}{\,=\,}\penalty-1p^{\prime\prime}\ \ {\vee}\penalty-2\ {{p^{\prime}}\,{\parallel}\,{p^{\prime\prime}}}\ )}$ (#$\mathchar 258\relax$2) and $\forall\bar{w}{\,\in\,}{\mathcal{T}}{.}\penalty-1\,\,\bar{w}\mu_{1}^{\prime}{\,=\,}\penalty-1{\bar{w}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 258\relax_{\bar{w}}\,]}}}.$ (#$\mathchar 258\relax$3) Note that for $\mathchar 259\relax:=\mathchar 258\relax_{l_{1}}{\setminus}\mathchar 261\relax^{\prime}$ we have $\mathchar 258\relax_{l_{1}}=\mathchar 261\relax^{\prime}\uplus\mathchar 259\relax.$ (#2) By (#$\mathchar 258\relax$1) and (#2) we get $\forall p^{\prime}\in\mathchar 261\relax^{\prime}\cup\mathchar 259\relax\cup\mathchar 261\relax^{\prime\prime}{.}\penalty-1\,\,l_{1}\mu_{1}/p^{\prime}{\,=\,}\penalty-1s$ (#3) and by (#$\mathchar 258\relax$2) and (#2) $\forall p^{\prime},p^{\prime\prime}{\,\in\,}\mathchar 261\relax^{\prime}{\cup}\mathchar 259\relax{.}\penalty-1\,\,{(\ p^{\prime}{\,=\,}\penalty-1p^{\prime\prime}\ \ {\vee}\penalty-2\ \ {{p^{\prime}}\,{\parallel}\,{p^{\prime\prime}}}\ )}.$ (#4) Since $\forall p^{\prime}{\,\in\,}\mathchar 261\relax^{\prime}{\cup}\mathchar 259\relax{.}\penalty-1\,\,{{(p^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{1}})}\ \ {\Rightarrow}\penalty-2\ \ l_{1}/p^{\prime}\in{{\rm V}})}};$ $\forall p^{\prime\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}{.}\penalty-1\,\,{(\ p^{\prime\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{1}})}\ \ {\wedge}\penalty-2\ \ l_{1}/p^{\prime\prime}\notin{{\rm V}}\ )}$ (#5) we get by (#3) $\forall p^{\prime}{\,\in\,}\mathchar 261\relax^{\prime}{\cup}\mathchar 259\relax{.}\penalty-1\,\,\forall p^{\prime\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}{.}\penalty-1\,\,{{p^{\prime\prime}}\,{\parallel}\,{p^{\prime}}}$ (#6) and then together with (#2) and (#4) $\forall p^{\prime},p^{\prime\prime}\in\mathchar 261\relax^{\prime}\uplus\mathchar 259\relax\uplus\mathchar 261\relax^{\prime\prime}{.}\penalty-1\,\,{(\ p^{\prime}{\,=\,}\penalty-1p^{\prime\prime}\ \ {\vee}\penalty-2\ \ {{p^{\prime}}\,{\parallel}\,{p^{\prime\prime}}}\ )}.$ (#7) Now due to (#2) and (#$\mathchar 258\relax$3) we have $l_{1}\mu_{1}^{\prime}{\,=\,}\penalty-1{l_{1}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax^{\prime}{\cup}\mathchar 259\relax\,]}}}$ (#8) and then by (#6) and (#3) $\forall p^{\prime\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}{.}\penalty-1\,\,l_{1}\mu_{1}^{\prime}/p^{\prime\prime}{\,=\,}\penalty-1s.$ (#9) Summing up and defining we have: $\rule{0.0pt}{8.43889pt}\begin{array}[t]{@{}l@{}lrl@{}ll}&\check{u}_{0}&:=&{u\penalty-1{[\,q\leftarrow{l_{1}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax^{\prime}\,]}}}\,]}}&{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 260\relax\,]}}&;\\\ &\check{u}_{1}&:=&{u\penalty-1{[\,q\leftarrow{l_{1}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax^{\prime}{\cup}\mathchar 259\relax\,]}}}\,]}}&{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 260\relax\,]}}&\\\ \mbox{(by (\\#8))}&&=&{u\penalty-1{[\,q\leftarrow l_{1}\mu_{1}^{\prime}\,]}}&{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 260\relax\,]}}&;\\\ &\check{u}_{2}&:=&{u\penalty-1{[\,q\leftarrow{l_{1}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax^{\prime}{\cup}\mathchar 261\relax^{\prime\prime}\,]}}}\,]}}&{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 260\relax\,]}}&;\\\ &\check{u}_{3}&:=&{u\penalty-1{[\,q\leftarrow{l_{1}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax^{\prime}{\cup}\mathchar 259\relax{\cup}\mathchar 261\relax^{\prime\prime}\,]}}}\,]}}&{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 260\relax\,]}}&\\\ \mbox{(by (\\#6), (\\#8))}&&=&{u\penalty-1{[\,q\leftarrow{l_{1}\mu_{1}^{\prime}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}\,]}}}\,]}}&{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 260\relax\,]}}&;\\\ &u^{\prime}&=&{u\penalty-1{[\,q\leftarrow r_{1}\mu_{1}\,]}}&&;\\\ &\hat{u}_{0}&:=&{u\penalty-1{[\,q\leftarrow r_{1}\mu_{1}^{\prime}\,]}}&{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 260\relax\,]}}&\\\ \mbox{(by Claim~{}2)}&&=&{u\penalty-1{[\,q\leftarrow\bar{u}_{0}\,]}}&{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 260\relax\,]}}&;\\\ &\hat{u}_{i+1}&:=&{u\penalty-1{[\,q\leftarrow\bar{u}_{i+1}\,]}}&{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 260\relax\,]}}&.\\\ \end{array}$ Due to (#3) we have $\check{u}_{2}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+\omega,(q\mathchar 261\relax^{\prime\prime})}}\check{u}_{0}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+\omega,(q\mathchar 259\relax)}}\check{u}_{1}.$ Thus by (#6): $\check{u}_{2}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+\omega,(q\mathchar 259\relax)}}\check{u}_{3}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+\omega,(q\mathchar 261\relax^{\prime\prime})}}\check{u}_{1}.$ We get $\check{u}_{1}{{\longrightarrow}_{{}_{\\!\omega+\omega,q}}}\hat{u}_{0}$ by Lemma 2.7 and Claim 3: $C_{1}\mu_{1}^{\prime}$ is fulfilled. Moreover, we get $\hat{u}_{0}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{0}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v$ for some $w_{0}$ by (#$\mathchar 258\relax$1), (#$\mathchar 258\relax$2), (#$\mathchar 258\relax$3), and $P{{(v,u^{\prime},s,s^{\prime},\mathchar 260\relax\cup(q\mathchar 258\relax_{r_{1}}))}},$ which is smaller in the second or third position of the measure. Claim 1: We may assume that there is some $p{\,\in\,}\mathchar 261\relax^{\prime\prime}$ with ${l_{1}\mu_{1}^{\prime}\penalty-1{[\,p\leftarrow s^{\prime}\,]}}{\,\not=\,}r_{1}\mu_{1}^{\prime}.$ Claim 2: There are some $\bar{n}{\,\in\,}{{\bf N}}$; ${{{\bar{p}}:{{{\\{0,\ldots,\bar{n}{-}1\\}}\rightarrow{{{\bf N}}^{\ast}}}}}};$ ${{{\bar{u}}:{{{\\{0,\ldots,\bar{n}\\}}\rightarrow{{{\mathcal{T}}({{\rm sig},{{\rm X}}})}}}}}};$ such that ${l_{1}\mu_{1}^{\prime}\penalty-1{{[\,p^{\prime\prime}\leftarrow s^{\prime}\ \|\ p^{\prime\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}\,]}}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\bar{u}_{n};$ $\forall i{\,\prec\,}n{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\bar{u}_{i+1}{\,=\,}\penalty-1{\bar{u}_{i}\penalty-1{[\,\bar{p}_{i}\leftarrow\bar{u}_{i+1}/\bar{p}_{i}\,]}}\\\ {\wedge}&\bar{u}_{i+1}/\bar{p}_{i}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\bar{u}_{i}/\bar{p}_{i}<s\\\ \end{array}}}\right)}};$ and $\bar{u}_{0}{\,=\,}\penalty-1r_{1}\mu_{1}^{\prime}.$ Inductively for $i\prec n$ we now get some $w_{i+1}$ with $\hat{u}_{i+1}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{i+1}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}w_{i}$ due to Claim 2 and $P(w_{i},\hat{u}_{i},\bar{u}_{i}/\bar{p}_{i},\bar{u}_{i+1}/\bar{p}_{i},\\{q\bar{p}_{i}\\})$ which is smaller in the first position of the measure by Claim 2. Finally by Claim 2 we get $\check{u}_{3}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}{u\penalty-1{[\,q\leftarrow\bar{u}_{n}\,]}}{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 260\relax\,]}}{\,=\,}\penalty-1\hat{u}_{n}.$ This completes the proof of Claim due to ${u\penalty-1{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 261\relax\,]}}}{\,=\,}\penalty-1\check{u}_{2}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{n}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v.$ Proof of Claim 1: In case of $p,p^{\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}$ with ${l_{1}\mu_{1}^{\prime}\penalty-1{[\,p\leftarrow s^{\prime}\,]}}{\,=\,}\penalty-1r_{1}\mu_{1}^{\prime}$ and ${l_{1}\mu_{1}^{\prime}\penalty-1{[\,p^{\prime}\leftarrow s^{\prime}\,]}}{\,=\,}\penalty-1r_{1}\mu_{1}^{\prime}$ we cannot have ${{p}\,{\parallel}\,{p^{\prime}}}$ because then by (#9) we would get the contradiction $s{\,=\,}\penalty-1l_{1}\mu_{1}^{\prime}/p{\,=\,}\penalty-1{l_{1}\mu_{1}^{\prime}\penalty-1{[\,p^{\prime}\leftarrow s^{\prime}\,]}}/p{\,=\,}\penalty-1r_{1}\mu_{1}^{\prime}/p{\,=\,}\penalty-1{l_{1}\mu_{1}^{\prime}\penalty-1{[\,p\leftarrow s^{\prime}\,]}}/p{\,=\,}\penalty-1s^{\prime}{<}s.$ Therefore, if Claim 1 does not hold, i.e. if $\forall p^{\prime\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}{.}\penalty-1\,\,{l_{1}\mu_{1}^{\prime}\penalty-1{[\,p^{\prime\prime}\leftarrow s^{\prime}\,]}}{\,=\,}\penalty-1r_{1}\mu_{1}^{\prime},$ by (#7) must have we have ${\,\|{\mathchar 261\relax^{\prime\prime}}\|\,}{\,\preceq\,}1.$ In case $\mathchar 261\relax^{\prime\prime}{\,=\,}\penalty-1\emptyset,$ we have $\check{u}_{3}{\,=\,}\penalty-1\check{u}_{1}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{0}.$ Otherwise, in case of $\mathchar 261\relax^{\prime\prime}{\,=\,}\penalty-1\\{p\\}$ and ${l_{1}\mu_{1}^{\prime}\penalty-1{[\,p\leftarrow s^{\prime}\,]}}{\,=\,}\penalty-1r_{1}\mu_{1}^{\prime},$ we have ${l_{1}\mu_{1}^{\prime}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}\,]}}}{\,=\,}\penalty-1{l_{1}\mu_{1}^{\prime}\penalty-1{[\,p\leftarrow s^{\prime}\,]}}{\,=\,}\penalty-1r_{1}\mu_{1}^{\prime},$ and then $\check{u}_{3}{\,=\,}\penalty-1\hat{u}_{0}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{0}.$ In both cases we have shown Claim due to ${u\penalty-1{{[\,o\leftarrow s^{\prime}\ \|\ o\in\mathchar 261\relax\,]}}}{\,=\,}\penalty-1\check{u}_{2}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\check{u}_{3}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}w_{0}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v.$ Q.e.d. (Claim 1) Proof of Claim 2: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{l_{0}{=}r_{0}{\longleftarrow}C_{0}}})}]\cap{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}=\emptyset.$ Let $\varrho$ be given by ($x\in{{\rm V}}$): $x\varrho:=\left\\{\begin{array}[c]{@{}l@{}l@{}}x\mu_{1}^{\prime}&\mbox{ if }x\in{{{\mathcal{V}}}({{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})}\\\ x\xi^{-1}\mu_{0}&\mbox{ otherwise}\\\ \end{array}\right\\}$. By (#9) and (#5) for the $p$ of Claim 1 we have $l_{0}\xi\varrho{\,=\,}\penalty-1l_{0}\xi\xi^{-1}\mu_{0}{\,=\,}\penalty-1s{\,=\,}\penalty-1l_{1}\mu_{1}^{\prime}/p{\,=\,}\penalty-1(l_{1}/p)\varrho$ and $l_{1}/p{\,\not\in\,}{{\rm V}}.$ Thus, let ${\rm Y}:={{{\mathcal{V}}}({({l_{0}{=}r_{0}{\longleftarrow}C_{0}})\xi,{l_{1}{=}r_{1}{\longleftarrow}C_{1}}})};$ $\sigma:={{\rm mgu}({\\{(l_{0}\xi,l_{1}/p)\\}},{{\rm Y}})};$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}={{{}_{{\rm Y}}{\upharpoonleft}\varrho}}.$ Let $t_{0}:={l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}$ and $t_{1}:=r_{1}$. By Claim 1 we may assume $t_{0}\sigma{\,\not=\,}t_{1}\sigma$ (since otherwise ${l_{1}\mu_{1}^{\prime}\penalty-1{[\,p\leftarrow s^{\prime}\,]}}{\,=\,}\penalty-1{l_{1}\mu_{1}^{\prime}\penalty-1{[\,p\leftarrow r_{0}\mu_{0}\,]}}{\,=\,}\penalty-1t_{0}\sigma\varphi{\,=\,}\penalty-1t_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\mu_{1}^{\prime}$). Thus $((t_{0},C_{0}\xi,\dots),\ (t_{1},C_{1},\ldots),\ l_{1},\ \sigma,\ p)$ is a critical peak. By Lemma 2.12, $(C_{0}\xi\,C_{1})\sigma\varphi$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!\omega+\omega}}$. Since $(l_{1}/p)\sigma\varphi{\,=\,}\penalty-1s$ it makes sense to define $\mathchar 257\relax:={{\\{\ }p^{\prime}{\,\in\,}{{{\mathcal{POS}}}({l_{1}})}{\setminus}\\{p\\}}~{}{\|}\penalty-9\,\ {l_{1}/p^{\prime}{\,\not\in\,}{{\rm V}}\ {\wedge}\penalty-2\ (l_{1}/p^{\prime})\sigma\varphi{\,=\,}\penalty-1s{\ \\}}}.$ Then by (#5) and (#9) we get $\mathchar 261\relax^{\prime\prime}\subseteq\\{p\\}{\cup}\mathchar 257\relax.$ Thus by $p{\,\in\,}\mathchar 261\relax^{\prime\prime}$ we get $\mathchar 261\relax^{\prime\prime}{\cup}\mathchar 257\relax=\\{p\\}{\cup}\mathchar 257\relax$ and therefore $\begin{array}[t]{lcl@{}l@{}ll}{l_{1}\mu_{1}^{\prime}\penalty-1{{[\,p^{\prime\prime}\leftarrow s^{\prime}\ \|\ p^{\prime\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}\,]}}}&{\,=\penalty-1}&{l_{1}\sigma\varphi}{{[\,p^{\prime\prime}\leftarrow s^{\prime}\ \|\ p^{\prime\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}\,]}}&{{[\,p^{\prime\prime}\leftarrow s\ \|\ p^{\prime\prime}{\,\in\,}\mathchar 257\relax{\setminus}\mathchar 261\relax^{\prime\prime}\,]}}\\\ &{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}&{l_{1}\sigma\varphi}{{[\,p^{\prime\prime}\leftarrow s^{\prime}\ \|\ p^{\prime\prime}{\,\in\,}\mathchar 261\relax^{\prime\prime}\,]}}&{{[\,p^{\prime\prime}\leftarrow s^{\prime}\ \|\ p^{\prime\prime}{\,\in\,}\mathchar 257\relax{\setminus}\mathchar 261\relax^{\prime\prime}\,]}}\\\ &{\,=\penalty-1}&{l_{1}\sigma\varphi}{{[\,p^{\prime\prime}\leftarrow s^{\prime}\ \|\ p^{\prime\prime}{\,\in\,}\\{p\\}{\cup}\mathchar 257\relax\,]}}\\\ &{\,=\penalty-1}&{l_{1}\penalty-1{[\,p\leftarrow r_{0}\xi\,]}}&{{[\,p^{\prime\prime}\leftarrow r_{0}\xi\ \|\ p^{\prime\prime}{\,\in\,}\mathchar 257\relax\,]}}&\sigma\varphi\\\ &{\,=\penalty-1}&{t_{0}}&{{[\,p^{\prime\prime}\leftarrow t_{0}/p\ \|\ p^{\prime\prime}{\,\in\,}\mathchar 257\relax\,]}}&\sigma\varphi.\\\ \end{array}$ Moreover, for $w$ with $w\,{{({\longleftarrow}{\cup}\,\lhd)}^{\scriptscriptstyle+}}\,\,(l_{1}/p)\sigma\varphi$ due to $(l_{1}/p)\sigma\varphi{\,=\,}\penalty-1s$ we have $w{<}s$ and therefore ${\longrightarrow}$ is confluent below $w$ due to $P{{(?,w,w,?,\\{\emptyset\\})}}$ which is smaller in the first position of the measure. Finally, by Claim 0 we get $\forall p^{\prime\prime}{\,\in\,}{{{\mathcal{POS}}}({(l_{1}/p)\sigma\varphi})}{\setminus}\\{\emptyset\\}{.}\penalty-1\,\,(l_{1}/p)\sigma\varphi/p^{\prime\prime}{\,\not\in\,}{{\rm dom}({{{\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}}})}.$ Thus, by $\rhd$-quasi overlay joinability, there are some $\bar{n}{\,\in\,}{{\bf N}};$ ${{{\bar{p}}:{{{\\{0,\ldots,\bar{n}{-}1\\}}\rightarrow{{{\bf N}}^{\ast}}}}}};$ ${{{\bar{u}}:{{{\\{0,\ldots,\bar{n}\\}}\rightarrow{{\mathcal{T}}}}}}};$ with $t_{0}{{[\,p^{\prime\prime}\leftarrow t_{0}/p\ \|\ p^{\prime\prime}{\,\in\,}\mathchar 257\relax\,]}}\sigma\varphi{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\bar{u}_{\bar{n}};$ $\forall i{\,\prec\,}\bar{n}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&\bar{u}_{i+1}{\,=\,}\penalty-1{\bar{u}_{i}\penalty-1{[\,\bar{p}_{i}\leftarrow\bar{u}_{i+1}/\bar{p}_{i}\,]}}\\\ {\wedge}&\bar{u}_{i+1}/\bar{p}_{i}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\bar{u}_{i}/\bar{p}_{i}\;{{({\longleftarrow}\cup{\lhd})}^{\scriptscriptstyle+}}\;(l_{1}/p)\sigma\varphi{\,=\,}\penalty-1s\\\ \end{array}}}\right)}}$ and $\bar{u}_{0}{\,=\,}\penalty-1t_{1}\sigma\varphi{\,=\,}\penalty-1r_{1}\mu_{1}^{\prime}.$ Finally, for all $\bar{v}$ due to $s{\,\in\,}{\trianglerighteq}{[{\rm T}]}$ we know that $s{{({{\longrightarrow}}\cup{\rhd})}^{\scriptscriptstyle+}}\bar{v}$ implies $s{>}\bar{v}.$ Q.e.d. (Claim 2) Proof of Claim 3: For $(\bar{u}{=}\bar{v})$ in $C_{1}$ we have $\bar{u}\mu_{1}{\downarrow_{{}_{\beta}}}\bar{v}\mu_{1}.$ In case of $\beta{\,=\,}\penalty-10,$ due to (#$\mathchar 258\relax$1), (#$\mathchar 258\relax$2), and (#$\mathchar 258\relax$3), we have ${\bar{u}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 258\relax_{\bar{u}}\,]}}}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\mathchar 258\relax_{\bar{u}}}}\penalty-1\bar{u}\mu_{1}{\,=\,}\penalty-1\bar{v}\mu_{1}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\mathchar 258\relax_{\bar{v}}}}\penalty-1{\bar{v}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 258\relax_{\bar{v}}\,]}}}$ and then $\bar{u}\mu_{1}^{\prime}{\,=\,}\penalty-1{\bar{u}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 258\relax_{\bar{u}}\,]}}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\mathchar 258\relax_{\bar{v}}\setminus\mathchar 258\relax_{\bar{u}}}}$ ${\bar{u}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 258\relax_{\bar{u}}{\cup}\mathchar 258\relax_{\bar{v}}\,]}}}{\,=\,}\penalty-1{\bar{v}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 258\relax_{\bar{v}}{\cup}\mathchar 258\relax_{\bar{u}}\,]}}}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\mathchar 258\relax_{\bar{u}}\setminus\mathchar 258\relax_{\bar{v}}}}{\bar{v}\mu_{1}\penalty-1{{[\,p^{\prime}\leftarrow s^{\prime}\ \|\ p^{\prime}{\,\in\,}\mathchar 258\relax_{\bar{v}}\,]}}}{\,=\,}\penalty-1\bar{v}\mu_{1}$. Otherwise, in case of $0{\,\prec\,}\beta,$ we have for the sort $\bar{s}\in{{\mathbb{S}}}$ of $\bar{u}$: $\bot{{{\stackrel{{\scriptstyle\scriptscriptstyle+}}{{{\longleftarrow}}}}}_{{}_{\\!\beta}}}({{{\mathsf{eq}}_{\bar{s}}}{(}{\bar{u}}{,\,}{\bar{v}}{)}})\mu_{1}$. We get $\bot\downarrow({{{\mathsf{eq}}_{\bar{s}}}{(}{\bar{u}}{,\,}{\bar{v}}{)}})\mu_{1}^{\prime}$ by $P{{(\bot,\ ({{{\mathsf{eq}}_{\bar{s}}}{(}{\bar{u}}{,\,}{\bar{v}}{)}})\mu_{1},\ s,\ s^{\prime}\ ,\mathchar 258\relax_{{{{\mathsf{eq}}_{\bar{s}}}{(}{\bar{u}}{,\,}{\bar{v}}{)}}})}}$ which is smaller in the second position. Since there are no rules for $\bot$ and only one for ${\mathsf{eq}}_{\bar{s}}$, this means $\bar{u}\mu_{1}^{\prime}\downarrow\bar{v}\mu_{1}^{\prime}.$ For $(\mbox{{\rm Def}}\,\bar{u})$ in $C_{1}$ we know the existence of some $\vec{\bar{u}}\in{{\mathcal{GT}}({{\rm cons}})}$ with $\vec{\bar{u}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\beta}}}\bar{u}\mu_{1}$. We get some $\hat{\bar{u}}$ with $\vec{\bar{u}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{\bar{u}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\bar{u}\mu_{1}^{\prime}$ by $P{{(\vec{\bar{u}},\bar{u}\mu_{1},s,s^{\prime},\mathchar 258\relax_{\bar{u}})}}$ which is smaller in the second position. By Lemma 2.10 we get $\hat{\bar{u}}\in{{\mathcal{GT}}({{\rm cons}})}$. Finally, for $(\bar{u}{\not=}\bar{v})$ in $C_{1}$ we have some $\vec{\bar{u}},\vec{\bar{v}}\in{{\mathcal{GT}}({{\rm cons}})}$ with $\bar{u}\mu_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\beta}}}\vec{\bar{u}}{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}\vec{\bar{v}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\beta}}}\bar{v}\mu_{1}$ (by Lemma 2.11 and $\omega{\,\preceq\,}\beta$). By applying the same procedure as before twice we get $\hat{\bar{u}},\hat{\bar{v}}\in{{\mathcal{GT}}({{\rm cons}})}$ with $\bar{u}\mu_{1}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{\bar{u}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\vec{\bar{u}}{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}\vec{\bar{v}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{\bar{v}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\bar{v}\mu_{1}^{\prime}$, i.e. $\bar{u}\mu_{1}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}\hat{\bar{u}}{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}\hat{\bar{v}}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\bar{v}\mu_{1}^{\prime}$. Q.e.d. (Claim 3) Q.e.d. (Lemma B.8) Proof of Lemma C.3 Claim 0: ${\longrightarrow}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ are commuting. Proof of Claim 0: By the assumed strong commutation assumption and Lemma 3.3 ${\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ and ${{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}$ are commuting. Since by Corollary 2.14 we have ${{\longrightarrow}}\subseteq{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\subseteq{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}},$ now ${\longrightarrow}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ are commuting, too. Q.e.d. (Claim 0) Claim 1: If ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}$, then ${\longrightarrow}$ is confluent. Proof of Claim 1: ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ and ${\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}$ are commuting by Lemma 3.3. Since by Corollary 2.14 we have ${{\longrightarrow}}\subseteq{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\subseteq{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}},$ now ${\longrightarrow}$ and ${\longrightarrow}$ are commuting, too. Q.e.d. (Claim 1) We are going to show the following property: $w_{0}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+\omega,\,\mathchar 261\relax_{0}}}u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+\omega,\,\mathchar 261\relax_{1}}}w_{1}\quad\ {\Rightarrow}\penalty-2\ \quad w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}w_{1}.$ Claim 2: The above property implies that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}$ and that ${\longrightarrow}$ is confluent. Proof of Claim 2: First we show the strong commutation. By Lemma 3.3 it suffices to show that ${{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}$ strongly commutes over ${\longrightarrow}$. Assume $u^{\prime\prime}{\longleftarrow}u^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}u{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}$ (cf. diagram below). By the strong commutation assumed for our lemma and Corollary 2.14, there are $w_{0}$ and $w_{0}^{\prime}$ with $u^{\prime\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{0}{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}u.$ By the above property there are some $w_{3}$, $w_{1}^{\prime}$ with $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{1}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}w_{1}.$ By Claim 0 we can close the peak $w_{1}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}w_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}$ according to $w_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}w_{2}$ for some $w_{2}^{\prime}$. By the assumed confluence of ${\longrightarrow}_{{}_{\\!\omega}}$, we can close the peak $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ according to $w_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{3}$ for some $w_{3}^{\prime}$. To close the whole diagram, we only have to show that we can close the peak $w_{3}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{3}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime}$ according to $w_{3}^{\prime}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}w_{2}^{\prime},$ which is possible due to the strong commutation assumed for our lemma. Finally, confluence of ${\longrightarrow}$ follows from Claim 1. Q.e.d. (Claim 2) W.l.o.g. let the positions of $\mathchar 261\relax_{i}$ be maximal in the sense that for any $p\in\mathchar 261\relax_{i}$ and $\mathchar 260\relax\subseteq{{{\mathcal{POS}}}({u})}{\cap}(p{{\bf N}}^{+})$ we do not have $u{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+\omega,(\mathchar 261\relax_{i}\setminus\\{p\\})\cup\mathchar 260\relax}}w_{i}$ anymore. Then for each $i\prec 2$ and $p\in\mathchar 261\relax_{i}$ there are ${((l_{i,p},r_{i,p}),C_{i,p})}\in{\rm R}$ and $\mu_{i,p}\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $u/p{\,=\,}\penalty-1l_{i,p}\mu_{i,p},$ $r_{i,p}\mu_{i,p}{\,=\,}\penalty-1w_{i}/p,$ $C_{i,p}\mu_{i,p}$ fulfilled w.r.t. ${\longrightarrow}$. Finally, for each $i\prec 2$: $w_{i}{\,=\,}\penalty-1{u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}\,]}}}.$ Claim 5: We may assume $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,l_{i,p}{\,\not\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}.$ Proof of Claim 5: Define $\mathchar 260\relax_{i}:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{i}}~{}{\|}\penalty-9\,\ {l_{i,p}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}{\ \\}}}$ and $u_{i}^{\prime}:={u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}{\setminus}\mathchar 260\relax_{i}\,]}}}$. If we have succeeded with our proof under the assumption of Claim 5, then we have shown $u_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{0}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{1}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}u_{1}^{\prime}$ for some $v_{0}$, $v_{1}$ (cf. diagram below). By Lemma 13.2 (matching both its $\mu$ and $\nu$ to our $\mu_{i,p}$) we get $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 260\relax_{i}{.}\penalty-1\,\,l_{i,p}\mu_{i,p}{{\longrightarrow}_{{}_{\\!\omega}}}r_{i,p}\mu_{i,p}$ and therefore $\forall i{\,\prec\,}2{.}\penalty-1\,\,u_{i}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}w_{i}.$ Thus from $v_{1}\penalty-1{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\penalty-1u_{1}^{\prime}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\penalty-1w_{1}$ we get $v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{2}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}w_{1}$ for some $v_{2}$ by Claim 0. Due to the assumed confluence of ${\longrightarrow}_{{}_{\\!\omega}}$, we can close the peak $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}u_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{0}$ according to $w_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{0}$ for some $v_{0}^{\prime}$. By the strong commutation assumption of our lemma, from $v_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{0}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{2}$ we can finally conclude $v_{0}^{\prime}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{2}.$ Q.e.d. (Claim 5) Define the set of inner overlapping positions by $\displaystyle\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1}):=\bigcup_{i\prec 2}{{\\{\ }p{\,\in\,}\mathchar 261\relax_{1-i}}~{}{\|}\penalty-9\,\ {\exists q{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}},$ and the length of a term by $\lambda({{f}{(}{t_{0}}{,\,}\ldots{,\,}{t_{m-1}}{)}}):=1+\sum_{j\prec m}\lambda(t_{j}).$ Now we start an induction on $\displaystyle\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})$ in $\,\prec\,$. Define the set of top positions by $\displaystyle\mathchar 258\relax:={{\\{\ }p{\,\in\,}\mathchar 261\relax_{0}{\cup}\mathchar 261\relax_{1}}~{}{\|}\penalty-9\,\ {\neg\exists q{\,\in\,}\mathchar 261\relax_{0}{\cup}\mathchar 261\relax_{1}{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{+}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}{\ \\}}}.$ Since the prefix ordering is wellfounded we have $\forall i{\,\prec\,}2{.}\penalty-1\,\,\forall p{\,\in\,}\mathchar 261\relax_{i}{.}\penalty-1\,\,\exists q{\,\in\,}\mathchar 258\relax{.}\penalty-1\,\,\exists q^{\prime}{\,\in\,}{{\bf N}}^{\ast}{.}\penalty-1\,\,p{\,=\,}\penalty-1qq^{\prime}.$ Then $\forall i{\,\prec\,}2{.}\penalty-1\,\,w_{i}{\,=\,}\penalty-1{w_{i}\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{{u\penalty-1{{[\,p\leftarrow r_{i,p}\mu_{i,p}\ \|\ p{\,\in\,}\mathchar 261\relax_{i}\,]}}}\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{i}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}.$ Thus, it now suffices to show for all $q\in\mathchar 258\relax$ $\rule{0.0pt}{8.43889pt}\raisebox{-4.52083pt}{\rule{0.0pt}{1.50694pt}}w_{0}/q{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}w_{1}/q$ because then we have $w_{0}{\,=\,}\penalty-1{u\penalty-1{{[\,q\leftarrow w_{0}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}{u\penalty-1{{[\,q\leftarrow w_{1}/q\ \|\ q{\,\in\,}\mathchar 258\relax\,]}}}{\,=\,}\penalty-1w_{1}.$ Therefore we are left with the following two cases for $q\in\mathchar 258\relax$: $q{\,\not\in\,}\mathchar 261\relax_{1}$: Then $q{\,\in\,}\mathchar 261\relax_{0}.$ Define $\mathchar 261\relax_{1}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{1}{\ \\}}}$. We have two cases: “The variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}{.}\penalty-1\,\,l_{0,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{0,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}{\ \\}}}.$ Claim 7: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\mu_{0,q}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}x\nu\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,x\nu{\,=\,}\penalty-1{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\\\ \end{array}}}\right)}}.$ Proof of Claim 7: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{0,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{0,q}&{\,=\penalty-1}&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}\\\ &&{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{0,q}$ is not linear in $x$. By the conditions of our lemma and Claim 5 this implies $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Since there is some $(p^{\prime},p^{\prime\prime})\in\mathchar 256\relax(x)$ with $x\mu_{0,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}$ this implies $l_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then $l_{1,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which contradicts Claim 5. Q.e.d. (Claim 7) Claim 8: $l_{0,q}\nu{\,=\,}\penalty-1w_{1}/q.$ Proof of Claim 8: By Claim 7 we get $w_{1}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{0,q}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{0,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{1,qp^{\prime}p^{\prime\prime}}\mu_{1,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{0,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{0,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{0,q}\nu.$ Q.e.d. (Claim 8) Claim 9: $w_{0}/q{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}r_{0,q}\nu.$ Proof of Claim 9: Since $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q},$ this follows directly from Claim 7. Q.e.d. (Claim 9) By claims 8 and 9 it now suffices to show $l_{0,q}\nu{{\longrightarrow}}r_{0,q}\nu,$ which again follows from Lemma C.4 since ${\longrightarrow}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ are commuting by Claim 0 and since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{0,q}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}x\nu$ by Claim 7 and Corollary 2.14. Q.e.d. (“The variable overlap (if any) case”) “The critical peak case”: There is some $p\in\mathchar 261\relax_{1}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{0,q}})}$ with $l_{0,q}/p{\,\not\in\,}{{\rm V}}$: Claim 10: $p{\,\not=\,}\emptyset.$ Proof of Claim 10: If $p{\,=\,}\penalty-1\emptyset,$ then $\emptyset{\,\in\,}\mathchar 261\relax_{1}^{\prime},$ then $q{\,\in\,}\mathchar 261\relax_{1},$ which contradicts our global case assumption. Q.e.d. (Claim 10) Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{1,qp},r_{1,qp}),C_{1,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{0,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{0,q},r_{0,q}),C_{0,q})}})}\\\ x\xi^{-1}\mu_{1,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{1,qp}\xi\varrho{\,=\,}\penalty-1l_{1,qp}\xi\xi^{-1}\mu_{1,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{0,q}\mu_{0,q}/p{\,=\,}\penalty-1l_{0,q}\varrho/p{\,=\,}\penalty-1(l_{0,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{1,qp}\xi},{l_{0,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{0,q}\mu_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}$. We get $\begin{array}[]{l@{}l}u^{\prime}{\,=\penalty-1}&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{1,qp}\mu_{1,qp}\,]}}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+\omega,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}\\\ &{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{1,qp^{\prime}}\mu_{1,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{1}^{\prime}\,]}}}{\,=\,}\penalty-1w_{1}/q.\end{array}$ If ${l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{0,q}\sigma,$ then the proof is finished due to $w_{0}/q{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1r_{0,q}\sigma\varphi{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+\omega,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q.$ Otherwise we have $(\,({l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma,C_{1,qp}\xi\sigma,1),\penalty-1\,(r_{0,q}\sigma,C_{0,q}\sigma,1),\penalty-1\,l_{0,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $p{\,\not=\,}\emptyset$ (due to Claim 10); $C_{1,qp}\xi\sigma\varphi=C_{1,qp}\mu_{1,qp}$ is fulfilled w.r.t. ${\longrightarrow}$; $C_{0,q}\sigma\varphi=C_{0,q}\mu_{0,q}$ is fulfilled w.r.t. ${\longrightarrow}$. Due to Claim 0 and our assumed $\omega$-coarse level parallel closedness we have $u^{\prime}{\,=\,}\penalty-1{l_{0,q}\penalty-1{[\,p\leftarrow r_{1,qp}\xi\,]}}\sigma\varphi\penalty-1{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}\penalty-1v_{1}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}r_{0,q}\sigma\varphi{\,=\,}\penalty-1r_{0,q}\mu_{0,q}{\,=\,}\penalty-1w_{0}/q$ for some $v_{1}$, $v_{2}$. We then have $v_{1}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+\omega,\mathchar 261\relax^{\prime\prime}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+\omega,\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}}w_{1}/q$ for some $\mathchar 261\relax^{\prime\prime}$. By $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 266\relax(\mathchar 261\relax^{\prime\prime},\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\})}\lambda(u^{\prime}/p^{\prime\prime})\ \ \preceq\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}\lambda(u^{\prime}/p^{\prime\prime})\ \ =\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}\setminus\\{p\\}}\lambda(u/qp^{\prime\prime})\ \ \prec$ $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 261\relax_{1}^{\prime}}\lambda(u/qp^{\prime\prime})\ \ =\sum_{p^{\prime}\in q\mathchar 261\relax_{1}^{\prime}}\lambda(u/p^{\prime})\ \ =\sum_{p^{\prime}\in\mathchar 266\relax(\\{q\\},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})\ \ \preceq\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime}),$ due to our induction hypothesis we get some $v_{1}^{\prime}$, $v_{3}$ with $v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{3}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{1}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}w_{1}/q.$ By confluence of ${\longrightarrow}_{{}_{\\!\omega}}$ we can close the peak at $v_{1}$ according to $v_{2}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{4}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{3}$ for some $v_{4}$. Finally by the strong commutation assumption of our lemma, the peak at $v_{3}$ can be closed according to $v_{4}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!\omega}}}v_{1}^{\prime}.$ Q.e.d. (“The critical peak case”) Q.e.d. (“$q{\,\not\in\,}\mathchar 261\relax_{1}$”) $q{\,\in\,}\mathchar 261\relax_{1}$: Define $\mathchar 261\relax_{0}^{\prime}:={{\\{\ }p}~{}{\|}\penalty-9\,\ {qp{\,\in\,}\mathchar 261\relax_{0}{\ \\}}}$. We have two cases: “The second variable overlap (if any) case”: $\forall p{\,\in\,}\mathchar 261\relax_{0}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{1,q}})}{.}\penalty-1\,\,l_{1,q}/p{\,\in\,}{{\rm V}}$: Define a function $\mathchar 256\relax$ on ${\rm V}$ by ($x{\,\in\,}{{\rm V}}$): $\mathchar 256\relax(x):={{\\{\ }(p^{\prime},p^{\prime\prime})}~{}{\|}\penalty-9\,\ {l_{1,q}/p^{\prime}{\,=\,}\penalty-1x\ \wedge\ p^{\prime}p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}{\ \\}}}.$ Claim 11: There is some $\nu\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with $\forall x\in{{\rm V}}{.}\penalty-1\,\,{{\left({{\begin{array}[]{ll}&x\nu{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}x\mu_{1,q}\\\ {\wedge}&\forall p^{\prime}{\,\in\,}{{\rm dom}({\mathchar 256\relax(x)})}{.}\penalty-1\,\,{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu\\\ \end{array}}}\right)}}.$ Proof of Claim 11: In case of ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\emptyset$ we define $x\nu:=x\mu_{1,q}.$ If there is some $p^{\prime}$ such that ${{\rm dom}({\mathchar 256\relax(x)})}{\,=\,}\penalty-1\\{p^{\prime}\\}$ we define $x\nu:={x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}.$ This is appropriate since due to $\forall(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x){.}\penalty-1\,\,x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p^{\prime}p^{\prime\prime}{\,=\,}\penalty-1u/qp^{\prime}p^{\prime\prime}{\,=\,}\penalty-1l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}$ we have $\begin{array}[]{l@{}l@{}l}x\mu_{1,q}&{\,=\penalty-1}&{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}\\\ &&{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1x\nu.\end{array}$ Finally, in case of ${\,\|{{{\rm dom}({\mathchar 256\relax(x)})}}\|\,}\succ 1,$ $l_{1,q}$ is not linear in $x$. By the conditions of our lemma and Claim 5 this implies $x{\,\in\,}{{{\rm V}}\\!_{{\mathcal{C}}}}.$ Since there is some $(p^{\prime},p^{\prime\prime})\in\mathchar 256\relax(x)$ with $x\mu_{1,q}/p^{\prime\prime}{\,=\,}\penalty-1l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}$ this implies $l_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then $l_{0,qp^{\prime}p^{\prime\prime}}{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{{\rm V}}\\!_{{\rm SIG}}}\\!\uplus\\!{{{\rm V}}\\!_{{\mathcal{C}}}}}})}$ which contradicts Claim 5. Q.e.d. (Claim 11) Claim 12: $w_{0}/q{\,=\,}\penalty-1l_{1,q}\nu.$ Proof of Claim 12: By Claim 11 we get $w_{0}/q{\,=\,}\penalty-1{u/q\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {{l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow x\mu_{1,q}\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}\penalty-1{{[\,p^{\prime}p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ \exists x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,(p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow{x\mu_{1,q}\penalty-1{{[\,p^{\prime\prime}\leftarrow r_{0,qp^{\prime}p^{\prime\prime}}\mu_{0,qp^{\prime}p^{\prime\prime}}\ \|\ (p^{\prime},p^{\prime\prime}){\,\in\,}\mathchar 256\relax(x)\,]}}}\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1\\\ {l_{1,q}\penalty-1{{[\,p^{\prime}\leftarrow x\nu\ \|\ l_{1,q}/p^{\prime}{\,=\,}\penalty-1x{\,\in\,}{{\rm V}}\,]}}}{\,=\,}\penalty-1l_{1,q}\nu.$ Q.e.d. (Claim 12) Claim 13: $r_{1,q}\nu{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}w_{1}/q.$ Proof of Claim 13: Since $r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q,$ this follows directly from Claim 11. Q.e.d. (Claim 13) By claims 12 and 13 using Corollary 2.14 it now suffices to show $l_{1,q}\nu{{\longrightarrow}}r_{1,q}\nu,$ which again follows from Lemma C.4 since ${\longrightarrow}$ and ${\longrightarrow}_{{}_{\\!\omega}}$ are commuting by Claim 0 and since $\forall x{\,\in\,}{{\rm V}}{.}\penalty-1\,\,x\mu_{1,q}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}x\nu$ by Claim 11 and Corollary 2.14. Q.e.d. (“The second variable overlap (if any) case”) “The second critical peak case”: There is some $p\in\mathchar 261\relax_{0}^{\prime}{\cap}{{{\mathcal{POS}}}({l_{1,q}})}$ with $l_{1,q}/p{\,\not\in\,}{{\rm V}}$: Let $\xi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\rm V}}})}$ be a bijection with $\xi[{{{\mathcal{V}}}({{((l_{0,qp},r_{0,qp}),C_{0,qp})}})}]\cap{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}=\emptyset.$ Define ${\rm Y}:=\xi[{{{\mathcal{V}}}({{((l_{0,qp},r_{0,qp}),C_{0,qp})}})}]\cup{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}.$ Let $\varrho\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ be given by $\ x\varrho=\left\\{\begin{array}[]{@{}l@{}l@{}}x\mu_{1,q}&\mbox{ if }x\in{{{\mathcal{V}}}({{((l_{1,q},r_{1,q}),C_{1,q})}})}\\\ x\xi^{-1}\mu_{0,qp}&\mbox{ else}\\\ \end{array}\right\\}\>(x{\,\in\,}{{\rm V}})$. By $l_{0,qp}\xi\varrho{\,=\,}\penalty-1l_{0,qp}\xi\xi^{-1}\mu_{0,qp}{\,=\,}\penalty-1u/qp{\,=\,}\penalty-1l_{1,q}\mu_{1,q}/p{\,=\,}\penalty-1l_{1,q}\varrho/p{\,=\,}\penalty-1(l_{1,q}/p)\varrho$ let $\sigma:={{\rm mgu}({\\{(l_{0,qp}\xi},{l_{1,q}/p)\\},{\rm Y}})}$ and $\varphi\in{{{\mathcal{SUB}}}({{{\rm V}}},{{{\mathcal{T}}({{{\rm X}}})}})}$ with ${{{}_{{\rm Y}}{\upharpoonleft}{(\sigma\varphi)}}}{\,=\,}\penalty-1{{{}_{{\rm Y}}{\upharpoonleft}\varrho}}$. Define $u^{\prime}:={l_{1,q}\mu_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\mu_{0,qp}\,]}}$. We get $\begin{array}[]{l@{}l@{}l}w_{0}/q&{\,=\penalty-1}&{u/q\penalty-1{{[\,p^{\prime}\leftarrow r_{0,qp^{\prime}}\mu_{0,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{0}^{\prime}\,]}}}{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+\omega,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}\\\ &&{{u/q\penalty-1{{[\,p^{\prime}\leftarrow l_{0,qp^{\prime}}\mu_{0,qp^{\prime}}\ \|\ p^{\prime}{\,\in\,}\mathchar 261\relax_{0}^{\prime}{\setminus}\\{p\\}\,]}}}\penalty-1{[\,p\leftarrow r_{0,qp}\mu_{0,qp}\,]}}{\,=\,}\penalty-1u^{\prime}.\end{array}$ If ${l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma{\,=\,}\penalty-1r_{1,q}\sigma,$ then the proof is finished due to $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+\omega,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}u^{\prime}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q.$ Otherwise we have $(\,({l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma,C_{0,qp}\xi\sigma,1),\penalty-1\,(r_{1,q}\sigma,C_{1,q}\sigma,1),\penalty-1\,l_{1,q}\sigma,\penalty-1\,p\,)\in{\rm CP}({\rm R})$ (due to Claim 5); $C_{0,qp}\xi\sigma\varphi=C_{0,qp}\mu_{0,qp}$ is fulfilled w.r.t. ${\longrightarrow}$; $C_{1,q}\sigma\varphi=C_{1,q}\mu_{1,q}$ is fulfilled w.r.t. ${\longrightarrow}$. Due to Claim 0 and our assumed $\omega$-coarse level parallel joinability we have $u^{\prime}{\,=\,}\penalty-1{l_{1,q}\penalty-1{[\,p\leftarrow r_{0,qp}\xi\,]}}\sigma\varphi{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}\penalty-1v_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\penalty-1v_{2}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}\penalty-1r_{1,q}\sigma\varphi{\,=\,}\penalty-1r_{1,q}\mu_{1,q}{\,=\,}\penalty-1w_{1}/q$ for some $v_{1}$, $v_{2}$. We then have $w_{0}/q{{\mathchoice{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-14.6pt\shortparallel}\hskip 7.0pt}{{{\longleftarrow}\hskip-8.0pt\shortparallel}\hskip 5.0pt}{{{\longleftarrow}\hskip-7.0pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.25pt\scriptscriptstyle\omega+\omega,\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}}u^{\prime}{{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}_{\hskip-1.5pt\scriptscriptstyle\omega+\omega,\mathchar 261\relax^{\prime\prime}}}v_{1}$ for some $\mathchar 261\relax^{\prime\prime}$. Since $\displaystyle\sum_{p^{\prime\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\},\mathchar 261\relax^{\prime\prime})}\lambda(u^{\prime}/p^{\prime\prime})\ \ \preceq\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}\lambda(u^{\prime}/p^{\prime\prime})\ \ =\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}\setminus\\{p\\}}\lambda(u/qp^{\prime\prime})\ \ \prec\sum_{p^{\prime\prime}\in\mathchar 261\relax_{0}^{\prime}}\lambda(u/qp^{\prime\prime})\ \ =\sum_{p^{\prime}\in q\mathchar 261\relax_{0}^{\prime}}\lambda(u/p^{\prime})\ \ =\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\\{q\\})}\lambda(u/p^{\prime})\ \ \preceq\sum_{p^{\prime}\in\mathchar 266\relax(\mathchar 261\relax_{0},\mathchar 261\relax_{1})}\lambda(u/p^{\prime})$ due to our induction hypothesis we get some $v_{1}^{\prime}$ with $w_{0}/q{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}{\circ}{\mathchoice{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-16.0pt\shortparallel}\hskip 8.5pt}{{{\longrightarrow}\hskip-8.5pt\shortparallel}\hskip 6.0pt}{{{\longrightarrow}\hskip-7.5pt\shortparallel}\hskip 5.0pt}}{\circ}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}v_{1}^{\prime}{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v_{1}.$ Finally the peak at $v_{1}$ can be closed according to $v_{1}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!\omega}}}\circ{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}v_{2}$ by Claim 0. Q.e.d. (“The second critical peak case”) Q.e.d. (Lemma C.3) Proof of Lemma C.4 By Lemma 2.7 it suffices to show for each literal $L$ in $C$ that $L\nu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$. Note that we already know that $L\mu$ is fulfilled w.r.t. ${\longrightarrow}_{{}_{\\!{\rm R},{{\rm X}}}}$. Since ${{{\mathcal{V}}}({C})}{\subseteq}{{{\rm V}}\\!_{{\mathcal{C}}}},$ for all $x$ in ${{\mathcal{V}}}({C})$ we have $x\mu{\,\in\,}{{\mathcal{T}}({{\rm cons},{{{\rm V}}\\!_{{\mathcal{C}}}}})}$ and then by Lemma 2.10 $x\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}y\mu.$ $L=(s_{0}{=}s_{1})$: We have $s_{0}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}s_{0}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t_{0}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}s_{1}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}s_{1}\nu$ for some $t_{0}.$ By the inclusion assumption of the lemma we get some $v$ with $s_{0}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t_{0}$ and then (due to $v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}s_{1}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}s_{1}\nu)$ $v{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}\circ{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}s_{1}\nu.$ $L=({{\rm Def}\>}s)$: We know the existence of $t\in{{\mathcal{GT}}({{\rm cons}})}$ with $s\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}s\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t.$ By the above inclusion property again, there is some $t^{\prime}$ with $s\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t.$ By Lemma 2.10 we get $t^{\prime}{\,\in\,}{{\mathcal{GT}}({{\rm cons}})}.$ $L=(s_{0}{\not=}s_{1})$: There exist some $t_{0},t_{1}\in{{\mathcal{GT}}({{\rm cons}})}$ with $\forall i{\,\prec\,}2{.}\penalty-1\,\,s_{i}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{{\rm R},{{\rm X}}},\omega}}}s_{i}\mu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t_{i}$ and $t_{0}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{\rm R},{{\rm X}}}}}t_{1}.$ Just like above we get $t_{0}^{\prime},\ t_{1}^{\prime}\in{{\mathcal{GT}}({{\rm cons}})}$ with $\forall i{\,\prec\,}2{.}\penalty-1\,\,s_{i}\nu{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}\penalty-1t_{i}^{\prime}\penalty-1{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t_{i}.$ Finally $t_{0}^{\prime}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longleftarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t_{0}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{\rm R},{{\rm X}}}}}t_{1}{{{\stackrel{{\scriptstyle\scriptstyle\ast}}{{{\longrightarrow}}}}}_{{}_{\\!{\rm R},{{\rm X}}}}}t_{1}^{\prime}$ implies $t_{0}^{\prime}{{\mathchoice{{\hskip 1.5pt\nmid\hskip-4.69754pt\downarrow}}{{\hskip 1.5pt\nmid\hskip-4.65pt\downarrow}}{{\hskip 1.0pt\nmid\hskip-3.494pt\downarrow\hskip 1.0pt}}{{\hskip 1.0pt\nmid\hskip-3.01pt\downarrow\hskip 0.5pt}}}_{{}_{{\rm R},{{\rm X}}}}}t_{1}^{\prime}.$	arxiv-papers	2009-02-20T16:26:52	2024-09-04T02:49:00.741555	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Claus-Peter Wirth", "submitter": "Claus-Peter Wirth", "url": "https://arxiv.org/abs/0902.3614" }
0902.3711	††thanks: Corresponding author. Email address: flyan@mail.hebtu.edu.cn # Probabilistic Dense Coding Using Non-Maximally Entangled Three-Particle State ZHANG Guo-Hua, YAN Feng-Li College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang 050016, China; Hebei Advanced Thin Films Laboratory, Shijiazhuang 050016, China ###### Abstract We present a scheme of probabilistic dense coding via a quantum channel of non-maximally entangled three-particle state. The quantum dense coding will be succeeded with a certain probability if the sender introduces an auxiliary particle and performs a collective unitary transformation. Furthermore, the average information transmitted in this scheme is calculated. ###### pacs: 03.65.Ta, 03.67.Hk, 03.67.Lx One of the essential features of quantum information is its capacity for entanglement. The present-day entanglement theory has its roots in the key discoveries: quantum teleportation,[1] quantum cryptography with Bell theorem,[2] and quantum dense coding.[3] Entanglement has also played important role in development of quantum computation and quantum communication. [4-7] Holevo has shown that one qubit can carry at most only one bit of classical information. [8] In 1992, Bennett and Wiesner discovered a fundamental primitive, quantum dense coding, [3] which allows to communicate two classical bits by sending one a priori entangled qubit. Quantum dense coding is one of many surprising applications of quantum entanglement. In 1996, quantum dense coding was experimentally demonstrated with polarization entangled photons for the case of discrete variables by Mattle et al. [9] Recent years, some schemes for quantum dense coding using multi-particle entangled states via local measurements,[10] GHZ state,[11] non-symmetric quantum channel [12] were proposed. Liu et al.[13] presented the general scheme for dense coding between multi-parties using a high-dimensional state. All these cases deal with maximally entangled states. Recently, Hao et al.[14] gave a probabilistic dense coding scheme using the two-qubit pure state $\|\phi\rangle=a\|00\rangle+b\|11\rangle$. A general probabilistic dense coding scheme was put forward by Wang et al [15]. In this Letter, we suggest a scheme of probabilistic dense coding via a quantum channel of non-maximally entangled three-particle state. The average information transmitted in the scheme is calculated. Furthermore, the scheme is generalized to $d$-level $(d>3)$ for three parties. We first consider the dense coding between three-parties (Alice, Bob and Charlie) via a maximally entangled three-particle state. Suppose Alice and Bob are the senders, Charlie is the receiver, a maximally entangled three-particle state $\|\psi_{00}\rangle_{ABC}=\frac{1}{\sqrt{3}}(\|000\rangle+\|111\rangle+\|222\rangle)_{ABC}$ (1) is shared by them, and the three particles $A$, $B$ and $C$ are held by Alice, Bob and Charlie, respectively. Let us introduce the nine single-particle operations as follows: $\begin{array}[]{ll}U_{00}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right),&U_{01}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&e^{2\pi i/3}&0\\\ 0&0&e^{4\pi i/3}\end{array}\right),\\\ U_{02}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&e^{4\pi i/3}&0\\\ 0&0&e^{2\pi i/3}\end{array}\right),&U_{10}=\left(\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\ 0&1&0\end{array}\right),\\\ U_{11}=\left(\begin{array}[]{ccc}0&0&e^{4\pi i/3}\\\ 1&0&0\\\ 0&e^{2\pi i/3}&0\end{array}\right),&U_{12}=\left(\begin{array}[]{ccc}0&0&e^{2\pi i/3}\\\ 1&0&0\\\ 0&e^{4\pi i/3}&0\end{array}\right),\\\ U_{20}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ 1&0&0\end{array}\right),&U_{21}=\left(\begin{array}[]{ccc}0&e^{2\pi i/3}&0\\\ 0&0&e^{4\pi i/3}\\\ 1&0&0\end{array}\right),\\\ U_{22}=\left(\begin{array}[]{ccc}0&e^{4\pi i/3}&0\\\ 0&0&e^{2\pi i/3}\\\ 1&0&0\end{array}\right).\end{array}$ (2) It is easy to prove that $\displaystyle U_{00}(A)\otimes U_{00}(B)\|\psi_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{3}}(\|000\rangle+\|111\rangle+\|222\rangle)_{ABC}\equiv\|\psi^{0}_{00}\rangle_{ABC},$ $\displaystyle U_{00}(A)\otimes U_{10}(B)\|\psi_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{3}}(\|010\rangle+\|121\rangle+\|202\rangle)_{ABC}\equiv\|\psi^{0}_{01}\rangle_{ABC},$ $\displaystyle U_{00}(A)\otimes U_{20}(B)\|\psi_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{3}}(\|020\rangle+\|101\rangle+\|212\rangle)_{ABC}\equiv\|\psi^{0}_{02}\rangle_{ABC},$ $\displaystyle U_{01}(A)\otimes U_{00}(B)\|\psi_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{3}}(\|000\rangle+e^{2\pi i/3}\|111\rangle+e^{4\pi i/3}\|222\rangle)_{ABC}\equiv\|\psi^{0}_{10}\rangle_{ABC},$ $\displaystyle U_{01}(A)\otimes U_{10}(B)\|\psi_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{3}}(\|010\rangle+e^{2\pi i/3}\|121\rangle+e^{4\pi i/3}\|202\rangle)_{ABC}\equiv\|\psi^{0}_{11}\rangle_{ABC},$ $\displaystyle\cdots,$ $\displaystyle U_{22}(A)\otimes U_{20}(B)\|\psi_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{3}}(\|220\rangle+e^{4\pi i/3}\|001\rangle+e^{2\pi i/3}\|112\rangle)_{ABC}\equiv\|\psi^{2}_{22}\rangle_{ABC}$ and $\\{\|\psi^{0}_{00}\rangle,\|\psi^{0}_{01}\rangle,\|\psi^{0}_{02}\rangle,\|\psi^{0}_{10}\rangle,\|\psi^{0}_{11}\rangle,\cdots,\|\psi^{2}_{22}\rangle\\}$ is a basis of the Hilbert space of the particles $A$, $B$ and $C$. Alice performs one of the nine unitary transformations stated in Eq.(2) on particle $A$, Bob operates one of three unitary transformations $U_{00},U_{10},U_{20}$, on particle $B$. Then they send their particles $A$ and $B$ to the receiver Charlie. After receiving the particles $A$ and $B$, Charlie takes only one measurement in the basis $\\{\|\psi^{0}_{00}\rangle,\|\psi^{0}_{01}\rangle,\|\psi^{0}_{02}\rangle,\|\psi^{0}_{10}\rangle,\|\psi^{0}_{11}\rangle,\cdots,\|\psi^{2}_{22}\rangle\\}$, and she will know what operation Alice and Bob have carried out, that is, what messages are that Alice and Bob have encoded in the quantum state. Then Charlie can obtain ${\rm log}_{2}27$ bits of information through only one measurement. So the dense coding is realized successfully. In the following we will discuss a scheme of probabilistic dense coding between three parties via a non-maximally entangled three-particle state. We suppose that Alice, Bob and Charlie share a non-maximally entangled three- particle state $\|\varphi\rangle_{ABC}=(x_{0}\|000\rangle+x_{1}\|111\rangle+x_{2}\|222\rangle)_{ABC},$ (4) where $x_{0},x_{1},x_{2}$ are real numbers, and $\|x_{0}\|^{2}+\|x_{1}\|^{2}+\|x_{2}\|^{2}=1$. Without loss of generality, we can suppose that $\|x_{0}\|\leq\|x_{1}\|\leq\|x_{2}\|$. The scheme of probabilistic dense coding is composed of four steps. Firstly, Alice introduces an auxiliary three-level particle $a$ in the quantum state $\|0\rangle_{a}$. Then she performs a unitary transformation $U_{sim}$ on her particle A and auxiliary particle $a$ under the basis $\\{\|00\rangle_{Aa},\|01\rangle_{Aa},\|02\rangle_{Aa},\|10\rangle_{Aa},\|11\rangle_{Aa},\|12\rangle_{Aa},\|20\rangle_{Aa},$ $\|21\rangle_{Aa},\|22\rangle_{Aa}\\}$: $U_{sim}=\left(\begin{array}[]{ccccccccc}1&0&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0\\\ 0&0&0&m_{01}&m&0&0&0&0\\\ 0&0&0&m&-m_{01}&0&0&0&0\\\ 0&0&0&0&0&m_{02}&0&M&-N\\\ 0&0&0&0&0&0&m_{02}&N&M\\\ 0&0&0&0&0&M&N&-m_{02}&0\\\ 0&0&0&0&0&-N&M&0&-m_{02}\\\ \end{array}\right),$ (5) where $M=\sqrt{(x^{2}_{2}-x^{2}_{1})/x_{2}^{2}},N=\sqrt{(x^{2}_{1}-x^{2}_{0})/x_{2}^{2}},m=\sqrt{1-x^{2}_{0}/x_{1}^{2}},m_{01}=x_{0}/x_{1},m_{02}=x_{0}/x_{2}$. The collective unitary operator $U_{sim}\otimes I_{BC}$ (where $I_{BC}$ is a $9\times 9$ identity matrix) transforms the state $\|\varphi\rangle_{ABC}\otimes\|0\rangle_{a}$ into $\displaystyle\|\varphi^{\prime}\rangle_{ABCa}$ $\displaystyle=$ $\displaystyle U_{sim}\otimes I_{BC}\|\varphi\rangle_{ABC}\otimes\|0\rangle_{a}$ $\displaystyle=$ $\displaystyle\sqrt{3}x_{0}\frac{1}{\sqrt{3}}(\|000\rangle+\|111\rangle+\|222\rangle)_{ABC}\|0\rangle_{a}$ $\displaystyle+\sqrt{2}\sqrt{x_{1}^{2}-x_{0}^{2}}\frac{1}{\sqrt{2}}(\|111\rangle+\|222\rangle)_{ABC}\|1\rangle_{a}$ $\displaystyle+\sqrt{x_{2}^{2}-x_{1}^{2}}\|222\rangle_{ABC}\|2\rangle_{a}$ $\displaystyle\equiv$ $\displaystyle\sqrt{3}x_{0}\|\varphi_{00}\rangle_{ABC}\|0\rangle_{a}+\sqrt{2}\sqrt{x_{1}^{2}-x_{0}^{2}}\|\varphi^{\prime}_{00}\rangle_{ABC}\|1\rangle_{a}$ $\displaystyle+\sqrt{x_{2}^{2}-x_{1}^{2}}\|\varphi^{\prime\prime}_{00}\rangle_{ABC}\|2\rangle_{a}.$ Then Alice makes a measurement on the auxiliary particle $a$ and tells Bob and Charlie her measurement result via a classical channel. If she gets the result $\|0\rangle_{a}$, she ensures that the three particles $A$, $B$ and $C$ are in the maximally entangled three-particle state $\|\varphi_{00}\rangle=\frac{1}{\sqrt{3}}(\|000\rangle+\|111\rangle+\|222\rangle)$, the probability of obtaining $\|0\rangle_{a}$ is $3x_{0}^{2}$ according to Eq.(6); if the result $\|1\rangle_{a}$ is obtained, she ensures that the three particles $A$, $B$ and $C$ are in the state $\|\varphi^{\prime}_{00}\rangle=\frac{1}{\sqrt{2}}(\|111\rangle+\|222\rangle)$, and the probability of getting this result is $2(x_{1}^{2}-x_{0}^{2})$; if she gets the result $\|2\rangle_{a}$, she knows that the three particles are in the product state $\|\varphi^{\prime\prime}_{00}\rangle=\|222\rangle$, and the probability of obtaining this result is $x_{2}^{2}-x_{1}^{2}$. Secondly, Alice and Bob encode classical information by performing the unitary transformations on their particle $A$ and $B$ respectively. If the three particles $A$, $B$ and $C$ are in the state $\|\varphi_{00}\rangle=\frac{1}{\sqrt{3}}(\|000\rangle+\|111\rangle+\|222\rangle)$, one uses the dense coding protocol stated above. Here we do not recite any more. If the three particles $A$, $B$ and $C$ are in the state $\|\varphi^{\prime}_{00}\rangle=\frac{1}{\sqrt{2}}(\|111\rangle+\|222\rangle)$, Alice encodes her message by performing one of six single-particle operations $\begin{array}[]{ll}U^{\prime}_{00}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right),&U^{\prime}_{01}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&-1\end{array}\right),\\\ U^{\prime}_{10}=\left(\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\ 0&1&0\end{array}\right),&U^{\prime}_{11}=\left(\begin{array}[]{ccc}0&0&-1\\\ 1&0&0\\\ 0&1&0\end{array}\right),\\\ U^{\prime}_{20}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ 1&0&0\end{array}\right),&U^{\prime}_{21}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&-1\\\ 1&0&0\end{array}\right)\end{array}$ (7) on particle $A$. However, Bob encodes his message only by operating one of three single-particle operations $U^{\prime}_{00}$, $U^{\prime}_{10}$, $U^{\prime}_{20}$ on particle $B$. Through simple calculation, we can prove that $\displaystyle U^{\prime}_{00}(A)\otimes U^{\prime}_{00}(B)\|\varphi^{\prime}_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|111\rangle+\|222\rangle)_{ABC}\equiv\|\varphi^{\prime 0}_{00}\rangle_{ABC},$ $\displaystyle U^{\prime}_{00}(A)\otimes U^{\prime}_{10}(B)\|\varphi^{\prime}_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|121\rangle+\|202\rangle)_{ABC}\equiv\|\varphi^{\prime 1}_{00}\rangle_{ABC},$ $\displaystyle U^{\prime}_{00}(A)\otimes U^{\prime}_{20}(B)\|\varphi^{\prime}_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|101\rangle+\|212\rangle)_{ABC}\equiv\|\varphi^{\prime 2}_{00}\rangle_{ABC},$ $\displaystyle U^{\prime}_{01}(A)\otimes U^{\prime}_{00}(B)\|\varphi^{\prime}_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|111\rangle-\|222\rangle)_{ABC}\equiv\|\varphi^{\prime 0}_{01}\rangle_{ABC},$ $\displaystyle\cdots,$ $\displaystyle U^{\prime}_{21}(A)\otimes U^{\prime}_{20}(B)\|\varphi^{\prime}_{00}\rangle_{ABC}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}(\|001\rangle-\|112\rangle)_{ABC}\equiv\|\varphi^{\prime 2}_{21}\rangle_{ABC}.$ It is easy to see that the states in the set $\\{\|\varphi^{\prime k}_{mn}\rangle,m,k=0,1,2;n=0,1\\}$ are orthogonal each other. If the three particles $A$, $B$ and $C$ are in the product state $\|\varphi^{\prime\prime}_{00}\rangle=\|222\rangle$, Alice and Bob can encode their classical information by performing one of three single-particle operations on particles $A$ and $B$ independently: $\begin{array}[]{ll}U^{\prime\prime}_{00}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right),&U^{\prime\prime}_{10}=\left(\begin{array}[]{ccc}0&0&1\\\ 1&0&0\\\ 0&1&0\end{array}\right),\\\ U^{\prime\prime}_{20}=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ 1&0&0\end{array}\right).\end{array}$ (9) The state $\|\varphi^{\prime\prime}_{00}\rangle$ will be transformed into the corresponding state respectively: (10) $\displaystyle U^{\prime\prime}_{00}(A)\otimes U^{\prime\prime}_{00}(B)\|\varphi^{\prime\prime}_{00}\rangle_{ABC}=\|222\rangle_{ABC}\equiv\|\varphi^{\prime\prime}_{00}\rangle_{ABC},$ $\displaystyle U^{\prime\prime}_{00}(A)\otimes U^{\prime\prime}_{10}(B)\|\varphi^{\prime\prime}_{00}\rangle_{ABC}=\|202\rangle_{ABC}\equiv\|\varphi^{\prime\prime}_{01}\rangle_{ABC},$ $\displaystyle U^{\prime\prime}_{00}(A)\otimes U^{\prime\prime}_{20}(B)\|\varphi^{\prime\prime}_{00}\rangle_{ABC}=\|212\rangle_{ABC}\equiv\|\varphi^{\prime\prime}_{02}\rangle_{ABC},$ $\displaystyle U^{\prime\prime}_{10}(A)\otimes U^{\prime\prime}_{00}(B)\|\varphi^{\prime\prime}_{00}\rangle_{ABC}=\|022\rangle_{ABC}\equiv\|\varphi^{\prime\prime}_{10}\rangle_{ABC},$ $\displaystyle\cdots,$ $\displaystyle U^{\prime\prime}_{20}(A)\otimes U^{\prime\prime}_{20}(B)\|\varphi^{\prime\prime}_{00}\rangle_{ABC}=\|112\rangle_{ABC}\equiv\|\varphi^{\prime\prime}_{22}\rangle_{ABC}.$ Evidently, the states in the set $\\{\|\varphi^{\prime\prime}_{mn}\rangle,m,n=0,1,2\\}$ are orthogonal each other. Thirdly, Alice and Bob send their particles $A$ and $B$ independently to Charlie. Finally, After Charlie receives particle $A$ and $B$, she takes only one measurement on the three particle $A$, $B$ and $C$. The measurement basis is determined by Alice’s measurement result. According to Charlie’s measurement result, Charlie will know what operators Alice and Bob have carried out, i.e. he can obtain the classical information that Alice and Bob have encoded. Apparently, the average information transmitted in this procedure is $I_{aver}=3x_{0}^{2}{\rm log}_{2}27+2(x_{1}^{2}-x_{0}^{2}){\rm log}_{2}18+(x_{2}^{2}-x_{1}^{2}){\rm log}_{2}9.$ (11) In fact, the above protocol needs $2{\rm log}_{2}3$ bits of classical information for Alice to tell Bob and Charlie her measurement result on the auxiliary particle. Obviously, when $x_{0}=x_{1}=x_{2}=\frac{1}{\sqrt{3}}$, the three particles $A$, $B$ and $C$ is in the maximally entangled three- particle state, and the success probability of dense coding is one. The average information transmitted is ${\rm log}_{2}27$ bits. Now we would like to generalize the above protocol to $d$-level for three parties. Suppose that Alice, Bob and Charlie share a non-maximally entangled three-particle state $\|\varphi\rangle_{ABC}=(x_{0}\|000\rangle+x_{1}\|111\rangle+\cdots+x_{d-1}\|d-1d-1d-1\rangle)_{ABC},$ (12) where $x_{0},x_{1},\cdots,x_{d-1}$ are real numbers and satisfy $\|x_{0}\|\leq\|x_{1}\|\leq\cdots\leq\|x_{d-1}\|$. The scheme of the probabilistic dense coding can be accomplished by four steps. (1) Alice introduces an auxiliary $d$-level particle in the quantum state $\|0\rangle_{a}$. Then she performs a proper unitary transformation on her particle $A$ and the auxiliary particle. The collective unitary transformation $U_{sim}\otimes I_{BC}$ (where $I_{BC}$ is a $d^{2}\times d^{2}$ identity matrix) transforms the state $\|\varphi\rangle_{ABC}\otimes\|0\rangle_{a}$ into the state $\displaystyle\|\varphi\rangle_{ABCa}$ $\displaystyle=$ $\displaystyle x_{0}(\|000\rangle+\|111\rangle+\cdots+\|d-1d-1d-1\rangle)_{ABC}\|0\rangle_{a}$ $\displaystyle+\sqrt{x_{1}^{2}-x_{0}^{2}}(\|111\rangle+\cdots+\|d-1d-1d-1\rangle)_{ABC}\|1\rangle_{a}$ $\displaystyle+\cdots+\sqrt{x_{d-1}^{2}-x_{d-2}^{2}}\|d-1d-1d-1\rangle_{ABC}\|d-1\rangle_{a}.$ After that Alice performs a measurement on the auxiliary particle. The resulting state of the particles $A$, $B$ and $C$ will be respectively $\frac{1}{\sqrt{d}}(\|000\rangle+\|111\rangle+\cdots+\|d-1d-1d-1\rangle),$ $\frac{1}{\sqrt{d-1}}(\|111\rangle+\cdots+\|d-1d-1d-1\rangle),$ $\cdots,$ $\|d-1d-1d-1\rangle$. The probability of obtaining each resulting state is $dx_{0}^{2}$, $(d-1)(x_{1}^{2}-x_{0}^{2}),\cdots,(x_{d-1}^{2}-x_{d-2}^{2})$, respectively. (2) Alice tells Bob and Charlie her measurement result, then Alice and Bob encode classical information by making a unitary transformation on particle $A$ and $B$ respectively. (3) Alice and Bob send particle $A$ and $B$ to Charlie respectively. (4) After Charlie receives the particles $A$ and $B$, she takes a measurement in the basis determined by Alice’s measurement result. According to his measurement result, Charlie can obtain the classical information that Alice and Bob have encoded. The average information transmitted is $\displaystyle I_{aver}$ $\displaystyle=$ $\displaystyle dx_{0}^{2}{\rm log}_{2}d^{3}+(d-1)(x_{1}^{2}-x_{0}^{2}){\rm log}_{2}d^{2}(d-1)$ $\displaystyle+(d-2)(x_{2}^{2}-x_{1}^{2}){\rm log}_{2}d^{2}(d-2)+\cdots$ $\displaystyle+(x_{d-1}^{2}-x_{d-2}^{2}){\rm log}_{2}d^{2}.$ Obviously, this probabilistic dense coding scheme needs $2{\rm log}_{2}d$ bits of information to transmit Alice’s measurement results on the auxiliary particle to Bob and Charlie. It is easy to see that if the non-zero coefficients in Eq.(12) are totally equal, Alice does not need to introduce the auxiliary particle and makes unitary transformation $U_{sim}$. The classical information can be encoded directly by performing single-particle operations on particle $A$ and $B$ respectively. In this case the quantum state is called a deterministic quantum channel; otherwise, a probabilistic one if the coefficients are not equal totally. Obviously, a non-maximally entangled three-particle state is not equivalent to the probabilistic quantum channel. For example, in the $3\otimes 3\otimes 3$-dimensional case, the quantum state $\frac{1}{\sqrt{2}}(\|111\rangle+\|222\rangle)$ is not a maximally entangled three-particle state, but it is a deterministic quantum channel. So the first step in our protocol is to extract a series of deterministic quantum channels from a probabilistic one. In summary, we have presented a scheme of probabilistic dense coding via a quantum channel of non-maximally entangled three-particle state. The average information transmitted in this scheme is explicitly given. We also generalize this scheme to the more general case. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant No: 10671054, Hebei Natural Science Foundation of China under Grant No: 07M006 and the Key Project of Science and Technology Research of Education Ministry of China under Grant No:207011. ## References * (1) Bennett C H et al 1993 _Phys. Rev. Lett._ 70 1895 * (2) Ekert A K 1991 _Phys. Rev. Lett._ 67 661 * (3) Bennett C H and Wiesner S J 1992 _Phys. Rev. Lett._ 69 2881 * (4) Raussendorf R and Briegel H J 2001 _Phys. Rev. Lett._ 86 5188 * (5) Wang X B, Hiroshima T, Tomita A and Hayashi M 2007 _Phys. Rep._ 448 1 * (6) Long G L, Deng F G, Wang C, Li X H, Wen K and Wang W Y 2007 _Front. Phys. China_ 2 251 * (7) Gao T, Yan F L and Li Y C 2008 _Europhys. Lett._ 84 50001 * (8) Holevo A S 1973 _Probl. Peredachi Inf._ 9 3 * (9) Mattle K, Weinfurter H, Kwiat P G and Zeilinger A 1996 _Phys. Rev. Lett._ 76 4656 * (10) Chen J L and Kuang L M 2004 _Chin. Phys. Lett._ 21 12 * (11) Hao J C, Li C F and Guo G C 2000 _Phys. Rev. A_ 63 054301 * (12) Yan F L and Wang M Y 2004 _Chin. Phys. Lett._ 21 1195 * (13) Liu X S, Long G L, Tong D M and Li F 2002 _Phys. Rev. A_ 65 022304 * (14) Hao J C, Li C F and Guo G C 2000 _Phys.Lett. A_ 278 113 * (15) Wang M Y, Yang L G and Yan F L 2005 _Chin. Phys. Lett._ 22 1053	arxiv-papers	2009-02-21T02:41:18	2024-09-04T02:49:00.782053	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Zhang Guo-Hua, Yan Feng-Li", "submitter": "Ting Gao", "url": "https://arxiv.org/abs/0902.3711" }
0902.3729	Uncertainty relation of mixed states by means of Wigner-Yanase-Dyson information D. Lia111email address:dli@math.tsinghua.edu.cn, X. Lib, F. Wangc, H. Huangd, X. Lie, L. C. Kwekf a Dept of mathematical sciences, Tsinghua University, Beijing 100084 CHINA b Department of Mathematics, University of California, Irvine, CA 92697-3875, USA c Insurance Department, Central University of Finance and Economics, Beijing 100081, CHINA d Electrical Engineering and Computer Science Department University of Michigan, Ann Arbor, MI 48109, USA e Dept. of Computer Science, Wayne State University, Detroit, MI 48202, USA f National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 Institute of Advanced Studies (IAS), Nanyang Technological University, 60 Nanyang View Singapore 639673 Abstract The variance of an observable in a quantum state is usually used to describe Heisenberg uncertainty relation. For mixed states, the variance includes quantum uncertainty and classical uncertainty. By means of the skew information and the decomposition of the variance, a stronger uncertainty relation was presented by Luo in [Phys. Rev. A 72, 042110 (2005)]. In this paper, by using Wigner-Yanase-Dyson information which is a generalization of the skew information, we propose a general uncertainty relation of mixed states. PACS 03.65.Ta Keywords: Heisenberg uncertainty relation, the skew information, Dyson information ## 1 Introduction In quantum measurement theory, the Heisenberg uncertainty principle provides a fundamental limit for the measurements of incompatible observables. On the other hand, as dictated by Cramer-Rao’s lower bound, there is also an ultimate limit for the resolution of any unbiased parameter (see for instance, [1]), and this lower bound is given by a quantity called Fisher information. A long time ago, Wigner demonstrated that it is more difficult to measure observables that do not commute with some additive conserved quantity. Thus, observables not commuting with some conserved quantity cannot be measured exactly and only approximate measurement is possible. This trade-off in measurement forms the basis of the well-known Wigner-Araki-Yanase theorem. In their study of quantum measurement theory, Wigner and Yanase introduced a quantity called the skew information. As shown in [2], the skew information is essentially a form of Fisher information. The skew information for a mixed state $\rho$ relative to a self-adjoint “observable”, $A$, is defined as $I(\rho$, $A)=$ $-\frac{1}{2}\mbox{Tr}\rho^{1/2}$, $A]^{2}$. This definition was subsequently generalized by Dyson as $I_{\alpha}(\rho,A)=-\frac{1}{2}\mbox{Tr}([\rho^{\alpha},A][\rho^{1-\alpha},A])$, where $0<\alpha<1$ [3]. When $\alpha=1/2$, $I_{\alpha}(\rho$, $X)$ is reduced to the skew information. The convexity of $I_{\alpha}(\rho,A)$ was finally resolved by Lieb[4, 5]. The von Neumann entropy of $\rho$, defined as $S(\rho)=-tr\rho\ln\rho$, has been widely used as a measure of the uncertainty of a mixed state. This quantity, profoundly rooted in quantum statistical mechanics, possesses several remarkable and satisfactory properties. Like all measures, the von Neumann entropy, together with its classical analog called the Shannon entropy, is not always the best measure under certain contexts. In [6, 7, 2, 8], the skew information was proposed as means to unify the study of Heisenberg uncertainty relation for mixed states. It is well know in the standard textbooks that the Heisenberg uncertainty relation for any two self-adjoint operators $X$ and $Y$ is given by $V(\rho,X)V(\rho,Y)\geq\frac{1}{4}\|\|\mbox{Tr}(\rho[X,Y]\|\|^{2}.$ (1) Note that $[$,$]$ is commutator, i.e. $[A$, $B]=AB-BA$ and the variance of the observable $X$ with respect to $\rho$ is $V(\rho,X)=\mbox{Tr}(\rho X^{2})-(\mbox{Tr}(\rho X))^{2}.$ (2) A similar definition applies to $V(\rho,Y)$. When $\rho$ is a mixed state, Luo showed that the variance comprises of two terms: a quantum uncertainty term and a classical uncertainty term[6, 7]. He separated the variance into its quantum and classical part by using the skew information. He interpreted $I(\rho$, $X)$ as the quantum uncertainty of $X$ in $\rho$ by the Bohr complementary principle and $V(\rho,X)-I(\rho$, $X)$ as the classical uncertainty of the mixed state. He then considered $U(\rho,X)=\sqrt{V^{2}(\rho,X)-[V(\rho,X)-I(\rho,X)]^{2}}$ as a measure of quantum uncertainty. Thus, he obtained the following two inequalities for the uncertainty relation. $I(\rho,X)J(\rho,Y)\geq\frac{1}{4}\|\|\mbox{Tr}(\rho[X,Y]\|\|^{2}.$ (3) $U(\rho,X)U(\rho,Y)\geq\frac{1}{4}\|\|\mbox{Tr}(\rho[X,Y]\|\|^{2}.$ (4) where $J(\rho$, $Y)=\frac{1}{2}\mbox{Tr}\\{\rho^{1/2}$, $Y_{0}\\}^{2}$, and $Y_{0}=Y-\mbox{Tr}(\rho Y)$. The notation $\\{$ $\\}$ is the anticommutator, i.e. $\\{A$, $B\\}=AB+BA$. This article is organized as follows: In section 2, we discuss various properties of the Wigner-Yanase-Dyson information. We show using a counter example that it need not satisfy the uncertainty relation obtained from the skew information. In section 3, we formulate an uncertainty relation for Wigner-Yanase-Dyson information. Finally, in section 4, we reiterate our main results. We have also provided two appendices concerning the proof of the new uncertainty principle and additivity of the Wigner-Yanase-Dyson information. ## 2 Wigner-Yanase-Dyson information violates Heisenberg uncertainty relation In this paper, we extend the above discussion to Wigner-Yanase-Dyson information. The skew information proposed by Dyson can also be written as $\displaystyle I_{\alpha}(\rho,X)$ $\displaystyle=$ $\displaystyle\mbox{Tr}(\rho X^{2})-\mbox{Tr}(\rho^{\alpha}X\rho^{1-\alpha}X)$ (5) $\displaystyle=$ $\displaystyle\mbox{Tr}(\rho X_{0}^{2})-\mbox{Tr}(\rho^{\alpha}X_{0}\rho^{1-\alpha}X_{0})\text{, }$ where $X_{0}=X-\mbox{Tr}(\rho X)$. $I_{\alpha}(\rho,X)$ is positive from Eq. (LABEL:q-info-2). Similarly, we define $J_{\alpha}(\rho,Y)=\frac{1}{2}tr(\\{\rho^{\alpha}$, $Y_{0}\\}\\{\rho^{1-\alpha}$, $Y_{0}\\})$. When $\alpha=1/2$, $J_{\alpha}(\rho$, $Y)$ is reduced to $J(\rho$, $Y)$. As well, we can define $J_{\alpha}(\rho,X)$, $J_{\alpha}(\rho,A)$, and $J_{\alpha}(\rho,B)$. By calculating, $\displaystyle J_{\alpha}(\rho,Y)=$ $\displaystyle\mbox{Tr}(\rho Y_{0}^{2})+\mbox{Tr}(\rho^{\alpha}Y_{0}\rho^{1-\alpha}Y_{0})=$ $\displaystyle\mbox{Tr}(\rho Y^{2})+\mbox{Tr}(\rho^{\alpha}Y\rho^{1-\alpha}Y)-2(\mbox{Tr}\rho Y)^{2}.$ (6) $J_{\alpha}(\rho$, $Y)$ is also positive from Eq. (A9) in this paper. Adopting the Luo’s interpretations, by the following properties of Wigner- Yanase-Dyson information we interpret $I_{\alpha}(\rho,X)$ as quantum uncertainty of $X$ in $\rho$, $V(\rho,X)-I_{\alpha}(\rho$, $X)$ as the classical mixing uncertainty, and $U_{\alpha}(\rho,X)=\sqrt{V^{2}(\rho,X)-[V(\rho,X)-I_{\alpha}(\rho,X)]^{2}}$ as a measure of quantum uncertainty. Lieb studied the properties of Wigner- Yanase-Dyson information in [4]. Wigner-Yanase-Dyson information satisfies the following requirements. (1). Wigner-Yanase-Dyson conjecture about the convexity of $I_{\alpha}(\rho,X)$ with respect to $\rho$ was proved by Lieb [4]. (2). Wigner-Yanase-Dyson information $I_{\alpha}(\rho,X)$ is additive under the following sense (See [2] and [4]). Let $\rho_{1}$ and $\rho_{2}$ be two density operators of two subsystems, and $A_{1}$ (resp. $A_{2}$) be a self- adjoint operator on $H^{1}$ (resp. $H^{2}$). $I_{\alpha}(\rho,X)$ is additive if $I_{\alpha}(\rho_{1}\otimes\rho_{2}$, $A_{1}\otimes I_{2}+I_{1}\otimes A_{2})=I_{\alpha}(\rho_{1}$, $A_{1})+I_{\alpha}(\rho_{2}$, $A_{2})$, where $I_{1}$ and $I_{2}$ are the identity operators for the first and second systems, respectively. For the proof see Appendix B. (3). $J_{\alpha}(\rho$, $Y)$ is also additive under the above sense. For the proof see Appendix B. (4). However, Hansen showed that Wigner-Yanase-Dyson information is not subadditive [11]. For the definition of subadditivity see [4] and [11]. (5). $J_{\alpha}(\rho$, $Y)$ is concave with respect to $\rho$. This is because $tr(\rho Y_{0}^{2})$ is linear operator with respect to $\rho$ and $tr(\rho^{\alpha}Y_{0}\rho^{1-\alpha}Y_{0})$ is concave with respect to $\rho$. (6). When $\rho$ is pure, $V(\rho,X)=I_{\alpha}(\rho$, $X)$. Thus, Wigner- Yahase-Dyson information reduces to the variance. That is, the variance $V(\rho,X)$ does not include the classical mixing uncertainty because of no mixing. In other words, the variance only includes the quantum uncertainty of $X$ in $\rho$. The case in which $\alpha=1/2$ was discussed in [7]. The above fact can be argued as follows. When $\rho$ is pure, $tr(\rho^{\alpha}X_{0}\rho^{1-\alpha}X_{0})=$ $(tr(\rho X_{0}))^{2}=0$. Thus, $I_{\alpha}(\rho,X)=tr(\rho X_{0}^{2})=V(\rho,X)$. (7). When $\rho$ is a mixed state, $V(\rho,X)\geq I_{\alpha}(\rho$, $X)$. This is because $tr(\rho^{\alpha}X\rho^{1-\alpha}X)$ $=$ $tr((\rho^{\alpha/2}X\rho^{(1-\alpha)/2})$ $(\rho^{\alpha/2}X\rho^{(1-\alpha)/2})^{\dagger})\geq 0$. Also, see Eq. (A3) in this paper. The case in which $\alpha=1/2$ was discussed in [7]. (8). When $\rho$ and $A$ commute, according to the discussion for the skew information in [6, 8], the quantum uncertainty should vanish and thus, the variance only includes the classical uncertainty. We can argue that the above conclusion is also true for Wigner-Yanase-Dyson information. When $\rho$ and $A$ commute, it is well known that $\rho$ and $A$ have the same orthonormal eigenvector basis [9]. Hence, $\rho^{\alpha}$ and $A$ also commute. By the definition in Eq. (5), Wigner-Yanase-Dyson information $I_{\alpha}(\rho,X)$ vanishes. However, $I_{\alpha}(\rho,X)$ and $J_{\alpha}(\rho$, $Y)$ do not satisfy Eq. (3). We give the following counter example for Eq. (3). Let $n=2$, $\alpha=1/4$, and $\rho$ have the eigenvalues $\lambda_{1}=1/4$ and $\lambda_{2}=3/4$. Since $A$ and $B$ are self-adjoint, then we write $A=\left(\begin{tabular}[]{ll}$x$&$u+iv$\\\ $u-iv$&$y$\end{tabular}\right)$, $B=\left(\begin{tabular}[]{ll}$a$&$c+di$\\\ $c-di$&$b$\end{tabular}\right)$. In this example, $u=4$, $v=2$, $a=b=0$, $c=1$, and $d=-5$. By calculating $I_{\alpha}(\rho,A)$ in Eq. (LABEL:q-info-2) and $J_{\alpha}(\rho,B)$ in Eq. (A8), $I_{\alpha}(\rho,A)J_{\alpha}(\rho,B)=[1-(\lambda_{1}^{\alpha}\lambda_{2}^{1-\alpha}+\lambda_{2}^{\alpha}\lambda_{1}^{1-\alpha})^{2}](u^{2}+v^{2})(c^{2}+d^{2})=99.83$. By calculating $\mbox{Tr}(\rho[A,B]$ in Eq. (A11), $\frac{1}{4}\|\mbox{Tr}(\rho[A,B]\|^{2}=(\lambda_{1}-\lambda_{2})^{2}(cv- du)^{2}=121$. Hence, it violates Eq. (3). It implies that the bound on the right side of the inequality in Eq. (3) is too large in this example. We need to get the appropriate lower bound for Wigner-Yanase-Dyson information, i.e., we need to modify the term on RHS of the inequality. ## 3 The general uncertainty relation We replace $\mbox{Tr}(\rho[X,Y]$ with $l_{\alpha}(\rho$, $X$, $Y)$ which is defined as follows: $l_{\alpha}(\rho,X,Y)=\mbox{Tr}(\rho[X,Y])-\mbox{Tr}\rho^{\left\|2\alpha-1\right\|}[X,Y].$ (7) When $\alpha=1/2$, $l_{\alpha}(\rho$, $X$, $Y)$ reduces to $\mbox{Tr}(\rho[X$, $Y])$. In [6], Luo defined $k=\mathrm{i}[\rho^{1/2}$, $X_{0}]t+\\{\rho^{1/2},Y_{0}\\}$, where $t\in R$ and $\mathrm{i}$ is an imaginary number. From $\mbox{Tr}(kk^{\dagger})\geq 0$, by expanding $\mbox{Tr}(kk^{\dagger})$, he derived $\mbox{Tr}(kk^{\dagger})=2(I[\rho$, $X]t^{2}+\mathrm{i}(tr(\rho[X$, $Y])t+J[\rho$, $Y])\geq 0$. Since the above inequality is true for any real $t$, Luo obtained the inequality in Eq. (3). However, unlike his previous case, the form of $I_{\alpha}(\rho,X)$ does not allow us to employ the trick $k=\mathrm{i}[\rho^{\alpha}$, $X_{0}]t+\\{\rho^{\alpha},Y_{0}\\}$ nor $k=\mathrm{i}[\rho^{1-\alpha}$, $X_{0}]t+\\{\rho^{1-\alpha},Y_{0}\\}$ to derive the uncertainty relation from $\mbox{Tr}(kk^{\dagger})\geq 0$. The proof becomes more involved and one needs to modify the RHS of the previous uncertainty relation. In Appendix A, we see that if $A$ and $B$ are self-adjoint observables, then $I_{\alpha}(\rho,A)J_{\alpha}(\rho,B)\geq\frac{1}{4}\|\|l_{\alpha}(\rho,A,B)\|\|^{2}\text{, }$ (8) and $I_{\alpha}(\rho,B)J_{\alpha}(\rho,A)\geq\frac{1}{4}\|\|l_{\alpha}(\rho,A,B)\|\|^{2}\text{. }$ (9) If we denote $U_{\alpha}(\rho,\mathcal{O})$ as $\sqrt{V^{2}(\rho,\mathcal{O})-[V(\rho,\mathcal{O})-I_{\alpha}(\rho,\mathcal{O})]^{2}}$, we see that by Eq. (2) and Eq.(5) (and the analogous form for $J_{\alpha}(\rho,\mathcal{O})$), $U_{\alpha}(\rho,\mathcal{O})=\sqrt{I_{\alpha}(\rho,\mathcal{O})J_{\alpha}(\rho,\mathcal{O})}$, where $\mathcal{O}$ is either the operator $A$ or $B$. Thus, we obtain our main result from Eqs. (8) and (9), $U_{\alpha}(\rho,A)U_{\alpha}(\rho,B)\geq\frac{1}{4}\|\|l_{\alpha}(\rho,X,Y)\|\|^{2}.$ (10) For the counter example in Sec. 2, a direct calculation of Eq. (A13) yields $\frac{1}{4}\|\|l_{\alpha}(\rho,A,B)\|\|^{2}=$ $8.\,\allowbreak 687\,4$. Therefore, the inequality in Eq. (8) holds in this case. ## 4 Summary In [6], Luo presented a refined Heisenberg uncertainty relation. In this paper, we demonstrate some properties of Wigner-Yanase-Dyson information and provide a counter example to show that Wigner-Yanase-Dyson information does not in general satisfy Heisenberg uncertainty relation. We have also proposed a new general uncertainty relation of mixed states based on Wigner-Yanase- Dyson information. Bell-type inequalities based on the skew information have been proposed as nonlinear entanglement witnesses [12]. We note here that similar Bell-type inequalities with the advantage of an additional $\alpha$ parameter for fine adjustments could also be constructed from the uncertainty principle derived from the Wigner-Yanase-Dyson information. ## Appendix A. Proof of uncertainty relation By spectral decomposition, there exists an orthonormal basis $\\{x_{1}$,…, $x_{n}\\}$ consisting of eigenvectors of $\rho$. Let $\lambda_{1}$, …, $\lambda_{n}$ be the corresponding eigenvalues, where $\lambda_{1}+...+\lambda_{n}=1$ and $\lambda_{i}\geq 0$. Thus, $\rho$ has a spectral representation $\rho=\lambda_{1}\|x_{1}\rangle\langle x_{1}\|+....+\lambda_{n}\|x_{n}\rangle\langle x_{n}\|.$ (A1) ### 1\. Calculating $I_{\alpha}(\rho$, $A)$ By Eq. (A1), $\rho A^{2}=\lambda_{1}\|x_{1}\rangle\langle x_{1}\|A^{2}+....+\lambda_{n}\|x_{n}\rangle\langle x_{n}\|A^{2}$ and $\displaystyle\mbox{Tr}\rho A^{2}$ $\displaystyle=$ $\displaystyle\lambda_{1}\langle x_{1}\|A^{2}\|x_{1}\rangle+....+\lambda_{n}\langle x_{n}\|A^{2}\|x_{n}\rangle$ (A2) $\displaystyle=$ $\displaystyle\lambda_{1}\|\|A\|x_{1}\|\|^{2}+....+\lambda_{n}\|\|A\|x_{n}\|\|^{2}.$ Moreover, since $\rho^{\alpha}A=\lambda_{1}^{\alpha}\|x_{1}\rangle\langle x_{1}\|A+....+\lambda_{n}^{\alpha}\|x_{n}\rangle\langle x_{n}\|A$ and $\rho^{1-\alpha}A=\lambda_{1}^{1-\alpha}\|x_{1}\rangle\langle x_{1}\|A+....+\lambda_{n}^{1-\alpha}\|x_{n}\rangle\langle x_{n}\|A$, we have, $\rho^{\alpha}A\rho^{1-\alpha}A=\sum_{i,j=1}\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}\|x_{i}\rangle\langle x_{i}\|A\|x_{j}\rangle\langle x_{j}\|A$. Thus $\displaystyle\mbox{Tr}\rho^{\alpha}A\rho^{1-\alpha}A$ $\displaystyle=$ $\displaystyle\sum_{i,j=1}\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}\langle x_{i}\|A\|x_{j}\rangle\langle x_{j}\|A\|x_{i}\rangle$ (A3) $\displaystyle=$ $\displaystyle\sum_{i,j=1}\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}\|\|\langle x_{i}\|A\|x_{j}\rangle\|\|^{2}.$ From Eqs. (5), (A2) and (A3), $I_{\alpha}(\rho,A)=\sum_{i=1}\lambda_{i}\|\|A\|x_{i}\|\|^{2}-\sum_{i,j=1}\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}\|\|\langle x_{i}\|A\|x_{j}\rangle\|\|^{2}.$ (A4) Let $A=\\{A_{ij}\\}$ (resp. $B=\\{B_{ij}\\}$) be the matrix representation of the operator $A$ (resp. $B$) corresponding to the orthonormal basis $\\{x_{1}$,…, $x_{n}\\}$. Then $\langle x_{i}\|A\|x_{j}\rangle=A_{ij}$, and $\displaystyle I_{\alpha}(\rho,A)$ $\displaystyle=$ $\displaystyle\sum_{i\neq j}(\lambda_{i}-\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha})\left\|\left\|A_{ij}\right\|\right\|^{2}$ $\displaystyle=$ $\displaystyle\sum_{i<j}(\lambda_{i}+\lambda_{j}-\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}-\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha})\left\|\left\|A_{ij}\right\|\right\|^{2}\text{.}$ ### 2\. Calculating $J_{\alpha}(\rho,B)$ Similarly, from Eqs. (6) and (A1), we can obtain $\displaystyle J_{\alpha}(\rho,B)$ $\displaystyle=$ $\displaystyle\sum_{i=1}\lambda_{i}\|\|B\|x_{i}\|\|^{2}+\sum_{i,j=1}\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}\|\|\langle x_{i}\|B\|x_{j}\rangle\|\|^{2}$ (A6) $\displaystyle-2(\sum\lambda_{i}\langle x_{i}\|B\|x_{i}\rangle)^{2}.$ Let $\langle x_{i}\|B\|x_{j}\rangle=B_{ij}$. Then, from Eq. (A6), $\displaystyle J_{\alpha}(\rho,B)$ $\displaystyle=$ $\displaystyle 2\sum_{i=1}\lambda_{i}\left\|B_{ii}\right\|^{2}-2(\sum_{i=1}\lambda_{i}B_{ii})^{2}$ (A7) $\displaystyle+\sum_{i\neq j}(\lambda_{i}+\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha})\left\|\left\|B_{ij}\right\|\right\|^{2}\text{.}$ By simplifying, $\displaystyle J_{\alpha}(\rho,B)$ $\displaystyle=$ $\displaystyle 2\sum_{i=1}\lambda_{i}\left\|B_{ii}\right\|^{2}-2(\sum_{i=1}\lambda_{i}B_{ii})^{2}$ (A8) $\displaystyle+\sum_{i<j}(\lambda_{i}+\lambda_{j}+\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}+\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha})\left\|\left\|B_{ij}\right\|\right\|^{2}\text{.}$ Since $x^{2}$ is convex, $(\sum_{i=1}\lambda_{i}B_{ii})^{2}\leq\sum_{i=1}\lambda_{i}\left\|B_{ii}\right\|^{2}$. So from Eq. (A8), $J_{\alpha}(\rho,B)\geq\sum_{i<j}(\lambda_{i}+\lambda_{j}+\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}+\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha})\left\|\left\|B_{ij}\right\|\right\|^{2}\text{.}$ (A9) ### 3\. Calculating $l_{\alpha}(\rho$, $A$, $B)$ First we calculate $\mbox{Tr}(\rho[A$, $B])$. By Eq. (A1), $\rho[A,B]=\lambda_{1}\|x_{1}\rangle\langle x_{1}\|[A$, $B]+....+\lambda_{n}\|x_{n}\rangle\langle x_{n}\|[A$, $B]$ and $\mbox{Tr}(\rho[A$, $B])=\lambda_{1}\langle x_{1}\|[A$, $B]\|x_{1}\rangle+....+\lambda_{n}\langle x_{n}\|[A$, $B]\|x_{n}\rangle$. It is well known that $Re\langle x_{i}\|[A$, $B]\|x_{i}\rangle=0$ and $\langle x_{i}\|[A$, $B]\|x_{i}\rangle=\mathrm{i}(2Im\langle x_{i}\|AB\|x_{i}\rangle)$, where $\mathrm{i}$ is an imaginary number. Consequently, $\mbox{Tr}(\rho[A$, $B])=2\mathrm{i}(\lambda_{1}Im\langle x_{1}\|AB\|x_{1}\rangle+....+\lambda_{n}Im\langle x_{n}\|AB\|x_{n}\rangle)$. Therefore we obtain $\displaystyle\mbox{Tr}(\rho[A,B])$ $\displaystyle=$ $\displaystyle 2\mathrm{i}Im(\lambda_{1}\langle x_{1}\|AB\|x_{1}\rangle+...+\lambda_{n}\langle x_{n}\|AB\|x_{n}\rangle)$ (A10) $\displaystyle=$ $\displaystyle 2\mathrm{i}Im\sum_{j\neq i}\lambda_{i}A_{ij}B_{ji}.$ Note that in Eq. (A10)$\ A_{ii}$ and $B_{ii}$ are real because $A$ and $B$ are self-adjoint. Since $A_{ij}B_{ji}=(A_{ji}B_{ij})^{\ast}$, $\Im\sum_{j\neq i}\lambda_{i}A_{ij}B_{ji}=Im\sum_{i<j}(\lambda_{i}-\lambda_{j})A_{ij}B_{ji}$. Thus, by simplifying, $\mbox{Tr}(\rho[A,B])=2\mathrm{i}Im\sum_{i<j}(\lambda_{i}-\lambda_{j})A_{ij}B_{ji}.$ (A11) Moreover, $\mbox{Tr}\rho^{\left\|2\alpha-1\right\|}[A,B]=2\mathrm{i}Im\sum_{i<j}(\lambda_{i}^{\left\|2\alpha-1\right\|}-\lambda_{j}^{\left\|2\alpha-1\right\|})A_{ij}B_{ji}.$ (A12) Hence, from Eqs. (7), (A11) and (A12), $l_{\alpha}(\rho,A,B)=2\mathrm{i}\sum_{i<j}(\lambda_{i}-\lambda_{j}-(\lambda_{i}^{\left\|2\alpha-1\right\|}-\lambda_{j}^{\left\|2\alpha-1\right\|}))Im(A_{ij}B_{ji}).$ (A13) ### 4\. The proof of the uncertainty relation From Eqs. (LABEL:q-info-2), (A9) and (A13), for Eq. (8) we need to show $\displaystyle[\sum_{i<j}(\lambda_{i}+\lambda_{j}-\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}-\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha})\left\|\left\|A_{ij}\right\|\right\|^{2}][\sum_{i<j}(\lambda_{i}+\lambda_{j}+\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}+\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha})\left\|\left\|B_{ij}\right\|\right\|^{2}]$ (A14) $\displaystyle\geq$ $\displaystyle\\{\sum_{i<j}[\lambda_{i}-\lambda_{j}-(\lambda_{i}^{\left\|2\alpha-1\right\|}-\lambda_{j}^{\left\|2\alpha-1\right\|})]Im(A_{ij}B_{ji})\\}^{2}.$ It is easy to know $[Im(A_{ij}B_{ji})]^{2}\leq\left\|\left\|A_{ij}\right\|\right\|^{2}\left\|\left\|B_{ij}\right\|\right\|^{2}$. Note that $\lambda_{i}+\lambda_{j}-\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}-\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha}=(\lambda_{i}^{\alpha}-\lambda_{j}^{\alpha})(\lambda_{i}^{1-\alpha}-\lambda_{j}^{1-\alpha})\geq 0$. By the Cauchy-Schwartz inequality, the LHS of the inequality in Eq. (A14) $\geq\\{\sum[(\lambda_{i}+\lambda_{j})^{2}-(\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}+\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha})^{2}]^{1/2}Im(A_{ij}B_{ji})\\}^{2}$. Finally, what needs to be shown is $\displaystyle(\lambda_{i}+\lambda_{j})^{2}-(\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}+\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha})^{2}$ (A15) $\displaystyle\geq$ $\displaystyle\|(\lambda_{i}-\lambda_{j})-(\lambda_{i}^{2\alpha-1}-\lambda_{j}^{2\alpha-1})\|^{2}\text{.}$ It is easy to see that $\displaystyle(\lambda_{i}+\lambda_{j})^{2}-(\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}+\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha})^{2}$ $\displaystyle=$ $\displaystyle(\lambda_{i}-\lambda_{j})^{2}-(\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}-\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha})^{2}.$ When $\alpha\geq 1/2$, $\displaystyle(\lambda_{i}-\lambda_{j})^{2}-(\lambda_{i}^{\alpha}\lambda_{j}^{1-\alpha}-\lambda_{i}^{1-\alpha}\lambda_{j}^{\alpha})^{2}$ $\displaystyle=$ $\displaystyle(\lambda_{i}-\lambda_{j})^{2}-\lambda_{i}^{2(1-\alpha)}\lambda_{j}^{2(1-\alpha)}(\lambda_{i}^{2\alpha-1}-\lambda_{j}^{2\alpha-1})^{2}$ $\displaystyle\geq$ $\displaystyle(\lambda_{i}-\lambda_{j})^{2}-(\lambda_{i}^{2\alpha-1}-\lambda_{j}^{2\alpha-1})^{2}$ $\displaystyle=$ $\displaystyle\|(\lambda_{i}-\lambda_{j})-(\lambda_{i}^{2\alpha-1}-\lambda_{j}^{2\alpha-1})\|$ $\displaystyle\times\left\|(\lambda_{i}-\lambda_{j})+(\lambda_{i}^{2\alpha-1}-\lambda_{j}^{2\alpha-1})\right\|$ $\displaystyle\geq$ $\displaystyle\|(\lambda_{i}-\lambda_{j})-(\lambda_{i}^{2\alpha-1}-\lambda_{j}^{2\alpha-1})\|^{2}.$ Note that the last inequality holds because $(\lambda_{i}-\lambda_{j})$ and $(\lambda_{i}^{2\alpha-1}-\lambda_{j}^{2\alpha-1})$ have the same sign. Also, when $0<\alpha\leq 1/2$, we can prove the inequality in Eq. (A15) as follows: Let $\beta=1-\alpha$ with $1/2\leq\beta<1$. Replacing $\alpha$ in Eq. (A15) with $1-$ $\beta$, we obtain $(\lambda_{i}+\lambda_{j})^{2}-(\lambda_{i}^{1-\beta}\lambda_{j}^{\beta}+\lambda_{i}^{\beta}\lambda_{j}^{1-\beta})^{2}\geq\|(\lambda_{i}-\lambda_{j})-(\lambda_{i}^{2\beta-1}-\lambda_{j}^{2\beta-1})\|^{2}$. This ends the proof. ## Appendix B. Additivity The quantity $J_{\alpha}(\rho$, $B)$ is additive in the following sense: $J_{\alpha}(\rho_{1}\otimes\rho_{2}$, $B_{1}\otimes I_{2}+I_{1}\otimes B_{2})=J_{\alpha}(\rho_{1}$, $B_{1})+J_{\alpha}(\rho_{2}$, $B_{2})$. Using the notation in [4], the proof proceeds by letting $\rho_{12}=\rho_{1}\otimes\rho_{2}$ and $L=B_{1}\otimes I_{2}+I_{1}\otimes B_{2}$. Setting $\rho_{12}^{\alpha}=\rho_{1}^{\alpha}\otimes\rho_{2}^{\alpha}$, we have $\rho_{12}^{\alpha}L\rho_{12}^{1-\alpha}L$ $=\rho_{1}^{\alpha}B_{1}\rho_{1}^{1-\alpha}B_{1}\otimes\rho_{2}+\rho_{1}^{\alpha}B_{1}\rho_{1}^{1-\alpha}\otimes\rho_{2}B_{2}$ $+\rho_{1}B_{1}\otimes\rho_{2}^{\alpha}B_{2}\rho_{2}^{1-\alpha}+\rho\otimes\rho_{2}^{\alpha}B_{2}\rho_{2}^{1-\alpha}B_{2}$, and $\displaystyle\mbox{Tr}(\rho_{12}^{\alpha}L\rho_{12}^{1-\alpha}L)$ (B1) $\displaystyle=$ $\displaystyle\mbox{Tr}(\rho_{1}^{\alpha}B_{1}\rho_{1}^{1-\alpha}B_{1})+2\mbox{Tr}(\rho_{1}B_{1})\mbox{Tr}(\rho_{2}B_{2})+\mbox{Tr}(\rho_{2}^{\alpha}B_{2}\rho_{2}^{1-\alpha}B_{2}).$ Similarly, $\mbox{Tr}(\rho_{12}^{\alpha}L^{2})=\mbox{Tr}(\rho_{1}B_{1}^{2})+2\mbox{Tr}(\rho_{1}B_{1})\mbox{Tr}(\rho_{2}B_{2})+\mbox{Tr}(\rho_{2}B_{2}^{2}).$ (B2) From the above Eqs. (B1) and (B2), we can derive $I_{\alpha}(\rho$, $B)$ is additive. Similarly, $\mbox{Tr}(\rho_{12}^{\alpha}L)=\mbox{Tr}(\rho_{1}B_{1})+\mbox{Tr}(\rho_{2}B_{2}).$ (B3) By Eqs. (B1), (B2), and (B3), and the definition of $J_{\alpha}(\rho$, $B)$ in Eq. (6), we can conclude that $J_{\alpha}(\rho$, $B)$ is additive. Acknowledgments The first author wants to thank Prof. Jinwen Chen for his helpful discussion about the inequality in Eq. (A15) and Mr Qin Zhang for the discussion about the idea for $\mbox{Tr}(KK^{\prime})\geq 0$. The paper was supported by NSFC(Grants No.10875061, 60673034. KLC would like to acknowledge financial support by the National Research Foundation & Ministry of Education, Singapore, for his visit and collaboration at Tsinghua University. ## References * [1] Fisher information has been discussed extensively in literature on statistical estimation theory. A particularly insightful survey of the connection between Fisher information and the Heisenberg uncertainty principle can be found in arXiv: quant-ph//0309184 * [2] S. Luo, Phys. Rev. Lett., 91, 180403 (2003). * [3] E.P. Wigner and M. M. Yanase, Proc. Nat. Acad. Sci. U.S.A. 49, 910-918 (1963). * [4] E.H. Lieb, Adv. Math. 11, 267 (1973). * [5] E.H. Lieb and M.M. Ruskai, Phys. Rev. Lett. 30, 434 (1973). * [6] S. Luo, Phys. Rev. A 72, 042110 (2005). * [7] S. Luo, Phys. Rev. A 73, 022324 (2006). * [8] S. Luo, Theor. Math. Phys. 143, 681 (2005). * [9] M. Hirvensalo, Quantum Computing, Springer-Verlag, Berlin, (2001). * [10] M. A. Nielsen and I. L. Chuang, see p.89, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge (2000). * [11] F. Hansen, J. Stat. Phys., 126, 643 (2007). * [12] Z. Chen, Phys. Rev. A 71, 052302 (2005).	arxiv-papers	2009-02-21T09:41:03	2024-09-04T02:49:00.787273	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "D. Li, X. Li, F. Wang, X. Li, H. Huang, L. C. Kwek", "submitter": "Dafa Li", "url": "https://arxiv.org/abs/0902.3729" }
0902.3760	# Inelastic Neutron Scattering Studies of the Spin and Lattice Dynamics in Iron Arsenide Compounds R. Osborn rosborn@anl.gov S. Rosenkranz E. A. Goremychkin A. D. Christianson Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA ISIS Pulsed Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot OX11 0QX, UK Neutron Scattering Science Division, Oak Ridge Naitonal Laboratory, Oak Ridge, TN 37831, USA ###### Abstract Although neutrons do not couple directly to the superconducting order parameter, they have nevertheless played an important role in advancing our understanding of the pairing mechanism and the symmetry of the superconducting energy gap in the iron arsenide compounds. Measurements of the spin and lattice dynamics have been performed on non-superconducting ‘parent’ compounds based on the LaFeAsO (‘1111’) and BaFe2As2 (‘122’) crystal structures, and on electron and hole-doped superconducting compounds, using both polycrystalline and single crystal samples. Neutron measurements of the phonon density-of- state, subsequently supported by single crystal inelastic x-ray scattering, are in good agreement with ab initio calculations, provided the magnetism of the iron atoms is taken into account. However, when combined with estimates of the electron-phonon coupling, the predicted superconducting transition temperatures are less than 1 K, making a conventional phononic mechanism for superconductivity highly unlikely. Measurements of the spin dynamics within the spin density wave phase of the parent compounds show evidence of strongly dispersive spin waves with exchange interactions consistent with the observed magnetic order and a large anisotropy gap. Antiferromagnetic fluctuations persist in the normal phase of the superconducting compounds, but they are more diffuse. Below Tc, there is evidence in three ‘122’ compounds that these fluctuations condense into a resonant spin excitation at the antiferromagnetic wavevector with an energy that scales with Tc. Such resonances have been observed in the high-Tc copper oxides and a number of heavy fermion superconductors, where they are considered to be evidence of $d$-wave symmetry. In the iron arsenides, they also provide evidence of unconventional superconductivity, but a comparison with ARPES and other measurements, which indicate that the gaps are isotropic, suggests that the symmetry is more likely to be extended-$s_{\pm}$ wave in character. ###### keywords: iron pnictide , superconductivity , magnetism , inelastic neutron scattering ###### PACS: 74.20.Mn , 78.70.Nx , 74.25.Kc , 75.30.Fv ## 1 Introduction Since the discovery of superconductivity in iron arsenide compounds [1, 2, 3], neutron scattering experiments have made significant contributions to our understanding of the underlying physics. Early neutron diffraction results generated considerable excitement because they revealed remarkable similarities with the high-temperature copper oxide superconductors. For example, in both the iron arsenides and the cuprates, superconductivity arises when an antiferromagnetically ordered phase has been suppressed by chemical doping [4]. Neutron scattering continues to be essential in determining the magnetic and structural phase diagrams of these materials as a function of dopant concentration or applied pressure [5]. On the other hand, neutrons have also identified important differences with the cuprates, such as the reduced size of the ordered moments and their extreme sensitivity to structural modifications [4, 6, 7]. Elastic neutron scattering, which probes static magnetic and structural correlations, is discussed in more detail in another article in this issue [5]. The purpose of this review is to summarize the results of inelastic neutron scattering, which probes dynamic correlations involving phonons and spin fluctuations, both of which are candidates for binding the superconducting electron pairs. With the discovery of any new family of superconductors, the first task is to determine whether the critical temperature can be explained by electron-phonon coupling within a conventional BCS theory. This question is usually addressed within the formalism of Eliashberg theory [8], in which the superconducting energy gap is expressed in terms of a spectral density function derived from the phonon density-of-states (PDOS) weighted by electron-phonon matrix elements. We will review inelastic neutron scattering, mostly on polycrystalline samples, that have been used to estimate the PDOS [9] and validate the results of first principles density functional calculations [10, 11]. In broad terms, the agreement between theory and experiment is very good, although some modes are extremely sensitive to the spin state assumed in the theoretical estimates. Although there are subtle, so far unexplained, anomalies that will require further single crystal measurements to resolve fully, the early consensus is that the electron-phonon coupling is too weak by a factor of about five to explain the observed critical temperatures. If phonons are not responsible for the superconductivity, spin fluctuations offer an alternative bosonic spectrum to mediate the electron pairing. As first shown by de la Cruz et al [4], the ground state of the non- superconducting parent compounds is a spin density wave, whose transition occurs close to a tetragonal-orthorhombic structural transition. Both chemical doping and, in contrast to the cuprates, pressure [12] can be used to suppress antiferromagnetic order and induce superconductivity. One of the key questions to resolve is what drives the magnetism. Band structure calculations show that hole pockets at the $\Gamma$-point and the electron pockets at the $M$-point can show strong nesting with a sharp peak in the Lindhard susceptibility at Q=($\pi$,$\pi$), using tetragonal notation [13]. Since this is also the wavevector of magnetic order, it is natural to propose that the undoped arsenides are itinerant spin density waves like chromium, which would provide an explanation for the reduced size of the ordered moment. However, some have argued that the pnictides are in fact close to a Mott insulating phase and that the reduced moments are due to frustration caused by competing superexchange interactions [14, 15], so the strength of electron correlations remains an important issue. Inelastic neutron scattering experiments have shown that the spin waves are strongly dispersive, with velocities that are not inconsistent with an itinerant spin density wave, and more three- dimensional than the cuprates. They also show substantial energy gaps that are not fully understood. Finally, we review three reports of a spin resonant excitation seen so far only in the ‘122’ compounds [16, 17, 18]. With chemical doping, the spin fluctuations become more diffuse in the normal state, though still centered at the antiferromagnetic wavevectors. However, below Tc, these fluctuations condense into a resonant excitation that is localized in both momentum transfer, $Q$, and energy transfer, $\omega$. Such resonant excitations have been observed in a wide range of high-temperature copper oxide superconductors [19] as well as, more recently, several heavy fermion superconductors [20, 21, 22, 23], where they are considered to be evidence of $d$-wave superconductivity [24, 25]. In the case of the iron arsenides, a comparison with ARPES data suggests that the resonance is evidence of extended-$s_{\pm}$ wave symmetry [26, 27], in which the disconnected hole and electron pockets have energy gaps of opposite sign. Since there is no angular anisotropy of the gap in this symmetry, it is difficult to verify by the techniques used in the copper oxides. Inelastic neutron scattering is so far the only probe that has provided phase-sensitive information about the unconventional symmetry of the energy gap in the iron arsenide superconductors. ## 2 Lattice Dynamics Within Eliashberg theory, the superconducting gap equation is expressed in terms of an electron-phonon coupling, $\lambda$, and a Coulomb repulsion, $\mu^{\ast}$ [8]. The electron-phonon coupling is derived from the spectral density function $\alpha^{2}F(\omega)$, in which the phonon modes, represented by the bare phonon density-of-states (PDOS), $F(\omega)$, are weighted by electron-phonon matrix elements. In many superconductors, there is little difference in the functional forms of $\alpha^{2}F(\omega)$ and $F(\omega)$, i.e., the phonons are all coupled to the electronic states equally strongly, but there are exceptions. For example, in MgB2, the electron-phonon coupling of the $E_{2g}$ modes is particularly strong because their energies are governed by strongly covalent boron-boron $\sigma$-bonds. In principle, $\alpha^{2}F(\omega)$ can be determined by inverting tunneling data, but this is not always available so neutron scattering measurements of $F(\omega)$ play a valuable role in providing initial estimates of Tc and in validating ab initio calculations. In BCS superconductors, numerical calculations based on the Eliashberg equations have shown that the critical temperature can often be approximated using the Allen-Dynes equation [28] $k_{B}T_{c}=\frac{\hbar\omega_{ln}}{1.2}\exp{\left[-\frac{1.04(1+\lambda)}{\lambda-\mu^{}(1+0.62\lambda)}\right]}$ (1) where $\omega_{ln}$ is a logarithmic phonon average defined in equation 2.16 of ref. [8], $\mu^{\ast}$ is the Coulomb repulsion, and the electron-phonon coupling, $\lambda$, is given by $\lambda=2\int_{0}^{\infty}{d\omega\frac{\alpha^{2}F(\omega)}{\omega}}$ (2) The Coulomb repulsion, $\mu^{\ast}$, is normally treated as a phenomenological parameter but it is possible to make accurate predictions of the critical temperature from first principles calculations without any adjustable parameters [29]. There were two ab initio calculations of the phonon density-of-states in LaFeAsO before the first neutron measurements were published, and both are in broad agreement with each other [11, 10]. Singh and Du established the main features of the band structure in the Local Spin Density Approximation, showing that the low-lying electronic states arise from five iron $d$-bands spanning an energy range of -2.1 eV to 2.0 eV, with the oxygen and arsenic $p$-states well below the Fermi level [10]. The Fermi surface consists of two- dimensional cylinders, with two hole pockets at the $\Gamma$-point and two electron pockets at the M-point, derived mainly from $d_{xz}$ and $d_{yz}$ states, along with a fifth more three-dimensional hole pocket, centered at Z, derived from $d_{z^{2}}$ states hybridized with As $p$-states and La orbitals. The resulting phonon excitations are spread over 70 meV, with those over 40 meV representing oxygen modes. Those at lower frequency are of mixed lanthanum, iron, and arsenic character. As several of the electron bands are two-dimensional, it is not surprising that the optic modes show little dispersion along the $\Gamma$-Z direction, but the acoustic phonons are fairly three-dimensional in character with a Debye temperature of 340 K. The calculated phonon density-of-states has three main peaks at approximately 12, 21, and 34 meV. On the basis of such band structure calculations, Boeri et al calculate the electron-phonon coupling, which they find to be evenly distributed among all the modes [11]. When they integrate this coupling over all frequencies, they derive $\lambda=0.21$, which is much smaller than any other known electron- phonon superconductors. In the Allen-Dynes approximation, they estimate Tc = 0.5 K, from their calculated value of $\omega_{ln}$ = 205 K, assuming $\mu^{\ast}=0$. A more accurate numerical Migdal-Eliashberg calculation only increases this to 0.8 K, and they estimate that $\lambda$ would have to be a factor five stronger to generate the observed Tc of 26 K. Their calculations are based on the non-superconducting parent compound, LaFeAsO, but adding electrons through fluorine doping would, in the rigid band approximation, reduce the electronic density-of-states at the Fermi level, and so tend to reduce Tc even further. Orbitals that might be expected to produce an enhanced electron-phonon coupling, such as the $d_{x^{2}-y^{2}}$ orbitals that are directed along the Fe-Fe bond, are too far from the Fermi level to have a strong influence. Figure 1: Inelastic neutron scattering from LaFeAsO0.89F0.11 as a function of momentum transfer, Q, and energy transfer, $\Delta$E (denoted as $\omega$ in the text), measured at 10 K with an incident neutron energy of 100 meV [9]. The units of the intensity scale are arbitrary. Figure 2: Generalized phonon density of states, $G(\omega)$, of LaFeAsO1-xFx measured with an incident neutron energy of 130 meV and normalized to an integral of 1.0 after correction for the Bose population and Debye-Waller factors, detector efficiency, and multiphonon scattering [9]. (a) Experimental $G(\omega)$ measured at 35 and 300 K for $x=0.1$. (b) Experimental $G(\omega)$ for $x=0$ and $x=0.1$ measured at 300 K. (c) First-principles calculation of the phonon density-of-states of LaFeAsO based on the band structure of Singh and Du [10]. Even before any measurements were reported, it therefore seemed unlikely that a conventional BCS mechanism could explain the elevated transition temperatures in the iron arsenides. Nevertheless, it was important to validate these predictions from experiment. Although inelastic x-ray scattering can also be used to measure phonon dispersion relations (and has been used in the iron pnictides [30, 31]), neutron scattering is an extremely efficient method of determining the phonon density-of-states. In a multicomponent system, the inelastic neutron scattering law in polycrystalline samples is given by $\begin{split}{\rm S}(Q,\omega)=\sum_{i}&{\sigma_{i}\frac{\hbar Q^{2}}{2M_{i}}\exp{(-2W_{i}(Q))}}\\\ &\frac{{\rm G}_{i}(\omega)}{\omega}[n(\omega)+1]\end{split}$ (3) where $\sigma_{i}$ and $M_{i}$ are the neutron scattering cross section and atomic mass of the $i^{th}$ atom and $n(\omega)=[\exp{(\hbar\omega/k_{B}T)}-1]^{-1}$ is the Bose population factor. The generalized PDOS, ${\rm G}(\omega)=\sum_{i}{{\rm G}_{i}(\omega)}$, where Gi($\omega$) is defined as ${\rm G}_{i}(\omega)=\frac{1}{3N}\sum_{j{\bf q}}{\|{\bf e}_{i}(j,{\bf q})\|^{2}\delta[\omega-\omega(j,{\bf q})]}$ (4) which, in turn, defines the Debye-Waller factor, Wi(Q) through $W_{i}(Q)=\frac{\hbar Q^{2}}{2M_{i}}\int_{0}^{\infty}{d\omega\frac{G_{i}(\omega)}{\omega}[2n(\omega)+1]}$ (5) $\omega(j,{\bf q})$ and $e_{i}(j,{\bf q})$ are the frequencies and eigenvectors, respectively, of the phonon modes. Inelastic neutron scattering therefore measures a sum of the partial phonon density-of-states of each constituent element weighted by $\sigma_{i}$/Mi. In LaFeAsO, these weights, relative to the value for oxygen, are 0.23, 0.76, and 0.27, for La, Fe, and As, respectively [9], so the low energy scattering is dominated by iron modes. The generalized PDOS also differs from the base PDOS, F($\omega$), because of additional weighting by the eigenvectors (see Equation 4). Osborn et al argued that this makes the use of G($\omega$) preferable to F($\omega$) as an approximation to $\alpha^{2}$F($\omega$), because these eigenvectors also enter into electron-phonon matrix elements [32]. Nevertheless, the safest way to compare ab initio calculations to the neutron data is to calculate the generalized PDOS directly as was done by Bohnen et al in the case of MgB2 [33]. Strictly speaking, Equation 3 only applies to materials in which the neutron cross sections of the constituent elements, $\sigma_{i}$, are entirely incoherent, whereas both iron and arsenic have strongly coherent cross sections. This means that there will be strong deviations from the simple $Q^{2}$ dependence of the scattering intensity caused by variations in the eigenvectors within each Brillouin zone. Nevertheless, if the data are taken on polycrystalline samples, so that the measured scattering is spherically averaged, and integrated over a sufficiently broad range of $Q$, they can be used to derive a good approximation to the generalized PDOS even when the $\sigma_{i}$ are coherent. This is known as the incoherent approximation, which must be satisfied for the analysis to be reliable. In MgB2, for example, there were reports of a low-energy peak that were then shown to be an artifact of insufficient Q-averaging [32]. However, most of the reported PDOS measurements in the iron arsenides were taken on neutron spectrometers using relatively high incident energies and summed over a large range of momentum transfers, ensuring that the conditions for the incoherent approximation are met. Figure 3: The experimental phonon spectra of Sr0.6K0.4Fe2As2 and Ca0.6Na0.4Fe2As2, measured in neutron energy gain with an incident neutron energy of 3.1 meV [35], compared to BaFe2As2, measured in energy loss with an incident energy of 57.5 meV [34]. All the phonon spectra are normalized to unity. The first reported phonon measurements in the ‘1111’ system covered a limited energy range up to 20 meV [36], but subsequent experiments have covered the entire phonon spectrum [9]. Christianson et al used samples of both non- superconducting LaFeAsO and superconducting LaFeAsO1-xFx, with $x\approx 0.1$ prepared by two different synthesis groups, based at Oak Ridge National Laboratory and Ames Laboratory. The data were collected on the newly commissioned Fermi chopper spectrometer, ARCS, at the Spallation Neutron Source in Oak Ridge. Using incident neutron energies of 130, 60, and 30 meV, supplemented by some triple-axis measurements, they observed peaks at 12, 25, 31, 40, and 60 meV. These features seem to be unaffected either by temperature reduction, apart from a sharpening of the peaks, or by dopant concentration. In particular, there does not appear to be any strong phonon renormalization either at the orthorhombic transition in the non-superconducting samples, or at Tc in the superconducting samples. A comparison of their data with the ab initio calculations showed good qualitative agreement in the overall energy scale of the phonons and the energies of most of the peaks. The most notable discrepancy was in the location of the 31 meV peak, which is distinctly softer than the theoretical prediction. A similar discrepancy was noted in inelastic x-ray scattering, where it was attributed it to a 30% reduction in the Fe-As force constants[30]. The length of the Fe-As bond shows a remarkable sensitivity to the iron spin state; calculations that underestimate the iron magnetism also substantially underestimate the bond lengths, and therefore overestimate the bond energies. The fact that there is such little variation in the phonon peak energies is evidence therefore that, locally, the iron spin state is remarkably robust, persisting above the SDW transition temperature in LaFeAsO and surviving the destruction of SDW order in LaFeAsO0.9F0.1. This is borne out by measurements of the spin dynamics discussed in the next section. There have also been phonon measurements performed on the ‘122’ compounds, firstly with experiments on non-superconducting BaFe2As2 [34] and then on superconducting Sr0.6K0.4Fe2As2 (Tc=32 K) and Ca0.6Na0.4Fe2As2 (T$c$=21K) [35]. The measurements were taken on the IN4 spectrometer, using an incident neutron energy of about 60 meV, and the IN6 spectrometer, using an incident energy of 3.1 meV, both at the Institut Laue Langevin, France. The IN6 data were measured in neutron energy gain, which requires an elevated temperature. With the absence of light oxygen ions in this structure, the maximum phonon energy is just under 40 meV. However, in other respects, the measured phonon spectra of the different ‘122’ compounds look similar to the ‘1111’ compounds and to each other. There are peaks at 12, 22, 27, and 34 meV in the barium and strontium samples but, in the calcium sample, the 22 meV peak appears to have shifted down to below 18 meV. Since the bond lengths are shorter in Ca0.6Na0.4Fe2As2, which would normally increase the mode energies, there must be some change in the bonding characteristics. Mittal et al note that there is a slight stiffening from 300 K to 140 K of the higher energy peaks in the IN6 data, but a softening of the acoustic modes, which they suggest is a sign of electron-phonon coupling [35]. In summary, neutron scattering studies of the lattice dynamics of the iron arsenides are broadly consistent with ab initio calculations, although there are some unexplained anomalies that will require more detailed single crystal measurements before they are explained satisfactorily. The integrated electron-phonon coupling is estimated to be far too weak to be responsible for the superconductivity. However, the sensitivity of the Fe-As bond length to the iron spin state shows that there are potential sources of electron-phonon coupling that may need to be investigated more fully before some contribution from electron-phonon coupling is definitively ruled out. ## 3 Spin Dynamics The non-superconducting parent compounds, such as LaFeAsO or BaFe2As2, undergo two phase transition with decreasing temperature [4, 6, 37]. The first is a structural transition from the high-temperature tetragonal phase to a low- temperature orthorhombic phase, which is closely followed by (or sometimes coincident with) a second magnetic transition [5]. The low-temperature antiferromagnetic structure is not the checkerboard order that would be expected from nearest-neighbor antiferromagnetic interactions, but a stripe phase, which implies the presence of competing interactions (see Fig. 4). Yildirim has shown that this stripe structure results from an inherent frustration produced by a strong next-nearest-neighbor antiferromagnetic exchange between iron spins on the square planar lattice [38]. The observed structure is stable provided the nearest-neighbor exchange, $J_{1}$, along the square edges, and next-nearest-neighbor exchange, $J_{2}$, along the square diagonals, satisfy, $J_{2}>J_{1}/2$. He postulates that the structural transition is a means of relieving this frustration, which implies the existence of strong short-range spin correlations above the antiferromagnetic transition temperature, TSDW, consistent with the discussion in the previous section. Figure 4: The crystal and magnetic structure of BaFe2As2 (taken from Ref. [17]). The unit cell contains two layers of Fe2As2 tetrahedra (Fe, blue spheres; As, yellow spheres), separated by planes of barium atoms (red spheres). The blue arrows show the observed ordering of the iron spins. The red arrow shows the spacing of the antiferromagnetic stripes. As stated in the introduction, there are two alternative explanations for the origin of the magnetic interactions. The first is that the antiferromagnetism is produced by a nesting instability coupling the $\Gamma$-centered hole pockets and the M-centered electron pockets. This assumes an itinerant picture of weakly interacting electrons, that is consistent with the reduced size of the magnetic moments (0.36$\mu_{B}$ in LaFeAsO [4] or 0.87$\mu_{B}$ in BaFe2As2 [37]), which can be explained by density functional theory provided the experimental lattice parameters are used [39]. However, there are alternative models, which assume much stronger electron correlations, with moments reduced by frustration [14, 15]. Inelastic neutron scattering is the most direct way of determining the spin wave excitations of the antiferromagnetically ordered compounds. Whatever the origin of the magnetic interactions, it is usually possible to analyze the measured dispersion using a Heisenberg model including single-ion anisotropy terms that are required to produce the observed energy gaps [40, 41] $\begin{split}H=&\sum_{ij}{J_{ij}\mathbf{S_{i}}\cdot\mathbf{S_{j}}}+\\\ &\sum_{i}{K_{c}(S^{z}_{i})^{2}+K_{ab}\left[(S^{x}_{i})^{2}-(S^{y}_{i})^{2}\right]}\end{split}$ (6) Ewings et al derive explicit solutions of this Hamiltonian, giving both the energies and spin wave cross sections [40], but, at low energies, the spin wave dispersion can be approximated by [42] $\begin{split}\hbar\omega(\mathbf{q})=\sqrt{\Delta^{2}+v^{2}_{xy}(q^{2}_{x}+q^{2}_{y})+v^{2}_{z}q^{2}_{z}}\end{split}$ (7) where q is the reduced wavevector relative to the antiferromagnetic zone center, $\Delta$ is the anisotropy gap, and $v_{xy}$ and $v_{z}$ are the in- plane and $c$-axis spin wave velocities. Figure 5: Neutron scattering spectra from a polycrystalline sample of BaFe2As2 at 7 K [40]. The data were taken on the MERLIN (ISIS) spectrometer using incident neutron energies of (a) 200 meV and (b) 50 meV. The pillars of scattering show the steep spin wave excitations emerging from the ($\frac{1}{2}$,$\frac{1}{2}$,$l$) and ($\frac{1}{2}$,1,$l$) positions (using the tetragonal Brillouin zone) at Q=1.2 Å and 2.6 Å, respectively. Figure 6: Constant energy scans performed on triple-axis spectrometers on SrFe2As2 at 160 K, showing the broadening as a function of increasing energy resulting from the spin wave dispersion [43]. Spin wave measurements have been reported in a number of the ‘122’ compounds, SrFe2As2 [43], CaFe2As2 [42, 44], and BaFe2As2 [40, 45]. Most of these are single crystal measurements using triple-axis spectrometers, although Ewings et al [40] and Diallo et al [44] used pulsed neutron source Fermi chopper spectrometers, MERLIN and MAPS respectively, at ISIS, to extend the energy range to 100 meV and above. Broadly speaking, all these experiments are consistent. They observe extremely steep spin waves emerging from the SDW wavevector with a substantial energy gap, ranging from 6.9 meV in SrFe2As2 [43] to 9.8 meV in BaFe2As2 [45]. The value of the spin wave velocity is more uncertain because it is difficult to resolve the propagating modes in constant energy scans when the dispersions are so steep, except at very high energy [44]. However, estimates can be derived by performing model simulations including the instrumental resolution. The in-plane estimates vary from 280 meVÅ in BaFe2As2 [40, 45] to 420 meVÅ in CaFe2As2 [42]. Low-energy measurements predicted zone boundary energies of about 175 meV [40, 45], which is consistent with the first single crystal measurements using pulsed neutrons [44], although the spin waves above 100 meV appear to be heavily damped. The out-of-plane velocities are smaller but still substantial, reduced from the in-plane values by a factor of 2 [42] to 5 [45], so the magnetic order is truly three-dimensional. The temperature dependence of the spin wave scattering is consistent with Bose statistics, with the energy gap renormalizing to zero at TSDW. However, there is evidence of the persistence of two-dimensional short-range spin correlations in BaFe2As2 above TSDW, in the form of quasielastic scattering centered on ($\frac{1}{2}$,$\frac{1}{2}$,$l$) rods [45]. If the Heisenberg model includes next-nearest-neighbors, as required to stabilize the stripe structure, there are four exchange constants: $J_{1a}$ and $J_{1b}$ couple antiferromagnetic and ferromagnetic nearest neighbors, respectively; $J_{2}$ couples next-nearest neighbors within the plane; and $J_{z}$ is the out-of-plane coupling. In SrFe2As2, Zhao et al estimated the exchange interactions to be $J_{1a}+2J_{2}\sim 100$ meV, with $J_{z}\sim 5$ meV [43]. These values are consistent with McQueeney et al’s analysis of CaFe2As2, where they estimate $J_{1a}$ = 41 meV, $J_{1b}$ = 10 meV, $J_{2}$ = 21 meV, and $J_{z}$ = 3 meV. Note that the value of $J_{2}$ is sufficiently large to stabilize stripe antiferromagnetism. In an itinerant SDW, the spin wave velocity produced by the excitation of particle-hole pairs across the nested surfaces is given by $\sqrt{v_{e}v_{h}}$, where $v_{e}$ and $v_{h}$ are the electron and hole velocities of their respective bands [46]. In LaFeAsO, band structure calculations estimate that the in-plane electron velocity is a factor of 3 greater than the in-plane hole velocity, which is, in turn, a factor of 2.5 greater than both the $c$-axis hole and electron velocities [10]. The absolute values are uncertain since the calculated bandwidths are greater than ARPES measurements [47]. They do, however, give a ratio of in-plane to $c$-axis spin wave velocities of about 4, which is within the experimental range. Zhao et al go further and show that the absolute values are in reasonable agreement with band structure, after renormalizing the bandwidths [43]. The damping of the high-energy excitations seen in CaFe2As2 could be due to the decay of spin waves into electron-hole pairs within a Stoner continuum [44]. More speculatively, Matan et al report the existence of excess scattering above 20 meV, which is more two-dimensional than the spin waves, and which they suggest is consistent with the existence of excess scattering in the Stoner continuum. However, the data from Ewings et al do not show evidence for this anomaly [40], so it is difficult to draw any firm conclusions. Finally, there are claims that an unusual temperature dependence of the excitations in LaFeAsO is also consistent with itinerant spin density waves [48]. In summary, the neutron measurements in the parent compounds of the iron arsenides provide a consistent picture of spin wave dispersions, that are extremely steep and anisotropic, although clearly three-dimensional, with a very large energy gap. The exchange parameters are consistent with the observed stripe phase, and are not inconsistent with itinerant SDW models, although it is too early to rule out more localized magnetic models based on strong but frustrated superexchange interactions. ## 4 Resonant Spin Excitations Figure 7: Inelastic neutron scattering on Ba0.6K0.4Fe2As2, measured using an incident neutron energy of 60 meV below and above Tc, i.e., 7 K (a) and 50 K (b) respectively, and 15 meV below and above Tc, (c) and (d) respectively. The data shows the transfer of spectral weight from diffuse spin fluctuations centered at 1.15 Å-1 in the normal state into a resonant spin excitation at an energy transfer of 15 meV in the superconducting state. The color scale is in units of mbarns/sr/meV/mol [17]. The previous two sections concerned inelastic neutron scattering results that provide the context within which superconductivity develops in the iron arsenides, but do not directly address the superconducting state itself. That has been the traditional role for neutron scattering since, in conventional superconductors, neutrons do not couple directly to the superconducting order parameter. However, neutron scattering results on other unconventional superconductors over the past twenty years has shown that they can provide direct information concerning the superconducting energy gap, in particular, its symmetry. Surprisingly, in the case of the iron arsenides, it is currently the only technique to give phase-sensitive evidence of the gap symmetry, since many of the techniques commonly used in other superconductors do not apply. Inelastic neutron scattering measures directly the dynamic magnetic susceptibility of the band electrons. Normally, this signal is too diffuse for useful measurements, but there are cases where the magnetic response is strongly enhanced at particular energies and wavevectors making such experiments feasible. The most common example is in transition metal magnets such as iron and cobalt, where the RPA susceptibility of the itinerant $d$-electrons has poles corresponding to propagating spin wave modes [49]. In an itinerant SDW model, the spin waves described in the previous section fall into this category. No such divergences exist in the dynamic magnetic susceptibility of conventional $s$-wave superconductors, but it turns out that they can exist when the superconducting energy gap changes sign, either within a single Fermi surface or between two disconnected Fermi surfaces. Specifically, there is a strong enhancement of the dynamic magnetic susceptibility due to coherence factors that appear because of the anomalous Green function of the superconducting phase. The following term appears in the non-interacting susceptibility [50, 51, 25, 24]. $\Big{(}1-\frac{\xi_{\mathbf{k}+\mathbf{q}}\xi_{\mathbf{k}}+\Delta_{\mathbf{k}}\Delta_{\mathbf{k}+\mathbf{q}}}{E_{\mathbf{k}+\mathbf{q}}E_{\mathbf{k}}}\Big{)}$ (8) where the energy of the superconducting quasiparticles, $E_{\mathbf{k}}=\sqrt{\xi_{\mathbf{k}}^{2}+\Delta_{\mathbf{k}}^{2}}$. Here, $\xi_{\mathbf{k}}$ are the single electron energies of the normal-state and $\Delta_{\mathbf{k}}$ are the values of the energy gap at points $\mathbf{k}$ on the Fermi surface. Equation 8 defines the principal characteristics of the resonant spin excitations. If $\Delta_{\mathbf{k}+\mathbf{Q}}=\Delta_{\mathbf{k}}$, as in a conventional $s$-wave superconductor, Equation 8 vanishes on the Fermi surface ($\xi_{\mathbf{k}}=0$). In this case, there is no pole when this non- interacting susceptibility is introduced into an RPA expression for the interacting susceptibility. However, if $\Delta_{\mathbf{k}+\mathbf{Q}}=-\Delta_{\mathbf{k}}$, i.e. when Q connects two Fermi surface points whose superconducting order parameters have opposite sign, then Equation 8 is maximal, resulting in a pole at Q for some energy less than $2\Delta$. Such resonant excitations are now believed to be a universal feature of the high-temperature copper oxide superconductors [19], with observations in a large number of different systems [52, 53, 51, 54, 55, 56]. The resonance energy scales approximately as $\omega_{0}\sim 5k_{B}$Tc. They are commonly taken as evidence of $d_{x^{2}-y^{2}}$ symmetry, in which the energy gap changes sign within a single Fermi surface. Strikingly, there are now several reports that similar resonant excitations are present in heavy fermion superconductors [20, 21, 22, 23], where Tc can be as low as 0.7 K. Nevertheless, the energy of the resonance appears to obey the same scaling relations, with $\omega_{0}/2\Delta$, where $\Delta$ is the maximum value of the superconducting energy gap, taking values between 0.62 and 0.74, a remarkable observation given that Tc varies by over two orders of magnitude. Figure 8: The inelastic neutron scattering from Ba0.6K0.4Fe2As2 integrated over a $Q$-range of 1.0 to 1.3 Å-1 and an $\omega$ range of 12.5 to 17.5 meV. The integration range corresponds to the region of maximum intensity of the resonant excitation observed below Tc (see Fig. 7). The dashed line is a guide to the eye below Tc and shows the average value of the integrals above Tc [17]. Figure 9: (a) Calculated imaginary part of the RPA spin susceptibility at the SDW wave vector $Q_{AFM}$ as a function of frequency in the normal and superconducting states. The red, dotted blue, and solid blue curves correspond to the total RPA susceptibility for non-superconducting, $d_{x^{2}-y^{2}}$ symmetry and extended-$s_{\pm}$ symmetry models.. The thin (black) curves refer to the partial RPA contributions for the interband and intraband transitions in the $s_{\pm}$ superconducting state. (b) Calculated imaginary part of the total RPA spin susceptibility in the $s_{\pm}$ state as a function of frequency and momentum along Q=($h$,$h$) [58]. Figure 7 shows inelastic neutron scattering data taken on the MERLIN spectrometer, at the ISIS Pulsed Neutron Source. The polycrystalline sample of Ba0.6K0.4Fe2As2 used in these measurements was optimally doped with a Tc of 38 K. The intense scattering at high Q and low energy results from phonon and elastic nuclear scattering, respectively, but the remaining scattering results from spin fluctuations. In the normal phase, there is an echo of the sharp spin waves seen in Fig. 5, with a pillar of scattering at the antiferromagnetic wavevector, Q=1.15Å . However, the scattering is considerably more diffuse representing short-range spin fluctuations rather than propagating spin waves. It also persists down to lower energy. As the temperature is lowered through Tc, there is a transfer of spectral weight into the resonant spin excitation at an energy transfer of 15 meV. Since the maximum gap seen in ARPES data is 12 meV [57], $\omega_{0}/2\Delta\sim 0.58$ in good agreement with the scaling in other unconventional superconductors. The temperature dependence of this resonant spin excitation is shown in Fig. 8, where it behaves like an order parameter, as also observed in the other unconventional superconductors. The observation of a spin resonance does not necessarily imply $d$-wave symmetry. In fact, ARPES data on Ba0.6K0.4Fe2As2 shows purely isotropic gaps around each surface [57], which is inconsistent with a $d$-wave model. Although such models have been discussed in connection with the iron arsenides [26], most interest has been focussed on extended-$s_{\pm}$ models. In fact, explicit calculations of the neutron scattering cross section predict the existence of a resonant spin excitation with extended-$s_{\pm}$, but not $d$-wave symmetry [58, 59]. The small Fermi surfaces seen in the iron arsenides would not intersect any of the nodal lines in these models, so the gaps on each surface would be nearly constant. However, the prediction is that the sign of the energy gap on the hole pockets at the $\Gamma$-point and the electron pockets at the M-point would be opposite, reflecting an electron- electron repulsion at short-range but an attractive interaction for electrons on neighboring iron atoms. The wavevector connecting the hole and electron pockets is precisely where the resonant spin excitation has been observed. More recently, resonant spin excitations have also been seen in single crystals of BaFe1.84Co0.16As2 using both pulsed source and triple-axis instruments at Oak Ridge [18]. This has confirmed that the excitation is centered at Q=($\frac{1}{2}$,$\frac{1}{2}$) within the plane, but it is two- dimensional, with intensity spread out along Q=$l$ following a single-ion Fe2+ form-factor. Since Tc = 22 K, which is somewhat lower than in Ba0.6K0.4Fe2As2, the energy of the resonance is also lower at 9.6 meV, in agreement with the previous scaling. There is a third report of a spin resonance in BaFe1.9Ni0.1A2 (Tc = 20 K) [16], again observed at Q=($\frac{1}{2}$,$\frac{1}{2}$,$l$). Chi et al observe an energy dispersion of about 2 meV; the resonance is at 9.1 meV at $l=0$ but at 7.0 meV at $l=\pm 1$. They attribute this to antiferromagnetic coupling between the FeAs layers, although it could also reflect some three-dimensional modulation of the Fermi surfaces. Although this work shows evidence of three- dimensionality in the superconducting order, it is still much more two- dimensional than the spin wave excitations of the parent compounds. ## 5 Conclusions It is far too soon since the discovery of these fascinating compounds to declare that any of the important issues surrounding their superconductivity have been settled. As shown in this review, there are many unanswered questions concerning the origin of the magnetic interactions, the strength of the electron-phonon coupling, or the dimensionality of the superconducting order parameter. Nevertheless, it is remarkable how much progress has been made in such a short time. This reflects the experience that the scientific community has gained over the past thirty years of studying unconventional superconductors, first the heavy fermions and then the copper oxide superconductors. Many of the insights gained in those investigations have been directly applied to the new iron arsenide superconductors, and, in some cases, theories considered but then rejected in other systems have become useful here. This is particularly true for neutron scattering. It took several years for the significance of the resonant spin excitations in the copper oxides to be appreciated. Now our theoretical understanding is well advanced, particularly now that they have been seen in other unconventional superconductors. This has allowed the new measurements on the iron arsenides to be incorporated into theories of the superconductivity much more rapidly than before. Although the new methods of detecting extended $s_{\pm}$-wave symmetry have been proposed [60], inelastic neutron scattering is until now the only phase-sensitive technique to provide direct evidence in these compounds. In spite of the many similarities to other superconductors, the iron arsenides are not carbon copies. If, as seems likely, the superconducting pairing is mediated by spin fluctuations, they are much more itinerant than in the copper oxides or the heavy fermions and the superconducting symmetry appears to be quite different. The existence of the resonance at the same wavevector as the SDW does lend some support to the idea that the transition from antiferromagnetism to superconductivity occurs when the changes in the Fermi surface with doping or pressure suppress the nesting instability. Although the conditions for magnetic order are no longer satisfied, the susceptibility is still sufficiently enhanced to favor spin fluctuation-mediated superconductivity as discussed by Singh in his review [13]. The fact that the wavevectors characterizing both the spin density wave and superconductivity are the same is therefore no coincidence. Inelastic neutron scattering experiments are only just beginning in these systems. The improvements in sample quality and, particularly important for neutrons, size will allow more detailed studies over a wider range of wavevectors and energies, with single crystal results superceding the earlier polycrystalline data, as has already begun to happen. We have also not addressed the many materials science issues resulting from phase separation, both intrinsic and extrinsic. Nevertheless, progress is likely to be rapid if the experience of the past year is an accurate guide. ## 6 Acknowledgements We acknowledge valuable conversations with Michael Norman, David Singh, and Taner Yildirim in writing this review, and express gratitude for our many collaborators over the past year. This work was supported by the Division of Materials Sciences and Engineering Division and the Scientific User Facilities Division of the Office of Basic Energy Sciences, U.S. Department of Energy Office of Science, under Contract Nos. DE-AC02-06CH11357 and DE- AC05-00OR22725. ## References [1] Y. Kamihara, T. Watanabe, M. Hirano, H. Hosono, J Am Chem Soc 130 (2008) 3296–3297. * [2] H. Takahashi, K. Igawa, K. Arii, Y. Kamihara, M. Hirano, H. Hosono, Nature 453 (2008) 376–378. * [3] Z.-A. Ren, G.-C. Che, X.-L. Dong, J. Yang, W. Lu, W. Yi, X.-L. Shen, Z.-C. Li, L.-L. Sun, F. Zhou, Z.-X. Zhao, Europhys Lett 83 (2008) 17002. * [4] C. D. L. Cruz, Q. Huang, J. Lynn, J. Li, W. R. Ii, J. Zarestky, H. A. Mook, G. F. Chen, J. L. Luo, N. Wang, P. Dai, Nature 453 (2008) 899–902. * [5] J. W. Lynn, P. Dai, arXiv:0902.0091. * [6] M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W. Hermes, R. Pöttgen, Phys Rev B 78 (2008) 020503. * [7] T. Yildirim, arXiv:0807.3936. * [8] J. P. Carbotte, Rev Mod Phys 62 (1990) 1027. * [9] A. D. Christianson, M. D. Lumsden, O. Delaire, M. B. Stone, D. L. Abernathy, M. A. Mcguire, A. S. Sefat, R. Jin, B. C. Sales, D. Mandrus, E. D. Mun, P. C. Canfield, J. Y. Y. Lin, M. Lucas, M. Kresch, J. B. Keith, B. Fultz, E. Goremychkin, R. McQueeney, Phys Rev Lett 101 (2008) 157004. * [10] D. Singh, M.-H. Du, Phys Rev Lett 100 (2008) 237003. * [11] L. Boeri, O. V. Dolgov, A. A. Golubov, Phys Rev Lett 101 (2008) 026403. * [12] T. Park, E. Park, H. Lee, T. Klimczuk, E. D. Bauer, F. Ronning, J. Thompson, J Phys-Cond Mat 20 (2008) 322204. * [13] D. Singh, arXiv:0901.2149. * [14] Q. Si, E. Abrahams, Phys Rev Lett 101 (2008) 076401. * [15] C. Fang, H. Yao, W.-F. Tsai, J. Hu, S. Kivelson, Phys Rev B 77 (2008) 224509. * [16] S. Chi, A. Schneidewind, J. Zhao, L. W. Harriger, Y. Luo, G. Cao, M. Loewenhaupt, P. Dai, arXiv:0812.1354. * [17] A. Christianson, E. Goremychkin, R. Osborn, S. Rosenkranz, M. D. Lumsden, C. Malliakas, lllya Todorov, H. Claus, D. Y. Chung, M. Kanatzidis, R. Bewley, T. Guidi, Nature 456 (2008) 930–932. * [18] M. D. Lumsden, A. Christianson, D. Parshall, M. B. Stone, S. E. Nagler, H. A. Mook, K. Lokshin, T. Egami, D. L. Abernathy, E. Goremychkin, R. Osborn, M. A. Mcguire, A. S. Sefat, R. Jin, B. C. Sales, D. Mandrus, arXiv:0811.4755. * [19] S. Hüfner, M. A. Hossain, A. Damascelli, G. Sawatzky, Rep Prog Phys 71 (2008) 2501. * [20] N. Metoki, Y. Haga, Y. Koike, Y. O nuki, Phys Rev Lett 80 (1998) 5417–5420. * [21] N. K. Sato, N. Aso, K. Miyake, R. Shiina, P. Thalmeier, G. Varelogiannis, C. Geibel, F. Steglich, P. Fulde, T. Komatsubara, Nature 410 (2001) 340–343. * [22] C. Stock, C. Broholm, J. Hudis, H. J. Kang, C. Petrovic, Phys Rev Lett 100 (2008) 087001. * [23] O. Stockert, J. Arndt, A. Schneidewind, H. Schneider, H. S. Jeevan, C. Geibel, F. Steglich, M. Loewenhaupt, Physica B 403 (2008) 973. * [24] J. Chang, I. Eremin, P. Thalmeier, P. Fulde, Phys Rev B 75 (2007) 24503. * [25] M. Norman, Phys Rev B 75 (2007) 184514. * [26] K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, H. Aoki, Phys Rev Lett 101 (2008) 087004. * [27] I. Mazin, D. Singh, M. D. Johannes, M. H. Du, Phys Rev Lett 101 (2008) 057003. * [28] P. B. Allen, R. C. Dynes, Phys. Rev. B 12 (1975) 905–922. * [29] A. Floris, G. Profeta, N. Lathiotakis, M. Luders, M. Marques, C. Franchini, E. Gross, A. Continenza, S. Massidda, Phys Rev Lett 94 (2005) 037004. * [30] T. Fukuda, A. Baron, S.-I. Shamoto, M. Ishikado, H. Nakamura, M. Machida, H. Uchiyama, S. Tsutsui, A. Iyo, H. Kito, J. Mizuki, M. Arai, H. Eisaki, H. Hosono, J Phys Soc Jpn 77 (2008) 103715. * [31] D. Reznik, K. Lokshin, D. C. Mitchell, D. Parshall, W. Dmowski, D. Lamago, R. Heid, K. P. Bohnen, A. S. Sefat, M. A. McGuire, B. C. Sales, D. G. Mandrus, A. Asubedi, D. Singh, A. Alatas, M. H. Upton, A. H. Said, Y. Shvyd’ko, T. Egami, arXiv:0810.4941. * [32] R. Osborn, E. Goremychkin, A. Kolesnikov, D. G. Hinks, Phys Rev Lett 87 (2001) 017005\. * [33] K. P. Bohnen, R. Heid, B. Renker, Phys Rev Lett 86 (2001) 5771. * [34] R. Mittal, Y. Su, S. Rols, T. Chatterji, S. L. Chaplot, H. Schober, M. Rotter, D. Johrendt, T. Brueckel, Phys Rev B 78 (2008) 104514. * [35] R. Mittal, Y. Su, S. Rols, M. Tegel, S. L. Chaplot, H. Schober, T. Chatterji, D. Johrendt, T. Brueckel, Phys Rev B 78 (2008) 224518. * [36] Y. Qiu, M. Kofu, W. Bao, S.-H. Lee, Q. Huang, T. Yildirim, J. R. D. Copley, J. Lynn, T. Wu, G. Wu, X. H. Chen, Phys Rev B 78 (2008) 052508. * [37] Q. Huang, Y. Qiu, W. Bao, M. A. Green, J. W. Lynn, Y. C. Gasparovic, T. Wu, G. Wu, X. H. Chen, Phys Rev Lett 101 (2008) 257003. * [38] T. Yildirim, Phys Rev Lett 101 (2008) 057010. * [39] I. I. Mazin, M. D. Johannes, K. Koepernik, D. Singh, Phys Rev B 78 (2008) 085104\. * [40] R. A. Ewings, T. G. Perring, R. I. Bewley, T. Guidi, M. J. Pitcher, D. R. Parker, S. J. Clarke, A. T. Boothroyd, Phys Rev B 78 (2008) 220501. * [41] D.-X. Yao, E. Carlson, arXiv:0804.4115. * [42] R. McQueeney, S. O. Diallo, V. P. Antropov, G. D. Samolyuk, C. Broholm, N. Ni, S. Nandi, M. Yethiraj, J. L. Zarestky, J. J. Pulikkotil, A. Kreyssig, M. D. Lumsden, B. N. Harmon, P. C. Canfield, A. I. Goldman, Phys Rev Lett 101 (2008) 227205. * [43] J. Zhao, D.-X. Yao, S. Li, T. Hong, Y. Chen, S. Chang, W. Ratcliff, J. Lynn, H. A. Mook, G. F. Chen, J. L. Luo, N. L. Wang, E. W. Carlson, J. Hu, P. Dai, Phys Rev Lett 101 (2008) 167203. * [44] S. O. Diallo, V. P. Antropov, C. Broholm, T. G. Perring, J. J. Pulikkotil, S. L. Bud’ko, P. C. Canfield, A. Kreyssig, A. I. Goldman, R. J. McQueeney, arXiv:0901.3784. * [45] K. Matan, R. Morinaga, K. Iida, T. J. Sato, arXiv:0810.4790. * [46] E. Fawcett, Rev Mod Phys 60 (1988) 209–283. * [47] H. Liu, W. Zhang, L. Zhao, X. Jia, J. Meng, G. Liu, X. Dong, G. F. Chen, J. L. Luo, N. L. Wang, W. Lu, G. Wang, Y. Zhou, Y. Zhu, X. Wang, Z. Xu, C. Chen, X. J. Zhou, Phys Rev B 78 (2008) 184514. * [48] M. Ishikado, R. Kajimoto, S. ichi Shamoto, M. Arai, A. Iyo, K. Miyazawa, P. M. Shirage, H. Kito, H. Eisaki, S. Kim, H. Hosono, T. Guidi, R. Bewley, S. Bennington, arXiv:0809.5128. * [49] J. Cooke, J. Lynn, H. Davis, Phys. Rev. B 21 (1980) 4118–4131. * [50] J. R. Schrieffer, Theory of Superconductivity (Benjamin, Reading, MA, 1964). * [51] H. F. Fong, B. Keimer, P. W. Anderson, F. Doğan, I. A. Aksay, Phys Rev Lett 75 (1995) 316–319. * [52] J. Rossat-Mignod, L. Regnault, C. Vettier, P. Bourges, P. Burlet, J. Bossy, J. Y. Henry, G. Lapertot, Physica C 185 (1991) 86. * [53] H. A. Mook, M. Yethiraj, G. Aeppli, T. E. Mason, T. Armstrong, Phys Rev Lett 70 (1993) 3490–3493. * [54] H. F. Fong, P. Bourges, Y. Sidis, L. Regnault, A. Ivanov, G. D. Gu, N. Koshizuka, B. Keimer, Nature 398 (1999) 588–591. * [55] P. Dai, H. A. Mook, G. Aeppli, S. Hayden, F. Doğan, Nature 406 (2000) 965. * [56] H. He, P. Bourges, Y. Sidis, C. Ulrich, L. Regnault, S. Pailhès, N. S. Berzigiarova, N. N. Kolesnikov, B. Keimer, Science 295 (2002) 1045–1047. * [57] H. Ding, P. Richard, K. Nakayama, K. Sugawara, T. Arakane, Y. Sekiba, A. Takayama, S. Souma, T. J. Sato, T. Takahashi, Z. Wang, X. Dai, Z. Fang, G. F. Chen, J. L. Luo, N. Wang, Europhys Lett 83 (2008) 47001. * [58] M. M. Korshunov, I. Eremin, Phys Rev B 78 (2008) 140509. * [59] T. A. Maier, D. Scalapino, Phys Rev B 78 (2008) 020514. * [60] D. Inotani, Y. Ohashi, arXiv:0901.1718.	arxiv-papers	2009-02-21T22:16:11	2024-09-04T02:49:00.791959	{ "license": "Public Domain", "authors": "R. Osborn, S. Rosenkranz, E. A. Goremychkin, A. D. Christianson", "submitter": "Ray Osborn", "url": "https://arxiv.org/abs/0902.3760" }
0902.3780	2010561-572Nancy, France 561 Dániel Marx Barry O’Sullivan Igor Razgon # Treewidth reduction for constrained separation and bipartization problems D.Marx Tel Aviv University dmarx@cs.bme.hu , B.O’Sullivan and I.Razgon Cork Constraint Computation Centre, University College Cork b.osullivan,i.razgon@cs.ucc.ie ###### Abstract. We present a method for reducing the treewidth of a graph while preserving all the minimal $s-t$ separators. This technique turns out to be very useful for establishing the fixed-parameter tractability of constrained separation and bipartization problems. To demonstrate the power of this technique, we prove the fixed-parameter tractability of a number of well-known separation and bipartization problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set). These results answer a number of open questions in the area of parameterized complexity. ###### Key words and phrases: fixed-parameter algorithms, graph separation problems, treewidth ###### 1991 Mathematics Subject Classification: G.2.2. Graph Theory, Subject: Graph Algorithms ## 1\. Introduction Finding cuts and separators is a classical topic of combinatorial optimization and in recent years there has been an increase in interest in the fixed- parameter tractability of such problems [MarxTCS, 1132573, DBLP:conf/iwpec/Guillemot08a, DBLP:conf/csr/Xiao08, DBLP:journals/eor/GuoHKNU08, MR2330167, DBLP:conf/wads/ChenLL07, marxrazgon- esa2009]. Recall that a problem is fixed-parameter tractable (or FPT) with respect to a parameter $k$ if it can be solved in time $f(k)\cdot n^{O(1)}$ for some function $f(k)$ depending only on $k$ [MR2001b:68042, MR2238686, MR2223196]. In typical parameterized separation problems, the parameter $k$ is the size of the separator we are looking for, thus fixed-parameter tractability with respect to this parameter means that the combinatorial explosion is restricted to the size of the separator, but otherwise the running time depends polynomially on the size of the graph. The main technical contribution of the present paper is a theorem stating that given a graph $G$, two terminal vertices $s$ and $t$, and a parameter $k$, we can compute in a fpt-time a graph $G^{}$ having its treewidth bounded by a function of $k$ while (roughly speaking) preserving all the minimal $s-t$ separators of size at most $k$. Combining this theorem with the well-known Courcelle’s Theorem, we obtain a powerful tool for proving the fixed parameter tractability of constrained separation and bipartization problems. We demonstrate the power of the methodology with the following results. • We prove that the minimum stable $s-t$ cut problem (Is there an independent set $S$ of size at most $k$ whose removal separates $s$ and $t$?) is fixed- parameter tractable. This problem received some attention in the community. Our techniques allow us to prove various generalizations of this result very easily. First, instead of requiring that $S$ is independent, we can require that it induces a graph that belongs to a hereditary class $\mathcal{G}$; the problem remains fpt. Second, in the multicut problem a list of pairs of terminals are given $(s_{1},t_{1})$, $\dots$, $(s_{\ell},t_{\ell})$ and the solution $S$ has to be a set of at most $k$ vertices that induces a graph from $\mathcal{G}$ and separates $s_{i}$ from $t_{i}$ for every $i$. We show that this problem is fpt parameterized by $k$ and $\ell$, which is a very strong generalization of previous results [MarxTCS, DBLP:conf/csr/Xiao08]. Third, the results generalize to the multicut-uncut problem, where two sets $T_{1}$, $T_{2}$ of pairs of terminals are given, and $S$ has to separate every pair of $T_{1}$ and should not separate any pair of $T_{2}$. * • We prove that the exact stable bipartization problem (Is there an independent set of size _exactly_ $k$ whose removal makes the graph bipartite?) is fixed- parameter tractable (fpt) answering an open question posed in 2001 by Díaz et al. [MR1907021]. We establish this result by proving that the stable bipartization problem (Is there an independent set of size _at most_ $k$ whose removal makes the graph bipartite?) is fpt, answering an open question posed by Fernau [demaine_et_al:DSP:2007:1254]. * • We show that the edge-induced vertex cut (Are there at most $k$ edges such that the removal of their endpoints separates two given terminals $s$ and $t$?) is fpt, answering an open problem posed in 2007 by Samer [demaine_et_al:DSP:2007:1254]. The motivation behind this problem is described in [DBLP:journals/corr/abs-cs-0607109]. We believe that the above results nicely demonstrate the message of the paper. Slightly changing the definition of a well-understood cut problem usually makes the problem NP-hard and determining the parameterized complexity of such variants directly is by no means obvious. On the other hand, using our techniques, the fixed-parameter tractability of many such problems can be shown with very little effort. Let us mention (without proofs) three more variants that can be treated in a similar way: (1) separate $s$ and $t$ by the deletion of at most $k$ edges and at most $k$ vertices, (2) in a 2-colored graph, separate $s$ and $t$ by the deletion of at most $k$ black and at most $k$ white vertices, (3) in a $k$-colored graph, separate $s$ and $t$ by the deletion of one vertex from each color class. As the examples above show, our method leads to the solution of several independent problems; it seems that the same combinatorial difficulty lies at the heart of these problems. Our technique manages to overcome this difficulty and it is expected to be of use for further problems of similar flavor. Note that while designing fpt-time algorithms for bounded-treewidth graphs and in particular the use of Courcelle’s Theorem is a fairly standard technique, we use this technique for problems where there is no bound on the treewidth of the graph appearing in the input. (Multiterminal) cut problems [MarxTCS, DBLP:journals/eor/GuoHKNU08, MR2330167, DBLP:conf/wads/ChenLL07] play a mysterious, and not yet fully understood, role in the fixed-parameter tractability of certain problems. Proving that bipartization [ReedSmithVetta-OddCycle], directed feedback vertex set [DBLP:journals/jacm/ChenLLOR08], and almost 2-sat [ROicalp] are fpt answered longstanding open questions, and in each case the algorithm relies on a non- obvious use of separators. Furthermore, edge multicut has been observed to be equivalent to fuzzy cluster editing, a correlation clustering problem [DBLP:conf/mfcs/BodlaenderFHMPR08, DBLP:journals/tcs/DemaineEFI06, DBLP:journals/ml/BansalBC04]. Thus aiming for a better understanding of separators in a parameterized setting seems to be a fruitful direction of research. Our results extend our understanding of separators by showing that various additional constraints can be accommodated. It is important to point out that our algorithm is very different from previous parameterized algorithms for separation problems [MarxTCS, DBLP:journals/eor/GuoHKNU08, MR2330167, DBLP:conf/wads/ChenLL07]. Those algorithms in the literature exploit certain nice properties of separators, and hence it seems impossible to generalize them for the problems we consider here. On the other hand, our approach is very robust and, as demonstrated by our examples, it is able to handle many variants. The paper assumes the knowledge of the definition of treewidth and its algorithmic use, including Courcelle’s Theorem (see the surveys [DBLP:conf/wg/Bodlaender06, GroheLGA]). ## 2\. Treewidth Reduction The main combinatorial result of the paper is presented in this section. We start with some preliminary definitions. Two slightly different notions of separation will be used in the paper: ###### Definition 2.1. We say that a set $S$ of vertices separates sets of vertices $A$ and $B$ if no component of $G\setminus S$ contains vertices from both $A\setminus S$ and $B\setminus S$. If $s$ and $t$ are two distinct vertices of $G$, then an $s-t$ separator is a set $S$ of vertices disjoint from $\\{s,t\\}$ such that $s$ and $t$ are in different components of $G\setminus S$. In particular, if $S$ separates $A$ and $B$, then $A\cap B\subseteq S$. Furthermore, given a set $W$ of vertices, we say that a set $S$ of vertices is a balanced separator of $W$ if $\|W\cap C\|\leq\|W\|/2$ for every connected component $C$ of $G\setminus S$. A $k$-separator is a separator $S$ with $\|S\|=k$. The treewidth of a graph is closely connected with the existence of balanced separators: ###### Lemma 2.2 ([ree97], [MR2238686, Section 11.2]). 1. (1) If $G(V,E)$ has treewidth greater than $3k$, then there is a set $W\subseteq V$ of size $2k+1$ having no balanced $k$-separator. 2. (2) If $G(V,E)$ has treewidth at most $k$, then every $W\subseteq V$ has a balanced $(k+1)$-separator. Note that the contrapositive of (1) in Lemma 2.2 says that if every set $W$ of vertices has a balanced $k$-separator, then the treewidth is at most $3k$. This observation, and the following simple extension, will be convenient tools for showing that a certain graph has low treewidth. ###### Lemma 2.3. Let $G$ be a graph, $C_{1}$,$\dots$, $C_{r}$ subsets of vertices, and let $C:=\bigcup_{i=1}^{r}C_{i}$. Suppose that every $W_{i}\subseteq C_{i}$ has a balanced separator $S_{i}\subseteq C_{i}$ of size at most $w$. Then every $W\subseteq C$ has a balanced separator $S\subseteq C$ of size $wr$. If we are interested in separators of a graph $G$ contained in a subset $C$ of vertices, then each component of $G\setminus C$ (or the neighborhood of each component in $C$) can be replaced by a clique, since there is no way to disconnect these components with separators in $C$. The notion of torso and Proposition 2.5 formalize this concept. ###### Definition 2.4. Let $G$ be a graph and $C\subseteq V(G)$. The graph $\textup{torso}(G,C)$ has vertex set $C$ and vertices $a,b\in C$ are connected by an edge if $\\{a,b\\}\in E(G)$ or there is a path $P$ in $G$ connecting $a$ and $b$ whose internal vertices are not in $C$. ###### Proposition 2.5. Let $C_{1}\subseteq C_{2}$ be two subsets of vertices in $G$ and let $a,b\in C_{1}$ be two vertices. A set $S\subseteq C_{1}$ separates $a$ and $b$ in $\textup{torso}(G,C_{1})$ if and only if $S$ separates these vertices in $\textup{torso}(G,C_{2})$. In particular, by setting $C_{2}=V(G)$, we get that $S\subseteq C_{1}$ separates $a$ and $b$ in $\textup{torso}(G,C_{1})$ if and only if it separates them in $G$. Analogously to Lemma 2.3, we can show that if we have a treewidth bound on $\textup{torso}(G,C_{i})$ for every $i$, then these bounds add up for the union of the $C_{i}$’s. ###### Lemma 2.6. Let $G$ be a graph and $C_{1}$,$\dots$, $C_{r}$ be subsets of $V(G)$ such that for every $1\leq i\leq r$, the treewidth of $\textup{torso}(G,C_{i})$ is at most $w$. Then the treewidth of $\textup{torso}(G,C)$ for $C:=\bigcup_{i=1}^{r}C_{i}$ is at most $3r(w+1)$. If the minimum size of an $s-t$ separator is $\ell$, then the excess of an $s-t$ separator $S$ is $\|S\|-\ell$ (which is always nonnegative). Note that if $s$ and $t$ are adjacent, then no $s-t$ separator exists, and in this case we say that the minimum size of an $s-t$ separator is $\infty$. The aim of this section is to show that, for every $k$, we can construct a set $C^{\prime}$ covering all the $s-t$ separators of size at most $k$ such that $\textup{torso}(G,C^{\prime})$ has treewidth bounded by a function of $k$. Equivalently, we can require that $C^{\prime}$ covers every $s-t$ separator of excess at most $e:=k-\ell$, where $\ell$ is the minimum size of an $s-t$ separator. If $X$ is a set of vertices, we denote by $\delta(X)$ the set of those vertices in $V(G)\setminus X$ that are adjacent to at least one vertex of $X$. The following result is folklore; it can be proved by a simple application of the uncrossing technique (see the proof below) and it can be deduced also from the observations of [MR592081] on the strongly connected components of the residual graph after solving a flow problem. ###### Lemma 2.7. Let $s,t$ be two vertices in graph $G$ such that the minimum size of an $s-t$ separator is $\ell$. Then there is a collection $\mathcal{X}=\\{X_{1},\dots,X_{q}\\}$ of sets where $\\{s\\}\subseteq X_{i}\subseteq V(G)\setminus(\\{t\\}\cup\delta(\\{t\\}))$ ($1\leq i\leq q$), such that 1. (1) $X_{1}\subset X_{2}\subset\dots\subset X_{q}$, 2. (2) $\|\delta(X_{i})\|=\ell$ for every $1\leq i\leq q$, and 3. (3) every $s-t$ separator of size $\ell$ is a subset of $\bigcup_{i=1}^{q}\delta(X_{i})$. Furthermore, such a collection $\mathcal{X}$ can be found in polynomial time. ###### Proof 2.8. Let $\mathcal{X}=\\{X_{1},\dots,X_{q}\\}$ be a collection of sets such that (2) and (3) holds. Let us choose the collection such that $q$ is the minimum possible, and among such collections, $\sum_{i=1}^{q}\|X_{i}\|^{2}$ is the maximum possible. We show that for every $i,j$, either $X_{i}\subset X_{j}$ or $X_{j}\subset X_{i}$ holds, thus the sets can be ordered such that (1) holds. Suppose that neither $X_{i}\subset X_{j}$ nor $X_{j}\subset X_{i}$ holds for some $i$ and $j$. We show that after replacing $X_{i}$ and $X_{j}$ in $\mathcal{X}$ with the two sets $X_{i}\cap X_{j}$ and $X_{i}\cup X_{j}$, properties (2) and (3) still hold, and the resulting collection $\mathcal{X}^{\prime}$ contradicts the optimal choice of $\mathcal{X}$. The function $\delta$ is well-known to be submodular, i.e., $\|\delta(X_{i})\|+\|\delta(X_{j})\|\geq\|\delta(X_{i}\cap X_{j})\|+\|\delta(X_{i}\cup X_{j})\|.$ Both $\delta(X_{i}\cap X_{j})$ and $\delta(X_{i}\cup X_{j})$ are $s-t$ separators (because both $X_{i}\cap X_{j}$ and $X_{i}\cup X_{j}$ contain $s$) and hence have size at least $k$. The left hand side is $2\ell$, hence there is equality and $\|\delta(X_{i}\cap X_{j})\|=\|\delta(X_{i}\cup X_{j})\|=\ell$ follows. This means that property (2) holds after the replacement. Observe that $\delta(X_{i}\cap X_{j})\cup\delta(X_{i}\cup X_{j})\subseteq\delta(X_{i})\cup\delta(X_{j})$: any edge that leaves $X_{i}\cap X_{j}$ or $X_{i}\cup X_{j}$ leaves either $X_{i}$ or $X_{j}$. We show that there is equality here, implying that property (3) remains true after the replacement. It is easy to see that $\delta(X_{i}\cap X_{j})\cap\delta(X_{i}\cup X_{j})\subseteq\delta(X_{i})\cap\delta(X_{j})$, hence we have $\|\delta(X_{i}\cap X_{j})\cup\delta(X_{i}\cup X_{j})\|=2\ell-\|\delta(X_{i}\cap X_{j})\cap\delta(X_{i}\cup X_{j})\|\geq 2\ell-\|\delta(X_{i})\cap\delta(X_{j})\|=\|\delta(X_{i})\cup\delta(X_{j})\|,$ showing the required equality. If $X_{i}\cap X_{j}$ or $X_{i}\cup X_{j}$ was already present in $\mathcal{X}$, then the replacement decreases the size of the collection, contradicting the choice of $\mathcal{X}$. Otherwise, we have that $\|X_{i}\|^{2}+\|X_{j}\|^{2}<\|X_{i}\cap X_{j}\|^{2}+\|X_{i}\cup X_{j}\|^{2}$ (to verify this, simply represent $\|X_{i}\|$ as $\|X_{i}\cap X_{j}\|+\|X_{i}\setminus X_{j}\|$, $\|X_{j}\|$ as $\|X_{i}\cap X_{j}\|+\|X_{j}\setminus X_{i}\|$, $\|X_{i}\cup X_{j}\|$ as $\|X_{i}\cap X_{j}\|+\|X_{i}\setminus X_{j}\|+\|X_{j}\setminus X_{i}\|$ and do direct calculation having in mind that both $\|X_{i}\setminus X_{j}\|$ and $\|X_{j}\setminus X_{i}\|$ are greater than $0$), again contradicting the choice of $\mathcal{X}$. Thus an optimal collection $\mathcal{X}$ satisfies (1) as well. To construct $\mathcal{X}$ in polynomial time, we proceed as follows. It is easy to check in polynomial time whether a vertex $v$ is in a minimum $s-t$ separator, and if so to produce such a separator $S_{v}$. Let $X_{v}$ be the set of vertices reachable from $s$ in $G\setminus S_{v}$. It is clear that $X_{v}$ satisfies (2) and if we take the collection $\mathcal{X}$ of all such $X_{v}$’s, then together they satisfy (3). If (1) is not satisfied, then we start doing the replacements as above. Each replacement either decreases the size of the collection or increases $\sum_{i=1}^{t}\|X_{i}\|^{2}$ (without increasing the collection size), thus the procedure terminates after a polynomial number of steps. ∎ Lemma 2.7 shows that the union $C$ of all minimum $s-t$ separators can be covered by a chain of minimum $s-t$ separators. It is not difficult to see that this chain can be used to define a tree decomposition (in fact, a path decomposition) of $\textup{torso}(G,C)$. This observation solves the problem for $e=0$. For the general case, we use induction on $e$. ###### Lemma 2.9. Let $s,t$ be two vertices of graph $G$ and let $\ell$ be the minimum size of an $s-t$ separator. For some $e\geq 0$, let $C$ be the union of all minimal $s-t$ separators having _excess_ at most $e$ (i.e. of size at most $k=\ell+e$). Then, for some constant $d$, there is an $O(f(\ell,e)\cdot\|V(G)\|^{d})$ time algorithm that returns a set $C^{\prime}\supseteq C\cup\\{s,t\\}$ such that the treewidth of $\textup{torso}(G,C^{\prime})$ is at most $g(\ell,e)$, where functions $f$ and $g$ depend only on $\ell$ and $e$ .	arxiv-papers	2009-02-22T09:02:26	2024-09-04T02:49:00.797780	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "D\\'aniel Marx (1), Barry O'Sullivan (2), Igor Razgon (2) ((1) Budapest\n University of Technology and Economics, (2) Cork Constraint Computation\n Centre, University College Cork)", "submitter": "D\\'aniel Marx", "url": "https://arxiv.org/abs/0902.3780" }
0902.3781	# Measuring of fissile isotopes partial antineutrino spectra in direct experiment at nuclear reactor V.V. Sinev Institute for Nuclear Research RAS, Moscow ###### Abstract The direct measuring method is considered to get nuclear reactor antineutrino spectrum. We suppose to isolate partial spectra of the fissile isotopes by using the method of antineutrino spectrum extraction from the inverse beta decay positron spectrum applied at Rovno experiment. This admits to increase the accuracy of partial antineutrino spectra forming the total nuclear reactor spectrum. It is important for the analysis of the reactor core fuel composition and could be applied for non-proliferation purposes. ## Introduction Energy spectrum of antineutrinos from nuclear reactor is a fundamental characteristic of a reactor. When outgoing from a reactor antineutrinos penetrate through any shielding. These particles carry out the information concerning the chain reaction in the reactor core. Their spectrum is unique for every reactor type and depends on the reactor fuel composition. That is why just when the difference in energy spectra of the fuel components became clear there appeared an idea of reactor control through the neutrino emission [1]. There are mainly four fissile isotopes which undergo the fission in the core, 235U, 239Pu, 238U and 241Pu. Others give an input less than 1% and may be neglected. Detector at some distance can detect the total flux from all these components. But during the reactor operational run the shape of total spectrum changes because of the burn up effect. So, one can know the fuel composition of the reactor by fitting the total spectrum with the sum of four partial spectra. But what are the uncertainties of these partial spectra. At the beginning of the era of experiments with reactor neutrinos the partial spectra of individual isotopes were calculated [2, 3], but the accuracy of the calculations was not so good. Mainly because of insufficient knowledge of fission fragments antineutrino spectra. Later the situation becomes better, while the base of fragments was growing [4, 5, 6]. It becomes much better when the first experimental spectra appeared. They were obtained by converting measured beta-spectra from fissile isotopes [7, 8, 9]. Electron spectra were measured at Grenoble in ILL by using magnetic spectrometer. These spectra are accounted as the best. Their uncertainty in main spectrum part (2$-$7 MeV) is 3.8$-$4.2% at 90% CL. In spite of high enough accuracy this is not sufficient, reactor control demands at least 1control problem this accuracy needs to be improved. Just now several International experiments are under preparation. They are Double Chooz [10] in France, Daya Bay [11] in China and RENO [12] in Korea. In all of them the detectors of a new generation will be used. These detectors can give a possibility of obtaining high statistics while measuring antineutrino spectrum. And the high statistics admits to isolate individual fissile isotopes spectra and compare them with measured by conversion technique. The appearing estimated uncertainty could be close to the uncertainty of ILL spectra. We know about some projects with a goal to measure spectra ratio for beta particles of 235U and 239Pu [13], which can also help to understand better spectra behavior. In the article we consider the method of isotopes spectra isolation by using the direct measurement of positron spectrum from inverse beta decay reaction. As a result we hope to obtain 235U and 239Pu antineutrino spectra, which produce about 90% of total antineutrino flux of nuclear reactor. ## I Antineutrino registration Antineutrino can be registered through the inverse beta decay reaction on proton which has the largest cross section $\bar{\nu_{e}}+p\rightarrow n+e^{+}.$ (1) The positron appeared as a result of the reaction carries out practically all antineutrino energy [14, 15]. Its kinetic energy is linearly connected with antineutrino energy $T=E-\Delta-r_{n},$ (2) where $T$ \- positron kinetic energy, $E$ \- antineutrino energy, $\Delta$ \- the reaction threshold equals to 1.806 MeV and $r_{n}$ is neutron recoil energy. So, the positron spectrum is the same as antineutrino’s, but shifted on 1.8 MeV and convoluted with cross section. Recoil energy in the first approximation can be neglected. ## II Antineutrino spectrum Antineutrino spectrum is being formed inside the reactor core from a number of beta-decays of fission fragments. Fragments are produced by several fissile isotopes, not only 235U as it was considered at the beginning of the period of searching neutrino. We know that the most part of fissions is produced by four isotopes, they are 235U, 239Pu, 241Pu and 238U. Figure 1: Antineutrino spectra 235U (1), 239Pu (2), 241Pu (3) from [7-9] and 238U (4) from [18]. Each isotope antineutrino spectrum can be calculated using data base for fissile fragments $\rho_{\nu}(E)=\sum_{i}{y_{i}A_{i}(E)},$ (3) where $y_{i}$ \- yield of $i$-th fragment in fission and $A(E)$ is its antineutrino spectrum. But the accuracy of evaluation the spectrum (3) is limited by the number of nuclei with unknown decay schemes (about 25% of total number). They have half-life periods less then 0.3 seconds. One can take antineutrino spectrum also by another method, by converting experimentally measured beta-spectrum from fissile isotope. This method is accounted for the present moment as the most exact. It was used in experiments in ILL (Grenoble). In [7, 8, 9] the beta-spectrum was measured with high level of accuracy for three isotopes 235U, 239Pu and 241Pu , which undergo fission through absorption of thermal neutrons. Fission of 238U goes on only by capturing fast neutrons and has a small fission cross section. Its high enough rate of fissions is explained by the great mass of this isotope in the content of nuclear fuel. They fit the measured beta spectra by a set of 30 hypothetical beta-spectra with boarding energies uniformly distributed from 0 up to high energy of experimental spectrum. The coefficients found (similar to $y_{i}$ in (3)) were used in formula (3) for calculating the total antineutrino spectrum. These antineutrino spectra are accounted as a standard for the moment for data analysis in experiments with reactor antineutrinos. They are presented at figure 1. There is also the third method of taking antineutrino spectrum, it is a method of direct measuring. When one measures spectrum of positrons from reaction (1) and extracts antineutrino spectrum. This method was realized in experiments of Rovno group [16, 17]. The authors have obtained the antineutrino spectrum as an exponential function with polynomial of 10-th power while solving the equation $S_{e}(T)=\int{\rho_{\nu}(E){\sigma}_{{\nu}p}(E)R(E,T)dE},$ (4) where $S_{e}(T)$ $-$ positron spectrum from reaction (1), $\rho_{\nu}(E)$ $-$ antineutrino spectrum, $\sigma_{{\nu}p}(E)$ $-$ inverse beta decay reaction cross section and $R(E,T)$ $-$ response function of detector. As a result we have a formula for antineutrino spectrum $\rho_{\nu}(E)=5.09\cdot exp(-0.648E-0.0273E^{2}-1.411(E/8)^{10}),$ (5) This spectrum corresponds to some reactor fuel compositions like the following one ${}^{235}{\rm U}-0.586,^{239}{\rm Pu}-0.292,^{238}{\rm U}-0.075,^{241}{\rm Pu}-0.047.$ (6) Response function was simulated by Monte Carlo method. This function transforms positron energy spectrum appeared in (1) in experimentally observed one. Each detector has its individual response function depending on detector features. At figure 2 one can see response functions for some values of positron kinetic energy for detector RONS used at Rovno experiment. The control for simulated function was done by comparing the spectra measured and calculated from some gamma sources (60Co and 24Na) which were placed in the center and the periphery of the detector. Calibration of the detector was made by the beta-source 144Ce$-^{144}$Pr with boarding energy 2997 keV. ## III The extraction of 235U and 239Pu antineutrino spectra from measured positron spectrum During the reactor operational run antineutrino spectrum changes its shape. At the beginning of run the spectrum is formed mainly by 235U ($\sim$60-70%) and at the end of run the largest or equal with 235U part of fissions comes from 239Pu ($\sim$40-50%). This is a foundation for proposed method of extraction the partial spectra. Let’s regard the positron spectrum (1), which could be measured in a detector placed in the vicinity of some nuclear reactor. At figure 3 one can see positron spectra produced by pure isotopes of uranium and plutonium and real spectrum corresponding to (6), which one can observe during reactor operational run. This real spectrum takes place between spectra of 235U and 241Pu. But at the beginning of the run it will be closer to 235U and at the end closer to 239Pu.. Figure 2: Simulated response function for detector RONS, which was used in Rovno experiments. So, one can divide all the data measured during the reactor run into two parts - “beginning” and “ending”. And we can use them for extracting individual spectra. Spectrum “beginning” contains 235U in larger proportion. Let’s account the input of other fissile isotopes as a background and remove it. The main question will concern the value of uncertainty of the spectrum. We can suppose that the total uncertainty may be about 1% including 0.1-0.2% statistics, as they account to achieve in modern experiments like Double Chooz. Expected statistics may be about 106 events. In this case the main error will come from the background where the greatest part will appear from the spectra of other fissile isotopes. We can write the total uncertainty of three background spectra like $\sigma_{b}=\sum_{i=8,9,1}{\alpha_{i}\sigma_{i}}.$ (7) While experimental spectrum is known with high accuracy ${\sigma}_{e}\sim$1%, for extracted 235U spectrum we will get the error $\sigma_{5}=\frac{1}{\alpha_{5}}\sqrt{\sigma_{e}^{2}+\sigma_{b}^{2}}.$ (8) In (7) and (8) the letter $\alpha_{i}$ is assigned for individual parts of fission. One can find values of $\alpha_{i}$, which we have used for the estimations of uncertainties, in table 1. Table 1: Parts of isotope fission at the beginning and end of reactor operational run Isotope \| 235U \| 239Pu \| 238U \| 241Pu ---\|---\|---\|---\|--- “beginning” \| 0.65 \| 0.25 \| 0.07 \| 0.03 “ending” \| 0.35 \| 0.50 \| 0.08 \| 0.04 It is important to note that the experimental error is not a constant for a whole spectrum, it varies from bin to bin. At figure 4 one can see the standard behavior of experimental uncertainty for restored spectrum. At minimum it is equal to 1% as we supposed to get. In table 2 we show achievable values of uncertainties for235U. One can compare these errors with uncertainties for 235U spectrum from ILL shown in the second column. To obtain the standard spectrum we can have an average of the both spectra and this is shown in the fourth column. Table 2: Expected relative uncertainty for 235U antineutrino spectrum (fission part 65%, see table 1) $E$m MeV \| ${\delta}S_{5}$ (ILL 68% CL) \| ${\delta}S_{5}(experim)$ (68% CL) \| ${\delta}S_{5}(average)$ (68% CL) ---\|---\|---\|--- 2.0 \| 0.026 \| 0.923 \| 0.026 2.5 \| 0.024 \| 0.155 \| 0.024 3.0 \| 0.023 \| 0.035 \| 0.019 3.5 \| 0.021 \| 0.025 \| 0.016 4.0 \| 0.020 \| 0.027 \| 0.016 4.5 \| 0.020 \| 0.028 \| 0.017 5.0 \| 0.024 \| 0.028 \| 0.018 5.5 \| 0.026 \| 0.029 \| 0.019 6.0 \| 0.030 \| 0.029 \| 0.021 6.5 \| 0.035 \| 0.031 \| 0.023 7.0 \| 0.040 \| 0.055 \| 0.032 7.5 \| 0.047 \| 0.109 \| 0.040 8.0 \| 0.061 \| 0.138 \| 0.056 8.5 \| 0.134 \| 0.276 \| 0.120 9.0 \| 0.486 \| 0.843 \| 0.421 9.5 \| 0.608 \| 1.610 \| 0.569 Similar table 3 is constructed for 239Pu, where we used uncertainty from the last column of table 2, while applying the same procedure for this isotope for experimental spectrum marked “ending”. Table 3: Expected relative uncertainty for 239Pu antineutrino spectrum (fission part 50%, see table 1) $E$m MeV \| ${\delta}S_{9}$ (ILL 68% CL) \| ${\delta}S_{9}(experim)$ (68% CL) \| ${\delta}S_{9}(average)$ (68% CL) ---\|---\|---\|--- 2.0 \| 0.027 \| 1.200 \| 0.027 2.5 \| 0.026 \| 0.204 \| 0.026 3.0 \| 0.026 \| 0.055 \| 0.024 3.5 \| 0.026 \| 0.048 \| 0.023 4.0 \| 0.027 \| 0.057 \| 0.024 4.5 \| 0.029 \| 0.064 \| 0.026 5.0 \| 0.032 \| 0.074 \| 0.029 5.5 \| 0.036 \| 0.074 \| 0.032 6.0 \| 0.041 \| 0.090 \| 0.038 6.5 \| 0.045 \| 0.100 \| 0.041 7.0 \| 0.067 \| 0.183 \| 0.063 7.5 \| 0.116 \| 0.229 \| 0.103 8.0 \| 0.213 \| 0.438 \| 0.191 8.5 \| 0.486 \| 0.266 \| 0.234 9.0 \| 0.578 \| 1.320 \| 0.529 9.5 \| 0.608 \| 2.340 \| 0.588 Figure 3: Positron spectra: 1 - 238U, 2 - 235U, 3 - 241Pu, 4 - 239Pu. Dashed line shows spectrum corresponding to the fuel composition (6). The last table 4 demonstrates what the values can be achieved while applying the procedure to 238U. This spectrum we can get to know only from calculations. There may be a good chance to take it experimentally. Table 4: Expected relative uncertainty for 238U antineutrino spectrum $E$m MeV \| ${\delta}S_{8}$ (Vogel 68% CL) \| ${\delta}S_{8}(experim)$ (68% CL) \| ${\delta}S_{8}(average)$ (68% CL) ---\|---\|---\|--- 2.0 \| 0.05 \| $-$ \| 0.05 2.5 \| 0.06 \| 1.270 \| 0.06 3.0 \| 0.06 \| 0.301 \| 0.06 3.5 \| 0.08 \| 0.192 \| 0.074 4.0 \| 0.10 \| 0.188 \| 0.088 4.5 \| 0.10 \| 0.187 \| 0.088 5.0 \| 0.10 \| 0.186 \| 0.088 5.5 \| 0.10 \| 0.195 \| 0.089 6.0 \| 0.10 \| 0.197 \| 0.089 6.5 \| 0.10 \| 0.195 \| 0.089 7.0 \| 0.20 \| 0.294 \| 0.165 7.5 \| 0.20 \| 0.508 \| 0.186 8.0 \| 0.30 \| 0.778 \| 0.280 8.5 \| 0.40 \| 0.914 \| 0.366 9.0 \| 0.70 \| $-$ \| 0.696 9.5 \| 1.00 \| $-$ \| 0.997 Figure 4: Experimental uncertainty of antineutrino spectrum after transforming from positron spectrum. Suppose that systematic and statistical errors in total do not exceed 1% in the most part of spectrum. ## IV Conclusion The discussed method of obtaining 235U and 239Pu antineutrino spectra can improve uncertainties in their spectra. It is important for neutrino control method. The using of different methods of spectra obtaining increases the reliability of standard spectra. The accuracy in 3% in not enough today for measuring the fuel composition by neutrino method, 1% seems desirable. In that case statistics 2000$-$3000 events/day may be enough to have uncertainty 5% for 235U part of fission per month of measurement. From one hand this method is additional to other methods of spectra obtaining, from the other hand it is direct and is not affected by some procedures of recalculating like converting method. Also it does not depend on knowledge of decay schemes as calculating method because it accounts all decays automatically. In collaboration with ILL spectra there may be achieved better accuracy for standard spectra. Also there may be directly determined the 238U spectrum in spite of the high uncertainty and compared to the calculated one. Of cause the direct method is additional to other methods. These calculations demonstrate the importance of continuation of the experiments on measuring beta-spectra of the fissile isotopes and improving the conversation procedure because it is the most exact. Direct measurement may serve as an indirect test for conversion spectra. Calculations are very important for searching time evolution of nuclear reactor antineutrino spectrum. All three methods improve the complete knowledge of reactor antineutrino spectrum and its behavior during the operational run. Altogether, the usage of several methods increases reliability of partial spectra while proposing the standard spectra for the future analysis. As an example we can attract the attention to hard part of ILL spectra (higher than 8 MeV) where all spectra become the same, while calculated spectra demonstrate growing difference in this region. Proposed method on base of Rovno experience shows its applicability in the analysis of the data in future Double Chooz experiment. ## Acknowledgments Author thanks L.A. Mikaelyan and A.Ya. Balysh for useful discussions and friendly criticism. ## References * (1) L.A. Mikaelyan, in Proceedings of International Conference Neutrino-77 (NAUKA, Moscow), v2, p.383 (1978). * (2) F. Reines, C.L. Cowan, Phys. Rev., 113, 273 (1959). * (3) F. Avignone et al, Phys. Rev. D2, 2609 (1970). * (4) Klapdor H.V., Metsinger J., Phys. Rev. Lett., 48, 127 (1982). * (5) Davis R., Vogel P., Mann F.M., Schenter R.E., Phys. Rev. C 19, 2259 (1979). * (6) P.M. Rubtsov., P.A. Ruzhansky, V.G. Aleksankin, S.V. Rodichev, Physics of Atomic Nuclei, 46, issue10, 1028 (1987). * (7) F. v. Feilitzsch, A.A. Hahn and K. Schreckenbach, Phys. Lett. B118, 162 (1982). * (8) K. Schreckenbach, G. Colvin, W. Gelletly and F. v. Feilitzsch, Phys. Lett. B160, 325 (1985). * (9) A.A. Hahn and K. Schreckenbach, Phys. Lett. B218, 365 (1989). * (10) F. Ardellier, I. Barabanov, J.C. Barriere et al. (Double Chooz Collaboration), arXiv:hep-ex/0606025v4 (2006). * (11) Daya Bay Collaboration proposal, arXiv:hep-ex/0701029v1 (2007). * (12) S.-B. Kim (RENO Collaboration), in Proceedings of International Conference TAUP-2007 (2007). * (13) V.I. Kopeikin, A.E. Makeenkov, L.A. Mikaelyan et al., preprint IAE6520/2 (2008). * (14) P. Vogel, Phys.Rev. D29, 1918 (1984). * (15) A. Strumia and F. Vissani, Phys. Lett. B564, 42 (2003); astro-ph/0302055 (2003). * (16) A.I. Afonin, S.N. Ketov, V.I. Kopeikin et al., JETP, 94, 1 (1988). * (17) Yu.V. Klimov, V.I. Kopeikin, A.A. Labzov et al., Physics of Atomic Nuclei, 52, issue 6, 1574 (1990). * (18) P. Vogel, R.E. Schenter, F. M. Mann and G.K. Schenter, Phys. Rev. C24, 1543 (1981).	arxiv-papers	2009-02-22T08:58:55	2024-09-04T02:49:00.802542	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "V.V. Sinev", "submitter": "V. V. Sinev", "url": "https://arxiv.org/abs/0902.3781" }
0902.3944	# On Stochastic Model Predictive Control with Bounded Control Inputs††thanks: This research was partially supported by the Swiss National Science Foundation under grant 200021-122072 Peter Hokayem, Debasish Chatterjee, John Lygeros The authors are with the Automatic Control Laboratory, Electrical Engineering, ETH Zurich, Switzerland hokayem,chatterjee,lygeros@control.ee.ethz.ch ###### Abstract This paper is concerned with the problem of Model Predictive Control and Rolling Horizon Control of discrete-time systems subject to possibly unbounded random noise inputs, while satisfying hard bounds on the control inputs. We use a nonlinear feedback policy with respect to noise measurements and show that the resulting mathematical program has a tractable convex solution in both cases. Moreover, under the assumption that the zero-input and zero-noise system is asymptotically stable, we show that the variance of the state, under the resulting Model Predictive Control and Rolling Horizon Control policies, is bounded. Finally, we provide some numerical examples on how certain matrices in the underlying mathematical program can be calculated off-line. ## I Introduction Model Predictive Control (MPC) for deterministic systems has received a considerable amount of attention over the last few decades, and significant advancements have been realized in terms of theoretical analysis as well as industrial applications. The motivation for such research thrust comes primarily from tractability of calculating optimal control laws for constrained systems. In contrast, the counterpart of this development for stochastic systems is still in its infancy. The deterministic setting is dominated by worst-case analysis relying on robust control methods. The central idea is to synthesize a controller based on the bounds of the noise such that a certain target set becomes invariant with respect to the closed-loop dynamics. However, such an approach usually leads to rather conservative controllers and to large infeasibility regions, and although disturbances are not likely to be unbounded in practice, assigning an a priori bound to them seems to demand considerable insight. A stochastic model of the disturbance is a natural alternative approach to this problem: the conservatism of the worst-case analysis may be circumvented, and one need not impose any a priori bounds on the maximum magnitude of the noise. However, since in practice control inputs are almost always bounded, it is of great importance to consider hard bounds on the control inputs as essential ingredients of the controller synthesis; probabilistic constraints on the controllers naturally raise difficult questions on what actions to take when such constraints are violated (see however [1] for one possible approach to answer these questions). In this paper we aim to provide answers to the following questions: Given a linear system that is affected by (possibly unbounded) stochastic noise, to be controlled by applying predictive-type bounded control inputs, (i) is the associated optimization problem tractable? (ii) under what conditions is stability (in a suitable stochastic sense) of the closed-loop system guaranteed? (iii) is stability retained both in the case of MPC implementation and the case of Rolling Horizon Control (RHC) implementation? In the deterministic setting, there exists a plethora of literature that settles tractability and stability of model-based predictive control, see, for example, [2, 3, 4, 5] and the references therein. However, there are fewer results in the stochastic case, some of which we outline next. In [6], the authors reformulate the stochastic programming problem as a deterministic one with bounded noise and solve a robust optimization problem over a finite horizon, followed by estimating the performance when the noise can take unbounded values, i.e., when the noise is unbounded, but takes high values with low probability (as in the Gaussian case). In [7, 8] a slightly different problem is addressed in which the noise enters in a multiplicative manner into the system, and hard constraints on the state and control input are relaxed to probabilistic ones. Similar relaxations of hard constraints to soft probabilistic ones have also appeared in [9] for both multiplicative and additive noise inputs, as well as in [10]. There are also other approaches, for example those employing randomized algorithms as in [11, 12]. Finally, a related line of research can be found in [13], and a novel convex analysis dealing with chance and integrated chance constraints can be found in [14]. In this paper we restrict attention to linear time-invariant controlled systems with affine stochastic disturbance inputs. Our approach has three main features. Firstly, for the finite-horizon optimal control subproblem we adopt a feedback control strategy that is affine in certain bounded nonlinear functions of the past noise inputs. Secondly, instead of following the usual trend of adding element-wise constraints to the control input in the optimization, we propose a new approach that entails saturating the utilized noise measurements first and then optimizing over the feedback gains, ensuring that the hard constraints on the input will be satisfied by construction. This novel approach does not require artificially relaxing the hard constraints on the control input to soft probabilistic ones to ensure large feasible sets, and still provides a solution to the problem for a wide class of noise input distributions. In fact, we demonstrate that our strategy (without state constraints) leads to global feasibility. The effect of the noise appears in the finite-horizon optimal control problem as certain covariance matrices, and these matrices may be computed off-line and stored. Thirdly, the measurement saturation functions are only required to be elementwise _bounded_ in order to ensure tractability of the optimization problem while maintaining hard constraints on the control input; therefore, these measurement saturation functions may be picked from among the wide class of saturation functions, the standard sigmoidal functions and their piecewise affine approximations, etc. Once tractability of the finite-horizon underlying optimization problem is insured, it is possible to implement the resulting optimal solution using an MPC approach or an RHC approach. In the former case [2], the optimization problem is resolved at each step and only the first control input is implemented. In the latter case [15], the optimization problem is resolved every $N$ steps (with $N$ being the horizon length) and the entire sequence of $N$ input vectors is implemented. Both of these approaches are shown to provide stability under the assumption that the zero-input and zero-noise system is asymptotically stable, which translates into the condition that the state matrix $A$ is Schur stable. At a first glance, this assumption might seem restrictive. However, the problem of ensuring bounded variance of linear Gaussian systems with bounded control inputs is, to our knowledge, still open, and here we are considering the problem of controlling a linear system with bounded control input and possibly unbounded noise. It is known that for discrete-time systems without any noise acting on the system it is possible to achieve global stability if and only if the matrix $A$ is neutrally stable [16]. This paper unfolds as follows. In §II we state the main problem to be tackled with the underlying assumptions. In §III, we provide a tractable approach to the finite horizon optimization problem with hard constraints on the control input, as well as some examples in §III-A. Stability of the MPC and RHC implementations is shown in §IV, and hints onto the input-to-state stable properties of this result are provided in §IV-C. Finally, we provide a numerical example in §V and conclude in §VI. ### Notation Hereafter, $\mathbb{N}:=\\{1,2,\ldots\\}$ is the set of natural numbers, $\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$, and $\mathbb{R}_{\geqslant 0}$ is the set of nonnegative real numbers. We let $\mathbf{1}_{A}(\cdot)$ denote the indicator function of a set $A$, and $\mathbf{I}_{n\times n}$ and $\mathbf{0}_{n\times n}$ denote the $n$-dimensional identity and zeros matrices, respectively. Also, let $\mathbb{E}_{x_{0}}[\cdot]$ denote the expected value given $x_{0}$, and $\mathbf{tr}\\!\left(\cdot\right)$ denote the trace of a matrix. For a given symmetric $n$-dimensional matrix $M$ with real entries, let $\\{\lambda_{i}(M)\mid i=1,\ldots,n\\}$ be the set of eigenvalues of $M$, and let $\lambda_{\rm max}(M):=\max_{i}\lambda_{i}(M)$ and $\lambda_{\text{min}}(M):=\min_{i}\lambda_{i}(M)$. Let $\left\lVert{\cdot}\right\rVert_{p}$ denote standard $\ell_{p}$ norm. Finally, the mean and covariance matrix of any vector $v$ are denoted by $\Sigma_{v}$ and $\mu_{v}$, respectively. ## II Problem Statement Consider the following general affine discrete-time stochastic dynamical model: $x_{t+1}=Ax_{t}+Bu_{t}+Fw_{t}+r,\qquad t\in\mathbb{N}_{0},$ (1) where $x_{t}\in\mathbb{R}^{n}$ is the state, $u_{t}\in\mathbb{R}^{m}$ is the control input, $w_{t}\in\mathbb{R}^{n}$ is a stochastic noise input vector, $A$, $B$ and $F$ are known matrices, and $r\in\mathbb{R}^{n}$ is a known constant vector. We assume that the initial condition $x_{0}$ is given and that, at any time $t$, $x_{t}$ is observed exactly. We shall assume further that the noise vectors $w_{t}$ are i.i.d. and that the control input vector is bounded at each instant of time $t$, i.e., $u_{t}\in\mathbb{U}:=\bigl{\\{}u\in\mathbb{R}^{m}\big{\|}\left\lVert{u}\right\rVert_{\infty}\leq U_{\rm max}\bigr{\\}}\quad\forall\,t\in\mathbb{N}_{0},$ (2) where $U_{\mathrm{max}}>0$ is some given element-wise saturation bound. Note that the model (1) with constraints (2) can handle a wide range of convex polytopic constraints. In particular, any system $x_{t+1}=Ax_{t}+\hat{B}v_{t}+F\hat{w}_{t}+\hat{r}$ (3) with input constraints $v_{t}\in\mathbb{V}$ that can be transformed to the form (2) by an affine transformation $v_{t}=Su_{t}+l$ is amenable to our approach by setting $B=\hat{B}S$ and $r=\hat{B}l+\hat{r}$ in (1). Note that the set $\mathbb{V}$ need not necessarily be a hypercube, or even contain the origin. Note also that we can assume that $w_{t}$ is zero mean in (1) without loss of generality; given a system of the form (3) where $\hat{w}_{t}$ is not zero mean, we can replace it by a system in the form (1) with zero mean in which $w_{t}=\hat{w}_{t}-\mathbb{E}[w_{t}]$ by setting $r=\hat{r}+F\mathbb{E}[w_{t}]$. Fix a horizon $N\in\mathbb{N}$ and set $t=0$. The _MPC_ procedure can be described as follows. * (a) Determine an admissible optimal feedback control policy, say $\pi^{\star}_{t:t+N-1}\in\Pi$, for an $N$-stage cost function starting from time $t$, given the (measured) initial condition $x_{t}$; * (b) increase $t$ to $t+1$, and go back to step (a). On the other hand, the _RHC_ procedure simply replaces (b) above by * (b′) apply the entire sequence $\pi^{\star}_{t:t+N-1}$ of control inputs, update the state $x_{t+N}$ at the $(t+N-1)$-th step, increase $t$ to $t+N$ and go back to step (a). Accordingly, the $t$-th step of this procedure consists of minimizing the stopped $N$-period cost function starting at time $t$, namely, the objective is to find a feedback control policy that attains $\displaystyle\inf_{\pi\in\Pi}V_{t,t+N-1}(\pi,x):=$ $\displaystyle\;\inf_{\pi\in\Pi}\mathsf{E}^{\pi}_{x_{t}}\\!\biggl{[}\sum_{i=t}^{t+N-1}\\!\\!\bigl{(}x_{i}^{\mathsf{T}}Q_{i}x_{i}+u_{i}^{\mathsf{T}}R_{i}u_{i}\bigr{)}$ $\displaystyle\qquad\qquad+x_{t+N}^{\mathsf{T}}Q_{t+N}x_{t+N}\biggr{]}.$ (4) Since both the system (1) and cost (4) are time-invariant, it is enough to consider the problem of minimizing the cost for $t=0$, i.e., the problem of minimizing $V_{0,N-1}(\pi,x)$ over $\pi\in\Pi$. In view of the above we consider the problem $\displaystyle\min_{\pi\in\Pi}$ $\displaystyle\mathbb{E}_{x_{0}}\left[\sum\limits_{t=0}^{N-1}\bigl{(}x_{t}^{\mathsf{T}}Q_{t}x_{t}+u_{t}^{\mathsf{T}}R_{t}u_{t}\bigr{)}+x_{N}^{\mathsf{T}}Q_{N}x_{N}\right],$ (5) s.t. $\displaystyle\mathrm{dynamics}\,(\ref{eq:system}),\,\mathrm{and\,\,constraints}\,(\ref{eq:bddu})$ where $Q_{t}>0$ and $R_{t}>0$ are some given symmetric matrices of appropriate dimension. If feasible with respect to (2), Problem (5) generates an optimal sequence of feedback control laws $\pi^{}=\\{u^{}_{0},\cdots,u^{}_{N-1}\\}$. The evolution of the system (1) over a single optimization horizon $N$ can be described in compact form as follows: $\bar{x}=\bar{A}x_{0}+\bar{B}\bar{u}+\bar{D}\bar{F}\bar{w}+\bar{D}\bar{r},$ (6) where $\bar{x}:=\begin{bmatrix}x_{0}\\\ x_{1}\\\ \vdots\\\ x_{N}\end{bmatrix},\,\bar{u}:=\begin{bmatrix}u_{0}\\\ u_{1}\\\ \vdots\\\ u_{N-1}\end{bmatrix},\,\bar{r}:=\left[\begin{matrix}r\\\ \vdots\\\ r\end{matrix}\right],\,\bar{w}:=\begin{bmatrix}w_{0}\\\ w_{1}\\\ \vdots\\\ w_{N-1}\end{bmatrix},$ $\bar{A}:=\begin{bmatrix}\mathbf{I}_{n\times n}\\\ A\\\ \vdots\\\ A^{N}\end{bmatrix},\,\bar{B}:=\begin{bmatrix}\mathbf{0}_{n\times m}&\cdots&\cdots&\mathbf{0}_{n\times m}\\\ B&\ddots&&\vdots\\\ AB&B&\ddots&\vdots\\\ \vdots&&\ddots&\mathbf{0}_{n\times m}\\\ A^{N-1}B&\cdots&AB&B\end{bmatrix},$ ${\small\bar{D}:=\begin{bmatrix}\mathbf{0}_{n\times n}&\cdots&\cdots&\mathbf{0}_{n\times n}\\\ \mathbf{I}_{n\times n}&\ddots&&\vdots\\\ A&\mathbf{I}_{n\times n}&\ddots&\vdots\\\ \vdots&&\ddots&\mathbf{0}_{n\times n}\\\ A^{N-1}&\cdots&A&\mathbf{I}_{n\times n}\end{bmatrix},\,\bar{F}:=\left[\begin{matrix}F&\ldots&\mathbf{0}\\\ \vdots&\ddots&\vdots\\\ \mathbf{0}&\ldots&F\end{matrix}\right]}$ where the input $\bar{u}\in\bar{\mathbb{U}}:=\bigl{\\{}\xi\in\mathbb{R}^{Nm}\big{\|}\left\lVert{\xi}\right\rVert_{\infty}\leq U_{\rm max}\bigr{\\}}.$ (7) Using the compact notation above, the optimization Problem (5) can be rewritten as follows: $\displaystyle\min_{\pi\in\Pi}$ $\displaystyle\mathbb{E}_{x_{0}}\bigl{[}\bar{x}^{\mathsf{T}}\bar{Q}\bar{x}+\bar{u}^{\mathsf{T}}\bar{R}\bar{u}\bigr{]},$ (8) s.t. $\displaystyle{\rm dynamics}\,(\ref{eq:compactdyn}),\,\mathrm{and\,\,constraints}\,(\ref{eq:bddu2}),$ where $\bar{Q}=\left[\begin{matrix}Q_{0}&\ldots&\mathbf{0}_{n\times n}\\\ \vdots&\ddots&\vdots\\\ \mathbf{0}_{n\times n}&\ldots&Q_{N}\end{matrix}\right],\,\bar{R}=\left[\begin{matrix}R_{0}&\ldots&\mathbf{0}_{m\times m}\\\ \vdots&\ddots&\vdots\\\ \mathbf{0}_{m\times m}&\ldots&R_{N-1}\end{matrix}\right].$ The solution to Problem (8) is difficult to obtain in general. In order to obtain an optimal solution to Problem (8) over the class of feedback policies, we need to solve the Dynamic Programming equations. This generally requires using some gridding technique, making the problem extremely difficult to solve computationally. Another approach is to restrict attention to a specific class of state feedback policies. This will result in a suboptimal solution to our problem, but may yield a tractable optimization problem. It is the track we pursue in the next section. ## III Tractable Solution under Bounded Control Inputs By the hypothesis that the state is observed without error, one may reconstruct the noise sequence from the sequence of observed states and inputs by the formula $Fw_{t}=x_{t+1}-Ax_{t}-Bu_{t}-r,\qquad t\in\mathbb{N}_{0}.$ (9) In the light of this, and inspired by the works [17, 18], we shall consider feedback policies of the form: $u_{t}=\sum_{i=0}^{t-1}G_{t,i}Fw_{i}+d_{t},$ (10) where the feedback gains $G_{t,i}\in\mathbb{R}^{m\times n}$ and the affine terms $d_{t}\in\mathbb{R}^{m}$ must be chosen based on the control objective, while observing the constraints (2). With this definition, the value of $u$ at time $t$ depends on the values of $w$ up to time $t-1$. Using (9) we see that $u_{t}$ is a function of the observed states up to time $t$. It was shown in [18] that there exists a one-to-one (nonlinear) mapping between control policies in the form (10) and the class of affine state feedback policies. That is, provided one is interested in affine state feedback policies, parametrization (9) constitutes no loss of generality. Of course, this choice is generally suboptimal, but it will ensure the tractability of a large class of optimal control problems. In compact notation, the control sequence up to time $N-1$ is given by $\bar{u}=\bar{G}\bar{F}\bar{w}+\bar{d},$ (11) where $\bar{d}:=\left[\begin{matrix}d_{0}^{\mathsf{T}}&d_{2}^{\mathsf{T}}&\ldots&d^{\mathsf{T}}_{N-1}\end{matrix}\right]^{\mathsf{T}}$, and $\bar{G}:=\begin{bmatrix}\mathbf{0}_{m\times n}\\\ G_{1,0}&\mathbf{0}_{m\times n}\\\ \vdots&\ddots&\ddots\\\ G_{N-1,0}&\cdots&G_{N-1,N-2}&\mathbf{0}_{m\times n}\end{bmatrix}.$ Since the elements of the noise vector $\bar{w}$ are not assumed to be bounded, there can be no guarantee that the control input (11) will meet the constraint (7). This is a problem in practical applications, and has traditionally been circumvented by assuming that the noise input lies within a compact set [18], and designing a worst-case controller. In this article we propose to use the controller $\bar{u}=\bar{G}\bar{\varphi}(\bar{F}\bar{w})+\bar{d},$ (12) instead of (11), where $\bar{\varphi}(\bar{F}\bar{w})=\left[\begin{matrix}\varphi_{0}(F{w}_{0})\\\ \vdots\\\ \varphi_{N-1}(F{w}_{N-1})\end{matrix}\right],$ $\varphi_{i}(Fw_{i})$ is a shorthand for the vector $\bigl{[}\varphi_{i}^{1}(F_{1}w_{i}),\ldots,\varphi_{i}^{n}(F_{n}w_{i})\bigr{]}^{\mathsf{T}}$, $F_{j}$ is the $j$-th row of the matrix $F$, and $\varphi_{i}^{j}:\mathbb{R}\to\mathbb{R}$ is any function with $\sup\limits_{s\in\mathbb{R}}\|\varphi_{i}^{j}(s)\|\leq\phi_{\max}\leq U_{\rm max}$. In other words, we have chosen to saturate the measurements that we obtain from the noise input vector before inserting them into our control vector. This way we do not assume that the noise distribution is defined over a compact domain, which is an advantage over other approaches [6, 18]. Moreover, the choice of element-wise saturation functions $\varphi_{i}(\cdot)$ is left open. As such, we can accommodate standard saturation, piecewise linear, and sigmoidal functions, to name a few. ###### Remark 1 Our choice of saturating the measurement from the noise vectors renders the optimization problem tractable as opposed to just calculating the whole input vector $\bar{u}$ and then saturating it afterwards, which tends to an intractable optimization problem.$\vartriangleleft$ ###### Remark 2 Note that the choices of control inputs in (11) and (12) are both _non Markovian_ ; however, they differ in the fact that the former depends affinely on previous noise inputs $\bar{w}$, whereas the latter is a nonlinear feedback due to passing noise measurements through the function $\bar{\varphi}(.)$.$\vartriangleleft$ ###### Proposition 3 Assume that $\mathbb{E}_{x_{0}}\left[\bar{\varphi}(\bar{F}\bar{w})\right]=0$, $\forall x_{0}\in\mathbb{R}^{n}$. Then, Problem (8) with the input (12) is a convex optimization problem, with respect to the decision variables $(\bar{G},\bar{d})$, which is given by $\displaystyle\min\limits_{(\bar{G},\bar{d})}$ $\displaystyle b^{\mathsf{T}}\bar{d}+\bar{d}^{\mathsf{T}}M_{1}\bar{d}+\mathbf{tr}\\!\left(\bar{G}^{\mathsf{T}}M_{1}\bar{G}\Lambda_{1}+M_{2}\bar{G}\Lambda_{2}\right)$ (13) $\displaystyle\mathrm{s.t.}$ $\displaystyle\|\bar{d}_{i}\|+\left\lVert{\bar{G}_{i}}\right\rVert_{1}\phi_{\rm max}\leq U_{\max},\quad\forall i=1,\cdots,Nm$ where $G_{i}$ is the $i$-th row of $G$, $\displaystyle b^{T}$ $\displaystyle=2(\bar{A}x_{0}+\bar{D}\bar{F}\mu_{\bar{w}}+\bar{r})^{\mathsf{T}}\bar{Q}\bar{B},\quad M_{1}=\bar{R}+\bar{B}^{\mathsf{T}}\bar{Q}\bar{B},$ $\displaystyle M_{2}$ $\displaystyle=2\bar{F}^{\mathsf{T}}\bar{D}^{\mathsf{T}}\bar{Q}\bar{B},$ $\displaystyle\Lambda_{1}$ $\displaystyle=\mathrm{diag}\bigl{\\{}\mathbb{E}\bigl{[}\varphi_{0}(Fw_{0})\varphi_{0}(Fw_{0})^{\mathsf{T}}\bigr{]},\cdots,\bigr{.}$ $\displaystyle\qquad\qquad\bigl{.}\mathbb{E}\bigl{[}\varphi_{N-1}(Fw_{N-1})\varphi_{N-1}(Fw_{N-1})^{\mathsf{T}}\bigr{]}\bigr{\\}},$ $\displaystyle\Lambda_{2}$ $\displaystyle=\mathrm{diag}\bigl{\\{}\mathbb{E}\bigl{[}\varphi_{0}(Fw_{0})w_{0}^{\mathsf{T}}\bigr{]},\cdots,\bigr{.}$ $\displaystyle\qquad\qquad\bigl{.}\mathbb{E}\bigl{[}\varphi_{N-1}(Fw_{N-1})w_{N-1}^{\mathsf{T}}\bigr{]}\bigr{\\}}.$ ###### Proof: Let us first consider the cost function in Problem 8. After substituting the system equations, we obtain $\displaystyle\mathbb{E}_{x_{0}}\bigl{[}\bar{x}^{\mathsf{T}}\bar{Q}\bar{x}+\bar{u}^{\mathsf{T}}\bar{R}\bar{u}\bigr{]}=$ (14) $\displaystyle\mathbb{E}_{x_{0}}[\left(\bar{A}x_{0}+\bar{B}\bar{u}+\bar{D}\bar{F}\bar{w}+\bar{r}\right)^{\mathsf{T}}\bar{Q}\left(\bar{A}x_{0}+\bar{B}\bar{u}+\bar{D}\bar{F}\bar{w}+\bar{r}\right)$ $\displaystyle\qquad+\bar{u}^{\mathsf{T}}\bar{R}\bar{u}]$ $\displaystyle=(\bar{A}x_{0}+\bar{r})^{\mathsf{T}}\bar{Q}(\bar{A}x_{0}+\bar{r})+2(\bar{A}x_{0}+\bar{r})^{\mathsf{T}}\bar{Q}\bar{D}\bar{F}\mathbb{E}_{x_{0}}\bigl{[}\bar{w}\bigr{]}$ $\displaystyle\quad+2(\bar{A}x_{0}+\bar{r})^{\mathsf{T}}\bar{Q}\bar{B}\mathbb{E}_{x_{0}}\bigl{[}\bar{u}\bigr{]}+2\mathbb{E}_{x_{0}}\bigl{[}\bar{w}^{\mathsf{T}}\bar{F}^{\mathsf{T}}\bar{D}^{\mathsf{T}}\bar{Q}\bar{B}\bar{u}\bigr{]}$ $\displaystyle\quad+\mathbb{E}_{x_{0}}\bigl{[}\bar{w}^{\mathsf{T}}\bar{F}^{\mathsf{T}}\bar{D}^{\mathsf{T}}\bar{Q}\bar{D}\bar{F}\bar{w}\bigr{]}+\mathbb{E}_{x_{0}}\bigl{[}\bar{u}^{\mathsf{T}}(\bar{R}+\bar{B}^{\mathsf{T}}\bar{Q}\bar{B})\bar{u}\bigr{]}.$ Note that since $\mathbb{E}_{x_{0}}\left[\bar{\varphi}(\bar{F}\bar{w})\right]=0$, we have that $\mathbb{E}_{x_{0}}\bigl{[}\bar{u}\bigr{]}=\bar{d}$. Accordingly, using the definitions of $b$, $M_{1}$, $M_{2}$, and $\Lambda_{2}$, $\displaystyle\mathbb{E}_{x_{0}}\bigl{[}\bar{x}^{\mathsf{T}}\bar{Q}\bar{x}+\bar{u}^{\mathsf{T}}\bar{R}\bar{u}\bigr{]}$ $\displaystyle=b^{\mathsf{T}}\bar{d}+\mathbf{tr}\\!\left(M_{2}\bar{G}\Lambda_{2}\right)+c$ $\displaystyle\quad+\mathbb{E}_{x_{0}}\bigl{[}\bar{u}^{\mathsf{T}}M_{1}\bar{u}\bigr{]},$ (15) where $c=(\bar{A}x_{0}+\bar{r})^{\mathsf{T}}\bar{Q}(\bar{A}x_{0}+\bar{r})+\mathbf{tr}\\!\left(\bar{F}^{\mathsf{T}}\bar{D}^{\mathsf{T}}\bar{Q}\bar{D}\bar{F}\Sigma_{\bar{w}}\right)+2(\bar{A}x_{0}+\bar{r})^{\mathsf{T}}\bar{Q}\bar{D}\bar{F}\mu_{\bar{w}}$ is a constant that we omit as it does not change the optimization problem, and we have used the following intermediate step $\displaystyle\mathbb{E}_{x_{0}}\bigl{[}\bar{w}^{\mathsf{T}}\bar{F}^{\mathsf{T}}\bar{D}^{\mathsf{T}}\bar{Q}\bar{B}\bar{u}\bigr{]}=\mathbb{E}_{x_{0}}\bigl{[}\bar{w}^{\mathsf{T}}\bar{F}^{\mathsf{T}}\bar{D}^{\mathsf{T}}\bar{Q}\bar{B}(\bar{G}\bar{\varphi}(\bar{F}\bar{w})+\bar{d})\bigr{]}$ $\displaystyle\qquad=\mathbf{tr}\\!\left(\bar{F}^{\mathsf{T}}\bar{D}^{\mathsf{T}}\bar{Q}\bar{B}\bar{G}\Lambda_{2}\right)+\mu_{\bar{w}}^{\mathsf{T}}\bar{F}^{\mathsf{T}}\bar{D}^{\mathsf{T}}\bar{Q}\bar{B}\bar{d}.$ Using again the assumption that $\mathbb{E}_{x_{0}}\left[\bar{\varphi}(\bar{F}\bar{w})\right]=0$, we have that $\displaystyle\mathbb{E}_{x_{0}}\bigl{[}\bar{u}^{\mathsf{T}}$ $\displaystyle M_{1}\bar{u}\bigr{]}=\mathbb{E}_{x_{0}}\bigl{[}(\bar{G}\bar{\varphi}(\bar{F}\bar{w})+\bar{d})^{\mathsf{T}}M_{1}(\bar{G}\bar{\varphi}(\bar{F}\bar{w})+\bar{d})\bigr{]}$ $\displaystyle=\mathbb{E}_{x_{0}}\bigl{[}\bar{\varphi}(\bar{F}\bar{w})^{\mathsf{T}}\bar{G}^{\mathsf{T}}M_{1}\bar{G}\bar{\varphi}(\bar{F}\bar{w})\bigr{]}+\bar{d}^{\mathsf{T}}M_{1}\bar{d}$ $\displaystyle=\mathbf{tr}\\!\left(\bar{G}^{\mathsf{T}}M_{1}\bar{G}\mathbb{E}_{x_{0}}\bigl{[}\bar{\varphi}(\bar{F}\bar{w})\bar{\varphi}(\bar{F}\bar{w})^{\mathsf{T}}\bigr{]}\right)+\bar{d}^{\mathsf{T}}M_{1}\bar{d}$ $\displaystyle=\mathbf{tr}\\!\left(\bar{G}^{\mathsf{T}}M_{1}\bar{G}\Lambda_{1}\right)+\bar{d}^{\mathsf{T}}M_{1}\bar{d}.$ (16) Finally, combining (15) and (16), we obtain the cost in Problem 13, which is convex. Let us look at the constraints in Problem 8. The proposed control input (12) satisfies the hard constraints (7) as long as the following condition is satisfied: $\left\lVert{\bar{d}+\bar{G}\bar{\varphi}(\bar{w})}\right\rVert_{\infty}\leq U_{\max}$, $\forall\bar{\varphi}(\bar{w})$ such that $\left\lVert{\bar{\varphi}(\bar{w})}\right\rVert_{\infty}\leq\phi_{\max}$. This is equivalent to the following conditions: $\forall i=1,\cdots,Nm$, $\|\bar{d}_{i}+\bar{G}_{i}\bar{\varphi}(\bar{w})\|\leq U_{\max}$, $\forall\bar{\varphi}(\bar{w})$ such that $\left\lVert{\bar{\varphi}(\bar{w})}\right\rVert_{\infty}\leq\phi_{\max}$. As these conditions should hold for any permissible value of the function $\bar{\varphi}(\bar{w})$, we can eliminate the dependence of the constraints on $\bar{\varphi}(\bar{w})$ through the following optimization problems $\max\limits_{\left\lVert{\bar{\varphi}(\bar{w})}\right\rVert_{\infty}\leq\phi_{\max}}\|\bar{d}_{i}+\bar{G}_{i}\bar{\varphi}(\bar{w})\|\leq U_{\max},\,\forall i=1,\cdots,Nm$. It is straightforward now to show, using Hölder’s inequality [19, p. 29], that $\max\limits_{\left\lVert{\bar{\varphi}(\bar{w})}\right\rVert_{\infty}\leq\phi_{\max}}\|\bar{d}_{i}+\bar{G}_{i}\bar{\varphi}(\bar{w})\|=\|\bar{d}_{i}\|+\left\lVert{\bar{G}_{i}}\right\rVert_{1}\phi_{\rm max}$, and the result follows. ∎ ###### Remark 4 Problem (13) is a quadratic program in the optimization parameters $\theta:=(\bar{G},\bar{d})$ [20, p. 111], and can be solved efficiently by standard solvers such as cvx [21].$\vartriangleleft$ ### III-A Examples An important step in the solvability of Problem (13) is being able to calculate the matrices $\Lambda_{1}$ and $\Lambda_{2}$. In general, these matrices can be calculated off-line by numerical integration. However, in some instances these matrices can be given in terms of explicit formulas; two of these instances are given in the following examples. Recall the following standard special mathematical functions: the _standard error function_ $\operatorname{erf}(z):=\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\textstyle\sqrt{\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptstyle\sqrt{\pi\,}$}\lower 0.4pt\hbox{\vrule height=2.10971pt,depth=-1.68779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\pi\,}$}\lower 0.4pt\hbox{\vrule height=1.50694pt,depth=-1.20557pt}}}}\int_{0}^{z}\mathrm{e}^{-\frac{t^{2}}{2}}\mathrm{d}t$ and the _complementary error function_ [22, p. 297] defined by $\operatorname{erfc}(z):=1-\operatorname{erf}(z)$ for $z\in\mathbb{R}$, the _incomplete Gamma function_ [22, p. 260] defined by $\Gamma(a,z):=\int_{z}^{\infty}t^{a-1}\mathrm{e}^{-t}\mathrm{d}t$ for $z,a>0$, the _confluent hypergeometric function_ [22, p. 505] defined by $U(a,b,z):=\frac{1}{\Gamma(a)}\int_{0}^{\infty}\mathrm{e}^{-zt}t^{a-1}(1+t)^{b-a-1}\mathrm{d}t$ for $a,b,z>0$ and $\Gamma$ is the standard Gamma function. We collect a few facts in the following ###### Proposition 5 For $\sigma^{2}>0$ we have 1. 1. $\displaystyle{\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma}\int_{z}^{\infty}\mathrm{e}^{-\frac{t^{2}}{2\sigma^{2}}}\mathrm{d}t=\frac{1}{2}\Bigl{(}1+\operatorname{erf}\Bigl{(}\frac{z}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma}\Bigr{)}\Bigr{)}}$; 2. 2. $\displaystyle{\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma}\int_{0}^{\infty}\frac{t^{2}}{1+t^{2}}\mathrm{e}^{-\frac{t^{2}}{2\sigma^{2}}}\mathrm{d}t}$ $\displaystyle{\qquad=\frac{1}{2}\Bigl{(}\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma-\pi\mathrm{e}^{-\frac{1}{2\sigma^{2}}}\operatorname{erfc}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma}\Bigr{)}\Bigr{)}}$; 3. 3. $\displaystyle{\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma}\int_{0}^{1}t^{2}\mathrm{e}^{-\frac{t^{2}}{2\sigma^{2}}}\mathrm{d}t}$ $\displaystyle{\qquad=\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{\pi}{2}\,}$}\lower 0.4pt\hbox{\vrule height=7.52776pt,depth=-6.02223pt}}}{{\hbox{$\textstyle\sqrt{\frac{\pi}{2}\,}$}\lower 0.4pt\hbox{\vrule height=5.26944pt,depth=-4.21558pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{\pi}{2}\,}$}\lower 0.4pt\hbox{\vrule height=3.76387pt,depth=-3.01111pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{\pi}{2}\,}$}\lower 0.4pt\hbox{\vrule height=3.76387pt,depth=-3.01111pt}}}\sigma^{3}\operatorname{erf}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma}\Bigr{)}-\sigma^{2}\mathrm{e}^{-\frac{1}{2\sigma^{2}}}}$; 4. 4. $\displaystyle{\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma}\int_{1}^{\infty}t\mathrm{e}^{-\frac{t^{2}}{2\sigma^{2}}}\mathrm{d}t=\frac{\sigma}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}}\Gamma(2\sigma^{2},1)}$; 5. 5. $\displaystyle{\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma}\int_{0}^{\infty}\frac{t^{2}}{\mathchoice{{\hbox{$\displaystyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\textstyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\scriptstyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=5.59444pt,depth=-4.47557pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=4.36427pt,depth=-3.49144pt}}}}\mathrm{e}^{-\frac{t^{2}}{2\sigma^{2}}}\mathrm{d}t=\frac{\sigma}{2\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}}U\Bigl{(}\frac{1}{2},0,\frac{1}{2\sigma^{2}}\Bigr{)}}$. ###### Example 6 Let us consider (1) with Gaussian noise and sigmoidal bounds on the control input. More precisely, suppose that the noise process $(w_{t})_{t\in\mathbb{N}_{0}}$ is an independent and identically distributed (i.i.d) sequence of Gaussian random vectors of mean $0$ and covariance $\Sigma$. Let the components of $w_{t}$ be mutually independent, which implies that $\Sigma$ is a diagonal matrix $\operatorname{diag}\\{\sigma_{1}^{2},\ldots,\sigma_{n}^{2}\\}$. Suppose further that the matrix $F=I$ and that the function $\varphi$ is a standard sigmoid, i.e., $\varphi(t):=t/\mathchoice{{\hbox{$\displaystyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\textstyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\scriptstyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=5.59444pt,depth=-4.47557pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=4.36427pt,depth=-3.49144pt}}}$. Then from Proposition 5 we have for $i=1,\ldots,n$ and $j=0,\ldots,N-1$, $\displaystyle\mathbb{E}[\varphi(w_{j}^{i})^{2}]$ $\displaystyle=2\cdot\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{0}^{\infty}\frac{t^{2}}{1+t^{2}}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}$ $\displaystyle=\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}-\pi\mathrm{e}^{-\frac{1}{2\sigma_{i}^{2}}}\operatorname{erfc}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\Bigr{)}.$ This shows that the matrix $\Lambda_{1}$ in Proposition 3 is equal to $\operatorname{diag}\\{\Sigma^{\prime},\ldots,\Sigma^{\prime}\\}$, where $\Sigma^{\prime}:=\operatorname{diag}\left\\{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{1}-\pi\mathrm{e}^{-\frac{1}{2\sigma_{1}^{2}}}\operatorname{erfc}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=2.25555pt,depth=-1.80446pt}}}\sigma_{1}}\Bigr{)},\right.$ $\left.\ldots,\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{n}-\pi\mathrm{e}^{-\frac{1}{2\sigma_{n}^{2}}}\operatorname{erfc}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=2.25555pt,depth=-1.80446pt}}}\sigma_{n}}\Bigr{)}\right\\}.$ Similarly, since $\displaystyle\mathbb{E}[\varphi(w_{j}^{i})w_{j}^{i}]$ $\displaystyle=\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{-\infty}^{\infty}\frac{t^{2}}{\mathchoice{{\hbox{$\displaystyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\textstyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=7.95523pt,depth=-6.36421pt}}}{{\hbox{$\scriptstyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=5.59444pt,depth=-4.47557pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=4.36427pt,depth=-3.49144pt}}}}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}}}\mathrm{d}t$ $\displaystyle=\frac{\sigma_{i}}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}}U\Bigl{(}\frac{1}{2},0,\frac{1}{2\sigma_{i}^{2}}\Bigr{)},$ the matrix $\Lambda_{2}$ in Proposition 3 is $\operatorname{diag}\\{\Sigma^{\prime\prime},\ldots,\Sigma^{\prime\prime}\\}$, where $\Sigma^{\prime\prime}:=\operatorname{diag}\left\\{\frac{\sigma_{1}}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=2.25555pt,depth=-1.80446pt}}}}U\Bigl{(}\frac{1}{2},0,\frac{1}{2\sigma_{1}^{2}}\Bigr{)},\ldots,\frac{\sigma_{n}}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=2.25555pt,depth=-1.80446pt}}}}U\Bigl{(}\frac{1}{2},0,\frac{1}{2\sigma_{n}^{2}}\Bigr{)}\right\\}.$ Therefore, given the system (1), the control policy (10), and the description of the noise input as above, the matrices $\Lambda_{1}$ and $\Lambda_{2}$ derived above complete the set of hypotheses of Proposition 3. The problem (5) can now be solved as a quadratic program (13).$\triangle$ Note that we have chosen to use the standard sigmoidal functions in Example 6. However, the result still holds for more general sigmoidal functions of the form $\tilde{\phi}(t)=M\frac{\alpha t}{\mathchoice{{\hbox{$\displaystyle\sqrt{1+\alpha^{2}t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=5.56866pt,depth=-4.45496pt}}}{{\hbox{$\textstyle\sqrt{1+\alpha^{2}t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=5.56866pt,depth=-4.45496pt}}}{{\hbox{$\scriptstyle\sqrt{1+\alpha^{2}t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=3.9161pt,depth=-3.1329pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1+\alpha^{2}t^{2}\,}$}\lower 0.4pt\hbox{\vrule height=3.055pt,depth=-2.44402pt}}}}$, where $M\in\mathbb{R}$ is some given magnitude and $\alpha\in\mathbb{R}$ is some given slope. This slight change is reflected in the entries of the matrices $\Lambda_{1}$ and $\Lambda_{2}$, i.e., for $i=1,\ldots,n$ and $j=0,\ldots,N-1$, $\displaystyle\mathbb{E}[\varphi(w_{j}^{i})^{2}]$ $\displaystyle=M\left(\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}\alpha-\pi\mathrm{e}^{-\frac{1}{2\sigma_{i}^{2}\alpha^{2}}}\operatorname{erfc}\left(\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}\alpha}\right)\right),$ and $\mathbb{E}[\varphi(w_{j}^{i})w_{j}^{i}]=M\frac{\sigma_{i}\alpha}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.15778pt,depth=-2.52625pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=2.25555pt,depth=-1.80446pt}}}}U\Bigl{(}\frac{1}{2},0,\frac{1}{2\sigma_{i}^{2}\alpha^{2}}\Bigr{)}$. ###### Example 7 Consider the system (1) as in Example 6, with $\varphi$ being the standard saturation function defined as $\varphi(t)=\operatorname{sat}(t):=\operatorname{sgn}(t)\min\\{\|t\|,1\\}$. From Proposition 3 we have for $i=1,\ldots,n$ and $j=0,\ldots,N-1$, $\displaystyle\xi_{i}^{\prime}$ $\displaystyle:=\mathbb{E}[\varphi(w_{j}^{i})^{2}]=\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{-\infty}^{\infty}\varphi(t)^{2}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t$ $\displaystyle=\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{0}^{1}t^{2}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t+\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{1}^{\infty}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t$ $\displaystyle=\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}^{3}\operatorname{erf}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\Bigr{)}-2\sigma_{i}^{2}\mathrm{e}^{-\frac{1}{2\sigma_{i}^{2}}}+1+\operatorname{erf}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\Bigr{)}$ and $\displaystyle\xi_{i}^{\prime\prime}$ $\displaystyle:=\mathbb{E}[\varphi(w_{j}^{i})w_{j}^{i}]=\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{-\infty}^{\infty}t\varphi(t)\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t$ $\displaystyle=\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{0}^{1}t^{2}\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t+\frac{2}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\int_{1}^{\infty}t\mathrm{e}^{-\frac{t^{2}}{2\sigma_{i}^{2}}}\mathrm{d}t$ $\displaystyle=\mathchoice{{\hbox{$\displaystyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\pi\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}^{3}\operatorname{erf}\Bigl{(}\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower 0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}\sigma_{i}}\Bigr{)}-2\sigma_{i}^{2}\mathrm{e}^{-\frac{1}{2\sigma_{i}^{2}}}+\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{2}{\pi}\,}$}\lower 0.4pt\hbox{\vrule height=8.59721pt,depth=-6.8778pt}}}{{\hbox{$\textstyle\sqrt{\frac{2}{\pi}\,}$}\lower 0.4pt\hbox{\vrule height=6.01805pt,depth=-4.81447pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{2}{\pi}\,}$}\lower 0.4pt\hbox{\vrule height=4.2986pt,depth=-3.4389pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{2}{\pi}\,}$}\lower 0.4pt\hbox{\vrule height=4.2986pt,depth=-3.4389pt}}}\sigma_{i}\Gamma(2\sigma_{i}^{2},1).$ Therefore, in this case the matrix $\Lambda_{1}$ in Proposition 3 is $\operatorname{diag}\\{\Sigma^{\prime},\ldots,\Sigma^{\prime}\\}$ with $\Sigma^{\prime}:=\operatorname{diag}\\{\xi_{1}^{\prime},\ldots,\xi_{n}^{\prime}\\}$, and the matrix $\Lambda_{2}$ is $\operatorname{diag}\\{\Sigma^{\prime\prime},\ldots,\Sigma^{\prime\prime}\\}$ with $\Sigma^{\prime\prime}:=\operatorname{diag}\\{\xi_{1}^{\prime\prime},\ldots,\xi_{n}^{\prime\prime}\\}$. These information complete the set of hypotheses of Proposition 3, and the problem (5) can now be solved as a quadratic program (13).$\triangle$ ## IV Stability Analysis In this section, we assume that the matrix $A$ is Schur stable, i.e., $\left\lvert{\lambda_{i}(A)}\right\rvert<1$, $\forall\,i$. Accordingly, and since the control is bounded, it is intuitively evident that the closed-loop system is stable in some sense. Indeed, we shall show that the variance of the state is uniformly bounded both in the MPC and RHC cases, the only difference being a choice of implementation based on available memory. First we need the following Lemma. It is a standard variant of the Foster- Lyapunov condition [23]; we include a proof here for completeness. The hypotheses of this Lemma are stronger than usual, but are sufficient for our purposes; see e.g., [24] for more general conditions. ###### Lemma 8 Let $(x_{t})_{t\in\mathbb{N}_{0}}$ be an $\mathbb{R}^{n}$-valued Markov process. Let $V:\mathbb{R}^{n}\to\mathbb{R}_{\geqslant 0}$ be a continuous positive definite and radially unbounded function, integrable with respect to the probability distribution function of $w$. Suppose that there exists a compact set $K\subseteq\mathbb{R}^{n}$ and a number $\lambda\in\;]0,1[$ such that $\mathbb{E}\bigl{[}V(x_{1})\big{\|}x_{0}=x\bigr{]}\leqslant\lambda V(x),\qquad\forall x\not\in K.$ Then $\sup\limits_{t\in\mathbb{N}_{0}}\mathbb{E}_{x}\bigl{[}V(x_{t})\bigr{]}<\infty$. ###### Proof: From the conditions it follows immediately that $\mathbb{E}_{x}\bigl{[}V(x_{1})\bigr{]}\leqslant\lambda V(x)+b\mathbf{1}_{K}(x),\qquad\forall\,x\in\mathbb{R}^{n}$ where $b:=\sup\limits_{x\in K}\mathbb{E}_{x}\bigl{[}V(x_{1})\bigr{]}$. We then have $\displaystyle\mathbb{E}_{x}\bigl{[}V(x_{t})\bigr{]}$ $\displaystyle=\mathbb{E}_{x}\bigl{[}\mathbb{E}\bigl{[}V(x_{t})\big{\|}x_{t-1}\bigr{]}\bigr{]}$ (17) $\displaystyle\leqslant\mathbb{E}_{x}\bigl{[}\mathbb{E}\bigl{[}\lambda V(x_{t-1})+b\mathbf{1}_{K}(x_{t-1})\bigr{]}\bigr{]}$ $\displaystyle\leqslant\lambda^{t}V(x)+\sum_{i=0}^{t-1}\lambda^{t-1-i}b\;\mathbb{E}_{x}\bigl{[}\mathbf{1}_{K}(x_{i})\bigr{]}$ $\displaystyle\leqslant\lambda^{t}V(x)+\frac{b(1-\lambda^{t})}{1-\lambda},$ (18) which shows that $\sup\limits_{t\in\mathbb{N}_{0}}\mathbb{E}_{x}\bigl{[}V(x_{t})\bigr{]}\leqslant V(x)+b/(1-\lambda)<\infty$ as claimed. ∎ We shall utilize Lemma 8 in order to show that the implementation of either the MPC or the RHC strategy generated by the solution of Problem (13) results in a uniformly bounded state variance. ### IV-A MPC Case The MPC implementation corresponding to our input (12) and optimization program (13) consists of the following steps: Given a fixed optimization horizon $N$, set the initial time $t=0$, calculate the optimal control gains $(\bar{G}^{},\bar{d}^{})$ using the program (13), apply the first optimal control input $\pi_{0\|t}^{}=u_{0\|t}^{}=\bar{d}_{0\|t}^{}$, increase $t$ to $t+1$, and iterate. Of course, the optimal gain depends implicity on the current given initial state, i.e., $\bar{d}_{0\|t}^{}=\bar{d}_{0\|t}^{}(x_{t})$, which in turn gives rise to a stationary infinite horizon optimal policy given by $\mathbf{\pi}^{\rm MPC}:=\bigl{(}\pi_{0\|0}^{},\pi_{0\|1}^{},\ldots\bigr{)}=\bigl{(}\bar{d}_{0\|t}^{},\bar{d}_{0\|t}^{},\ldots\bigr{)}$. The closed-loop system is thus given by $x_{t+1}=Ax_{t}+B\bar{d}^{}_{0\|t}+Fw_{t}+r,\qquad t\in\mathbb{N}_{0}.$ (19) ###### Proposition 9 Assume that the matrix $A$ is Schur stable and the assumptions of Proposition 3 hold. Then, under the control policy $\pi^{\rm MPC}$ defined above, the closed loop system (19) satisfies $\sup_{t\in\mathbb{N}_{0}}\mathbb{E}_{x_{0}}\Bigl{[}\left\lVert{x_{t}}\right\rVert^{2}\Bigr{]}<\infty$. ###### Proof: Since by assumption the matrix $A$ is Schur stable, there exists a positive definite and symmetric matrix with real entries, say $P$, such that $A^{\mathsf{T}}PA-P\leqslant-\mathbf{I}_{n\times n}$. Using the system (19), at each time instant $t\in\mathbb{N}_{0}$ we have $\displaystyle\mathbb{E}_{x_{t}}\bigl{[}x_{t+1}^{\mathsf{T}}Px_{t+1}\bigr{]}=$ $\displaystyle\mathbb{E}_{x_{t}}\bigl{[}(Ax_{t}+B\bar{d}^{}_{0\|t}+Fw_{t}+r)^{\mathsf{T}}P(Ax_{t}+B\bar{d}^{}_{0\|t}+Fw_{t}+r)\bigr{]}$ $\displaystyle=x_{t}^{\mathsf{T}}A^{\mathsf{T}}PAx_{t}+2x_{t}^{\mathsf{T}}A^{\mathsf{T}}P(B\bar{d}^{}_{0\|t}+F\mu_{w_{t}}+r)$ $\displaystyle\quad+\bar{d}^{\mathsf{T}}_{0\|t}B^{\mathsf{T}}PB\bar{d}^{}_{0\|t}+r^{\mathsf{T}}Pr+2(F\mu_{w_{t}}+r)^{\mathsf{T}}PB\bar{d}^{}_{0\|t}$ $\displaystyle\quad+2r^{\mathsf{T}}PF\mu_{w_{t}}+\mathbf{tr}\\!\left(F^{\mathsf{T}}PF\Sigma_{w_{t}}\right).$ Using the fact that $\left\lVert{\bar{d}^{}_{0\|t}}\right\rVert_{\infty}\leqslant U_{\text{max}}$ (from (13)), we obtain the following bound $\displaystyle\mathbb{E}_{x_{t}}\bigl{[}x_{t+1}^{\mathsf{T}}Px_{t+1}\bigr{]}$ $\displaystyle\leq x_{t}^{\mathsf{T}}A^{\mathsf{T}}PAx_{t}+2c_{1}\left\lVert{x_{t}}\right\rVert_{\infty}+c_{2},$ where $c_{1}:=\left\lVert{A^{\mathsf{T}}P(F\mu_{w_{t}}+r)}\right\rVert_{1}+m\left\lVert{A^{\mathsf{T}}PB}\right\rVert_{\infty}U_{\max}$ and $c_{2}:=r^{\mathsf{T}}Pr+2\left\lVert{B^{\mathsf{T}}P(F\mu_{w_{t}}+r)}\right\rVert_{1}U_{\max}+m\left\lVert{B^{\mathsf{T}}PB}\right\rVert_{\infty}U_{\max}^{2}+2\|r^{\mathsf{T}}PF\mu_{w_{t}}\|+\mathbf{tr}\\!\left(F^{\mathsf{T}}PF\Sigma_{w_{t}}\right)$. Since $x_{t}^{\mathsf{T}}A^{\mathsf{T}}PAx_{t}\leqslant x_{t}^{\mathsf{T}}Px_{t}-x_{t}^{\mathsf{T}}x_{t}$, we have that $\displaystyle\mathbb{E}_{x_{t}}\bigl{[}x_{t+1}^{\mathsf{T}}Px_{t+1}\bigr{]}\leq x_{t}^{\mathsf{T}}Px_{t}-\left\lVert{x_{t}}\right\rVert^{2}+2c_{1}\left\lVert{x_{t}}\right\rVert_{\infty}+c_{2}.$ (20) For $\theta\in\;]\max\\{0,1-\lambda_{\max}(P)\\},1[$ we know that $\displaystyle-\theta\left\lVert{x_{t}}\right\rVert_{\infty}^{2}+2c_{1}\left\lVert{x_{t}}\right\rVert_{\infty}+c_{2}\leqslant 0,\quad\forall\left\lVert{x_{t}}\right\rVert_{\infty}>r,$ where $r:=\frac{1}{\theta}\bigl{(}c_{1}+\mathchoice{{\hbox{$\displaystyle\sqrt{c_{1}^{2}+c_{2}\theta\,}$}\lower 0.4pt\hbox{\vrule height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\textstyle\sqrt{c_{1}^{2}+c_{2}\theta\,}$}\lower 0.4pt\hbox{\vrule height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\scriptstyle\sqrt{c_{1}^{2}+c_{2}\theta\,}$}\lower 0.4pt\hbox{\vrule height=4.8611pt,depth=-3.8889pt}}}{{\hbox{$\scriptscriptstyle\sqrt{c_{1}^{2}+c_{2}\theta\,}$}\lower 0.4pt\hbox{\vrule height=3.47221pt,depth=-2.77779pt}}}\bigr{)}$. From (20) it now follows that $\mathbb{E}_{x_{t}}\bigl{[}x_{t+1}^{\mathsf{T}}Px_{t+1}\bigr{]}\leqslant x_{t}^{\mathsf{T}}Px_{t}-(1-\theta)\left\lVert{x_{t}}\right\rVert^{2},\forall\left\lVert{x_{t}}\right\rVert_{\infty}>r,$ whence $\mathbb{E}_{x_{t}}\bigl{[}x_{t+1}^{\mathsf{T}}Px_{t+1}\bigr{]}\leqslant\Bigl{(}1-\frac{1-\theta}{\lambda_{\text{max}}(P)}\Bigr{)}x_{t}^{\mathsf{T}}Px_{t},\,\quad\forall\left\lVert{x_{t}}\right\rVert_{\infty}>r.$ We see that the hypotheses of Lemma 8 are satisfied with $V(x):=x^{\mathsf{T}}Px$, $\lambda:=\left(1-\frac{1-\theta}{\lambda_{\text{max}}(P)}\right)$, and $K:=\bigl{\\{}x\in\mathbb{R}^{n}\big{\|}\left\lVert{x}\right\rVert_{\infty}\leqslant r\bigr{\\}}$. Since $\lambda_{\text{min}}(P)\left\lVert{x}\right\rVert^{2}\leqslant x^{\mathsf{T}}Px$, it follows that $\sup_{t\in\mathbb{N}_{0}}\mathbb{E}_{x_{0}}\Bigl{[}\left\lVert{x_{t}}\right\rVert^{2}\Bigr{]}\leqslant\frac{1}{\lambda_{\text{min}}(P)}\sup\limits_{t\in\mathbb{N}_{0}}\mathbb{E}_{x_{0}}\bigl{[}V(x_{t})\bigr{]}<\infty,$ which completes the proof. ∎ ### IV-B RHC Case In the RHC implementation is also iterative in nature, however instead of recalculating the gains at each time instant the optimization problem is solved every $kN$ steps, where $k\in\mathbb{N}_{0}$. The resulting optimal control policy (applied over a horizon $N$) is given by $\pi_{kN}^{}=\bar{G}_{kN}^{}\bar{\varphi}(\bar{F}\bar{w})+\bar{d}_{kN}^{}$, where again the control gains depend implicitly on the initial condition $x_{kN}$, i.e., $\bar{G}_{kN}^{}=\bar{G}_{kN}^{}(x_{kN})$ and $\bar{d}_{kN}^{}=\bar{d}_{kN}^{}(x_{kN})$. Therefore, the optimal policy is given by $\pi^{\rm RHC}=\left(\pi_{0}^{},\pi_{N}^{},\cdots\right)$. For $\ell=1,\cdots,N$, the resulting closed-loop system over horizon $N$ is given by $x_{kN+\ell}=A^{\ell}x_{kN}+\bar{B}_{\ell}\bar{G}^{}_{kN}\bar{\varphi}(\bar{F}\bar{w})+\bar{B}_{\ell}\bar{d}^{}_{kN}+\bar{D}_{\ell}\bar{F}\bar{w}+\bar{D}_{\ell}\bar{r},$ (21) where $k\in\mathbb{N}_{0}$, and $\bar{B}_{\ell}$ and $\bar{D}_{\ell}$ are suitably defined matrices that are extracted from $\bar{B}$ and $\bar{D}$, respectively. ###### Proposition 10 Assume that the matrix $A$ is Schur stable and the assumptions of Proposition 3 hold. Then, under the control policy $\pi^{RHC}$ defined above, the closed loop system (21) satisfies $\sup_{t\in\mathbb{N}_{0}}\mathbb{E}_{x_{0}}\Bigl{[}\left\lVert{x_{t}}\right\rVert^{2}\Bigr{]}<\infty$. ###### Proof: Using (21) and the fact that $\mathbb{E}_{x}\left[\bar{\varphi}(\bar{F}\bar{w})\right]=0$, $\forall x\in\mathbb{R}^{n}$, we have that $\forall\,\ell=1,\cdots,N$ $\displaystyle\mathbb{E}_{x_{kN}}\bigl{[}x_{kN+\ell}^{\mathsf{T}}P_{\ell}x_{kN+l}\bigr{]}=x_{kN}^{\mathsf{T}}(A^{\ell})^{\mathsf{T}}P_{\ell}A^{\ell}x_{kN}$ $\displaystyle\,+2x_{kN}^{\mathsf{T}}(A^{\ell})^{\mathsf{T}}P_{\ell}(\bar{B}_{\ell}\bar{d}^{}_{kN}+\bar{D}_{\ell}\bar{F}\mu_{\bar{w}}+\bar{D}_{\ell}\bar{r})+\bar{r}^{\mathsf{T}}\bar{D}_{\ell}^{\mathsf{T}}P_{\ell}\bar{D}_{\ell}\bar{r}$ $\displaystyle\,+(\bar{d}^{}_{kN})^{\mathsf{T}}\bar{B}_{\ell}^{\mathsf{T}}P_{\ell}\bar{B}_{\ell}\bar{d}^{}_{kN}+2(\bar{d}^{}_{kN})^{\mathsf{T}}\bar{B}_{\ell}^{\mathsf{T}}P_{\ell}\bar{D}_{\ell}(\bar{F}\mu_{\bar{w}}+\bar{r})$ $\displaystyle\,+2\mu_{\bar{w}}^{\mathsf{T}}\bar{F}^{\mathsf{T}}\bar{D}_{\ell}^{\mathsf{T}}P_{\ell}\bar{D}_{\ell}\bar{r}+\mathbf{tr}\\!\left((\bar{G}^{}_{kN})^{\mathsf{T}}\bar{B}_{\ell}^{\mathsf{T}}P_{\ell}\bar{B}_{\ell}\bar{G}_{kN}^{}\Lambda_{1}\right)$ $\displaystyle\,+2\mathbf{tr}\\!\left((\bar{G}^{}_{kN})^{\mathsf{T}}\bar{B}_{\ell}^{\mathsf{T}}P_{\ell}\bar{D}_{\ell}\bar{F}\Lambda_{2}\right)+\mathbf{tr}\\!\left(\bar{F}^{\mathsf{T}}D_{\ell}^{\mathsf{T}}P_{\ell}D_{\ell}\bar{F}\Sigma_{\bar{w}}\right).$ Using the fact that $\left\lVert{\bar{d}^{}_{kN}}\right\rVert_{\infty}\leq U_{\rm max}$ and $\left\lVert{\bar{G}_{kN}^{}}\right\rVert_{\infty}\leq U_{\rm max}/\phi_{\rm max}$ (from (13)), we obtain the following bound $\displaystyle\mathbb{E}_{x_{kN}}\bigl{[}x_{kN+\ell}^{\mathsf{T}}P_{\ell}x_{kN+\ell}\bigr{]}$ $\displaystyle\quad\leq x_{kN}^{\mathsf{T}}(A^{\ell})^{\mathsf{T}}P_{\ell}A^{\ell}x_{kN}+2c_{1\ell}\left\lVert{x_{kN}}\right\rVert_{\infty}+c_{2\ell},$ where $c_{1\ell}:=\left\lVert{(A^{\ell})^{\mathsf{T}}P_{\ell}\bar{D}_{\ell}(\bar{F}\mu_{\bar{w}}+\bar{r})}\right\rVert_{1}+m\left\lVert{(A^{\ell})^{\mathsf{T}}P_{\ell}\bar{B}_{\ell}}\right\rVert_{\infty}U_{\rm max}$ and $c_{2\ell}:=\bar{r}^{\mathsf{T}}\bar{D}_{\ell}^{\mathsf{T}}P\bar{D}_{\ell}\bar{r}+2\left\lVert{(\bar{B}_{\ell})^{\mathsf{T}}P_{\ell}\bar{D}_{\ell}(\bar{F}\mu_{\bar{w}}+\bar{r})}\right\rVert_{1}U_{\max}+m\left\lVert{\bar{B}_{\ell}^{\mathsf{T}}P_{\ell}\bar{B}_{\ell}}\right\rVert_{\infty}U_{\max}^{2}+2\|\bar{r}^{\mathsf{T}}\bar{D}_{\ell}^{\mathsf{T}}P_{\ell}\bar{D}_{\ell}\bar{F}\mu_{\bar{w}}\|+\mathbf{tr}\\!\left(\bar{F}^{\mathsf{T}}\bar{D}_{\ell}^{\mathsf{T}}P_{\ell}\bar{D}_{\ell}\bar{F}\Sigma_{\bar{w}}\right)+\max\limits_{\left\lVert{\bar{G}^{}_{kN}}\right\rVert_{\infty}\leq U_{\max}/\phi_{\max}}\big{[}\mathbf{tr}\\!\left(\bar{G}^{\mathsf{T}}_{kN}\bar{B}_{\ell}^{\mathsf{T}}P_{\ell}\bar{B}_{\ell}\bar{G}^{}_{kN}\Lambda_{1}\right)+2\mathbf{tr}\\!\left(\bar{G}^{\mathsf{T}}_{kN}\bar{B}_{\ell}^{\mathsf{T}}P_{\ell}\bar{D}_{\ell}\bar{F}\Lambda_{2}\right)\big{]}$. Again, since $A$ is a Schur stable matrix (and hence $A^{\ell}$) there exists a matrix $P_{\ell}=P_{\ell}^{\mathsf{T}}>0$ with real valued entries that satisfies $(A^{\ell})^{\mathsf{T}}P_{\ell}A^{\ell}-P_{\ell}\leq-\mathbf{I}_{n\times n}$, and its eigenvalues are real. Then we have $x_{kN}^{\mathsf{T}}(A^{\ell})^{\mathsf{T}}P_{\ell}A^{\ell}x_{kN}\leqslant x_{kN}^{\mathsf{T}}P_{\ell}x_{kN}-x_{kN}^{\mathsf{T}}x_{kN}$. Therefore, $\displaystyle\mathbb{E}_{x_{kN}}\bigl{[}x_{kN+\ell}^{\mathsf{T}}P_{\ell}x_{kN+\ell}\bigr{]}$ $\displaystyle\leq x_{kN}^{\mathsf{T}}P_{\ell}x_{kN}-\left\lVert{x_{kN}}\right\rVert^{2}$ $\displaystyle+2c_{1\ell}\left\lVert{x_{kN}}\right\rVert_{\infty}+c_{2\ell}.$ (22) For $\theta_{\ell}\in\;]\max\\{0,1-\lambda_{\max}(P_{\ell})\\},1[$ we know that $\displaystyle-\theta_{\ell}\left\lVert{x_{kN}}\right\rVert_{\infty}^{2}+2c_{1\ell}\left\lVert{x_{kN}}\right\rVert_{\infty}+c_{2\ell}\leqslant 0,\,\forall\left\lVert{x_{kN}}\right\rVert_{\infty}>r_{\ell},$ where $r_{\ell}:=\frac{1}{\theta_{\ell}}\bigl{(}c_{1\ell}+\mathchoice{{\hbox{$\displaystyle\sqrt{c_{1\ell}^{2}+c_{2\ell}\theta_{\ell}\,}$}\lower 0.4pt\hbox{\vrule height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\textstyle\sqrt{c_{1\ell}^{2}+c_{2\ell}\theta_{\ell}\,}$}\lower 0.4pt\hbox{\vrule height=6.94444pt,depth=-5.55559pt}}}{{\hbox{$\scriptstyle\sqrt{c_{1\ell}^{2}+c_{2\ell}\theta_{\ell}\,}$}\lower 0.4pt\hbox{\vrule height=4.8611pt,depth=-3.8889pt}}}{{\hbox{$\scriptscriptstyle\sqrt{c_{1\ell}^{2}+c_{2\ell}\theta_{\ell}\,}$}\lower 0.4pt\hbox{\vrule height=3.47221pt,depth=-2.77779pt}}}\bigr{)}$. From (22) it now follows that $\mathbb{E}_{x_{kN}}\bigl{[}x_{kN+\ell}^{\mathsf{T}}P_{\ell}x_{kN+\ell}\bigr{]}\leqslant x_{kN}^{\mathsf{T}}P_{\ell}x_{kN}-(1-\theta_{\ell})\left\lVert{x_{kN}}\right\rVert^{2},\forall\left\lVert{x_{kN}}\right\rVert_{\infty}>r_{\ell},$ whence $\displaystyle\mathbb{E}_{x_{kN}}\bigl{[}x_{kN+\ell}^{\mathsf{T}}P_{\ell}x_{kN+\ell}\bigr{]}\leqslant\lambda_{\ell}x_{kN}^{\mathsf{T}}P_{\ell}x_{kN},\,\forall\left\lVert{x_{kN}}\right\rVert_{\infty}>r_{\ell},$ (23) where $\lambda_{\ell}:=\Bigl{(}1-\frac{1-\theta}{\lambda_{\text{max}}(P_{\ell})}\Bigr{)}$. Define $\lambda:=\max\limits_{\ell=1,\cdots,N-1}\lambda_{\ell}$, $r^{\prime}:=\max\limits_{\ell=1,\cdots,N-1}r_{\ell}$, $\overline{\lambda}:=\max\limits_{\ell=1,\dots,N-1}\lambda_{\max}(P_{\ell})$, $\underline{\lambda}:=\min\limits_{\ell=1,\dots,N-1}\lambda_{\min}(P_{\ell})$, then we can obtain using (23) the conservative bound $\mathbb{E}_{x_{kN}}\bigl{[}x_{kN+\ell}^{\mathsf{T}}P_{N}x_{kN+\ell}\bigr{]}\leqslant\lambda^{\prime}x_{kN}^{\mathsf{T}}P_{N}x_{kN},\,\forall\left\lVert{x_{kN}}\right\rVert_{\infty}>r^{\prime}$ for every $\ell=1,\ldots,N-1$, where $\lambda^{\prime}:=\lambda\frac{\overline{\lambda}\lambda_{\max}(P_{N})}{\underline{\lambda}\lambda_{\min}(P_{N})}$, and the $N$-step bound $\displaystyle\mathbb{E}_{x_{kN}}\bigl{[}x_{(k+1)N}^{\mathsf{T}}P_{N}x_{(k+1)N}\bigr{]}$ $\displaystyle\leqslant\lambda_{N}x_{kN}^{\mathsf{T}}P_{N}x_{kN},$ $\displaystyle\qquad\forall\left\lVert{x_{kN}}\right\rVert_{\infty}>r_{N}.$ (24) Let $V_{N}(x):=x^{\mathsf{T}}P_{N}x$. Now, following the same reasoning as in Lemma 8, we can establish the following bound (for $k\in\mathbb{N}_{0}$, $\ell=1,\dots,N-1$) $\displaystyle\mathbb{E}_{x}\bigl{[}V_{N}(x_{kN+\ell})\bigr{]}=\mathbb{E}_{x}\bigl{[}\mathbb{E}[V_{N}(x_{kN+\ell})\|x_{kN}]\bigr{]}$ $\displaystyle\qquad\leq\mathbb{E}_{x}\bigl{[}\mathbb{E}[\lambda^{\prime}V_{N}(x_{kN})+b^{\prime}\mathbf{1}_{K^{\prime}}(x_{kN})]\bigr{]}$ $\displaystyle\qquad\leq\mathbb{E}_{x}\bigl{[}\mathbb{E}[\lambda^{\prime}\mathbb{E}[V_{N}(x_{kN})\|x_{(k-1)N}]+b^{\prime}\mathbf{1}_{K^{\prime}}(x_{kN})]\bigr{]}$ $\displaystyle\qquad\leq\mathbb{E}_{x}\bigl{[}\mathbb{E}[\lambda^{\prime}\mathbb{E}[\lambda_{N}V_{N}(x_{(k-1)N})+b\mathbf{1}_{K_{N}}(x_{(k-1)N})]$ $\displaystyle\qquad\quad+b^{\prime}\mathbf{1}_{K^{\prime}}(x_{kN})]\bigr{]}$ $\displaystyle\qquad\leq\lambda^{\prime}\lambda_{N}^{k}V_{N}(x)+\sum_{i=0}^{k-1}\lambda_{N}^{k-1-i}b\mathbb{E}_{x}\bigl{[}\mathbf{1}_{K_{N}}(x_{iN})\bigr{]}$ $\displaystyle\qquad\quad+b^{\prime}\mathbb{E}_{x}\bigl{[}\mathbf{1}_{K^{\prime}}(x_{kN})\bigr{]}$ $\displaystyle\qquad\leq\lambda^{\prime}\lambda_{N}^{k}V_{N}(x)+\frac{b(1-\lambda_{N}^{k})}{1-\lambda_{N}}+b^{\prime},$ (25) where $b:=\sup\limits_{x\in K}\mathbb{E}_{x}\bigl{[}V_{N}(x_{N})\bigr{]}$, $b^{\prime}:=\sup\limits_{x\in K^{\prime}}\mathbb{E}_{x}\bigl{[}V_{N}(x_{l})\bigr{]}$ for $\ell=1,\cdots,N-1$, $K_{N}:=\bigl{\\{}\xi\in\mathbb{R}^{n}\big{\|}\left\lVert{\xi}\right\rVert_{\infty}\leq r_{N}\bigr{\\}}$, and $K^{\prime}:=\bigl{\\{}\xi\in\mathbb{R}^{n}\big{\|}\left\lVert{\xi}\right\rVert_{\infty}\leq r^{\prime}\bigr{\\}}$. Note that the conditioning in the steps of (25) is done every $N$ steps as the problem is _not Markovian_ except then. Therefore, it follows from (25) that, $\forall\,t:=kN+\ell$, $\displaystyle\sup\limits_{t\in\mathbb{N}_{0}}\mathbb{E}_{x}\bigl{[}\left\lVert{x_{t}}\right\rVert^{2}\bigr{]}$ $\displaystyle\leq\frac{1}{\lambda_{\min}(P_{N})}\sup\limits_{t\in\mathbb{N}_{0}}\mathbb{E}_{x}\bigl{[}V_{N}(x_{kN+l})\bigr{]},$ $\displaystyle\leq\frac{1}{\lambda_{\min}(P_{N})}\left(\lambda^{\prime}\lambda_{N}^{k}V_{N}(x)+\frac{b}{1-\lambda_{N}}+b^{\prime}\right)<\infty$ (26) which completes the proof. ∎ ### IV-C Input-to-state Stability Input-to-state stability (iss) is an interesting and important qualitative property of systems, dealing with input-output behavior. In the deterministic context [25] it generalizes the well-known bounded input bounded output (BIBO) property of linear systems [26, p. 490]. iss provides a description of the behavior of a system subjected to bounded inputs. Here we are interested in a stochastic variant of input-to-state stability; see e.g., [27, 28] for other possible definitions and ideas (primarily in continuous-time). One possible way to measure the strength of stochastic inputs is in terms of their covariances; sometimes their moment generating functions are also employed. For Gaussian noise it is customary to consider a suitable norm of the covariance matrix as a measure of its strength. The deterministic version of input-to-state stability deals with $\mathcal{L}_{\infty}$-to-$\mathcal{L}_{\infty}$ gain from the input to the state of a system. We consider the linear system (1), and establish a natural iss-type property from the control and the noise inputs to the state of the system (1), under both the MPC and the RHC strategies. ###### Definition 11 The system (1) is _input-to-state stable in $\mathcal{L}_{1}$_ if there exist functions $\beta\in\mathcal{KL}$ and $\alpha,\gamma_{1},\gamma_{2}\in\mathcal{K}_{\infty}$ such that for every initial condition $x_{0}\in\mathbb{R}^{n}$ and $\forall t\in\mathbb{N}_{0}$ we have $\mathbb{E}_{x_{0}}\bigl{[}\alpha(\left\lVert{x_{t}}\right\rVert)\bigr{]}\leqslant\beta(\left\lVert{x_{0}}\right\rVert,t)+\gamma_{1}\Bigl{(}\sup_{s\in\mathbb{N}_{0}}\left\lVert{u_{s}}\right\rVert_{\infty}\Bigr{)}+\gamma_{2}\bigl{(}\left\lVert{\Sigma}\right\rVert^{\prime}\bigr{)},$ (27) where $\left\lVert{\cdot}\right\rVert^{\prime}$ is an appropriate matrix norm.$\Diamond$ One difference with the deterministic definition of iss is immediately evident, namely, the presence of the function $\alpha$ inside the expectation in (27). It turns out that often it is more natural to arrive at an estimate of $\mathbb{E}_{x_{0}}\bigl{[}\alpha(\left\lVert{x_{t}}\right\rVert)\bigr{]}$ for some $\alpha\in\mathcal{K}_{\infty}$ than an estimate of $\mathbb{E}_{x_{0}}[\left\lVert{x_{t}}\right\rVert]$. Moreover, in case $\alpha$ is convex, Jensen’s inequality [29, p. 348] shows that such an estimate implies an estimate of $\mathbb{E}_{x_{0}}[\left\lVert{x_{t}}\right\rVert]$. The following proposition can be easily established with the aid of Proposition 9 and Proposition 10. ###### Proposition 12 The closed-loop systems (19) and (21) are input-to-state stable in $\mathcal{L}_{1}$. $\blacksquare$ The proof is omitted for space limitations. ## V Numerical Example Let us consider the system (1) with some generic matrices $A=\left[\begin{matrix}0.8&0.1&0.01\\\ 0.3&0.3&0.06\\\ 0.09&0.02&0.5\end{matrix}\right]$, $B=\left[\begin{matrix}1\\\ 2\\\ 0.5\end{matrix}\right]$, $F=\mathbf{I}_{3\times 3}$, and $r=\mathbf{0}_{3\times 1}$. We simulate the system starting from $50$ different initial conditions, all of which are sampled according to a uniform distribution over $[-50,50]^{3}$. The noise inputs are independent and identically sampled according to a normal distribution, $w\sim\mathcal{N}(0,4\mathbf{I}_{3\times 3})$, the noise saturation function is chosen as in Example 6 with $\phi_{\max}=5$, and the input saturation bound $U_{\max}=10$. The optimization gain matrices are chosen to be $Q_{i}=3\mathbf{I}_{3\times 3}$ and $R_{i}=2\mathbf{I}_{1\times 1}$, $\forall i$, and the optimization horizon $N=6$. The optimization matrices are given by $\Lambda_{1}=3.3024\mathbf{I}_{9\times 9}$, and $\Lambda_{2}=0.7846\mathbf{I}_{9\times 9}$. We used the cvx solver [21] to handle the optimization problem (13). The results for the MPC implementation are shown in Figure 1(a), and those for the RHC implementation are shown in Figure 1(b), for the full state evolution over a horizon of 40 time steps. Finally, it is interesting to note that the MPC and RHC _average_ performance indices over the 50 different runs are given by 3985 and 4327, respectively. (a) MPC implementation (b) RHC implementation Figure 1: MPC and RHC algorithms corresponding to the system in §V. The plots correspond to the aforementioned algorithms each run from 50 identical initial conditions distributed uniformly over $[-50,50]$. ## VI Conclusions In this paper, we provided a tractable optimization program that solves the stochastic Model Predictive Control and Rolling Horizon Control problems, while guaranteeing the satisfaction of hard bounds on the control input. We have showed that in both cases the resulting closed-loop process has bounded variance. We demonstrated that both implementations enjoy some qualitative notion of stochastic input-to-state stability. We provided several examples in which crucial matrices in our optimization program can be calculated off-line. Future direction for this research is aimed at lifting the current feedback strategy onto general vector spaces. ## References * [1] D. Chatterjee, E. Cinquemani, G. Chaloulos, and J. Lygeros, “Stochastic optimal control up to a hitting time: optimality and rolling-horizon implementation,” 2008. [Online]. Available: http://arxiv.org/abs/0806.3008 * [2] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: stability and optimality,” _Automatica_ , vol. 36, no. 6, pp. 789–814, Jun 2000. * [3] A. Bemporad and M. Morari, “Robust model predictive control: A survey,” _Robustness in Identification and Control_ , vol. 245, pp. 207–226, 1999\. * [4] J. M. Maciejowski, _Predictive Control with Constraints_. Prentice Hall, 2001. * [5] F. Blanchini, “Set invariance in control,” _Automatica_ , vol. 35, no. 11, pp. 1747–1767, 1999. * [6] D. Bertsimas and D. B. Brown, “Constrained stochastic LQC: a tractable approach,” _IEEE Transactions on Automatic Control_ , vol. 52, no. 10, pp. 1826–1841, 2007. * [7] J. Primbs, “A soft constraint approach to stochastic receding horizon control,” in _Proceedings of the 46th IEEE Conference on Decision and Control_ , 2007, pp. 4797 – 4802. * [8] J. A. Primbs and C. H. Sung, “Stochastic receding horizon control of constrained linear systems with state and control multiplicative noise,” _IEEE Trans. Automatic Control_ , 2008, to appear. * [9] M. Cannon, B. Kouvaritakis, and X. Wu, “Probabilistic constrained MPC for systems with multiplicative and additive stochastic uncertainty,” in _IFAC World Congress_ , Seoul, Korea, July 2008. * [10] F. Oldewurtel, C. Jones, and M. Morari, “A tractable approximation of chance constrained stochastic MPC based on affine disturbance feedback,” in _Conference on Decision and Control, CDC_ , Cancun, Mexico, Dec. 2008. [Online]. Available: http://control.ee.ethz.ch/index.cgi?page=publications;action=details;id%=3118 * [11] I. Batina, “Model predictive control for stochastic systems by randomized algorithms,” Ph.D. dissertation, Technische Universiteit Eindhoven, 2004. * [12] M. Maciejowski, A. Lecchini, and J. Lygeros, “NMPC for complex stochastic systems using Markov Chain Monte Carlo,” in _International Workshop on Assessment and Future Directions of Nonlinear Model Predictive Control_ , ser. Lecture Notes in Control and Information Sciences, vol. 358/2007. Stuttgart, Germany: Springer, 2005, pp. 269–281. * [13] A. A. Stoorvogel, A. Saberi, and S. Weiland, “On external semi-global stochastic stabilization of linear systems with input saturation,” 2006, submitted. [Online]. Available: http://homepage.mac.com/a.a.stoorvogel/subm03.pdf * [14] M. Agarwal, E. Cinquemani, D. Chatterjee, and J. Lygeros, “On convexity of stochastic optimization problems with constraints,” in _European Control Conference_ , 2009, submitted. [Online]. Available: http://control.ee.ethz.ch/index.cgi?page=publications;action=details;id%=3271 * [15] J. M. Alden and R. L. Smith, “Rolling horizon procedures in nonhomogeneous Markov decision processes,” _Operations Research_ , vol. 40, no. suppl. 2, pp. S183–S194, May-Jun. 1992. * [16] Y. D. Yang, E. D. Sontag, and H. J. Sussmann, “Global stabilization of linear discrete-time systems with bounded feedback,” _Systems and Control Letters_ , vol. 30, no. 5, pp. 273–281, 1997. * [17] A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski, “Adjustable robust solutions of uncertain linear programs,” _Mathematical Programming_ , vol. 99, no. 2, pp. 351–376, 2004. * [18] P. J. Goulart, E. C. Kerrigan, and J. M. Maciejowski, “Optimization over state feedback policies for robust control with constraints,” _Automatica J. IFAC_ , vol. 42, no. 4, pp. 523–533, 2006. * [19] D. Luenberger, _Optimization by Vector Space Methods_. J. Wiley & Sons, 1969. * [20] S. Boyd and L. Vandenberghe, _Convex Optimization_. Cambridge: Cambridge University Press, 2004, sixth printing with corrections, 2008. * [21] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming (web page and software),” http://stanford.edu/~boyd/cvx, December 2000. * [22] M. Abramowitz and I. A. Stegun, _Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables_ , ser. National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964, vol. 55. * [23] S. P. Meyn and R. L. Tweedie, _Markov Chains and Stochastic Stability_. London: Springer-Verlag, 1993\. * [24] S. Foss and T. Konstantopoulos, “An overview of some stochastic stability methods,” _Journal of Operations Research Society of Japan_ , vol. 47, no. 4, pp. 275–303, 2004. * [25] Z.-P. Jiang and Y. Wang, “Input-to-state stability for discrete-time nonlinear systems,” _Automatica_ , vol. 37, no. 6, pp. 857–869, June 2001. * [26] P. J. Antsaklis and A. N. Michel, _Linear Systems_. Boston, MA: Birkhäuser Boston Inc., 2006. * [27] V. S. Borkar, “Uniform stability of controlled Markov processes,” in _System theory: modeling, analysis and control (Cambridge, MA, 1999)_ , ser. Kluwer International Series in Engineering Computer Science. Boston, MA: Kluwer Academic Publishers, 2000, vol. 518, pp. 107–120. * [28] J. Spiliotis and J. Tsinias, “Notions of exponential robust stochastic stability, ISS and their Lyapunov characterization,” _International Journal of Robust and Nonlinear Control_ , vol. 13, no. 2, pp. 173–187, 2003. * [29] R. M. Dudley, _Real Analysis and Probability_ , ser. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2002, vol. 74, revised reprint of the 1989 original.	arxiv-papers	2009-02-23T16:17:57	2024-09-04T02:49:00.809498	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Peter Hokayem, Debasish Chatterjee, John Lygeros", "submitter": "Debasish Chatterjee", "url": "https://arxiv.org/abs/0902.3944" }
0902.3986	SLAC-TN-09-002 B A B AR Note # 327 October 1996 Noise in a Calorimeter Readout System Using Periodic Sampling Walter R. Innes SLAC National Accelerator Laboratory Fourier transform analysis of the calorimeter noise problem gives quantitative results on a) the time-height correlation, b) the effect of background on optimal shaping and on the ENC, c) sampling frequency requirements, and d) the relation between sampling frequency and the required quantization error. ## 1 Introduction Noise in calorimeter readout electronics has been treated in some detail [Radeka, Haller, Dow]. However, since each author builds the details of their particular situation into their analysis, it is worthwhile to review the subject in light of our own experiment. Most previous studies assume that a single measurement will be made at a well determined time. Furthermore they assume that filtering preceding the digitization is limited to relatively simple analog devices. Since we have chosen to solve our lack of knowledge of the event time by recording periodic samples, applying digital signal processing techniques is natural. This also raises questions about the required sampling rate and the quantization error that don’t occur in the usual scheme. Electronics noise and filters are usually analyzed using Fourier transforms [Ambrozny, Humphreys, Papoulis]. In the frequency domain, filtering, differentiation, integration, and time shifting are all represented by multiplications. Stochastic noise is most easily represented by its power spectral density (which is the Fourier transform of the more complicated auto- correlation function in the time domain). Many questions such as the signal to noise ratio for particular parameters and filter design can be handled entirely in the frequency domain. ## 2 The Input Circuit Model Figure 1: Simplified photodiode-preamp circuit Figure 2: Model for the front end of the calorimeter readout. $i_{s}$ is the current due to the signal and is a function of time. $I_{d}$ is the spectral current noise generator corresponding to the shot noise of the photodiode. $I_{b}$ is the equivalent current noise caused by background photons from lost particles. $I_{f}$ is the FET input current noise, and $V_{n}$ is the FET input voltage noise. $C_{s}$ is the capacitance of the photodiode and $C_{iss}$ is the common source input capacitance of the FET. Figure 1 shows a simplified version of the calorimeter input circuitry. Figure 2 shows a noise equivalent representation of the same circuit. The right hand portion represents the input FET. This is one of many possible equivalent representations [vanderZiel, page 29][Ambrozny, page 137]. This particular one has a relatively simple connection between the physical processes which generate noise and the elements of the representation. The spectral densities of the noise generators will correspond to the those given in manufacturers specification sheets. ### 2.1 The photo-diode $i_{s}$ is a current generator corresponding to the signal generated by the interactions in the calorimeter. It will be represented by the function: $i_{s}(t)=\frac{1}{\tau_{s}}e^{-t/\tau_{s}}H(t).$ (1) $\tau_{s}$ is the decay time of the CsI(Tl) scintillation and $H(t)$ is the Heaviside unit step function. $\tau_{s}$ is 0.94$\,\mu\mbox{s}$ for CsI(Tl). The Fourier transform of $i(t)$ is $I_{s}(\omega)=\int_{-\infty}^{\infty}{i(t)e^{-j\omega t}\,dt}=\frac{1}{1+j\omega\tau_{s}}\ .$ (2) The spectral power density of the signal is: $I_{s}^{}(\omega)I_{s}(\omega)=\frac{1}{1+\omega^{2}\tau_{s}^{2}}.$ (3) For characteristic times represented as $\tau_{x}$, I will use the notation that the associated radial velocity is $\omega_{x}\equiv 1/\tau_{x}$, the associated frequency is $f_{x}\equiv\omega_{x}/(2\pi)$, and the associated period is $T_{x}\equiv 1/f_{x}$. Using this notation we can write: $I_{s}(\omega)=\frac{-j\omega_{s}}{\omega-j\omega_{s}}\ \ {\rm and}\ \ I_{s}^{}(\omega)I_{s}(\omega)=\frac{\omega_{s}^{2}}{\omega^{2}+\omega_{s}^{2}}.$ (4) $I_{d}$ is the current generator corresponding to the shot noise caused by the leakage current in the photodiode. This has a “white” (independent of $\omega$) spectrum with the value $I_{d}^{2}=2eI_{leak}$ (units of current squared per bandwidth). The order of magnitude for $I_{leak}$ is nanoamperes. We’ll assume our diode has a leakage of 4 nA. Since we average two diodes we can use the equivalent leakage of 2 nA for a single diode giving $I_{d}=25\mbox{\,fA}/\rm\sqrt{\mbox{\,Hz}}$. $C_{s}$ is the capacitance of the back-biased photodiode. A typical value is 80 pF, which we’ll use in our estimates. A photodiode with these properties is the 1N2744. As much as possible, the discussion will be restricted to the electronic regime. However, some effects depend on the efficiency of the conversion of calorimeter shower energy into charge. When this number is required, I will use the value of 3000 photo-electrons per MeV in each diode. ### 2.2 The FET $C_{iss}$ is the FET common source input capacitance. It can range from a few picofarads to tens of picofarads depending on the choice of FET. We will return to this choice later. In the circuit it functions in parallel with $C_{s}$. Not shown in the figure since they are much less than $C_{s}$ are the feedback capacitance $C_{f}$ and the calibration injection capacitance $C_{cal}$. I will call the sum $C_{s}+C_{iss}+C_{f}+C_{cal}$ simply $C$. $I_{f}$ is a noise generator corresponding the input current noise of the FET. It is primarily due to the gate leakage current, although for very large gate areas the effect the thermal noise of the real part of the input impedance should be checked. The shot noise acts just like the diode shot noise and the two leakage currents may be added. In practice the FET gate leakage current is of the order of picoamperes and is negligible compared to the diode leakage current. Let $I_{n}^{2}=I_{d}^{2}+I_{f}^{2}\approx I_{d}^{2}$. $V_{n}$ is a voltage noise generator representing the input voltage noise of the FET. This is primarily due the thermal noise in the FET channel. Many authors put a formula, which shows the dependence of this noise on the temperature and the forward transconductance, into their equations [vanderZiel, page 75]. I prefer to keep $V_{n}$ as an explicit term. The value of $V_{n}$ is usually given on specification sheets and is also easy to measure. For junction FETs, $V_{n}$ is approximately independent of frequency over the range of interest to us. Typical values are 1 to 2 nV/$\rm\sqrt{\mbox{\,Hz}}$. In order to be more specific I’ll take as an example the SANYO 2SK932 JFET. $C_{iss}$ is 20 pF and $V_{n}$ is $\approx 0.7\mbox{\,nV}/\rm\sqrt{\mbox{\,Hz}}$. Because we average the outputs two diode-FET-preamp combinations on each crystal, we divide by $\sqrt{2}$ to get 0.5 nV/$\rm\sqrt{\mbox{\,Hz}}$. This is a good FET but not optimally matched to our problem. We’ll return to this issue after we have developed the signal to noise formulas. Figure 3: The amplitude spectra of the various noise sources expressed as input current densities. For the background, nominal conditions are assumed and no energy cutoff is used. Note that the signal has the same spectrum as the background noise. ### 2.3 Background $I_{b}$ is a current noise generator representing the sea of photons generated by particles lost from the beam. This noise resembles shot noise in that near impulsive elements arrive at random times. It differs in that the pulse shape is not a delta function but has the shape of the signal and there is a distribution of pulse areas instead of just the electron charge.111In radar theory, noise with these properties is called “clutter.” If we return to the derivation of the shot noise formula [Ambrozny, page 81], we see that the first difference can be treated by giving this noise the power spectrum of the signal (the Fourier transform of each pulse is the same as that of the signal except for phase and magnitude). The second is handled by noting that the shot noise formula is really a pulse area squared times a rate: $I_{d}^{2}=2e^{2}f_{leak}$. $e$ is the size of the pulse and $f_{leak}$ is the mean rate of their occurrence. Therefore we can get the low frequency limit of the noise spectral density of the background by doing the integral: $I_{b}^{2}=\int_{0}^{\infty}{2q^{2}B(q)\,dq}$ (5) where $B$ is the rate of background events per pulse area interval per time interval. The value of this integral can be inferred from the statistics of histograms such as those in B A B AR Technical Design Report Figure 12-9 [BaBar]. Here I use more recent calculations of the background [Marsiske]. Assuming $3000\rm\,pe/\mbox{\,Me\kern-0.80002ptV}/diode$, the result for an average crystal is $I_{b}=460\mbox{\,fA}/\rm\sqrt{\mbox{\,Hz}}$ at nominal background. For convenience I’ll use the symbol $I_{b}$ to represent this low frequency limit and put the frequency dependence into the formulas explicitly. Since $I_{n}$ and $V_{n}$ are approximately independent of frequency, I can then treat $I_{b}$, $I_{n}$, and $V_{n}$ as constants. Note that individual crystals can differ from the average by factors as large as $4$. In particular, compared an average barrel crystal, an average endcap crystal has $5/7$th the hit rate, twice the average energy deposit, and an $I_{b}$ twice as large. Figure 3 summarizes all the noise sources. #### 2.3.1 Limitations The treatment of background as noise is a good description if the background signal after all processing contains significant contributions from many background events. If this is not the case, the data may fall into two classes, those showers little affected by background, and those with a large background contribution. The former will have a distribution of measured values consistent with electronic noise, while the latter may have a wider, highly skewed distribution. While the background as noise treatment may give a near optimal procedure for minimizing the rms error, it will widen the distribution for that class events unaffected by background. Depending on the physics objective, treating the background affected events as an inefficiency rather than events with errors may give the better result. How well does our case satisfy this many hit condition? In the background calculation in the preceding paragraph, an average crystal had $0.11\,\rm hits/crystal/\,\mu\mbox{s}$. Assuming a window of $2\,\mu\mbox{s}$ and a sum of 25 crystals, this implies 5.5 background hits contributing to the measurement during nominal conditions. This is enough to give reasonably Guassian behavior. However, much of $I_{b}$ is generated by the high energy tail of the background distribution. If we count down from high energies until get a mean number of hits of at least 2, the cut would be in the vicinity of $1\mbox{\,Me\kern-0.80002ptV}$. At 10 times nominal this cut would be close to $3\mbox{\,Me\kern-0.80002ptV}$. Another consideration is that an approach which tries to separate a background tail from other noise is hardly likely to succeed if the magnitude of the background contribution is of the same size as other errors. The size of the intrinsic error depends on the size of the signal. For a $100\mbox{\,Me\kern-0.80002ptV}$ shower the expected error is $\approx 2\mbox{\,Me\kern-0.80002ptV}$. Assuming that background contributions must be at least $3\mbox{\,Me\kern-0.80002ptV}$ before they can be separated from noise seems conservative. If only background hits with energies below $3\mbox{\,Me\kern-0.80002ptV}$ are included, $I_{b}$ falls to $160\mbox{\,fA}/\rm\sqrt{\mbox{\,Hz}}$. The background rate falls to $0.10\,\rm hits/crystal/\,\mu\mbox{s}$. ## 3 Theorems ### 3.1 $S/N$ The Radar Handbook [Skolnik, page 4-11] reviews the theory of finding a known signal in noise from the starting point of doing a least square fit to a finite set of measurements. Under fairly general conditions this is equivalent to the maximum likelihood fit and is the best that can be done. The treatment proceeds by taking the continuous infinite time limit which turns the usual matrix equations into integral equations. These are then solved by applying Fourier transforms. For a signal $i(t,a_{1},...,a_{n})$ with Fourier transform $I(\omega,a_{1},...,a_{n})$ the elements of the inverse of the error matrix for $a_{i}$ are given by: $[{\bf V}^{-1}]_{ik}=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\left(\left(\frac{\partial I(\omega)}{\partial a_{i}}\frac{\partial I^{}(\omega)}{\partial a_{k}}\right)/N(\omega)\right)d\omega},$ (6) where $N$ is the double sided noise power spectral density. This matrix is known as the information matrix and also as the signal to noise ratio ($S/N$) matrix. In our case the signal has two parameters, amplitude and time offset: $i(t)=ai_{0}(t-t_{0}).$ (7) This has the Fourier transform $I(\omega)=aI_{0}(\omega)e^{-j\omega t_{0}}.$ (8) If we define $I_{0}$ such that the actual value of $a$ is 1 we find $\displaystyle\frac{\partial I}{\partial a}$ $\displaystyle=$ $\displaystyle I(\omega)\mbox{,\rm\hskip 72.26999pt and}$ (9) $\displaystyle\frac{\partial I}{\partial t_{0}}$ $\displaystyle=$ $\displaystyle-j\omega I(\omega).$ (10) Thus the matrix of integrands in the $S/N$ formula is $\left[\begin{array}[]{cc}~{}~{}~{}1{}{}{}&~{}~{}j\omega{}{}\\\ -j\omega&\omega^{2}\end{array}\right]\frac{\|I(\omega)\|^{2}}{N(\omega)}$ (11) Because both $\|I(\omega)\|^{2}$ and $N(\omega)$ are even in $\omega$, the integral of the off-diagonal elements is 0. Since the matrix is diagonal, inverting it to get the error matrix ${\bf V}$ is trivial. If $i(t)$ is normalized to unit area, the $aa$ element of ${\bf V}$ is the square of the equivalent noise charge (ENC) and the $t_{0}t_{0}$ element is the square of product of the pulse height and the time error (ENTC). Since there is no correlation between these errors, a priori knowledge of the time does not improve the determination of the amplitude. The final formulas for the signal to noise are: $\left[\frac{S}{N}\right]_{aa}=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\frac{\|I(\omega)\|^{2}}{N(\omega)}\,d\omega},\\\ $ (12) and $\left[\frac{S}{N}\right]_{tt}=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\frac{\|I(\omega)\|^{2}}{N(\omega)}\,\omega^{2}d\omega}$ (13) ### 3.2 Matched filter This result for the error in the amplitude is the same that is given in many texts as the best that can be achieved using an optimal matched filter [Humphreys, page 65][Papoulis, page 135]. Assume there exists a filter such that the output peaks at $t=0$. The square of that peak value is given by: $o^{2}(0)=\left\|\frac{1}{2\pi}\int_{-\infty}^{\infty}{I(\omega)F(\omega)\,d\omega}\right\|^{2},$ (14) where $F(\omega)$ is the transfer function of the filter. The mean square noise after filtering is the same at all times and is given by $n_{o}^{2}=\frac{1}{2\pi}\int_{-\infty}^{\infty}{N(\omega)\|F(\omega)\|^{2}\,d\omega}.$ (15) The signal to noise is the ratio of $o^{2}(0)$ to $n_{o}^{2}$. Multiplying the integrand of the numerator by $\sqrt{N(\omega)}/\sqrt{N(\omega)}$ and using Schwarz’s inequality proves that the signal to noise is less than or equal to the $S/N$ derived from the least square fit method (equation 12). Furthermore, inspection of the $S/N$ equation before this manipulation shows that the equality can be achieved using a filter with a transfer function of $F(\omega)\propto\frac{I^{}(\omega)}{N(\omega)}\ .$ (16) From this I conclude that the best possible result can be achieved with a filter technique and that the required filter is readily calculated given knowledge of the shape of the signal and the spectrum of the noise. The impulse response to the optimal filter is $f(t)\propto\frac{1}{2\pi}\int_{-\infty}^{\infty}{\frac{I^{}(\omega)}{N(\omega)}\,e^{j\omega t}d\omega}\ ,$ (17) and the shape of a noiseless signal after filtering (the convolution of the signal with impulse response) is $o(t)\propto\frac{1}{2\pi}\int_{-\infty}^{\infty}{\frac{\|I(\omega)\|^{2}}{N(\omega)}\,e^{j\omega t}d\omega}\ .$ (18) A useful normalization for the filter can be achieved by dividing by the signal to noise ratio (equation 12). This yields $o(0)=1$. In the following discussion, $F$, $f$, and $o$ are so normalized. ## 4 Application to our problem We have now collected all the necessary input information and the tools we need to design an optimal filter and calculate the signal to noise. Before tackling the full problem lets explore some special cases so that we can get an understanding of the effect of each input, can compare our results with previous work, and gain confidence in the method. Some of the formulas are followed by a bracketed reference to the source of the evaluation of the previous integral. Actually most of the integrals are straightforward (but sometimes tedious) to do using contour integration. This is a consequence of the fact that the signals, backgrounds, and analog filters all have exponential shapes in the time domain. More general signal shapes are less tractable. ### 4.1 No background ($I_{b}=0$) and $\tau_{s}=0$ This is the usual treatment when the “ballistic deficit” is ignored. $i_{s}(t)$ is a delta function, $I_{s}(\omega)=1$, and $2N(\omega)=I_{n}^{2}+V_{n}^{2}\omega^{2}C^{2}$. Before starting lets evaluate some useful constants related to the inputs so that we can use them to evaluate the results as we go: $\tau_{n}\equiv V_{n}C/I_{n}=2.0\,\mu\mbox{s}$, $\omega_{n}\equiv 1/\tau_{n}=0.50\rm\,M\,radians/s$, and $f_{n}=80\mbox{\,kHz}$. $\displaystyle\left[\frac{S}{N}\right]_{aa}$ $\displaystyle=$ $\displaystyle\frac{\omega_{n}^{2}}{\pi I_{n}^{2}}\int_{-\infty}^{\infty}{\frac{1}{\omega_{n}^{2}+\omega^{2}}\,d\omega}$ (19) $\displaystyle=$ $\displaystyle\frac{\omega_{n}}{I_{n}^{2}}$ (20) $\displaystyle ENC^{2}$ $\displaystyle=$ $\displaystyle I_{n}^{2}\tau_{n}=I_{n}V_{n}C$ (21) $\displaystyle\left[\frac{S}{N}\right]_{tt}$ $\displaystyle=$ $\displaystyle\frac{\omega_{n}^{2}}{\pi I_{n}^{2}}\int_{-\infty}^{\infty}{\frac{\omega^{2}}{\omega_{n}^{2}+\omega^{2}}\,d\omega}$ (22) $\displaystyle=$ $\displaystyle\infty$ (23) $\displaystyle ENTC^{2}$ $\displaystyle=$ $\displaystyle 0$ (24) $\displaystyle F(\omega)$ $\displaystyle=$ $\displaystyle\frac{2\omega_{n}}{\omega_{n}^{2}+\omega^{2}}$ (25) $\displaystyle f(t)$ $\displaystyle=$ $\displaystyle e^{-\|t\|/\tau_{n}}$ (26) $\displaystyle o(t)$ $\displaystyle=$ $\displaystyle e^{-\|t\|/\tau_{n}}$ (27) Substituting our sample values, we find that $ENC=223\,e$, and the corner frequency of the optimal filter is 80 kHz. ### 4.2 No background ($I_{b}=0$) and $\tau_{s}>0$ This is the case usually treated, ballistic deficit included. $\displaystyle I_{s}(\omega)$ $\displaystyle=$ $\displaystyle\frac{-j\omega_{s}}{\omega-j\omega_{s}}$ (28) $\displaystyle 2N(\omega)$ $\displaystyle=$ $\displaystyle(I_{n}^{2}+V_{n}^{2}\omega^{2}C^{2})=I_{n}^{2}(\omega^{2}+\omega_{n}^{2})/\omega_{n}^{2}$ (29) $\displaystyle\left[\frac{S}{N}\right]_{aa}$ $\displaystyle=$ $\displaystyle\frac{\omega_{s}^{2}\omega_{n}^{2}}{\pi I_{n}^{2}}\int_{-\infty}^{\infty}{\frac{1}{(\omega^{2}+\omega_{s}^{2})(\omega^{2}+\omega_{n}^{2})}\,d\omega}$ (30) $\displaystyle=$ $\displaystyle\frac{\omega_{s}\omega_{n}}{I_{n}^{2}(\omega_{s}+\omega_{n})}\mbox{\hskip 72.26999pt \rm[Dwight~{}856.31]}$ (31) $\displaystyle ENC^{2}$ $\displaystyle=$ $\displaystyle I_{n}^{2}(\tau_{n}+\tau_{s})=I_{n}V_{n}C+I_{n}^{2}\tau_{s}$ (32) $\displaystyle\left[\frac{S}{N}\right]_{tt}$ $\displaystyle=$ $\displaystyle\frac{\omega_{s}^{2}\omega_{n}^{2}}{\pi I_{n}^{2}}\int_{-\infty}^{\infty}{\frac{\omega^{2}}{(\omega^{2}+\omega_{s}^{2})(\omega^{2}+\omega_{n}^{2})}\,d\omega}$ (33) $\displaystyle ENTC^{2}$ $\displaystyle=$ $\displaystyle I_{n}^{2}\tau_{n}\tau_{s}(\tau_{n}+\tau_{s})=ENC^{2}\tau_{n}\tau_{s}\mbox{\hskip 36.135pt \rm[Maple]}$ (34) $\displaystyle F(\omega)$ $\displaystyle=$ $\displaystyle\frac{2j\omega_{n}(\omega_{n}+\omega_{s})}{(\omega+j\omega_{s})(\omega^{2}+\omega_{n}^{2})}$ (35) $\displaystyle f(t)$ $\displaystyle=$ $\displaystyle\frac{\omega_{n}(\omega_{n}+\omega_{s})}{\pi}\int_{-\infty}^{\infty}{\frac{(\omega_{s}+j\omega)}{(\omega^{2}+\omega_{s}^{2})(\omega^{2}+\omega_{n}^{2})}e^{j\omega t}d\omega}$ (36) $\displaystyle=$ $\displaystyle\frac{\omega_{n}(\omega_{n}+\omega_{s})}{(\omega_{n}^{2}-\omega_{s}^{2})}\left(e^{-\|t\|/\tau_{s}}-\frac{\omega_{s}}{\omega_{n}}e^{-\|t\|/\tau_{n}}-\rm signum(t)\left(e^{-\|t\|/\tau_{s}}-e^{-\|t\|/\tau_{n}}\right)\right)$ (37) $\displaystyle=$ $\displaystyle\left(e^{-\|t\|/\tau_{n}}+H(-t)\frac{2\tau_{s}}{\tau_{n}-\tau_{s}}\left(e^{-\|t\|/\tau_{n}}-e^{-\|t\|/\tau_{s}}\right)\right)$ (38) $\displaystyle o(t)$ $\displaystyle=$ $\displaystyle\frac{\omega_{n}\omega_{s}(\omega_{n}+\omega_{s})}{\pi}\int_{-\infty}^{\infty}{\frac{e^{j\omega t}}{(\omega^{2}+\omega_{s}^{2})(\omega^{2}+\omega_{n}^{2})}\,d\omega}$ (39) $\displaystyle=$ $\displaystyle\frac{1}{(\tau_{n}-\tau_{s})}\left(\tau_{n}e^{-\|t\|/\tau_{n}}-\tau_{s}e^{-\|t\|/\tau_{s}}\right)$ (40) Substituting our sample values, we find that $ENC=239\,e$, $ENTC=327\,\mu\mbox{s}\,e$ (1.1 ns for a 100 MeV deposit), and the corner frequency of the optimal filter is still 80 kHz. The ENC corresponds to and equivalent noise energy (ENE) of 80 keV. ### 4.3 $I_{b}>0$, $V_{n}=0$, and $I_{n}=0$ This special case of no electronic noise is explored to give us confidence that the filter technique does separate the desired signal from the background noise. $\displaystyle I_{s}(\omega)$ $\displaystyle=$ $\displaystyle\frac{-j\omega_{s}}{\omega-j\omega_{s}}$ (41) $\displaystyle 2N(\omega)$ $\displaystyle=$ $\displaystyle\frac{\omega_{s}^{2}I_{b}^{2}}{\omega_{s}^{2}+\omega^{2}}$ (42) $\displaystyle\frac{\|I_{s}\|^{2}}{N}$ $\displaystyle=$ $\displaystyle\frac{2}{I_{b}^{2}}$ (43) $\displaystyle\left[\frac{S}{N}\right]_{aa}$ $\displaystyle=$ $\displaystyle\infty\ ,\ ENC^{2}=0$ (44) $\displaystyle\left[\frac{S}{N}\right]_{tt}$ $\displaystyle=$ $\displaystyle\infty\ ,\ ENCT^{2}=0$ (45) $\displaystyle f(t)$ $\displaystyle\propto$ $\displaystyle\frac{1}{\pi}\int_{-\infty}^{\infty}{(\omega_{s}-j\omega)e^{j\omega t}d\omega}$ (46) $\displaystyle\propto$ $\displaystyle 2\left(\delta(t)-\tau_{s}\delta^{\prime}(t)\right)$ (47) $\displaystyle o(t)$ $\displaystyle\propto$ $\displaystyle\frac{1}{\pi}\int_{-\infty}^{\infty}{e^{j\omega t}d\omega}\,\,\propto\,\,2\delta(t)$ (48) Because the signal and (background) noise have the same spectrum, the signal to noise ratio is independent of frequency. This results in infinite signal to noise integrals and no error on the pulse height and time determinations. In the absence of other noise, the matched filter turns both the signal and background events into delta functions which are always separable. ### 4.4 The general case Now we will treat our general case, with all noise and width processes active. The signal is as for the previous two cases. The noise is the sum of the noise in those cases. $\displaystyle I_{s}(\omega)$ $\displaystyle=$ $\displaystyle\frac{-j\omega_{s}}{\omega-j\omega_{s}}$ (49) $\displaystyle\|I_{s}(\omega)\|^{2}$ $\displaystyle=$ $\displaystyle\frac{\omega_{s}^{2}}{\omega_{s}^{2}+\omega^{2}}$ (50) $\displaystyle 2N(\omega)$ $\displaystyle=$ $\displaystyle\frac{I_{n}^{2}}{\omega_{n}^{2}}(\omega_{n}^{2}+\omega^{2})+\frac{\omega_{s}^{2}I_{b}^{2}}{\omega_{s}^{2}+\omega^{2}}$ (51) $\displaystyle\left[\frac{S}{N}\right]_{aa}$ $\displaystyle=$ $\displaystyle\frac{\omega_{s}^{2}\omega_{n}^{2}}{\pi I_{n}^{2}}\int_{-\infty}^{\infty}{\frac{1}{\omega^{4}+(\omega_{s}^{2}+\omega_{n}^{2})\omega^{2}+\omega_{n}^{2}\omega_{s}^{2}(1+I_{b}^{2}/I_{n}^{2})}\,d\omega}$ (52) $\displaystyle b^{2}$ $\displaystyle\equiv$ $\displaystyle 1+I_{b}^{2}/I_{n}^{2}\mbox{\hskip 36.135pt\rm and\hskip 36.135pt }\omega_{b}\equiv\omega_{n}b$ (53) $\displaystyle\left[\frac{S}{N}\right]_{aa}$ $\displaystyle=$ $\displaystyle\frac{\omega_{s}\omega_{b}}{(I_{n}^{2}+I_{b}^{2})\sqrt{2\omega_{b}\omega_{s}+(\omega_{n}^{2}+\omega_{s}^{2})}}\mbox{\hskip 72.26999pt \rm[Dwight~{}857.11]}$ (54) $\displaystyle ENC^{2}$ $\displaystyle=$ $\displaystyle(I_{n}^{2}+I_{b}^{2})\sqrt{\tau_{b}\tau_{s}}\sqrt{2+\tau_{b}/\tau_{s}+\tau_{s}/(b^{2}\tau_{b})}$ (55) $\displaystyle\left[\frac{S}{N}\right]_{tt}$ $\displaystyle=$ $\displaystyle\frac{\omega_{s}^{2}\omega_{n}^{2}}{\pi I_{n}^{2}}\int_{-\infty}^{\infty}{\frac{\omega^{2}}{\omega^{4}+(\omega_{s}^{2}+\omega_{n}^{2})\omega^{2}+\omega_{n}^{2}\omega_{s}^{2}(1+I_{b}^{2}/I_{n}^{2})}\,d\omega}$ (56) $\displaystyle ENTC^{2}$ $\displaystyle=$ $\displaystyle I_{n}^{2}\tau_{n}\tau_{s}\sqrt{\tau_{n}^{2}+\tau_{s}^{2}+2b\tau_{n}\tau_{s})}=ENC^{2}\tau_{b}\tau_{s}\mbox{\hskip 25.29494pt\rm[Prudnikov 2.2.10 \\#4 page 313]}\hfil$ (57) $\displaystyle F(\omega)$ $\displaystyle=$ $\displaystyle ENC^{2}\,\,\frac{2\omega_{s}\omega_{n}^{2}(\omega_{s}+j\omega)}{I_{n}^{2}(\omega^{4}+(\omega_{s}^{2}+\omega_{n}^{2})\omega^{2}+\omega_{s}^{2}\omega_{n}^{2}b^{2})}$ (58) $\displaystyle f(t)$ $\displaystyle=$ $\displaystyle ENC^{2}\,\,\frac{\omega_{s}\omega_{n}^{2}}{\pi I_{n}^{2}}\int_{-\infty}^{\infty}{\frac{(\omega_{s}+j\omega)e^{j\omega t}}{(\omega^{4}+(\omega_{s}^{2}+\omega_{n}^{2})\omega^{2}+\omega_{s}^{2}\omega_{n}^{2}b^{2})}d\omega}$ (59) $\displaystyle=$ $\displaystyle ENC^{2}\,\,\frac{2\omega_{s}\omega_{n}^{2}}{\pi I_{n}^{2}}\left(\int_{0}^{\infty}{\frac{\omega_{s}\cos(\omega t)-\omega\sin(\omega t)}{(\omega^{4}+(\omega_{s}^{2}+\omega_{n}^{2})\omega^{2}+\omega_{s}^{2}\omega_{n}^{2}b^{2})}d\omega}\right)$ (60) $\displaystyle\omega_{+}$ $\displaystyle\equiv$ $\displaystyle\sqrt{(\omega_{b}\omega_{s}+(\omega_{s}^{2}+\omega_{n}^{2})/2)/2}$ (61) $\displaystyle\omega_{-}$ $\displaystyle\equiv$ $\displaystyle\sqrt{(\omega_{b}\omega_{s}-(\omega_{s}^{2}+\omega_{n}^{2})/2)/2}$ (62) $\displaystyle f(t)$ $\displaystyle=$ $\displaystyle ENC^{2}\,\,\frac{\omega_{n}\omega_{s}}{2I_{n}^{2}}e^{-\omega_{+}\|t\|}\left(\frac{1}{b\omega_{+}}\cos(\omega_{-}t)+\frac{1}{b\omega_{-}}\sin(\omega_{-}\|t\|)-\frac{\omega_{n}}{\omega_{+}\omega_{-}}\sin(\omega_{-}t)\right)$ [Prudnikov 2.5.10 #15 and#17 page 397] $\displaystyle=$ $\displaystyle e^{-\|t\|/\tau_{+}}\left(\cos(t/\tau_{-})+\left(\frac{\tau_{-}}{\tau_{+}}\rm signum(t)-\frac{\tau_{+}}{\tau_{b}}\right)\sin(t/\tau_{-})\right)$ (64) $\displaystyle o(t)$ $\displaystyle=$ $\displaystyle ENC^{2}\,\frac{\omega_{n}^{2}\omega_{s}^{2}}{\pi I_{n}^{2}}\int_{-\infty}^{\infty}{\frac{e^{j\omega t}}{(\omega^{4}+(\omega_{s}^{2}+\omega_{n}^{2})\omega^{2}+\omega_{s}^{2}\omega_{n}^{2}b^{2})}\,d\omega}$ (65) $\displaystyle=$ $\displaystyle e^{-\|t\|/\tau_{+}}\left(\cos(t/\tau_{-})+\frac{\tau_{-}}{\tau_{+}}\sin(\|t\|/\tau_{-})\right)$ (66) The filter output for a noiseless signal pulse is shown in Figure 4. Figure 4: A noiseless signal after passing through the optimal filter. The peak gives the estimate for the signal charge and time. In this case t=0 and the charge is unity. ### 4.5 $I_{n}=0$, $\omega_{s}<<\omega_{b}$ The results in the previous section are a bit complex, but can be simplified for our case if we note that $I_{b}$ is more than 10 times $I_{n}$ even at nominal background levels. In addition, $\omega_{b}$ is more than five times $\omega_{s}$. Setting $I_{n}=0$, using the approximation $\omega_{b}>>\omega_{s}$, and defining $\omega_{sb}\equiv\sqrt{\omega_{b}\omega_{s}/2}\approx\omega_{-}\approx\omega_{+}$, we find: $\displaystyle I_{s}(\omega)$ $\displaystyle=$ $\displaystyle\frac{-j\omega_{s}}{\omega-j\omega_{s}}$ (67) $\displaystyle\|I_{s}(\omega)\|^{2}$ $\displaystyle=$ $\displaystyle\frac{\omega_{s}^{2}}{\omega^{2}+\omega_{s}^{2}}$ (68) $\displaystyle 2N(\omega)$ $\displaystyle\approx$ $\displaystyle\frac{I_{b}^{2}(\omega^{4}+4\omega_{sb}^{4})}{\omega_{b}^{2}(\omega^{2}+\omega_{s}^{2})}$ (69) $\displaystyle\|I_{s}(\omega)\|^{2}/N(\omega)$ $\displaystyle\approx$ $\displaystyle\frac{8\omega_{sb}^{4}}{I_{b}^{2}(\omega^{4}+4\omega_{sb}^{4})}$ (70) $\displaystyle ENC^{2}$ $\displaystyle\approx$ $\displaystyle I_{b}^{2}\tau_{sb}=\sqrt{2\tau_{s}I_{b}^{3}V_{n}C}$ (71) $\displaystyle ENTC^{2}$ $\displaystyle\approx$ $\displaystyle ENC^{2}\tau_{s}\tau_{b}=\sqrt{2I_{b}\tau_{s}^{3}V_{n}^{3}C^{3}}$ (72) $\displaystyle F(\omega)$ $\displaystyle\approx$ $\displaystyle\frac{j\omega_{b}\omega_{sb}(\omega-j\omega_{s})}{(\omega^{4}+4\omega_{sb}^{4})}$ (73) $\displaystyle f(t)$ $\displaystyle\approx$ $\displaystyle e^{-\|t\|/\tau_{sb}}\left(\cos(t/\tau_{sb})+\sin(\|t\|/\tau_{sb})-\sqrt{2\tau_{s}/\tau_{b}}\sin(t/\tau_{sb})\right)$ (74) $\displaystyle o(t)$ $\displaystyle\approx$ $\displaystyle e^{-\|t\|/\tau_{sb}}\left(\cos(t/\tau_{sb})+\sin(\|t\|/\tau_{sb})\right)$ (75) Figure 5: The optimal filter in the frequency domain. Figure 6: The optimal filter in the time domain. The optimal filter is shown in the frequency and time domains in Figures 5 and 6. Now for some values: $\tau_{b}=V_{n}C/I_{b}=0.11\,\mu\mbox{s}$, the filter corner $\sqrt{\tau_{s}\tau_{b}}=0.32\,\mu\mbox{s}$, $ENC=1930$ e, and $ENTC=620$$\,\mu\mbox{s}$ e or 2.1 ns for a 100 MeV deposit. The frequency corner of the optimum filter is 0.50 MHz. For our worst case of times $10\times$nominal background, $I_{b}$ goes up $\times 3.1$, $\tau_{b}$ goes down $\times 3.1$, and $\sqrt{\tau_{s}\tau_{b}}$ goes down $\times 1.78$ to 0.17$\,\mu\mbox{s}$. $ENC=4460\,e$, $ENTC=760\,\mu\mbox{s}\,e$, and the corner frequency goes to $0.90\mbox{\,MHz}$. If the $3\mbox{\,Me\kern-0.80002ptV}$ cut were used to calculate $I_{b}$, $I_{b}$ would be lower by a factor of $2.4$ and the values for $\times 10$ would be about the same as the uncut nominal values. ## 5 Picking the best FET We see that the $ENC$ depends on the FET properties only in the combination $V_{n}$($C_{iss}$+$C_{s}$). The smaller this value the better. For a given FET design, these parameters can be varied by changing the gate area $A$ (or equivalently connecting FETs in parallel). $V_{n}$ decreases as $1/\sqrt{A}$ while $C_{iss}$ is proportional to $A$. $V_{n}$$C$ is a minimum with respect to $A$ when $C_{iss}=C_{s}$. This implies that our sample FET should be scaled to 4 times its original gate area. Making this change would decrease $V_{n}$$C$ by a factor of 1.25. Since $ENC$ goes as the fourth root of $V_{n}$$C$, $ENC$ would decrease only 5%. On the other hand, $ENTC$ would decrease by 16%. ## 6 Sampling Requirements So far we have assumed that the signals are ideally and continuously measured. In fact they will be sampled with limited precision. What restrictions on the sampling rate, length, and accuracy will prevent the loss of information? ### 6.1 Sampling frequency The sampling frequency must be high enough to capture all frequencies with useful $S/N$. It must also be high enough to avoid misrepresenting noise at higher frequencies as noise in the signal region (aliasing). On the other hand the sampling frequency should be no higher than necessary. What is the highest useful frequency? We note that $S/N$ is approximately constant up to $\omega=\omega_{sb}$ and then decreases as $1/\omega^{4}$. If we throw away all information above $\omega=\omega_{c}$ the fraction of information lost is approximately $(3/4)(\omega_{sb}/\omega_{c})^{3}$. For a 20% loss of information this implies that $\omega_{c}\geq 1.5\omega_{sb}$, and that $f_{c}>0.6\mbox{\,MHz}$. The fractional loss in time signal noise is somewhat larger, being given approximately by $(3/4)(\omega_{sb}/\omega_{c})$. Thus our choice based on the amplitude error will lose us half of our time information and increase our time error by 25%. This is probably acceptable. The exact formula for the fractional loss of amplitude information is $1-\frac{1}{\pi}\left(\frac{1}{2}\log\left(\frac{\omega_{c}^{2}+2\omega_{sb}\omega_{c}+2\omega_{sb}^{2}}{\omega_{c}^{2}-2\omega_{sb}\omega_{c}+2\omega_{sb}^{2}}\right)+\tan^{-1}\left(\frac{2\omega_{c}\omega_{sb}}{(2\omega_{sb}^{2}-\omega_{c}^{2})}\right)\right)$ (76) Assume that we have one stage of near ideal integration before the digitizer (the charge sensitive preamp) followed by a near ideal differentiator. This pair gives a band pass filter with a pass band from a low frequency to $f_{c}$. The fall in the stop band is $1/\omega$. Further assume that there is an additional two pole low pass filter with a corner at $f_{c}$ (drop off like $(\omega_{c}/\omega)^{2}$ for $\omega>\omega_{c}$) between the preamp and the digitizer. If we sample at frequency $f_{d}$, noise at frequencies above $f_{d}/2$ will appear at $f_{obs}=f_{d}-f$. We are interested in studying the contribution to $N$ from $V_{n}^{2}$ which drops off as $(\omega_{c}/w)^{4}$ after all the analog filters. We have sufficient protection against aliasing if $V_{n}^{2}$ at $f_{d}/2-f_{c}\ll V_{n}^{2}$ at $f_{c}$. This is satisfied if $(f_{c}/(f_{d}/2-f_{c}))^{4}<4$. This implies $f_{d}>3.4f_{c}$, which is $3.4(1.5)/(2\pi\sqrt{\tau_{s}\tau_{b}})=0.8/\sqrt{\tau_{s}\tau_{b}}$. This is 2.5 MHz at nominal background and 4.7 MHz at $\times 10$ nominal. 3.7 MHz is adequate adequate for nominal backgrounds and marginal for $\times 10$. ### 6.2 Sample length The number of samples required is related to the lowest frequency of interest. So far we have assumed that all integrals go from zero frequency. Pile up makes this impractical as does the fact that we can’t use all past history and wait forever to get an answer. Lets look at the pile up problem first. Allowing for variations from crystal to crystal, and warm diodes, we should plan for at least 20 nA of total current in a diode. If the offset from this current is to stay within 10% of full scale, the product of this current and the integration time must be less than 10% of the charge of a 14 GeV shower, i.e., $<0.5\rm\,pC$. This limits the integration time to 200$\,\mu\mbox{s}$. This sets the minimum value for the frequency corner of the first integrator. This gives 740 samples as the maximum useful number at 3.7 MHz. This is a large number. What is the smallest number of samples we can use without significant loss of information? For the amplitude parameter, the $S/N$ is flat up to the corner frequency and then drops rapidly. For our lowest noise case ($I_{b}=0$ during source calibration for example), we have a corner frequency of 80 kHz. If we permit a 20% loss of information, we can cut off the integral at the low frequency side at 13 kHz. This is less than three times the previous integration limit leaving us little choice and many samples. The problem here is that the sample rate is much higher than needed for this case. Once filtering has been performed, the sample set could be decimated, but most of the work is done by then. In the nominal background case the corner frequency is 400 kHz, and the integral could be cut off at 80 kHz for a sample length of 12.5$\,\mu\mbox{s}$ or 46 samples. This would be the normal operating mode. If the sampling frequency were optimized for this nominal background level, the number of samples would be less than 25. ### 6.3 Sampling summary We can summarize the conclusions of the last two sections by noting that the number samples that must be dealt with is proportional to the ratio of the highest and lowest frequencies of used. The proportionality constant lies somewhere between 2.5 and 5 depending of the details of the system and how “highest” is defined. If the signal to noise ratio is approximately constant from low frequencies up to the highest frequency of interest then the fraction of the available information retained (the efficiency) of the system is given by $1-f_{lowest}/f_{highest}$. An interesting but not necessarily relevant observation: the maximum information per sample is obtained when $f_{lowest}$ is one half of $f_{highest}$ and the efficiency is 50%. ## 7 Quantization Error ### 7.1 With no net integration or differentiation With the same assumptions about filtering as in the previous section we can estimate the requirements on the quantization error. We assume there is a low pass filter such that there is no aliasing, no loss of information, and there are offsetting integrators and differentiators such that there is no net effect in the pass band. Under such conditions, quantization error may be referred back to the input where it appears as another current noise source. According to the sampling theorem [Papoulis, page 141], if the anti-aliasing conditions are met, the original signal may be reconstructed from the samples by the interpolation formula $i_{s}(t)=\sum_{n=-\infty}^{\infty}i_{s}(nT_{d})\frac{\sin((\omega_{d}/2)(t-nT_{d}))}{(\omega_{d}/2)(t-nT_{d})}$ (77) where $T_{d}$ is the sampling period, and $\omega_{d}$ is related to it in the usual way. An error in a sample can be represented as a function proportional to one element of this sum. The Fourier transform of such a function is flat out to $f_{d}/2$. The amplitude of this noise pulse is drawn from a square distribution whose width is given by the bit resolution (or the effective bit resolution for a non-ideal ADC). The rms area of the pulse is then $T_{d}i_{bit}/\sqrt{12}$. The rate of pulses with effectively random phase is $f_{d}\equiv 1/T_{d}$, and the noise spectral density is flat with a value of $I_{q}^{2}=i_{bit}^{2}/(12f_{d})$. This is to be added to the FET input current noise. We should compare this to $I_{n}^{2}$ for very quiet conditions, and to $I_{b}^{2}$ for nominal conditions. First lets treat the quiet case for which $ENC^{2}$ goes as $\sqrt{I_{n}^{2}}$. If we wish to restrict our error increase to less than 5% due to this source we require that $I_{q}^{2}<I_{n}^{2}/5$. Therefore $i_{bit}$ should be less than $\sqrt{2f_{d}}\,I_{n}$. For the example diode and our sampling rate, this is $i_{bit}<68\mbox{\,pA}$. The current at the signal peak is $q/\tau_{s}$ if there is no analog shaping. For a triple RC filter with a time constant of 0.25$\,\mu\mbox{s}$, this is reduced by a factor of 2. Assuming that 12 GeV somehow gets into one crystal, the maximum peak current would be 3 $\mu$A. This implies a dynamic range requirement of 43,000 or 15.4 bits. Since the signal is not calibrated, we need to add a bit for gain variations suggesting that 16.5 effective bits of dynamic range are required during quiet conditions. Note that better light collection imposes a greater dynamic range requirement. During nominal conditions, $ENC^{2}$ goes as $\sqrt{I_{b}^{3}}$. The 5% error increase condition implies that $I_{q}^{2}<I_{b}^{2}/7$, and that $i_{bit}$ be equivalent to less than $\sqrt{1.7f_{d}}\,I_{b}$. For nominal background conditions and our sampling rate this is $i_{bit}<710\mbox{\,pA}$. The dynamic range requirement is 4,200 or 12 bits. With an extra bit for calibration differences, this is 13 bits. In practice, many crystals are summed to measure a shower. Both the electronics and background noise increase as $\sqrt{m}$, where $m$ is the number of crystals included. The dynamic range required decreases as $\sqrt{m}$. For a 9 crystal sum and nominal background conditions, the number of bits required is 10.5. ### 7.2 With one net integration If we assume that instead of offsetting integrators and differentiators, we have only one integrator and then the low pass filter, the quantization error behaves as a voltage noise when referred to the input. The rms area of the error pulse is $T_{d}v_{bit}/\sqrt{12}$, and the noise spectral density is flat with a value of $V_{q}^{2}=v_{bit}^{2}/(12f_{d})$. This is to be added to the FET input voltage noise $V_{n}^{2}$. Recall that in our case $ENC^{2}$ goes as $\sqrt{V_{n}^{2}}$ in the quiet case and as $\sqrt[4]{V_{n}^{2}}$ in the nominal case. If we wish to restrict our error increase to less than 5% due to this source we require that $V_{q}^{2}<V_{n}^{2}/10$ in the quiet case and $V_{q}^{2}<V_{n}^{2}/5$ in the nominal case. This implies that $v_{bit}$ be equivalent to less than $\sqrt{1.2f_{d}}\,V_{n}$ and $\sqrt{2.4f_{d}}\,V_{n}$, respectively. For the example FET and our sampling rate this is $v_{bit}<1.05\mbox{\,$\mu$V}$ and $v_{bit}<1.5\mbox{\,$\mu$V}$, which given the source capacitance corresponds to a charges of $<660$ and $<940$ e, and to a shower energies of $<0.22$ and $<0.31\mbox{\,Me\kern-0.80002ptV}$. Assuming that 12 GeV somehow gets into one crystal, the dynamic ranges required are 55,000 and 42,000, or 15.7 bits for quiet conditions and 15.4 bits for nominal conditions. Since the signal is not calibrated, we need to add a bit for gain variations suggesting that 17 effective bits of dynamic range may be sufficient. All of the above dynamic range calculations address the sufficient conditions for which the quantization error will not contribute to the electronic (and background) noise. They do not treat whether or not this dynamic range is necessary given the inherent energy fluctuations of the shower process. The considerations for multi-range digitizing have been addressed by Al Eisner and Gunther Haller and remain unchanged. For larger pulses, sources of error other than electronics dominate. ## 8 Filter Implementation and Interpolation There is still the problem of signal extraction. Because it is non-causal, a matched filter cannot be implemented as a simple analog filter. A procedure for directly calculating the coefficients of a tapped delay line approximation to the optimum matched filter is described by Papoulis [Papoulis, page 327]. The result is exactly parallel to the usual least square fit formulas. The autocorrelation function, $n_{f}$, is the inverse Fourier transform of the noise power spectral density after the analog filter. The elements of $\rm r$, the error matrix for the samples, are given by $n_{f}(\|l-k\|T)$. The expected values $\bf a$ for the samples is given by $i_{sf}((l-m)T+t)$, where $i_{sf}$ is the expected signal after analog filtering and $t$ is the time offset of the signal from sample $m$. The coefficients of the tapped delay line filter are given by $\rm r^{-1}\bf a$. The optimally filtered estimate for the signal at time $t$ is given by $o(t)=\bf d\rm r^{-1}\bf a$, where $\bf d$ is the set of samples. For the case of the triple RC analog filter, the coefficients each have four terms with different $t$ dependencies. The estimate for $o(t)$ also has four such terms and its derivative with respect to $t$ has three. Calculating $o(t)$ takes four multiply and adds for each sample used. Finding the root of the expression for the derivative with Newton’s method should not take many iterations. Each iteration involves calculating one exponential and approximately a dozen multiply and adds. The presence of multiple peaks in the time window or the absence of any peaks would complicate this last step. A possibility is to not try to find the peak, but to report the coefficients of the four terms instead. Knowing the event time would reduce the time to get an estimate for $o(t)$ by a factor of four, and there would no need to search for the maximum. ## 9 Less Than the Best The previous sections deal largely with optimum solutions, albeit with some practical limitations. This section will examine the loss of information if other that optimal filtering is used. It may be less than optimal in that it is not matched, e.g., an all analog filter, or in that the pass band is optimized for another condition. Figure 7: Raw input current signal and the signal after triple RC filters with time constants of $0.25\,\mu\mbox{s}$ and $2.0\,\mu\mbox{s}$. Figure 8: The peak signal current after a triple RC filter vs. the filter time constant Let us look at the case of an all analog filter. In this case all filtering is done prior to digitizing and the only digital processing performed is interpolation. I will take the B A B AR TDR design (described in subsection 6.1). This has a charge sensitive preamp followed by a CR and then two RC filters. Together these give the equivalent of three RC filters, all with the corner frequency $f_{c}$. This has the transfer function $F(\omega)=\left(\frac{-j\omega_{c}}{\omega-j\omega_{c}}\right)^{3}.$ (78) The signal is $I(\omega)=\left(\frac{-j\omega_{s}}{\omega-j\omega_{s}}\right).$ (79) Figure 9: The Equivalent Noise Energy (electronic only) of a single sample at the peak after a triple RC filter versus the radial velocity of the filter corner ($1/RC$). Figure 10: The efficiency vs. 1/RC for nominal background conditions. The time dependence of the output signal is given by the inverse Fourier transform of their product. This integral can be done by contour integration. If the contour is completed by a semi-circle at infinity around the upper half plane (for $t>0$), the integral can be deduced from the residues at the poles $j\omega_{c}$ and $j\omega_{s}$. The result is $o(t)=\frac{\omega_{c}^{3}\omega_{s}}{(\omega_{s}-\omega_{c})^{3}}\left\\{\left(\frac{(\omega_{s}-\omega_{c})^{2}}{2}t^{2}-(\omega_{s}-\omega_{c})t+1\right)e^{-\omega_{c}t}-e^{-\omega_{s}t}\right\\}.$ (80) This is shown for several time constants in figure 8. The peak value vs. time constant is shown in figure 8. Since there are no poles in the lower half plane, the integral is $0$ for $t<0$. Figure 11: The efficiency vs. 1/RC for low background conditions. Figure 12: The efficiency vs. 1/RC for $10\times$ nominal background conditions. The mean square noise is given by the inverse Fourier transform of the noise evaluated at $t=0$, i.e., the autocorrelation for no time difference. The noise before filtering is given by equation 51. The mean square noise is $n=\frac{1}{\pi}\int_{0}^{\infty}{N(\omega)\|F(\omega)\|^{2}\,d\omega}.$ (81) The contribution of the electronic noise is $n_{elec}=\omega_{c}(V_{n}^{2}C^{2}\omega_{c}^{2}+3I_{n}^{2})/32.$ (82) The peak of $\sqrt{n_{elec}/o^{2}(t)}$ gives the ENC for the electronic noise after analog filtering. Appropriate scaling gives the ENE shown in Figure 10. The background noise contribution is $n_{back}=\frac{(8\omega_{c}^{2}+9\omega_{s}\omega_{c}+3\omega_{s}^{2})\omega_{c}\omega_{s}I_{b}^{2}}{32(\omega_{c}+\omega_{s})^{3}}$ (83) $n$ is given by their sum. The signal to noise ratio for a single sample taken at time $t$ is given by $o^{2}(t)/n$. If we divide this by the optimum signal to noise given by equation 55 we get the “efficiency” for this single sample versus the sample time. Figure 10 shows this function for the B A B AR calorimeter during nominal operating conditions for several values of the corner frequency. The highest efficiency achieved is 68% for a shaping time constant of 0.2$\,\mu\mbox{s}$. For low background conditions (Figure 12) the peak efficiency is 80% at a shaping time of 1.5$\,\mu\mbox{s}$. At ten times nominal background, (Figure 12) the efficiency drops to 50% at a shaping time of 0.12$\,\mu\mbox{s}$. The loss of efficiency for nominal and better background conditions is not significant. The loss at during high background conditions hits just when noise is the causing the most problems. Matched filters utilize knowledge of the signal shape. In low background conditions, the optimum shaping time is longer than the signal. The shape of the signal is not seen and matched filters offer little advantage. The situation and conclusions are reversed in high background conditions. The cost of using an analog filter optimized for conditions other than those encountered is more dramatic. If the corner frequency is set to the 0.2$\,\mu\mbox{s}$ which is optimal for nominal conditions, but the backgrounds are actually at $10\times$ nominal levels, the efficiency is 44%. If the corner frequency were optimal for low background conditions, the efficiency at $10\times$ nominal background would be 12%. ## 10 Summary of Significant Findings Since the important conclusions may have gotten lost in the derivations, I’ll review them here. • Machine background can be treated as a current noise with the spectrum of the signal. * • The best possible amplitude and time measurements can be obtained with a matched filter. * • The errors in the amplitude and time are not correlated, therefore knowing the time does not improve the ultimate precision of the amplitude determination, although such knowledge may reduce the computation required. * • The signal to noise methodology used here reproduces previous results. * • The background current noise dominates the photodiode shot noise even at nominal backgrounds and in the quietest part of the calorimeter. * • For nominal background where $I_{b}\gg I_{n}$: $\displaystyle ENC^{2}$ $\displaystyle\approx$ $\displaystyle I_{b}^{2}\sqrt{2\tau_{s}\tau_{b}+\tau_{b}^{2}}=\sqrt{2\tau_{s}I_{b}^{3}V_{n}C+I_{b}^{2}V_{n}^{2}C^{2}}$ $\displaystyle ENTC^{2}$ $\displaystyle\approx$ $\displaystyle ENC^{2}\tau_{s}\tau_{b}=\sqrt{2I_{b}\tau_{s}^{3}V_{n}^{3}C^{3}+\tau_{s}^{2}V_{n}^{4}C^{4}}$ * • The sampling frequency of 3.7 MHz is adequate at nominal backgraounds, but it is marginal for $10\times$nominal background levels. * • At this frequency, 64 samples will have to be processed under nominal conditions. * • The step of an ADC bit (on the most sensitive range) should be set so that there are at least 13 bits to full scale (12 GeV) during nominal background. For quiet conditions and with a differentiating stage in the analog filter, 16.5 bits may be required. * • Simple analog filters can achieve near optimal results in low to moderate background conditions if the shaping time is optimized for the actual conditions. ## 11 Acknowledgements I thank my BaBar colleagues for consultations and for providing the data used in the examples. This work was supported by the U.S. Department of Energy under contract number DE-AC02-76SF00515. ## References * [Ambrozny] A. Ambrózny, Electronics Noise (McGraw-Hill, NY, NY, 1982). * [BaBar] B A B AR Collaboration, Technical Design Report, SLAC-R-95-457, March 1995. * [Bracewell] R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, NY, NY, 1978). * [Dow] S. Dow,et al., A CMOS Front End for the CsI(Tl)Calorimeter, B A B AR Note # 139 (1994). * [Dwight] H. Dwight, Tables of Integrals and Other Mathematical Data (MacMillan, NY, NY, 1961). * [Haller] D. Haller, D. Freytag, and J. Hoeflich, Proposal for the Electronics System for the B A B ARCsI(Tl)Calorimeter B A B AR Note # 184 (1994). * [Humphreys] D. Humphreys, The Analysis, Design, and Synthesis of Electrical Filters (Prentice-Hall, Englewood Cliffs, NJ, 1970). * [Maple] B. Char, et al., Maple V release 3, A symbolic algebra program. Maple is the trademark of Waterloo Maple Software. * [Marsiske] H. Marsiske, private communication. This background calculation dates from April 1996. It includes modified beam optics and the expected vacuum profile. The lost particle background is calculated. This is then scaled up by a factor of 1.5 in the barrel and 2.0 in the endcap to account radiative Bhabha induced background. * [Papoulis] A. Papoulis, Signal Analysis (McGraw-Hill, NY, NY, 1977). * [Prudnikov] A. Prudnikov, Yu. Brychov, and O. Marichev Integrals and Series, Vol.I (, , , 19). * [Radeka] V. Radeka, High Resolution Liquid Argon Total Absorption Detectors: Electronic Noise and Electrode Configuration, IEEE Trans.Nucl.Sci.Ns24 (1977) 293. * [Skolnik] M. Skolnik, Radar Handbook (McGraw-Hill, NY, NY, 1970). * [vanderZiel] A. van der Ziel, Noise in Solid State Devices and Circuits (Wiley, NY, NY, 1986).	arxiv-papers	2009-02-23T20:31:18	2024-09-04T02:49:00.817200	{ "license": "Public Domain", "authors": "Walter R. Innes", "submitter": "Walter Innes", "url": "https://arxiv.org/abs/0902.3986" }
0902.3992	###### Abstract Let $R$ be a ring and $\sigma$ an endomorphism of $R$. In this note, we study the transfert of the symmetry ($\sigma$-symmetry) and reversibility ($\sigma$-reversibility) from $R$ to its skew power series ring $R[[x;\sigma]]$. Moreover, we study on the relationship between the Baerness, quasi-Baerness and p.p.-property of a ring $R$ and these of the skew power series ring $R[[x;\sigma]]$ in case $R$ is right $\sigma$-reversible. As a consequence we obtain a generalization of [10]. A note on $\sigma$-reversibility and $\sigma$-symmetry of skew power series rings L’moufadal Ben Yakoub and Mohamed Louzari Department of mathematics Abdelmalek Essaadi University B.P. 2121 Tetouan, Morocco benyakoub@hotmail.com, mlouzari@yahoo.com Mathematics Subject Classification: 16S36; 16W20; 16U80 Keywords: Armendariz rings; Baer rings; p.p.-rings; quasi-Baer rings; skew power series rings; reversible rings; symmetric rings ## 1 Introduction Throughout this paper $R$ denotes an associative ring with identity and $\sigma$ denotes a nonzero non identity endomorphism of a given ring. Recall that a ring is reduced if it has no nonzero nilpotent elements. Lambek [16], called a ring $R$ symmetric if $abc=0$ implies $acb=0$ for $a,b,c\in R$. Every reduced ring is symmetric ([19, Lemma 1.1]) but the converse does not hold by [1, Example II.5]. Cohen [8], called a ring $R$ reversible if $ab=0$ implies $ba=0$ for $a,b\in R$. It is obvious that commutative rings are symmetric and symmetric rings are reversible, but the converse does not hold by [1, Examples I.5 and II.5] and [17, Examples 5 and 7]. From [3], a ring $R$ is called right $($left$)$ $\sigma$-reversible if whenever $ab=$ for $a,b\in R$, $b\sigma(a)=0$ ($\sigma(b)a=0$). $R$ is called $\sigma$-reversible if it is both right and left $\sigma$-reversible. Also, by [15], a ring $R$ is called right $($left$)$ $\sigma$-symmetric if whenever $abc=0$ for $a,b,c\in R$, $ac\sigma(b)=0$ ($\sigma(b)ac=0$). $R$ is called $\sigma$-symmetric if it is both right and left $\sigma$-symmetric. Clearly right $\sigma$-symmetric rings are right $\sigma$-reversible. Rege and Chhawchharia [18], called a ring $R$ an Armendariz if whenever polynomials $f=\sum_{i=0}^{n}a_{i}x^{i},\;g=\sum_{j=0}^{m}b_{j}x^{j}\in R[x]$ satisfy $fg=0$, then $a_{i}b_{j}=0$ for each $i,j$. The Armendariz property of a ring was extended to one of skew polynomial ring in [11]. For an endomorphism $\sigma$ of a ring $R$, a skew polynomial ring (also called an Ore extension of endomorphism type) $R[x;\sigma]$ of $R$ is the ring obtained by giving the polynomial ring over $R$ with the new multiplication $xr=\sigma(r)x$ for all $r\in R$. Also, a skew power series ring $R[[x;\sigma]]$ is the ring consisting of all power series of the form $\sum_{i=0}^{\infty}a_{i}x^{i}\;(a_{i}\in R)$, which are multiplied using the distributive law and the Ore commutation rule $xa=\sigma(a)x$, for all $a\in R$. According to Hong et al. [11], a ring $R$ is called $\sigma$-skew Armendariz if whenever polynomials $f=\sum_{i=0}^{n}a_{i}x^{i}$ and $g=\sum_{j=0}^{m}b_{j}x^{j}$ $\in R[x;\sigma]$ satisfy $fg=0$ then $a_{i}\sigma^{i}(b_{j})=0$ for each $i,j$. Baser et al. [4], introduced the concept of $\sigma$-(sps) Armendariz rings. A ring $R$ is called $\sigma$-(sps) Armendariz if whenever $pq=0$ for $p=\sum_{i=0}^{\infty}a_{i}x^{i},\;q=\sum_{j=0}^{\infty}b_{j}x^{j}\in R[[x;\sigma]]$, then $a_{i}b_{j}=0$ for all $i$ and $j$. According to Krempa [14], an endomorphism $\sigma$ of a ring $R$ is called rigid if $a\sigma(a)=0$ implies $a=0$ for all $a\in R$. We call a ring $R$ $\sigma$-rigid if there exists a rigid endomorphism $\sigma$ of $R$. Note that any rigid endomorphism of a ring $R$ is a monomorphism and $\sigma$-rigid rings are reduced by Hong et al. [10]. Also, by [15, Theorem 2.8(1)], a ring $R$ is $\sigma$-rigid if and only if $R$ is semiprime right $\sigma$-symmetric and $\sigma$ is a monomorphisme, so right $\sigma$-symmetric ($\sigma$-reversible) rings are a generalization of $\sigma$-rigid rings. In this note, we introduce the notion of $\sigma$-skew $($sps$)$ Armendariz rings which is a generalization of $\sigma$-(sps) Armendariz rings, and we study the transfert of the symmetry ($\sigma$-symmetry) and reversibility ($\sigma$-reversibility) from $R$ to its skew power series ring $R[[x;\sigma]]$. Also we show that $R$ is $\sigma$-(sps) Armendariz if and only if $R$ is $\sigma$-skew (sps) Armendariz and $a\sigma(b)=0$ implies $ab=0$ for $a,b\in R$. Moreover, we study on the relationship between the Baerness, quasi-Baerness and p.p.-property of a ring $R$ and these of the skew power series ring $R[[x;\sigma]]$ in case $R$ is right $\sigma$-reversible. As a consequence we obtain a generalization of [10]. ## 2 $\sigma$-Reversibility and $\sigma$-Symmetry of Skew Power Series Rings We introduce the next definition. ###### Definition 2.1. Let $R$ be a ring and $\sigma$ an endomorphism of $R$. A ring $R$ is called $\sigma$-skew $($sps$)$ Armendariz if whenever $pq=0$ for $p=\sum_{i=0}^{\infty}a_{i}x^{i},\;q=\sum_{j=0}^{\infty}b_{j}x^{j}\in R[[x;\sigma]]$, then $a_{i}\sigma^{i}(b_{j})=0$ for all $i$ and $j$. Every subring $S$ with $\sigma(S)\subseteq S$ of an $\sigma$-skew (sps) Armendariz ring is a $\sigma$-skew (sps) Armendariz ring. In the next, we give an example of a ring $R$ which is $\sigma$-skew (sps) Armendariz but not $\sigma$-(sps) Armendariz. ###### Example 2.2. Let $R$ be the polynomial ring $\mathbb{Z}_{2}[x]$ over $\mathbb{Z}_{2}$, and let the endomorphism $\sigma\colon R\rightarrow R$ be defined by $\sigma(f(x))=f(0)$ for $f(x)\in\mathbb{Z}_{2}[x]$. $(i)$ $R$ is not $\sigma$-$(sps)$ Armendariz because $\sigma$ is not a monomorphism. $(ii)$ $R$ is an $\sigma$-skew $($sps$)$ Armendariz ring $($as in [11, Example 5]$)$. Consider $R[[y;\sigma]]=\mathbb{Z}_{2}[x][[y;\sigma]]$. Let $p=\sum_{i=0}^{\infty}f_{i}y^{i}$ and $q=\sum_{j=0}^{\infty}g_{j}y^{j}\in R[[y;\sigma]]$. We have $pq=\sum_{\ell\geq 0}\sum_{\ell=i+j}f_{i}\sigma^{i}(g_{j})y^{\ell}=0$. If $pq=0$ then $\sum_{\ell=i+j}f_{i}\sigma^{i}(g_{j})y^{\ell}=0$, for each $\ell\geq 0$. Suppose that there is $f_{s}\neq 0$ for some $s\geq 0$ and $f_{0}=f_{1}=\cdots=f_{s-1}=0$, then $\sum_{i+j=s}f_{i}\sigma^{i}(g_{j})y^{i+j}=0\Rightarrow f_{s}\sigma^{s}(g_{0})=0$, since $R$ is a domain then $g_{0}(0)=0$. Also $\sum_{i+j=s+1}f_{i}\sigma^{i}(g_{j})y^{i+j}=0\Rightarrow f_{s}\sigma^{s}(g_{1})+f_{s+1}\sigma^{s+1}(g_{0})=0$, since $g_{0}(0)=0$ then $f_{s}\sigma^{s}(g_{1})=0$ and so $g_{1}(0)=0$ by the same method as above. Continuing this process, we have $g_{j}(0)=0$ for all $j\geq 0$. Thus $f_{i}\sigma^{i}(g_{j})=0$ for all $i,j$. We say that $R$ satisfies the condition $(\mathcal{C_{\sigma}})$, if whenever $a\sigma(b)=0$ for $a,b\in R$, then $ab=0$. By [4, Theorem 3.3(3iii)], if $R$ is $\sigma$-(sps) Armendariz then it satisfies $(\mathcal{C_{\sigma}})$ (so $\sigma$ is a monomorphism). If $R$ is an $\sigma$-skew (sps) Armendariz ring satisfying the condition $(\mathcal{C_{\sigma}})$ then $R$ is $\sigma$-(sps) Armendariz. ###### Theorem 2.3. A ring $R$ is $\sigma$-$(sps)$ Armendariz ring if and only if it is $\sigma$-skew $(sps)$ Armendariz and satisfies the condition $(\mathcal{C_{\sigma}})$. ###### Proof. $(\Leftarrow)$. It is clear. $(\Rightarrow)$. If $R$ is $\sigma$-$(sps)$ Armendariz then it satisfies the condition $(\mathcal{C_{\sigma}})$. It suffices to show that if $R$ is $\sigma$-$(sps)$ Armendariz then it is $\sigma$-skew (sps) Armendariz. The proof is similar as of [12, Theorem 1.8]. Let $p=\sum_{i=0}^{\infty}a_{i}x^{i}$ and $q=\sum_{j=0}^{\infty}b_{j}x^{j}\in R[[x;\sigma]]$ with $pq=0$. Note that $a_{j}b_{j}=0$ for all $i$ and $j$. We claim that $a_{i}\sigma^{i}(b_{j})=0$ for all $i$ and $j$. We have $(a_{0}+a_{1}x+\cdots)(b_{0}+b_{1}x+\cdots)=0$, then $a_{0}(b_{0}+b_{1}x+\cdots)+(a_{1}x+a_{2}x^{2}\cdots)(b_{0}+b_{1}x+\cdots)=0$. Since $a_{0}b_{j}=0$ for all $j$, we get $\;0=(a_{1}x+a_{2}x^{2}+\cdots)(b_{0}+b_{1}x+\cdots)$ $0=(a_{1}+a_{2}x+\cdots)x(b_{0}+b_{1}x+\cdots)$ $\qquad\quad\;0=(a_{1}+a_{2}x+\cdots)(\sigma(b_{0})x+\sigma(b_{1})x^{2}+\cdots).$ Put $p_{1}=a_{1}+a_{2}x+\cdots$ and $q_{1}=\sigma(b_{0})x+\sigma(b_{1})x^{2}+\cdots$. Since $p_{1}q_{1}=0$ then $a_{i}\sigma(b_{j})=0$ for all $i\geq 1$ and $j\geq 0$. We have, also $0=a_{1}(\sigma(b_{0})x+\sigma(b_{1})x^{2}+\cdots)+(a_{2}x+a_{3}x^{2}+\cdots)(\sigma(b_{0})x+\sigma(b_{1})x^{2}+\cdots).$ Since $a_{1}\sigma(b_{j})=0$ for all $j$, then $0=(a_{2}x+a_{3}x^{2}+\cdots)(\sigma(b_{0})x+\sigma(b_{1})x^{2}+\cdots)$ $\;\;0=(a_{2}+a_{3}x+\cdots)(\sigma^{2}(b_{0})x^{2}+\sigma^{2}(b_{1})x^{3}+\cdots).$ Put $p_{2}=a_{2}+a_{3}x+a_{4}x^{2}+\cdots$ and $q_{2}=\sigma^{2}(b_{0})x^{2}+\sigma^{2}(b_{1})x^{3}+\cdots$, and then $p_{2}q_{2}=0$ implies $a_{i}\sigma^{2}(b_{j})=0$ for all $i\geq 2$ and $j\geq 0$. Continuing this process, we can show that $a_{i}\sigma^{i}(b_{j})=0$ for all $i\geq 0$ and $j\geq 0$. Thus $R$ is $\sigma$-skew $(sps)$ Armendariz. ∎ ###### Lemma 2.4. Let $R$ be an $\sigma$-$(sps)$ Armendariz ring. Then for $f=\sum_{i=0}^{\infty}a_{i}x^{i}$, $g=\sum_{j=0}^{\infty}b_{j}x^{j}$ and $h=\sum_{k=0}^{\infty}c_{k}x^{k}\in R[[x;\sigma]]$, if $fgh=0$ then $a_{i}b_{j}c_{k}=0$ for all $i,j,k$. ###### Proof. Note that, if $fg=0$ then $a_{i}g=0$ for all $i$. Suppose that $fgh=0$ then $a_{i}(gh)=0$ for all $i$, and so $(a_{i}g)h=0$ for all $i$. Therefore $a_{i}b_{j}c_{k}=0$ for all $i,j,k$. ∎ ###### Proposition 2.5. Let $R$ be an $\sigma$-$(sps)$ Armendariz ring. Then $(1)$ $R$ is reversible if and only if $R[[x;\sigma]]$ is reversible. $(2)$ $R$ is symmetric if and only if $R[[x;\sigma]]$ is symmetric. ###### Proof. If $R[[x;\sigma]]$ is symmetric (reversible) then $R$ is symmetric (reversible). Conversely, $(1)$. Let $f=\sum_{i=0}^{\infty}a_{i}x^{i}$ and $g=\sum_{j=0}^{\infty}b_{j}x^{j}\in R[[x;\sigma]]$, if $fg=0$ then $a_{i}b_{j}=0$ for all $i$ and $j$. By [4, Theorem 3.3 (3ii)], we have $\sigma^{j}(a_{i})b_{j}=0$ for all $i$ and $j$. Since $R$ is reversible, we obtain $b_{j}\sigma^{j}(a_{i})=0$ for all $i$ and $j$. Thus $gf=\sum_{\ell=0}^{\infty}\sum_{\ell=i+j}b_{j}\sigma^{j}(a_{i})x^{\ell}=0$. $(2)$. We will use freely [4, Theorem 3.3 (3ii)], reversibility and symmetry of $R$. Let $f=\sum_{i=0}^{\infty}a_{i}x^{i}$, $g=\sum_{j=0}^{\infty}b_{j}x^{j}$ and $h=\sum_{k=0}^{\infty}c_{k}x^{k}\in R[[x;\sigma]]$, if $fgh=0$ then $a_{i}b_{j}c_{k}=0$ for all $i$, $j$ and $k$, by Lemma 2.4. Then for all $i,j,k$ we have $b_{j}c_{k}a_{i}=0\Rightarrow\sigma^{k}(b_{j})c_{k}a_{i}=0\Rightarrow a_{i}\sigma^{k}(b_{j})c_{k}=0\Rightarrow a_{i}c_{k}\sigma^{k}(b_{j})=0\Rightarrow c_{k}\sigma^{k}(b_{j})a_{i}=0\Rightarrow\sigma^{i}[c_{k}\sigma^{k}(b_{j})]a_{i}=0\Rightarrow a_{i}\sigma^{i}[c_{k}\sigma^{k}(b_{j})]=0$. Thus $fhg=0$. ∎ The next Lemma gives a relationship between $\sigma$-reversibility ($\sigma$-symmetry) and reversibility (symmetry). ###### Lemma 2.6 ([5, Lemma 3.1]). Let $R$ be a ring and $\sigma$ an endomorphism of $R$. If $R$ satisfies the condition $(\mathcal{C_{\sigma}})$. Then $(1)$ $R$ is reversible if and only if $R$ is $\sigma$-reversible; $(2)$ $R$ is symmetric if and only if $R$ is $\sigma$-symmetric. ###### Theorem 2.7. Let $R$ be an $\sigma$-$(sps)$ Armendariz ring. The following statements are equivalent: $(1)$ $R$ is reversible $($symmetric$)$; $(2)$ $R$ is $\sigma$-reversible $($$\sigma$-symmetric$)$; $(3)$ $R$ is right $\sigma$-reversible $($right $\sigma$-symmetric$)$; $(4)$ $R[[x;\sigma]]$ is reversible $($symmetric$)$. ###### Proof. We prove the reversible case (the same for the symmetric case). $(1)\Leftrightarrow(4)$. By Proposition 2.5. $(1)\Rightarrow(2)$ and $(2)\Rightarrow(3)$. Immediately from Lemma 2.6. $(3)\Rightarrow(1)$. Let $a,b\in R$, if $ab=0$ then $b\sigma(a)=0$ (right $\sigma$-reversibility), so $ba=0$ (condition $(\mathcal{C_{\sigma}})$). ∎ ## 3 Related Topics In this section we turn our attention to the relationship between the Baerness, quasi-Baerness and p.p.-property of a ring $R$ and these of the skew power series ring $R[[x;\sigma]]$ in case $R$ is right $\sigma$-reversible. For a nonempty subset $X$ of $R$, we write $r_{R}(X)=\\{c\in R\|dc=0\;\mathrm{for\;any}\;d\in X\\}$ which is called the right annihilator of $X$ in $R$. ###### Lemma 3.1. If $R$ is a right $\sigma$-reversible ring with $\sigma(1)=1$. Then $(1)$ $\sigma(e)=e$ for all idempotent $e\in R$; $(2)$ $R$ is abelian. ###### Proof. (1) Let $e$ an idempotent of $R$. We have $e(1-e)=(1-e)e=0$ then $(1-e)\sigma(e)=e\sigma((1-e))=0$, so $\sigma(e)-e\sigma(e)=e-e\sigma(e)=0$, therefore $\sigma(e)=e$. (2) Let $r\in R$ and $e$ an idempotent of $R$. We have $e(1-e)=0$ then $e(1-e)r=0$, since $R$ is right $\sigma$-reversible then $(1-e)r\sigma(e)=0=(1-e)re=0$, so $re=ere$. Since $(1-e)e=0$, we have also $er=ere$. Then $R$ is abelian. ∎ ###### Lemma 3.2. Let $R$ be a right $\sigma$-reversible ring with $\sigma(1)=1$, then the set of all idempotents in $R[[x;\sigma]]$ coincides with the set of all idempotents of $R$. In this case $R[[x;\sigma]]$ is abelian. ###### Proof. We adapt the proof of [3, Theorem 2.13(iii)] for $R[[x;\sigma]]$. Let $f^{2}=f\in R[[x;\sigma]]$, where $f=f_{0}+f_{1}x+f_{2}x^{2}+\cdots$. Then $\sum_{\ell=0}^{\infty}\sum_{\ell=i+j}f_{i}\sigma^{i}(f_{j})x^{\ell}=\sum_{\ell=0}^{\infty}f_{\ell}x^{\ell}.$ For $\ell=0$, we have $f_{0}^{2}=f_{0}$. For $\ell=1$, we have $f_{0}f_{1}+f_{1}\sigma(f_{0})=f_{1}$, but $f_{0}$ is central and $\sigma(f_{0})=f_{0}$, so $f_{0}f_{1}+f_{1}f_{0}=f_{1}$, a multiplication by $(1-f_{0})$ on the left hand gives $f_{1}=f_{0}f_{1}$, and so $f_{1}=0$. For $\ell=2$, we have $f_{0}f_{2}+f_{1}\sigma(f_{1})+f_{2}\sigma^{2}(f_{0})=f_{2}$, so $f_{0}f_{2}+f_{2}f_{0}=f_{2}$ (because $f_{1}=0$ and $\sigma^{2}(f_{0})=f_{0}$), a multiplication by $(1-f_{0})$ on the left hand gives $f_{0}f_{2}=f_{2}=0$. Continuing this procedure yields $f_{i}=0$ for all $i\geq 1$. Consequently, $f=f_{0}=f_{0}^{2}\in R$. Since $R$ is abelian then $R[[x;\sigma]]$ is abelian. ∎ Kaplansky [13], introduced the concept of Baer rings as rings in which the right (left) annihilator of every nonempty subset is generated by an idempotent. According to Clark [7], a ring $R$ is called quasi-Baer if the right annihilator of each right ideal of $R$ is generated (as a right ideal) by an idempotent. It is well-known that these two concepts are left-right symmetric. A ring $R$ is called a right (left) p.p.-ring if the right (left) annihilator of an element of $R$ is generated by an idempotent. $R$ is called a p.p.-ring if it is both a right and left p.p.-ring. ###### Theorem 3.3. Let $R$ be a right $\sigma$-reversible ring with $\sigma(1)=1$. Then $(1)$ $R$ is a Baer ring if and only if $R[[x;\sigma]]$ is a Baer ring; $(2)$ $R$ is a quasi-Baer ring if and only if $R[[x;\sigma]]$ is a quasi-Baer ring. ###### Proof. $(\Rightarrow)$. Suppose that $R$ is Baer. Let $A$ be a nonempty subset of $R[[x;\sigma]]$ and $A^{}$ be the set of all coefficients of elements of $A$. Then $A^{}$ is a nonempty subset of $R$ and so $r_{R}(A^{})=eR$ for some idempotent element $e\in R$. Since $e\in r_{R[[x;\sigma]]}(A)$ by Lemma 3.1. We have $eR[[x;\sigma]]\subseteq r_{R[[x;\sigma]]}(A)$. Now, let $0\neq q=b_{0}+b_{1}x+b_{2}x^{2}+\cdots\in r_{R[[x;\sigma]]}(A)$. Then $Aq=0$ and hence $pq=0$ for any $p\in A$. Let $p=a_{0}+a_{1}x+a_{2}x^{2}+\cdots$, then $pq=\sum_{\ell\geq 0}\sum_{\ell=i+j}a_{i}\sigma^{i}(b_{j})x^{\ell}=0.$ • $\ell=0$ implies $a_{0}b_{0}=0$ then $b_{0}\in r_{R}(A^{})=eR$. • $\ell=1$ implies $a_{0}b_{1}+a_{1}\sigma(b_{0})=0$, since $b_{0}=eb_{0}$ and $\sigma(e)=e$ then $a_{0}b_{1}+a_{1}e\sigma(b_{0})=0$, but $a_{1}e=0$ so $a_{0}b_{1}=0$ and hence $b_{1}\in r_{R}(A^{})$. • $\ell=2$ implies $a_{0}b_{2}+a_{1}\sigma(b_{1})+a_{2}\sigma^{2}(b_{0})=0$, then $a_{0}b_{2}+a_{1}e\sigma(b_{1})+a_{2}e\sigma^{2}(b_{0})=0$, but $a_{1}e\sigma(b_{1})=a_{2}e\sigma^{2}(b_{0})=0$, hence $a_{0}b_{2}=0$. Then $b_{2}\in r_{R}(A^{})$. Continuing this procedure yields $b_{0},b_{1},b_{2},b_{3},\cdots\in r_{R}(A^{})$. So, we can write $q=eb_{0}+eb_{1}x+eb_{2}x^{2}+\cdots\in eR[[x;\sigma]]$. Therefore $eR[[x;\sigma]]=r_{R[[x;\sigma]]}(A)$. Consequently, $R[[x;\sigma]]$ is a Baer ring. Conversely, Suppose that $R[[x;\sigma]]$ is Baer. Let $B$ be a nonempty subset of $R$. Then $r_{R[[x;\sigma]]}(B)=eR[[x;\sigma]]$ for some idempotent $e\in R$ by Lemma 3.2. Thus $r_{R}(B)=r_{R[[x;\sigma]]}(B)\cap R=eR[[x;\sigma]]\cap R=eR$. Therefore $R$ is Baer. The proof for the case of the quasi-Baer property follows in a similar fashion; In fact, for any right ideal $A$ of $R[[x;\sigma]]$, take $A^{}$ as the right ideal generated by all coefficients of elements of $A$. ∎ From [10, Example 20], $R=M_{2}(\mathbb{Z})$ is a Baer ring and $R[[x]]$ is not Baer. Clearly $R$ is not reversible. So that, the “right $\sigma$-reversibility” condition in Theorem 3.3(1) is not superfluous. According to Annin [2], a ring $R$ is $\sigma$-compatible if for each $a,b\in R$, $a\sigma(b)=0$ if and only if $ab=0$. Hashemi and Moussavi [9, Corollary 2.14] have proved Theorem 3.3(2), when $R$ is $\sigma$-compatible. Consider $R$ and $\sigma$ as in Example 2.2. Since $R$ is a domain then it is right $\sigma$-reversible (with $\sigma(1)=1$). Also $R$ is not $\sigma$-compatible (so $R$ does not satisfy the condition $(\mathcal{C_{\sigma}})$), because $\sigma$ is not a monomorphism. Therefore Theorem 3.3(2) is not a consequence of [9, Corollary 2.14]. On other hand, if $R$ is reversible then $\sigma$-compatibility implies right $\sigma$-reversibility. But, if $R$ is not reversible, we can easily see that this implication does not hold. ###### Theorem 3.4. Let $R$ be a right $\sigma$-reversible ring with $\sigma(1)=1$. If $R[[x;\sigma]]$ is a p.p.-ring then $R$ is a p.p.-ring. ###### Proof. Suppose that $R[[x;\sigma]]$ is a right p.p.-ring. Let $a\in R$, then there exists an idempotent $e\in R$ such that $r_{R[[x;\sigma]]}(a)=eR[[x;\sigma]]$ by Lemma 3.2. Hence $r_{R}(a)=eR$, and therefore $R$ is a right p.p.-ring. ∎ Also, in Example 2.2, $R$ is not $\sigma$-(sps) Armendariz. So Theorem 3.3 and Theorem 3.4 are not consequences of [4, Theorem 3.2]. Since $\sigma$-rigid rings are right $\sigma$-reversible [15, Theorem 2.8 (1)], we have the following Corollaries. ###### Corollary 3.5 ([10, Theorem 21]). Let $R$ be an $\sigma$-rigid ring. Then $R$ is a Baer ring if and only if $R[[x;\sigma]]$ is a Baer ring. ###### Corollary 3.6 ([10, Corollary 22]). Let $R$ be an $\sigma$-rigid ring. Then $R$ is a quasi-Baer ring if and only if $R[[x;\sigma]]$ is a quasi-Baer ring. ACKNOWLEDGEMENTS. The second author wishes to thank Professor Amin Kaidi of University of Almería for his generous hospitality. This work was supported by the project PCI Moroccan-Spanish A/011421/07. ## References [1] D.D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra, 27(6) (1999), 2847-2852. * [2] S. Annin, Associated primes over skew polynomials rings, Comm. Algebra, bf 30 (2002), 2511-2528. * [3] M. Baser, C.Y. Hong and T.K. Kwak, On extended reversible rings, Algebra Colloq., 16(1) (2009), 37-48. * [4] M. Baser, A. Harmanci and T.K. Kwak, Generalized semicommutative rings and their extensions, Bull. Korean Math. Soc., 45(2) (2008), 285-297. * [5] L. Ben Yakoub and M. Louzari, Ore extensions of extended symmetric and reversible rings, Inter. J. of Algebra, (to appear). * [6] G.F. Birkenmeier, J.Y. Kim, J.K. Park, Principally quasi-Baer rings, Comm. Algebra, 29(2) (2001), 639-660. * [7] W.E. Clark, Twisted matrix units semigroup algebras, Duke Math.Soc., 35 (1967), 417-424. * [8] P.M. Cohen, Reversible rings, Bull. London Math. Soc., 31(6) (1999), 641-648. * [9] E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta. Math. Hungar., 107 (3) (2005), 207-224. * [10] C.Y. Hong, N.K. Kim and T.K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure and Appl. Algebra, 151(3) (2000), 215-226. * [11] C.Y. Hong, N.K. Kim and T.K. Kwak, On Skew Armendariz Rings, Comm. Algebra, 31(1) (2003), 103-122. * [12] C.Y. Hong, T.K. Kwak and S.T. Rezvi, Extensions of generalized Armendariz rings, Algebra Colloq., 13(2) (2006), 253-266. * [13] I. Kaplansky, Rings of operators, Math. Lecture Notes series, Benjamin, New York, 1965. * [14] J. Krempa, Some examples of reduced rings, Algebra Colloq., 3(4) (1996), 289-300. * [15] T.K. Kwak, Extensions of extended symmetric rings, Bull. Korean Math.Soc., 44 (2007), 777-788. * [16] J. Lambek, On the reprentation of modules by sheaves of factor modules, Canad. Math. Bull., 14 (1971), 359-368. * [17] G. Marks, Reversible and symmetric rings, J. Pure Appl. Algebra, 174(3) (2002), 311-318. * [18] M.B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan. Acad. Ser. A Math. Sci., 73 (1997), 14-17. * [19] J.Y. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc., 184 (1973), 43-60.	arxiv-papers	2009-02-23T21:32:04	2024-09-04T02:49:00.824051	{ "license": "Public Domain", "authors": "Mohamed Louzari and L'moufadal Ben Yakoub", "submitter": "Louzari Mohamed", "url": "https://arxiv.org/abs/0902.3992" }
0902.4012	[labelstyle=] # When is the diagonal functor Frobenius? Alexandru Chirvăsitu Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, RO-70109 Bucharest 1, Romania chirvasitua@gmail.com ###### Abstract. Given a complete, cocomplete category $\mathcal{C}$, we investigate the problem of describing those small categories $I$ such that the diagonal functor $\Delta:\mathcal{C}\to{\rm Functors}(I,\mathcal{C})$ is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors $I\to\mathcal{C}$ are naturally isomorphic. We find necessary conditions on $I$ for a certain class of categories $\mathcal{C}$, and, as an application, we give both necessary and sufficient conditions in the two special cases $\mathcal{C}={\bf Set}$ or ${}_{R}\mathcal{M}$, the category of left modules over a ring $R$. ###### Key words and phrases: diagonal functor, Frobenius functor, complete, cocomplete, limit, colimit ###### 2000 Mathematics Subject Classification: 18A30, 18A35, 18A40, 18B05, 18B40 ## Introduction Functors having a left adjoint which is also a right adjoint were investigated by Morita in [10], where it is shown that given a ring morphism $R\to S$, the restriction of scalars functor has this property if and only if $R\to S$ is a Frobenius extension: $S$ is finitely generated and projective in ${}_{R}\mathcal{M}$, and $S\cong\ _{R}{\rm Hom}(S,R)$ as $(S,R)$-bimodules. Pairs of functors $F,G$ (between module categories) with the property that both $(F,G)$ and $(G,F)$ are adjunctions are called by Morita strongly adjoint pairs of functors. Later, a functor $F$ having a left adjoint which is also a right adjoint came to be referred to as a Frobenius functor ([3]), and Morita’s strongly adjoint pairs of functors are now known as Frobenius pairs. The natural question arises of when various well-known and extensively used functors are Frobenius. Examples include the already mentioned case of the restrictions of scalars functor for a ring extension ([9, 10]), forgetful functor from Doi-Hopf (or Doi-Koppinen) modules to modules ([3]), forgetful functor from $G$-graded modules over a $G$-graded ring to modules, where $G$ is a group ([4]), corestriction of scalars through an $A$-coring map $C\to D$ ([7], or [12] in the more general setting of a map from an $A$-coring $C$ to a $B$-coring $D$), and many more. In this paper the point of view is the following one: we fix a complete, cocomplete category $\mathcal{C}$, and seek to characterize those small categories $I$ for which the functors $\mathcal{C}^{I}\to C$ sending a functor in $\mathcal{C}^{I}$ to its limit and colimit are naturally isomorphic. We call such a category $\mathcal{C}$-Frobenius. The connection to Frobenius functors (hence the name $\mathcal{C}$-Frobenius) is highlighted by the following observation: the functor $\varprojlim:\mathcal{C}^{I}\to\mathcal{C}$ is right adjoint to the diagonal functor $\Delta:\mathcal{C}\to\mathcal{C}^{I}$, whereas the colimit functor is the left adjoint to $\Delta$. Hence our question can be rephrased as follows: for which small categories $I$ (depending on $\mathcal{C}$) is the diagonal functor $\Delta:\mathcal{C}\to\mathcal{C}^{I}$ a Frobenius functor? This question is investigated in [6], for discrete small categories $I$ (i.e. sets), and categories $\mathcal{C}$ enriched over the category of commutative monoids (referred to as ${\bf AMon}$ categories), and having a zero object. In that setting the problem is to find those sets $I$ for which direct sums and direct products in $\mathcal{C}$ indexed by $I$ are naturally isomorphic. The main result [6, Proposition 1.3] says that under reasonably mild conditions, this is equivalent to $I$ being finite. Here, on the other hand, we focus mainly on connected categories $I$. The structure of the paper is as follows: In Section 1 we introduce some conventions and prove Lemma 1.4, which allows us later on to break up the main problem into the two cases when $I$ is discrete (a set) or connected. In Section 2 we introduce the class of categories $\mathcal{C}$ we will be concerned with, which we call admissible, and also turn our attention to the case when $I$ is connected. Two general results, Theorem 2.7 and Proposition 2.8, are proven in this setting. In Section 3 necessary and sufficient conditions on $I$ are found in order that it be ${\bf Set}$-Frobenius or ${}_{R}\mathcal{M}$-Frobenius, where $R$ is a ring and ${}_{R}\mathcal{M}$ is the category of left $R$-modules. Since both ${\bf Set}$ and ${}_{R}\mathcal{M}$ are admissible in the sense of Section 2, the results proven there can be applied to the two particular cases. The conditions on $I$ appearing in the main results of Section 3 (Theorems 3.1 and 3.2) are of a combinatorial nature. The full description of the statements of these theorems requires some preparation (Definition 2.6), but they immediately imply, for instance, the characterization of ${\bf Set}$ or ${}_{R}\mathcal{M}$-Frobenius monoids $I$ (as usual, we regard a monoid as a category with a single object). A consequence of Theorem 3.1 is that the ${\bf Set}$-Frobenius monoids $I$ are precisely those containing an element $a\in I$ which is a fixed point for all left and right multiplications: $xa=ax=a,\ \forall x\in I$. Similarly, Theorem 3.2 implies that a monoid $I$ is ${}_{R}\mathcal{M}$-Frobenius if and only if it contains a finite (non-empty) set $S$ on which all multiplications, left or right, act as permutations, and such that the cardinality $\|S\|$ of $S$ is invertible in the ring $R$. The full description of connected Frobenius categories $I$ in the two cases is a natural generalization of this discussion. Finally, in Section 4 we finish with some open problems for the reader. ## 1\. Preliminaries Throughout this paper, $\mathcal{C}$ will denote a complete, cocomplete category, while $I$ stands for a small category. In general, for notions pertaining to category theory, we refer to [8]. The convention for composing morphisms is the usual one: given two morphisms $f:x\to y$ and $g:y\to z$ in a category, their composition is $gf:x\to z$. In order to keep the notation simple, if $i$ is an object of $I$ we write $i\in I$ (rather than $i\in{\rm Ob}(I)$, for example). Sometimes, in order to make it easier to keep track of the objects involved in morphisms, we shall denote $f\in{\rm Hom}(i,j)$ by $f_{i}^{j}$. Similarly, we might denote a subset $S\subseteq{\rm Hom}(i,j)$ by $S_{i}^{j}$. Given a set $S\subseteq{\rm Hom}(i,j)$ and a morphism $f\in{\rm Hom}(j,k)$, $fS$ stands for the set of all morphisms $fg,\ g\in S$; similarly for $Sf$, when the composition makes sense. Given categories $X,Y$, we denote the category ${\rm Functors}(X,Y)$ simply by $Y^{X}$. All functors are covariant, except when explicitly mentioned otherwise. ###### Definition 1.1. Let $\mathcal{C}$ be a complete, cocomplete category. A small category $I$ is said to be $\mathcal{C}$-Frobenius if the diagonal functor $\Delta:\mathcal{C}\to\mathcal{C}^{I}$ is a Frobenius functor. ###### Remark 1.2. The left adjoint to $\Delta$ is the functor $\mathcal{C}^{I}\to\mathcal{C}$, sending $F\in\mathcal{C}^{I}$ to its colimit $\varinjlim F$. Similarly, the right adjoint to $\Delta$ is the functor sending $F\in\mathcal{C}^{I}$ to its limit $\varprojlim F$ ([8, Chapter IV $\S$2]). Consequently, saying that $\Delta$ is Frobenius is the same as saying that $\varprojlim$ and $\varinjlim$ are naturally isomorphic. This means that we can find, for each functor $F\in\mathcal{C}^{I}$, an isomorphism $\psi_{F}:\varprojlim F\to\varinjlim F$ such that for every natural transformation $\eta:F\to G$ one has the commutative diagram $\begin{diagram}$ ###### Remark 1.3. Notice that the empty category is $\mathcal{C}$-Frobenius if and only if $\mathcal{C}$ has a zero object. In order to avoid splitting the arguments into cases, we assume from now on that all our categories are non-empty. We remarked earlier that we would be concerned primarily with the case when $I$ is connected. In fact, as the following lemma shows, the general problem of finding the $\mathcal{C}$-Frobenius small categories $I$ for a given $\mathcal{C}$ breaks up into the connected and the discrete case under certain conditions which do occur in the cases of interest. ###### Lemma 1.4. Let $\mathcal{C}$ be a complete, cocomplete category and $I$ a small category with connected components $I_{j},\ j\in J$. Then: 1. (a) If each component $I_{j}$ is $\mathcal{C}$-Frobenius and the set $J$, regarded as a discrete category, is $\mathcal{C}$-Frobenius, then $I$ is $\mathcal{C}$-Frobenius. 2. (b) If $I$ is $\mathcal{C}$-Frobenius, then $J$ is $\mathcal{C}$-Frobenius. 3. (c) The converse of ${\rm(a)}$ holds if $\mathcal{C}$ has a zero object. ###### Proof. Before proving the three assertions, we make some observations useful in all three arguments. Fix a functor $F\in\mathcal{C}^{I}$, and consider the contravariant functor $T_{F}:\mathcal{C}\to{\bf Set}$ defined by sending each object $c$ to the set of cones $\tau:c\stackrel{{\scriptstyle\cdot}}{{\rightarrow}}F$ (Mac Lane’s terminology and notation; see [8, Chapter III $\S$3]). The set of cones can also be defined as the object set of the comma category $c\downarrow F$ ([8, Chapter II $\S$6]). Since $I$ is small, the comma category is indeed small, so it makes sense to talk about its object set. Notice that the limit $\varprojlim F$ is precisely the representing object of $T_{F}$. Moreover, $F\mapsto T_{F}$ is natural in $F$. On the other hand, again having fixed $F\in\mathcal{C}^{I}$, consider the functor $S_{F}:\mathcal{C}\to{\bf Set}$ sending $c$ to collections of cones $\tau_{j}:c\stackrel{{\scriptstyle\cdot}}{{\rightarrow}}(F\|_{I_{j}}),\ j\in J$ from $c$ to the restrictions of $F$ to the connected components $I_{j}$. By the definition of limits, the representing object for $S_{F}$ is $\displaystyle\prod_{j\in J}(\varprojlim F\|_{I_{j}})$. Notice however that, since there are no morphisms between distinct connected components, the functors $T_{F}$ and $S_{F}$ actually coincide. In conclusion, the representing objects $\varprojlim F$ and $\displaystyle\prod_{j\in J}(\varprojlim F\|_{I_{j}})$ are in fact isomorphic; the isomorphism exhibited here is natural in $F$, because $F\mapsto T_{F}$ is. Similarly, $\varinjlim F\cong\displaystyle\coprod_{j\in J}(\varinjlim F\|_{I_{j}})$. We are now ready for the proof proper. (a) We have just seen that $\varprojlim F\cong\displaystyle\prod_{j\in J}(\varprojlim F\|_{I_{j}})$ naturally in $F$. Each $I_{j}$ is Frobenius, so the latter is isomorphic to $\displaystyle\prod_{j\in J}(\varinjlim F\|_{I_{j}})$ (naturally in $F$). The component set $J$ is Frobenius, so this is isomorphic to $\displaystyle\coprod_{j\in J}(\varinjlim F\|_{I_{j}})$ (again, naturally in $F$). Finally, the above discussion shows that this is isomorphic to $\varinjlim F$. (b) Instead of looking at the whole of $\mathcal{C}^{I}$, consider only those functors $I\to\mathcal{C}$ which restrict to constants on each component $I_{j}$. These are precisely the functors factoring through the canonical functor $\nu:I\to J$, which sends each $I_{j}$ to $j$. Again, use the isomorphism $\varprojlim F\cong\displaystyle\prod(\varprojlim F\|_{I_{j}})$: the limit of a constant functor on a connected category is easily seen to be precisely the image object (with all structural morphisms equal to the identity); it follows that in the case at hand, when $F$ restricts to a constant on each component, $\varprojlim F$ is naturally isomorphic to the product of the objects $F(I_{j})$. The same discussion applies to colimits: $\displaystyle\varinjlim F\cong\coprod F(I_{j})$. The desired conclusion that $J$ must be $\mathcal{C}$-Frobenius follows. (c) In view of (b), we must show that given the additional hypothesis of a zero object, each $I_{j}$ is $\mathcal{C}$-Frobenius. Fix some index $k\in J$, and consider only those functors $I\to\mathcal{C}$ which send each component $I_{j},\ j\neq k$ to the zero object $0$. Using once more the discussion at the beginning of the proof, we conclude that for these functors, the limit is naturally isomorphic to the product $\displaystyle\left(\varprojlim F\|_{I_{k}}\right)\times\prod_{j\neq k}0$. Since in any complete category product with the final object is naturally isomorphic to the identity, we conclude that $\varprojlim F\cong\varprojlim F\|_{I_{k}}$, naturally in $F$. Similarly, the colimit of $F$ is isomorphic to that of $F\|_{I_{k}}$, so $I_{k}$ must indeed be $\mathcal{C}$-Frobenius if $I$ is. ∎ ## 2\. Admissible categories, free objects, and some general results In the end, we are going to find the small categories $I$ which are ${\bf Set}$-Frobenius and those which are ${}_{R}\mathcal{M}$-Frobenius for a given ring $R$. Part of that proof will be unified by the results in this section, dealing with a certain class of categories $\mathcal{C}$ which contains both ${\bf Set}$ and ${}_{R}\mathcal{M}$, and many more familiar categories. We introduce this class below: ###### Definition 2.1. A category $\mathcal{C}$ is called admissible if: 1. (1) it is complete and cocomplete 2. (2) there is a faithful functor $U:\mathcal{C}\to{\bf Set}$ which has a left adjoint $T$ 3. (3) for at least one object $c$ of $\mathcal{C}$, the set $Uc$ has $\geq 2$ elements 4. (4) for any set $X$ and any element $t$ of the set $UT(X)$, there is a smallest finite subset $Y\subseteq X$ such that $t$ belongs to the set $UT(Y)$. For a set $X$, we denote the free object $T(X)$ by $T_{X}$. The faithful functor $U:\mathcal{C}\to{\bf Set}$ makes $\mathcal{C}$ into what is usually called a concrete category. Most of the time we will simply omit $U$, and regard $\mathcal{C}$ as a category whose objects are sets (with “additional structure”; that is, we keep in mind that the same set might correspond to different objects), and whose morphisms are functions between these sets. ###### Remark 2.2. Condition (3) implies that for each set $X$, the component $\psi_{X}:X\to UT_{X}$ of the unit of our adjunction $(T,U)$ is mono. Indeed, if $c$ is an object of $\mathcal{C}$ such that the set $Uc$ has at least two elements and $X$ is any set, then any two different elements of $X$ can be mapped to different elements of $Uc$, meaning that any two different elements of $X$ must have different images in the set $UT_{X}$. Hence, from now on we will regard $X$ as a subset of $UT_{X}$ (or of $T_{X}$, with the convention in the previous paragraph). Also, condition (3) implies that $T_{\emptyset}$ is not isomorphic to any of the other free objects, a fact that will be useful at some point: $T_{\emptyset}$ is initial, whereas any other free object admits at least two morphisms to any object $c\in\mathcal{C}$ such that $Uc$ has at least two elements. ###### Remark 2.3. Another observation which will be used tacitly from now on is this: inclusions of sets $X\to Y$ induce inclusions of sets $T_{X}\to T_{Y}$ (we omit $U$ in this remark). When $X\neq\emptyset$ this is clear: every monomorphism of sets $X\to Y$ is then a coretraction, and functors preserve coretractions. When $X=\emptyset$, on the other hand, $T_{X}$ is the initial object of $\mathcal{C}$. The initial object can be constructed, in any complete category, as a subobject of any weakly initial object (see [8, Chapter V $\S$6, proof of Theorem 1]). More precisely, it is the equalizer of all endomorphisms of any such object. By weakly initial we mean object admitting a morphism (not necessarily unique) to any object. Free objects are all weakly initial (unless $T_{\emptyset}=\emptyset$, in which case there is nothing left to prove), so $T_{\emptyset}$ is a subobject of each of them. Right adjoints (such as $U$) preserve monomorphisms, so, given a subset $X$ of $Y$, we will regard $T_{X}$ as a subset of $T_{Y}$; the inclusion is always the one induced by $X\to Y$. Here we make a short digression to identify many familiar categories which are, in fact, admissible. These are the so-called varieties of algebras, in the sense of Universal Algebra. For definitions and a detailed treatment we refer to [2, Chapter II]. Also, there is some discussion on the topic, from a more category theoretical point of view, in [8, Chapter V $\S$6]; here the main definitions are given, and the proof for the existence of free objects is sketched, using Freyd’s Adjoint Functor Theorem ([8, Chapter V $\S$6, Theorem 2]). We will not give complete proofs or definitions here. Given an $\mathbb{N}$-graded set $\Omega$ whose elements are called operations, an action of $\Omega$ on a set $A$ is a map assigning to each $\omega\in\Omega$ of degree $n\in\mathbb{N}$ a function $\omega_{A}:A^{n}\to A$. The degree $n$ is also called the arity of $\omega$. From the operations in $\Omega$, named fundamental operations, others can be derived, by composition and substitution; see the reference from Mac Lane. A set $E$ of equational identities is a set of pairs $(\mu,\nu)$ of derived operations having the same arity. A set $A$ with an $\Omega$ action is then said to satisfy the equations $E$ if $\mu_{A}=\nu_{A}$ for all $(\mu,\nu)\in E$. The class of all sets with an $\Omega$ action and satisfying the identities $E$ will be denoted by $\langle\Omega,E\rangle-{\bf Alg}$, and a member of this class will be called an $\langle\Omega,E\rangle$-algebra. A morphism between algebras $A,B\in\langle\Omega,E\rangle-{\bf Alg}$ is a map $f:A\to B$ which, for each $\omega\in\Omega$, makes the following diagram commutative: $\begin{diagram}$ We now have a category $\langle\Omega,E\rangle-{\bf Alg}$. Examples include the categories of sets (no operations at all), monoids, groups, rings, modules over a ring $R$ (these are abelian groups with some unary operations describing multiplications with scalars in $R$), Lie algebras, etc. Notice that we allow the underlying set of an algebra to be empty, although the authors of [2] do not. A variety contains the empty set if and only if there are no nullary operations (i.e. operations of arity $0$). The definitions allow for a variety of algebras not to satisfy condition (3) of Definition 2.1. Assuming it does, however, it can be shown that $\langle\Omega,E\rangle-{\bf Alg}$ is admissible, with $U$ (from Definition 2.1) being the forgetful functor, which sends an algebra to its underlying set, and a morphism of algebras to the corresponding map of sets. We will not give the complete proof here. As mentioned above, Mac Lane proves the existence of free objects indirectly, using the Adjoint Functor Theorem. In [2, Chapter II $\S$10] an explicit construction of free objects is given. Condition (4) follows from the fact that a filtered union of $\langle\Omega,E\rangle$-algebras is again such with an obvious structure; it is easy to check the required universality property for the union of all $T_{Y}$ as $Y$ ranges through the finite subsets of $X$, which makes it into the free object on $X$; hence, elements of free objects are contained in finitely generated free subobjects. Finally, (4) is proven by noticing that for varieties of algebras, one always has $T_{Y}\cap T_{Z}=T_{Y\cap Z}$ (including the case when there are no nullary operations, and the second set in this equality happens to be empty). This can be seen by constructing the free objects explicitly. For completeness and cocompleteness one can mimic the usual constructions of products, coproducts, equalizers and coequalizers from group theory, for example. In particular, ${\bf Set}$ and ${}_{R}\mathcal{M}$ are admissible. Of course, this can be seen directly. We will need the next lemma in the proof of Theorem 2.7. Recall that a directed graph (digraph) is said to be strongly connected if for any two vertices $i,j$ there is a directed path from $i$ to $j$. A digraph is said to be transitive if whenever we have directed paths $i\to j$ and $j\to k$ we also have a directed path $i\to k$. The underlying graph of a category is transitive, for instance. If a digraph is transitive then strong connectedness is equivalent to having an edge $i\to j$ for any pair of distinct vertices $i,j$. ###### Lemma 2.4. Let $\mathcal{C}$ be an admissible category, and $I$ a small, connected, $\mathcal{C}$-Frobenius category. Then $I$ is in fact strongly connected, i.e. ${\rm Hom}(i,j)\neq\emptyset$ for all pairs of objects $i,j\in I$. ###### Proof. We will make use of the following well-known combinatorial result: if a connected directed graph is not strongly connected, then its vertex set can be partitioned into two non-empty subsets $A,\ B$ such that all the arrows connecting them go from $A$ to $B$. Moreover, $A$ can be chosen to be connected. Assuming that $I$ is not strongly connected, apply this to the underlying graph of $I$. We get non-empty, full subcategories $A,\ B$ of $I$ with $A$ connected, which partition its object set, and such that all morphisms between $A$ and $B$ go from $A$ to $B$. Now consider the functor $F\in\mathcal{C}^{I}$ which restricts to the constant functor $T_{\emptyset}$ on $A$, to the constant $T_{1}$ on $B$, and sends all morphisms $A\to B$ onto the unique morphism $T_{\emptyset}\to T_{1}$: $\begin{diagram}$ An argument very similar to the one used in the proof of Lemma 1.4 (the beginning of that proof) shows that the limit of $F$ is $T_{\emptyset}$. On the other hand, the colimit is the coproduct of one copy of $T_{1}$ for each connected component $B_{j},\ j\in J$ of $B$; here $J$ is simply the (non- empty) set of connected components. $T$ is a left adjoint by definition, so it preserves coproducts; this means that $\displaystyle\coprod_{J}T_{1}\cong T_{J}$. We have already remarked, in the discussion after Definition 2.1, that $T_{\emptyset}$ cannot be isomorphic to a free object $T_{J},\ J\neq\emptyset$, so $I$ is not $\mathcal{C}$-Frobenius. We have reached a contradiction. ∎ ###### Remark 2.5. Notice that in the above proof, instead of the unique arrow $T_{\emptyset}\to T_{1}$ we could just as well have taken the unique arrow from an initial object to a non-initial object. Hence the statement holds for any (complete, cocomplete) category $\mathcal{C}$ having at least one object which is not initial. The following definition is crucial in subsequent results. $I$ stands for a small category. ###### Definition 2.6. A left invariant system (LS) of $I$ is a collection of finite, non-empty sets $S_{i}^{j}\subseteq{\rm Hom}(i,j)$, one for each pair $i,j\in I$, such that composition to the left with any $f_{j}^{k}\in{\rm Hom}(j,k)$ sends $S_{i}^{j}$ bijectively onto $S_{i}^{k}$ for all $i,j,k\in I$. A right invariant system (RS) of $I$ is a collection of finite, non-empty sets $S_{i}^{j}\subseteq{\rm Hom}(i,j)$, one for each pair $i,j\in I$, such that composition to the right with any $f_{k}^{i}\in{\rm Hom}(k,i)$ sends $S_{i}^{j}$ bijectively onto $S_{k}^{j}$ for all $i,j,k\in I$. An invariant system (IS) of $I$ is an LS which is also an RS. The main result of this section follows: ###### Theorem 2.7. Let $\mathcal{C}$ be an admissible category, and let $I$ be a small, connected, $\mathcal{C}$-Frobenius category. Then $I$ has an IS. ###### Proof. The functors in $\mathcal{C}^{I}$ we will work with are $i^{}=T_{{\rm Hom}(i,-)}$ for objects $i\in I$. $T$ being a left adjoint, it preserves colimits. In other words, $\varinjlim i^{}\cong T_{\varinjlim{\rm Hom}(i,-)}$. By the description of colimits in ${\bf Set}$ one sees immediately that $\varinjlim{\rm Hom}(i,-)$ is a singleton. In conclusion, $\varinjlim i^{}\cong T_{1}$. By the $\mathcal{C}$-Frobenius property we can identify $\varprojlim i^{}$ with $T_{1}$ as well. We will denote by $1$ the element generating $T_{1}$; in the present context it corresponds to the image of any morphism in ${\rm Hom}(i,j)$ through the canonical map ${\rm Hom}(i,j)\to T_{\varinjlim{\rm Hom}(i,-)}\cong\varprojlim i^{}$. Let $\psi_{i}^{j}:T_{1}\cong\varprojlim i^{}\to T_{{\rm Hom}(i,j)}$ be the structure map of the limit, and denote by $x_{i}^{j}$ the element $\psi_{i}^{j}(1)\in T_{{\rm Hom}(i,j)}$ (keep in mind the convention made after Definition 2.1: we regard the objects of $\mathcal{C}$ simply as sets, omitting the faithful functor $U:\mathcal{C}\to{\bf Set}$). By condition (4) of Definition 2.1, there is a smallest finite set $S\subseteq{\rm Hom}(i,j)$ such that $x_{i}^{j}\in T_{S}$. Denote it by $S_{i}^{j}$; as the notation suggests, these will be the components of our IS. $(S_{i}^{j})$ is an LS. For all $j,k\in I$ and all $f_{j}^{k}$ we have a commutative diagram $\begin{diagram}$ It follows that $(i^{}f_{j}^{k})(x_{i}^{j})=x_{i}^{k}$, so, by the definition of the sets $S_{i}^{j}$, we have $f_{j}^{k}S_{i}^{j}\supseteq S_{i}^{k}$. In other words, composition to the left maps $S_{i}^{j}$ onto a set containing $S_{i}^{k}$. A consequence of this is that $\|S_{i}^{k}\|\leq\|S_{i}^{j}\|$ whenever the hom set ${\rm Hom}(j,k)$ is non-empty. However, we know from Lemma 2.4 that all hom sets are nonempty, so all $S_{i}^{j}$ have the same cardinality. Moreover, composition to the left with any morphism must be a bijection. All we need to do now in order to conclude that $S=(S_{i}^{j})$ is an LS is to show that the sets $S_{i}^{j}$ are non-empty. Assume they are. Then $\psi_{i}^{j}$ maps $\varprojlim i^{}\cong T_{1}$ into $T_{\emptyset}\subset T_{{\rm Hom}(i,j)}=i^{}(j)$ for all $j$. This means that the limiting cone $\varprojlim i^{}\stackrel{{\scriptstyle\cdot}}{{\rightarrow}}i^{}$ factors through $T_{\emptyset}\stackrel{{\scriptstyle\cdot}}{{\rightarrow}}i^{}$ which, in turn, implies that $T_{1}\cong\varprojlim i^{}\cong T_{\emptyset}$. This is impossible by condition (3) in Definition 2.1 (see Remark 2.2). $(S_{i}^{j})$ is an RS. This is where the naturality of $\eta:\varprojlim\cong\varinjlim$ comes in. More pecisely, consider any morphism $f=f_{i}^{j}\in{\rm Hom}(i,j)$. It induces a natural transformation $f^{}$ from $j^{}$ to $i^{}$. The corresponding transformations $\varprojlim j^{}\to\varprojlim i^{}$ and $\varinjlim j^{}\to\varinjlim i^{}$ will again be denoted by $f^{}$. For each $k\in I$ we have the following commutative diagram: $\begin{diagram}$ The horizontal arrows of the left square are the components of the natural isomorphism $\eta:\varinjlim\cong\varprojlim$. Notice that $1\in T_{1}\cong\varinjlim j^{}$ gets mapped onto $1\in T_{1}\cong\varinjlim i^{}$ (see the description of $1$ in the first paragraph of the proof). Since we have identified $\varprojlim j^{}$ to $T_{1}$ through $\eta$, it follows from this diagram that $f^{}(x_{j}^{k})=x_{i}^{k}$. By the definiton of the sets $S$, this means that $S_{j}^{k}f_{i}^{j}\supseteq S_{i}^{k}$. Now we continue as in the proof for left invariance, using the fact that all hom sets are non-empty ($I$ is strongly connected). ∎ Let $I$ be a small, connected category with an IS $(S_{i}^{j})$ (in particular, $I$ will be strongly connected). Consider a set $S_{i}^{i}$ for some object $i\in I$. Composition of morphisms gives such a set a structure of finite semigroup in which all multiplications, left or right, act as permutations. It is not difficult to see that such a semigroup is in fact a group. Indeed, since all multiplications act as permutations of a finite set, some power of any element acts as an identity; hence the semigroup is a monoid. Since every element permutes the monoid both by right and by left multiplication, every element has both a left and a right inverse, and so the monoid must be a group. All our $S_{i}^{i}$ are then finite groups (their identites may not coincide with the identity $1_{i}$ in the category $I$). Denote by $e_{i}^{i}$ the identity of this group structure on $S_{i}^{i}$; it is the unique idempotent morphism in $S_{i}^{i}$. In fact, $e_{i}^{i}$ acts as the identity not only on $S_{i}^{i}$, but on all $S_{i}^{j}$ by right multiplication and on all $S_{j}^{i}$ by left multiplication. This is easily seen from the fact that these actions are permutations and the idempotence of $e_{i}^{i}$. Now consider the subgraph of the underlying graph of $I$ whose vertices are all the objects of $I$ and whose arrows are those belonging to the sets $S_{i}^{j}$. Composition of arrows in $I$ gives this graph a structure of category, with identities $e_{i}^{i}$; this follows from the discussion in the previous paragraph. In fact, this category is a groupoid: given $s_{i}^{j}\in S_{i}^{j}$, take any $s_{j}^{i}\in S_{j}^{i}$. Then the composition $s_{j}^{i}s_{i}^{j}$ belongs to the group $S_{i}^{i}$, so it must be invertible. This means that any morphism $s_{i}^{j}$ in our new category is left invertible, so all morphisms are invertible. We will denote this groupoid by $\mathcal{G}_{I}$. Notice that it is connected, and the automorphism groups of the vertices are the groups $S_{i}^{i}$. In particular, all these groups are isomorphic. We denote this unique finite group by $G_{I}$. Of course, when regarded as a category with only one object, $G_{I}$ is equivalent to $\mathcal{G}_{I}$. The groupoid $\mathcal{G}_{I}$ is embedded in $I$ graph-theoretically, but the embedding is not necessarily a functor, since it need not preserve identities. There is, however, a canonical functor $\tau:I\to\mathcal{G}_{I}$ which is a left inverse to the embedding of graphs $\mathcal{G}_{I}\to I$, and which makes $\mathcal{G}_{I}$ into the enveloping groupoid of $I$. We do not require this last fact, but we will define the mentioned functor $\tau$; it is simply the map which acts on morphisms as follows: $\begin{diagram}$ The properties of $e_{i}^{i}$ noted above prove that the restriction of $\tau$ to the subgraph $\mathcal{G}_{I}\subset I$ is the identity, and also that $\tau$ is indeed a functor. The following result will be useful in dealing with the categories ${\bf Set}$ and ${}_{R}\mathcal{M}$ in the next section. ###### Proposition 2.8. Let $I$ be a small connected category with an IS consisting of the sets $(S_{i}^{j})$, and let $\mathcal{C}$ be any complete, cocomplete category. Then $I$ is $\mathcal{C}$-Frobenius if and only if the group $G_{I}$ (regarded as a category) is $\mathcal{C}$-Frobenius. Before embarking on the proof, we need some preparations. Denote by $M$ the two-element monoid $\\{1,e\\}$, where $1$ is the identity and $e$ is idempotent. Then, regarding $M$ as a one-object category, we have the following simple result: ###### Lemma 2.9. $M$ is $\mathcal{C}$-Frobenius for any complete, cocomplete category $\mathcal{C}$. ###### Proof. A functor $M\to\mathcal{C}$ is an action of $M$ on some object $c\in\mathcal{C}$, i.e. a monoid morphism $M\to{\rm Hom}(c,c)$. For such a functor $F$, glue the limting and the colimiting cone into the following commutative diagram: $\begin{diagram}$ Because $e$ is idempotent, we get a cone $\begin{diagram}$ which induces a unique morphism $\xi:c\to\varprojlim F$ such that $Fe=\phi\xi$. From the uniqueness of $\xi$ we get $\xi\circ Fe=\xi$. Now the commutative diagram $\begin{diagram}$ and the universality of the limit prove that $\xi\phi$ is the identity of $\varprojlim F$. Dually, one finds $\eta:\varinjlim F\to c$ through which $Fe$ factors, with the properties $Fe\circ\eta=\eta$ and $\psi\eta=1_{\varinjlim F}$. Putting all of this together we see that the composition $\xi\eta:\varinjlim F\to\varprojlim F$ is the inverse of the natural morphism $\psi\phi:\varprojlim F\to\varinjlim F$ (bottom row of the first diagram above). All the constructions used above are natural with respect to $F$, so we get a natural isomorphism $\varprojlim\cong\varinjlim$, as desired. ∎ Let $G$ be a semigroup, and denote by $G^{+}$ the monoid obtained by adjoining an identity to $G$. As a set, it consists of $G$ together with an element $1$; multiplication on $G$ is the one inherited from the semigroup structure of $G$, and $1$ acts as a unit on $G^{+}=G\cup\\{1\\}$. When $G$ was a group to begin with (or more generally a monoid), we denote its unit by $e$. Notice that $e$ is an idempotent in $G^{+}$, but it is no longer the unit for the multiplication in $G^{+}$. In the proof of Proposition 2.8 we make use of the following lemma: ###### Lemma 2.10. Let $\mathcal{C}$ be any complete and cocomplete category, and let $G$ be a $\mathcal{C}$-Frobenius group. Then the monoid $G^{+}$ is also $\mathcal{C}$-Frobenius. ###### Proof. The two-element monoid $M$ from the previous lemma is embedded in $G^{+}$ as $\\{1,e\\}$, where $1$ is the identity of $G^{+}$ and $e$ is the identity of $G$. A functor $F:G^{+}\to\mathcal{C}$ is an action of the monoid $G^{+}$ on some object $c\in C$. Restrict this action to the submonoid $M\leq G^{+}$, and let $\phi:d\to c$ be the limiting cone of the restriction $F\|_{M}$. We construct an action $F^{}$ of $G$ on $d$ as follows: for every $s\in G$ we have a commutative diagram $\begin{diagram}$ which induces a unique endomorphism $F^{}s$ of $d$ making the following diagram commutative: $\begin{diagram}$ That $F^{}$ is indeed a functor is easily checked; it must preserve composition by uniqueness because $F$ does, and $F^{}e$ is the identity because $\phi:d\to c$ is a cone from $d$ to $F\|_{M}$, and $e$ is a morphism in $M$. I claim now that $\varprojlim F$ is naturally isomorphic to $\varprojlim F^{}$. Indeed, because $\phi:d\to c$ is limiting, any cone $\varphi:t\stackrel{{\scriptstyle\cdot}}{{\rightarrow}}c$ which (by definition) makes commutative the diagrams $\begin{diagram},\qquad\forall s\in G$ must factor through $d$: $\begin{diagram},\qquad\forall s\in G$ Dually, one constructs an action $F_{}$ of $G$ on $\varinjlim F\|_{M}$, and we have a natural isomorphism $\varinjlim F\cong\varinjlim F_{}$. Now, because $M$ is always $\mathcal{C}$-Frobenius (Lemma 2.9), $\varprojlim F\|_{M}\cong\varinjlim F\|_{M}$ naturally. Moreover, recall from the proof of Lemma 2.9 that the isomorphism between $\varprojlim F\|_{M}$ and $\varinjlim F\|_{M}$ we have exhibited was precisely the composition of natural maps $\varprojlim F\|_{M}\to c\to\varinjlim F\|_{M}$. The actions $F^{}$ and $F_{}$ were constructed such that the following diagrams are commutative: $\begin{diagram}\qquad\forall s\in G$ Hence, upon identifying the limit and colimit of $F\|_{M}$ by the given isomorphism, the action $F_{}$ is identified to $F^{}$. The conclusion now follows from the hypothesis that $G$ is $\mathcal{C}$-Frobenius. ∎ Finally, we are ready to prove Proposition 2.8 ###### Proof of Proposition 2.8. We have noticed in the discussion above that $G_{I}$ and $\mathcal{G}_{I}$ are equivalent categories, so we can replace $G_{I}$ with $\mathcal{G}_{I}$ in the statement of the proposition. Assume first that $I$ is $\mathcal{C}$-Frobenius. Since $\tau:I\to\mathcal{G}_{I}$ is a retraction onto the subgraph $\mathcal{G}_{I}\to I$, it is bijective on objects and surjective on morphisms. From this it follows immediately that for every $c\in\mathcal{C}$ the cones $c\stackrel{{\scriptstyle\cdot}}{{\rightarrow}}F$ coincide with the cones $c\stackrel{{\scriptstyle\cdot}}{{\rightarrow}}F\tau$. Consequently, the canonical morphism $\varprojlim F\to\varprojlim F\tau$ is an isomorphism. Similarly, $\varinjlim F$ is isomorphic to $\varinjlim F\tau$, naturally in $F$. Applying the $\mathcal{C}$-Frobenius property to the functors in $\mathcal{C}^{I}$ of the form $F\tau$, this discussion implies that $\mathcal{G}_{I}$ and hence $G_{I}$ must be Frobenius as well. Conversely, assume that $\mathcal{G}_{I}$ (and so $G_{I}$) is $\mathcal{C}$-Frobenius. For each object $i\in I$, denote by $M_{i}$ the submonoid of ${\rm Hom}(i,i)$ consisting of the elements of $S_{i}^{i}$ together with the identity. If $S_{i}^{i}$ already contains the identity, then $M_{i}$ is isomorphic to the group $G_{I}\cong S_{i}^{i}$. Otherwise, it will be isomorphic to the monoid denoted above by $G_{I}^{+}$. Either way, we know (Lemma 2.10) that $M_{i}$ is a $\mathcal{C}$-Frobenius monoid. Given an object $i\in I$ and a functor $F\in\mathcal{C}^{I}$, let $F_{i}$ be the restriction $F\|_{M_{i}}$. If we manage to prove that $\varprojlim F\cong\varprojlim F_{i}$ naturally (for a fixed $i\in I$), then the dual argument would apply to show that $\varinjlim F\cong\varinjlim F_{i}$; from the fact that $M_{i}$ is Frobenius it would then follow that $I$ is also. Hence it remains to prove that there is a natural isomorphism $\varprojlim F\cong\varprojlim F_{i}$. Let $\phi_{i}:d_{i}\to F(i)$ be the limiting cones for $F_{i}$. for objects $i,j\in I$, consider an arbitrary morphism $f_{i}^{j}\in{\rm Hom}(i,j)$. I claim that there is a unique morphism $\phi_{i}^{j}:d_{i}\to d_{j}$ making the following diagram commutative, and that moreover, it does not depend on the morphism $f_{i}^{j}$: $\begin{diagram}$ Independence of $f_{i}^{j}$ is immediate: since $\phi_{i}:d_{i}\to F_{i}$ is a cone from $d_{i}$ to $F_{i}=F\|_{M_{i}}$, we have $Fs_{i}^{i}\circ\phi_{i}=\phi_{i}$ for every morphism $s_{i}^{i}\in S_{i}^{i}\subseteq M_{i}$. Composing to the left with $Ff_{i}^{j}$ and using the invariance properties of the IS $(S_{i}^{j})$, we get $Ff_{i}^{j}\circ\phi_{i}=Fs_{i}^{j}\circ\phi_{i}$ for any $s_{i}^{j}\in S_{i}^{j}$. The existence of $\phi_{i}^{j}$ also follows from this discussion, for it follows that composition to the right with any $Fs_{j}^{j},\ s_{j}^{j}\in S_{j}^{j}$ fixes $Ff_{i}^{j}\circ\phi_{i}$, so this latter morphism gives a cone $d_{i}\stackrel{{\scriptstyle\cdot}}{{\rightarrow}}F_{j}$. From the uniqueness of all $\phi_{i}^{j}$ (including the cases $i=j$) it follows that they are isomorphisms; more precisely, for every $i,j\in I,\ \phi_{j}^{i}$ is the inverse of $\phi_{i}^{j}$. From the universality of $\phi_{i}:d_{i}\to F(i)$ and the construction of $\phi_{i}^{j}$ it follows that every cone $c\stackrel{{\scriptstyle\cdot}}{{\rightarrow}}F$ factors through maps $\psi_{i}:c\to d_{i}$ making commutative the triangles $\begin{diagram}\qquad i,j\in I$ Now, since $\phi_{i}^{j}$ are isomorphisms, this says that $\varprojlim F$ is naturally isomorphic to $d_{i}=\varprojlim F_{i}$ (the constructions appearing above are natural in $F$ once we fix an object $i\in I$). We have thus reached the desired conclusion. ∎ ## 3\. Special cases: sets and modules In this section we characterize those small $I$ (not necessarily connected) which are ${\bf Set}$-Frobenius and ${}_{R}\mathcal{M}$-Frobenius for a ring $R$. Section 1 and Section 2 will allow us to obtain both necessary and sufficient conditions on $I$ in order that it be Frobenius for these categories. We have already remarked in the discussion on varieties of algebras above that ${\bf Set}$ and ${}_{R}\mathcal{M}$ are admissible categories, so the results in Section 2 apply in both cases. Remember that all our categories are non-empty. The following theorem describes the Set-Frobenius categories: ###### Theorem 3.1. A small category $I$ is ${\bf Set}$-Frobenius if and only if it is connected and it has an IS consisting of singletons $S_{i}^{j}$. ###### Proof. Assume $I$ satisfies the conditions in the statement. Then the group $G_{I}\cong S_{i}^{i},\ \forall i\in I$ introduced in the discussion before Proposition 2.8 is the trivial group. From Proposition 2.8 we know that in order to conclude that $I$ is Frobenius, it suffices to check that $G_{I}$ is. It is clear that the trivial group is $\mathcal{C}$-Frobenius for any complete, cocomplete category $\mathcal{C}$, and the proof of this implication is finished. Conversely, suppose $I$ is ${\bf Set}$-Frobenius. Lemma 1.4 (b) then tells us that the set $J$ of connected components of $I$, viewed as a discrete category, must be ${\bf Set}$-Frobenius. The only non-empty ${\bf Set}$-frobenius discrete category is the singleton: notice for instance that the product of a non-empty set and at least one copy of the empty set is empty, whereas the disjoint union of all these sets is non-empty. Hence $J$ is a singleton, i.e. $I$ is connected. Now Theorem 2.7 applies to show that $I$ has an IS consisting of finite non- empty sets $S_{i}^{j}$. Now we go once more through the argument in the first paragraph, in reverse: Proposition 2.8 says that $I$ is ${\bf Set}$-Frobenius if and only if the finite group $G_{I}$ is, so we have to prove that the only ${\bf Set}$-Frobenius finite group is the trivial group. Functors from $G_{I}$ to ${\bf Set}$ are actions of $G_{I}$ on a set. They have easily described limits and colimits: the limiting cone of an action of $G_{I}$ on the set $c$ is the inclusion of the set of points in $c$ fixed by all elements of $G_{I}$. The colimiting cone, on the other hand, is the canonical projection of $c$ onto the set of orbits of the action (sending each element onto its orbit). In particular, we see that the colimit of an action on a non-empty set is always a non-empty set, whereas one can always find actions on non-empty sets with no fixed points whenever $G_{I}$ is non- trivial: simply make $G_{I}$ act on itself by left multiplication, for example. ∎ For $R$-modules, the result reads as follows: ###### Theorem 3.2. Let $R$ be a ring. A small category $I$ is ${}_{R}\mathcal{M}$-Frobenius if and only if it has finitely many components, each of which has an IS consisting of finite sets $S_{i}^{j}$ such that $\|S_{i}^{j}\|$ is invertible in the ring $R$. In the course of the proof we will make use of the following result regarding discrete categories: ###### Proposition 3.3. Let $R$ be a ring, and $J$ a set. The discrete category $J$ is ${}_{R}\mathcal{M}$-Frobenius if and only if it is finite. ###### Proof. This is [6, Theorem 2.7]. In that paper it is both an immediate consequence of the main result [6, Theorem 1.4], and proved separately using a finiteness result on Frobenius corings ([6, Theorem 2.3]; see also [1, $\S$27] for definitions and relevant results on Frobenius corings). We give here a different proof, relying on another proposition found in [6]. On the one hand, it is well-known that finite sets are ${}_{R}\mathcal{M}$-Frobenius. In fact, products and coproducts are canonically isomorphic in any additive category. Conversely, assume that $J$ is ${}_{R}\mathcal{M}$-Frobenius. Now [6, Proposition 1.2] says that the canonical map $\displaystyle\bigoplus_{J}\to\prod_{J}$ is a natural isomorphism. Consider the composition $\displaystyle\begin{diagram}$ in which the first arrow is the map with all components equal to the identity on $R$, while the second arrow is the inverse of the canonical isomorphism $\displaystyle\bigoplus_{J}\to\prod_{J}$. It is a morphism from $R$ to $\displaystyle\bigoplus_{J}R$ having the property that the image of $R$ is not contained in any $\displaystyle\bigoplus_{J^{\prime}}R$ for $J^{\prime}\subsetneq J$. As $R$ is a finitely generated $R$-module, however, its image is certainly contained in a finite direct sum. Hence $J$ must be finite. ∎ ###### Remark 3.4. It is clear that the direct sum of infinitely many non-zero modules is strictly smaller than their direct product. However, note that the proof, arranged as above, applies to all (complete, cocomplete) abelian categories having a non-zero small object. We say that an object $x$ in a category with coproducts is small if any morphism of $x$ to a coproduct factors through a finite coproduct. Indeed, [6, Proposition 1.2] covers this situation as well (and in fact holds for all categories enriched over the category of commutative monoids and having a zero object), and all we need to do is replace $R$ in the above proof with a small, non-zero object. A cocomplete abelian category with a small projective generator is equivalent to some ${}_{R}\mathcal{M}$ ([5, Chapter 4, exercises E and F]). There are, however, examples of complete, cocomplete abelian categories with a non-zero small object and which are not equivalent to some ${}_{R}\mathcal{M}$. We give such an example below. ###### Example 3.5 (Torsion modules). Let $R$ be a DVR (discrete valuation ring), and let $\mathcal{C}$ be the full subcategory of ${}_{R}\mathcal{M}$ consisting of torsion modules. $\mathcal{C}$ is an abelian category, because kernels, cokernels, finite direct sums, etc. of morphisms of torsion modules are morphisms of torsion modules. Completeness and cocompleteness are, again, easily checked: the direct sum in ${}_{R}\mathcal{M}$ is also the direct sum in $\mathcal{C}$, and the direct product in $\mathcal{C}$ is the torsion of the direct product in ${}_{R}\mathcal{M}$. Finally, the category has non-zero small objects: any non-zero finitely generated torsion module will do. A small projective object in $\mathcal{C}$ must be finitely generated, and the structure theorem for finitely generated modules over a PID now easily shows that $\mathcal{C}$ has no non-zero small projectives, hence cannot be equivalent to some ${}_{S}\mathcal{M}$. ###### Remark 3.6. Although we will not prove this here, with a little more work, it can be shown that the previous example still works if $R$ is taken to be any noetherian local integral domain (which is not a field). At the other end of the spectrum, when working with connected categories, we will need the following characterization of ${}_{R}\mathcal{M}$-Frobenius groups: ###### Proposition 3.7. Let $R$ be a ring and $G$ a group, regarded as a one-object category. $G$ is ${}_{R}\mathcal{M}$-Frobenius if and only if it is finite, and the natural number $\|G\|$ is invertible in $R$. ###### Proof. Functors $G\to\ _{R}\mathcal{M}$ are precisely $R$-modules with a $G$ action, or, in other words, $R[G]$-modules. The diagonal functor ${}_{R}\mathcal{M}\to(_{R}\mathcal{M})^{G}$ associates to each $R$-module the same module with trivial $G$ action. This means that one can identify the diagonal functor with the restriction of scalars from $R$ to $R[G]$ through the augmentation $\varepsilon:R[G]\to R$ (the unique ring morphism sending each element of $G\subset R[G]$ to the identity $1_{R}\in R$). The problem has now been reduced to the classical question of deciding when a restriction of scalars is Frobenius. By a well-known result of Morita ([10] or [9, Theorem 3.15]), restriction of scalars through a ring morphism $A\to B$ is Frobenius if and only if $B$ is left $A$-projective and finitely generated, and $B\cong\ _{A}{\rm Hom}(B,A)$ as $(B,A)$-bimodules. We are going to apply this characterization to the ring extension $\varepsilon:R[G]\to R$. $R$ is left $R[G]$-projective if and only if the augmentation $\varepsilon:R[G]\to R$ splits through some left $R[G]$-module map $\eta:R\to R[G]$. For any such splitting, $\eta(1)$ is some element $\displaystyle\sum_{g\in G}a_{g}g$ of $R[G]$ fixed by left multiplication with any element of $G$. This shows at once that $G$ must be finite, and that $\displaystyle\eta(1)=a\sum_{g\in G}g$. Finally, from $\varepsilon\circ\eta={\rm id}_{R}$ we find that $a\in R$ must in fact be the inverse of $\|G\|$. Conversely, if $\|G\|<\infty$ is invertible in $R$, simply consider the $R$-module map sending $1\in R$ to $\displaystyle\|G\|^{-1}\sum_{g\in G}g\in R[G]$; clearly, it is a splitting for $\varepsilon$. We still have to prove that when $G$ satisfies the condtitions in the statement of the proposition (and hence, as we have just seen, $R$ is left $R[G]$-projective), we also have an isomorphism $\displaystyle R\cong\ _{R[G]}{\rm Hom}(R,R[G])$ in ${}_{R}\mathcal{M}_{R[G]}$. The second term is canonically isomorphic to the $(R,R[G])$-sub-bimodule of $R[G]$ generated by the central idempotent $e=\displaystyle\|G\|^{-1}\sum_{G}g$; there is an obvious $(R,R[G])$-bimodule isomorphism of $R$ onto this bimodule, sending $1$ to $e$. ∎ We are now ready to prove the theorem. ###### Proof of Theorem 3.2. Since ${}_{R}\mathcal{M}$ is a complete, cocomplete category with a zero object, points (a) and (c) of Lemma 1.4 show that $I$ is Frobenius if and only if (i) its set of connected components $J$ is Frobenius, and (ii) each connected component is Frobenius. Hence the problem breaks up into the discrete and the connected case. Proposition 3.3 says that the component set is ${}_{R}\mathcal{M}$-Frobenius if and only if it is finite. In the connected case we can apply the results in Section 2. Theorem 2.7 and Proposition 2.8 together imply that a connected category is ${}_{R}\mathcal{M}$-Frobenius if and only if it has an IS such that the group $G_{I}$ is ${}_{R}\mathcal{M}$-Frobenius. Finally, apply Proposition 3.7 to finish the proof. ∎ ## 4\. Some open problems The problem posed here, of finding the $\mathcal{C}$-Frobenius categories $I$ for a fixed complete and cocomplete $\mathcal{C}$, has variations which would make interesting topics for further inquiry. We give only a few examples. For one thing, we would like to extend the results obtained in this paper to various categories (or perhaps large classes of categories) which were not covered here. One conspicuous example is that of the category of (left or right) comodules over some $R$-coring $C$. This would cover the case of $R$-modules, since these are the simply the comodules over the Sweedler coring $R$ over $R$ ([1, Examples 17.3 and 18.5]). Choose right comodules, in order to fix the notation. Because we want the category $\mathcal{M}^{C}$ of right comodules to be complete and cocomplete, we impose the condition that ${}_{R}C$ be flat (see [1, Theorem 18.13]). ###### Problem 1. Given a ring $R$ and an $R$-coring $C$ which is flat as a left $R$-module, find the $\mathcal{M}^{C}$-Frobenius small categories $I$. Even within the realm of admissible categories, treated here, the results we have proven give rise to some interesting questions. For example, Theorem 2.7 and Proposition 2.8 together reduce the problem of finding the connected $\mathcal{C}$-Frobenius categories to that of finding the $\mathcal{C}$-Frobenius finite groups, whenever $\mathcal{C}$ is admissible. We have already seen two classes of groups arising as the class of $\mathcal{C}$-Frobenius finite groups for various $\mathcal{C}$: the trivial group if $\mathcal{C}={\bf Set}$, and the finite groups whose cardinality is invertible in $R$ for $\mathcal{C}=\ _{R}\mathcal{M}$. Can all such classes of finite groups be described? ###### Problem 2. Which classes of finite groups arise as the class of $\mathcal{C}$-Frobenius finite groups for some admissible category $\mathcal{C}$? We can turn this question around, and ask for a characterization of those admissible categories $\mathcal{C}$ having the property that the only $\mathcal{C}$-Frobenius finite group is the trivial group. We have already seen in Theorem 3.1 that ${\bf Set}$ is such a category. Although we do not prove this here, it is not difficult to see that ${\bf Grp}$, the category of groups, is another example. Note that Grp is a variety of algebras, so it is indeed admissible. ###### Problem 3. Find simple necessary and sufficient (or, alternatively, only sufficient) conditions on an admissible category $\mathcal{C}$ in order that the only $\mathcal{C}$-Frobenius finite group be the trivial group. ## Aknowledgement The author wishes to thank Professor Gigel Militaru, who posed the problem and suggested this line of inquiry, for the insight gained through countless discussions on the topic, as well as the referee for valuable suggestions on how to revise an initial version of this paper. ## References * [1] Brzeziński, T. and Wisbauer, R. - Corings and comodules, Cambridge University Press (2003) * [2] Burris, S. and Sankappanavar, H. P. - A course in universal algebra, Springer-Verlag (1981) * [3] Caenepeel, S., militaru, G. and Shenglin Zhu - Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties, Trans. Am. Math. Soc., 349 (1997), pp. 4311 - 4342 * [4] Dăscălescu, S., Năstăsescu, C., A. Del Rio and F. van Oystaeyen - Gradings of finite support. Applications to injective objects, J. Pure and Appl. Algebra, 107 (1996), pp. 193 - 206 * [5] Freyd, Peter J. - Abelian Categories - An introduction to the theory of functors, Harper & Row (1964) * [6] Iovanov, M. C. - When is the product isomorphic to the coproduct?, Comm. Algebra, 34 (2006), pp. 4551 - 4562 * [7] \- Frobenius extensions of corings, Comm. Algebra, 36 (2008), pp. 869 - 892 * [8] Mac Lane, S. - Categories for the working mathematician, Springer-Verlag (1971) * [9] Menini, C. and Năstăsescu, C. - When are the induction and coinduction functors isomorphic?, Bull. Belg. Math. Soc., 1 (1994), pp. 521 - 558 * [10] Morita, K - Adjoint pairs of functors and Frobenius extensions, Sci. Rep. Tokyo Kyoiku Daigaku (Sect. A), 9 (1965), pp. 40 - 71 * [11] Wisbauer, R. - Foundations of module and ring theory, Gordon and Breach (1991) * [12] Zarouali Darkaoui, M. - Adjoint and Frobenius pairs of functors for corings, Comm. Algebra, 35 (2007), pp. 689 - 724	arxiv-papers	2009-02-23T21:16:19	2024-09-04T02:49:00.829052	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Alexandru Chirvasitu", "submitter": "Alexandru Chirv{\\ba}situ L.", "url": "https://arxiv.org/abs/0902.4012" }
0902.4231	# New approach to radiation reaction in classical electrodynamics Richard T. Hammond rhammond@email.unc.edu Department of Physics, University of North Carolina at Chapel Hill, and the Army Research Office, Research Triangle Park, North Carolina, 27703 ###### Abstract The problem of self forces and radiation reaction is solved by conservation of energy methods. The longstanding problem of constant acceleration is solved, and it is shown that the self force does indeed affect the particle’s motion, as expected on physical grounds. The relativistic generalization is also presented. radiation reaction, self force ###### pacs: 41.60.-m, 03.50.De The classical problem of self forces due to the radiation field of an accelerating charged particle goes back over a century, to the nonrelativistic derivation of Lorentz.lorentz Soon after, Abraham used a shell model to develop an equation of motion that was a terminated version of an infinite series in terms of the radius of the shell.abraham Dirac re-derived that result, but did it for a point particle, did it relativistically, and did not have the remaining series.dirac Recently a new urgency has been given to this problem. Laser intensities of $10^{22}$ W cm-2, corresponding to an energy density over $3\times 10^{17}$ J m-3, have been reached,bahk and this is expected to increase by two orders of magnitude in the near future.mourou Traditionally, it had been thought that the observation of radiation reaction effects would have to wait until there were pulses of the characteristic time $\tau_{0}$, but with these extreme intensities, and the associated time dilation, radiation reaction effects are important now,hammond08nc and might even dominate the interactions expected in the near future. The equation derived by Dirac, mentiuoned above, is called the LAD equation and is given by (I use $ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$ and cgs units), $\frac{dv^{\mu}}{d\tau}=\frac{e}{mc}F^{\mu\sigma}v_{\sigma}+\tau_{0}\left(\frac{v^{\mu}}{c^{2}}\dot{v}_{\sigma}\dot{v}^{\sigma}+\ddot{v}^{\mu}\right)$ (1) where $\tau_{0}=2e^{2}/3mc^{3}$, which is $\sim 10^{-23}$s. The main problem with this equation is the Schott term, $\tau_{0}\ddot{v}^{\mu}$, which leads to unphysical runaway solutions.jackson Landau and Lifshitz found a way around this difficulty by using an iterative approach, and derivedlandau $\frac{dv^{\mu}}{d\tau}=(e/mc)F^{\mu\sigma}v_{\sigma}+\tau_{0}\left((e/mc)\dot{F}^{\mu\sigma}v_{\sigma}+(e/mc)^{2}(F^{\mu\gamma}F_{\gamma}^{\ \phi}v_{\phi}+F^{\nu\gamma}v_{\gamma}F_{\nu}^{\ \phi}v_{\phi}v^{\mu})\right).$ (2) This equation was used extensively over the years, but if the LAD equation, its progenitor, is wrong, then one must question the validity of the LL equation. Inspired by the unsolved problem, over the years several authors have put forward solutions of their own, most notably, that of Mo and Papas,mo Steiger and Woods,steiger Ford and O’Connell (FO)ford (which appears in Jackson’s third edition and was derived again by a different formalism),hammond08nc , Hartemann and Luhmannhartemann and through the years, Rohrlich.rohrlich All of these are based on series expansions or some other approximations, sometimes invoking a finite radius electron. For example, a drawback of the nonrelativistic FO equation, $m\bm{\dot{\bm{V}}}={\bm{F}}+\tau_{0}\frac{d}{dt}{\bm{F}}$, is that, in a uniform field, it cannot account for radiation reaction. The LL equation suffers the same problem. A fuller discussion may be found elsewhere.hammond08ejtp Before proceeding, let us examine what the LAD equation has to say about energy. To do this, we integrate the time component of the LAD equation (1) with respect to proper time. This gives, $mc^{2}(\gamma-\gamma_{\mbox{\scriptsize inc}})=\int{\bm{F}}\cdot d{\bm{x}}-\int Pdt+\tau_{0}(\dot{v}^{0}-\dot{v}_{\mbox{\scriptsize inc}}^{0}).$ (3) where ${\bm{F}}=e{\bm{E}}$, $\gamma=v^{0}/c$, $\gamma_{\mbox{\scriptsize inc}}$ is the incident value of $\gamma$, and $P=m\tau_{0}\dot{v}_{\sigma}\dot{v}^{\sigma}$. Although the LAD equation is covariant, we have now chosen a component of this equation, and therefore we must specify the reference frame, which is taken to be the lab frame in which the electric field has the value used above. In this frame we measure the particle to move through a distance $d{\bm{x}}$ in the time $dt$, which appear (3). The physical interpretation of (3) is easy to see: It reads, the change in kinetic energy is equal to the work done by the external field minus the energy radiated away plus something else. The something else seems to destroy our concept of what conservation energy should be, but we may assess its damage by noting that $\dot{v}^{0}$ vanishes when $\dot{v}^{n}$ does, so that if we integrate over a pulse this term vanishes. We do expect this to be valid in the case of a uniform electric field or in an extended magnetic field, which explains the long suffering debate about the constant force problem. To find an equation that may derived with no approximations, we assume that, corresponding to the power scalar, there is an scalar, say $W$, from which the force is derived accoring to $f_{\sigma}\equiv W,_{\sigma}$. This may be viewed as the relativisitc generalization of assuming that the force is derived from a scalar potential. With this we have a covariant equation, assuming the Lorentz force, $m\frac{dv^{\mu}}{d\tau}=\frac{e}{c}F^{\mu\sigma}v_{\sigma}-f^{\mu}.$ (4) If we integrate (4) with respect to proper time we find, $mc^{2}(\gamma-\gamma_{\mbox{\scriptsize inc}})=\int{\bm{F}}\cdot d{\bm{x}}-c\int W^{{{}^{,}}^{0}}d\tau.$ (5) Conservation of energy implies that $W,_{0}=\gamma P/c$ (6) The orthogonality of the four velocity and acceleration implies that $v_{\mu}W^{{{}^{,}}^{\mu}}=0$, so that $dW/dt=0.$ (7) This tells us that $\gamma W,_{t}=-v^{n}W,_{n}.$ (8) Thus, (4), with (6) and (8), gives a complete solution to the self force problem. Since $\tau_{0}$ is so small, it is sometimes useful to consider the series, $v^{\sigma}={{}_{0}v}^{\sigma}+\tau_{0}({{}_{1}v}^{\sigma}).$ (9) With this, we can consider the age old problem of the constant force. However, a problem arises if we naively use the above equation withour due regard to the initial condition. Conventionally one would take the extrnal force to be constant and assume the initial velocity is zero (or any value). Physically this corresponds to holding a particle fixed and at $t=0$ giving it an acceleration. Thus, this acceleration is discontinuous. Normally this is not a problem, but when the power is computed, it produces a singularity at $t=0$. To overcome this let us assume that the external electric field is given by ${\cal E}E$ where $E$ is the constant electric field and ${\cal E}=\frac{1+\mbox{Tanh t/T}}{2}.$ (10) As $T\rightarrow 0$, we obtain the step function, but in the following $T$ is taken to be unity. In addition, we shall rescale to dimensionless coordinates so that $t\rightarrow\Omega t$ and $x\rightarrow x\Omega/c$, where $\Omega=eE/mc$. To zero order the equations are ${{}_{0}\dot{v}}^{0}={\cal E}{{}_{0}}v^{1}$ (11) and ${{}_{0}\dot{v}}^{1}={\cal E}{{}_{0}}v^{0}$ (12) which imply, ${{}_{0}}v^{0}=\frac{e^{-t/2}\left(2+e^{2t}\right)}{2\sqrt{2}\sqrt{\cosh(t)}}$ (13) and ${{}_{0}}v^{1}=\frac{e^{3t/2}}{2\sqrt{2}\sqrt{\cosh(t)}}.$ (14) To ${\cal O}(\tau_{0})$ we have, using $S\equiv\dot{v}^{\sigma}\dot{v}_{\sigma}$, ${{}_{1}\dot{v}}^{0}={\cal E}{{}_{1}v}^{1}+\tau_{0}\Omega{{}_{0}v}^{0}S$ (15) and ${{}_{0}v}^{1}{{}_{1}\dot{v}}^{1}={{}_{1}v}^{0}{{}_{0}v}^{1}+({{}_{0}v}^{0})^{2}S,$ (16) although it is easier to use $v_{\sigma}v^{\sigma}=1$ to find ${{}_{1}v}^{1}=\frac{{{}_{0}v}^{0}}{{{}_{0}v}^{1}}{{}_{1}v}^{0},$ (17) and use this in (15) to get $\displaystyle{{}_{1}v}^{0}=\frac{be^{3\tau/2}}{8\left(1+e^{2\tau}\right)^{3/2}\sqrt{\cosh(\tau)}}\times\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ $\displaystyle\left(e^{\tau/2}\sqrt{\cosh(\tau)}(-1+\log(4))\left(1+e^{2\tau}\right)+\sqrt{2}\left(1-\left(1+e^{2\tau}\right)\log\left(1+e^{2\tau}\right)\right)\sqrt{1+e^{2\tau}}\right)$ (18) Figure 1: ${{}_{1}v}^{0}$ (top) and ${{}_{1}v}^{1}$, divided by $b$, vs. dimensionless proper time This solves the constant force problem. The results (13) and (14) quickly approach their asymptotic values of Cosh$t$ and Sinh$t$. The solutions (New approach to radiation reaction in classical electrodynamics) and (17) show how the energy and velocity are reduced due to the radiation. Now we may look at the realistic and practical problem of an electron in a uniform magnetic field (we revert to cgs). For a two or three dimensional problem we may find the spatial part of the radiation force, $f^{n}$, by making the ansatz $f^{n}=\xi v^{n}$ which implies that $\xi=v_{0}^{2}P/c^{2}/(v_{0}^{2}-c^{2})$, in cgs. We assume that the magnetic field $\bm{B}$ is in the $z$ direction and the charged particle has an initial four velocity $u$ in the $x$ direction, i.e., $v^{1}(0)=u$. Using (4) we have $\dot{v}^{0}=-f^{0}/m$ (19) $\dot{v}^{1}=\omega v^{2}-f^{1}/m$ (20) $\dot{v}^{2}=-\omega v^{1}-f^{2}/m$ (21) where $\omega=eB/mc$. To order $\tau_{0}$ the solution to the spatial equations is, $v^{1}=u\cos\omega\tau(1-b\tau)$ (22) $v^{2}=-u\sin\omega\tau(1-b\tau)$ (23) where $b=\tau_{0}\omega^{2}(1+u^{2}/c^{2})$. These can be integrated to find the position as a function of proper time and are plotted in Fig. 2. Figure 2: Parametric plot of $x$ and $y$ versus proper time, showing the electron spiraling in due to radiation reaction. For illustrative purposes, I set $u=1$, $\omega$=1, and $b=0.01$ (which, of course, corresponds to a huge and false value of $\tau_{0}$). The zero component of the equation of motion is an energy balance equation. Integrating (19) with respect to proper time gives $v^{0}-v_{\mbox{\scriptsize inc}}^{0}=-\frac{1}{mc}\int Pdt,$ (24) which was engineered from the start (the magnetic field does no work on the particle). In particular, using the expression for kinetic energy, $K=mc^{2}(\gamma-\gamma_{\mbox{\scriptsize inc}})$, (20) and (21) show that the change in kinetic energy, which is negative, is negative of the energy radiated, $W_{R}=-\int Pdt$. Another way of looking at this is to use $E^{2}=p^{2}c^{2}+m^{2}c^{4}$ (25) which implies for small changes, $\Delta E=\frac{{\bm{p}}\cdot\Delta{\bm{p}}}{\gamma m}.$ (26) In this equation we use (22) and (23) to obtain ${\bm{p}}$ and $\Delta{\bm{p}}$. The piece without the $\tau_{0}$ term is used to find $p$ while the $\Delta p$ is obtained from the $\tau_{0}$ piece. With this, the above yields, $\Delta K=-\tau_{0}mu^{2}\omega^{2}\gamma\tau.$ (27) To check, we integrate $P$, which gives the same result (one may note that $\gamma=\gamma_{\mbox{\scriptsize inc}}+{\cal O}(\tau_{0})$, so that to this order $\gamma\tau=t$. Thus, by generalizing the simple equation of motion along with the equation expressing conservation of energy, equations of motion with radiation reaction have been derived that do not suffer from the unphysical behavior of, for example, the LAD equation, or the problem of uniform fields of the FO and LL equations. Solutions for a few special cases were given, and the age old problem of a charged particle in a uniform field was solved. ## References * (1) H. A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat (Leipzig, New York 1909). * (2) M. Abraham, Theorie der Elektrizita t, Vol. II (Teubner, Leipzig, 1905). * (3) P. A. M. Dirac, Proc. Roy. Soc. A 167, 148 (1938). * (4) J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York 1998, 3rd ed.). * (5) L. D. Landau and E. M. Lifshitz The Classical Theory of Fields (Pergamon Press, Addison-Welsey, Reading, MA, 1971), equation 76.1. This appeard in the first edition in 1951. * (6) S.-W. Bahk et al, Opt. Lett. 29, 2837 (2004); V.Yanovsky et al Optics Express, 16, 2109 (2008). * (7) G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006). * (8) R. T. Hammond, Il Nuovo Cim. 123, 567 (2008). * (9) T. C. Mo and C. H. Papas, Phys. Rev. D 4, 3566 (1971). * (10) A. D. Steiger and C. H. Woods, Phys. Rev. A 5, 1467 (1972). * (11) A. D. Steiger and C. H. Woods, Phys. Rev. D 5, 2927 (1972). * (12) G. W. Ford and R. F. O’Connell, Phys. Lett. A 157, 217 (1991); G. W. Ford and R. F. O’Connell, Phys. Lett. A 174, 182 (1993); Phys. Rev. A 44, 6386 (1991); G. W. Ford and R. F. O’Connell, Phys. Lett. A 158, 31 (1991). * (13) F. V. Hartemann and N. C. Luhmann, Jr. Phys. Rev. Lett. 74, 1107 (1995). * (14) F. Rohrlich, Phys. Rev. E 77, 046609 (2008); Physics Letters A 283, 276 (2001); Physics Letters A 303, 307 (2002); Am. J. of Physics 68, 1109 (2000). * (15) R. T. Hammond, EJTP, 5 17, 17, (2008)	arxiv-papers	2009-02-24T20:43:26	2024-09-04T02:49:00.840091	{ "license": "Public Domain", "authors": "Richard T Hammond", "submitter": "Richard T. Hammond", "url": "https://arxiv.org/abs/0902.4231" }
0902.4269	# How well can one resolve the state space of a chaotic map? Domenico Lippolis and Predrag Cvitanović Center for Nonlinear Science, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430 ###### Abstract All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. We propose to determine the ‘finest attainable’ partition for a given hyperbolic dynamical system and a given weak additive white noise, by computing the local eigenfunctions of the adjoint Fokker-Planck operator along each periodic point, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. Numerical tests of such ‘optimal partition’ of a one-dimensional repeller support our hypothesis. ###### pacs: 05.45.-a, 45.10.db, 45.50.pk, 47.11.4j The effect of noise on the behavior of a nonlinear dynamical system is a fundamental problem in many areas of science van Kampen (1992); Lasota and MacKey (1994); Risken (1996), and the interplay of noise and chaotic dynamics is of particular current interest Gaspard (2002); Fogedby (2005, 2006). The purpose of this letter is two-fold. First, and conceptually the most important, we point out an effect of noise that has not been addressed in literature: weak noise limits the attainable resolution of the state space (‘phase space’) of a chaotic system. We formulate the ‘optimal partition’ hypothesis whose implementation requires only integration of a small set of solutions of the deterministic equations of motion. Second, more technical point; we show that the optimal partition hypothesis replaces the Fokker- Planck PDEs by finite, low-dimensional Fokker-Planck matrices, whose eigenvalues give good estimates of long-time observables (escape rates, Lyapunov exponents, etc.). A chaotic trajectory explores a strange attractor, and evaluation of long-time averages requires effective partitioning of the state space into smaller regions. The set of unstable periodic orbits forms a ‘skeleton’ that can be used to partition the state space into such smaller regions, each region a neighborhood of a periodic point Ruelle (1978); Cvitanović (1988) (i.e., a point on a periodic orbit). The number of periodic orbits grows exponentially with period length, yielding finer and finer partitions, with the neighborhood of each periodic orbit shrinking exponentially. As there is an infinity of periodic orbits, with each neighborhood shrinking asymptotically to a point, a deterministic chaotic system can - in principle - be resolved arbitrarily finely. However, any physical system suffers background noise, any numerical prediction suffers computational roundoff noise, and any set of equations models nature up to a given accuracy, since degrees of freedom are always neglected. If the noise is weak, the short-time dynamics is not altered significantly: short periodic orbits of the deterministic flow still partition coarsely the state space. Intuitively, the noise smears out the neighborhood of a periodic point, whose size is now determined by the interplay between the diffusive spreading parameterized Dekker and Kampen (1979); Gaspard et al. (1995) by the diffusion constant $D$, and its exponentially shrinking deterministic neighborhood. As the periods of periodic orbits increase, the diffusion always wins, and successive refinements of a deterministic partition of the state space stop at the finest attainable partition, beyond which the diffusive smearing exceeds the size of any deterministic subpartition. The smearing width differs from trajectory to trajectory, so there is no one single time beyond which noise takes over; rather, as we shall show here, the optimal partition has to be computed for a given dynamical system and given noise. This effort brings a handsome practical reward: as the optimal partition is finite, the Fokker-Planck operator can be represented by a finite matrix. While the general idea is intuitive, nonlinear dynamics interacts with noise in a nonlinear way, and methods for implementing the optimal partition for a given noise still need to be developed. In this letter we propose a new approach to this partitioning. We compute the width of the leading eigenfunction of the linearized adjoint Fokker-Planck operator on each periodic point. The optimal partition is then obtained by tracking the diffusive widths of unstable periodic orbits until they start to overlap. We describe here the approach as applied to 1$\,d$ expanding maps; higher- dimensional hyperbolic maps and flows require a separate treatment for contracting directions, a topic for a future publication Lippolis and Cvitanović . As the simplest application of the method, consider the orbit $\\{\ldots,x_{-1},x_{0},x_{1},x_{2},\ldots\\}$ of a 1$\,d$ map $x_{n+1}=f(x_{n})$, and the associated discrete Langevin equation Lasota and MacKey (1994) $x_{n+1}=f(x_{n})+\xi_{n}\,,$ (1) where the $\xi_{n}$ are independent Gaussian random variables of mean 0 and variance $2D$ (the method can be applied to continuous time flows as well, but a 1$\,d$ map suffices to illustrate the optimal partition algorithm). The corresponding Fokker-Planck operator Risken (1996), ${\cal L}\circ{\rho}_{n}({y})=\int\frac{dx}{\sqrt{4\pi D}}\,e^{-\frac{(y-f(x))^{2}}{4D}}{\rho}_{n}({x})$ (2) carries the density of Langevin trajectories ${\rho}_{n}(x)$ forward in time to ${\rho}_{n+1}={\cal L}\circ{\rho}_{n}$. Since a density concentrated at point $x_{n}$ is carried into a density concentrated at $x_{n+1}$, we introduce local coordinate systems ${z}_{a}$ centered on the orbit points $x_{a}$, together with a notation for the map (1), its derivative, and, by the chain rule, the derivative of the $k$th iterate $f^{k}$ evaluated at the point $x_{a}$, $\displaystyle x$ $\displaystyle=$ $\displaystyle x_{a}+{z}_{a}\,,\quad f_{a}({z}_{a})=f(x_{a}+{z}_{a})$ $\displaystyle{f^{\prime}_{a}}$ $\displaystyle=$ $\displaystyle f^{\prime}(x_{a})\,,\;\;f_{a}^{k}{}^{\prime}={f^{\prime}_{a+k-1}}\cdots{f^{\prime}_{a+1}}{f^{\prime}_{a}}\,,\;\;k\geq 2\,.$ (3) Here $a$ is the label of point $x_{a}$, and the label $a\\!+\\!1$ is a shorthand for the next point $b$ on the orbit of $x_{a}$, $x_{b}=x_{a+1}=f(x_{a})$. For example, a period-3 periodic point might have label $a=001$, and by $x_{010}=f(x_{001})$ the next point label is $b=010$. If the noise is weak, we can approximate (to leading order in $D$) the Fokker- Planck operator, ${\cal L}_{a}\circ{\rho}_{n}(x_{a+1}+{z}_{a+1})=\int d{z}_{a}{\cal L}_{a}({z}_{a+1},{z}_{a}){\rho}_{n}(x_{a}+{z}_{a})$, by linearization centered on $x_{a}$, the $a$th point along the orbit, $\displaystyle{\cal L}_{a}({z}_{a+1},{z}_{a})$ $\displaystyle=$ $\displaystyle(4\pi D)^{-1/2}\,e^{-\frac{({z}_{a+1}-{f^{\prime}_{a}}{z}_{a})^{2}}{4D}}\,.$ (4) ${\cal L}_{a}$ maps a Gaussian density ${\rho}_{n}(x_{a}+{z}_{a})=c_{a}\exp\left\\{-{{z}_{a}^{2}}/{2\sigma_{a}^{2}}\right\\}$, of variance $\sigma_{a}^{2}$, into a Gaussian density ${\rho}_{n+1}(x)$ of variance $\sigma_{a+1}^{2}=({f^{\prime}_{a}}\sigma_{a})^{2}+2D$. This variance is an interplay of the Brownian noise contribution $2D$ and the nonlinear contracting/amplifying contribution $(\sigma{f^{\prime}})^{2}$. The diffusive dynamics of a nonlinear system are thus fundamentally different from Brownian motion, as the map induces a history dependent effective noise. In order to determine the smallest noise-resolvable state space partition along the trajectory of $x_{a}$, we need to determine the effect of noise on the points preceding $x_{a}$. This is achieved by the adjoint Fokker-Planck operator ${\cal L}^{\dagger}\circ\tilde{{\rho}}_{n}({x})=\int\frac{dy}{\sqrt{4\pi D}}\,e^{-\frac{(y-f(x))^{2}}{4D}}\tilde{{\rho}}_{n}({y})\,,$ (5) which relates a density $\tilde{{\rho}}_{n}$ concentrated around $x_{a}$ to $\tilde{{\rho}}_{n-1}={\cal L}^{\dagger}\circ\tilde{{\rho}}_{n}$, a density concentrated around the previous point $x_{a-1}$, the variance transforming as $({f^{\prime}_{a-1}}\sigma_{a-1})^{2}=\sigma_{a}^{2}+2D$. For an unstable (expanding) map, these variances shrink. After $n$ steps the variance is given by $(f_{a-n}^{n^{\prime}}\sigma_{a-n})^{2}=\sigma_{a}^{2}+2D(1+(f_{a-1}^{\prime})^{2}+\cdots+(f_{a-n+1}^{n-1^{\prime}})^{2})\,.$ (6) From the dynamical point of view, a good state space partition encodes the recurrent dynamics; here we shall seek a partition in terms of neighborhoods of periodic points Cvitanović (1988); Cvitanović et al. (2009) of short periods. For the linearized ${\cal L}^{\dagger}_{a}$ acting on a fixed point $x_{a}=f(x_{a})$, the $n\to\infty$ sum (6) converges to a Gaussian of variance $\sigma_{a}^{2}={2D}/{(\Lambda_{a}^{2}-1)}\,,$ (7) where $\Lambda_{a}=f_{a}^{\prime}$, and for a periodic point $x_{a}\in p$ to a Gaussian of variance $\sigma_{a}^{2}=\frac{2D}{1-\Lambda_{p}^{-2}}\left(\frac{1}{(f_{a}^{\prime})^{2}}+\cdots+\frac{1}{\Lambda_{p}^{2}}\right)\,,$ (8) where $\Lambda_{p}=f_{a}^{{n_{p}}}{}^{\prime}$ is the Floquet multiplier (eigenvalue of the Jacobian linearized flow) of an unstable ($\|\Lambda_{p}\|>1$) periodic orbit $p$ of period ${n_{p}}$. This is the key formula; note that its evaluation requires no Fokker-Planck formalism, it depends only on the deterministic orbit and its linear stability. We can now state the main result of this letter, _‘the best possible of all partitions’_ hypothesis, as an algorithm: assign to each periodic point $x_{a}$ a neighborhood of finite width $[x_{a}-\sigma_{a},x_{a}+\sigma_{a}]$. Consider periodic orbits of increasing period ${n_{p}}$, and stop the process of refining the state space partition as soon as the adjacent neighborhoods overlap. Figure 1: $f_{0},f_{1}$: branches of the deterministic map (9) for $\Lambda_{0}=8$ and $b=0.6$. The local eigenfunctions $\tilde{{\rho}}_{a,0}$ with variances given by (8) provide a state space partitioning by neighborhoods of periodic points of period 3. These are computed for noise variance ($D$ = diffusion constant) $2D=0.002$. The neighborhoods ${\cal M}_{000}$ and ${\cal M}_{001}$ already overlap, so ${\cal M}_{00}$ cannot be resolved further. For periodic points of period 4, only ${\cal M}_{011}$ can be resolved further, into ${\cal M}_{0110}$ and ${\cal M}_{0111}$. As a concrete application to the Langevin map (1) consider map Cvitanović et al. (2009) $f(x)=\Lambda_{0}x(1-x)(1-bx)$ (9) plotted in figure 1; this figure also shows the local eigenfunctions $\tilde{{\rho}}_{a,0}$ with variances given by (8). Each Gaussian is labeled by the $\\{f_{0},f_{1}\\}$ branches visitation sequence of the corresponding deterministic periodic point (a symbolic dynamics, however, is not a prerequisite for implementing the method). Figure 2: Transition graph (graph whose links correspond to the nonzero elements of a transition matrix $T_{ba}$) describes which regions $b$ can be reached from the region $a$ in one time step. The 7 nodes correspond to the 7 regions of the optimal partition (10). Dotted links correspond to symbol $0$, and the full ones to 1, indicating that the next region is reached by the $f_{0}$, respectively $f_{1}$ branch of the map plotted in figure 1. We find that in this case the state space (the unit interval) can be resolved into 7 neighborhoods $\\{{\cal M}_{00},{\cal M}_{011},{\cal M}_{010},{\cal M}_{110},{\cal M}_{111},{\cal M}_{101},{\cal M}_{100}\\}\,.$ (10) Evolution in time maps the optimal partition interval ${\cal M}_{011}\to\\{{\cal M}_{110},{\cal M}_{111}\\}$, ${\cal M}_{00}\to\\{{\cal M}_{00},{\cal M}_{011},{\cal M}_{010}\\}$, etc., as compactly summarized by the transition graph of figure 2. Next we show that the optimal partition enables us to replace Fokker-Planck PDEs by finite-dimensional matrices. The variance (8) is stationary under the action of ${\cal L}^{\dagger{n_{p}}}_{a}$, and the corresponding Gaussian is thus an eigenfunction. Indeed, for the linearized flow the entire eigenspectrum is available analytically, and will be a key ingredient in what follows. For a periodic point $x_{a}\in p$, the ${n_{p}}$th iterate ${\cal L}^{{n_{p}}}_{a}$ of the linearization (4) is the discrete time version of the Ornstein-Uhlenbeck process Uhlenbeck and Ornstein (1930), with left $\tilde{{\rho}}_{0}$, $\tilde{{\rho}}_{1}$, $\cdots$, respectively right ${{\rho}}_{0}$, ${{\rho}}_{1}$, $\cdots$ mutually orthogonal eigenfunctions Risken (1996) given by $\displaystyle\tilde{{\rho}}_{a,k}({z})$ $\displaystyle=$ $\displaystyle\frac{\beta^{k+1}}{\sqrt{\pi}2^{k}k!}H_{k}(\beta{z})e^{-(\beta{z})^{2}}$ $\displaystyle{\rho}_{a,k}({z})$ $\displaystyle=$ $\displaystyle\frac{1}{\beta^{k}}H_{k}(\beta{z})\,,$ (11) where $H_{k}(x)$ is the $k$th Hermite polynomial, $1/\beta=\sqrt{2}\sigma_{a}$, and the $k$th eigenvalue is ${1}/{\|\Lambda\|\Lambda^{k}}$. Partition (10) being the finest possible partition, the Fokker-Planck operator now acts as [$7\\!\times\\!7$] matrix with non-zero $a\to b$ entries expanded in the Hermite basis, $\displaystyle[{{\bf L}}_{ba}]_{kj}$ $\displaystyle=$ $\displaystyle\left\langle\tilde{{\rho}}_{b,k}\|{\cal L}\|{{\rho}}_{a,j}\right\rangle$ (12) $\displaystyle=$ $\displaystyle\int\frac{d{z}_{b}d{z}_{a}\,\beta}{2^{j+1}j!\pi\sqrt{D}}e^{-(\beta{z}_{b})^{2}-\frac{({z}_{b}-f_{a}({z}_{a}))^{2}}{4D}}$ $\displaystyle\qquad\times\,H_{k}(\beta{z}_{b})H_{j}(\beta{z}_{a})\,,$ where $1/\beta=\sqrt{2}\sigma_{a}$, and ${z}_{a}$ is the deviation from the periodic point $x_{a}$. It is the number of resolved periodic points that determines the dimensionality of the Fokker-Planck matrix. Periodic orbit theory Cvitanović et al. (2009); Gaspard (1997) expresses the long-time dynamical averages, such as Lyapunov exponents, escape rates, and correlations, in terms of the leading eigenvalues of the Fokker-Planck operator ${\cal L}$. In our ‘optimal partition’ approach, ${\cal L}$ is approximated by the finite-dimensional matrix ${{\bf L}}$, and its eigenvalues are determined from the zeros of $\det(1-z{{\bf L}})$, expanded as a polynomial in $z$, with coefficients given by traces of powers of ${{\bf L}}$. As the trace of the $n$th iterate of the Fokker-Planck operator ${\cal L}^{n}$ is concentrated on periodic points $f^{n}(x_{a})=x_{a}$, we evaluate the contribution of periodic orbit $p$ to $\mbox{\rm tr}\,{{\bf L}}^{n_{p}}$ by centering ${{\bf L}}$ on the periodic orbit, $t_{p}=\mbox{\rm tr}\,_{p}\,{\cal L}^{{n_{p}}}=\mbox{\rm tr}\,{{\bf L}_{ad}}\cdot\cdot\cdot{{\bf L}_{cb}}{{\bf L}_{ba}}\,,$ (13) where $x_{a},x_{b},\cdots x_{d}\in p$ are successive periodic points. To leading order in the noise variance $2D$, $t_{p}$ takes the deterministic value $t_{p}=1/\|\Lambda_{p}-1\|$. We illustrate the method by calculating the escape rate $\gamma=-\ln z_{0}$, where $z_{0}^{-1}$ is the leading eigenvalue of Fokker-Planck operator ${\cal L}$, for the repeller plotted in figure 1. The spectral determinant can be read off the transition graph of figure 2, $\displaystyle\det(1-z{{\bf L}})=1-(t_{0}+t_{1})z-(t_{01}-t_{0}t_{1})\,z^{2}$ $\displaystyle\quad-(t_{001}+t_{011}-t_{01}t_{0}-t_{01}t_{1})\,z^{3}-\cdots$ $\displaystyle\quad-(t_{0010111}+t_{0011101}-\cdots+t_{001}t_{011}t_{1})\,z^{7}.$ (14) The polynomial coefficients are given by products of non-intersecting loops of the transition graph Cvitanović et al. (2009), with the escape rate given by the leading root $z_{0}^{-1}$ of the polynomial. Twelve periodic orbits $\overline{0}$, $\overline{1}$, $\overline{01}$, $\overline{001}$, $\overline{011}$, $\overline{0011}$, $\overline{0111}$, $\overline{00111}$, $\overline{001101}$, $\overline{001011}$, $\overline{0010111}$, $\overline{0011101}$ up to period 7 (out of the 41 contributing to the noiseless, deterministic cycle expansion up to cycle period 7) suffice to fully determine the spectral determinant of the Fokker-Planck operator. Figure 3: (left scale) the escape rate of the repeller (9) vs. the noise strength $D$, calculated using ($\square$) the ‘optimal partition’, and ($\color[rgb]{0,0,1}\times$) a uniform discretization (16) in $N=128$ intervals; (right scale) the Lyapunov exponent of the same repeller vs. $D$, estimated using ($\bullet$) the ‘optimal partition’, and ($\diamond$) the average (17). The ‘optimal partition’ estimate of the Lyapunov exponent is given Cvitanović et al. (2009) by $\lambda=\left<\ln\,\Lambda\right>/\left<n\right>$, where the cycle expansion average of an observable $A$ $\displaystyle\left<A\right>$ $\displaystyle=$ $\displaystyle A_{0}t_{0}+A_{1}t_{1}+(A_{01}t_{01}-(A_{0}+A_{1})t_{0}t_{1})+$ (15) $\displaystyle\quad(A_{001}t_{001}-(A_{01}+A_{0})t_{01}t_{0})+\cdots$ is the finite sum over cycles contributing to (14), and $\ln\Lambda_{p}=\sum\ln\|f^{\prime}(x_{a})\|$ sum over the points of cycle $p$ is the cycle Lyapunov exponent. Since our ‘optimal partition’ algorithm is based on a sharp overlap criterion, small changes in noise strength $D$ can lead to transition graphs of different topologies. We assess the accuracy of our finite Fokker-Planck matrix approximations by discretizing the Fokker-Planck operator ${\cal L}$ with a piecewise-constant approximation on a uniform mesh on the unit interval Ulam (1960), $[{\cal L}]_{ij}\,=\,\frac{1}{\sqrt{4\pi D}}\int_{{\cal M}_{i}}\frac{dx}{\|{\cal M}_{i}\|}\int_{f^{-1}({\cal M}_{j})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!dy\,e^{-\frac{1}{4D}(y-f(x))^{2}}\,,$ (16) where ${\cal M}_{i}$ is the $i$th interval in equipartition of the unit interval into $N$ pieces. Empirically, $N=128$ intervals suffice to compute the leading eigenvalue of the discretized $[128\times 128]$ matrix $[{\cal L}]_{ij}$ to four significant digits. The Lyapunov exponent is evaluated as the average $\lambda=\int dx\,e^{\gamma}\rho(x)\ln\|f^{\prime}(x)\|$ (17) where $\rho(x)$ is the leading eigenfunction of (16), $\gamma$ is the escape rate, and $e^{\gamma}\rho$ is the normalized repeller measure, $\int dx\,e^{\gamma}\rho(x)=1$. The numerical results are summarized in figure 3, with the estimates of the ‘optimal partition’ method within $1\%$ of those given by the uniform discretization of Fokker-Planck. In summary, we have presented a new method for partitioning the state space of a chaotic repeller in the presence of noise. The key idea is that the width of the linearized adjoint Fokker-Planck operator ${\cal L}^{\dagger}_{a}$ eigenfunction computed on a periodic point $x_{a}$ provides the scale beyond which no further local refinement of state space is possible. This computation enables us to systematically determine the optimal partition, of the finest state space resolution attainable for a given chaotic dynamical system and a given noise. Once the optimal partition is determined, we use the associated transition graph to describe the stochastic dynamics by a finite dimensional Fokker-Planck matrix. While an expansion of the Fokker-Planck operator about periodic points was already introduced in ref. Cvitanović et al. (1999), the novel aspect of this work is its representation in terms of the eigenfunctions of the linearized Fokker-Planck operator (4), ie. the Hermite basis Lippolis and Cvitanović ; Gaspard et al. (1995). We test our optimal partition hypothesis by applying it to evaluation of the escape rate and the Lyapunov exponent of a $1d$ repeller in presence of additive noise. Numerical tests indicate that, the ‘optimal partition’ method can be as accurate as a $128$-interval discretization of the Fokker-Planck operator. The success of the optimal partition hypothesis in a 1-dimensional setting is encouraging. However, higher-dimensional hyperbolic maps and flows, for which an effective optimal partition algorithm would be very useful, present new challenges due to the subtle interactions between expanding, marginal and contracting directions. The nonlinear diffusive effects (weak stochastic corrections Cvitanović et al. (1999)) need to be accounted for as well. These issues will be addressed in a future publication Lippolis and Cvitanović . ###### Acknowledgements. We are indebted to C.P. Dettmann, W.H. Mather, A. Grigo and G. Vattay for many stimulating discussions, and S.A. Solla for a critical reading of the manuscript. P.C. thanks Glen P. Robinson and NSF grant DMS-0807574 for partial support. ## References * van Kampen (1992) N. G. van Kampen, _Stochastic Processes in Physics and Chemistry_ (North-Holland, Amsterdam, 1992). * Lasota and MacKey (1994) A. Lasota and M. MacKey, _Chaos, Fractals, and Noise; Stochastic Aspects of Dynamics_ (Springer-Verlag, Berlin, 1994). * Risken (1996) H. Risken, _The Fokker-Planck Equation_ (Springer-Verlag, 1996). * Gaspard (2002) P. Gaspard, J. Stat. Phys. 106, 57 (2002). * Fogedby (2005) H. C. Fogedby, Phys. Rev. Lett. 94, 195702 (2005). * Fogedby (2006) H. C. Fogedby, Phys. Rev. E 73, 031104 (2006). * Ruelle (1978) D. Ruelle, _Statistical Mechanics, Thermodynamic Formalism_ (Addison-Wesley, Reading, MA, 1978). * Cvitanović (1988) P. Cvitanović, Phys. Rev. Lett. 61, 2729 (1988). * Dekker and Kampen (1979) H. Dekker and N. V. Kampen, Physics Lett. 73A, 374 (1979). * Gaspard et al. (1995) P. Gaspard, G. Nicolis, A. Provata, and S. Tasaki, Phys. Rev. E 51, 74 (1995). * (11) D. Lippolis and P. Cvitanović, in preparation, 2009. * Cvitanović et al. (2009) P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, _Chaos: Classical and Quantum_ (Niels Bohr Institute, Copenhagen, 2009), ChaosBook.org. * Uhlenbeck and Ornstein (1930) G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930). * Gaspard (1997) P. Gaspard, _Chaos, Scattering and Statistical Mechanics_ (Cambridge Univ. Press, Cambridge, 1997). * Ulam (1960) S. M. Ulam, _A Collection of Mathematical Problems_ (Interscience Publishers, New York, 1960). * Cvitanović et al. (1999) P. Cvitanović, N. Søndergaard, G. Palla, G. Vattay, and C. Dettmann, Phys. Rev. E 60, 3936 (1999).	arxiv-papers	2009-02-25T00:20:16	2024-09-04T02:49:00.844551	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Domenico Lippolis and Predrag Cvitanovic", "submitter": "Domenico Lippolis", "url": "https://arxiv.org/abs/0902.4269" }
0902.4276	# Magnetic properties of a spin-3 Chromium condensate Liang He and Su Yi Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China ###### Abstract We study the ground state properties of a spin-3 Cr condensate subject to an external magnetic field by numerically solving the Gross-Piteavskii equations. We show that the widely adopted single-mode approximation is invalid under a finite magnetic field. In particular, a phase separation like behavior may be induced by the magnetic field. We also point out the possible origin of the phase separation phenomenon. ###### pacs: 03.75.Mn, 03.75.Hh ## I Introduction Since the realization of Bose-Einstein condensate of chromium atoms Pfau05 , there have been considerable experimental and theoretical efforts in exploring physical properties of chromium condensates. Owing to the large magnetic dipole moment of chromium atoms, the dipolar effects was first identified experimentally from its expansion dynamics stuh . More remarkably, with the precise control of the short-range interaction using Feshbach resonance, the $d$-wave collapse of a pure dipolar condensate has been observed laha . In the context of spinor condensates, chromium atom has an electronic spin $s=3$, which provides an ideal platform for exploring even richer quantum phases as compared to those offered by the spin-1 and spin-2 atoms Ho98 ; Ohmi98 ; Law98 ; Ciobanu00 ; Koashi00 ; Stenger1998 ; Barret01 ; Schmaljohann04 ; Chang04 ; Kuwamoto04 . To date, theorists have mapped out the detailed phase diagram of a spin-3 chromium condensate Santos06 ; Ho06 ; Makela07 . In particular, a more exotic biaxial nematic phase was also predicted Ho06 . The possible quantum phases and defects of spin-3 condensates were also classified based on the symmetry considerations barnett ; yip . Other work on spin-3 chromium condensates includes theoretically studying the strongly correlated states of spin-3 bosons in optical lattices bernier and the Einstein-de Haas effect in chromium condensates Santos06 ; Ueda . Nevertheless, all the previous work concerning the ground state and the magnetic properties of spin-3 Cr condensates has adopted the so-called single- mode approximation (SMA), which assumes that all spin components share a common density profile. However, the studies on spin-1 case show that, for an antiferromagnetic spinor condensate, SMA is invalid in the presence of magnetic field for antiferromagnetic spin exchange interaction Yi02(SMA) . One would naturally question the validity of SMA for spin-3 condensate since the short-range interactions involved here are more complicated than those in spin-1 system. In the present paper, we study the ground state properties of a spin-3 chromium condensate subject to a uniform axial magnetic field by numerically solving the Gross-Pitaevskii equations. We show that even though SMA is still valid in the absence of an external magnetic field, it fails when the magnetic field is switched on. More remarkably, we find that when the undetermined scattering length corresponding to total spin zero channel falls into a certain region, the magnetic field may induce a phase separation like behavior such that the peak densities of certain spin components do not occur at the center of the trapping potential. This paper is organized as follows. In Sec. II, we introduce our model for numerical calculation. The results for the ground state structure of a spin-3 condensate under an external magnetic field are presented in Sec. III. Finally, we conclude in Sec. IV. ## II Formulation We consider a condensate of $N$ spin $s=3$ chromium atoms subject to a uniform magnetic field ${\mathbf{B}}=B{\mathbf{z}}$. In mean-field treatment, the system is described by the condensate wave functions $\psi_{m}$ ($m=-3,-2,\ldots,3$). The total energy functional of the system, $E[\psi_{m},\psi_{m}^{}]$, can be decomposed into two parts $E=E_{0}+E_{1}$ with $E_{0}$ and $E_{1}$ being, respectively, the single-body and interaction energies. Adopting the summation convention over repeated indices, the single- body energy can be expressed as $\displaystyle E_{0}\\!=\\!\int\\!d{\mathbf{r}}\psi_{m}^{}\left[\left(-\frac{\hbar^{2}\nabla^{2}}{2M}+V_{\mathrm{ext}}\right)\delta_{mm^{\prime}}+g\mu_{B}Bs^{z}_{mm^{\prime}}\right]\psi_{m^{\prime}},$ where $M$ is the mass of the atom, the trapping potential $V_{\mathrm{ext}}({\mathbf{r}})=\frac{1}{2}M\omega_{\perp}^{2}(x^{2}+y^{2}+\eta^{2}z^{2})$ is assumed to be axially symmetric with $\eta$ being the trap aspect ratio, ${\mathbf{s}}=(s^{x},s^{y},s^{z})$ are the spin-3 matrices, $g=2$ is the Landé $g$-factor of 52Cr atoms, and $\mu_{B}$ is Bohr magneton. Figure 1: (Color online) Left panel: phase diagram of spin-3 Cr condensate in the $a_{0}$-$B$ parameter space. The shaded region indicates the region where phase separation occurs (see text for details). Right panel: the main characteristics of the quantum phases. The collisional interaction between two spin-3 atoms takes the form Ho98 ; Ohmi98 $\displaystyle V_{\mathrm{int}}(\mathbf{r},\mathbf{r}^{\prime})=\delta(\mathbf{r}-\mathbf{r}^{\prime})\sum_{S=0}^{2s}g_{S}\mathcal{P}_{S},$ (2) where $\mathcal{P}_{S}$ projects onto the state with total spin $S$ and $g_{S}=4\pi\hbar^{2}a_{S}/M$ with $a_{S=0,2,4,6}$ being the scattering lengths for the combined symmetric channel $S$. For 52Cr, it was determined experimentally that $a_{6}=112\,a_{B}$, $a_{4}=58\,a_{B}$, and $a_{2}=-7\,a_{B}$ with $a_{B}$ being Bohr radius stuh , while the value of $a_{0}$ is unknown, and we shall treat it as a free parameter in the results presented below. Making use of the relations Ho98 $\displaystyle 1$ $\displaystyle=$ $\displaystyle\sum_{S}\mathcal{P}_{S},$ $\displaystyle{\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2}$ $\displaystyle=$ $\displaystyle\sum_{S}\frac{\mathcal{P}_{S}}{2}[S(S+1)-24],$ $\displaystyle({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2})^{2}$ $\displaystyle=$ $\displaystyle\sum_{S}\frac{\mathcal{P}_{S}}{4}[S(S+1)-24]^{2},$ we may replace $\mathcal{P}_{2}$, $\mathcal{P}_{4}$, and $\mathcal{P}_{6}$ in Eq. (2) by $1$, ${\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2}$, and $({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2})^{2}$, such that the interaction energy functional becomes $\displaystyle E_{1}=\frac{1}{2}\int d{\mathbf{r}}n^{2}\left[C+\alpha\|\Theta\|^{2}+\beta{\rm Tr}{\mathcal{N}}^{2}+\gamma\langle{\mathbf{s}}\rangle^{2}\right],$ (3) where the interaction parameters are $C=-\frac{1}{7}g_{4}+\frac{81}{77}g_{4}+\frac{1}{11}7g_{6}$, $7\alpha=g_{0}-\frac{5}{3}g_{2}+\frac{9}{11}g_{4}-\frac{5}{33}g_{6}$, $\beta=\frac{1}{126}g_{2}-\frac{1}{77}g_{4}+\frac{1}{198}g_{6}$, and $\gamma=-\frac{5}{84}g_{2}+\frac{1}{154}g_{4}+\frac{7}{132}g_{6}$. Furthermore, $n({\mathbf{r}})=\psi_{m}^{}\psi_{m}$ is the total density, $\Theta({\mathbf{r}})=\frac{1}{n}\sqrt{7}\langle 00\|3m;3m^{\prime}\rangle\psi_{m}\psi_{m^{\prime}}$ is the singlet amplitude, and $\langle{\mathbf{s}}\rangle({\mathbf{r}})=\frac{1}{n}\psi_{m}^{}{\mathbf{s}}_{mm^{\prime}}\psi_{m^{\prime}}$ is the density of spin. Finally, $\mathcal{N}_{ij}({\mathbf{r}})=\frac{1}{2n}\psi^{}_{m}(s^{i}s^{j}+s^{j}s^{i})_{mm^{\prime}}\psi_{m^{\prime}},\quad i,j=x,y,z$ is the nematic tensor, and to obtain it, we have utilized the relation $\langle({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2})^{2}\rangle={\rm Tr}{\cal N}^{2}-\frac{1}{2}\langle{\mathbf{s}}_{1}\rangle\cdot\langle{\mathbf{s}}_{2}\rangle.$ The nematic tensor was first introduced in the liquid crystal physics as the order parameter $\mathcal{N}$ de Gennes to describe the orientation order of the liquid crystal molecules. Since $\mathcal{N}$ is Hermitian, it can be diagonalized with all eigenvalues $\lambda_{a=1,2,3}$ (ordered as $\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}$) being real and the corresponding principle axes $\hat{\mathbf{e}}_{a}$ being mutually orthogonal. Unless all three eigenvalues are equal, the systems with two identical eigenvalues are usually refer to as uniaxial nematics, while those with three unequal eigenvalues are biaxial ones. More importantly, $\lambda_{a}$ can be determined by performing Stern-Gerlach experiments along $\hat{\mathbf{e}}_{a}$ Ho06 . As it can be seen from Eq. (3), different quantum phases originate from the competition of $\Theta$, $\langle\mathbf{s}\rangle$, and $\mathcal{N}$. Following the discussion of Diener and Ho Ho06 , we shall characterize phases in a spin-3 Cr condensate using the condensate wave functions $\psi_{m}$, singlet amplitude $\Theta$, spin $\langle{\mathbf{s}}\rangle$, and nematic tensor $\mathcal{N}$. Figure 2: (Color online) The typical $\rho$ dependence of the densities for all spin components in the phase separation region. From (a) to (d), the magnetic field strengths (in units of mG) are, respectively, $B=0.0211$, $0.0633$, $0.0844$, and $0.1689$. The scattering length $a_{0}=5.47a_{B}$ is the same for all figures. The densities of those components not shown in the figures are too small to be seen. To simplify the numerical calculations, we shall focus on highly oblate trap geometries ($\eta\gg 1$) such that the condensate can be regarded as quasi- two-dimensional whose motion along the $z$-axis is frozen to the ground state of the axial harmonic oscillator. The condensate wave functions can then be decomposed into $\displaystyle\psi_{m}({\mathbf{r}})=(\eta/\pi)^{1/4}e^{-\eta z^{2}/2}\phi_{m}({\bm{\rho}})$ (4) with ${\bm{\rho}}=(x,y)$ and $\phi_{m}$ being normalized to the total number of atoms $N$, i.e., $\int d{\mathbf{r}}\phi_{m}^{}\phi_{m}=N$. After integrating out the $z$ variable, $E_{0}$ gives an extra constant, while the interaction parameters $C$, $\alpha$, $\beta$, and $\gamma$ are all rescaled by a factor $(\eta/2\pi)^{1/2}$. The mean-field wave functions $\\{\psi_{m}\\}$ are obtained by minimizing the total energy functional numerically using imaginary time evolution. We shall focus our study on the Cr line Ho06 , namely only the scattering length $a_{0}$ is allowed to changed freely, since experimentally, it is the most relevant case. For all results presented in the present work, we have chosen $N=10^{5}$, $\omega_{\perp}=2\pi\times 100\,\mathrm{Hz}$, and $\eta=10$. Correspondingly, the dimensionless length unit $a_{\perp}=\sqrt{\hbar/(M\omega_{\perp})}$ is adopted in throughout this paper. We remark that we have neglected the magnetic dipole-dipole interaction energy in Eq. (3) for simplicity, as in the present work, we are concentrating on investigating how short-range interaction and magnetic field affect the ground state wave function. The ignorance of dipolar interaction in spinor Cr condensate was also justified in Ref. Ho06 . Moreover, we have numerically confirmed that for the parameter used in this paper, the dipole-dipole interaction energy is much smaller than short-ranged spin-dependent interaction energy when the condensate is not polarized by the magnetic field. ## III Results Figure 1 summarizes the main results of this paper. In the left panel of Fig. 1, we present the phase diagram of spin-3 Cr condensate in the $a_{0}$-$B$ parameter space, here we have adopted the similar notations for different phases as in Ref. Ho06 . In the right panel, we tabulate the major characters of each phase. We remark that the resolution of phase diagram is limited by step sizes of $a_{0}$ and $B$ when we numerically scan the parameter plane, therefore, it is possible that more details may emerge by reducing the step sizes. ### III.1 Condensate wave functions The numerical results indicate that the condensate wave functions can always be expressed as $\displaystyle\phi_{m}({\bm{\rho}})=\sqrt{n_{m}({\rho})}e^{i\vartheta_{m}},$ (5) where the density of $m$th component $n_{m}({\rho})$ is an axially symmetric function and the corresponding phase $\vartheta_{m}$ is a constant independent of the spatial coordinates. In case the external magnetic field is completely switched off, we find that all wave functions $\phi_{m}({\mathbf{r}})$ have the same density profile, indicating that the SMA is valid for spin-3 condensates in the absence of magnetic field. We note that this conclusion also holds true for spin-$1$ and -$2$ condensates. Once the external magnetic field is applied, SMA quickly becomes invalid. More remarkably, as shown in Fig. 2, when control parameters $a_{0}$ and $B$ fall into the shaded region in the left panel of Fig. 1, the peak densities of at least one of the spin components among $m=-3$, $\pm 2$, and $0$ do not occur at the center of the trap, in analogy to the phase separation in a two- component condensate binexp ; binary . In the absence of magnetic field, the system is symmetric under SO(3) rotation of the spin, and $m=-3$ component can be populated. However, immediately after we switch on the magnetic field, this SO(3) symmetry is broken such that $m=\pm 1$ spin components are highly populated under a very weak magnetic field. As one continuously increases the magnetic field, the occupation number in $m=-3$ component increases with density in the margin of the trap growing faster than that in the center, which induces the phase separation like behavior. When the population in $m=-3$ component dominates, the peak densities of all spin component occur at the center of the trap. We remark that similar behavior of the wave functions also appears outside the Cr line Ho06 . Figure 3: (color online) The instable region (enclosed by solid line) of a homogeneous Cr condensate and the phase separation region (enclosed by dashed line) of a trapped Cr condensate (Same as that in Fig. 1). To gain more insight into the origin of the phase separation like behavior, we consider a homogeneous Cr condensate where each spin component has already condensed into the zero momentum mode. The wave function $\psi_{m}({\mathbf{r}})$ for phase unseparated state is then replaced by a uniform $c$-number $\displaystyle\bar{\psi}_{m}=\sqrt{n}\xi_{m},$ (6) where $n$ is a real constant and $\xi_{m}$ are complex constants. The ground state can be obtained by minimizing the total energy $E$ subject to the normalization condition $\xi_{m}^{}\xi_{m}=1$. In such a way, we have reproduce the phase diagram in Ref. Ho06 . To confirm that those phases are indeed the ground states, we introduce a new set of variables, $\zeta_{p}$ and $\zeta_{p+1}$, corresponding to, respectively, the real and imaginary parts of the wave function $\xi_{m}$ as $\displaystyle\zeta_{p=2(3-m)+1}={\rm Re}[\xi_{m}]\mbox{ and }\zeta_{p=2(3-m)+2}={\rm Im}[\xi_{m}].$ We then construct the Hessian matrix ${\bm{H}}=\left[\frac{\partial^{2}E}{\partial\zeta_{p}\partial\zeta_{q}}\right]$. For a solution to be stable, the Hessian matrix must be positive definite timmer . In Fig. 3, we present the unstable region of a homogeneous Cr condensate on $a_{0}$-$B$ plane. To obtain it, we have chosen the density to be $n=3.3\times 10^{14}\,{\rm cm}^{-3}$ which is the peak density of the trapped system in our numerical calculations. One immediately sees that the unstable region of a homogeneous condensate roughly agrees with the phase separation region of the trapped system, which suggests that the possible origin of the phase separation behavior is the instability of the phase unseparated solution. We emphasize that, unlike in a binary Bose-Einstein condensate where the emergence of phase separation is determined by the strengths of intra- and inter-species interactions, here for a given scattering length $a_{0}$, the phase separation like behavior is induced by the magnetic field. ### III.2 Singlet amplitude Since the spatial independence of $\Theta({\mathbf{\rho}})$ is a necessary condition for SMA, it can also be used as a criterion to check the validity of SMA. As shown in Fig. 4, $\|\Theta\|$ is a constant when $B=0$; while immediately after the magnetic field is turned on, $\|\Theta\|$ becomes spatially dependent. In addition, the peak value of $\|\Theta\|$ decreases continuously as one increases the magnetic field until it completely vanish. In Fig. 4 (a), $\|\Theta(\rho)\|$ becomes zero only after the condensate is completely polarized, while in (b) and (c), it vanishes once the system enters the ${\rm H}_{1}$ phase. Therefore, using singlet amplitude, we may map out the phase boundaries between ${\rm A}_{1}$ and ${\rm FF}$, ${\rm Z}$ and ${\rm H}_{1}$, and ${\rm B}_{1}$ and ${\rm H}_{1}$. However, $\Theta$ alone is incapable of determining other phase boundaries. Finally, we note that, for $a_{0}>8.9a_{B}$, the value of $\|\Theta\|$ drops much faster with the increasing magnetic field than that corresponding to $a_{0}<8.9a_{B}$, as shown below this behavior has a direct impact on the magnetization curve of the system. Figure 4: (Color online) The typical behaviors of $\|\Theta(\rho)\|$ for $a_{0}=-8.27a_{B}$ (a), $5.47a_{B}$ (b), and $12.35a_{B}$ (c). In descending order of central value, the lines in (a) correspond to the magnetic field (in units of mG) $B=0$, $0.0244$, $0.1689$, $0.2533$, and $0.3377$; those in (b) correspond to $B=0$, $0.0422$, $0.0844$, $0.1266$, and $0.19$; and finally, those in (c) correspond to $B=0$, $0.0211$, $0.0422$, and $0.0633$. Figure 5: (Color online) The field dependence of the magnetization for $a_{0}=-59.82a_{B}$ (solid line), $-25.45a_{B}$ (dashed line), $5.47a_{B}$ (dash-dotted line), and $12.35a_{B}$ (dotted line). For $a_{0}>8.9a_{B}$ ($\alpha>0$), the $\mathcal{M}(p)$ curves corresponding to different $a_{0}$’s are indistinguishable. ### III.3 Magnetization We now turn to study magnetic field dependence of the total magnetization. To this end, we define the reduced magnetization as $\displaystyle\mathcal{M}=N^{-1}\int d{\mathbf{\rho}}\langle s^{z}\rangle.$ (7) Unlike in Ref. Makela07 where the total magnetization is conserved, here we allow it to change freely. Therefore, the transverse components of the spin, $\langle s^{x}\rangle$ and $\langle s^{y}\rangle$, are always zero. Figure 5 shows the field dependence of the reduced magnetization, which approaches $-3$ when $B$ reaches the saturation field. We note that, for $a_{0}<8.9a_{B}$, the behavior of $\mathcal{M}(B)$ slightly depends on the value of $a_{0}$; while for $a_{0}>8.9a_{B}$, the magnetization curves corresponding to different $a_{0}$’s become indistinguishable. Consequently, as shown in Fig. 1, the saturation field for the former case is a decreasing function of $a_{0}$, while for the latter one, it becomes a constant. The $a_{0}$ independence of the magnetization for $a_{0}>8.9a_{B}$ case can be qualitatively understood as follows. The scattering length $a_{0}$ only contributes to the total energy through singlet amplitude $\Theta$. As shown in Fig. 4, for $a_{0}>8.9a_{B}$, $\Theta$ vanishes quickly as one increases the magnetic field, such that varying $a_{0}$ only yields a negligible effect on magnetization curve. With the help magnetization, we can further identify the phase boundary between ${\rm H_{1}}$ and ${\rm FF}$ phases. Figure 6: (Color online) The spatial dependence of $\lambda_{1}$ (dash-dotted lines), $\lambda_{2}$ (dashed lines), and $\lambda_{3}$ (solid lines) for $a_{0}=5.47a_{B}$ (left panels) and $12.35a_{B}$ (right panels). The magnetic field strength (in units of mG) is denoted in each figure. ### III.4 Nematic tensor To determine other phase boundaries, we have to rely on the nematic tensor. In Fig. 6, we plot typical behavior of the eigenvalues of nematic tensor under different $a_{0}$ and magnetic fields. As a consequence of the failure of SMA, $\lambda_{a}$’s are generally spatially dependent. However, some of their characteristics obtained under SMA remain to be true. The full ferromagnet (FF) phase occurs when the magnetic field exceeds the saturation field such that only $m=-3$ component is occupied. The nematic tensor takes a diagonal form with $\lambda_{1}=\lambda_{2}=\frac{3}{2}$ and $\lambda_{3}=9$. For the wave functions of A1 and H1 phases, only two spin states are populated: other than a common $m=-3$ state, $m=3$ and $2$ are also occupied for, respectively, A1 and H1 phases. One can easily deduce that the nematic tensors of phases A1 and H1 are, respectively, ${\rm diag}\\{\frac{3}{2},\frac{3}{2},9\\}$ and $n^{-1}{\rm diag}\\{\frac{3}{2}n_{-3}+4n_{2},\frac{3}{2}n_{-3}+4n_{2},9n_{-3}+4n_{2}\\}$. Since for both cases, ${\cal N}_{zz}$ is the largest eigenvalue, we have $\hat{\mathbf{e}}_{3}\parallel\hat{\mathbf{z}}$. Moreover, $\hat{\mathbf{e}}_{1}$ and $\hat{\mathbf{e}}_{2}$ are spatially independent. For G1 phase, $m=\pm 2$ and $0$ states are unoccupied, consequently, ${\cal N}$ becomes a block diagonal matrix with one of the eigenvalues being ${\cal N}_{zz}=n^{-1}\left[9(n_{3}+n_{-3})+n_{1}+n_{-1}\right]$. Furthermore, from numerical results, we find that ${\cal N}_{zz}$ is always the smaller eigenvalue and $\lambda_{2}({\bm{\rho}})\neq\lambda_{3}({\bm{\rho}})$, which suggests that G1 is a biaxial nematic phase with $\hat{\mathbf{e}}_{1}\parallel\hat{\mathbf{z}}$. All spin components of ${\rm B}_{1}$ and ${\rm Z}$ phases are populated, they are all are biaxial nematic with three unequal and spatially dependent $\lambda$’s. The principal axes for these phases are also spatially dependents. The only difference is that for ${\rm B}_{1}$ phase we can identify that either ${\mathbf{e}}_{1}$ or ${\mathbf{e}}_{2}$ is perpendicular to $z$-axis. ## IV Conclusions To conclude, we have mapped out the phase diagram of a spin-3 Cr condensate subject to an external magnetic field based on the numerical calculation of the ground state wave function. In particular, we show that SMA becomes invalid for Cr condensates under a finite magnetic field. More remarkably, if the unknown scattering length $a_{0}$ falls into the region $[-0.37,8.2]a_{B}$, a phase separation like behavior may be induced by the magnetic field. We also point out that such behavior might originate from the instability of a phase unseparated solution. As a future work, we shall investigate the ground state structure of a spin-3 Cr condensate by including the magnetic dipole-dipole interaction. ###### Acknowledgements. We thank Han Pu for useful discussion. This work is supported by NSFC (Grant No. 10674141), National 973 program (Grant No. 2006CB921205), and the “Bairen” program of the Chinese Academy of Sciences. ## References (1) A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Phys. Rev. Lett. 94, 160401 (2005). * (2) J. Stuhler, A. Griesmaier, T. Koch, M. Fattori, T. Pfau, S. Giovanazzi, P. Pedri, and L. Santos, Phys. Rev. Lett. 95, 150406 (2005). * (3) T. Lahaye, J. Metz, B. Fröhlich, T. Koch, M. Meister, A. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi and M. Ueda, Phys. Rev. Lett. 101, 080401 (2008). * (4) J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, and W. Ketterle, Nature (London) 396, 345 (1998) * (5) M. Barrett, J. Sauer, and M. S. Chapman, Phys. Rev. Lett. 87, 010404 (2001). * (6) T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998). * (7) T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998). * (8) C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett. 81, 5257 (1998). * (9) M. Koashi and M. Ueda, Phys. Rev. Lett. 84, 1066 (2000). * (10) C. V. Ciobanu, S.-K. Yip, and T.-L. Ho, Phys. Rev. A, 61, 033607 (2000). * (11) H. Schmaljohann, M. Erhard, J. Kronjäger, M. Kottke, S. van Staa, L. Cacciapuoti, J. J. Arlt, K. Bongs, and K. Sengstock, Phys. Rev. Lett. 92, 040402 (2004). * (12) M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, K. M. Fortier, W. Zhang, L. You, and M. S. Chapman, Phys. Rev. Lett. 92, 140403 (2004). * (13) T. Kuwamoto, K. Araki, T. Eno, and T. Hirano, Phys. Rev. A 69, 063604 (2004). * (14) R. B. Diener and T.-L. Ho, Phys. Rev. Lett. 96, 190405 (2006). * (15) L. Santos and T. Pfau, Phys. Rev. Lett. 96, 190404 (2006). * (16) H. Mäkelä and K.-A. Suominen, Phys. Rev. A. 75, 033610 (2007). * (17) R. Barnett, A. Turner, and E. Demler, Phys. Rev. Lett. 97, 180412 (2006); ibid. Phys. Rev. A 76, 013605 (2007). * (18) S.-K. Yip, Phys. Rev. A 75, 023625 (2007). * (19) J.-S. Bernier, K. Sengupta, and Y.-B. Kim, Phys. Rev. B 76, 014502 (2007). * (20) Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett. 96, 080405 (2006). * (21) S. Yi, Ö. E. Müstecaplıoğlu, C. P. Sun, and L. You, Phys. Rev. A 66, 011601(R) (2002); * (22) P. G. de Gennes and J. Prost, _The Physics of Liquid Crystals_ , 2nd ed (Oxford University Press, London, 1993). * (23) C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 78, 586 (1997); D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 81, 1539 (1998). * (24) T. L. Ho and V. B. Shenoy, Phys. Rev. Lett. 77, 3276 (1996); H. Pu and N. P. Bigelow, Phys. Rev. Lett. 80, 1130 (1998). * (25) E. Timmermans, Phys. Rev. Lett. 81, 5718 (1998)	arxiv-papers	2009-02-25T02:56:02	2024-09-04T02:49:00.849475	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Liang He and Su Yi", "submitter": "Liang He", "url": "https://arxiv.org/abs/0902.4276" }
0902.4313	# Semileptonic Decays of $B$ Meson Transition Into $D$-wave Charmed Meson Doublets Long-Fei Gan lfgan@nudt.edu.cn Ming-Qiu Huang Department of Physics, National University of Defense Technology, Hunan 410073, China ###### Abstract We use QCD sum rules to estimate the leading-order universal form factors describing the semileptonic $B$ decay into orbital excited $D$-wave charmed doublets, including the ($1^{-}$, $2^{-}$) states ($D_{1}^{}$, $D^{\prime}_{2}$) and the ($2^{-}$, $3^{-}$) states ($D_{2}$, $D_{3}^{}$). The decay rates we predict are $\Gamma_{B\rightarrow D^{}_{1}\ell\overline{\nu}}=\Gamma_{B\rightarrow D^{\prime}_{2}\ell\overline{\nu}}=2.4\times 10^{-18}\mbox{GeV}$, $\Gamma_{B\rightarrow D_{2}\ell\overline{\nu}}=6.2\times 10^{-17}\mbox{GeV}$, and $\Gamma_{B\rightarrow D_{3}^{}\ell\overline{\nu}}=8.6\times 10^{-17}\mbox{GeV}$. The branching ratios are $\mathcal{B}(B\rightarrow D^{}_{1}\ell\overline{\nu})=\mathcal{B}(B\rightarrow D^{\prime}_{2}\ell\overline{\nu})=6.0\times 10^{-6}$, $\mathcal{B}(B\rightarrow D_{2}\ell\overline{\nu})=1.5\times 10^{-4}$, and $\mathcal{B}(B\rightarrow D_{3}^{}\ell\overline{\nu})=2.1\times 10^{-4}$, respectively. ###### pacs: 14.40.-n, 11.55.Hx, 12.38.Lg, 12.39.Hg ## I Introduction Higher excitations than $D^{()}$ play an important role in the understanding of semileptonic $B$ decays. Knowledge of these processes is important to reduce the uncertainties of the measurements on other semileptonic $B$ decays, and thus the determination of the Cabibbo-Kobayashi-Maskawa matrix elements, such as $\|V_{cb}\|$. Theoretically, the semileptonic decay processes are described by some form factors. The challenge for theory is the calculation of these decay form factors. Fortunately, the heavy quark effective theory (HQET) Wis , with an expansion in terms of $1/m_{Q}$ for hadrons containing a single heavy quark, provides a systematic method for investigating such processes. In HQET the approximate symmetries allow one to organize the spectrum of heavy mesons according to parity $P$ and total angular momentum $s_{l}$ of the light degree of freedom. Coupling the spin of the light degrees of freedom $s_{l}$ with the spin of a heavy quark $s_{Q}=1/2$ yields a doublet of meson states with a total spin $s=s_{l}\pm 1/2$. For charmed mesons, the lowest lying states $(0^{-},1^{-})$ doublet ($D$, $D^{}$) are $S$-wave states with the spin of light degrees $s_{l}=1/2$. The $P$-wave excitation corresponds to two series of states, one is the $s_{l}=1/2$ series, the $(0^{+},1^{+})$ doublet ($D_{0}^{}$, $D^{\prime}_{1}$); the other is the $s_{l}=3/2$ series, the $(1^{+},2^{+})$ doublet ($D_{1}$, $D^{}_{2}$). For $D$-wave states, those are $(1^{-},2^{-})$ and $(2^{-},3^{-})$ doublets (($D_{1}^{}$, $D^{\prime}_{2}$) and ($D_{2}$, $D^{}_{3}$)), corresponding to the spin of light degrees of freedom $s_{l}=3/2$ and $s_{l}=5/2$. The early study of the heavy-light mesons can be found in Ref. God . The $S$-wave and $P$-wave charmed states have been observed so far. The properties of these states have been extensively studied using different approaches during the past few years, including masses Ebe ; Dai , decay constants Neu ; Cap ; Cve , and decay widths Dai3 ; Dai2 ; Eic ; Xia . For the $D$-wave charmed mesons, their properties were investigated with the potential model Eic and QCD sum rules Wei . Semileptonic $B$ decay into an excited heavy meson has been observed in experiments Cle ; Ale . Recently, BABAR has measured semileptonic $B$ decays into orbitally excited charmed mesons $D_{1}(2420)$ and $D^{}_{2}(2460)$ Aub1 . They also reported two new $D_{s}$ states $D_{sJ}(2860)$ and $D_{sJ}(2690)$ in the $DK$ channel, which may fit in the $D$-wave charm-strange doublets Aub . A similar state $D_{sJ}(2715)$ has also been observed by Belle Bel . It is expected that the nonstrange $D$-wave charmed mesons will be found, and the measurements of the semileptonic $B$ decays into these states become available in the near future. To this end we study the predictions of HQET for semileptonic $B$ decays to $D$-wave charmed mesons. The semileptonic decay rate of a B meson transition into an charmed meson is determined by the corresponding matrix elements of the weak axial-vector and vector currents. In the heavy quark limit these elements are described, respectively, by one universal Isgur-Wise function at the leading order of heavy quark expansion Wis1 . The universal Isgur-Wise function is a nonperturbtive parameter. It must be calculated in some nonperturbative approaches. The main theoretical approaches are QCD sum rules Shi , constituent quark models, and lattice QCD. The investigations of semileptonic $B$ decays into charmed mesons can be found in Refs. Neu ; Hua1 ; Ebe1 ; Dea ; Wis1 with different methods. In this work, we estimate the leading-order Isgur-Wise functions describing the decays $B\rightarrow(D_{1}^{},D^{\prime}_{2})\ell\overline{\nu}$ and $B\rightarrow(D_{2},D_{3}^{})\ell\overline{\nu}$ and give a prediction for the widths of the decays. The remainder of this paper is organized as follows. In Sec. II we present the formulas of weak current matrix elements and decay rates. In Sec. III we give the relevant sum rules for two-point correlators, and then deduce the three- point sum rules for the Isgur-Wise functions. Section IV is devoted to numerical results and discussions. ## II Analytic formulations for semileptonic decay amplitudes $B\rightarrow(D_{1}^{},D^{\prime}_{2})\ell\overline{\nu}$ and $B\rightarrow(D_{2},D_{3}^{})\ell\overline{\nu}$ The heavy-light meson doublets can be expressed conveniently by effective operators Fal . For the ground doublet, the operator is $H_{a}=\frac{1+\hbox to0.0pt{/\hss}v}{2}[D^{}_{\mu}\gamma^{\mu}-D\gamma_{5}].$ (1) The effective operators describing the meson doublets $D(1^{-},2^{-})$ and $D(2^{-},3^{-})$ are given by $X^{\mu}=\frac{1+\hbox to0.0pt{/\hss}v}{2}[D^{\prime\mu\nu}_{2}\gamma_{5}\gamma_{\nu}-D^{}_{1\nu}\sqrt{\frac{3}{2}}(g^{\mu\nu}-\frac{1}{3}\gamma^{\nu}(\gamma^{\mu}+v^{\mu}))],$ (2) and $H^{\mu\nu}=\frac{1+\hbox to0.0pt{/\hss}v}{2}[D^{\mu\nu\sigma}_{3}\gamma_{\sigma}-\sqrt{\frac{3}{5}}\gamma_{5}D^{\alpha\beta}_{2}(g^{\mu}_{\alpha}g^{\nu}_{\beta}-\frac{\gamma_{\alpha}}{5}g^{\nu}_{\beta}(\gamma^{\mu}-v^{\mu})-\frac{\gamma_{\beta}}{5}g^{\mu}_{\alpha}(\gamma^{\nu}-v^{\nu}))].$ (3) In these operators, $D^{}_{\mu}$, $D$, $D^{\prime\mu\nu}_{2}$, $D^{}_{1\nu}$, $D^{\mu\nu\sigma}_{3}$, and $D^{\alpha\beta}_{2}$ separately represent annihilation operators of the $Q\overline{q}$ mesons with appropriate quantum numbers and $\hbox to0.0pt{/\hss}v=v\cdot\gamma$, $v$ is the heavy meson velocity. The theoretical description of semileptonic decays involves the matrix elements of vector and axial-vector currents ($V^{\mu}=\overline{c}\gamma^{\mu}b$ and $A^{\mu}=\overline{c}\gamma^{\mu}\gamma_{5}b$) between $B$ mesons and excited $D$ mesons. For the processes $B\rightarrow(D^{}_{1},D^{\prime}_{2})\ell\overline{\nu}$ and $B\rightarrow(D_{2},D_{3}^{})\ell\overline{\nu}$, these matrix elements can be parametrized through applying the trace formalism as follows Fal : $\displaystyle\langle D^{}_{1}(v^{{}^{\prime}},\varepsilon)\|(V-A)^{\mu}\|B(v)\rangle$ $\displaystyle=$ $\displaystyle\sqrt{\frac{3}{2}}\sqrt{m_{B}m_{D^{}_{1}}}\tau_{1}(y)[\varepsilon^{}\cdot v(v^{\mu}-\frac{y+2}{3}v^{\prime\mu})$ (4) $\displaystyle-$ $\displaystyle i\frac{1-y}{3}\epsilon^{\mu\alpha\beta\sigma}\varepsilon^{}_{\alpha}v^{\prime}_{\beta}v_{\sigma}],$ $\langle D^{\prime}_{2}(v^{{}^{\prime}},\varepsilon)\|(V-A)^{\mu}\|B(v)\rangle=\sqrt{m_{B}m_{D^{\prime}_{2}}}\tau_{1}(y)\varepsilon^{}_{\rho\nu}v^{\rho}[g^{\mu\nu}(y-1)-v^{\nu}v^{{}^{\prime}\mu}+i\epsilon^{\alpha\beta\nu\mu}v^{{}^{\prime}}_{\alpha}v_{\beta}],$ (5) and $\displaystyle\langle D_{2}(v^{{}^{\prime}},\varepsilon)\|(V-A)^{\mu}\|B(v)\rangle$ $\displaystyle=$ $\displaystyle\sqrt{\frac{5}{3}}\sqrt{m_{B}m_{D_{2}}}\tau_{2}(y)\varepsilon^{}_{\alpha\beta}v^{\alpha}[\frac{2(1-y^{2})}{5}g^{\mu\beta}-v^{\beta}v^{\mu}+\frac{2y-3}{5}v^{\beta}v^{{}^{\prime}\mu}$ (6) $\displaystyle+$ $\displaystyle i\frac{2(1+y)}{5}\epsilon^{\mu\lambda\beta\rho}v_{\lambda}v^{{}^{\prime}}_{\rho}],$ $\langle D_{3}^{}(v^{{}^{\prime}},\varepsilon)\|(V-A)^{\mu}\|B(v)\rangle=\sqrt{m_{B}m_{D_{3}^{}}}\tau_{2}(y)\varepsilon^{}_{\alpha\beta\lambda}v^{\alpha}v^{\beta}[g^{\mu\lambda}(1+y)-v^{\lambda}v^{{}^{\prime}\mu}+i\epsilon^{\mu\lambda\rho\tau}v_{\rho}v^{{}^{\prime}}_{\tau}],$ (7) where $(V-A)^{\mu}=\overline{c}\gamma^{\mu}(1-\gamma_{5})b$ is the weak current, $y=v\cdot v^{{}^{\prime}}$ and $\tau_{1}(y)$, $\tau_{2}(y)$ are the universal form factors, and $\varepsilon^{}_{\alpha}$, $\varepsilon^{}_{\alpha\beta}$, $\varepsilon^{}_{\alpha\beta\lambda}$ are the polarization tensors of these mesons. The differential decay rates are calculated by making use of the formulas (4) to (7) given above: $\frac{d\Gamma}{dy}(B\rightarrow D^{}_{1}\ell\overline{\nu})=\frac{G^{2}_{F}V^{2}_{cb}m^{2}_{B}m^{3}_{D^{}_{1}}}{72\pi^{3}}(\tau_{1}(y))^{2}(y-1)^{\frac{5}{2}}(y+1)^{\frac{3}{2}}[(1+r_{1}^{2})(2y+1)-2r_{1}(y^{2}+y+1)],$ (8) $\frac{d\Gamma}{dy}(B\rightarrow D^{\prime}_{2}\ell\overline{\nu})=\frac{G^{2}_{F}V^{2}_{cb}m^{2}_{B}m^{3}_{D^{\prime}_{2}}}{72\pi^{3}}(\tau_{1}(y))^{2}(y-1)^{\frac{5}{2}}(y+1)^{\frac{3}{2}}[(1+r_{2}^{2})(4y-1)-2r_{2}(3y^{2}-y+1)],$ (9) $\frac{d\Gamma}{dy}(B\rightarrow D_{2}\ell\overline{\nu})=\frac{G^{2}_{F}V^{2}_{cb}m^{2}_{B}m^{3}_{D_{2}}}{360\pi^{3}}(\tau_{2}(y))^{2}(y-1)^{\frac{5}{2}}(y+1)^{\frac{7}{2}}[(1+r_{3}^{2})(7y-3)-2r_{3}(4y^{2}-3y+3)],$ (10) $\frac{d\Gamma}{dy}(B\rightarrow D_{3}^{}\ell\overline{\nu})=\frac{G^{2}_{F}V^{2}_{cb}m^{2}_{B}m^{3}_{D_{3}^{}}}{360\pi^{3}}(\tau_{2}(y))^{2}(y-1)^{\frac{5}{2}}(y+1)^{\frac{7}{2}}[(1+r_{4}^{2})(11y+3)-2r_{4}(8y^{2}+3y+3)],$ (11) with $r_{i}=\frac{m_{D_{i}}}{m_{B}}$ ($D_{i}=D^{}_{1},D^{\prime}_{2},D_{2},D^{}_{3}$ for $i=1,2,3,4$ ). In the equations above, we have presented the decay rates of B semileptonic decay processes $B\rightarrow(D^{}_{1},D^{\prime}_{2})\ell\overline{\nu}$ and $B\rightarrow(D_{2},D^{}_{3})\ell\overline{\nu}$ in terms of the universal form factors $\tau_{1}(y)$ and $\tau_{2}(y)$, respectively. The only unknown factors in these equations are $\tau_{1}(y)$ and $\tau_{2}(y)$, which need to be determined by nonperturbative methods. ## III Sum rules for Isgur-Wise functions In the calculation of Isgur-Wise functions in HQET by means of QCD sum rule, the interpolating currents are potentially important. In Ref. Dai , two series of interpolating currents with nice propertties were proposed: $J^{{\dagger}\alpha_{1}\ldots\alpha_{j}}_{j,P,i}=\overline{h}_{v}(x)\Gamma^{\\{\alpha_{1}\ldots\alpha_{j}\\}}_{j,P,i}(D_{x_{t}})q(x)$ (12) or $J^{{}^{\prime}{\dagger}\alpha_{1}\ldots\alpha_{j}}_{j,P,i}=\overline{h}_{v}(x)\Gamma^{\\{\alpha_{1}\ldots\alpha_{j}\\}}_{j,P,i}(D_{x_{t}})(-i)\hbox to0.0pt{/\hss}D_{x_{t}}q(x)$ (13) where $i=1,2$ corresponding to two series of doublets of the spin-parity $[j^{(-1)^{j+1}},(j+1)^{(-1)^{j+1}}]$ and $[j^{(-1)^{j}},(j+1)^{(-1)^{j}}]$, respectively. $D_{t\mu}=D_{\mu}-v_{\mu}(v\cdot D)$ is the transverse component of the covariant derivative with respect to the velocity of the meson and $\Gamma^{\\{\alpha_{1}\ldots\alpha_{j}\\}}(D_{x_{t}})=\text{symmetrize}\\{\Gamma^{\alpha_{1}\ldots\alpha_{j}}(D_{x_{t}})-\frac{1}{3}g^{\alpha_{1}\alpha_{2}}_{t}g^{t}_{\alpha^{{}^{\prime}}_{1}\alpha^{{}^{\prime}}_{2}}\Gamma^{\alpha^{{}^{\prime}}_{1}\alpha^{{}^{\prime}}_{2}\alpha_{3}\cdots\alpha_{j}}\\}$ (14) with the transverse metric $g^{\alpha\beta}_{t}=g^{\alpha\beta}-v^{\alpha}v^{\beta}$. For the doublets of spin-parity $[j^{(-1)^{j+1}},(j+1)^{(-1)^{j+1}}]$ and $[j^{(-1)^{j}},(j+1)^{(-1)^{j}}]$, the expressions for $\Gamma^{\alpha_{1}\ldots\alpha_{j}}(D_{x_{t}})$ have been explicitly given in Dai as $\Gamma(D_{x_{t}})=\left\\{\begin{tabular}[]{ll}$\sqrt{\frac{2j+1}{2j+2}}\gamma^{5}(-i)^{j}D^{\alpha_{2}}_{x_{t}}\cdots D^{\alpha_{j}}_{x_{t}}(D^{\alpha_{1}}_{x_{t}}-\frac{j}{2j+1}\gamma^{\alpha_{1}}_{t}\hbox to0.0pt{/\hss}D_{x_{t}})$,&for $j^{(-1)^{j+1}}$\\\ $\frac{1}{\sqrt{2}}\gamma^{\alpha_{1}}_{t}(-i)^{j}D^{\alpha_{2}}_{x_{t}}\cdots D^{\alpha_{j}}_{x_{t}}$,&for $(j+1)^{(-1)^{j+1}}$\end{tabular}\right.$ $\Gamma(D_{x_{t}})=\left\\{\begin{tabular}[]{ll}\ $\frac{1}{\sqrt{2}}\gamma^{5}(-i)^{j}\gamma^{\alpha_{1}}_{t}D^{\alpha_{2}}_{x_{t}}\cdots D^{\alpha_{j+1}}_{x_{t}}$,&for $(j+1)^{(-1)^{j}}$\\\ $\sqrt{\frac{2j+1}{2j+2}}(-i)^{j}D^{\alpha_{2}}_{x_{t}}\cdots D^{\alpha_{j}}_{x_{t}}(D^{\alpha_{1}}_{x_{t}}-\frac{j}{2j+1}\gamma^{\alpha_{1}}_{t}\hbox to0.0pt{/\hss}D_{x_{t}})$,&for $j^{(-1)^{j}}$\end{tabular}\right.$ where $\gamma_{t\mu}=\gamma_{\mu}-\hbox to0.0pt{/\hss}vv_{\mu}$ is the transverse component of $\gamma_{\mu}$ with respect to the heavy quark velocity. For the $D$-wave meson doublets with $s_{l}=\frac{3}{2}^{-}$ and $s_{l}=\frac{5}{2}^{-}$, where $j=1$ and $j=2$, the currents are given by the following expressions: $J^{{\dagger}\alpha}_{1,-,3/2}=-i\sqrt{\frac{3}{4}}\overline{h}_{v}(D^{\alpha}_{t}-\frac{1}{3}\gamma^{\alpha}_{t}\hbox to0.0pt{/\hss}D_{t})q,$ (15) $J^{{\dagger}\alpha\beta\lambda}_{2,-,3/2}=-i\frac{1}{\sqrt{2}}T^{\alpha\beta,\mu\nu}\overline{h}_{v}\gamma_{5}\gamma_{t\mu}D_{t\nu}q,$ (16) and $J^{{\dagger}\alpha\beta}_{2,-,5/2}=-\sqrt{\frac{5}{6}}T^{\alpha\beta,\mu\nu}\overline{h}_{v}\gamma_{5}(D_{t\mu}D_{t\nu}-\frac{2}{5}D_{t\mu}\gamma_{t\nu}\hbox to0.0pt{/\hss}D_{t})q,$ (17) $J^{{\dagger}\alpha\beta\lambda}_{3,-,5/2}=-\frac{1}{\sqrt{2}}T^{\alpha\beta\lambda,\mu\nu\sigma}\overline{h}_{v}\gamma_{t\mu}D_{t\nu}D_{t\sigma}q,$ (18) which correspond to Eq. (12), and corresponding to Eq. (13) are $J^{{\dagger}\alpha}_{1,-,3/2}=-\sqrt{\frac{3}{4}}\overline{h}_{v}(D^{\alpha}_{t}-\frac{1}{3}\gamma^{\alpha}_{t}\hbox to0.0pt{/\hss}D_{t})\hbox to0.0pt{/\hss}D_{t}q,$ (19) $J^{{\dagger}\alpha\beta\lambda}_{2,-,3/2}=-\frac{1}{\sqrt{2}}T^{\alpha\beta,\mu\nu}\overline{h}_{v}\gamma_{5}\gamma_{t\mu}D_{t\nu}\hbox to0.0pt{/\hss}D_{t}q,$ (20) and $J^{{\dagger}\alpha\beta}_{2,-,5/2}=-\sqrt{\frac{5}{6}}T^{\alpha\beta,\mu\nu}\overline{h}_{v}\gamma_{5}(D_{t\mu}D_{t\nu}-\frac{2}{5}D_{t\mu}\gamma_{t\nu}\hbox to0.0pt{/\hss}D_{t})(-i)\hbox to0.0pt{/\hss}D_{t}q,$ (21) $J^{{\dagger}\alpha\beta\lambda}_{3,-,5/2}=-\frac{1}{\sqrt{2}}T^{\alpha\beta\lambda,\mu\nu\sigma}\overline{h}_{v}\gamma_{t\mu}D_{t\nu}D_{t\sigma}(-i)\hbox to0.0pt{/\hss}D_{t}q,$ (22) where $h_{v}$ is the generic velocity-dependent heavy quark effective field in HQET and $q$ denotes the light quark field. The tensors $T^{\alpha\beta,\mu\nu}$ and $T^{\alpha\beta\lambda,\mu\nu\sigma}$ are used to symmetrize indices and are given by Dai $T^{\alpha\beta,\mu\nu}=\frac{1}{2}(g^{\alpha\mu}_{t}g^{\beta\nu}_{t}+g^{\alpha\nu}_{t}g^{\beta\mu}_{t})-\frac{1}{3}g^{\alpha\beta}_{t}g^{\mu\nu}_{t},$ (23) $\displaystyle T^{\alpha\beta\lambda,\mu\nu\sigma}$ $\displaystyle=$ $\displaystyle\frac{1}{6}(g^{\alpha\mu}_{t}g^{\beta\nu}_{t}g^{\lambda\sigma}_{t}+g^{\alpha\mu}_{t}g^{\beta\sigma}_{t}g^{\lambda\nu}_{t}+g^{\alpha\nu}_{t}g^{\beta\mu}_{t}g^{\lambda\sigma}_{t}+g^{\alpha\nu}_{t}g^{\beta\sigma}_{t}g^{\lambda\mu}_{t}+g^{\alpha\sigma}_{t}g^{\beta\nu}_{t}g^{\lambda\mu}_{t}+g^{\alpha\sigma}_{t}g^{\beta\mu}_{t}g^{\lambda\nu}_{t})$ (24) $\displaystyle-$ $\displaystyle\frac{1}{15}(g^{\alpha\beta}_{t}g^{\mu\nu}_{t}g^{\lambda\sigma}_{t}+g^{\alpha\beta}_{t}g^{\mu\sigma}_{t}g^{\lambda\nu}_{t}+g^{\alpha\beta}_{t}g^{\nu\sigma}_{t}g^{\lambda\mu}_{t}+g^{\alpha\lambda}_{t}g^{\mu\nu}_{t}g^{\beta\sigma}_{t}+g^{\alpha\lambda}_{t}g^{\mu\sigma}_{t}g^{\beta\nu}_{t}$ $\displaystyle+$ $\displaystyle g^{\alpha\lambda}_{t}g^{\nu\sigma}_{t}g^{\beta\mu}_{t}+g^{\beta\lambda}_{t}g^{\mu\nu}_{t}g^{\alpha\sigma}_{t}+g^{\beta\lambda}_{t}g^{\mu\sigma}_{t}g^{\alpha\nu}_{t}+g^{\beta\lambda}_{t}g^{\nu\sigma}_{t}g^{\alpha\mu}_{t}).$ Usually the currents with derivatives of the lowest order (12) are used in the QCD sum rule approach. However, currents with derivatives of one order higher (13) are also used in some conditions because in the nonrelativistic quark model there is a corresponding relation between the orbital angular momenta and the orders of derivatives in the space wave functions. As for the orbital D-wave mesons, which corresponding to derivatives of order two, it is reasonable to use the currents (17), (18), (19) and (20). These currents have nice properties, they have nonvanishing projection only to the corresponding states of the HQET in the $m_{Q}\rightarrow\infty$ limit, without mixing with states of the same quantum number but different $s_{l}$. Thus we can define one-particle-current couplings as follows: $J^{P}=1^{-}:\langle D^{}_{1}(v,\varepsilon)\|J^{\alpha}\|0\rangle=f_{1}\sqrt{m_{D^{}_{1}}}\varepsilon^{\alpha},$ (25) $J^{P}=2^{-}:\langle D^{\prime}_{2}(v,\varepsilon)\|J^{\alpha\beta}\|0\rangle=f^{\prime}_{2}\sqrt{m_{D^{\prime}_{2}}}\varepsilon^{\alpha\beta},$ (26) $J^{P}=2^{-}:\langle D_{2}(v,\varepsilon)\|J^{\alpha\beta}\|0\rangle=f_{2}\sqrt{m_{D_{2}}}\varepsilon^{\alpha\beta},$ (27) $J^{P}=3^{-}:\langle D^{}_{3}(v,\varepsilon)\|J^{\alpha\beta\lambda}\|0\rangle=f_{3}\sqrt{m_{D^{}_{3}}}\varepsilon^{\alpha\beta\lambda}.$ (28) The couplings $f_{i}$ are low-energy parameters which are determined by the dynamics of the light degree of freedom. Since the pairs ($f_{1}$, $f^{\prime}_{2}$) and ($f_{2}$, $f_{3}$) are related by the spin symmetry, we will consider $f_{1}$ and $f_{2}$ hereafter. The decay constants $f_{i}$ can be estimated from two-point sum rules, therefore we list the sum rules after the Borel transformation. For the ground-state heavy mesons, the sum rule for the correlator of two heavy-light currents is well known. It is Hua1 $f^{2}_{-,\frac{1}{2}}e^{-2\bar{\Lambda}_{-,\frac{1}{2}}/T}=\frac{3}{16\pi^{2}}\int_{0}^{\omega_{c0}}\omega^{2}e^{-\omega/T}d\omega-\frac{1}{2}\langle\bar{q}q\rangle(1-\frac{m^{2}_{0}}{4T^{2}}).$ (29) For the $s_{l}^{P}=\frac{3}{2}^{-}$ doublet, when the currents (19) and (20) are used, the corresponding sum rule is : $f^{2}_{-,\frac{3}{2}}e^{-2\bar{\Lambda}_{-,\frac{3}{2}}/T}=\frac{1}{2^{8}\pi^{2}}\int_{0}^{\omega_{c1}}\omega^{6}e^{-\omega/T}d\omega-\frac{5}{3\times 2^{8}}\int_{0}^{\omega_{c1}}\omega^{2}e^{-\omega/T}d\omega\langle\frac{\alpha_{s}}{\pi}GG\rangle.$ (30) For the $s_{l}^{P}=\frac{5}{2}^{-}$ doublet, when the currents (17) and (18) are used, the corresponding sum rule is : $\displaystyle f^{2}_{-,\frac{5}{2}}e^{-2\bar{\Lambda}_{-,5/2}/T}=\frac{1}{5\times 2^{7}\pi^{2}}\int_{0}^{\omega_{c2}}\omega^{6}e^{-\omega/T}d\omega-\frac{5}{3\times 2^{6}}\int_{0}^{\omega_{c2}}\omega^{2}e^{-\omega/T}d\omega\langle\frac{\alpha_{s}}{\pi}GG\rangle.$ (31) As we have just mentioned, for the amplitudes of the semileptonic decays into excited states in the infinite mass limit, the only unknown quantities in (8), (9), (10) and (11) are the universal functions $\tau_{1}(y)$ and $\tau_{2}(y)$. In Ref. Col the form factors $\tau_{1}(y)$ and $\tau_{2}(y)$ were estimated through QCD sum rule by using currents with derivatives of lower order, (15) to (18). Considering that the corresponding relation between the orbital angular momentum and the order of the derivative mentioned above, we use the currents (19) and (20) instead of (15) and (16) for the ($D_{1}^{}$, $D^{\prime}_{2}$) doublet. As for the ($D_{2}$, $D^{}_{3}$) doublet, we also use the currents (17) and (18). In order to calculate this two form factors by QCD sum rules, we study the analytic properties of three-point correlators: $i^{2}\int d^{4}xd^{4}ze^{i(k^{{}^{\prime}}\cdot x-k\cdot z)}\langle 0\|T[J^{\alpha}_{1,-}(x)J^{\mu(v,v^{{}^{\prime}})}_{V,A}(0)J^{{\dagger}}_{0,-}(z)\|0\rangle=\Gamma(\omega,\omega^{{}^{\prime}},y)\mathcal{L}^{\mu\alpha}_{V,A},$ (32) $i^{2}\int d^{4}xd^{4}ze^{i(k^{{}^{\prime}}\cdot x-k\cdot z)}\langle 0\|T[J^{\alpha\beta}_{2,-}(x)J^{\mu(v,v^{{}^{\prime}})}_{V,A}(0)J^{{\dagger}}_{0,-}(z)\|0\rangle=\Gamma^{\prime}(\omega,\omega^{{}^{\prime}},y)\mathcal{L}^{\mu\alpha\beta}_{V,A},$ (33) where $J^{\mu(v,v^{{}^{\prime}})}_{V}=h(v^{{}^{\prime}})\gamma^{\mu}h(v)$ and $J^{\mu(v,v^{{}^{\prime}})}_{A}=h(v^{{}^{\prime}})\gamma^{\mu}\gamma_{5}h(v)$. The variables $k$($=P-m_{b}v$) and $k^{{}^{\prime}}$($=P^{\prime}-m_{c}v^{\prime}$) denote residual “off-shell” momenta of the initial and final meson states, respectively. For heavy quarks in bound states they are typically of order $\Lambda_{QCD}$ and remain finite in the heavy quark limit. $\Gamma(\omega,\omega^{{}^{\prime}},y)$ and $\Gamma^{\prime}(\omega,\omega^{{}^{\prime}},y)$ are analytic functions in the “off-shell” energies $\omega=2v\cdot k$ and $\omega^{\prime}=2v^{\prime}\cdot k^{\prime}$ with discontinuities for positive values of these variables. They also depend on the velocity transfer $y=v\cdot v^{\prime}$, which is fixed in a physical region. $\mathcal{L}_{V,A}$ are Lorentz structures. Following the standard QCD sum rule procedure, the calculations of $\Gamma(\omega,\omega^{{}^{\prime}},y)$ and $\Gamma^{\prime}(\omega,\omega^{{}^{\prime}},y)$ are straightforward. First, we saturate Eqs.(32) and (33) with physical intermediate states in HQET and find that the hadronic representations of the correlators as follows: $\Gamma_{hadron}(\omega,\omega^{{}^{\prime}},y)=\frac{f_{-,\frac{1}{2}}f_{-,j_{l}}\tau_{i}(y)}{(2\bar{\Lambda}_{-,\frac{1}{2}}-\omega-i\varepsilon)(2\bar{\Lambda}_{-,j_{l}}-\omega^{{}^{\prime}}-i\varepsilon)}+\text{higher resonances},$ (34) where $f_{-,j_{l}}$ are the decay constants defined in Eqs.(25) and (27), $\overline{\Lambda}_{-,j_{l}}=m_{-,j_{l}}-m_{Q}$. Second, the functions can be approximated by a perturbative calculation supplemented by nonperturbative power corrections proportional to the vacuum condensates which are treated as phenomenological parameters. The perturbative contribution can be represented by a double dispersion integral in $\nu$ and $\nu^{{}^{\prime}}$ plus possible subtraction terms. So the theoretical expression for the correlator has the form $\Gamma_{theo}(\omega,\omega^{{}^{\prime}},y)\simeq\int d\nu d\nu^{{}^{\prime}}\frac{\rho^{pert}(\nu,\nu^{{}^{\prime}},y)}{(\nu-\omega-i\varepsilon)(\nu^{{}^{\prime}}-\omega^{{}^{\prime}}-i\varepsilon)}+\text{subtractions}+\Gamma^{cond}(\omega,\omega^{{}^{\prime}},y).$ (35) The perturbative part of the spectral density can be calculated straightforward. Confining us to the leading order of perturbation, the perturbative spectral densities of the two sum rules for $\tau_{1}(y)$ and $\tau_{2}(y)$ are $\displaystyle\rho_{pert}(\nu,\nu^{{}^{\prime}},y)=\frac{3}{2^{8}\pi^{2}}\frac{1}{(y+1)^{\frac{3}{2}}(y-1)^{\frac{5}{2}}}\nu^{{}^{\prime}}[(3\nu^{2}-(1+2y)(2\nu\nu^{\prime}-\nu^{\prime 2})]$ $\displaystyle\times\Theta(\nu)\Theta(\nu^{{}^{\prime}})\Theta(2y\nu\nu^{{}^{\prime}}-\nu^{2}-\nu^{{}^{\prime}2}),$ (36) and $\displaystyle\rho_{pert}(\nu,\nu^{{}^{\prime}},y)=\frac{3}{2^{8}\pi^{2}}\frac{1}{(y+1)^{\frac{7}{2}}(y-1)^{\frac{5}{2}}}[(5\nu-12y\nu^{{}^{\prime}}+3\nu^{{}^{\prime}})\nu^{2}+(3\nu+\nu^{{}^{\prime}})(2y^{2}-2y+1)\nu^{{}^{\prime}2}]$ $\displaystyle\times\Theta(\nu)\Theta(\nu^{{}^{\prime}})\Theta(2y\nu\nu^{{}^{\prime}}-\nu^{2}-\nu^{{}^{\prime}2}).$ (37) Following the arguments in Refs. Neu ; Blo , the perturbative and the hadronic spectral densities cannot be locally dual to each other, the necessary way to restore duality is to integrate the spectral densities over the “off-diagonal” variable $\nu_{-}=\nu-\nu^{{}^{\prime}}$, keeping the “diagonal” variable $\nu_{+}=\frac{\nu+\nu^{{}^{\prime}}}{2}$ fixed. It is in $\nu_{+}$ that the quark-hadron duality is assumed for the integrated spectral densities. The integration region can be expressed in terms of the variables $\nu_{-}$ and $\nu_{+}$ and we choose the triangular region defined by the bounds: $0\leq\nu_{+}\leq\omega_{c}$, $-2\sqrt{\frac{y-1}{y+1}}\nu_{+}\leq\nu_{-}\leq 2\sqrt{\frac{y-1}{y+1}}\nu_{+}$. As discussed in Refs. Blo ; Neu , the upper limit $\omega_{c}$ for $\nu_{+}$ in the region $\frac{1}{2}[(y+1)-\sqrt{y^{2}-1}]\omega_{c0}\leqslant\omega_{c}\leqslant\frac{1}{2}(\omega_{c0}+\omega_{c2})$ is reasonable. A double Borel transformation in $\omega$ and $\omega^{{}^{\prime}}$ is performed on both sides of the sum rules, in which for simplicity we take the Borel parameters equal Neu ; Hua1 ; Col : $T_{1}=T_{2}=2T$. In the calculation, we have considered the operators of dimension $D\leq 5$ in OPE. After adding the nonperturbative parts, we obtain the sum rules for $\tau_{1}$ and $\tau_{2}$ as follows: $\displaystyle\tau_{1}(y)f_{-,1/2}f_{-,3/2}e^{-(\bar{\Lambda}_{-,1/2}+\bar{\Lambda}_{-,3/2})/T}$ $\displaystyle=$ $\displaystyle\frac{1}{2^{4}\pi^{2}}\frac{1}{(1+y)^{3}}\int^{\omega^{\prime}_{c}}_{0}d\nu_{+}e^{-\frac{\nu_{+}}{T}}\nu^{4}_{+}$ (38) $\displaystyle-$ $\displaystyle\frac{T}{3\times 2^{5}}\frac{2y+3}{(y+1)^{2}}\langle\frac{\alpha_{s}}{\pi}GG\rangle,$ $\displaystyle\tau_{2}(y)f_{-,1/2}f_{-,5/2}e^{-(\bar{\Lambda}_{-,1/2}+\bar{\Lambda}_{-,5/2})/T}$ $\displaystyle=$ $\displaystyle\frac{3}{8\pi^{2}}\frac{1}{(1+y)^{4}}\int^{\omega_{c}}_{0}d\nu_{+}e^{-\frac{\nu_{+}}{T}}\nu^{4}_{+}$ (39) $\displaystyle-$ $\displaystyle\frac{T}{3\times 2^{4}}\frac{1}{(y+1)^{3}}\langle\frac{\alpha_{s}}{\pi}GG\rangle.$ We also derive the sum rule for $\tau_{2}$ by using the currents (21) and (22), which appears to be $\displaystyle\tau_{2}(y)f_{-,1/2}f_{-,5/2}e^{-(\bar{\Lambda}_{-,1/2}+\bar{\Lambda}_{-,5/2})/T}$ $\displaystyle=$ $\displaystyle\frac{21}{5\times 2^{4}\pi^{2}}\frac{1}{(1+y)^{4}}\int^{\omega_{c}}_{0}d\nu_{+}e^{-\frac{\nu_{+}}{T}}\nu^{5}_{+}$ (40) $\displaystyle+$ $\displaystyle\frac{T^{2}}{3\times 2^{4}}\frac{4y-25}{(y+1)^{3}}\langle\frac{\alpha_{s}}{\pi}GG\rangle.$ ## IV Numerical results and discussions We now evaluate the sum rules numerically. For the QCD parameters entering the theoretical expressions, we take the standard values: $\langle\overline{q}q\rangle=-(0.24)^{3}\mbox{GeV}^{3}$, $\langle\alpha_{s}GG\rangle=0.04\mbox{GeV}^{4}$, and $m^{2}_{0}=0.8\mbox{GeV}^{2}$. In the numerical calculations, we take $2.83\mbox{GeV}$ God ; Eic for the mass of the $s_{l}=5/2$ doublet and $2.78\mbox{GeV}$ for the $s_{l}=3/2$ doublet. For mass of initial $B$ meson, we use $m_{B}=5.279\mbox{GeV}$ Pdg . In order to obtain information of $\tau_{1}(y)$ and $\tau_{2}(y)$ with less systematic uncertainties in the calculation, we divide the three-point sum rules by the square roots of relevant two-point sum rules, as many authors did Neu ; Hua1 ; Col , to reduce the number of input parameters and improve stabilities. Then we obtain expressions for the $\tau_{1}(y)$ and $\tau_{2}(y)$ as functions of the Borel parameter $T$ and the continuum thresholds. Imposing usual criteria for the upper and lower bounds of the Borel parameter, we found they have a common sum rule “window”: $0.7\mbox{GeV}<T<1.5\mbox{GeV}$, which overlaps with those of two-point sum rules (29), (30) and (31) (see Fig. 1). Notice that the Borel parameter in the sum rules for three-point correlators is twice the Borel parameter in the sum rules for the two-point correlators. In the evaluation we have taken $2.0\mbox{GeV}<\omega_{c0}<2.4\mbox{GeV}$ Hua1 ; Neu , $2.8\mbox{GeV}<\omega_{c1}<3.2\mbox{GeV}$, and $3.2\mbox{GeV}<\omega_{c2}<3.6\mbox{GeV}$. The regions of these continuum thresholds are fixed by analyzing the corresponding two-point sum rules. According to the discussion in Sec. III, we can fix $\omega^{\prime}_{c}$ and $\omega_{c}$ in the regions $2.3\mbox{GeV}<\omega^{\prime}_{c}<2.6\mbox{GeV}$ and $2.5\mbox{GeV}<\omega_{c}<2.7\mbox{GeV}$. The results are showed in Fig. 2. \| \| ---\|---\|--- Figure 1: Dependence of $\tau_{1}(y)$ and $\tau_{2}(y)$ on Borel parameter $T$ at $y=1$. Figure 2: Prediction for the Isgur-Wise functions $\tau_{1}(y)$ and $\tau_{2}(y)$. The resulting curves for $\tau_{1}(y)$ and $\tau_{2}(y)$ can be parametrized by the linear approximation $\tau_{1}(y)=\tau_{1}(1)[1-\rho^{2}_{\tau_{1}}(y-1)],\text{ }\tau_{1}(1)=0.14\pm 0.03,\text{ }\rho^{2}_{\tau_{1}}=0.13\pm 0.02;$ (41) $\tau_{2}(y)=\tau_{2}(1)[1-\rho^{2}_{\tau_{2}}(y-1)],\text{ }\tau_{2}(1)=0.57\pm 0.09,\text{ }\rho^{2}_{\tau_{2}}=0.78\pm 0.13.$ (42) The errors mainly come from the uncertainty due to $\omega_{c}$’s and $T$. It is difficult to estimate these systematic errors which are brought in by the quark-hadron duality. The maximal values of $y$ are $y^{D^{}_{1}}_{max}=y^{D^{\prime}_{2}}_{max}=(1+r_{1,2}^{2})/2r_{1,2}\approx 1.213$ and $y^{D_{2}}_{max}=y^{D^{}_{3}}_{max}=(1+r_{3,4}^{2})/2r_{3,4}\approx 1.201$. By using the parameters $V_{cb}=0.04$, $G_{F}=1.166\times 10^{-5}\mbox{GeV}^{-2}$, we get the semileptonic decay rates of $B\rightarrow(D_{1}^{},D^{\prime}_{2})\ell\overline{\nu}$ and $B\rightarrow(D_{2},D_{3}^{})\ell\overline{\nu}$. Consider that $\tau_{B}=1.638\text{ps}$ Pdg , we get the branching ratios, respectively. All these results are listed in Table 1. Table 1: Predictions for the decay widths and branching ratios Decay mode \| \| Decay width $\Gamma$ (GeV) \| \| Branching ratio \| \| Branching ratio of Ref.Col ---\|---\|---\|---\|---\|---\|--- $B\rightarrow D^{}_{1}\ell\overline{\nu}$ \| \| $2.4\times 10^{-18}$ \| \| $6.0\times 10^{-6}$ \| \| $B\rightarrow D^{\prime}_{2}\ell\overline{\nu}$ \| \| $2.4\times 10^{-18}$ \| \| $6.0\times 10^{-6}$ \| \| $B\rightarrow D_{2}\ell\overline{\nu}$ \| \| $6.2\times 10^{-17}$ \| \| $1.5\times 10^{-4}$ \| \| $1\times 10^{-5}$ $B\rightarrow D^{}_{3}\ell\overline{\nu}$ \| \| $8.6\times 10^{-17}$ \| \| $2.1\times 10^{-4}$ \| \| $1\times 10^{-5}$ Because of the large background from $B\rightarrow D^{()}\ell\overline{\nu}$ decays, there is no experimental data available so far. As we can see from Table 1, the rates of semileptonic $B$ decay into the $s^{P}_{l}=\frac{3}{2}^{-}$ doublet are tiny and our results are larger than those predicted by Ref. Col in the $B$ to $s^{P}_{l}=\frac{5}{2}^{-}$ charmed doublet channels. The difference comes because the way in which we choose the parameters is different from theirs. They chose the parameters according to other theoretical approaches. In contrast, we choose the parameters following the way of Ref. Neu . In addition, we also estimate the universal form factor $\tau_{2}(y)$ with the sum rule (40) and we get almost the same result as (42). When trying to estimate the $\tau_{1}(y)$ by using the currents (15) and (16), we find that after the quark-hadron duality are assumed the integral over the perturbative spectral density becomes zero. As for the $P$-wave and the $F$-wave mesons, similar results can be obtained after the calculations above have been carefully repeated. The semileptonic and leptonic $B$ decay rate is about $10.9\%$ of the total $B$ decay rate, in which the $S$-wave charmed mesons $D$ and $D^{}$ contribute about $8.65\%$ Pdg and the $P$-wave charmed mesons contribute about $0.9\%$ Hua1 . Our results then suggest that the $D$-wave charmed mesons contribute about $0.04\%$ of the total $B$ decay rate. Sum up the branching ratios of these semileptonic $B$ decay processes, the eight lightest charmed mesons contribute about $9.59\%$ of the $B$ decay rate. Therefore, semileptonic decays into higher excited states and nonresonant multibody channels should be about $1.31\%$ of the $B$ decay rate. Whatsoever, our result is just a leading-order estimate of the contribution of the $D$-wave charmed mesons channels to the semileptonic $B$ decay. In summary, we estimate the leading-order universal form factors describing the $B$ meson of ground-state transition into orbital excited $D$-wave charmed resonances, the ($1^{-}$, $2^{-}$) states ($D_{1}^{}$, $D^{\prime}_{2}$), which belong to the $s_{l}^{P}=\frac{3}{2}^{-}$ heavy quark doublet and the ($2^{-}$, $3^{-}$) states ($D_{2}$, $D^{}_{3}$), which belong to the $s_{l}^{P}=\frac{5}{2}^{-}$ heavy quark doublet, by use of QCD sum rules within the framework of HQET. The semileptonic decay widths as well as the branching ratios we get are shown in Table 1. The predictions are larger than those predicted by Ref. Col . This needs future experiments for clarification. We also prove that when $s^{P}_{l}=\frac{5}{2}^{-}$ the interpolating currents (12) and (13) proposed in Ref. Dai are really equivalent. It is worth noting that in the estimate of the semileptonic $B$ decay form factors when the currents (12) with quantum numbers of light degree of freedom $s_{l}^{P}=\frac{1}{2}^{+},\frac{3}{2}^{-},\frac{5}{2}^{+}$ are used for the excited charmed mesons, we find the perturbative contributions vanish after the quark-hadron duality are assumed. In this case we should use the currents (13) which contain derivatives of one order higher. ###### Acknowledgements. L. F. Gan thanks M. Zhong for useful discussions. This work was supported in part by the National Natural Science Foundation of China under Contract No. 10675167. ## References (1) M. Neubert, Phys. Rep. 245, 259 (1994) and references therein; Aneesh V. Mannohar and Mark B. Wise, Heavy Quark Physics, Cambridge University Press (2000). * (2) S. Godfrey and R. Kokoski, Phys. Rev. D 43, 1679 (1991); S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985). * (3) D. Ebert, V. O. Galkin and R. N. Faustov, Phys. Rev. D 57, 5663 (1998). * (4) Y. B. Dai, C. S. Huang, M. Q. Huang, and C. Liu, Phys. Lett. B 390, 350 (1997); Y. B. Dai, C. S. Huang, and M. Q. Huang, Phys. Rev. D 55, 5719 (1997). * (5) M. Neubert, Phys. Rev. D 45, 2451 (1992); 46, 3914 (1992). * (6) S. Capstick and S. Godfrey, Phys. Rev. D 41, 2856 (1990). * (7) G. Cvetič , C. S. Kim, Guo-Li Wang and Wuk Namgung, Phys. Lett. B 596, 84 (2004); Guo-Li Wang, Phys. Lett. B 633, 492 (2006). * (8) Y. B. Dai, C. S. Huang, M. Q. Huang, H. Y. Jin, and C. Liu, Phys. Rev. D 58, 094032 (1998). * (9) Y. B. Dai, C. S. Huang, and H. Y. Jin, Z. Phys. C 60, 527-534 (1993); Phys. Lett. B 331, 174 (1994); Y. B. Dai and H. Y. Jin, Phys. Rev. D 52, 236 (1995). * (10) E. J. Eichten, C. T. Hill, and C. Quigg, Phys. Rev. Lett.71, 4116 (1993). * (11) X. H. Zhong and Q. Zhao, Phys. Rev. D 78, 014029 (2007). * (12) W. Wei, X. Liu, and S. L. Zhu, Phys. Rev. D 75, 014013 (2007). * (13) A. Anastassov et al. (CLEO Collaboration), Phys. Rev. Lett. 80, 4127 (1998). * (14) D. Buskulic et al. (ALEPH Collaboration), Phys. Lett. B 395, 373 (1997); Z. Phys. C 73, 601 (1997). * (15) B. Aubert et al. (BARBAR Collaboration), hep-ex/0808.0333. * (16) B. Aubert et al. (BARBAR Collaboration), Phys. Rev. Lett. 97, 222001 (2006). * (17) K. Abe et al. (BELLE Collaboration), hep-ex/0608031. * (18) A. K. Leibovich, Z. Ligeti, I. W. Stewart, and M. B. Wise, Phys. Rev. Lett.78, 3995 (1997); Phys. Rev. D 57, 308 (1998). * (19) M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979); 147, 448 (1979); V. A. Novikov, M. A. Shifman and A. I. Vainshtein, and V. I. Zakharov, Fortschr. Phys. 32, 585 (1984). * (20) M. Q. Huang and Y. B. Dai, Phys. Rev. D 59, 034018 (1999); 64, 014034 (2001). * (21) D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Rev. D 61, 014016 (1999); 75, 074008 (2007). * (22) A. Deandrea, N. Di Bartolomeo, R. Gatto, G. Nardulli and A. D. Polosa, Phys. Rev. D 58, 034004 (1998); V. Morénas, A. Le Yaouanc, L. Oliver, O. Pène and J. C. Raynal, Phys. Rev. D 56, 5668 (1997). * (23) A. F. Falk, Nul. Phys. B 378, 79 (1992); A. F. Falk and M. Luke, Phys. Lett. B 292, 119 (1992). * (24) P. Colangelo, F. De Fazio and G. Nardulli, Phys. Lett. B 478, 408 (2000). * (25) B. Blok and M. Shifman, Phys. Rev. D 47, 2949 (1993). * (26) Particle Data Group, C. Amsler et al., Phys. Lett. B 667, 1 (2008).	arxiv-papers	2009-02-25T08:18:54	2024-09-04T02:49:00.854744	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Long-Fei Gan and Ming-Qiu Huang", "submitter": "Long-Fei Gan", "url": "https://arxiv.org/abs/0902.4313" }
0902.4328	Vol.0 (2008) No.2, 000–000 Jianghui Ji 11institutetext: Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China 11email: jijh@pmo.ac.cn 22institutetext: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China 33institutetext: National Astronomical Observatory, Mitaka, Tokyo 181-8588, Japan 44institutetext: Department of Astronomy, Nanjing University, Nanjing 210093, China # The Dynamical Architecture and Habitable zones of the Quintuplet Planetary System 55 Cancri Jianghui JI 1122 Hiroshi Kinoshita 33 Lin LIU 44 Guangyu LI 1122aaaa (Received 2008/09/08) ###### Abstract We perform numerical simulations to study the secular orbital evolution and dynamical structure in the quintuplet planetary system 55 Cancri with the self-consistent orbital solutions by Fischer and coworkers (2008). In the simulations, we show that this system can be stable at least for $10^{8}$ yr. In addition, we extensively investigate the planetary configuration of four outer companions with one terrestrial planet in the wide region of 0.790 AU $\leq a\leq$ 5.900 AU to examine the existence of potential asteroid structure and Habitable Zones (HZs). We show that there are unstable regions for the orbits about 4:1, 3:1 and 5:2 mean motion resonances (MMRs) with the outermost planet in the system, and several stable orbits can remain at 3:2 and 1:1 MMRs, which is resemblance to the asteroidal belt in solar system. In a dynamical point, the proper candidate HZs for the existence of more potential terrestrial planets reside in the wide area between 1.0 AU and 2.3 AU for relatively low eccentricities. ###### keywords: celestial mechanics-methods:n-body simulations-planetary systems- stars:individual(55 Cancri) ## 1 Introduction The nearby star 55 Cancri is of spectral type K0/G8V with a mass of $0.92\pm 0.05M_{\odot}$ (Valenti & Fischer 2005). Marcy et al. (2002) reported a second giant planet with a long period of $\sim 14$ yr after the first planet discovered in 1997. The 55 Cnc system can be very attractive, because first it hosts a distant giant Jupiter-like planet about 5.5 AU resembling Jupiter in our solar system. The second interesting thing is that this system may be the only known planetary system in which two giant planets are close to the 3:1 orbital resonance, and the researchers have extensively studied the dynamics and formation of the 3:1 MMR in this system (see Beaugé et al. 2003; Ji et al. 2003; Zhou et al. 2004; Kley, Peitz, & Bryden 2004; Voyatzis & Hadjidemetriou 2006; Voyatzis 2008). Still, the additional collection of follow-up observations and the increasing of precision of measurements (at present $\sim 1$ ms-1 to $3$ ms-1) have indeed identified more planets. McArthur et al. (2004) reported the fourth planet with a small minimum mass $\sim 14$ $M_{\oplus}$ that orbits the host star with a short period of 2.8 day, by analyzing three sets of radial velocities. The improvement of the observations will actually induce additional discovery. Hence, it is not difficult to understand that more multiple planetary systems or additional planets in the multiple systems are to be dug out supplemental data. More recently, Fischer et al. (2008) (hereafter Paper I) reported the fifth planet in the 55 Cnc system with the Doppler shift observations over 18 yrs, and showed that all five planets are in nearly circular orbits and four have eccentricities under 0.10. It is really one of the most extreme goals for the astronomers devoted to searching for the extrasolar planets to discover a true solar system analog, which may hold one or two gas giants orbiting beyond 4 AU that can be compared to Jupiter and Saturn in our own solar system (Butler 2007, private communication; see also Gaudi et al. 2008). This indicates that several terrestrial planets may move in the so-called Habitable Zones (HZs) (Kasting et al. 1993; Jones et al. 2005), and the potential asteroidal structure can exist. Considering the probability of the coplanarity and nearly circular orbits for five planets (Paper I), the 55 Cnc system is suggested to be a comparable twin of the solar system. Hence, firstly, in a dynamical viewpoint, one may be concerned about the stability of the system over secular timescale. On the other hand, the small bodies as terrestrial objects may exist in this system and are to be detected with forthcoming space-based missions (Kepler, SIM). In this paper, we focus on understanding the dynamical structure and finding out suitable HZs for life-bearing terrestrial planets in this system. ## 2 Dynamical Analysis In this paper, we adopt the orbital parameters of the 55 Cancri system provided by Paper I (see their Table 4). For the convenience of narration, we re-label the planets according to the ascendant semi-major axes in the order from the innermost to the outermost planet (e.g., B, C, D, E, F), while the original names discovered in the chronological order are also accompanied but in braces (see Table 1). Furthermore, McArthur et al. (2004) derived the orbital inclination $i=53^{\circ}\pm 6^{\circ}.8$ with respect to the sight line for the outermost planet from HST astrometric data by measuring the apparent astrometric motion of the host star. In the simulations, we adopt this estimated orbital inclination of $53^{\circ}$ and further assume all the orbits to be coplanar. With the planetary masses $M\sin i$ reported in Table 1, then we obtain their true masses. Specifically, the masses of five planets are respectively, 0.03 $M_{Jup}$, 1.05 $M_{Jup}$, 0.21 $M_{Jup}$, 0.18 $M_{Jup}$ and 4.91 $M_{Jup}$, where $\sin i=\sin 53^{\circ}=0.7986$. Thus, we take the stellar mass $M_{c}$ of 0.94 $M_{\odot}$ (Paper I), and the planetary masses above-mentioned in the numerical study, except where noted. We utilize N-body codes (Ji, Li & Liu 2002) to perform numerical simulations by using RKF7(8) and symplectic integrators (Wisdom & Holman 1991) for this system. In the numerical runs, the adopted time stepsize is usually $\sim$ 2% \- 5% of the orbital period of the innermost planet. In addition, the numerical errors were effectively controlled over the integration timescale, and the total energy is generally conserved to $10^{-6}$ for the integrations. The typical timescale of simulations of the 55 Cnc system is from 100 Myr to 1 Gyr. ### 2.1 The Stability of the 55 Cancri Planetary System #### 2.1.1 case 1: 5-p for $10^{8}$ yr To explore the secular stability of this system, firstly, we numerically integrated the five-planet system on a timescale of $10^{8}$ yr, using the initials listed in Table 1. In Figure 1, a snapshot of the secular orbital evolution of all planets is illustrated, where $Q_{i}=a_{i}(1+e_{i})$, $q_{i}=a_{i}(1-e_{i})$ (the subscript $i=1-5$, individually, denoting Planet B, C, D, E, and F) are, respectively, the apoapsis and periapsis distances. In the secular dynamics, the semi-major axis $a_{1}$ and $a_{2}$ remain unchanged to be 0.0386 and 0.115 AU, respectively, for $10^{8}$ yr, while $a_{3}$, $a_{4}$ and $a_{5}$ slightly librate about 0.241, 0.786, and 6.0 AU with quite small amplitudes over the same timescale. The variations of eccentricities during long-term evolution are followed, where $0.23<e_{1}<0.28$, $0.0<e_{2}<0.03$, $0.034<e_{3}<0.069$, $0.0<e_{4}<0.013$, and $0.056<e_{5}<0.095$, implying that all the eccentricities undergo quasi- periodic modulations. In Fig.1, we note that the time behaviors of $Q_{i}$ and $q_{i}$ show regular motions of bounded orbits for all five planets and indicate their orbits are well separated during the secular evolution due to small mutual interactions, which again reflect the regular dynamics of the eccentricities over secular timescale. In the numerical study, we find the system can be dynamically stable and last at least for $10^{8}$ yr. Thus our numerical outcomes strengthen and verify those of Paper I for the integration of $10^{6}$ yr, which also showed the system can remain stable over 1 Myr and the variations of all planetary eccentricities are modest. Secondly, we further performed an extended integration for the planetary configuration simply consisting of four outer planets over timescale up to 1 Gyr (see Figure 2). The longer integration again reveals that the orbital evolution of four planets are quite similar to those exhibited in Fig.1, and then strongly supports the secular stability of this system. In a recent study, Gayon et al. (2008) show that the 55 Cnc system may remain a stable chaos state as the planetary eccentricities do not grow over longer timescale. Therefore, it is safely to conclude that the 55 Cnc system remain dynamically stable in the lifetime of the star. In order to assess the stability of 55 Cnc with respect to the variations of the planetary masses, we first fix $\sin i$ in increment of 0.1 from 0.3 to 0.9. In the additional numerical experiments, we simply vary the masses but keep all orbital parameters (Table 1), again restart new runs of integration for the five-planet system for 100 - 1000 Myr with the rescaled masses. As a result, we find the system could remain definitely stable for the above investigated timescale with slight vibrations in semi-major axes and eccentricities for all planets, indicating the present configuration is not so sensitive to the planetary masses. Subsequently, we again examine the stabilities of different orbital configurations within the error range of the Keplerian orbital fit given by Paper I. Herein 100 simulations are carried out for 10 Myr, and the numerical results show that all the runs are stable over the simulation timescale, indicating that this five-planet system is fairly robust with respect to the variational planetary configurations. #### 2.1.2 case 2: 7-p for $10^{8}$ yr However, Paper I argued that the 6th or more planets could exist and maintain dynamical stability in the large gap between Planets E and F in this system. Next, we also integrate the 55 Cnc system with additional planets (2 massive terrestrial planets, Earth at 1 AU and Mars at 1.52 AU) to mimic the situation of the inner solar system. In this runs, we examine the configuration consisting of 5 planets and 2 terrestrial bodies to study the coexistence of multiple objects. This means that we directly place Earth and Mars into the 55 Cnc system to simulate ”the inner solar system”, where the orbital elements for above terrestrial planets are calculated from JPL planetary ephemerides DE405 at Epoch JD 2446862.3081 corresponding to the outermost companion (see Table 2), e.g., the semi-major axes are respectively, 1.00 and 1.524 AU. The five planets are always assumed to be coplanar in the simulations, thus the inclinations for 2 terrestrial planets refer to the fundamental plane of their orbits. In this numerical experiment, we find that the 7-p system can remain dynamically stable and last at least for $10^{8}$ yr. In Figure 3 are shown the time behaviors of $Q$ (yellow line) and $q$ (black line) for Mars, Earth and Planet E. The numerical results show the regular bounded motions that their semi-major axes and eccentricities do not dramatically change in their secular orbital evolution, and this is also true for the other four planets in the 55 Cnc. It is not so surprised for one to realize that two additional terrestrial planets could exist for long time because the gravitational perturbations arise from other planets are much smaller. In the following section, we will extensively explore this issue on the dynamical architecture for the Earth-like planets in the system. ## 3 Dynamical Architecture and Potential HZs To investigate the dynamical structure and potential HZs in this system, we extensively performed additional simulations with the planetary configuration of coplanar orbits of four outer companions with one terrestrial planet. In this series of runs, the mass of the assumed terrestrial planet selected randomly in the range 0.1 $M_{\oplus}$ to 1.0 $M_{\oplus}$. The initial orbital parameters are as follows: the numerical investigations were carried out in [$a,e$] parameter space by direct integrations, and for a uniform grid of 0.01 AU in semi-major axis (0.790 AU $\leq a\leq$ 5.900 AU) and 0.01 in eccentricity ($0.0\leq e\leq 0.2$), the inclinations are $0^{\circ}<I<5^{\circ}$. The angles of the nodal longitude, the argument of periastron, and the mean anomaly are randomly distributed between $0^{\circ}$ and $360^{\circ}$ for each orbit. Then each terrestrial mass body was numerically integrated with four outer planets in the 55 Cnc system. In total, about 10,750 simulations were exhaustively run for typical integration time spans from $10^{5}$ to $10^{6}$ yr (about $10^{6}$ \- $10^{7}$ times the orbital period of Planet C). Then, our main results now follow. Figure 4 shows the contours of the surviving time for Earth-like planets (Upper) and the status of their final eccentricities (Lower) for the integration over $10^{5}$ yr, and the horizontal and vertical axes represent initial $a$ and $e$ of the orbits. Fig. 4 (Upper) displays that there are stable zones for the Earth-like planets in the region between 1.0 and 2.3 AU with final low eccentricities of $e<0.10$. The extended simulations ($10^{6}$ yr) for the objects of the above region also exhibit the same results. This zone may be strongly recommended to be one of the potential candidate HZs in the 55 Cnc, and our results coincide with those by Jones et al. (2005), who showed the possible HZs of 1.04 AU $<a<$ 2.07 AU. Still, the outcomes presented here have confirmed those in §2.1.2, where we show that the stable configuration of Earth at 1.00 AU and Mars at 1.523 AU in this five-planet system. The sixth planet or additional habitable bodies may be expected to revealed in this region by future observations111The semi-amplitude of wobble velocity $K\propto{M_{p}\sin i}/{\sqrt{a(1-e^{2})}}$ (with $M_{p}\ll M_{c}$), herein $M_{c}$, $M_{p}$, $a$, $e$ and $i$ are, respectively, the stellar mass, the planetary mass, the orbital semi-major axis, the eccentricity and the inclination of the orbit relative to the sky plane. This means that planets with larger masses and (or) smaller orbits could have larger $K$. For example, a planet of 1.0 $M_{\oplus}$ at 1 AU in a nearly circular orbit may cause stellar wobble about 0.10 m/s. In this sense, much higher Doppler precision is required to discover such Earth-like planets in future.. In general, the planetary embryos or planetesimals may be possibly captured into the mean motion resonance regions or thrown into HZs by a giant planet under migration due to the planet-disk interaction and could survive during the final planetary evolution over the secular timescale after complex scenarios of secular resonance sweeping, gravitationally scattering, and late heavy bombardments (Nagasawa et al. 2005; Thommes et al. 2008). We note that there are strongly unstable orbits 222We define an unstable orbit as an Earth- like planet is ejected far away or moves too close to the parent star or the giant planets, meeting the following criteria: (1) the eccentricity approaches unity, (2) the semi-major axis exceeds a maximum value, e.g., 1000 AU, (3) the assuming planet collides with the star or enters the mutual Hill sphere of the known giant planets. for the low-mass planets initially distributed in the region $3.9$ AU $<a<5.9$ AU, where the planetary embryos have very short dynamical surviving time. In the meantime, the eccentricities can be quickly pumped up to a high value $\sim 0.9$ (see Fig. 4, Lower). We note that the orbital evolution is not so sensitive to the initial masses. In fact, these planetary embryos are involved in many of MMRs with the outermost giant in the 55 Cnc system, e.g., 7:4 (4.063 AU) and 3:2 MMRs (4.503 AU). The overlapping resonance mechanism (Murray & Dermott 1999) can reveal their chaotic behaviors of being ejected from the system in short dynamical lifetime $\sim 10^{2}-10^{3}$ yr, furthermore the majority of orbits are within the sphere of 3 times Hill radius ($R_{H}={\left({M_{5}}/(3M_{c})\right)}^{1/3}a_{5}$, $3R_{H}\doteq 2.10$ AU) of the 14-yr planet. Using resonance overlapping criterion (MD99; Duncan et al. 1989), the separation in semi-major axis $\Delta{a}\approx 1.5{\left({M_{5}}/M_{c}\right)}^{2/7}a_{5}\doteq 1.95$ AU, then the inner boundary $R_{O}=a_{5}-\Delta{a}$ for Planet F is at $\sim 3.95$ AU. And the orbits in this zone become chaotic during the evolution because the planets are both within 3 $R_{H}$ and in the vicinity of $R_{O}$. Similarly, there exist unstable zones for the nearby orbits around Planet E (0.78 AU $<a<$ 0.90 AU), which may not be habitable in dynamical point. It is suggested that MMRs can play an important role in determining the orbital dynamics of the terrestrial bodies, which are either stabilized or destabilized in the vicinities of the MMRs. The outermost giant, like Jupiter, may shape and create the characteristic of dynamical structure of the small bodies. Most of the initial orbits for planetary embryos located about 3:1 (2.837 AU), 5:2 (3.204 AU), and 4:1 MMRs (2.342 AU), are quickly cleared off by the perturbing from Planet F. In the region of $2.4$ AU $<a<3.8$ AU, stable zones are separated by the mean motion resonance barriers, e.g., 3:1 and 5:2 MMRs. Note that the initial orbits for the relatively low eccentricity (under 0.06) for 4:1 MMR can remain stable over the simulation timescale. However, the terrestrial bodies about 7:3 MMR (3.354 AU) and 2:1 MMR (3.717 AU) are both on the edge of the stability, and the former are close to 5:2 MMR, while the latter just travel around the inner border of $3R_{H}$ at $\sim 3.80$ AU. The extended longer integrations show that their eccentricities can be further excited to a high value and a large fraction of them lose stabilities in the final evolution. The above gaps are apparently resembling those of the asteroidal belt in solar system. In the simulations, several stable orbits can be found about 3:2 MMR at 4.503 AU, which is analogous to the Hilda group for the asteroids in the solar system, surviving at least for $10^{6}$ yr. In addition, the other several stable cases are the so-called Trojan planets (1:1 MMR), residing at $\sim$ 5.9 AU. The studies show that the stable Trojan configurations may be possibly common in the extrasolar planetary systems (Dvorak et al. 2004; Ji et al. 2005; Gozdziewski & Konacki 2006). Indeed, terrestrial Trojan planets with circular orbits $\sim$ 1 AU could potentially be habitable, and are worthy of further investigation in future. ## 4 Summary and Discussions In this work, we have studied the secular stability and dynamical structure and HZs of the 55 Cnc planetary system. We now summarize the main results as follows: (1) In the simulations, we show that the quintuplet planetary system can remain dynamically stable at least for $10^{8}$ yr and that the stability would not be greatly influenced by shifting the planetary masses. Account for the nature of near-circular well-spaced orbits, the 55 Cnc system may be a close analog of the solar system. In addition, we extensively investigated the planetary configuration of four outer companions with one terrestrial planet in the region 0.790 AU $\leq a\leq$ 5.900 AU to examine the existence of potential Earth-like planets and further study the asteroid structure and HZs in this system. We show that unstable zones are about 4:1, 3:1 and 5:2 MMRs in the system, and several stable orbits can remain at 3:2 and 1:1 MMRs. The simulations not only present a clear picture of a resembling of the asteroidal belt in solar system, but also may possibly provide helpful information to identify the objects when modeling multi-planet orbital solutions (Paper I) by analyzing RV data. The dynamical examinations are helpful to search for best- fit stable orbital solutions to consider the actual role of the resonances, where some of best-fit solutions close to unstable islands of MMRs can be dynamically ruled out in the fitting process. As well-known, the extensive investigations in the planetary systems (Menou & Tabachnik 2003; Érdi et al. 2004; Ji et al. 2005, 2007; Pilat-Lohinger et al. 2008; Raymond et al. 2008) show that the dynamical structure is correlated with mean motion and secular resonances. The eccentricities of the planetesimals (or terrestrial planets) can be excited by sweeping secular resonance (Nagasawa & Ida 2000) as well as mean motion resonances, thus the orbits of the small bodies can undergo mutual crossings and then they are directly cleared up in the post-formation stage. In conclusion, the mentioned dynamical factors perturbing from the giant planets will influence and determine the characteristic distribution of the terrestrial planets in the late stage formation of the planetary systems, to settle down the remaining residents in the final system. (2) As the stellar luminosity of 55 Cnc is lower than that of the Sun, the HZ should shift inwards compared to our solar system. It seems that the newly- discovered planet at $\sim 0.783$ AU can reside in the HZs (Rivera & Haghighipour 2007), and this planet may be habitable provided that it bears surface atmosphere to sustain the necessary liquid water and other suitable life-developing conditions (Kasting et al. 1993). In a dynamical consideration, the proper candidate HZs for the existence of more potential terrestrial planets reside in the wide area between 1.0 AU and 2.3 AU for relatively low eccentricities, and the maintenance of low eccentricity can play a vital role in avoiding large seasonal climate variations (Menou & Tabachnik 2003) for the dynamical habitability of the terrestrial planets. Moreover, our numerical simulations also suggest that additional Earth-like planets (§2.1.2) can also coexist with other five known planets in this system over secular timescale. This should be carefully examined by abundant measurements and space missions (e.g. Kepler and TPF) for this system in future. ###### Acknowledgements. We would like to thank the anonymous referee for valuable comments and suggestions that help to improve the contents. We are grateful to G.W. Marcy and D. A. Fischer for sending us their manuscript and insightful discussions. This work is financially supported by the National Natural Science Foundations of China (Grants 10573040, 10673006, 10833001, 10203005) and the Foundation of Minor Planets of Purple Mountain Observatory. We are also thankful to Q.L. Zhou for the assistance of computer utilization. Part of the computations were carried out on high performance workstations at Laboratory of Astronomical Data Analysis and Computational Physics of Nanjing University. ## References * [Beaugé et al.(2003)] Beaugé, C., et al. 2003, ApJ, 593, 1124 * [Dvorak et al. (2004)] Dvorak, R., et al. 2004, A&A, 426, L37 * [Duncan et al.(1989)] Duncan, M., Quinn, T., & Tremaine, S. 1989, Icarus, 82, 402 * [Érdi et al.(2004)] Érdi, B., et al. 2004, MNRAS, 351, 1043 * [Fischer et al.(2008)] Fischer, D. A., et al. 2008, ApJ, 675, 790 (Paper I) * [Gaudi et al.(2008)] Gaudi, B. S., et al. 2008, Science, 319, 927 * [Gayon et al.(2008)] Gayon, J., Marzari, F., & Scholl, H. 2008, MNRAS, 389, L1 * [Gozdziewski(2006)] Gozdziewski, K., & Konacki, M. 2006, ApJ, 647, 573 * [Ji et al.(2002)] Ji, J. H., Li, G.Y., & Liu, L. 2002, ApJ, 572, 1041 * [Ji et al.(2003)] Ji, J. H., Kinoshita, H., Liu, L., & Li, G.Y. 2003, ApJ, 585, L139 * [Ji et al.(2005)] Ji, J. H., Liu, L., Kinoshita, H., & Li, G.Y. 2005, ApJ, 631, 1191 * [Ji et al.(2007)] Ji, J. H., Kinoshita, H., Liu, L., & Li, G.Y. 2007, ApJ, 657, 1092 * [Jones et al.(2005)] Jones, B. W., et al. 2005, ApJ, 622, 1091 * [Kasting et al.(1993)] Kasting, J. F., et al. 1993, Icarus, 101, 108 * [Kley et al.(2004)] Kley, W., Peitz, J., & Bryden, G. 2004, A&A, 414, 735 * [Marcy et al.(2002)] Marcy,G. W., et al. 2002, ApJ, 581, 1375 * [McArthur et al. (2004)] McArthur, B.E., et al. 2004, ApJ, 614, L81 * [Menou & Tabachnik(2003)] Menou, K., & Tabachnik, S. 2003, ApJ, 583, 473 * [Murray Dermott(1999)] Murray, C. D., & Dermott, S. F. 1999, Solar System Dynamics (New York: Cambridge Univ. Press) (MD99) * [Nagasawa & Ida(2000)] Nagasawa, M., & Ida, S. 2000, AJ, 120, 3311 * [Nagasawa (2005)] Nagasawa, M., Lin, D.N.C., & Thommes, E. 2005, ApJ, 635, 578 * [Pilat-Lohinger(2008)] Pilat-Lohinger, E., et al. 2008, ApJ, 681, 1639 * [Raymond et al.(2008)] Raymond, S. N., Barnes, R., & Gorelick, N. 2008, ApJ, 689, 478 * [Rivera & Haghighipour(2007)] Rivera, E., & Haghighipour, N. 2007, MNRAS, 374, 599 * [Thommes et al.(2008)] Thommes, E. W., et al. 2008, ApJ, 675, 1538 * [Valenti et al. (2005)] Valenti, J.A., & Fischer, D. A. 2005, ApJS, 159, 141 * [Voyatzis & Hadjidemetriou(2006)] Voyatzis, G., & Hadjidemetriou, J. D. 2006, CeMDA, 95, 259 * [Voyatzis(2008)] Voyatzis, G. 2008, ApJ, 675, 802 * [Wisdom Holman(1991)] Wisdom, J., & Holman, M. 1991, AJ, 102, 1528 * [Zhou et al.(2004)] Zhou, L.-Y., et al. 2004, MNRAS, 350, 1495 Table 1: The orbital parameters of 55 Cancri planetary system. Planet \| $M$sin$i$($M_{Jup}$) \| $P$(days) \| $a$(AU) \| $e$ \| $\varpi$(deg) \| $T_{p}$ ---\|---\|---\|---\|---\|---\|--- Planet B (e) \| 0.0241 \| 2.796744 \| 0.038 \| 0.2637 \| 156.500 \| 2447578.2159 Planet C (b) \| 0.8358 \| 14.651262 \| 0.115 \| 0.0159 \| 164.001 \| 2447572.0307 Planet D (c) \| 0.1691 \| 44.378710 \| 0.241 \| 0.0530 \| 57.405 \| 2447547.5250 Planet E (f) \| 0.1444 \| 260.6694 \| 0.785 \| 0.0002 \| 205.566 \| 2447488.0149 Planet F (d) \| 3.9231 \| 5371.8207 \| 5.901 \| 0.0633 \| 162.658 \| 2446862.3081 a. The parameters are taken from Table 4 of Fischer et al. (2008). The mass of the star is 0.94 $M_{\odot}$. Table 2: The orbital elements for 2 terrestrial planets at JD 2446862.3081 (From DE405). Planet \| $a$ (AU) \| $e$ \| $I$(deg) \| $\Omega$(deg) \| $\omega$(deg) \| M(deg) ---\|---\|---\|---\|---\|---\|--- Earth \| 1.000 \| 0.0164 \| 0.002 \| 348.33 \| 115.231 \| 61.647 Mars \| 1.524 \| 0.0935 \| 1.850 \| 49.60 \| 286.352 \| 85.614 Figure 1: Snapshot of the secular orbital evolution of all planets is illustrated, where $Q_{i}=a_{i}(1+e_{i})$, $q_{i}=a_{i}(1-e_{i})$ (the subscript $i=1-5$, each for Planet B, C, D, E, and F) are, respectively, the apoapsis and periapsis distances. $a_{1}$ and $a_{2}$ remain unchanged to be 0.0386 and 0.115 AU, respectively, while $a_{3}$, $a_{4}$ and $a_{5}$ slightly librate about 0.241, 0.786, and 6.0 AU with smaller amplitudes for $10^{8}$ yr. The simulations indicate the secular stability of the 55 Cnc. Figure 2: Numerical simulations for four outer planets for $10^{9}$ yr. The apoapsis and periapsis distances: $Q_{i}=a_{i}(1+e_{i})$, $q_{i}=a_{i}(1-e_{i})$ (the subscript $i=2-5$, each for Planet C, D, E, and F). The long-term simulation shows that the system can remain stable over 1 Gyr. Figure 3: Simulation for 7-p case. The system can be dynamically stable and last at least for $10^{8}$ yr, the time behaviors of $Q$ (yellow line) and $q$ (black line) each for Mars, Earth and Planet E. The results show that their semi-major axes and eccentricities do not dramatically change in the secular orbital evolution, and it is also true for the other four planets in the 55 Cnc. Figure 4: Upper: Contour of the surviving time for Earth-like planets for the integration of $10^{5}$ yr. Lower: Status of their final eccentricities. Horizontal and vertical axes are the initial $a$ and $e$. Stable zones for the Earth-like planets in the region between 1.0 and 2.3 AU with final low eccentricities of $e<0.10$. Unstable islands, e.g., 3:1 and 5:2 MMRs, have separated the region of $2.4$ AU $<a<3.8$ AU. Strongly chaos happen for the low-mass bodies initially distributed in $3.9$ AU $<a<5.9$ AU, and their eccentricities can be quickly pumped up to a high value $\sim 0.9$.	arxiv-papers	2009-02-25T09:51:50	2024-09-04T02:49:00.859706	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ji Jianghui (1,2), H. Kinoshita (3), Liu Lin (4), Li Guangyu (1,2)\n ((1)Purple Mountain Observatory, CAS (2)NAOC, (3)NAOJ, (4)Nanjing Univ.)", "submitter": "Jianghui Ji", "url": "https://arxiv.org/abs/0902.4328" }
0902.4479	# $\alpha$-Amenable Hypergroups Ahmadreza Azimifard ###### Abstract Let $K$ denote a locally compact commutative hypergroup, $L^{1}(K)$ the hypergroup algebra, and $\alpha$ a real-valued hermitian character of $K$. We show that $K$ is $\alpha$-amenable if and only if $L^{1}(K)$ is $\alpha$-left amenable. We also consider the $\alpha$-amenability of hypergroup joins and polynomial hypergroups in several variables as well as a single variable. Keywords. \| $\alpha$-Amenable hypergroups; ---\|--- \| Koornwinder, associated Legendre, Pollaczek, and disc polynomials AMS Subject Classification (2000). Primary 43A62, 43A07. Secondary 46H20. Introduction. Let $K$ denote a locally compact commutative hypergroup, $L^{1}(K)$ the hypergroup algebra, and $\alpha$ a hermitian character of $K$. It is shown in [7] that $K$ is $\alpha$-amenable if and only if either $K$ satisfies the modified Reiter’s condition of $P_{1}$-type in $\alpha$ or the maximal ideal in $L^{1}(K)$ generated by $\alpha$ has a bounded approximate identity. For instance, $K$ is always $(1-)$ amenable, and if $K$ is compact or $L^{1}(K)$ is amenable, then $K$ is $\alpha$-amenable in every character $\alpha$. It is worth noting, however, that there do exist hypergroups which are not $\alpha(\not=1)$-amenable; e.g. see [7, 15]. So, the amenability of a hypergroup in a character $\alpha$ cannot in general imply its amenability in other characters even if $\alpha$ is integrable, as illustrated in Section 2. In fact, this kind of amenability of hypergroups depends heavily on the asymptotic behavior of characters as well as Haar measures, as demonstrated in this paper and [2, 3, 7]. The paper is devoted to the character amenability of hypergroups. Sections 1 and 2 contain our main results. First we show that if the character $\alpha$ is real-valued, then $K$ is $\alpha$-amenable if and only if $L^{1}(K)$ is $\alpha$-left amenable; see Theorem 1.1. We then (Theorem 1.3) consider the $\alpha$-amenability of hypergroup joins. Section 2 is restricted to the polynomial hypergroups. Theorem 2.1 provides a necessary condition for the $\alpha$-amenability of hypergroups; and, subsequently we use this result to examine the $\alpha$-amenability of various polynomial hypergroups. In fact, we show that the majority of common examples of polynomial hypergroups are only $1$-amenable, and Example (VI) illustrates just how complicated hypergroups can be. Parts of this paper are taken from author’s dissertation at Technische Universit t M nchen. Preliminaries. Let $(K,p,\sim)$ denote a locally compact commutative hypergroup with Jewett’s axioms [10], where $p:K\times K\rightarrow M^{1}(K)$, $(x,y)\mapsto p(x,y)$, and $\sim:K\rightarrow K$, $x\mapsto\tilde{x}$, specify the convolution and involution on $K$ and $p{(x,y)}=p{(y,x)}$ for every $x,y\in K$. Here $M^{1}(K)$ denotes the set of all probability measures on $K$. Let us first recall required notions here, which are mainly from [4, 10]. Let $C_{c}(K)$, $C_{0}(K)$, and $C^{b}(K)$ be the spaces of all continuous functions, those which have compact support, vanishing at infinity, and bounded on $K$, respectively. Both $C^{b}(K)$ and $C_{0}(K)$ will be topologized by the uniform norm $\\|\cdot\\|_{\infty}$, and by Riesz’s theorem $C_{0}(K)^{\ast}\cong M(K)$, the space of all complex regular Radon measures on $K$. The translation of $f\in C_{c}(K)$ at the point $x\in K$, $T_{x}f$, is defined by $T_{x}f(y)=\int_{K}f(t)dp{(x,y)}(t)$, for every $y\in K$. Let $m$ denote the unique Haar measure of $K$ [16] and $(L^{p}(K),\\|\cdot\\|_{p})$ $(p\geq 1)$ the usual Banach space. If $p=1$, $(L^{1}(K),\\|\cdot\\|_{1})$ is a Banach $\ast$-algebra where the convolution and involution of $f,g\in L^{1}(K)$ are given by $fg(x)=\int_{K}f(y)T_{\tilde{y}}g(x)dm(y)$ ($m$-a.e.) and $f^{\ast}(x)=\overline{f(\tilde{x})}$ respectively. If $K$ is discrete, then $L^{1}(K)$ has an identity element; otherwise $L^{1}(K)$ has a bounded approximate identity (b. a. i.), i.e. there exists a net $\\{e_{i}\\}_{i}$ of functions in $L^{1}(K)$ with $\\|e_{i}\\|_{1}\leq M$, $M>0$, such that $\\|f\ast e_{i}-f\\|_{1}\rightarrow 0$ as $i\rightarrow\infty$. The dual of $L^{1}(K)$ can be identified with the usual Banach space $L^{\infty}(K)$, and its structure space is homeomorphic to the character space of $K$, i.e. $\mathfrak{X}^{b}(K):=\left\\{\alpha\in C^{b}(K):\alpha(e)=1,\;p(x,y)(\alpha)=\alpha(x)\alpha(y),\;\forall\;x,y\in K\right\\}$ equipped with the compact-open topology. $\mathcal{X}^{b}(K)$ is a locally compact Hausdorff space. Let $\widehat{K}$ denote the set of all hermitian characters $\alpha$ in $\mathcal{X}^{b}(K)$, i.e. $\alpha(\tilde{x})=\overline{\alpha(x)}$ for every $x\in K$, with a Plancherel measure $\pi$. In contrast to the case of groups, $\widehat{K}$ might not have the dual hypergroup structure and might properly contain $\mathcal{S}=\mbox{supp }{\pi}$. The Fourier-Stieltjes transform of $\mu\in M(K)$, $\widehat{\mu}\in C^{b}(\widehat{K})$, is given by $\widehat{\mu}(\alpha):=\int_{K}\overline{\alpha(x)}d\mu(x)$. Its restriction to $L^{1}(K)$ is called the Fourier transform. We have $\widehat{f}\in C_{0}(\widehat{K})$, for $f\in L^{1}(K)$, and $I(\alpha):=\\{f\in L^{1}(K):\widehat{f}(\alpha)=0\\}$ is the maximal ideal in $L^{1}(K)$ generated by $\alpha$ [5]. $K$ is called $\alpha$-amenable $(\alpha\in\widehat{K})$ if there exists $m_{\alpha}\in L^{\infty}(K)^{\ast}$ such that (i) $m_{\alpha}(\alpha)=1$ and (ii) $m_{\alpha}(T_{x}f)=\alpha(x)m_{\alpha}(f)$ for every $f\in L^{\infty}(K)$ and $x\in K$. $K$ is called amenable if the latter holds for $\alpha=1$. For the sake of completeness, we recall the modified Reiter’s condition of $P_{1}$-type in $\alpha\in\widehat{K}$ from [7] which is required in Theorem 1.2. By this condition we shall mean for every $\varepsilon>0$ and every compact subset $C$ of $K$ there exists $g\in L^{1}(K)$ with $\\|g\\|_{1}\leq M$ $(M>0)$ such that $\widehat{g}(\alpha)=1$ and $\\|T_{x}g-\alpha(x)g\\|_{1}<\varepsilon$ for all $x\in C.$ The condition is simply called Reiter’s condition if $\alpha=1$ [15]. ## 1 $\alpha$-Left Amenability of $L^{1}(K)$ Let $X$ be a Banach $L^{1}(K)$-bimodule and $\alpha\in\widehat{K}$. Then, in a canonical way, the dual space $X^{\ast}$ is a Banach $L^{1}(K)$-bimodule. The module $X$ is called a $\alpha$-left $L^{1}(K)$-module if the left module multiplication is given by $f\cdot x=\widehat{f}(\alpha)x$, for every $f\in L^{1}(K)$ and $x\in X$. In this case, $X^{\ast}$ turns out to be a $\alpha$-right $L^{1}(K)$-bimodule as well, i.e. $\varphi\cdot f=\widehat{f}(\alpha)\varphi$, for every $f\in L^{1}(K)$ and $\varphi\in X^{\ast}$. A continuous linear map $D:L^{1}(K)\rightarrow X^{\ast}$ is called a derivation if $D(f\ast g)=D(f)\cdot g+f\cdot D(g)$, for every $f,g\in L^{1}(K)$, and an inner derivation if $D(f)=f\cdot\varphi-\varphi\cdot f$, for some $\varphi\in X^{\ast}$. The algebra $L^{1}(K)$ is called $\alpha$-left amenable if for every $\alpha$-left $L^{1}(K)$-module $X$, every continuous derivation $D:L^{1}(K)\rightarrow X^{\ast}$ is inner; and, if the latter holds for every Banach $L^{1}(K)$-bimodule $X$, then $L^{1}(K)$ is called amenable. As shown in [7], $K$ is $\alpha$-amenable if and only if either $I(\alpha)$ has a b.a.i. or $K$ satisfies the modified Reiter’s condition of $P_{1}$-type in $\alpha$. In the following theorem we explore the connection between the $\alpha$-amenability of $K$ and $\alpha$-left amenability of $L^{1}(K)$. ###### Theorem 1.1. _Let $K$ be a hypergroup and $\alpha\in\widehat{K}$, real-valued. Then $K$ is $\alpha$-amenable if and only if $L^{1}(K)$ is $\alpha$-left amenable._ ###### Proof. Assume $K$ to be $\alpha$-amenable, choose $X$ to be an arbitrary $\alpha$-left $L^{1}(K)$-module, and suppose that $D:L^{1}(K)\rightarrow X^{\ast}$ is a continuous derivation. For fixed $x\in X$ define $\Phi_{x}\in L^{1}(K)^{\ast}$ by $\Phi_{x}(f)=D(f)(x)$ for $f\in L^{1}(K)$. Then for every $f,g\in L^{1}(K)$ $\displaystyle\Phi_{x}(f\ast g)$ $\displaystyle=D(f\ast g)(x)=(f\cdot D(g))(x)+(D(f)\cdot g)(x)$ $\displaystyle=D(g)(x\cdot f)+\widehat{g}(\alpha)D(f)(x)$ $\displaystyle=\Phi_{x\cdot f}(g)+\widehat{g}(\alpha)\Phi_{x}(f).$ (1) Moreover, $\Phi_{x+y}=\Phi_{x}+\Phi_{y}$, $\Phi_{\lambda\cdot x}=\lambda\Phi_{x}$, and $\\|\Phi_{x}\\|\leq\\|D_{\alpha}\\|\\|x\\|$ for $x,y\in X$ and $\lambda\in{\mathbb{C}}$. We identify $\Phi\in L^{1}(K)^{\ast}$ with $\eta\in L^{\infty}(K)$ by the relation $\Phi(g):=\int_{K}g(t)\eta(\tilde{t})dm(t)$ for $g\in L^{1}(K)$. Denote by $\eta_{x}$ and $\eta_{x\cdot f}$ the elements of $L^{\infty}(K)$ corresponding to $\Phi_{x}$ and $\Phi_{x\cdot f}$, respectively. Thus, $\eta_{x+y}=\eta_{x}+\eta_{y}$, $\eta_{\lambda x}=\lambda\eta_{x}$, and $\\|\eta_{x}\\|_{\infty}\leq\\|D_{\alpha}\\|\\|x\\|$ for $x,y\in X$ and $\lambda\in{\mathbb{C}}$. Now (1) can be rewritten as $\int_{K}f\ast g(t)\eta_{x}(\tilde{t})dm(t)=\int_{K}g(t)\eta_{x\cdot f}(\tilde{t})dm(t)+\widehat{g}(\alpha)\int_{K}f(t)\eta_{x}(\tilde{t})dm(t)$ (2) Applying Fubini’s theorem and [4, Thm.1.3.21] yield $\displaystyle\int_{K}f\ast g(t)\eta_{x}(\tilde{t})dm(t)$ $\displaystyle=\int_{K}\left(\int_{K}f(y)T_{\tilde{y}}g(t)dm(y)\right)\eta_{x}(\tilde{t})dm(t)$ $\displaystyle=\int_{K}\left(\int_{K}g(t)T_{y}\eta_{x}(\tilde{t})dm(t)\right)f(y)dm(y)$ $\displaystyle=\int_{K}\left(\int_{K}f(y)T_{y}\eta_{x}(\tilde{t})dm(y)\right)g(t)dm(t)$ and $\displaystyle\int_{K}g(t)\eta_{x\cdot f}(\tilde{t})dm(t)=$ $\displaystyle\int_{K}\left(\int_{K}f(y)T_{{y}}\eta_{x}(\tilde{t})dm(y)\right)g(t)dm(t)$ $\displaystyle-\int_{K}g(t)\left(\alpha(\tilde{t})\int_{K}f(y)\eta_{x}(\tilde{y})dm(y)\right)dm(t).$ (3) Now, by assumption there exists $m_{\alpha}\in L^{\infty}(K)^{\ast}$ such that $m_{\alpha}(\alpha)=1$ and $m_{\alpha}(T_{y}\eta)=\alpha(y)m_{\alpha}(\eta)$ for every $\eta\in L^{\infty}(K)$ and $y\in K$. Let $\varphi(x):=m_{\alpha}(\eta_{x}),\hskip 28.45274ptx\in X.$ Then $\varphi(x+y)=\varphi(x)+\varphi(y)$, $\varphi(\lambda x)=\lambda\varphi(x)$ and $\|\varphi(x)\|\leq\\|m_{\alpha}\\|\\|D_{\alpha}\\|\\|x\\|$. Hence $\varphi\in X^{\ast}$, and for $f\in L^{1}(K)$ and $x\in X$ it follows that $f\cdot\varphi(x)=\varphi(x\cdot f)=m_{\alpha}(\eta_{x\cdot f}).$ By Goldstein’s theorem [6], the functional $m_{\alpha}$ is the $w^{\ast}$-limit of a net of functions $g\in L^{1}(K)$, therefore from (3) we obtain that $m_{\alpha}(\eta_{x\cdot f})=\int_{K}f(y)m_{\alpha}(T_{y}\eta_{x})dm(y)-m_{\alpha}(\alpha)\Phi_{x}(f),$ and hence $\Phi_{x}(f)=\widehat{f}(\alpha)m_{\alpha}(\eta_{x})-m_{\alpha}(\eta_{x\cdot f})\hskip 14.22636pt\mbox{for }f\in L^{1}(K),x\in X.$ That means $D(f)(x)=\varphi\cdot f(x)-f\cdot\varphi(x)$, thence $D$ is an inner derivation, and this gives the $\alpha$-left amenability of $L^{1}(K)$. To prove the converse of the theorem we follow the method in [5, p.239]. Assume $L^{1}(K)$ to be $\alpha$-left amenable and consider $\alpha$-left $L^{1}(K)$-module $L^{\infty}(K)$ with the module multiplications $f\cdot\varphi=\widehat{f}(\alpha)\varphi$ and $\varphi\cdot f=f\ast\varphi$, for every $f\in L^{1}(K)$ and $\varphi\in L^{\infty}(K)$. Since $\alpha\cdot f=f\ast\alpha=\widehat{f}(\alpha)\alpha$, ${\mathbb{C}}\alpha$ is a closed $L^{1}(K)$-submodule of $L^{\infty}(K)$. Hence, $L^{\infty}(K)=X\oplus{\mathbb{C}}\alpha$ where $X$ is also a closed $L^{1}(K)$-submodule of $L^{\infty}(K).$ Choose $\nu\in L^{\infty}(K)^{\ast}$ such that $\nu(\alpha)=1$, and define $\delta:L^{1}(K)\rightarrow L^{\infty}(K)^{\ast},$ $\delta(f)=f\cdot\nu-\nu\cdot f$, for $f\in L^{1}(K).$ Then $\displaystyle\delta(f)(\alpha)$ $\displaystyle=f\cdot\nu(\alpha)-\nu\cdot f(\alpha)$ $\displaystyle=\nu(\alpha\cdot f)-\nu(f\cdot\alpha)$ $\displaystyle=\nu(f\ast\alpha)-\widehat{f}(\alpha)\nu(\alpha)$ $\displaystyle=\widehat{f}(\alpha)\nu(\alpha)-\widehat{f}(\alpha)\nu(\alpha)=0,\hskip 28.45274pt\mbox{ for every }f\in L^{1}(K).$ That means $\delta(f)\in\left({\mathbb{C}}\alpha\right)^{\bot}\subset L^{\infty}(K)^{\ast}$. Let $P:L^{\infty}(K)\rightarrow X$ denote the projection onto $X$ and $P^{\ast}:X^{\ast}\rightarrow L^{\infty}(K)^{\ast}$ the adjoint operator. $P^{\ast}$ is an injective $L^{1}(K)$-bimodule homomorphism; it follows that $({\mathbb{C}}\alpha)^{\bot}=(KerP)^{\bot}=(P^{\ast}(X^{\ast})_{\bot})^{\bot}=P^{\ast}(X^{\ast})$. Hence, for each $f\in L^{1}(K)$ there exists $D(f)\in X^{\ast}$ such that $P^{\ast}D(f)=\delta(f).$ Since $\delta$ is a continuous derivation on $L^{1}(K)$, the map $D:L^{1}(K)\rightarrow X^{\ast}$ is a continuous derivation as well. By assumption $D$ is inner, that is, there exists $\psi\in X^{\ast}$ such that $D(f)=f\cdot\psi-\psi\cdot f$ for all $f\in L^{1}(K)$. Define $m_{\alpha}:=\nu-P^{\ast}\psi$. Then $m_{\alpha}(\alpha)=\nu(\alpha)-P^{\ast}\psi(\alpha)=1-\psi(P\alpha)=1$ and $f\cdot(P^{\ast}\psi)(\varphi)=P^{\ast}\psi(\varphi\cdot f)=\psi(P(\varphi\cdot f))=f\cdot\psi(P\varphi)=P^{\ast}(f\cdot\psi)(\varphi)\hskip 8.5359pt(\varphi\in X).$ Similarly $(P^{\ast}\psi)\cdot f(\varphi)=P^{\ast}(\psi\cdot f)(\varphi)$, thus $\displaystyle f\cdot P^{\ast}\psi-P^{\ast}\psi\cdot f$ $\displaystyle=P^{\ast}(f\cdot\psi-\psi\cdot f)$ $\displaystyle=P^{\ast}Df=\delta(f)=f\cdot\nu-\nu\cdot f.$ Hence, $f\cdot m_{\alpha}=f\cdot\nu-f\cdot P^{\ast}\nu=\nu\cdot f-P^{\ast}\psi\cdot f=m_{\alpha}\cdot f$. This means $m_{\alpha}(\varphi\ast f)=m_{\alpha}(\varphi\cdot f)=f\cdot m_{\alpha}(\varphi)=m_{\alpha}\cdot f(\varphi)=m_{\alpha}(f\cdot\varphi)=\widehat{f}(\alpha)m_{\alpha}(\varphi),$ and hence $m_{\alpha}(T_{x}\varphi)=m_{\alpha}(\delta_{\tilde{x}}\ast\varphi)=\overline{\alpha(\tilde{x})}m_{\alpha}(\varphi)=\alpha(x)m_{\alpha}(\varphi),$ for every $\varphi\in L^{\infty}(K)$ and $x\in K$, giving the $\alpha$-amenability of $K$. ∎ The previous theorem combined with [7] yield Johnson-Reiter’s condition for hypergroups, in the $\alpha$-setting, which reads as follows. ###### Theorem 1.2. _Let $K$ be a hypergroup and $\alpha\in\widehat{K}$, real-valued. Then the following statements are equivalent:_ _(i)_ _$K$ is $\alpha$-amenable._ * _(ii)_ _$L^{1}(K)$ is $\alpha$-left amenable._ * _(iii)_ _$I(\alpha)$ has a b.a.i._ * _(iv)_ _$K$ satisfies the modified $P_{1}$-condition in $\alpha$._ ###### Corollary 1.2.1. _If $K$ is $\alpha$-amenable, then every functional $D:L^{1}(K)\rightarrow{\mathbb{C}}$ such that $D(f\ast g)=\widehat{f}(\alpha)D(g)+\widehat{g}(\alpha)D(f)$, $f,g\in L^{1}(K)$, is zero (see [2, 5.2]). The converse, however, is in general not true; see Example (II) or [2, 5.5]. The functional $D$ is called a $\alpha$-derivation._ ###### Remark 1.2.1. * _(i)_ _If $\alpha\in L^{1}(K)\cap L^{2}(K)$, then_ $m_{\alpha}(f):=\frac{1}{\\|\alpha\\|_{2}^{2}}\int_{K}f(x)\overline{\alpha(x)}dm(x),\hskip 28.45274ptf\in L^{\infty}(K),$ _is a $\alpha$-mean on $L^{\infty}(K)$. For example, if $K$ is a hypergroup of compact type [8], the functional $m_{\alpha}$ is an $\alpha$-mean on $L^{\infty}(K)$ for every $\alpha\in\widehat{K}\setminus{\\{1\\}}$; this holds also for $\alpha=1$ if $K$ is compact. We note that the $\alpha$-means $m_{\alpha}$, given as above, are unique [3]._ * _(ii)_ _Observe that $\widehat{K}$ might contain some positive characters $\alpha\not=1$ in which case $K$ is $\alpha$-amenable; see Example (VI)._ Our next topic is about the $\alpha$-amenability of hypergroup joins, and Theorem 1.3 generalizes [15, 3.12 ] to the $\alpha$-setting. For the sake of convenience, we first recall the definition of hypergroup joins and some known facts about their dual spaces. Let $(H,\ast)$ be a compact hypergroup with a normalized Haar measure $m_{H}$, $(J,\cdot)$ a discrete hypergroup with a Haar measure $m_{J}$, and suppose that $H\cap J=\\{e\\}$, where $e$ is the identity of both hypergroups. The hypergroup joins $(H\vee J,\odot)$ is the set $H\cup J$ with the unique topology for which $H$ and $J$ are closed subspace of, where the convolution $\odot$ is defined as follows: 1. 1. $\varepsilon_{x}\odot\varepsilon_{y}$ agrees with that on $H$ if $x,y\in H$, 2. 2. $\varepsilon_{x}\odot\varepsilon_{y}=\varepsilon_{x}\cdot\varepsilon_{y}$ if $x,y\in J$, $x\not=\tilde{y}$, 3. 3. $\varepsilon_{x}\odot\varepsilon_{y}=\varepsilon_{y}=\varepsilon_{y}\odot\varepsilon_{x}$ if $x\in H$, $y\in J\setminus{\\{e\\}}$, and 4. 4. if $y\in J$ and $y\not=e$, $\varepsilon_{\tilde{y}}\odot\varepsilon_{y}=c_{e}m_{H}+\sum_{w\in J\setminus{\\{e\\}}}c_{w}\varepsilon_{w}$ where $\varepsilon_{\tilde{y}}\cdot\varepsilon_{y}=\sum_{w\in J}c_{w}\varepsilon_{w}$, $c_{w}\geq 0$, only finitely many $c_{w}$ are nonzero, and $\sum_{w\in J}c_{w}\varepsilon_{w}=1$. If $m_{J}(\\{e\\})=1$, then $m_{K}:=m_{H}+1_{J\setminus{\\{e\\}}}m_{J}$ is a Haar measure for $K$. Observe that $K//H\cong J$ and $H$ is a subhypergroup of $K=H\vee J$ but that $J$ is not unless either $H$ or $J$ is trivial [22]. As proved in [4, p. 119], $\widehat{K}=\widehat{H}\cup\widehat{J}$, where $\widehat{H}\cap\widehat{J}=\\{1\\}$. The latter holds in the sense of hypergroup isomorphism, $\widehat{K}\cong\widehat{H}\vee\widehat{J}$, if $H$ and $J$ are strong hypergroups. In this case $K$ is a strong hypergroup as well. ###### Theorem 1.3. _Let $K$ be as above, $\|J\|\geq 2$, and $\alpha\in\widehat{J}$. Then $J$ is $\alpha$-amenable if and only if $K$ is $\alpha$-amenable. Moreover, if $H$ and $J$ are strong hypergroups, then $\widehat{H}$ is $\beta$-amenable if and only if $\widehat{K}$ is $\beta$-amenable ($\beta\in\widehat{\widehat{H}}$). _ ###### Proof. Let $x\in J^{\ast}:=J\setminus{\\{e\\}}$. By [15, 3.15] for $f\in L^{\infty}(K)$, we have $T_{x}f=T_{x}(f\|_{J^{\ast}})+T_{x}(1_{H})\int_{H}f(t)dm_{H}(t).$ (4) Now, take $\alpha\in\widehat{J}$ and assume $J$ to be $\alpha$-amenable. Then there exists $m_{\alpha}:\ell^{\infty}(J)\rightarrow{\mathbb{C}}$ such that $m_{\alpha}(\alpha)=1$ and $m_{\alpha}(T_{x}f)=\alpha(x)m_{\alpha}(f)$, for all $f\in\ell^{\infty}(J)$ and $x\in J$. The character $\alpha$ can be extended to $K$ by letting $\gamma(x):=1$ for all $x\in H$. Define $M_{\gamma}:L^{\infty}(K)\rightarrow{\mathbb{C}},\hskip 14.22636ptM_{\gamma}(f):=m_{\alpha}(f\|_{J^{\ast}}),\hskip 14.22636ptf\in L^{\infty}(K).$ We have $M_{\gamma}(\gamma)=m_{\alpha}(\gamma\|_{J^{\ast}})=m_{\alpha}(\alpha)=1$, and (4) implies that $\displaystyle M_{\gamma}(T_{x}f)$ $\displaystyle=M_{\gamma}(T_{x}(f\|_{J^{\ast}}))+M_{\gamma}(T_{x}(1_{H}))\int_{H}f(t)dm_{H}(t)$ $\displaystyle=m_{\alpha}(T_{x}(f\|_{J^{\ast}}))=\alpha(x)m_{\alpha}(f\|_{J^{\ast}})=\gamma(x)M_{\gamma}(f),\hskip 11.38092pt\mbox{ for all }f\in L^{\infty}(K),x\in J^{\ast}.$ (5) Since $\gamma\|_{H}=1$ and $(T_{x}f)\|_{J^{\ast}}=f\|_{J^{\ast}}$ for $x\in H$, the equality (5) is valid for all $x\in K$. Therefore, $K$ is $\gamma$-amenable. To prove the converse, let $\gamma\in\widehat{K}$ and assume $K$ to be $\gamma$-amenable. If $\gamma\|_{H}=1$, then by $\widehat{K}=\widehat{H}\cup\widehat{J}$ we have $\gamma\in\widehat{J}$. Define $m_{\gamma}:\ell^{\infty}(J)\rightarrow{\mathbb{C}},\hskip 14.22636ptm_{\gamma}(f):=M_{\gamma}(f),\hskip 14.22636ptf\in L^{\infty}(K),$ where $M_{\gamma}$ is a $\gamma$-mean on $L^{\infty}(K)$. Obviously $m_{\gamma}(\gamma)=1$, and $m_{\gamma}(T_{x}f)=M_{\gamma}(T_{x}f)=M_{\gamma}(T_{x}(f\|_{J^{\ast}}))=\gamma(x)M_{\gamma}(f)=\gamma(x)m_{\gamma}(f),$ for all $f\in\ell^{\infty}(J)$ and $x\in J^{\ast}.$ If $\gamma(x)\not=1$ for some $x\in H$, then $\gamma$ is a nontrivial character of $H$ and $\widehat{K}=\widehat{H}\cup\widehat{J}$ implies that $\gamma\|_{J}=1$. Since $J$ is commutative, $J$ must be amenable [15], and the amenability of $H$ in $\gamma$ follows from Remark 1.2.1. The proof of the second part can be obtained from the first part and the fact that $\widehat{K}\cong\widehat{H}\vee\widehat{J}$ if $H$ and $J$ are strong hypergroups. ∎ ## 2 $\alpha$-Amenability of polynomial hypergroups In this section we restrict our discussion to the polynomial hypergroups. First we consider polynomial hypergroups in several variables which have been already studied by several authors (e.g. see [12, 24]). The translation operators of these hypergroups seem to be complicated, and the study of their character amenability via the modified Reiter’s condition, in contrast to the one variable case [7], may require sophisticated calculations. In Theorem 2.1, however, we provide a necessary condition to the $\alpha$-amenability of these hypergroups. Hence we point out that the majority of common examples of polynomial hypergroups do not satisfy this condition. Let $\\{P_{\bf{n}}\\}_{{\bf{n}}\in\mathcal{K}}$ be a set of orthogonal polynomials on ${\mathbb{C}}^{d}$ with respect to a measure $\pi\in M^{1}({\mathbb{C}}^{d})$ such that $P_{\bf{n}}(u)=1$ for some $u\in{\mathbb{C}}^{d}$, where $\mathcal{K}:={\mathbb{N}}_{0}^{m}$ with the discrete topology, $m,d\in{\mathbb{N}}$, and ${\mathbb{N}}_{0}:={\mathbb{N}}\cup\\{0\\}$. Assume $\mathcal{P}_{n}$ denotes the set of all polynomials $P^{\prime}\in{\mathbb{C}}[z_{1},z_{2},...z_{d}]$ with degree less or equal than $n$ and $\mathcal{K}_{n}:=\\{{\bf{n}}\in\mathcal{K}:P_{\bf{n}}\in\mathcal{P}_{n}\\}$. Suppose that for every $n\in{\mathbb{N}}$ the set $\\{P_{\bf{n}}:{\bf{n}}\in\mathcal{K}_{n}\\}$ is a basis of $\mathcal{P}_{n}$, and for every ${\bf{n}},{\bf{m}}\in\mathcal{K}$ the product $P_{\bf{n}}\cdot P_{\bf{m}}$ admits the unique non-negative linearization formula, i.e. $P_{\bf{n}}\cdot P_{\bf{m}}:=\sum_{{\bf{t}}\in\mathcal{K}}g({\bf{n}},{\bf{m}},{\bf{t}})P_{\bf{t}}$ (6) where $g({\bf{n}},{\bf{m}},{\bf{t}})\geq 0$. Assume further that there exists a homeomorphism ${\bf{n}}\rightarrow\tilde{\bf{n}}$ on $\mathcal{K}$ such that $P_{\tilde{\bf{n}}}=\overline{P_{\bf{n}}}$ for every ${\bf{n}}\in\mathcal{K}$. In this case $\mathcal{K}$ with the convolution of two point measures defined by $\varepsilon_{\bf{n}}\ast\varepsilon_{\bf{m}}(\varepsilon_{\bf{t}}):=p({\bf{n}},{\bf{m}})(\varepsilon_{\bf{t}}):=g({\bf{n}},{\bf{m}},{\bf{t}})$ is a hypergroup which is called a polynomial hypergroup in $d$ variables. The hypergroup $\mathcal{K}$ is obviously commutative and the identity element $e$ is the constant polynomial $P_{0}\equiv 1.$ The character space $\widehat{\mathcal{K}}$ can be identified with the set $\\{{\bf{x}}\in{\mathbb{C}}^{d}:\|\alpha_{\bf{x}}({\bf{n}})\|\leq 1,\alpha_{\bf{x}}(\tilde{\bf{n}})=\overline{\alpha_{\bf{x}}({\bf{n}})}\hskip 5.69046pt\forall{\bf{n}}\in\mathcal{K}\\}$, where $\alpha_{\bf{x}}({\bf{n}}):=P_{{\bf{n}}}({\bf{x}})$ for ${\bf{x}}\in{\mathbb{C}}^{d}$ and ${\bf{n}}\in\mathcal{K}$. For more on polynomial hypergroups in several variables we refer the reader to e.g. [4, 12, 24]. ###### Theorem 2.1. _Let $\\{P_{\bf{n}}({\bf{x}})\\}_{{\bf{n}}\in\mathcal{K}}$ define a polynomial hypergroup in $d$ variables on $\mathcal{K}:={\mathbb{N}}_{0}^{m}$ and $\alpha_{\bf{x}}\in\widehat{\mathcal{K}}$ with $\pi(\\{\alpha_{\bf{x}}\\})=0$. If $\alpha_{\bf{x}}\in C_{0}(\mathcal{K})$, then $\mathcal{K}$ is not $\alpha_{\bf{x}}$-amenable. _ ###### Proof. Assume to the contrary that $\mathcal{K}$ is $\alpha_{\bf{x}}$-amenable and $m_{\alpha_{\bf{x}}}$ is a $\alpha_{\bf{x}}$-mean on $\ell^{\infty}(\mathcal{K})$. Due to $T_{\bf{n}}\varepsilon_{\bf{0}}({\bf{m}})=\sum_{\bf{t}\in\mathcal{K}}\varepsilon_{\bf{0}}({\bf{t}})p({\bf{n}},{\bf{m}})({\bf{t}})=p({\bf{n}},{\bf{m}})({\bf{0}})\varepsilon_{\tilde{{\bf{n}}}}({\bf{m}})=\frac{1}{h(\bf{n})}\varepsilon_{\tilde{{\bf{n}}}}(\bf{m}),$ we have $T_{\bf{n}}\varepsilon_{\bf{0}}=\frac{1}{h({\bf{n}})}\varepsilon_{\tilde{{\bf{n}}}}$ for every ${\bf{n}}\in\mathcal{K}$. Therefore, $m_{\alpha_{\bf{x}}}\left(\varepsilon_{\tilde{{\bf{n}}}}\right)=h({\bf{n}})m_{\alpha_{\bf{x}}}(T_{\bf{n}}\varepsilon_{\bf{0}})=h({\bf{n}})\alpha_{\bf{x}}({\bf{n}})m_{\alpha_{\bf{x}}}(\varepsilon_{\bf{0}}).$ (7) Let $M>0$ be a bound for $m_{\alpha_{\bf{x}}}$ and $\xi_{\bf{n}}=\frac{\overline{P_{\bf{n}}({\bf{x}})}}{\|P_{\bf{n}}({\bf{x}})\|}$ for $P_{\bf{n}}({\bf{x}})\not=0$. Then by the linearity of $m_{\alpha_{\bf{x}}}$ and (7) we have $\displaystyle M\geq\|m_{\alpha_{\bf{x}}}(\sum_{\bf{n}\in\mathcal{M}}\xi_{\bf{n}}\varepsilon_{\tilde{\bf{n}}})\|$ $\displaystyle=\|\sum_{\bf{n}\in\mathcal{M}}\xi_{\bf{n}}m_{\alpha_{\bf{x}}}(\varepsilon_{\tilde{\bf{n}}})\|=\|\sum_{\bf{n}\in\mathcal{M}}\|P_{\bf{n}}({\bf{x}})\|h({\bf{n}})m_{\alpha_{\bf{x}}}(\varepsilon_{\bf{0}})\|$ $\displaystyle\geq\sum_{\bf{n}\in\mathcal{M}}\|P_{\bf{n}}({\bf{x}})\|^{2}h({\bf{n}})\|m_{\alpha_{\bf{x}}}(\varepsilon_{\bf{0}})\|,$ where $\mathcal{M}$ is an arbitrary finite subset of $\mathcal{K}$. If $m_{\alpha_{\bf{x}}}(\varepsilon_{\bf{0}})\not=0$, then the provious inequalities show that $\alpha_{\bf{x}}\in\ell^{1}(\mathcal{K})\cap\ell^{2}(\mathcal{K})$, hence $\pi(\alpha_{\bf{x}})>0$ (see [4, Proposition 2.5.1]) which is a constradiction. If we now define $\\{\alpha_{\bf{x}}^{\bf{m}}\\}_{{\bf{m}}\in\mathcal{K}}$ by $\alpha_{\bf{x}}^{\bf{m}}(\bf{n}):=\left\\{\begin{array}[]{l @{\quad{\mbox{ }}\quad} l}0&n_{i}<m_{i}\hskip 5.69046pt(1\leq i\leq d),\\\ \alpha_{\bf{x}}({\bf{n}})&\mbox{other,}\\\ \end{array}\right.$ (8) then $\alpha_{\bf{x}}({\bf{n}})=(P_{\bf{n}}({\bf{x}}))_{{\bf{n}}\in\mathcal{K}}$ can be written as follows $\alpha_{\bf{x}}=\sum_{0\leq t_{i}\leq m_{i}}\varepsilon_{\bf{t}}P_{{\bf{t}}}({\bf{x}})+\alpha_{{\bf{x}}}^{\bf{m}}.$ Hence, $m_{\alpha_{\bf{x}}}(\alpha_{\bf{x}})=\sum_{0\leq t_{i}\leq m_{i}}m_{\alpha_{\bf{x}}}(\varepsilon_{\bf{t}})P_{{\bf{t}}}({\bf{x}})+m_{\alpha_{\bf{x}}}(\alpha_{\bf{x}}^{\bf{m}})$ which implies that $\left\|m_{\alpha_{\bf{x}}}(\alpha_{\bf{x}})\right\|=\left\|m_{\alpha_{\bf{x}}}(\alpha_{\bf{x}}^{\bf{m}})\right\|\leq M\\|\alpha_{\bf{x}}^{\bf{m}}\\|.$ The latter shows that if $\alpha_{\bf{x}}\in C_{0}(\mathcal{K})$, then $\alpha_{\bf{x}}^{\bf{m}}\in C_{0}(\mathcal{K})$ for all $m\in\mathcal{K}$, hence $m_{\alpha_{\bf{x}}}(\alpha_{\bf{x}})=0$ which is a contradiction . ∎ ###### Remark 2.1.1. __ 1. 1. Observe that in the preceding theorem neither of the assumptions $\pi(\\{\alpha_{\bf{x}}\\})=0$ nor $\alpha_{\bf{x}}\in C_{0}(\mathcal{K})$ can be omitted. For example, a hypergroup of compact type is $\alpha$-amenable in every character $\alpha$ while $1$ is the only character in $\widehat{K}$ with the vanishing Plancherel measure [8, 15]; see also Example (VI). 2. 2. Theorem 2.1 is known for $m=d=1$ in [7]. We continue the section by examining the $\alpha$-amenability of various polynomial hypergroups. Let us first start with polynomial hypergroups in two variables which have been extensively studied by T. H. Koornwinder in [12]. 1. (I) Koornwinder Class $V$ hypergroups: In this case $\mathcal{K}:=\\{(n,k)\in{\mathbb{N}}_{0}^{2}:n\geq k\\}$ and the characters are given by $P_{\bf{n}}(x,y):=P_{(n,k)}^{\alpha,\beta,\gamma,\eta}(x,y):=P_{n-k}^{(\alpha,\beta)}(x)P_{k}^{(\gamma,\eta)}(y),\hskip 14.22636pt{\bf{n}}=(n,k),$ where $P_{n}^{(\alpha,\beta)}$ denote the Jacobi polynomials, $(\alpha,\beta)$, $(\gamma,\eta)\in V$, $P_{\bf{n}}^{\alpha,\beta,\gamma,\eta}(1,1)=1$, and $\displaystyle V:=\\{(\alpha,\beta)\in{\mathbb{R}}^{2}:$ $\displaystyle\;\alpha\geq\beta>-1,(\alpha+\beta+1)(\alpha+\beta+4)^{2}(\alpha+\beta+6)$ $\displaystyle\geq(\alpha-\beta)^{2}\cdot(\alpha^{2}-2\alpha\beta+\beta^{2}-5\alpha-5\beta-30)\\}.$ The support of the Plancherel measure $d\pi(x,y)=(1-x)^{\alpha}(1+x)^{\beta}(1-y)^{\gamma}(1+y)^{\eta}dxdy$ is $D:=\\{(x,y)\|-1\leq x\leq 1,-1\leq y\leq 1\\}.$ Since $\|P_{n}^{(\alpha,\beta)}(y)\|=\mathcal{O}(n^{-\alpha-\frac{1}{2}})$ as $n\rightarrow\infty$ [9], we have $\|P_{(n,n)}^{\alpha,\beta,\gamma,\eta}(x,y)\|=\|P_{n}^{(\gamma,\eta)}(y)\|\rightarrow 0\hskip 28.45274pt(n\rightarrow\infty)$ when $(x,y)\in[-1,1]\times(-1,1)$ and $\alpha$, $\eta>-\frac{1}{2}$. So, from Theorem 2.1 it follows that $\mathcal{K}$ is not $\alpha_{(x,y)}$-amenable. For $(x,y)\in\\{(-1,1),(1,-1)(-1,-1)\\}$, if $\alpha>\beta$ and $\gamma>\eta$, $\alpha=\beta$ and $\gamma>\eta$, or $\alpha>\beta$ and $\gamma=\eta$ since $P_{n}^{(\alpha,\beta)}(-1)=(-1)^{n}\left(\begin{array}[]{c}n+\beta\\\ n\\\ \end{array}\right){\big{/}}\left(\begin{array}[]{c}n+\alpha\\\ n\\\ \end{array}\right),$ we have $\|P_{(2n,n)}^{\alpha,\beta,\gamma,\eta}(x,y)\|\rightarrow 0$ as $n\rightarrow\infty,$ hence $\mathcal{K}$ is not $\alpha_{(x,y)}$-amenable. The hypergroup $\mathcal{K}$ is, in fact, the product of two Jacobi polynomial hypergroups with parameters $(\alpha,\beta)$ and $(\gamma,\eta)$ on ${\mathbb{N}}_{0}$ [24]. Theorem 2.1 combined with [23] implies that $\ell^{1}({\mathbb{N}}_{0})$ is amenable if and only if $\alpha=\beta=\gamma=\eta=-\frac{1}{2}$. Thus, since $\ell^{1}(\mathcal{K})\cong\ell^{1}({\mathbb{N}}_{0})\otimes_{p}\ell^{1}({\mathbb{N}}_{0})$, the algebra $\ell^{1}(\mathcal{K})$ is amenable and its maximal ideals have b.a.i.; see [5, 11]. Consequently, Theorem 1.2 results in the $\alpha_{(x,y)}$-amenability of $\mathcal{K}$ for $(x,y)\in D$ and $x,y=\pm 1$. ###### Remark 2.1.2. __ 1. (i) Let $(x,y_{0})\in[-1,1]\times[-1,1]$ be as above fixed. For $\gamma>\frac{1}{2}$ one can show that the usual derivation of the Fourier transform gives a rise to a nonzero bounded $\alpha_{(x,y_{0})}$-derivation on $\ell^{1}(\mathcal{K})$. So, it follows from Remark 2.1.1 that $\mathcal{K}$ is not $\alpha_{(x,y_{0})}$-amenable and $\\{\alpha_{(x,y_{0})}\\}$ is not a spectral set. 2. (ii) Similar to the previous case, one can show that hypergroups of Koornwinder class III, VI, and some related hypergroups in two variables which are mentioned in [4, 3.1.16-20] are not $\alpha_{\bf{x}}$-amenable if $\alpha_{\bf{x}}\not=1$. 2. (II) Disc Polynomial Hypergroups: For $\alpha^{\prime}\geq 0$ the disc polynomials $P_{m,n}^{\alpha^{\prime}}(z,\bar{z})=\left\\{\begin{array}[]{ll}P_{n}^{(\alpha^{\prime},m-n)}(2z\bar{z}-1)z^{m-n},&\hbox{for $m\geq n$,}\\\ P_{m}^{(\alpha^{\prime},n-m)}(2z\bar{z}-1)z^{n-m},&\hbox{for $n\geq m$,}\end{array}\right.$ induce a hypergroup structure on $\mathcal{K}:={\mathbb{N}}_{0}^{2}$. The support of the Plancherel measure with the density $(z_{1},z_{2})\rightarrow c_{\alpha^{\prime}}(1-\|z_{1}\|^{2})^{\alpha^{\prime}}$ is $\mathcal{D}:=\\{(z_{1},z_{2})\in{\mathbb{C}}^{2}:\;z_{2}=\bar{z}_{1},\;\|z_{1}\|<1\\}$. From Theorem 2.1 and $\displaystyle P_{n,n}^{\alpha^{\prime}}(z,\bar{z})$ $\displaystyle=P_{n}^{(\alpha^{\prime},0)}(2z\bar{z}-1)=P_{n}^{(\alpha^{\prime},0)}(2\|z\|^{2}-1)$ $\displaystyle=\mathcal{O}(n^{-\alpha^{\prime}-1/2})\hskip 28.45274pt(z\in\mathcal{D}),$ as $n\rightarrow\infty$, we infer that $\mathcal{K}$ is $\alpha_{z}$-amenable if and only if $\alpha_{z}=1$. Observe that $\mathcal{H}:=\\{(n,n):\;n\in{\mathbb{N}}_{0}\\}$ is a supernormal subhypergroup of $\mathcal{K}$ which is isomorphic to the Jacobi hypergroup with the character set $\\{P_{n,n}^{\alpha^{\prime}}(x)\\}_{n\in{\mathbb{N}}_{0}}$. In this case we see also that $\mathcal{H}$ is $\alpha_{x}$-amenable if and only if $\alpha_{x}=1$ despite the fact that for every $x\in(-1,1)$ the singleton $\\{\alpha_{x}\\}$ is a spectral for $\mathcal{H}$ if $\alpha^{\prime}<\frac{1}{2}$; see [23]. In other words, if $\alpha^{\prime}<\frac{1}{2}$ then every bounded $\alpha_{x}$-derivation on $\ell^{1}(\mathcal{H})$ is zero, however $\mathcal{H}$ is only 1-amenable. In the rest of the section we deal with the polynomial hypergroups in one variable, i.e. the system $\\{P_{\bf{n}}\\}_{{\bf{n}}\in\mathcal{K}}$ consists of polynomials of one variable and the index set $\mathcal{K}$ is ${\mathbb{N}}_{0}$. The linearization formula in (6) can be expressed in the three term recursion formula $P_{1}(x)P_{n}(x)=a_{n}P_{n+1}(x)+b_{n}P_{n}(x)+c_{n}P_{n-1}(x),$ (9) for $n\in{\mathbb{N}}$ and $P_{0}(x)=1$, and we take $P_{n}(1)=1$, $P_{1}(x)=\frac{1}{a_{0}}(x-b_{0})$ with $a_{n}>0$, $b_{n}\in{\mathbb{R}}$, and $c_{n+1}>0$ for all $n\in{\mathbb{N}}_{0}$. The existence of the orthogonality measure is due to Favard’s theorem [9] and applying it to the relation (9) results in $a_{n}+b_{n}+c_{n}=1$ and $a_{0}+b_{0}=1$. The identity map defines an involution to these hypergroups and their Haar weights are given by $h(0)=1$ and $h(n)=\left(\int_{\mathbb{R}}P_{n}^{2}(x)d\pi(x)\right)^{-\frac{1}{2}}$ $(n\geq 1)$ [4, Theorem.1.3.26]. We consider the $\alpha$-amenability of following polynomial hypergroups. 1. (III) Associated Legendre hypergroups: For $\nu\in{\mathbb{R}}_{0}$, let $\gamma_{n}:=\frac{(\nu+1)_{n}}{2^{n}(\nu+\frac{1}{2})_{n}}\left(1+\sum_{k=1}^{n}\frac{\nu}{k+\nu}\right)$, $a_{n}:=\frac{\gamma_{n+1}}{\gamma_{n}}$, $b_{n}:=0$, and $c_{n}:=1-a_{n}$ if $n\geq 1$ and $\gamma_{0}=1$. The polynomial $P_{n}$ associated to the sequences $(a_{n})_{n\geq 1},(b_{n})_{n\geq 1},(c_{n})_{n\geq 1}$ in the recursion formula (9) is the $n$-th associated Legendre polynomial with parameter $\nu$. The Haar weights of the induced hypergroup on ${\mathbb{N}}_{0}$ are given by $h(0)=1$ and $h(n)=\frac{2\nu+2n+1}{2\nu+1}\left(1+\sum_{k=1}^{n}\frac{\nu}{(k+\nu)^{2}}\right)^{2}$, $n\geq 1$, and the support of the Plancherel measure can be identified with $[-1,1]$; see [4]. If $x\in(-1,1)$, $\pi(\\{\alpha_{x}\\})=0$ and $\alpha_{x}\in C_{0}({\mathbb{N}}_{0})$, so it follows from Theorem 2.1 that ${\mathbb{N}}_{0}$ is $\alpha_{x}$-amenable if and only if $\alpha_{x}=1$. 2. (IV) Pollaczek polynomials hypergroup: The Pollaczek polynomials $\\{P^{(\eta,\mu)}_{n}\\}_{n\in{\mathbb{N}}_{0}}$ depending on the parameters $\eta\geq 0$, $\mu>0$ or $-\frac{1}{2}<\eta<0$ and $0\leq\mu<\eta+\frac{1}{2}$ induce a hypergroup structure on ${\mathbb{N}}_{0}$ [13]. The Haar weights are given by $h(0)=1$ and $h(n)=\frac{(2n+2\eta+2\mu+1)(2\eta+1)_{n}}{(2\eta+2\nu+1)n!}\left(\sum_{k=0}^{n}\left(\begin{array}[]{c}n\\\ k\\\ \end{array}\right)\frac{(2\mu)^{k}}{(2\eta+1)_{k}}\right)^{2},$ and the Plancherel measure with the support $\mathcal{S}\cong[-1,1]$ is given by $d\pi(x)=A(x)dx$ where $A(\cos t)=(\sin t)^{2\eta}\|\Gamma(\eta+\frac{1}{2}+i\mu\cot(t))\|^{2}\exp((2t-\pi)\mu\cot(t))$, $0\leq t\leq\pi.$ Given $x\in(-1,1)$, since $\pi(\\{\alpha_{x}\\})=0$ and $\alpha_{x}\in C_{0}({\mathbb{N}}_{0})$, by Theorem 2.1 we see that ${\mathbb{N}}_{0}$ is $\alpha_{x}$-amenable if and only if $\alpha_{x}=1$. 3. (V) Generalized Soradi hypergroups: These are polynomial hypergroups of type [V] on ${\mathbb{N}}_{0}$ [4] with the characters $\displaystyle P_{n}(\cos\theta)=\frac{\sin(n+1)\theta-k\sin n\theta}{(nk+n+1)\sin\theta}\hskip 14.22636pt(n\geq 1),$ and the density of the Plancherel measure on the dual space $\widehat{\mathbb{N}}_{0}\cong[-1,1]$ is given by $p(x):=\frac{2(1-x^{2})^{1/2}}{\pi(1+k^{2}-2kx)}(k>1).$ For $x\in[-1,1)$, since $\pi(\\{\alpha_{x}\\})=0$ and $\alpha_{x}\in C_{0}({\mathbb{N}}_{0})$, Theorem 2.1 implies that ${\mathbb{N}}_{0}$ is $\alpha_{x}$-amenable if and only if $\alpha_{x}=1$. 4. (VI) Hypergroups associated with infinite distance-transitive graphs: They are polynomial hypergroups on ${\mathbb{N}}_{0}$ depending on $a,b\in{\mathbb{R}}$ with $a,b\geq 2$; and, one can associate them with infinite distance- transitive graphs if $a,b$ are integers. These hypergroups have been thoroughly studied by M. Voit [20]. For $b>a\geq 2$ (see below) they provide a rare and interesting case of $\alpha$-amenability of hypergroups. Their Haar weights and characters are given by $\displaystyle h^{(a,b)}(0):=1,\;h^{(a,b)}(n)=a(a-1)^{n-1}(b-1)^{n}\quad(n\geq 1),$ and $\displaystyle P_{n}^{(a,b)}(x)=\frac{a-1}{a\left((a-1)(b-1)\right)^{n/2}}\left(U_{n}(x)+\frac{b-2}{\left((a-1)(b-1)\right)^{1/2}}U_{n-1}(x)-\frac{1}{a-1}U_{n-2}(x)\right),$ respectively, where $U_{n}(\cos t)=\frac{\sin(n+1)t}{\sin t}$ are the Tchebychev polynomials of the second kind and $u_{-1}=u_{-2}:=0$. The dual space $\widehat{{\mathbb{N}}_{0}^{(a,b)}}$ can be identified with $[-s_{1},s_{1}]$, where $s_{1}:=\frac{ab-a-b+1}{2\sqrt{(a-1)(b-1)}}$. The normalized orthogonality measure $\pi\in M^{1}({\mathbb{R}})$ is $\displaystyle d\pi(x)=A(x)dx\|_{[-1,1]}\quad\quad\hskip 39.83368pt\text{for}\;a\geq b\geq 2,$ and $\displaystyle d\pi(x)=A(x)dx\|_{[-1,1]}+\frac{b-a}{b}ds_{0}\quad\text{for}\;b>a\geq 2$ with $A(x):=\frac{a}{2\pi}\frac{(1-x^{2})^{1/2}}{(s_{1}-x)(x-s_{0})},\;s_{0}=\frac{2-a-b}{2\sqrt{(a-1)(b-1)}}$. Note that $P_{n}^{(a,b)}(s_{1})=1\hskip 14.22636pt\mbox{ and }\hskip 14.22636ptP_{n}^{(a,b)}(s_{0})=(1-b)^{-n}\hskip 14.22636pt\mbox{ for }n\geq 0.$ ###### Proposition 2.1. _Let ${\mathbb{N}}_{0}^{(a,b)}$ denote the above hypergroup. Then _ * _(i)_ _for $a\geq b\geq 2$, ${\mathbb{N}}_{0}^{(a,b)}$is $\alpha_{x}$-amenable if and only if $x=s_{1}$. _ * _(ii)_ _for $b>a\geq 2$, ${\mathbb{N}}_{0}^{(a,b)}$is $\alpha_{x}$-amenable if and only if $x=s_{1}$ or $x=s_{0}$. _ ###### Proof. (i) If $x\in(-s_{1},s_{1})$, then $\pi(\\{\alpha_{x}\\})=0$ and $\alpha_{x}\in C_{0}({\mathbb{N}}_{0}^{(a,b)})$. So, applying Theorem 2.1 yields that ${\mathbb{N}}_{0}^{(a,b)}$ is $\alpha_{x}$\- amenable if and only if $x=s_{0}$, as $\alpha_{s_{0}}=1$. (ii) As in part (i), we can show that if $x\not=s_{0}$, then ${\mathbb{N}}_{0}^{(a,b)}$ is $\alpha_{x}$\- amenable if and only if $x=s_{1}$. In the case of $x=s_{0}$, obviously $\alpha_{s_{0}}\in\ell^{1}({\mathbb{N}}_{0}^{(a,b)})$ (see also [20, Remark 1.1]) which implies, by Remark 1.2.1 (ii), that ${\mathbb{N}}_{0}^{(a,b)}$ is $\alpha_{s_{0}}$-amenable. ∎ ###### Remark 2.1.3. _Notice that in the previous example $\widehat{K}$ contains two positive characters $\alpha_{s_{0}}$ and $\alpha_{s_{1}}$ with diverse behaviours. Indeed, Part (i) shows that ${\mathbb{N}}_{0}^{(a,b)}$ is $\alpha_{s_{1}}$-amenable but not $\alpha_{s_{0}}$-amenable if $a\geq b\geq 2$, whereas Part (ii) shows that ${\mathbb{N}}_{0}^{(a,b)}$ is $\alpha_{s_{1}}$ and $\alpha_{s_{0}}$-amenable for $b>a\geq 2$. In latter case, the functional $m_{\alpha_{s_{0}}}$, given in Remark 1.2.1 (ii), is a unique $\alpha_{s_{0}}$-mean on $\ell^{\infty}({\mathbb{N}}_{0}^{(a,b)})$ while the cardinality of $\alpha_{s_{1}}$-means on $\ell^{\infty}({\mathbb{N}}_{0}^{(a,b)})$ is infinity; see [3, 15]. _ ## References * [1] A. Azimifard, _ $\alpha$-Amenability of Banach algebras on commutative hypergroups_. Dissertation, Technische Universit t M nchen, 2006. * [2] A. Azimifard, On the $\alpha$-Amenability of Hypergroups. Montash. Math. (to appear 2008) * [3] A. Azimifard, Hypergroups with the unique $\alpha$-means. C. R. Math. Acad. Sci. Soc. R. Can. (to appear 2008) * [4] W. R. Bloom and H. Heyer, _Harmonic Analysis of Probability Measures on Hypergroups._ De Gruyter, 1994. * [5] F. F. Bonsall and J. Duncan, _Complete Normed Algebras._ Springer-Verlag, New York-Heidelberg, 1973. * [6] N. Dunford and J. T. Schwartz, _Linear Operators I._ Wiley & Sons, 1988. * [7] F. Filbir, R. Lasser and R. Szwarc, Reiter’s condition $P_{1}$ and approximate identities for hypergroups. _Monat. Math._ 143 (2004) 189–203. * [8] F. Filbir, R. Lasser and R. Szwarc, Hypergroups of compact type. _J. Comp. and App. Math._ 178 (1) (2005) 205–214. * [9] M. Ismail, _Classical and quantum orthogonal polynomials in one variables._ Cambridge University Press, 2005. * [10] R. I. Jewett, Spaces with an abstract convolution of measures. _Adv. in Math._ 18 (1975) 1–101. * [11] B. E. Johnson, Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 (1972). * [12] T. H. Koornwinder, Two-variable analogues of the classical orthogonal polynomials. In: Theory And Application Of Special Functions, (R. Askey, ed.), pp. 435 – 495, Academic Press, New York, 1975. * [13] R. Lasser, Orthogonal polynomials and hypergroups II-the symmetric case. _Trans. Amer. Math. Soc._ 341 (1994) 749–770. * [14] P. G. Nevai, _Orthogonal Polynomials_. Mem. Amer. Math. Soc. 213, 1979. * [15] M. Skantharajah, Amenable hypergroups. _Illinois J. Math._ 36 (1) (1992) 15–46. * [16] R. Spector, Aperçu de la théorie des hypergroupes. In _Anal. harmon. Groupes de Lie_ , _LNM 497_ (1975) 643–673. * [17] M. Voit, Positive characters on commutative hypergroups and some applications. _Math. Z._ 198 (1988) 405–421. * [18] M. Voit, On the dual space of a commutative hypergroup. _Arch. Math._ 56 (4) (1991) 380–385. * [19] M. Voit, Factorization of probability measures on symmetric hypergroups. J. Aust. Math. Soc. (Series A) 50 (1991) 417 – 467. * [20] M. Voit, A product formula for orthogonal polynomials associated with infinite distance-transitive graphs. _J. App. Theory._ 120 (2003) 337–354. * [21] M. Vogel, Spectral synthesis on algebras of orthogonal polynomial series. _Math. Z._ 194 (1) (1987) 99–116. * [22] R. C. Vrem, Hypergroups joins and their dual objects. _Pacific J. Math._ 111 (1984) 483– 495. * [23] S. Wolfenstetter, _Jacobi-Polynome und Bessel-Funktionen unter dem Gesichtspunkt der harmonischen Analyse_. Dissertation , Technische Universität München, 1984. * [24] H. Zeuner, Polynomial hypergroups in several variables. _Arch. Math._ 58 (1992) 425–434.	arxiv-papers	2009-02-25T22:54:22	2024-09-04T02:49:00.866947	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ahmadreza Azimifard", "submitter": "Ahmadreza Azimifard", "url": "https://arxiv.org/abs/0902.4479" }
0902.4746	# Massless Dirac Fermions in a Square Optical Lattice Jing-Min Hou1 jmhou@seu.edu.cn Wen-Xing Yang1,3 Xiong-Jun Liu2 1Department of Physics, Southeast University, Nanjing, 211189, China 2 Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA 3 Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan (January 5, 2009 ) ###### Abstract We propose a novel scheme to simulate and observe massless Dirac fermions with cold atoms in a square optical lattice. A $U(1)$ adiabatic phase is created by two laser beams for the tunneling of atoms between neighbor lattice sites. Properly adjusting the tunneling phase, we find that the energy spectrum has conical points in per Brillouin zone where band crossing occurs. Near these crossing points the quasiparticles and quasiholes can be considered as massless Dirac fermions. Furthermore, the anisotropic effects of massless Dirac fermions are obtained in the present square lattice model. The Dirac fermions as well as the anisotropic behaviors realizeded in our system can be experimentally detected with the Bragg spectroscopy technique. ###### pacs: 37.10.Jk, 03.75.Ss, 05.30.Fk ## I Introduction Realization of two-dimensional (2D) systems of massless Dirac fermions is of great fundamental importance, in the light of many exotic phenomena obtained in such systems, such as zero modes, fractional statistics, unconventional Landau levels, parity anomaly, chirality, and anomalous quantum Hall effects Semenoff ; Jackiw ; Haldane . However, two-dimensional massless Dirac field have not been observed untill the creation of graphene, a monolayer of graphite Novoselov2 ; Novoselov3 . Electrons in graphene, obeying a linear dispersion relation, behave like massless Dirac fermions Novoselov2 ; Novoselov3 ; Zhang ; Li ; Zheng ; Gusynin ; Hou ; Jachiw2 ; Pachos . Besides graphene, physicists also make efforts to search for other physical systems, e.g. patterned 2D electron gases Park and ultracold atoms in the honeycomb optical lattice Zhu ; Zhao ; Shao ; Wu , to simulate massless Dirac fermions. Realization of honeycomb optical lattice opens new possibility of studying Dirac fermions in cold atoms which provide an extremely clean environment and controllable fashion unique access to the study of complex physics Jaksch ; Greiner ; Lewenstein . Nevertheless, all of the above systems require the hexagonal symmetry. Then, it is very attractive to find a system without the hexagonal symmetry to observe massless Dirac fermions. Ultracold atom systems provide an ideal platform to study many interesting physics in condensed matters. To investigate the effects of gauge fields with ultracold atoms, several schemes have been proposed to create an artificial Abelian gauge field Jaksch2 ; Dum ; Juzeliunas1 ; Juzeliunas2 ; Juzeliunas3 ; Gunter or a non-Abelian gauge field Osterloh ; Ruseckas ; Lu for neutral atoms with laser fields. Many effects have been studied for cold atoms in an effective gauge field, e.g., Stern-Gerlach effect for chiral moleculesLi2 , Double and negative reflectionJuzeliunas4 , Landau levelsJacob , spin Hall effect Liu ; Zhu2 , induced spin-orbit couplingLiu2 , magnetic monopolePietila , spin field effect transistorsVaishnav . Furthermore, some groups have realized the light-induced gauge fields in experimentsDutta ; Lin . In this paper, we propose a scheme to generate a staggered gauge field with laser fields. A 2D square lattice model under this artificial gauge field has a spectrum behaving like massless Dirac fermions. Furthermore, our lattice model does not have the hexagonal symmetry. In our scheme, the energy bands of the system exhibit degeneracy points where the conduction and valence bands intersect. Near the these crossing points the dispersion relation is linearly dependent on the momentum, say, is of the Dirac type. The present scheme suggests a new direction to study Dirac fermions in the optical lattice without the hexagonal symmetry. Figure 1: (a) The atomic levels and the interactions between atoms and laser fields. (b) Schematic representation of the experimental setup with the two laser beams incident on the cloud of atoms. (c) Schematic of the square optical lattice and the designed phase factor (denoted by arrows). (d) The scheme of overlapping the two state-selective optical lattices. ## II Model We consider a system of ultracold fermionic atoms with four levels shown in FIG.1 (a). This atomic level configuration can be experimentally realized with alkali atom 6Li Fuchs . We choose the atomic states $2S_{1/2}(F=1/2,m_{F}=1/2)$, $2S_{1/2}(F=3/2,m_{F}=3/2)$, $2P_{1/2}(F=1/2,m_{F}=1/2)$ and $2P_{1/2}(F=1/2,m_{F}=-1/2)$ as $\|1\rangle,\|2\rangle,\|3\rangle$ and $\|4\rangle$, respectively. The cold atoms are trapped in two state-selective optical potentials as shown in FIG.1 (c) and (d). We assume that the states $\|1\rangle$ and $\|2\rangle$ have the same the state-selective optical potential, say sublattice $A$, and $\|4\rangle$ only perceives the other state-selective optical potential, say sublattice $B$. Here, for convenience, we assume that atoms in state $\|3\rangle$ also perceive sublattice $A$. However, this is unnecessary in our scheme, for the population of the quantum state $\|3\rangle$ is finally eliminated. The two sublattices have the lattice spacings $2l_{x}$ and $l_{z}$ in the $x$ and $z$ directions, respectively. The two sublattices make up a 2D rectangular lattice with the lattice spacings $l_{x}$ and $l_{z}$, especially a 2D square lattice for $l_{x}=l_{z}$, when overlapping together as shown in FIG.1 (c) and (d). Without loss of generality, we suppose that atoms with internal states $\|1\rangle$ and $\|2\rangle$ are trapped in odd columns and ones with internal states $\|4\rangle$ in even columns in the whole overlapped lattice. For convenience, we assume that the 2D square lattice considered here is in the $x-z$ plane as shown in FIG.1 (c). Two additional laser beams along the $y$ direction are added. When the potential barrier of the optical lattice along the $y$ direction is high enough, the tunneling along this direction between different planes is suppressed seriously, then every layer is an independent 2D lattice in $x-z$ plane. Using $\\{\|1\rangle,\|2\rangle,\|3\rangle,\|4\rangle\\}$ as the basis, the Hamiltonian of free ultracold fermions in the optical lattice can be written in the second quantized form as follows, $\displaystyle\hat{H}_{0}$ $\displaystyle=$ $\displaystyle\int d^{2}r\hat{\Psi}^{\dagger}\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V({\bf r})\right)\hat{\Psi},$ (1) where $\hat{\Psi}^{\dagger}=(\hat{\Psi}_{1}^{\dagger},\hat{\Psi}_{2}^{\dagger},\hat{\Psi}_{3}^{\dagger},\hat{\Psi}_{4}^{\dagger})$ and $\hat{\Psi}=(\hat{\Psi}_{1},\hat{\Psi}_{2},\hat{\Psi}_{3},\hat{\Psi}_{4})^{T}$ ($T$ denotes the matrix transposition) with $\hat{\Psi}_{i}({\bf r})$ and $\hat{\Psi}_{i}^{\dagger}({\bf r})$ being field operators corresponding to annihilating and creating an atom with the internal quantum state $\|i\rangle\ (i=1,2,3,4)$ at coordinate position ${\bf r}$ respectively. Here, $V({\bf r})$ is the trap potential matrix as $\displaystyle V({\bf r})=\left(\matrix{V_{A}({\bf r})&0&0&0\cr 0&V_{A}({\bf r})&0&0\cr 0&0&V_{A}({\bf r})&0\cr 0&0&0&V_{B}({\bf r})}\right),$ (2) where $V_{X}({\bf r})(X=A,B)$ are the two state-selective periodic potentials. The ground state $\|1\rangle$ is coupled to the excited state $\|3\rangle$ via a laser field with the corresponding Rabi frequency $\Omega_{1}e^{-iq_{1}z}$ and the state $\|2\rangle$ is coupled to the excited state $\|3\rangle$ via a laser field with the corresponding Rabi frequency $\Omega_{2}e^{iq_{2}z}$ as shown in FIG.1 (a) and (b). The corresponding light-atom interaction Hamiltonian is, $\displaystyle\hat{H}_{1}$ $\displaystyle=$ $\displaystyle\int d^{2}r\hat{\Psi}^{\dagger}M\hat{\Psi},$ (3) with $\displaystyle M=\hbar\left(\matrix{0&0&\Omega_{1}e^{iq_{1}z}&0\cr 0&0&\Omega_{2}e^{-iq_{2}z}&0\cr\Omega_{1}e^{-iq_{1}z}&\Omega_{2}e^{iq_{2}z}&0&0\cr 0&0&0&0}\right),$ (4) where $\Omega_{j}(j=1,2)$ are the Rabi frequencies. Additionally, the quantum state $\|1\rangle$ is coupled to the quantum state $\|4\rangle$ via a laser field propagating in the $y$ direction with Rabi frequency $\Omega_{3}e^{iq_{3}y}$. Because $e^{iq_{3}y}$ is a constant in the $x-z$ plane, we can omit this phase factor by supposing the two-dimensional lattice on the $y=0$ plane. The corresponding interaction Hamiltonian is $\displaystyle\hat{H}_{2}=\int d^{2}r\hat{\Psi}^{\dagger}N\hat{\Psi},$ (5) with $\displaystyle N=\hbar\left(\matrix{0&0&0&\Omega_{3}\cr 0&0&0&0\cr 0&0&0&0\cr\Omega_{3}&0&0&0}\right).$ (6) The total Hamiltonian can be written as $\hat{H}=\hat{H}_{0}+\hat{H}_{1}+\hat{H}_{2}$. The Hamiltonian (3) can be diagonalized by the matrix, $\displaystyle U=\left(\matrix{\cos\theta&-\sin\theta e^{iqz}&0&0\cr\frac{\sqrt{2}}{2}\sin\theta e^{-iqz}&\frac{\sqrt{2}}{2}\cos\theta&-\frac{\sqrt{2}}{2}e^{-iq_{2}z}&0\cr\frac{\sqrt{2}}{2}\sin\theta e^{-iqz}&\frac{\sqrt{2}}{2}\cos\theta&\frac{\sqrt{2}}{2}e^{-iq_{2}z}&0\cr 0&0&0&1}\right),$ (7) where $q=q_{1}+q_{2}$ and $\tan\theta=\|\Omega_{1}\|/\|\Omega_{2}\|$. Correspondingly, we obtain the dressed states as $\displaystyle\|\chi_{1}\rangle$ $\displaystyle=$ $\displaystyle\cos\theta\|1\rangle-\sin\theta e^{iqz}\|2\rangle,$ (8) $\displaystyle\|\chi_{2}\rangle$ $\displaystyle=$ $\displaystyle\frac{\sqrt{2}}{2}\sin\theta e^{-iqz}\|1\rangle+\frac{\sqrt{2}}{2}\cos\theta\|2\rangle-\frac{\sqrt{2}}{2}e^{-iq_{2}z}\|3\rangle,$ (9) $\displaystyle\|\chi_{3}\rangle$ $\displaystyle=$ $\displaystyle\frac{\sqrt{2}}{2}\sin\theta e^{-iqz}\|1\rangle+\frac{\sqrt{2}}{2}\cos\theta\|2\rangle+\frac{\sqrt{2}}{2}e^{-iq_{2}z}\|3\rangle,$ (10) $\displaystyle\|\chi_{4}\rangle$ $\displaystyle=$ $\displaystyle\|4\rangle,$ (11) with the energy eigenvalues $E_{i}=(0,-\hbar\Omega,\hbar\Omega,0)$ with $\Omega=\sqrt{\|\Omega_{1}\|^{2}+\|\Omega_{2}\|^{2}}$. Here, the state $\|\chi_{1}\rangle$ is a so-called dark state, which does not contain the component of the excited atomic state $\|3\rangle$, and $\|\chi_{2}\rangle,\|\chi_{3}\rangle$ are bright states. In the dressed state basis $\\{\|\chi_{1}\rangle,\|\chi_{2}\rangle,\|\chi_{3}\rangle,\|\chi_{4}\rangle\\}$, the vector field operator can be written as $\hat{\Phi}=({\hat{\Phi}_{1},\hat{\Phi}_{2}},\hat{\Phi}_{3},\hat{\Phi}_{4})^{T}=U({\hat{\Psi}_{1},\hat{\Psi}_{2}},\hat{\Psi}_{3},\hat{\Psi}_{4})^{T}$, where $\hat{\Phi}_{j}(j=1,2,3,4)$ represent destructing an atom in the dressed state $\|\chi_{j}\rangle(j=1,2,3,4)$. Thus, the Hamiltonian can be rewritten as $\displaystyle\hat{H}=\int d^{2}r\hat{\Phi}^{\dagger}\left[\frac{1}{2m}(-i\hbar\nabla-\tilde{\bf A})^{2}+\tilde{V}({\bf r})+\tilde{N}\right]\hat{\Phi},$ (12) where $\tilde{\bf A}=i\hbar U\nabla U^{\dagger}$, $\tilde{V}({\bf r})=UV({\bf r})U^{\dagger}+UMU^{\dagger}+\frac{\hbar^{2}}{2m}[(U\nabla U^{\dagger})^{2}+\nabla U\cdot\nabla U^{\dagger}]$ and $\tilde{N}=UNU^{\dagger}$. We straightforwardly calculate these matrices and obtain, $\displaystyle\tilde{\bf A}$ $\displaystyle=$ $\displaystyle-\hbar{\bf e}_{z}\left(\matrix{-q\sin^{2}\theta&\frac{\sqrt{2}}{2}q\sin\theta\cos\theta e^{iqz}&\frac{\sqrt{2}}{2}q\sin\theta\cos\theta e^{iqz}&0\cr\frac{\sqrt{2}}{2}q\sin\theta\cos\theta e^{-iqz}&\frac{1}{2}q\sin^{2}\theta+\frac{1}{2}q_{2}&\frac{1}{2}q\sin^{2}\theta-\frac{1}{2}q_{2}&0\cr\frac{\sqrt{2}}{2}q\sin\theta\cos\theta e^{-iqz}&\frac{1}{2}q\sin^{2}\theta-\frac{1}{2}q_{2}&\frac{1}{2}q\sin^{2}\theta+\frac{1}{2}q_{2}&0\cr 0&0&0&0}\right),$ (13) and $\displaystyle\tilde{V}({\bf r})$ $\displaystyle=$ $\displaystyle\left(\matrix{V_{A}({\bf r})&0&0&0\cr 0&V_{A}({\bf r})-\hbar\Omega&0&0\cr 0&0&V_{A}({\bf r})+\hbar\Omega&0\cr 0&0&0&V_{B}({\bf r})}\right),$ (14) and $\displaystyle\tilde{N}$ $\displaystyle=$ $\displaystyle\hbar\left(\matrix{0&0&0&\Omega_{3}\cos\theta\cr 0&0&0&\frac{\sqrt{2}}{2}\Omega_{3}\sin\theta e^{-iqz}\cr 0&0&0&\frac{\sqrt{2}}{2}\Omega_{3}\sin\theta e^{-iqz}\cr\Omega_{3}\cos\theta&\frac{\sqrt{2}}{2}\Omega_{3}\sin\theta e^{iqz}&\frac{\sqrt{2}}{2}\Omega_{3}\sin\theta e^{iqz}&0}\right).$ (15) In our scheme, we only consider the atoms in the dressed states $\|\chi_{1}\rangle$ and $\|\chi_{4}\rangle$. Thus, we have to adiabatically eliminate the populations of the dressed states $\|\chi_{2}\rangle$ and $\|\chi_{3}\rangle$ and to avoid the atoms decaying into these two dressed states. This can be realized in the steps. First, we start with the atoms in the atomic state $\|1\rangle$ and $\Omega_{1}=0$, $\Omega_{3}=0$ with $\Omega_{2}$ finite, then slowly turn $\Omega_{1}$, we will end up with the atoms in the dressed state $\|\chi_{1}\rangle$ Scully . During this process, the variation of $\Omega_{1}$ is slow enough to satisfy the adiabatic condition $\|\langle\chi_{j}\|\partial/\partial t\|\chi_{1}\rangle\|\ll\|E_{j}-E_{1}\|/\hbar=\Omega$ with $j=2,3$ Messiah ; jmhou . In the second step, we adiabatically turn the Rabi frequency $\Omega_{3}$ on, we will end up with atoms in the dressed states $\|\chi_{1}\rangle$ and $\|\chi_{4}\rangle$. To avoid the atoms decaying into the dressed states $\|\chi_{2}\rangle$ and $\|\chi_{3}\rangle$, the adiabatic conditions $\frac{1}{2m}\|\tilde{\bf A}_{j1}\|^{2}=\frac{\hbar^{2}}{2m}q^{2}\sin^{2}\theta\cos^{2}\theta\ll\|E_{j}-E_{1}\|=\hbar\Omega$ and $\|\tilde{N}_{j4}\|=\frac{\sqrt{2}}{2}\hbar\Omega_{3}\sin\theta\ll\|E_{j}-E_{4}\|=\hbar\Omega$ for $j=2,3$ are satisfied. This is to say, the off-diagonal elements of the Hamiltonian are small enough to avoid the atoms decaying into the dressed states $\|\chi_{2}\rangle$ and $\|\chi_{3}\rangle$. Since the atoms are only in the dressed states $\|\chi_{1}\rangle$ and $\|\chi_{4}\rangle$, we consider the reduced space with the dressed state basis $\\{\|\chi_{1}\rangle,\|\chi_{4}\rangle\\}$. Therefore, the total Hamiltonian can be reduced to $\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\int d^{2}r\hat{\Phi}_{1}^{\dagger}\left[\frac{1}{2m}(-i\hbar\nabla-{\bf{A}})^{2}+{V}_{A}({\bf r})\right]\hat{\Phi}_{1}$ (16) $\displaystyle+$ $\displaystyle\int d^{2}r\hat{\Phi}_{4}^{\dagger}\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+{V}_{B}({\bf r})\right]\hat{\Phi}_{4}$ $\displaystyle+$ $\displaystyle\hbar\Omega_{e}\int d^{2}r\left(\hat{\Phi}_{4}^{\dagger}\hat{\Phi}_{1}+\hat{\Phi}_{1}^{\dagger}\hat{\Phi}_{4}\right),$ where $\Omega_{e}=\tilde{N}_{14}/\hbar=\Omega_{3}\cos\theta$ and the $U(1)$ adiabatic gauge potential ${\bf A}=\tilde{\bf A}_{11}=\hbar q\sin^{2}\theta{\bf e}_{z}$. ## III Massless Dirac fermions Taking the tight-binding limit, we can superpose the Bloch states to get Wannier functions $w_{a}({\bf r}-{\bf r}_{i})$ and $w_{b}({\bf r}-{\bf r}_{j})$ for sublattice $A$ and $B$, respectively. In the present case, we can expand the field operator in the lowest band Wannier functions as, $\hat{\Phi}_{1}({\bf r})=\sum_{m(odd),n}\hat{a}_{m,n}e^{\frac{i}{\hbar}\int_{0}^{{\bf r}_{mn}}{\bf A}\cdot d{\bf r}}w_{a}({\bf r}-{\bf r}_{mn})$ and $\hat{\Phi}_{4}({\bf r})=\sum_{m(even),n}\hat{b}_{m,n}w_{b}({\bf r}-{\bf r}_{mn})$. Substituting the above expression into Eq.(16), we can rewrite the Hamiltonian as follows, $\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle-\sum_{(m(odd),n)}[t_{b}\hat{b}^{\dagger}_{m+1,n+1}\hat{b}_{m+1}+t_{a}e^{i\gamma}\hat{a}^{\dagger}_{m,n+1}\hat{a}_{m,n}$ (17) $\displaystyle+2t_{1}\hat{a}^{\dagger}_{m,n}\hat{b}_{m+1,n}+{\rm H.c.}]+\hat{H}_{K},$ with $H_{K}=\epsilon_{a}\sum_{(m(odd),n)}\hat{a}_{m,n}^{\dagger}\hat{a}_{m,n}+\epsilon_{b}\sum_{(m(even),n)}\hat{b}_{m,n}^{\dagger}\hat{b}_{m,n}$. Here, the parameters have the following forms: $t_{a}=\int d^{2}rw^{}_{a}({\bf r}-{\bf r}_{m,n+1})(-\hbar^{2}\nabla^{2}/2m+V_{A})w_{a}({\bf r}-{\bf r}_{mn})$, $t_{b}=\int d^{2}rw^{}_{b}({\bf r}-{\bf r}_{m,n+1})(-\hbar^{2}\nabla^{2}/2m+V_{B})w_{b}({\bf r}-{\bf r}_{mn})$, $t_{1}=\Omega_{e}\int d^{2}rw^{}_{b}({\bf r}-{\bf r}_{m+1,n})w_{a}({\bf r}-{\bf r}_{mn})$, $\epsilon_{a}=\int d^{2}rw^{}_{a}({\bf r}-{\bf r}_{m,n})(-\hbar^{2}\nabla^{2}/2m+V_{A})w_{a}({\bf r}-{\bf r}_{mn})$, $\epsilon_{b}=\int d^{2}rw^{}_{b}({\bf r}-{\bf r}_{m,n})(-\hbar^{2}\nabla^{2}/2m+V_{B})w_{b}({\bf r}-{\bf r}_{mn})$ and $\gamma=2\pi\sin^{2}\theta\hbar ql_{z}$ is the phase resulted from the adiabatic gauge potential. In our scheme, we consider $\epsilon_{a}=\epsilon_{b}$, so $\hat{H}_{K}$ in Eq. (17) can be dropped out as a constant term, which does not affect the physics considered here. First, we consider that the ideal conditions $\gamma=\pi$, $t_{a}=t_{b}=t_{1}=t$ and $l_{x}=l_{z}=l$ are satisfied. In experiments, these conditions can be achieved. Taking the Fourier transformation, $\hat{a}({\bf k})=\sum_{(m(odd),n)}\hat{a}_{m,n}\exp(-i{\bf k}\cdot{\bf r}_{m,n})$ and $\hat{b}({\bf k})=\sum_{(m(even),n)}\hat{b}_{m,n}\exp(-i{\bf k}\cdot{\bf r}_{m,n})$, we obtain the total Hamiltonian as $\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle-2t\sum_{k}[\cos(k_{z}l)\hat{b}^{\dagger}({\bf k})\hat{b}({\bf k})-\cos(k_{z}l)\hat{a}^{\dagger}({\bf k})\hat{a}({\bf k})$ (18) $\displaystyle+\cos(k_{x}l)\hat{a}^{\dagger}({\bf k})\hat{b}({\bf k})+\cos(k_{x}l)\hat{b}^{\dagger}({\bf k})\hat{a}({\bf k})].$ Diagonalizing the above Hamiltonian (18), we obtain the quasiparticle energy spectrum $E({\bf k})=2st\sqrt{\cos^{2}(k_{x}l)+\cos^{2}(k_{z}l)}$ with $s=\pm 1$ being the band index, which is similar to the spectrum of $\pi$ flux states in quantum spin liquids Wen . This energy spectrum has two energy bands and contains four zero-energy Dirac points, where the conduction and valence bands intersect, in the first Brillouin zone at ${\bf K}_{1}=\left({\pi}/{2l},{\pi}/{2l}\right),{\bf K}_{2}=\left(-{\pi}/{2l},{\pi}/{2l}\right),{\bf K}_{3}=\left({-\pi}/{2l},-{\pi}{2l}\right),{\bf K}_{4}=\left({\pi}/{2l},-{\pi}/{2l}\right)$. Near the Dirac points, the energy dispersion has standard cone-like shape as shown in Fig.2 (a) and (b) and the spectrum is linear. The low-energy state dynamics are described by linearizing their spectrum about the degeneracy points and are modeled by massless relativistic fermions. For simplicity, we only consider the part around the Dirac points ${\bf K}_{1}$, and the physics around the other Dirac points are similar. Setting ${\bf k}={\bf K}_{1}+{\bf p}$, we linearize the Hamiltonian around the Dirac point ${\bf K}_{1}$ as, $\hat{H}=\hbar v_{0}\sum_{p}[p_{z}\hat{b}^{\dagger}({\bf p})\hat{b}({\bf p})-p_{z}\hat{a}^{\dagger}({\bf p})\hat{a}({\bf p})+p_{x}\hat{a}^{\dagger}({\bf p})\hat{b}({\bf p})+p_{x}\hat{b}^{\dagger}({\bf p})\hat{a}({\bf p})]$ with $v_{0}=2tl/\hbar$, which can be rewritten in coordinate space as, $\hat{H}=\int d^{2}r\hat{\eta}^{\dagger}({\bf r})\hat{\cal H}\hat{\eta}({\bf r})$, where $\hat{\eta}=(\hat{\eta}_{b},\hat{\eta}_{a})^{T}$ with $\hat{\eta}_{b}({\bf r})=\int d^{2}pe^{-i{\bf p}\cdot{\bf r}}\hat{b}({\bf p})$ and $\hat{\eta}_{a}({\bf r})=\int d^{2}pe^{-i{\bf p}\cdot{\bf r}}\hat{a}({\bf p})$. Here, $\hat{\cal H}$ is the single-particle Hamiltonian as $\hat{\cal H}=\hbar v_{0}(\hat{p}_{x}\sigma_{x}+\hat{p}_{z}\sigma_{z})$, where $\sigma_{x}$ and $\sigma_{z}$ are Pauli matrixes. We obtain the eigenstates $\displaystyle\phi_{\bf p}^{s}=\frac{1}{\sqrt{2}}\left(\matrix{\cos\frac{\alpha}{2}+s\sin\frac{\alpha}{2}\cr s\cos\frac{\alpha}{2}-\sin\frac{\alpha}{2}}\right)e^{{i}{\bf p}\cdot{\bf r}},$ (19) where $s=\pm 1$ and $\tan\alpha=p_{z}/p_{x}$. The corresponding eigenenergies are $E^{s}({\bf p})=s\hbar v_{0}p$ with $p=\sqrt{p_{x}^{2}+p_{z}^{2}}$. When the wave vector is ${\bf p}=p_{x}\hat{x}+p_{z}\hat{z}$, the corresponding group velocity and pseudospin vector are ${\bf v}_{g}=sv_{0}(p_{x}\hat{x}+p_{z}\hat{z})/p$ and ${\bf c}=(p_{x}\hat{x}+p_{z}\hat{z})/p$, respectively. It is easy to find that the three vectors ${\bf v}_{g}$, ${\bf c}$ and ${\bf p}$ are collinear, i.e., they are parallel to each other. There is an intimate relation between the pseudospin and motion of the quasiparticle or quasihole: pseudospin can only be directed along the propagation direction (say, for quasiparticles) or only opposite to it (for quasiholes). As a result, quasiparticles or quasiholes exhibit a linear dispersion relation $E=\hbar v_{0}k$, as if they were massless relativistic particles but the role of the speed of light is played here by the Fermi velocity $v_{0}$. Figure 2: Energy dispersion for cold fermionic atoms in a square optical lattice. (a) shows the energy dispersion and (b) represents the profiles of the energy dispersion with $k_{z}=\pi/2l$ (blue line) and $k_{x}=\pi/2l$(red star), for the ideal case $t_{a}=t_{b}=t_{1}=t$, $l_{x}=l_{z}=l$, $\gamma=\pi$. (c) shows the energy dispersion and (d) represents the profiles of the energy dispersion with $k_{z}=\pi/2l$ (blue line) and $k_{x}=\pi/2l$ (red line), for the anisotropic case $t_{a}=t_{b}=2t_{1}/3=2t/3$, $l_{x}=l_{z}=l$, $\gamma=\pi$. (e) shows the energy dispersion and (f) represents the profiles of the energy dispersion with $k_{z}=2\pi/5l$ (blue line) and $k_{x}=\pi/2l$(red line), for the anisotropic case $t_{a}=t_{b}=t_{1}=t$, $l_{z}=5l_{x}/4=5l/4$, $\gamma=\pi$. (g) shows the energy dispersion and (h) represents the profiles of the energy dispersion with $k_{z}=2\pi/5l$ (blue line) and $k_{x}=\pi/2l$ (red line), for the anisotropic case with $t_{a}=t_{b}=t_{1}=t$, $l_{x}=l_{z}=l$, $\gamma=6\pi/5$. In practice, the parameters may have fluctuations around the ideal conditions considered above. Fortunately, even the parameters deviate from the ideal ones, the massless Dirac fermion spectrum persists and remarkably exhibit anisotropic behaviors, which are just pursued in References Park1 by adding external periodic potentials on graphene. Here, we provide alternative methods to exhibit anisotropic behaviors of massless Dirac fermions in a square optical lattice by setting the parameters deviated from the ideal situation. For simplicity, we only consider three cases with the existence of parameter deviation from the ideal situation as follow: (i) $t_{a}=t_{b}\neq t_{1}=t$, $l_{z}=l_{x}=l$, $\gamma=\pi$; (ii) $t_{a}=t_{b}=t_{1}=t$, $l_{z}\neq l_{x}=l$, $\gamma=\pi$; (iii) $t_{a}=t_{b}=t_{1}=t$, $l_{x}=l_{z}=l$, $\gamma=\pi+\delta$ with $\delta\neq 0$. The corresponding dispersion relations are $E_{\rm i}({\bf k})=2st\sqrt{\cos^{2}(k_{x}l)+(t_{a}/t)^{2}\cos^{2}(k_{z}l)}$, $E_{\rm ii}({\bf k})=2st\sqrt{\cos^{2}(k_{x}l_{x})+\cos^{2}(k_{z}l_{z})}$ and $E_{\rm iii}({\bf k})=t[\cos(k_{z}l+\delta)-\cos(k_{z}l)]+st\sqrt{4\cos^{2}(k_{x}l)+[\cos(k_{z}l+\delta)+\cos(k_{z}l)]^{2}}$ for cases (i), (ii) and (iii), respectively, which are shown in FIG.2 (c)-(h). For case (ii), the four Dirac points are ${\bf K}_{1}=\left({\pi}/{2l_{x}},{\pi}/{2l_{z}}\right),{\bf K}_{2}=\left(-{\pi}/{2l_{x}},{\pi}/{2l_{z}}\right),{\bf K}_{3}=\left({-\pi}/{2l_{x}},-{\pi}/{2l_{z}}\right),{\bf K}_{4}=\left({\pi}/{2l_{x}},-{\pi}/{2l_{z}}\right)$, which are dependent on the lattice spacing in the $x$ and $z$ direction, while the Dirac points for cases (i) are the same as those of the ideal case. For case (iii), the four Dirac points are ${\bf K}_{1}=\left({(\pi-\delta)}/{2l},{\pi}/{2l}\right),{\bf K}_{2}=\left({(-\pi-\delta)}/{2l},{\pi}/{2l}\right),{\bf K}_{3}=\left({(-\pi-\delta)}/{2l},-{\pi}/{2l}\right),{\bf K}_{4}=\left({(\pi-\delta)}/{2l},-{\pi}/{2l}\right)$. Around the Dirac points, these spectra can be linearized as $E_{\rm i}^{s}({\bf p})=s\hbar v_{0}p_{1}$ with $p_{1}=\sqrt{p_{x}^{2}+f_{1}^{2}p_{z}^{2}}$, $E_{\rm ii}^{s}({\bf p})=s\hbar v_{0}p_{1}$ with $p_{2}=\sqrt{p_{x}^{2}+f_{2}^{2}p_{z}^{2}}$ and $E_{\rm iii}^{s}({\bf p})=\pm 2t\sin(\delta/2)+s\hbar v_{0}p_{3}$ with $p_{3}=\sqrt{p_{x}^{2}+f_{3}^{2}p_{z}^{2}}$, where $f_{1}=t_{a}/t=t_{b}/t$, $f_{2}=l_{z}/l$ and $f_{3}=\cos(\delta/2)$. The corresponding single-particle Hamiltonian can be written as $\hat{\cal H}_{\rm i}=\hbar v_{0}(\hat{p}_{x}\sigma_{x}+f_{1}\hat{p}_{z}\sigma_{z})$, $\hat{\cal H}_{\rm ii}=\hbar v_{0}(\hat{p}_{x}\sigma_{x}+f_{2}\hat{p}_{z}\sigma_{z})$ and $\hat{\cal H}_{\rm iii}=\pm 2t\sin(\delta/2)+\hbar v_{0}(\hat{p}_{x}\sigma_{x}+f_{3}\hat{p}_{z}\sigma_{z})$ for cases (i),(ii),(iii), respectively. In all cases, the quasiparticles or quasiholes are still massless Dirac fermions and show chiral behavior. For the wave vector ${\bf p}=p_{x}\hat{x}+p_{z}\hat{z}$, the group velocity are pseudospin vector are ${\bf v}_{g}=sv_{t}(p_{x}\hat{x}+f_{j}^{2}p_{z}\hat{z})/p_{j}$ and ${\bf c}=(p_{x}\hat{x}+f_{j}p_{z}\hat{z})/p_{j}$ for $j=1,2,3$. Here, the three vectors ${\bf v}_{g},{\bf c}$ and ${\bf p}$ are not collinear and the dispersion relations near the Dirac points show anisotropic behaviors. Figure 3: The Bragg spectroscopies for the ideal case, (a) $t_{a}=t_{b}=t_{1}=t$, $l_{x}=l_{z}=l$, $\gamma=\pi$, and the anisotropic cases, (b) $t_{a}=t_{b}=2t_{1}/3=2t/3$, $l_{x}=l_{z}=l$, $\gamma=\pi$, (c) $t_{a}=t_{b}=t_{1}=t$, $l_{z}=5l_{x}/4=5l/4$, $\gamma=\pi$, (d) $t_{a}=t_{b}=t_{1}=t$, $l_{x}=l_{z}=l$, $\gamma=6\pi/5$ and $q=\pi/10l$. Here, we represent the Bragg spectroscopies with blue lines for the mentum difference ${\bf q}$ in the $x$ direction and with red stars or red lines for ${\bf q}$ in the $z$ direction. The frequency difference $\omega$ is expressed in units of $qv_{0}$ and the dynamic structure factor $S({\bf q},\omega)$ is expressed in units of $q/8\pi^{2}nv_{0}$ with $n$ being the number density of atoms in the system. ## IV Bragg spectroscopy Here, we propose to identify massless Dirac fermionic quasiparticles with Bragg spectroscopy Stamper-Kurn , which is extensively used to probe excitation spectra in condensed matter physics. In Bragg scattering, the atomic gas is exposed to two laser beams, with wavevectors ${\bf k}_{1}$ and ${\bf k}_{2}$ and a frequency difference $\omega$. The light-atom interaction Hamiltonian for Bragg scattering can be written as, $\hat{H}_{B}=\sum_{{\bf p}_{1},{\bf p}_{2}}\hbar\Omega_{B}e^{-i{\bf q}\cdot{\bf r}}\|\phi_{{\bf p}_{2}}^{f}\rangle\langle\phi_{{\bf p}_{1}}^{i}\|+{\rm H.c.}$ with ${\bf q}={\bf p}_{2}-{\bf p}_{1}$, where the initial state $\|\phi_{{\bf p}}^{i}\rangle$ is a filled state under Fermi surface and the final state $\|\phi_{{\bf p}}^{f}\rangle$ is an empty state above Fermi surface. From the Fermi’s golden rule, we obtain the dynamic structure factor as follows, $\displaystyle S({\bf q},\omega)$ $\displaystyle=$ $\displaystyle\frac{1}{N\hbar^{2}\Omega_{B}^{2}}\sum_{{\bf p}}\|\langle\phi_{{\bf p}+{\bf q}}^{f}\|\hat{H}_{B}\|\phi_{{\bf p}}^{i}\rangle\|^{2}$ (20) $\displaystyle\times\delta(\hbar\omega-E_{{\bf p}+{\bf q}}^{f}+E_{{\bf p}}^{i}),$ where $N$ is the total number of atoms in the system. Here, we consider the case of half filling of cold fermions in the optical lattice, i.e. the Fermi energy surface is at zero energy level, which is just at Dirac points for the cases except anisotropic case (iii). The numerical evaluation results of the dynamic structure factor is shown in FIG. 3. We note that there are lower cutoff frequencies $\omega_{r}$ for the fixed momentum difference $q$ and $S({\bf q},\omega)$ are approximately linear to the frequency difference $\omega$ for large frequency difference $\omega$ for FIG.3 (a),(b), (c). However, for FIG.3(d), the cutoff disappears for the Fermi surface is not at Dirac points for this case. FIG.3 (a) show that the bragg spectroscopy curves for the momentum difference ${\bf q}$ in the $x$ and $z$ directions are identical in the ideal case, which is just a consequence of the isotropy of the energy spectrum. From FIG.3 (b), (c) and (d), we clearly see that the Bragg spectroscopies are different for ${\bf q}$ in the $x$ and $z$ directions in the anisotropic cases, which features the anisotropic behaviors of those spectra. ## V Conclusion In summary, we have proposed a novel scheme to realize massless Dirac fermions in a 2D square optical lattice with assistance of laser fields. For massless Dirac fermions, the gap is zero and the linear dispersion law holds. Our scheme is very robust against perturbations. Even the experimental situation deviates from the ideal conditions, massless Dirac fermions persist and, furthermore, exhibit novel features, i.e., anisotropic behaviors. Due to the absence of hexagonal symmetry, our scheme suggests a new direction to study Dirac fermions in the optical lattice. ###### Acknowledgements. This work was supported by the Teaching and Research Foundation for the Outstanding Young Faculty of Southeast University. X. J. Liu acknowledges support from US NSF Grant No. DMR-0547875 and ONR under Grant No. ONR-N000140610122. ## References (1) G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984). * (2) R. Jackiw, Phys. Rev. D 29, 2375 (1984). * (3) F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). * (4) K. S. Novoselov, et al., Science 306, 666 (2004). * (5) K. S. Novoselov et al., Nature 438, 197 (2005). * (6) Y. Zhang, et al., Nature 438, 201 (2005). * (7) G. Li and E. Y. Andrei, Nature Phys. 3, 623 (2007). * (8) Y. Zheng and T. Ando, Phys. Rev. B 65, 245420 (2002). * (9) V. P. Gusynin et al., Phys. Rev. Lett. 95, 146801 (2005). * (10) C. Y. Hou et al., Phys. Rev. Lett. 98, 186809 (2007). * (11) R. Jackiw et al., Phys. Rev. Lett. 98, 266402 (2007). * (12) J. K. Pachos et al., Int. J. Mod. Phys. B, 21, 5113 (2007). * (13) P. H. Park and S. G. Louie, arXiv:0808.2127. * (14) S. L. Zhu et al., Phys. Rev. Lett. 98, 260402 (2007). * (15) E. Zhao et al., Phys. Rev. Lett. 97, 230404 (2006). * (16) L. B. Shao et al., arXiv:0804.1850. * (17) C. Wu et al., Phys. Rev. Lett. 99, 070401 (2007); C. Wu and S. Das Sarma, Phys. Rev. B 77, 235107 (2008). * (18) D. Jaksch et al., Phys. Rev. Lett. 81, 3108(1998). * (19) M. Greiner et al., Nature 415, 39, (2002). * (20) M. Lewenstein et al., Adv. Phys. 56, 243 (2007), and references therein. * (21) D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003). * (22) G. Juzeliūnas and P. Öhberg, Phys. Rev. Lett. 93, 033602 (2004). * (23) G. Juzeliūnas, P. Öhberg, J. Ruseckas, and A. Klein, Phys. Rev. A 71, 053614 (2005). * (24) G. Juzeliūnas, J. Ruseckas, P. Öhberg, and M. Fleischhauer, Phys. Rev. A 73, 025602 (2006). * (25) K. J. Günter, M. Cheneau, T. Yefsah, S. P. Rath, and J. Dalibard, Phys. Rev. A 79, 011604(R) (2009). * (26) R. Dum and M. Olshanii, Phys. Rev. Lett. 76, 1788 (1996). * (27) K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein, Phys. Rev. Lett. 95, 010403 (2005). * (28) J. Ruseckas, G. Juzeliūnas, P. Öhberg, and M. Fleischhauer, Phys. Rev. Lett. 95, 010404 (2005). * (29) L. H. Lu and Y. Q. Li, Phys. Rev. A 76, 023410 (2007). * (30) Y. Li, C. Bruder, and C. P. Sun, Phys. Rev. Lett. 99, 130403 (2007). * (31) G. Juzeliūnas, J. Ruseckas, A. Jacob, L. Santos, and P. Öhberg, Phys. Rev. Lett. 100, 200405 (2008). * (32) A. Jacob, P. Öhberg, G. Juzeliūnas, and L. Santos, New J. Phys. 10, 045022(2008) * (33) X. J. Liu, X. Liu, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 98, 026602 (2007). * (34) S. L. Zhu, H. Fu, C. J. Wu, S. C. Zhang, and L. M. Duan, Phys. Rev. Lett. 97, 240401 (2006). * (35) X. J. Liu, M. F. Borunda, X. Liu, and J. Sinova, Phys. Rev. Lett. 102, 046402 (2009). * (36) V. Pietilä and M. Möttönen, Phys. Rev. Lett. 102, 080403 (2009) * (37) J. Y. Vaishnav, J. Rusechas, C. W. Clark, and G. Juzeliūnas, Phys. Rev. Lett. 101, 265302 (2008). * (38) S. K. Dutta, B. K. Teo, and G. Raithel, Phys. Rev. Lett. 83. 1934 (1999). * (39) Y. J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, arXiv:0809.2976 (2008). * (40) J. Fuchs, G. J. Duffy, W. J. Rowlands, A. Lezama, P. Hannaford, and A. M. Akulshin, J. Phys. B: At. Mol. Opt. Phys. 40, 1117(2007). * (41) M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University Press, Cambridge, 1997). * (42) A. Messiah, Quantum Mechanics (North-Holland/ Elsevier Science, New York, 1962). * (43) J. M. Hou, L. J. Tian, and S. Jin, Phys. Rev. B 73, 134425 (2006). * (44) X. G. Wen, Quantum Field Theory of Many-Body Systems, (Oxford University Press, Oxford, 2004). * (45) C. H. Park et al., Nature Phys. 4, 213 (2008); C. H. Park et al., Phys. Rev. Lett. 101, 126804 (2008). * (46) D. M. Stamper-Kurn et al., Phys. Rev. Lett. 83, 2876 (1999).	arxiv-papers	2009-02-27T02:55:08	2024-09-04T02:49:00.874915	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jing-Min Hou, Wen-Xing Yang, Xiong-Jun Liu", "submitter": "Jing-Min Hou", "url": "https://arxiv.org/abs/0902.4746" }
0902.4748	# On Pointed Hopf Algebras with classical Weyl Groups Shouchuan Zhang Mathematics Department of Hunan University, Changsha China, 410082, E-mail: z9491@yahoo.com.cn ###### Abstract. Many cases of infinite dimensional Nichols algebras of irreducible Yetter- Drinfeld modules over classical Weyl groups are found. It is proved that except a few cases Nichols algebras of reducible Yetter-Drinfeld modules over classical Weyl groups are infinite dimensional. Some finite dimensional Nichols algebras of Yetter-Drinfeld modules over classical Weyl groups are given. ## 0\. Introduction This article is to contribute to the classification of finite-dimensional complex pointed Hopf algebras with Weyl groups of classical type. Weyl groups are very important in the theories of Lie groups, Lie algebras and algebraic groups. The classification of finite dimensional pointed Hopf algebra with finite abelian groups has been finished (see [He06, AS00]). Many cases of infinite dimensional Nichols algebras of irreducible Yetter-Drinfeld (YD) modules over symmetric group were discarded in [AFZ]. All $-1$-type pointed Hopf algebras with Weyl groups of exceptional type were found in [ZZWC] and it was showed that every non $-1$-type pointed Hopf algebra is infinite dimensional in [AZ07, ZZWC]. It was obtained that every Nichols algebra of reducible YD module over simple group and symmetric group is infinite dimensional in [HS]. Hopf subalgebras of co-path Hopf algebras was studied in [OZ04]. In this paper we discard many cases of infinite dimensional Nichols algebras of irreducible YD modules over classical Weyl groups by mean of co-path Hopf algebras and the results of [AFZ]. [HS] said that if Nichols algebra of reducible YD module is finite dimensional, then their conjugacy classes are square-commutative (see [HS, Theorem 8.6]). We obtain that except a few cases Nichols algebras of reducible YD modules over classical Weyl groups are infinite dimensional by applying result of [HS]. We also find some finite dimensional Nichols algebras of YD modules over classical Weyl groups. The main results in this paper are summarized in the following statements. ###### Theorem 1. Let $G=A\rtimes\mathbb{S}_{n}$ be a classical Weyl group with $A\subseteq(C_{2})^{n}$ and $n>2$. Assume that $M=M({\mathcal{O}}_{\sigma_{1}},\rho^{(1)})\oplus M({\mathcal{O}}_{\sigma_{2}},\rho^{(2)})\oplus\cdots\oplus M({\mathcal{O}}_{\sigma_{m}},\rho^{(m)})$ is a reducible YD module over $kG$ . (i) Assume that there exist $i\not=j$ such that $\sigma_{i}$, $\sigma_{j}$ $\notin A$. If ${\rm dim}\mathfrak{B}(M)<\infty$, then $n=4$, the type of $\sigma_{p}$ is $2^{2}$ and the sign of $\sigma_{p}$ is stable for any $1\leq p\leq m$ with $\sigma_{p}\notin A.$ (ii) If $\sigma_{i}=a:=(g_{2},g_{2},\cdots,g_{2})\in G$ and $\rho^{(i)}=\theta_{\chi^{(\nu_{i})},\mu^{(i)}}:=(\chi^{(\nu_{i})}\otimes\mu^{(i)})\uparrow_{G_{\chi{(\nu_{i})}}^{a}}^{G^{a}}\in\widehat{G^{a}}$ with odd $\nu_{i}$ for $i=1,2,\cdots,m$, then $\mathfrak{B}(M)$ is finite dimensional. ###### Theorem 2. Let $0\leq\nu\leq n$ and Let $G=A\rtimes\mathbb{S}_{n}$. Let $\sigma\in\mathbb{S}_{n}$ be of type $(1^{\lambda_{1}},2^{\lambda_{2}},\dots,n^{\lambda_{n}})$ and $\rho=\rho^{\prime}\otimes\rho^{\prime\prime}\in\widehat{({\mathbb{S}}_{n})^{\sigma}_{\chi^{(\nu)}}}$ with $\rho^{\prime}\in\widehat{\mathbb{S}_{\nu}^{\sigma}}$ and $\rho^{\prime\prime}\in\widehat{\mathbb{S}_{\\{\nu+1,\cdots,n\\}}^{\sigma}}$. Assume that $\mathfrak{B}({\mathcal{O}}_{\sigma}^{G},\theta_{\chi^{(\nu)},\rho})$ is matched with ${\rm dim}\mathfrak{B}({\mathcal{O}}_{\sigma}^{G},\theta_{\chi^{(\nu)},\rho})<\infty$. Let $\mu=\otimes_{1\leq i\leq n}\mu_{i}$ with $\mu_{i}:=\theta_{\chi^{{\bf t}_{i}},\rho_{i}}$ as in (2.2) denote $\rho^{\prime}$ when $\sigma\in{\mathbb{S}}_{\nu}$ and $\rho^{\prime\prime}$ when $\sigma\in{\mathbb{S}}_{\\{\nu+1,\nu+2,\cdots,n\\}}$, respectively. Let $\lambda_{1}^{\prime}=\lambda_{1}-(n-\nu)$ when $\sigma\in{\mathbb{S}}_{\nu}$; $\lambda_{1}^{\prime}=\lambda_{1}-\nu$ when $\sigma\in{\mathbb{S}}_{\\{\nu+1,\nu+2,\cdots,n\\}}$. Then some of the following hold: 1. (i) $(1^{\lambda_{1}^{\prime}},2)$, $\mu_{1}={\rm sgn}$ or $\epsilon$, $\mu_{2}=\chi_{(1;2)}$. 2. (ii) $(2,\sigma_{o})$, $\sigma_{o}:=\prod\limits_{1\leq i\leq n,1<i\hbox{ is odd}}\sigma_{i}$ $\neq{\rm id}$, $\mu_{2}=\chi_{(1;2)}$, $\mu_{j}=(\chi_{(0,\dots,0;j)}\otimes\rho_{j})\uparrow_{({\mathbb{S}}_{Y_{j}})_{\chi_{(0,\dots,0;j)}}^{\sigma_{j}}}^{({\mathbb{S}}_{Y_{j}})^{\sigma_{j}}}$, for all odd $j>1$. 3. (iii) $(1^{\lambda_{1}^{\prime}},2^{3})$, $\mu_{1}={\rm sgn}$ or $\epsilon$, $\mu_{2}=\chi_{(1,1,1;2)}\otimes\epsilon$ or $\chi_{(1,1,1;2)}\otimes{\rm sgn}$. Furthermore, if $\lambda_{1}^{\prime}>0$, then $\mu_{2}=\chi_{(1,1,1;2)}\otimes{\rm sgn}$. 4. (iv) $(2^{5})$, $\mu_{2}=\chi_{(1,1,1,1,1;2)}\otimes\epsilon$ or $\chi_{(1,1,1,1,1;2)}\otimes{\rm sgn}$. 5. (v) $(1^{\lambda_{1}^{\prime}},4)$, $\mu_{1}={\rm sgn}$ or $\epsilon$, $\mu_{4}=\chi_{(2;4)}$. 6. (vi) $(1^{\lambda_{1}^{\prime}},4^{2})$, $\mu_{1}={\rm sgn}$ or $\epsilon$, $\mu_{4}=\chi_{(1,1;4)}\otimes{\rm sgn}$ or $\chi_{(3,3;4)}\otimes{\rm sgn}$. 7. (vii) $(2,4)$, $\mu_{2}=\chi_{(1;2)}$ and $\mu_{4}=\epsilon$ or $\mu_{2}=\epsilon$ and $\mu_{4}=\chi_{(2;4)}$. 8. (viii) $(2,4^{2})$, $\mu_{2}=\epsilon$, $\mu_{4}=\chi_{(1,1;4)}\otimes{\rm sgn}$ or $\chi_{(3,3;4)}\otimes{\rm sgn}$. 9. (ix) $(2^{2},4)$, $\deg\mu_{2}=1$, $\mu_{4}=\chi_{(2;4)}$. Furthermore, $\mathfrak{B}({\mathcal{O}}_{\sigma}^{G},\theta_{\chi^{(\nu)},\rho})$ is $-1$-type under the cases above. Indeed, Theorem 1 follows Remark 3.8 and Remark 3.12. The proof of Theorem 2 is in subsection 3.1. ## Preliminaries And Conventions Let $k$ be the complex field and $G$ a finite group. Let $\hat{{G}}$ denote the set of all isomorphic classes of irreducible representations of group $G$, $G^{\sigma}$ the centralizer of $\sigma$, ${\mathcal{O}}_{\sigma}$ or ${\mathcal{O}}_{\sigma}^{G}$ the conjugacy class in $G$, $C_{j}$ the cycle group with order $j$, $g_{j}$ a generator of $C_{j}$ and $\chi_{j}$ a character of $C_{j}$ with order $j$. The Weyl groups of $A_{n}$, $B_{n}$, $C_{n}$ and $D_{n}$ are called the classical Weyl groups. Let $\rho\uparrow_{D}^{G}$ denote the induced representation of $\rho$ as in [Sa01]. A quiver $Q=(Q_{0},Q_{1},s,t)$ is an oriented graph, where $Q_{0}$ and $Q_{1}$ are the sets of vertices and arrows, respectively; $\sigma$ and $t$ are two maps from $Q_{1}$ to $Q_{0}$. For any arrow $a\in Q_{1}$, $s(a)$ and $t(a)$ are called its start vertex and end vertex, respectively, and $a$ is called an arrow from $s(a)$ to $t(a)$. For any $n\geq 0$, an $n$-path or a path of length $n$ in the quiver $Q$ is an ordered sequence of arrows $p=a_{n}a_{n-1}\cdots a_{1}$ with $t(a_{i})=s(a_{i+1})$ for all $1\leq i\leq n-1$. Note that a 0-path is exactly a vertex and a 1-path is exactly an arrow. In this case, we define $s(p)=s(a_{1})$, the start vertex of $p$, and $t(p)=t(a_{n})$, the end vertex of $p$. For a 0-path $x$, we have $s(x)=t(x)=x$. Let $Q_{n}$ be the set of $n$-paths. Let ${}^{y}Q_{n}^{x}$ denote the set of all $n$-paths from $x$ to $y$, $x,y\in Q_{0}$. That is, ${}^{y}Q_{n}^{x}=\\{p\in Q_{n}\mid s(p)=x,t(p)=y\\}$. A quiver $Q$ is finite if $Q_{0}$ and $Q_{1}$ are finite sets. A quiver $Q$ is locally finite if ${}^{y}Q_{1}^{x}$ is a finite set for any $x,y\in Q_{0}$. Let $G$ be a group. Let ${\mathcal{K}}(G)$ denote the set of conjugate classes in $G$. A formal sum $r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ of conjugate classes of $G$ with cardinal number coefficients is called a ramification (or ramification data ) of $G$, i.e. for any $C\in{\mathcal{K}}(G)$, $r_{C}$ is a cardinal number. In particular, a formal sum $r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ of conjugate classes of $G$ with non- negative integer coefficients is a ramification of $G$. For any ramification $r$ and a $C\in{\mathcal{K}}(G)$, since $r_{C}$ is a cardinal number, we can choice a set $I_{C}(r)$ such that its cardinal number is $r_{C}$ without loss of generality. Let ${\mathcal{K}}_{r}(G):=\\{C\in{\mathcal{K}}(G)\mid r_{C}\not=0\\}=\\{C\in{\mathcal{K}}(G)\mid I_{C}(r)\not=\emptyset\\}$. If there exists a ramification $r$ of $G$ such that the cardinal number of ${}^{y}Q_{1}^{x}$ is equal to $r_{C}$ for any $x,y\in G$ with $x^{-1}y\in C\in{\mathcal{K}}(G)$, then $Q$ is called a Hopf quiver with respect to the ramification data $r$. In this case, there is a bijection from $I_{C}(r)$ to ${}^{y}Q_{1}^{x}$, and hence we write ${\ }^{y}Q_{1}^{x}=\\{a_{y,x}^{(i)}\mid i\in I_{C}(r)\\}$ for any $x,y\in G$ with $x^{-1}y\in C\in{\mathcal{K}}(G)$. deg $\rho$ denotes the dimension of the representation space $V$ for a representation $(V,\rho).$ Recall the notation ${\rm RSR}$ in [ZCZ, Def. 1.1]. Let $\rho_{C}$ be a representation of $G^{u(C)}$ with irreducible decomposition $\rho=\oplus_{i\in I_{C}(r,u)}\rho_{C}^{(i)}$, where $I_{C}(r,u)$ is an index set. Let $\overrightarrow{\rho}$ denote $\\{\rho_{C}\\}_{C\in{\mathcal{K}}_{r}(G)}=$ $\\{\\{\rho_{C}^{(i)}\\}_{i\in I_{C}(r,u)}\\}_{C\in{\mathcal{K}}_{r}(G)}$. $(G,r,\overrightarrow{\rho},u)$ is called an ${\rm RSR}$ when ${\rm deg}(\rho_{C})=r_{C}$ for any $C\in{\mathcal{K}}_{r}(G)$, written as ${\rm RSR}(G,r,\overrightarrow{\rho},u)$. For any ${\rm RSR}(G,r,\overrightarrow{\rho},u)$, we obtain a co-path Hopf algebra $kQ^{c}(G,$ $r,\overrightarrow{\rho},u)$, a Hopf algebra $kG[kQ^{c}_{1},$ $G,r,$ $\overrightarrow{\rho},u]$ of one type, a $kG$-YD module $(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$ and a Nicolas algebra ${\mathfrak{B}(G,r,\overrightarrow{\rho},u)}:={\mathfrak{B}}(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$ (see [ZCZ]). If ramification $r=r_{C}C$ and $\mid\\!I_{C}(r,u)\\!\mid=1$ then we say that ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ is of bi-one since $r$ only has one conjugacy class $C$ and $\mid\\!I_{C}(r,u)\\!\mid=1$. Furthermore, if let $\sigma=u(C)$, $C={\mathcal{O}}_{\sigma}$, $r_{C}=m$ and $\rho_{C}^{(i)}=\rho$ for $i\in I_{C}(r,u)$, then bi-one ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ is denoted by ${\rm RSR}(G,m{\mathcal{O}}_{\sigma},\rho)$ ( or ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\rho)$), in short. ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ is called to be $-1$-type, if $u(C)$ is real (i.e. $u(C)^{-1}\in C$) and the order of $u(C)$ is even with $\rho_{C}^{(i)}(u(C))=-{\rm id}$ for any $C\in{\mathcal{K}}_{r}(G)$ and any $i\in I_{C}(r,u)$. In this case, the Nichols algebra ${\mathfrak{B}(G,r,\overrightarrow{\rho},u)}$ is called to be $-1$-type. For $s\in G$ and $(\rho,V)\in\widehat{G^{s}}$, here is a precise description of the YD module $M({\mathcal{O}}_{s},\rho)$, introduced in [Gr00, AZ07]. Let $t_{1}=s$, …, $t_{m}$ be a numeration of ${\mathcal{O}}_{s}$, which is a conjugacy class containing $s$, and let $g_{i}\in G$ such that $g_{i}\rhd s:=g_{i}sg_{i}^{-1}=t_{i}$ for all $1\leq i\leq m$. Then $M({\mathcal{O}}_{s},\rho)=\oplus_{1\leq i\leq m}g_{i}\otimes V$. Let $g_{i}v:=g_{i}\otimes v\in M({\mathcal{O}}_{s},\rho)$, $1\leq i\leq m$, $v\in V$. If $v\in V$ and $1\leq i\leq m$, then the action of $h\in G$ and the coaction are given by (0.1) $\displaystyle\delta(g_{i}v)=t_{i}\otimes g_{i}v,\qquad h\cdot(g_{i}v)=g_{j}(\gamma\cdot v),$ where $hg_{i}=g_{j}\gamma$, for some $1\leq j\leq m$ and $\gamma\in G^{s}$. Let $\mathfrak{B}({\mathcal{O}}_{s},\rho)$ denote $\mathfrak{B}(M({\mathcal{O}}_{s},\rho))$. By [ZZWC, Lemma 1.1], there exists a bi-one arrow Nichols algebra $\mathfrak{B}(G,r,\overrightarrow{\rho},u)$ such that $\mathfrak{B}({\mathcal{O}}_{s},\rho)\cong\mathfrak{B}(G,r,\overrightarrow{\rho},u)$ as graded braided Hopf algebras in ${}^{kG}_{kG}\\!{\mathcal{Y}D}$. If $D$ is a subgroup of $G$ and $C$ is a congugacy class of $D$, then $C_{G}$ denotes the conjugacy class of $G$ containing $C$. ## 1\. $G=A\rtimes D$ In this section we give the relation between Nichols algebras over group $A\rtimes D$ and group $D$. Let $G=A\rtimes D$ be a semidirect product of abelian group $A$ and group $D$. For any $\chi\in\hat{A},$ let $D_{\chi}:=\\{h\in D\mid h\cdot\chi=\chi\\}$; $G_{\chi}:=A\rtimes D_{\chi}$. For an irreducible representation $\rho$ of $D_{\chi}$, let $\theta_{\chi,\rho}:=(\chi\otimes\rho)\uparrow^{G}_{G_{\chi}}$, the induced representation of $\chi\otimes\rho$ on $G$. By [Se, Pro.25], every irreducible representation of $G$ is of the following form: $\theta_{\chi,\rho}$. Let $\epsilon\in\hat{A}$ with $\epsilon(a)=1$ for any $a\in A.$ Thus $D_{\epsilon}=D$ and $\theta_{\epsilon,\rho}$ is an irreducible representation of $G$. ###### Lemma 1.1. Let $G=A\rtimes D$ and $\sigma\in D.$ Then $G^{\sigma}=A^{\sigma}\rtimes D^{\sigma}.$ Proof. If $x=(a,d)\in G^{\sigma},$ then $x\sigma=\sigma x.$ Thus (1.1) $\displaystyle a=\sigma\cdot a\ \ \hbox{and }$ $\displaystyle d\sigma=\sigma d.$ This implies $d\in D^{\sigma}$ and $a\in A^{\sigma}$ since $\sigma\cdot a=\sigma a\sigma^{-1}$. Conversely, if $x=(a,d)\in A^{\sigma}\rtimes D^{\sigma}$, then (1.1) holds. This implies $x\sigma=\sigma x$ and $x\in G^{\sigma}.$ $\Box$ ###### Lemma 1.2. Let $D$ be a subgroup of $G$ with $\sigma\in D$ and let right coset decompositions of $D^{\sigma}$ in $D$ be (1.2) $\displaystyle D=\bigcup_{\theta\in\Theta}D^{\sigma}g_{\theta}.$ Then there exists a set $\Theta^{\prime}$ with $\Theta\subseteq\Theta^{\prime}$ such that (1.3) $\displaystyle G=\bigcup_{\theta\in\Theta^{\prime}}G^{\sigma}g_{\theta}$ is a right coset decompositions of $G^{\sigma}$ in $G.$ Proof. For any $h,g\in D$, It is clear that $hg^{-1}\in D^{\sigma}$ if and only if $hg^{-1}\in G^{\sigma}$, which prove the claim. $\Box$ ###### Lemma 1.3. If $kQ^{c}(G,r,\overrightarrow{\rho},u)$ is a co-path Hopf algebra (see [ZZC, ZCZ]), then $kG+kQ_{1}=(kG[kQ_{1}^{c}])_{1}$, where $kG[kQ_{1}^{c}]:=kG[kQ_{1}^{c},G,r,\overrightarrow{\rho},u]$ and $(kG[kQ_{1}^{c}])_{1}$ denotes the second term of the coradical filtration of $kG[kQ_{1}^{c}]$. Proof. By [ZCZ, Lemma 2.2], $R:={\rm diag}(kG[kQ_{1}^{c}])$ is a Nichols algebra. [AS98, Lemma 2.5] yields that $kG[kQ_{1}^{c}]$ is coradically graded. $\Box$ ###### Definition 1.4. Let $D$ be a subgroup of $G$; $r$ and $r^{\prime}$ ramifications of $D$ and $G$, respectively. If $r_{C}\leq r_{C_{G}}^{\prime}$ for any $C\in{\mathcal{K}}_{r}(D)$, then $r$ is called a subramification of $r^{\prime}$, written as $r\leq r^{\prime}.$ Furthermore, if $u(C)=u^{\prime}(C_{G})$, $I_{C}(r,u)\subseteq I_{C_{G}}(r^{\prime},u^{\prime})$ and $\rho_{C}^{(i)}$ is isomorphic to a subrepresentation of the restriction of $\rho^{\prime}{}_{C_{G}}^{(i)}$ on $D^{u(C)}$ for any $C\in{\mathcal{K}}_{r}(D)$, $i\in I_{C}(r,u)$, then ${\rm RSR}(D,r,\overrightarrow{\rho},u)$ is called a sub-RSR of ${\rm RSR}(G,r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$, written as ${\rm RSR}(D,r,\overrightarrow{\rho},u)$ $\leq$ ${\rm RSR}(G,r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$. ###### Lemma 1.5. Let $D$ be a subgroup of $G$. If $\sigma\in D$, then ${\rm RSR}(D,{\mathcal{O}}_{\sigma},\rho)\leq{\rm RSR}(G,{\mathcal{O}}_{\sigma},\rho^{\prime})$ if and only if $\rho$ is isomorphic to subrepresentation of the restriction of $\rho^{\prime}$ on $D^{\sigma}$. Proof. It follows from Definition 1.4. $\Box$ ###### Proposition 1.6. Let $D$ be a subgroup of $G$. If ${\rm RSR}(D,r,\overrightarrow{\rho},u)$ $\leq$ ${\rm RSR}(G,r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$, then (i) $kQ^{c}(D,r,\overrightarrow{\rho},u)$ is a Hopf subalgebra $kQ^{\prime}{}^{c}(G,r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$. (ii) $kD[kQ^{c},D,r,\overrightarrow{\rho},u]$ is a Hopf subalgebra $kG[kQ^{\prime}{}^{c},G,r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime}]$. (iii) If $\mathfrak{B}(G,r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ is finite dimensional with finite group $G$ then so is $\mathfrak{B}(D,r,\overrightarrow{\rho},u)$. Proof. (i) For any $C\in{\mathcal{K}}_{r}(D)$ and $i\in I_{C}(r,u)$, let $X_{C}^{(i)}$ be a representation space of $\rho_{C}^{(i)}$ with a basis $\\{x_{C}^{(i,j)}\mid j\in J_{C}(i)\\}$ and $X^{\prime}{}_{C_{G}}^{(i)}$ a representation space of $\rho^{\prime}{}_{C_{G}}^{(i)}$ with a basis $\\{x^{\prime}{}_{C_{G}}^{(i,j)}\mid j\in J_{C_{G}}(i)\\}$ and $J_{C}(i)\subseteq J_{C_{G}}(i).$ $\psi_{C}^{(i)}$ is a $kD^{u(C)}$-module monomorphism from $X_{C}^{(i)}$ to $X^{\prime}{}_{C_{G}}^{(i)}$ with $x^{\prime}{}_{C_{G}}^{(i,j)}=\psi_{C}^{(i)}(x_{C}^{(i,j)})$ for $i\in I_{C}(r,u)$, $j\in J_{C}(i)$. Let $\phi$ be an inclusion map from $kD$ to $kG$ and $\psi$ is a map from $kQ_{1}^{c}(D,r,\overrightarrow{\rho},u)$ to $kQ_{1}^{\prime}{}^{c}(G,r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ by sending $a_{y,x}^{(i,j)}$ to $a^{\prime}{}_{y,x}^{(i,j)}$ for any $y,x\in D$, $i\in I_{C}(r,u)$, $j\in J_{C}(i)$ with $x^{-1}y\in C\in{\mathcal{K}}_{r}(D)$. Now we show that $\psi$ is a $kD$-bimodule homomorphism from $kQ_{1}^{c}$ to ${}_{\phi}(kQ_{1}^{\prime}{}^{c})_{\phi}$ and a $kG$-bicomodule homomorphism from ${}^{\phi}(kQ_{1}^{c})^{\phi}$ to $kQ_{1}^{\prime}{}^{c}$. We only show this about right modules since the others are similar. For any $h\in D$, $C\in{\mathcal{K}}_{r}(D)$, $i\in I_{C}(r,u)$, $j\in J_{C}(i)$, $x,y\in D$ with $x^{-1}y=g_{\theta}^{-1}u(C)g_{\theta}$ and $g_{\theta}h=\zeta_{\theta}(h)g_{\theta^{\prime}}$, $\zeta_{\theta}(h)\in D^{\sigma}$, $\theta,\theta^{\prime}\in\Theta_{C}\subseteq\Theta_{C_{G}}$ (see Lemma 1.2), see $\displaystyle\psi(a_{y,x}^{(i,j)}\cdot h)$ $\displaystyle=$ $\displaystyle\psi(\sum_{s\in J_{C}(i)}k_{C,h}^{(i,j,s)}a^{(i,s)}_{yh,xh})\ \ (\hbox{by \cite[cite]{[\@@bibref{}{ZCZ08}{}{}, Pro.1.2]}})$ $\displaystyle=$ $\displaystyle\sum_{s\in J_{C}(i)}k_{C,h}^{(i,j,s)}a^{\prime}{}^{(i,s)}_{yh,xh}\ \ \hbox{and }$ $\displaystyle\psi(a_{y,x}^{(i,j)})\cdot h$ $\displaystyle=$ $\displaystyle a^{\prime}{}_{y,x}^{(i,j)}\cdot h$ $\displaystyle=$ $\displaystyle\sum_{s\in J_{C_{G}}(i)}k_{C_{G},h}^{(i,j,s)}a^{\prime}{}^{(i,s)}_{yh,xh}\ \ (\hbox{by \cite[cite]{[\@@bibref{}{ZCZ08}{}{}, Pro.1.2]}}),$ where $x_{C}^{(i,j)}\cdot\zeta_{\theta}(h)=\sum_{s\in J_{C}(i)}k_{C,h}^{(i,j,s)}x_{C}^{(i,s)}$, $x^{\prime}{}_{C_{G}}^{(i,j)}\cdot\zeta_{\theta}(h)=\sum_{s\in J_{C_{G}}(i)}k_{C_{G},h}^{(i,j,s)}x^{\prime}{}_{C_{G}}^{(i,s)}$. Since $\displaystyle x^{\prime}{}_{C_{G}}^{(i,j)}\cdot\zeta_{\theta}(h)$ $\displaystyle=$ $\displaystyle\psi_{C}^{(i)}(x_{C}^{(i,j)})\cdot\zeta_{\theta}(h)=\psi_{C}^{(i)}(x_{C}^{(i,j)}\cdot\zeta_{\theta}(h))$ $\displaystyle=$ $\displaystyle\psi_{C}^{(i)}(\sum_{s\in J_{C}(i)}k_{C,h}^{(i,j,s)}x{}_{C}^{(i,s)})=\sum_{s\in J_{C}(i)}k_{C,h}^{(i,j,s)}x^{\prime}{}_{C_{G}}^{(i,s)},$ which implies $\psi(a_{y,x}^{(i,j)}\cdot h)=\psi(a_{y,x}^{(i,j)})\cdot h$. By [ZZC, Lemma 1.5], $T_{kG}^{c}(\phi\pi_{0},\psi\pi_{1}):=\phi\pi_{0}+\sum_{n>0}T_{n}^{c}(\psi\pi_{1})\Delta_{n-1}$ is a graded Hopf algebra map from $T^{c}_{kD}(kQ_{1}^{c})$ to $T_{kG}^{c}(kQ_{1}^{\prime}{}^{c})$. By Lemma 1.3, $(T_{kD}^{c}(kQ_{1}^{c}))_{1}=kD+kQ_{1}^{c}$. Since the restriction of $T_{kG}^{c}(\phi\pi_{0},\psi\pi_{1})$ on $(T_{kD}^{c}(kQ_{1}^{c}))_{1}$ is $\phi+\psi$, we have that $T_{kG}^{c}(\phi\pi_{0},\psi\pi_{1})$ is injective by [Mo93, Theorem 5.3.1]. (ii) It follows from Part (i). (iii). By [ZCZ, Lemma 2.1] $kD[kQ^{c},$ $D,r,$ $\overrightarrow{\rho},$ $u]\cong\mathfrak{B}(D,r,$ $\overrightarrow{\rho},u)\\#kD$ and $kG[kQ^{\prime}{}^{c},G,r^{\prime},$ $\overrightarrow{\rho^{\prime}},u^{\prime}]\cong\mathfrak{B}(G,r^{\prime},$ $\overrightarrow{\rho^{\prime}},u^{\prime})\\#kG$. Applying Part (ii) we complete the proof. $\Box$ The relation $``\leq"$ has the transitivity, i.e. ###### Lemma 1.7. Assume that $G$ is a subgroup of $G^{\prime}$ and $G^{\prime}$ is a subgroup of $G^{\prime\prime}$. If ${\rm RSR}(G,r,\overrightarrow{\rho},u)\leq{\rm RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ and ${\rm RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})\leq{\rm RSR}(G^{\prime\prime},r^{\prime\prime},\overrightarrow{\rho^{\prime\prime}},u^{\prime\prime})$, then ${\rm RSR}(G,r,\overrightarrow{\rho},u)\leq{\rm RSR}(G^{\prime\prime},r,\overrightarrow{\rho^{\prime\prime}},u^{\prime\prime})$. Proof. Obviously, $G$ is a subgroup of $G^{\prime\prime}$ and $r\leq r^{\prime\prime}$. For any $C\in{\mathcal{K}}_{r}(G)$ and $i\in I_{C}(r,u)$, then $u(C)=u^{\prime}(C_{G^{\prime}})=u^{\prime\prime}(C_{G^{\prime\prime}})$. Let $s=u(C)$ and let $X_{C}^{(i)}$, $X^{\prime}{}_{C_{G^{\prime}}}^{(i)}$ and $X^{\prime\prime}{}_{C_{G^{\prime\prime}}}^{(i)}$ be representation spaces of $\rho_{C}^{(i)}$, $\rho^{\prime}{}_{C_{G^{\prime}}}^{(i)}$ and $\rho^{\prime\prime}{}_{C_{G^{\prime\prime}}}^{(i)}$, respectively. Let $(X_{C}^{(i)},\rho_{C}^{(i)})$ be isomorphic to a subrepresentation $(N,\rho^{\prime}{}_{C_{G^{\prime}}}^{(i)}\mid_{G^{s}})$ of $(X^{\prime}{}_{C_{G^{\prime}}}^{(i)},\rho^{\prime}{}_{C_{G^{\prime}}}^{(i)}\mid_{G^{s}})$. Considering $(X^{\prime}{}_{C_{G^{\prime}}}^{(i)},\rho^{\prime}{}_{C_{G^{\prime}}}^{(i)})$ is isomorphic to a subrepresentation of $(X^{\prime\prime}{}_{C_{G^{\prime\prime}}}^{(i)},\rho^{\prime\prime}{}_{C_{G^{\prime\prime}}}^{(i)}\mid_{G^{\prime}{}^{s}})$, we have that $(N,\rho^{\prime}{}_{C_{G^{\prime}}}^{(i)}\mid_{G^{s}})$ is isomorphic to a subrepresentation of $(X^{\prime\prime}{}_{C_{G^{\prime\prime}}}^{(i)},\rho^{\prime\prime}{}_{C_{G^{\prime\prime}}}^{(i)}\mid_{G{}^{s}})$. Consequently, $(X_{C}^{(i)},\rho_{C}^{(i)})$ is isomorphic to a subrepresentation of $(X^{\prime\prime}{}_{C_{G^{\prime\prime}}}^{(i)},\rho^{\prime\prime}{}_{C_{G^{\prime\prime}}}^{(i)}\mid_{G{}^{s}})$. $\Box$ ###### Lemma 1.8. Let $N$ be a subgroup of $G$ and $(X,\rho)$ be an irreducible representation of $N^{\sigma}$ with $\sigma\in N$. If the induced representation $\rho^{\prime}:=\rho\uparrow_{N^{\sigma}}^{G^{\sigma}}$ is an irreducible representation of $G^{\sigma}$, then ${\rm RSR}(N,{\mathcal{O}}_{\sigma},\rho)\leq{\rm RSR}(G,{\mathcal{O}}_{\sigma},\rho\uparrow_{N^{\sigma}}^{G^{\sigma}})$. Proof. Since $(X,\rho)\cong(X\otimes_{kN^{\sigma}}1,\rho^{\prime}\mid_{kN^{\sigma}})$ by sending $x$ to $x\otimes 1$ for any $x\in X$, the claim holds. $\Box$ ###### Proposition 1.9. Let $G=A\rtimes D$ with $\sigma\in D$, $\chi\in\widehat{A^{\sigma}},$ $G_{\chi}=A\rtimes D_{\chi}$, $\rho\in\widehat{D_{\chi}^{\sigma}}$ and $\theta_{\chi,\rho}:=(\chi\otimes\rho)\uparrow_{G_{\chi}^{\sigma}}^{G^{\sigma}}.$ Then (i) ${\rm RSR}(G_{\chi},{\mathcal{O}}_{\sigma},\chi\otimes\rho)\leq{\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi,\rho})$ (ii) ${\rm RSR}(G_{\chi},{\mathcal{O}}_{\sigma},\chi\otimes\rho)$ is $-1$-type if and only if ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi,\rho})$ is $-1$-type. (iii) ${\rm RSR}(D_{\chi},{\mathcal{O}}_{\sigma},\rho)\leq{\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi,\rho})$. (iv) ${\rm RSR}(D_{\chi},{\mathcal{O}}_{\sigma},\rho)$ is $-1$-type if and only if ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi,\rho})$ is $-1$-type. Proof. (i) It follows from Lemma 1.8. (ii) Let $P$ and $X$ be representation spaces of $\chi$ and $\rho$, respectively. Then $(P\otimes X)\otimes_{kG_{\chi}^{\sigma}}kG^{\sigma}$ is a representation space of $\rho^{\prime}:=\theta_{\chi,\rho}$. If $\rho(\sigma)=-id$, then for any $g\in G^{\sigma}$, $x\in X$, $p\in P,$ we have $((p\otimes x)\otimes_{kG_{\chi}^{\sigma}}g)\cdot\sigma=((p\otimes x)\cdot\sigma)\otimes_{kG_{\chi}^{\sigma}}g=-(p\otimes x)\otimes_{kG_{\chi}^{\sigma}}g$. Therefore $\rho^{\prime}(\sigma)=-id$. Conversely, if $\rho^{\prime}(\sigma)=-id$, then $((p\otimes x)\otimes_{kG_{\chi}^{\sigma}}1)\cdot\sigma=((p\otimes x)\cdot\sigma)\otimes_{kG_{\chi}^{\sigma}}1=-(p\otimes x)\otimes_{kG_{\chi}^{\sigma}}1$. Therefore $(p\otimes x)\cdot\sigma=-p\otimes x$ for any $x\in X.$ (iii) By (i), it is enough to show ${\rm RSR}(D_{\chi},{\mathcal{O}}_{\sigma},\rho)\leq{\rm RSR}(G_{\chi},{\mathcal{O}}_{\sigma},\chi\otimes\rho)$. Let $P$ and $X$ be the representation spaces of $\chi$ and $\rho$ on $A^{\sigma}$ and $D^{\sigma}_{\chi}$, respectively. Thus $(P\otimes X,\chi\otimes\rho)$ is an irreducible representation of $G^{\sigma}_{\chi}:=A^{\sigma}\rtimes(D^{\sigma})_{\chi}$. Considering Definition 1.4 we only need to show that $\rho$ is isomorphic to a submodule of the restriction of $\chi\otimes\rho$ on $D_{\chi}^{\sigma}$. Fix a nonzero $p\in P$ and define a map $\psi$ from $X$ to $P\otimes X$ by sending $x$ to $p\otimes x$ for any $x\in X$. It is clear that $\psi$ is a $kD^{\sigma}_{\chi}$-module isomorphism. (iv) Considering Part (ii), we only show that ${\rm RSR}(D_{\chi},{\mathcal{O}}_{\sigma},\rho)$ is $-1$-type if and only if ${\rm RSR}(G_{\chi},{\mathcal{O}}_{\sigma},\chi\otimes\rho)$ is $-1$-type. Since $\chi_{\rho}(\sigma)=\chi_{(\chi\otimes\rho)}(\sigma)$, the claim holds. $\Box$ If $D_{\chi}=D$, then it follows from the proposition above that ${\rm RSR}(D,{\mathcal{O}}_{\sigma},\rho)\leq{\rm RSR}(G,{\mathcal{O}}_{\sigma},$ $\theta_{\chi,\rho})$. Therefore we have ###### Corollary 1.10. Let $G=A\rtimes D.$ If $r\leq r^{\prime}$ and $\rho^{{}^{\prime}}{}^{(i)}_{C_{G}}=\theta_{\chi_{C}^{(i)},\rho_{C}^{(i)}}$ with $D_{\chi_{C}^{(i)}}=D$, $\chi_{C}^{(i)}\in\widehat{A^{u(C)}}$, $u(C)=u^{\prime}(C_{G})$ and $I_{C}(r,u)\subseteq I_{C_{G}}(r^{\prime},u^{\prime})$ for any $i\in I_{C}(r,u)$, $C\in{\mathcal{K}}_{r}(D)$, then ${\rm RSR}(D,r,$ $\overrightarrow{\rho},u)$ $\leq$ ${\rm RSR}(G,r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$. Furthermore, if $I_{C}(r,u)=I_{C_{G}}(r^{\prime},u^{\prime})$ for any $C\in{\mathcal{K}}_{r}(D)$ and ${\mathcal{K}}_{r^{\prime}}(G)=\\{C_{G}\mid C\in{\mathcal{K}}_{r}(D)\\}$, then ${\rm RSR}(D,r,\overrightarrow{\rho},u)$ is $-1$-type if and only if ${\rm RSR}(G,r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ is $-1$-type. ###### Lemma 1.11. Let $G=G_{1}\times G_{2}$. If $\sigma=(\sigma_{1},\sigma_{2})\in G$ with $\rho_{1}\in\widehat{G_{1}^{\sigma_{1}}}$ and $\rho_{2}\in\widehat{G_{2}^{\sigma_{2}}}$, then (i) $G^{\sigma}=G_{1}^{\sigma}\times G_{2}^{\sigma}$; $G_{1}^{\sigma}=G_{1}^{\sigma_{1}}$ and $G_{2}^{\sigma}=G_{2}^{\sigma_{2}}$. (ii) ${\mathcal{O}}_{\sigma}^{G}={\mathcal{O}}_{\sigma_{1}}^{G_{1}}\times{\mathcal{O}}_{\sigma_{2}}^{G_{2}}$, where ${\mathcal{O}}_{\sigma}^{G}$ denotes the conjugacy class containing $\sigma$ of $G$. (iii) ${\rm RSR}(G_{1},{\mathcal{O}}_{\sigma_{1}},\rho_{1})\leq{\rm RSR}(G,{\mathcal{O}}_{\sigma_{1}},\rho_{1}\otimes\rho_{2})$ when $\sigma_{2}=1$; ${\rm RSR}(G_{2},{\mathcal{O}}_{\sigma_{2}},\rho_{2})\leq{\rm RSR}(G,{\mathcal{O}}_{\sigma_{2}},\rho_{1}\otimes\rho_{2})$ when $\sigma_{1}=1$. Proof. (i) It is clear $G_{1}^{\sigma}=G_{1}^{\sigma_{1}}$ and $G_{2}^{\sigma}=G_{2}^{\sigma_{2}}$. For any $x=(a,h)\in G^{\sigma}$, then $x\sigma=\sigma x$, which implies that $a\sigma_{1}=\sigma_{1}a$ and $h\sigma_{2}=\sigma_{2}h.$ Thus $x\in G_{1}^{\sigma}\times G_{2}^{\sigma}$ and $G^{\sigma}\subseteq G_{1}^{\sigma}\times G_{2}^{\sigma}$. Similarly, we have $G_{1}^{\sigma}\times G_{2}^{\sigma}\subseteq G^{\sigma}$. (ii) It is clear. (iii) We only show the first claim. It is clear that $\rho_{1}$ is isomorphic to a subrepresentation of the restriction of $\rho_{1}\otimes\rho_{2}$ on the $G_{1}^{\sigma_{1}}$. Indeed, assume that $X$ and $Y$ are the representation spaces of $\rho_{1}$ and $\rho_{2}$, respectively. Obviously, $G_{1}^{\sigma_{1}}$-module $(X,\rho_{1})$ is isomorphic to a submodule of the restriction of $\rho_{1}\otimes\rho_{2}$ on $G_{1}^{\sigma_{1}}$ under isomorphism $\psi$ form $X$ to $X\otimes y_{0}$ by sending $x$ to $x\otimes y_{0}$ for any $x\in X$, where $y_{0}$ is a non-zero fixed element in $Y$. $\Box$ ###### Lemma 1.12. Let $G=G_{1}\times G_{2}$ and $\sigma=(\sigma_{1},\sigma_{2})\in G$ with $\rho_{1}\in\widehat{G_{1}^{\sigma_{1}}}$ and $\rho_{2}\in\widehat{G_{2}^{\sigma_{2}}}$. (i) If $\sigma_{2}=1$, then ${\rm RSR}(G,{\mathcal{O}}_{\sigma_{1}},\rho_{1}\otimes\rho_{2})$ is $-1$-type if and only if ${\rm RSR}(G_{1},{\mathcal{O}}_{\sigma_{1}},\rho_{1})$ is $-1$-type. (ii) If $\sigma_{1}=1$, then ${\rm RSR}(G,{\mathcal{O}}_{\sigma_{2}},\rho_{1}\otimes\rho_{2})$ is $-1$-type if and only if ${\rm RSR}(G_{2},{\mathcal{O}}_{\sigma_{2}},\rho_{2})$ is $-1$-type. Proof. (i) Considering $\chi_{\rho_{1}\otimes\rho_{2}}(\sigma_{1})=\chi_{\rho_{1}}(\sigma_{1}){\rm deg}(\rho_{2})$, we can complete the proof. (ii) It is similar. $\Box$ ###### Lemma 1.13. $\theta_{\chi,\rho}$ is a one dimensional representation of $G^{\sigma}=A^{\sigma}\rtimes D^{\sigma}$ if and only if $D_{\chi}^{\sigma}=D^{\sigma}$ and ${\rm deg}\rho=1$ Proof. Let $P$ and $X$ be the representation spaces of $\chi$ and $\rho$ on $A^{\sigma}$ and $D^{\sigma}_{\chi}$, respectively. $((P\otimes X)\otimes_{kG_{\chi}^{\sigma}}kG^{\sigma},\theta_{\chi,\rho})$ is a one dimensional representation of $G^{\sigma}=A^{\sigma}\rtimes D^{\sigma}$ if and only if $kG^{\sigma}=kG^{\sigma}_{\chi}$ and ${\rm dim}X=1$. However. $kG^{\sigma}=kG^{\sigma}_{\chi}$ if and only if $D_{\chi}^{\sigma}=D^{\sigma}$. $\Box$ Consequently, $\theta_{\chi,\rho}=\chi\otimes\rho$ when $\theta_{\chi,\rho}$ is one dimensional representation. ## 2\. Symmetric group $\mathbb{S}_{n}$ In this section we study the Nichols algebras over symmetric groups. Without specification, $\sigma\in\mathbb{S}_{n}$ is always of type $1^{\lambda_{1}}2^{\lambda_{2}}\cdots n^{\lambda_{n}}$. $g_{j}$ denotes the generator of cycle group $C_{j}$ with order $j$ for natural number $j$. We keep on the work in [Su78, Page 295-299 ]. Let $r_{j}:=\sum_{1\leq k\leq j-1}k\lambda_{k}$ and $\sigma_{j}:=\prod_{1\leq l\leq\lambda_{j}}$ $\Big{(}y_{r_{j}+(l-1)j+1},$ $\qquad y_{r_{j}+(l-1)j+2},\quad\cdots,\quad y_{r_{j}+lj}\Big{)}$, the multiplication of cycles of length $j$ in the independent cycle decomposition of $\sigma$, as well as $Y_{j}:=\\{y_{s}\mid s=r_{j}+1,\cdots,r_{j+1}\\}$. Therefore $\sigma=\prod\sigma_{i}$ and $({\mathbb{S}}_{n})^{\sigma}=\prod({\mathbb{S}}_{Y_{i}})^{\sigma_{i}}$ $=T_{1}\times\cdots\times T_{n}.$ It follows from [AFZ, subsection 2.2] that $T_{j}$ is generated by $A_{1,j},\dots,A_{\lambda_{j},j},B_{1,j},\dots,B_{\lambda_{j}-1,j}$, where $A_{1,1}=(y_{1}),\dots,A_{\lambda_{1},1}=(y_{\lambda_{1}})$, $A_{1,2}=(y_{\lambda_{1}+1}\,\,\,y_{\lambda_{1}+2})$,…, $A_{\lambda_{2},2}=(y_{\lambda_{1}+2\lambda_{2}-1}\,\,\,y_{\lambda_{1}+2\lambda_{2}})$, and so on. More precisely, if $1<j\leq n$, then $\displaystyle A_{l,j}$ $\displaystyle:=\Big{(}y_{r_{j}+(l-1)j+1},\qquad y_{r_{j}+(l-1)j+2},\quad\cdots,\quad y_{r_{j}+lj}\Big{)},$ $\displaystyle B_{h,j}$ $\displaystyle:=\Big{(}y_{r_{j}+(h-1)j+1},\quad y_{r_{j}+hj+1},\Big{)}\Big{(}y_{r_{j}+(h-1)j+2},\quad y_{r_{j}+hj+2}\Big{)}\cdots\Big{(}y_{r_{j}+hj},\quad y_{r_{j}+(h+1)j}\Big{)},$ for all $l$, $h$, with $1\leq l\leq\lambda_{j}$, $1\leq h\leq\lambda_{j}-1$. Notice that $\varphi(A_{l,j})=\big{(}\stackrel{{\scriptstyle l}}{{\overbrace{(1,\cdots,1,g_{j}^{j-1}}}},1\cdots,1),1\big{)}$ and $\varphi(B_{h,j})=\big{(}1,(h,h+1)\big{)}$, where $\varphi$ is an isomorphism from $G^{\sigma_{j}}$ to $(C_{j})^{\lambda_{j}}\rtimes$ $\mathbb{S}_{\lambda_{j}}$, defined in the proof of [ZWW, Pro. 2.10 ] (also see the below (2.1)). Furthermore, if $\cup_{i>1}Y_{i}\subseteq X\subseteq\\{1,2,\cdots,n\\}$, then $\sigma$ is said to be in $\mathbb{S}_{X}$. ###### Lemma 2.1. If $\sigma\in\mathbb{S}_{n}$ is the multiplication of $m$ independent cycles with the same length $l$, i.e. $\sigma$ is of type $l^{m}$, then (i) $\varphi(\sigma)=((g_{l}^{l-1},g_{l}^{l-1},\cdots,g_{l}^{l-1}),(1))$, where $\varphi$ is the isomorphism from $(\mathbb{S}_{n})^{\sigma}$ to $(C_{l})^{m}\rtimes$ $\mathbb{S}_{m}$ defined in the proof of [ZWW, Pro. 2.10 ]. (ii) $\theta_{\chi,\rho}(\varphi(\sigma))=$ $\chi((g_{l}^{l-1},g_{l}^{l-1},\cdots,g_{l}^{l-1}))\ {\rm id}$ for any $\rho\in\widehat{(\mathbb{S}_{m})_{\chi}}$ and $\chi\in\widehat{(C_{l})^{m}}$, where $G=(C_{l})^{m}\rtimes$ $\mathbb{S}_{m}$, $G_{\chi}=(C_{l})^{m}\rtimes$ $\mathbb{(}S_{m})_{\chi}$ and $\theta_{\chi,\rho}=(\chi\otimes\rho)\uparrow_{G_{\chi}}^{G}.$ Proof. (i) Assume that $\sigma=(a_{10}a_{11}\cdots a_{1,l-1})(a_{20}\cdots a_{2,l-1})\cdots(a_{m0}\cdots a_{m,l-1}).$ By [ZWW, Pro. 2.14 ] or [Su78], $\mathbb{S}_{n}^{\sigma}\stackrel{{\scriptstyle\varphi}}{{\cong}}(C_{l})^{m}\rtimes\mathbb{\mathbb{missing}}{S}_{m}$ where the map $\varphi$ is the same as in the proof of [ZWW, Pro. 2.10 ]. Indeed, here is precise definition of isomorphism $\varphi$. For any element $\tau$ of $({\mathbb{S}}_{n})^{\sigma}$, we will define $\theta(\tau)\in S_{m}$ and $f_{\tau}\in(C_{l})^{m}$ by $\tau^{-1}(a_{i0})=a_{jk},\qquad j=\theta(\tau)^{-1}(i),\qquad f_{\tau}(i)=g_{l}^{k},$ where $g_{l}$ is the generator of $C_{l}$, $1\leq i\leq m$. Let (2.1) $\displaystyle\varphi(\tau)=(f_{\tau},\theta(\tau)).$ Since $\sigma(a_{i,l-1})=a_{i,0}$, we have $\varphi(\sigma)=(f_{\sigma},\theta(\sigma))$ with $f_{\sigma}=(g_{l}^{l-1},g_{l}^{l-1},\cdots,g_{l}^{l-1})\in(C_{l})^{m}$ and $\theta(\sigma)=(1)\in\mathbb{S}_{m}$. (ii) Let $P$ and $X$ be representation spaces of $\chi$ and $\rho$, respectively. For any $0\not=p\in P$ and $0\not=x\in X$, see $\displaystyle((p\otimes x)\otimes_{kG_{\chi}}1)\cdot\varphi(\sigma)$ $\displaystyle=$ $\displaystyle((p\otimes x)\cdot\varphi(\sigma))\otimes_{kG_{\chi}}1$ $\displaystyle=$ $\displaystyle\chi((g_{l}^{l-1},g_{l}^{l-1},\cdots,g_{l}^{l-1}))((p\otimes x)\otimes_{kG_{\chi}}1)\ \ (\hbox{by Part (i)}).$ Since $\varphi(\sigma)$ is in the center of $C_{l}^{m}\rtimes\mathbb{S}_{m}$ and $\theta_{\chi,\rho}$ is irreducible, we have that Part (ii) holds. $\Box$ Obviously, every element in $\widehat{(C_{l})^{m}}$ can be denoted by $\chi_{(t_{1,l},t_{2,l},\cdots,t_{m,l};l)}:=\chi_{l}^{t_{1,l}}\otimes\chi_{l}^{t_{2,l}}\otimes\cdots\otimes\chi_{l}^{t_{m,l}}$ for $0\leq t_{j,l}\leq l-1$. For convenience, we denote $\chi_{(t_{1,l},t_{2,l},\cdots,t_{m,l};l)}$ by $\chi^{\bf t}$ when it does not cause mistake. ###### Lemma 2.2. (i) Every irreducible representation of $\mathbb{S}_{n}^{\sigma}=\prod_{1\leq i\leq n}\mathbb{S}_{Y_{i}}^{\sigma_{i}}$ is isomorphic to one of the following list: (2.2) $\displaystyle\otimes_{1\leq i\leq n}(\theta_{\chi^{{\bf t}_{i}},\rho_{i}}\varphi_{i}),$ where $\rho_{i}\in\widehat{(\mathbb{S}_{Y_{i}})_{\chi^{{\bf t}_{i}}}}$ and $\varphi_{i}$ is the isomorphism from $\mathbb{S}_{Y_{i}}^{\sigma_{i}}$ to $(C_{i})^{\lambda_{i}}\rtimes\mathbb{S}_{\lambda_{i}}$ as in (2.1). (ii) Let $\chi$ denote the character of $\otimes_{1\leq i\leq n}(\theta_{\chi^{{\bf t}_{i}},\rho_{i}}\varphi_{i})$. Then (2.3) $\displaystyle\chi(\sigma)=\prod_{1\leq i\leq n}(\prod_{1\leq j\leq\lambda_{i}}\chi_{i}(g_{i})^{(i-1)t_{j,i}}){\rm deg}(\theta_{\chi^{{\bf t}_{i}},\rho_{i}}\varphi_{i}).$ Proof. (i) By [ZWW, Pro. 2.10], $\mathbb{S}_{n}^{\sigma}=\prod_{1\leq i\leq n}\mathbb{S}_{Y_{i}}^{\sigma_{i}}\stackrel{{\scriptstyle\prod\varphi_{i}}}{{\cong}}\prod_{1\leq i\leq n}(C_{i})^{\lambda_{i}}\rtimes\mathbb{S}_{\lambda_{i}}$. It follows from [Se, Pro.25] that every irreducible representation of $(C_{i})^{\lambda_{i}}\rtimes\mathbb{S}_{\lambda_{i}}$ is isomorphic to $\theta_{\chi^{{\bf t}_{i}},\rho_{i}}$. This completes our proof. (ii) It follows from (i) and Lemma 2.1. $\Box$ ###### Definition 2.3. (2.4) $\displaystyle\xi_{{\bf t},\sigma}:=\sum_{1\leq k\leq n,1\leq j\leq\lambda_{k}}\frac{t_{j,k}}{k}+\frac{1}{2}$ is called the distinguished element of the representation (2.2) or ${\rm RSR}(\mathbb{S}_{n},{\mathcal{O}}_{\sigma},\otimes_{1\leq i\leq n}(\theta_{\chi^{{\bf t}_{i}},\rho_{i}}\varphi_{i}))$, where $0\leq t_{j,k}\leq k-1$. ###### Lemma 2.4. Let $\chi$ denote the character of $(\otimes_{1\leq k\leq n}(\theta_{\chi^{{\bf t}_{k}},\rho_{k}}\varphi_{k}))$. Then $\xi_{{\bf t},\sigma}$ is an integer if and only if $\chi(\sigma)=-{\rm deg}(\chi)$. Proof. Let $\omega_{j}:=e^{\frac{2\pi i}{j}}$, where $i:=\sqrt{-1}$ and $e$ is the Euler’s constant. For any $k$ with $\lambda_{k}\not=0$, since $(k,k-1)=1$, there exists $a_{k}$ with $(a_{k},k)=1$ such that $\omega_{k}^{a_{k}(k-1)}=\omega_{k}.$ Choice $\chi_{k}$ such that $\chi_{k}(g_{k})=\omega_{k}^{a_{k}}$. By formula (2.3), we have (2.5) $\displaystyle\chi(\sigma)$ $\displaystyle=$ $\displaystyle e^{(2\pi i)\sum_{1\leq k\leq n,1\leq j\leq\lambda_{k}}\frac{t_{j,k}}{k}}{\rm deg}(\otimes_{1\leq s\leq n}(\theta_{\chi^{{\bf t}_{s}},\rho_{s}}\varphi_{s})).$ Using the formula we complete the proof. $\Box$ Consequently, we have ###### Proposition 2.5. The distinguished element $\xi_{{\bf t},\sigma}$ of the representation $\otimes_{1\leq i\leq n}(\theta_{\chi^{{\bf t}_{i}},\rho_{i}}\varphi_{i})$, as in (2.2) is an integer and the order of $\sigma$ is even if and only if $\mathfrak{B}(,{\mathcal{O}}_{\sigma}^{\mathbb{S}_{n}},\otimes_{1\leq i\leq n}(\theta_{\chi^{{\bf t}_{i}},\rho_{i}}\varphi_{i}))$ is $-1$-type. ###### Lemma 2.6. If $\chi\in\widehat{(C_{l})^{m}}$, then $(\mathbb{S}_{m})_{\chi}=\mathbb{S}_{m}$ if and only if $\chi=\chi_{l}^{t}\otimes\chi_{l}^{t}\otimes\cdots\otimes\chi_{l}^{t}$ for some $0\leq t\leq l-1.$ Proof. Note that $\mathbb{S}_{m}$ acts $(C_{l})^{m}$ as follows: For any $a\in C_{l}^{m}$ with $a=(g_{l}^{a_{1}},g_{l}^{a_{2}},\cdots,g_{l}^{a_{m}})$ and $h\in\mathbb{S}_{m}$, (2.6) $\displaystyle h\cdot a=(g_{l}^{a_{h^{-1}(1)}},g_{l}^{a_{h^{-1}(2)}},\cdots,g_{l}^{a_{h^{-1}(m)}}).$ Let $\chi=\otimes_{i=1}^{m}\chi_{l}^{t_{i}}$ with $0\leq t_{i}\leq l-1$. If $(\mathbb{S}_{m})_{\chi}=\mathbb{S}_{m}$ and there exist $i\not=j$ such that $t_{i}\not=t_{j}$. Set $a=(g_{l}^{a_{1}},g_{l}^{a_{2}},\cdots,g_{l}^{a_{m}})\in C_{l}^{m}$ with $a_{i}=1$ and $a_{s}=0$ when $s\not=i$; $h=(i,j)\in\mathbb{S}_{m}$. See $\displaystyle(h\cdot\chi)(a)$ $\displaystyle=$ $\displaystyle\chi(h^{-1}\cdot a)=\chi(g_{l}^{a_{h(1)}},g_{l}^{a_{h(2)}},\cdots,g_{l}^{a_{h(m)}})$ $\displaystyle=$ $\displaystyle\chi_{l}(g_{l})^{t_{i}a_{j}+t_{j}a_{i}}=\chi_{l}(g_{l})^{t_{j}}$ and $\chi(a)=\chi_{l}(g_{l})^{t_{i}}$. This implies $\chi\not=(h\cdot\chi)$. We get a contradiction. Conversely, it is clear. $\Box$ ###### Proposition 2.7. Every one dimensional representation of $\mathbb{S}_{n}^{\sigma}$ is of the following form: (2.7) $\displaystyle\otimes_{1\leq i\leq n}({\chi_{(t_{i},\cdots,t_{i};i)}\otimes\rho_{i}})\varphi_{i},$ where $1\leq t_{i}\leq i-1$ and $\rho_{i}$ is a one dimensional representation for any $1\leq i\leq n$. Proof. By Lemma 2.2, every irreducible representation of $\mathbb{S}_{n}^{\sigma}$ is of form as (2.2). If (2.2) is one dimensional, then it follows from Lemma 1.13 that $\otimes_{1\leq i\leq n}(\theta_{\chi^{{\bf t}_{i}},\rho_{i}}\varphi_{i})=\otimes_{1\leq i\leq n}({\chi^{{\bf t}_{i}}\otimes\rho_{i}})\varphi_{i}$ and $(\mathbb{S}_{\lambda_{i}})_{\chi^{{\bf t}_{i}}}=\mathbb{S}_{\lambda_{i}}$. By Lemma 2.5, $\chi^{{\bf t}_{i}}=\chi_{i}^{t_{i}}\otimes\chi_{i}^{t_{i}}\otimes\cdots\otimes\chi_{i}^{t_{i}}$ $=\chi_{(t_{i},\cdots,t_{i};i)}$ for some $0\leq t_{i}\leq i-1$. $\Box$ Note that every one dimensional representation of $\mathbb{S}_{m}$ is $\chi_{2}$ or $\epsilon$. Therefore every one dimensional representation of $\mathbb{S}_{m}^{\sigma}$ can be denoted by $\otimes_{1\leq i\leq n}(\chi_{(t_{i},\cdots,t_{i};i)}\otimes\chi_{2}^{\delta_{i}})$ in short, where $\delta_{i}=1$ or $0$. Consequently, the distinguished element of one dimensional representation (2.7) becomes (2.8) $\displaystyle\xi_{{\bf t},\sigma}=\sum_{1\leq k\leq n}\frac{t_{k}\lambda_{k}}{k}+\frac{1}{2}\ \ \hbox{with }\ 0\leq t_{k}\leq k-1.$ For any $\mu=\otimes_{1\leq i\leq n}(\theta_{\chi^{{\bf t}_{i}},\rho_{i}}\varphi_{i})\in\widehat{\mathbb{S}_{n}^{\sigma}}$ as Lemma 2.2, let $\mu_{j}:=\theta_{\chi^{{\bf t}_{i}},\rho_{i}}\varphi_{i}$ be a representation of ${\mathbb{S}}_{Y_{i}}^{\sigma_{i}}$ and $\mu=\prod\limits_{1\leq j\leq n}\mu_{j}.$ We often omit $\varphi_{i}.$ ###### Proposition 2.8. Let $\sigma\in\mathbb{S}_{n}$ be of type $(1^{\lambda_{1}},2^{\lambda_{2}},\dots,n^{\lambda_{n}})$ and $\mu=\otimes_{1\leq i\leq n}\mu_{i}$ with $\mu_{i}:=(\theta_{\chi^{{\bf t}_{i}},\rho_{i}}\varphi_{i})$ as in (2.2). Then $\mathfrak{B}({\mathcal{O}}_{\sigma}^{{\mathbb{S}}_{n}},\mu)$ is $-1$-type in the following cases. 1. (i) $(1^{\lambda_{1}},2)$, $\mu_{1}={\rm sgn}$ or $\epsilon$, $\mu_{2}=\chi_{(1;2)}$. 2. (ii) $(2,\sigma_{o})$, $\sigma_{o}:=\prod\limits_{1\leq i\leq n,1<i\hbox{ is odd}}\sigma_{i}$ $\neq{\rm id}$, $\mu_{2}=\chi_{(1;2)}$, $\mu_{j}=(\chi_{(0,\dots,0;j)}\otimes\rho_{j})\uparrow_{({\mathbb{S}}_{Y_{j}})_{\chi_{(0,\dots,0;j)}}^{\sigma_{j}}}^{({\mathbb{S}}_{Y_{j}})^{\sigma_{j}}}$, for all odd $j>1$. 3. (iii) $(1^{\lambda_{1}},2^{3})$, $\mu_{1}={\rm sgn}$ or $\epsilon$, $\mu_{2}=\chi_{(1,1,1;2)}\otimes\epsilon$ or $\chi_{(1,1,1;2)}\otimes{\rm sgn}$. Furthermore, if $\lambda_{1}>0$, then $\mu_{2}=\chi_{(1,1,1;2)}\otimes{\rm sgn}$. 4. (iv) $(2^{5})$, $\mu_{2}=\chi_{(1,1,1,1,1;2)}\otimes\epsilon$ or $\chi_{(1,1,1,1,1;2)}\otimes{\rm sgn}$. 5. (v) $(1^{\lambda_{1}},4)$, $\mu_{1}={\rm sgn}$ or $\epsilon$, $\mu_{4}=\chi_{(2;4)}$. 6. (vi) $(1^{\lambda_{1}},4^{2})$, $\mu_{1}={\rm sgn}$ or $\epsilon$, $\mu_{4}=\chi_{(1,1;4)}\otimes{\rm sgn}$ or $\chi_{(3,3;4)}\otimes{\rm sgn}$. 7. (vii) $(2,4)$, $\mu_{2}=\chi_{(1;2)}$ and $\mu_{4}=\epsilon$ or $\mu_{2}=\epsilon$ and $\mu_{4}=\chi_{(2;4)}$. 8. (viii) $(2,4^{2})$, $\mu_{2}=\epsilon$, $\mu_{4}=\chi_{(1,1;4)}\otimes{\rm sgn}$ or $\chi_{(3,3;4)}\otimes{\rm sgn}$. 9. (ix) $(2^{2},4)$, $\deg\mu_{2}=1$, $\mu_{4}=\chi_{(2;4)}$. Proof. It is sufficient to show that their distinguished element $\xi_{{\bf t},\sigma}$, defined in Definition 2.3, is an integer by Lemma 2.4. (i) $\xi_{{\bf t},\sigma}=\frac{1}{2}+\frac{1}{2}=1$. (ii) $\xi_{{\bf t},\sigma}=\frac{1}{2}+\frac{1}{2}=1$. (iii) $\xi_{{\bf t},\sigma}=\frac{3}{2}+\frac{1}{2}=2$. (iv) $\xi_{{\bf t},\sigma}=\frac{5}{2}+\frac{1}{2}=3$. (v) $\xi_{{\bf t},\sigma}=\frac{2}{4}+\frac{1}{2}=1$. (vi) $\xi_{{\bf t},\sigma}=\frac{1}{4}+\frac{1}{4}+\frac{1}{2}=1$ or $\xi_{{\bf t},\sigma}=\frac{3}{4}+\frac{3}{4}+\frac{1}{2}=2$. (vii) $\xi_{{\bf t},\sigma}=\frac{1}{2}+\frac{1}{2}=1$ or $\xi_{{\bf t},\sigma}=\frac{2}{4}+\frac{1}{2}=1$. (viii) $\xi_{{\bf t},\sigma}=\frac{2}{4}+\frac{1}{2}=1$ or $\xi_{{\bf t},\sigma}=\frac{6}{4}+\frac{1}{2}=2$. (ix) Assume $\mu_{2}=\chi_{(t,t;2)}\otimes\rho_{2}.$ Thus $\xi_{{\bf t},\sigma}=\frac{2t}{2}+\frac{2}{4}+\frac{1}{2}=1+t$. We only show this for case (vi) since others is similar. Therefore the distinguished element $\xi_{{\bf t},\sigma}$ is an integer. $\Box$ In fact, it follows from [AFZ, Theorem 1] that if ${\rm dim}\mathfrak{B}{\mathcal{O}}_{\sigma}^{\mathbb{S}_{n}},\mu)<\infty$ then some of the case (i)–(ix) in proposition above hold. ## 3\. The classical Weyl groups By [Ca72], $(C_{2})^{n}\rtimes\mathbb{S}_{n}$ is isomorphic to the Weyl groups of $B_{n}$ and $C_{n}$ with $n>2$. Obviously, when $A=\\{a\in(C_{2})^{n}\mid\ a=(g_{2}^{a_{1}},g_{2}^{a_{2}},\cdots,g_{2}^{a_{n}})$ $\hbox{ with all }a_{i}=0\\}$, $A\rtimes\mathbb{S}_{n}$ is isomorphic to the Weyl groups of $A_{n-1}$ with $n>1$. The Weyl group $W(D_{n})$ of $D_{n}$ is a subgroup of $W(B_{n})$, Weyl groups of $B_{n}$. Without specification, $A\subseteq(C_{2})^{n}$, $\mathbb{S}_{n}\cdot A\subseteq A$ and $G=A\rtimes\mathbb{S}_{n}$ with $\sigma\in\mathbb{S}_{n}$. Note that $\mathbb{S}_{n}$ acts $A$ as follows: for any $a\in A$ with $a=(g_{2}^{a_{1}},g_{2}^{a_{2}},\cdots,g_{2}^{a_{n}})$ and $h\in\mathbb{S}_{n}$, define (3.1) $\displaystyle h\cdot a:=(g_{2}^{a_{h^{-1}(1)}},g_{2}^{a_{h^{-1}(2)}},\cdots,g_{2}^{a_{h^{-1}(n)}}).$ Let $G=A\rtimes\mathbb{S}_{n}$. $(a,\sigma)\in G$ is called a sign cycle if $\sigma=(i_{1},i_{2},\cdots,i_{r})$ is cycle and $a=(g_{2}^{a_{1}},\cdots,g_{2}^{a_{n}})$ with $a_{i}=0$ for $i\notin\\{i_{1},i_{2},\cdots,i_{r}\\}$. A sign cycle $(a,\sigma)$ is called positive ( or negative ) if $\sum_{i=1}^{n}a_{i}$ is even (or odd). $(a,\sigma)=(a^{(1)},\sigma_{1})(a^{(2)},\sigma_{2})\cdots(a^{(r)},\sigma_{r})$ is called an independent sign cycle decomposition of $(a,\sigma)$ if $\sigma=\sigma_{1}\sigma_{2}\cdots\sigma_{r}$ is an independent cycle decomposition of $\sigma$ in $\mathbb{S}_{n}$ and $(a^{(i)},\sigma_{i})$ is a sign cycle for $1\leq i\leq r$. Furthermore, $(a,\sigma)\in A\rtimes\mathbb{S}_{n}$ is called positive ( or negative ) if $\sum_{i=1}^{n}a_{i}$ is even (or odd). The type of $\sigma$ is said to be the type of $(a,\sigma)$. ### 3.1. Match ###### Lemma 3.1. Let $\chi^{(\nu)}$ denote the one dimensional representation $(\otimes_{j=1}^{\nu}\chi_{2})\otimes(\otimes_{j=\nu+1}^{n}\epsilon)$ $=\stackrel{{\scriptstyle\nu}}{{\overbrace{\chi_{2}\otimes\cdots\otimes\chi_{2}}}}\otimes\stackrel{{\scriptstyle n-\nu}}{{\overbrace{\epsilon\otimes\cdots\otimes\epsilon}}}$ of $A\subseteq(C_{2})^{n}$ for $\nu=0,1,\cdots,n$. Then $\\{\chi^{(\nu)}\mid\nu=0,1,\cdots,j\\}$ be a system of the representatives for the orbits of $\mathbb{S}_{n}$ on $\hat{A}$, where $j=0,n-1,n$ when $A\rtimes\mathbb{S}_{n}$ is isomorphic to the Weyl groups of $A_{n-1},$ $D_{n},$ and $B_{n}$, respectively. Furthermore, $(\mathbb{S}_{n})_{\chi^{(\nu)}}=\\{\sigma\in\mathbb{S}_{n}\mid 1\leq\sigma(i)\leq\nu$ when $1\leq i\leq\nu\\}$ $=\mathbb{S}_{\\{1,2,\cdots,\nu\\}}\times\mathbb{S}_{\\{\nu+1,\nu+2,\cdots,n\\}}$ for $\nu=0,1,\cdots,n.$ ###### Lemma 3.2. Let $1\leq\nu\leq n-1$, $\sigma\in\mathbb{S}_{n}$. Then (i) $\big{(}(\mathbb{S}_{n})_{\chi^{(\nu)}}\big{)}^{\sigma}=\big{(}(\mathbb{S}_{n})^{\sigma}\big{)}_{\chi^{(\nu)}}$; (ii) $\big{(}(\mathbb{S}_{n})_{\chi^{(\nu)}}\big{)}^{\sigma}=\mathbb{S}_{\\{1,2,\cdots,\nu\\}}^{\sigma}\times\mathbb{S}_{\\{1+\nu,2+\nu,\cdots,n\\}}^{\sigma}$ when $\sigma\in(\mathbb{S}_{n})_{\chi^{(\nu)}}$. (iii) If $\chi^{(\nu)}\in\widehat{A^{\sigma}}$ then $\sigma\in(\mathbb{S}_{n})_{\chi^{(\nu)}}.$ Proof. (i) It is clear. (ii) It follows from Lemma 1.11. (iii) Since $\chi^{(\nu)}\in\widehat{A^{\sigma}}$, $(\sigma\cdot\chi^{(\nu)})(a)$ $=\chi^{(\nu)}(\sigma^{-1}a\sigma)$$=\chi^{(\nu)}(a)$ for any $a\in A^{\sigma}$. Therefore $\sigma\cdot\chi^{(\nu)}=\chi^{(\nu)},$ i.e. $\sigma\in(\mathbb{S}_{n})_{\chi^{(\nu)}}.$ $\Box$ ###### Definition 3.3. Let $G=A\rtimes\mathbb{S}_{n}.$ If $\sigma\in\mathbb{S}_{\nu}$ or $\sigma\in\mathbb{S}_{\\{\nu+1,\cdots,n\\}}$, then ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi^{(\nu)},\rho})$ is called to be matched and the distinguished element of ${\rm RSR}(\mathbb{S}_{\nu},{\mathcal{O}}_{\sigma},\rho)$ (when $\sigma\in\mathbb{S}_{\nu}$) or ${\rm RSR}(\mathbb{S}_{\\{\nu+1,\cdots,n\\}}$ , ${\mathcal{O}}_{\sigma},\rho)$ (when $\sigma\in\mathbb{S}_{\\{\nu+1,\cdots,n\\}}$) is called the distinguished element of ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi,\rho})$. ###### Lemma 3.4. Let $D$ be a subgroup of group $G$ and $(X,\rho)$ a representation of $D$. If $\psi$ is a group isomorphism from $G$ to $G^{\prime}$, then $\rho\uparrow_{D}^{G}=((\rho\psi^{-1})\uparrow_{\psi(D)}^{\psi(G)})\psi.$ Proof. let left coset decompositions of $D$ in $G$ be $D=\bigcup_{i=1}^{n}t_{i}D$. Then $\psi(D)=\bigcup_{i=1}^{n}\psi(t_{i})\psi(D)$ is a left coset decompositions of $\psi(D)$ in $G^{\prime}.$ For any $g\in D$ , by [Sa01, Definition 1.12.2], $(\rho\uparrow_{D}^{G})(g)=(\rho(t_{i}^{-1}gt_{j}))_{n\times n}$ and $((\rho\psi^{-1})\uparrow_{\psi(D)}^{\psi(G)})\psi(g)=(\rho(t_{i}^{-1}gt_{j}))_{n\times n}$. Thus $\rho\uparrow_{D}^{G}=((\rho\psi^{-1})\uparrow_{\psi(D)}^{\psi(G)})\psi.$ $\Box$ ###### Lemma 3.5. Let $G:=A\rtimes\mathbb{S}_{n}$. If $\sigma\in\mathbb{S}_{n}$ and $\theta_{\chi,\rho}\in\widehat{G^{\sigma}}$, then there exist a natural number $\nu$, $0\leq\nu\leq n$, $\sigma^{\prime}\in\mathbb{S}_{n}$ and $\rho^{\prime}\in\widehat{(\mathbb{S}_{n})_{\chi^{(\nu)}}^{\sigma^{\prime}}}$ such that ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi,\rho})\cong{\rm RSR}(G,{\mathcal{O}}_{\sigma^{\prime}},\theta_{\chi^{(\nu)},\rho^{\prime}})$. Proof. We show this by following several steps. (i) Since $\chi$ $\in\widehat{A^{\sigma}}$, then there exists a $W\subseteq\\{1,2,\cdots,n\\}$ such that $\chi=\otimes_{i=1}^{n}\chi_{2}^{\delta_{i}}$ with $\delta_{i}=1$ when $i\in W$ and $\delta_{i}=0$ otherwise. (ii) Let $\nu=\mid\\!W\\!\mid$. Let $\phi\in\mathbb{S}_{n}$ with $\phi(W)=\\{1,2,\cdots,\nu\\}$ and $\sigma^{\prime}=\phi\sigma\phi^{-1}$. Define a map $\psi$ from $(\mathbb{S}_{n})^{\sigma}$ to $(\mathbb{S}_{n})^{\sigma^{\prime}}$ by sending $x$ to $\phi x\phi^{-1}$ for any $x\in(\mathbb{S}_{n})^{\sigma}$. Let $\beta$ be a bijective map from $A^{\sigma}$ to $A^{\psi(\sigma)}$ by sending $a=(g_{2}^{a_{1}},g_{2}^{a_{2}},\cdots,g_{2}^{a_{n}})$ to $b=(g_{2}^{a_{\phi^{-1}(1)}},g_{2}^{a_{\phi^{-1}(2)}},\cdots,g_{2}^{a_{\phi^{-1}(n)}})$. It is clear that $\beta$ is an isomorphism of groups. Note $\beta(a)=\phi\cdot a$. (iii) It is clear that $\chi\beta^{-1}=\chi^{(\nu)}\in\widehat{A^{\psi(\sigma)}}$. Indeed, for any $a=(g_{2}^{a_{1}},g_{2}^{a_{2}},\cdots,g_{2}^{a_{n}})\in A^{\psi(\sigma)}$, see $\displaystyle\chi^{(\nu)}(a)$ $\displaystyle=$ $\displaystyle\chi_{2}(g_{2})^{a_{1}+a_{2}+\cdots+a_{\nu}}\ \ \ \hbox{and}$ $\displaystyle\chi\beta^{-1}(a)$ $\displaystyle=$ $\displaystyle\chi_{2}(g_{2})^{\delta_{1}a_{\phi(1)}+\delta_{2}a_{\phi(2)}+\cdots+\delta_{n}a_{\phi(n)}}$ $\displaystyle=$ $\displaystyle\chi_{2}(g_{2})^{a_{1}+a_{2}+\cdots+a_{\nu}}.$ Thus $\chi\beta^{-1}=\chi^{(\nu)}$. (iv) $\rho\psi^{-1}\in\widehat{(\mathbb{S}_{n})_{\chi^{(\nu)}}^{\psi(\sigma)}}$. In fact, it is sufficient to show that $\psi^{-1}$ is a group isomorphism from ${(\mathbb{S}_{n})_{\chi^{(\nu)}}^{\psi(\sigma)}}$ to $(\mathbb{S}_{n})_{\chi}^{\sigma}.$ For any $h\in(\mathbb{S}_{n})_{\chi^{(\nu)}}^{\psi(\sigma)}$, we have $\psi(\sigma)h=h\psi(\sigma)$ and $h\cdot\chi^{(\nu)}=\chi^{(\nu)}$. Therefore $\sigma\psi^{-1}(h)=\psi^{-1}(h)\sigma$ and for any $a=(g_{2}^{a_{1}},g_{2}^{a_{2}},\cdots,g_{2}^{a_{n}})\in A^{\sigma},$ $\displaystyle(\psi^{-1}(h)\cdot\chi)(a)$ $\displaystyle=$ $\displaystyle\chi(\psi(h)\cdot a)$ $\displaystyle=$ $\displaystyle\chi(g_{2}^{a_{\psi^{-1}(h)(1)}},g_{2}^{a_{\psi^{-1}(h)(2)}},\cdots,g_{2}^{a_{\psi^{-1}(h)(n)}})$ $\displaystyle=$ $\displaystyle\chi_{2}(g_{2})^{\delta_{1}a_{\psi^{-1}(h)(1)}+\delta_{2}a_{\psi^{-1}(h)(2)}+\cdots+\delta_{n}a_{\psi^{-1}(h)(n)}}.$ Considering $\psi^{-1}(h)(i)=\phi^{-1}h\phi(i)\in W\ \hbox{ when }i\in W$ since $h\in\mathbb{(}S_{n})_{\chi^{(\nu)}}=\mathbb{S}_{\nu}\times\mathbb{S}_{\\{\nu+1,\cdots,n\\}}$, we have $\psi^{-1}(h)\cdot\chi=\chi.$ (v) $\beta\otimes\psi$ is a group isomorphism from $A^{\sigma}\rtimes(\mathbb{S}_{n})^{\sigma}_{\chi}$ to $A^{\psi(\sigma)}$ $\rtimes(\mathbb{S}_{n})^{\psi(\sigma)}_{\chi^{(\nu)}}$. In fact, since $\psi(h)\cdot\beta(a)=\beta(h\cdot a)$ for any $h\in(\mathbb{S}_{n})^{\sigma}_{\chi}$, $a\in A^{\sigma}$ we have that $\beta\otimes\psi$ is a group homomorphism. (vi) $\theta_{\chi^{(\nu)},\rho\psi^{-1}}(\beta\otimes\psi)=\theta_{\chi,\rho}$. This follows from Lemma 3.3. Consequently, ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi,\rho})\cong{\rm RSR}(G,{\mathcal{O}}_{\sigma^{\prime}},\theta_{\chi^{(\nu)},\rho^{\prime}})$ by [ZZWC, Lemma 1.8] and [ZCZ, Theorem 4], where $\rho^{\prime}=\rho\psi^{-1}$. $\Box$ By lemma above, we only consider $\chi^{(\nu)}$ in $\widehat{A^{\sigma}}$ from now on. ###### Lemma 3.6. Let $G=A\rtimes\mathbb{S}_{n}$ and $\sigma\in\mathbb{S}_{n}.$ If ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi^{(\nu)},\rho})$ is matched, then ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi^{(\nu)},\rho})$ is $-1$-type if and only if distinguished element $\xi_{{\bf t},\sigma}$ of ${\rm RSR}(G,{\mathcal{O}}_{\sigma},$ $\theta_{\chi^{(\nu)},\rho})$ is an integer and the order of $\sigma$ is even. Proof. ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi^{(\nu)},\rho})$ is $-1$-type if and only if so is ${\rm RSR}((\mathbb{S}_{n})_{\chi^{(\nu)}},{\mathcal{O}}_{\sigma},\rho)$ by Proposition 1.9. If $\sigma\in\mathbb{S}_{\nu}$, then $\rho=\rho^{\prime}\otimes\rho^{\prime\prime}$ with $\rho^{\prime}\in\widehat{\mathbb{S}_{\nu}^{\sigma}}$ and $\rho^{\prime\prime}\in\widehat{\mathbb{S}_{\\{\nu+1,\cdots,n\\}}}$. By Lemma 1.12, ${\rm RSR}((\mathbb{S}_{n})_{\chi^{(\nu)}},{\mathcal{O}}_{\sigma},\rho)$ is $-1$-type if and only if ${\rm RSR}(\mathbb{S}_{\nu},{\mathcal{O}}_{\sigma},\rho^{\prime})$ is $-1$-type. It follows from Proposition 2.5 that ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi^{(\nu)},\rho})$ is $-1$-type if and only if the distinguished element $\xi_{{\bf t},\sigma}$ of ${\rm RSR}(\mathbb{S}_{\nu},{\mathcal{O}}_{\sigma},\rho^{\prime})$ is an integer and the order of $\sigma$ is even. If $\sigma\in\mathbb{S}_{\\{\nu+1,\cdots,n\\}}$, then $\rho=\rho^{\prime}\otimes\rho^{\prime\prime}$ with $\rho^{\prime}\in\widehat{\mathbb{S}_{\nu}}$ and $\rho^{\prime\prime}\in\widehat{\mathbb{S}_{\\{\nu+1,\cdots,n\\}}^{\sigma}}$. By Lemma 1.12, ${\rm RSR}((\mathbb{S}_{n})_{\chi^{(\nu)}},{\mathcal{O}}_{\sigma},\rho)$ is $-1$-type if and only if ${\rm RSR}(\mathbb{S}_{\\{\nu+1,\cdots,n\\}},{\mathcal{O}}_{\sigma},\rho^{\prime\prime})$ is $-1$-type. It follows from Proposition 2.5 that ${\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi^{(\nu)},\rho})$ is $-1$-type if and only if the distinguished element $\xi_{{\bf t},\sigma}$ of ${\rm RSR}(\mathbb{S}_{\\{\nu+1,\cdots,n\\}},{\mathcal{O}}_{\sigma},\rho^{\prime\prime})$ is an integer and the order of $\sigma$ is even. $\Box$ The proof of Theorem 2: We only show this under the case of $\sigma\in{\mathbb{S}}_{\nu}$ since the other is similar. By Proposition 2.8 and Lemma 3.6, $\mathfrak{B}({\mathcal{O}}_{\sigma}^{G},\theta_{\chi^{(\nu)},\rho})$ is $-1$-type under the cases in Theorem 2. See $\displaystyle{\rm RSR}(\mathbb{S}_{\nu},{\mathcal{O}}_{\sigma},\rho^{\prime})$ $\displaystyle\leq$ $\displaystyle{\rm RSR}((\mathbb{S}_{n})_{\chi^{(\nu)}},{\mathcal{O}}_{\sigma},\rho^{\prime}\otimes\rho^{\prime\prime})\ \ (\hbox{by Lemma }\ref{1.11}(iii))$ $\displaystyle\leq$ $\displaystyle{\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi^{(\nu)},\rho})\ \ (\hbox{by Proposition }\ref{1.9}(iii)),$ which implies ${\rm RSR}(\mathbb{S}_{\nu},{\mathcal{O}}_{\sigma},\rho^{\prime})$ $\leq{\rm RSR}(G,{\mathcal{O}}_{\sigma},\theta_{\chi^{(\nu)},\rho})$. By Proposition 1.6 (iii), ${\rm dim}\mathfrak{B}(\mathbb{S}_{\nu},{\mathcal{O}}_{\sigma},$ $\mu)$ $<\infty$. Applying [AFZ, Theorem 1] we complete the proof. $\Box$ ###### Proposition 3.7. Let $G=A\rtimes\mathbb{S}_{n}$ with $\sigma\in\mathbb{S}_{n}$. Then $(\mathbb{S}_{n})^{\sigma}_{\chi^{(\nu)}}=(\mathbb{S}_{n})^{\sigma}$ if and only if $Y_{j}\subseteq\\{1,2,\cdots,\nu\\}$ or $Y_{j}\subseteq\\{\nu+1,\nu+2,\cdots,n\\}$ for $1\leq j\leq n$, where $Y_{j}$ is the same as in the begin of Section 2. Proof. If $Y_{j}\subseteq\\{1,2,\cdots,\nu\\}$, then $A_{l,j},B_{h,j}\in{\mathbb{S}}_{\\{1,2,\cdots,\nu\\}}$ for $1\leq l\leq\lambda_{j}$ and $1\leq h\leq\lambda_{j}-1$, where $A_{l,j}$, $B_{h,j}$ are the same as in the begin of Section 2. If $Y_{j}\subseteq\\{\nu+1,\nu+2,\cdots,n\\}$, then $A_{l,j},B_{h,j}\in{\mathbb{S}}_{\\{\nu+1,\nu+2,\cdots,n\\}}$ for $1\leq l\leq\lambda_{j}$ and $1\leq h\leq\lambda_{j}-1$. Consequently, $(\mathbb{S}_{n})^{\sigma}_{\chi^{(\nu)}}=(\mathbb{S}_{n})^{\sigma}$ Conversely, assume $(\mathbb{S}_{n})^{\sigma}_{\chi^{(\nu)}}=(\mathbb{S}_{n})^{\sigma}$. If there exists $1\leq j\leq n$ such that $Y_{j}\nsubseteq\\{1,2,\cdots,\nu\\}$ and $Y_{j}\nsubseteq\\{\nu+1,\nu+2,\cdots,n\\}$, then there exist $a,b\in Y_{j}$ with $a\in\\{1,2,\cdots,\nu\\}$ and $b\in\\{\nu+1,\nu+2,\cdots,n\\}$. Note $Y_{j}=\\{y_{s}\mid s=r_{j}+1,\cdots,r_{j+1}\\}$. If there exists $l$ such that $a,b\in\\{y_{r_{j}+(l-1)j+1},\qquad y_{r_{j}+(l-1)j+2},\quad\cdots,\quad y_{r_{j}+lj}\\}$, then $A_{l,j}\notin(\mathbb{S}_{n})^{\sigma}_{\chi^{(\nu)}}$, a contradiction. Thus there exist $l\not=l^{\prime}$ such that $a\in A_{l,j}$ and $b\in A_{l^{\prime},j}$. Let $a=y_{r_{j}+(l-1)j+s}$ and $b=y_{r_{j}+(l^{\prime}-1)j+s^{\prime}}$. Considering $B_{h,j}\in(\mathbb{S}_{n})^{\sigma}_{\chi^{(\nu)}}$, we have $y_{r_{j}+(l^{\prime}-1)j+s}\in\\{1,2,\cdots,\nu\\}$ which is a contradiction. $\Box$ ### 3.2. Central quantum linear space A central quantum linear space is a finite dimensional Nichols algebra, which was introduced in [ZZWC, Def. 2.12]. ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ is said to be a central quantum linear type if it is quantum symmetric and of the non-essentially infinite type with $C\subseteq Z(G)$ for any $C\in{\mathcal{K}}_{r}(G)$. In this case, $\mathfrak{B}(G,r,\overrightarrow{\rho},u)$ is called a central quantum linear space over $G$. We give the other main result. ###### Theorem 3. $\mathfrak{B}(G,r,\overrightarrow{\rho},u)$ is a central quantum linear space over classical Weyl group $G$ if and only if $C=\\{(g_{2},\cdots,g_{2})\\}\subseteq G$, $r=r_{C}C$, $\rho_{C}^{(i)}=\theta_{\chi_{C}^{(i)},\mu_{C}^{(i)}}:=(\chi_{C}^{(i)}\otimes\mu^{(i)}_{C})\uparrow_{G_{\chi_{C}^{(i)}}^{u(C)}}^{G^{u(C)}}\in\widehat{G^{u(C)}}$ with $\chi_{C}^{(i)}$ $\in\\{\chi_{2}^{\delta^{(i)}_{1}}\otimes\chi_{2}^{\delta^{(i)}_{2}}\otimes\cdots\otimes\chi_{2}^{\delta^{(i)}_{n}}$ $\mid$ $\delta^{(i)}_{1}+\delta^{(i)}_{2}+\cdots+\delta^{(i)}_{n}$ is odd } for any $i\in I_{C}(r,u)$. Proof. It is clear $\theta_{\chi_{C}^{(i)},\rho_{C}^{(i)}}((g_{2},\cdots,g_{2}))=\chi_{2}(g_{2})^{\delta^{(i)}_{1}+\delta^{(i)}_{2}+\cdots+\delta^{(i)}_{n}}\ {\rm id}$ as in the proof of Lemma 2.1 (ii). Applying [ZZWC, Remark 3.16], we complete the proof. $\Box$ In other words we have ###### Remark 3.8. Let $G=A\rtimes\mathbb{S}_{n}$. Assume that $a=(g_{2},g_{2},\cdots,g_{2})\in G$ and $M=M({\mathcal{O}}_{a},\rho^{(1)})\oplus M({\mathcal{O}}_{a},\rho^{(2)})\oplus\cdots\oplus M({\mathcal{O}}_{a},\rho^{(m)})$ is a YD module over $kG$ with $\rho^{(i)}=\theta_{\chi^{(\nu_{i})},\mu^{(i)}}:=(\chi^{(\nu_{i})}\otimes\mu^{(i)})\uparrow_{G_{\chi{(\nu_{i})}}^{a}}^{G^{a}}\in\widehat{G^{a}}$ and odd $\nu_{i}$ for $i=1,2,\cdots,m$. Then $\mathfrak{B}(M)$ is finite dimensional. ### 3.3. Reducible Yetter-Drinfeld modules ${\mathcal{O}}_{\sigma}$ and ${\mathcal{O}}_{\tau}$ are said to be square- commutative if $stst=tsts$ for any $s\in{\mathcal{O}}_{\sigma}$, $t\in{\mathcal{O}}_{\tau}$. Obviously, ${\mathcal{O}}_{\sigma}$ and ${\mathcal{O}}_{\tau}$ are square-commutative if and only if $sts\in G^{t}$ for any $s\in{\mathcal{O}}_{\sigma}$, $t\in{\mathcal{O}}_{\tau}$ if and only if $s\tau s\in G^{\tau}$ for any $s\in{\mathcal{O}}_{\sigma}$. ###### Lemma 3.9. Let $G=A\rtimes D$. Let $(a,\sigma),(b,\tau)\in G$ with $a,b\in A$, $\sigma,\tau\in D$. If ${\mathcal{O}}_{(a,\sigma)}^{G}$ and ${\mathcal{O}}_{(b,\tau)}^{G}$ are square-commutative then ${\mathcal{O}}_{\sigma}^{D}$ and ${\mathcal{O}}_{\tau}^{D}$ are square- commutative. Proof. It is clear that $(a,\sigma)^{-1}=(\sigma^{-1}\cdot a^{-1},\sigma^{-1})$ and (3.2) $\displaystyle(b,\tau)(a,\sigma)(b,\tau)^{-1}$ $\displaystyle=$ $\displaystyle(b(\tau\cdot a)(\tau\sigma\tau^{-1}\cdot b^{-1}),\tau\sigma\tau^{-1}).$ For any $x\in{\mathcal{O}}_{\sigma}^{D}$ and $y\in{\mathcal{O}}_{\tau}^{D}$, by (3.2), there exist $c,d\in A$ such that $(c,x)\in{\mathcal{O}}_{(a,\sigma)}^{G}$ and $(d,y)\in{\mathcal{O}}_{(b,\tau)}^{G}$. Since $(c,x)(d,y)(c,x)(d,y)=(d,y)(c,x)(d,y)(c,x)$, we have $xyxy=yxyx$, i.e. ${\mathcal{O}}_{\sigma}^{D}$ and ${\mathcal{O}}_{\tau}^{D}$ are square- commutative. $\Box$ ###### Lemma 3.10. Let $1\not=\sigma,1\not=\tau\in\mathbb{S}_{n}$ with $G=\mathbb{S}_{n}$ and $n>2$. If ${\mathcal{O}}_{\sigma}$ and ${\mathcal{O}}_{\tau}$ are square- commutative, then one of the following conditions holds. (i) $n=3$, ${\mathcal{O}}_{\sigma}={\mathcal{O}}_{\tau}={\mathcal{O}}_{(123)}$ or ${\mathcal{O}}_{\sigma}={\mathcal{O}}_{(12)}$ and ${\mathcal{O}}_{\tau}={\mathcal{O}}_{(123)}$. (ii) $n=4$, ${\mathcal{O}}_{\sigma}={\mathcal{O}}_{\tau}={\mathcal{O}}_{(12)(34)}$ or ${\mathcal{O}}_{\sigma}={\mathcal{O}}_{(12)(34)}$ and ${\mathcal{O}}_{\tau}={\mathcal{O}}_{(1234)}$ or ${\mathcal{O}}_{\sigma}={\mathcal{O}}_{(12)}$ and ${\mathcal{O}}_{\tau}={\mathcal{O}}_{(12)(34)}$. (iii) $n=2k$ with $k>2$, ${\mathcal{O}}_{\sigma}={\mathcal{O}}_{(12)}$ and ${\mathcal{O}}_{\tau}={\mathcal{O}}_{(12)(34)\cdots(n-1\ n)}$. Proof. We show this by following several steps. Assume that $1^{\lambda_{1}}$$2^{\lambda_{2}}\cdots n^{\lambda_{n}}$ and $1^{\lambda_{1}^{\prime}}$$2^{\lambda_{2}^{\prime}}\cdots n^{\lambda_{n}^{\prime}}$ are the types of $\sigma$ and $\tau$, respectively; ${\mathcal{O}}_{\sigma}$ and ${\mathcal{O}}_{\tau}$ are square-commutative. (i) Let $n=3$. Obviously, $\mathcal{O}_{(12)}$ and $\mathcal{O}_{(12)}$ are not square-commutative. Then ${\mathcal{O}}_{\sigma}={\mathcal{O}}_{\tau}={\mathcal{O}}_{(123)}$ or ${\mathcal{O}}_{\sigma}={\mathcal{O}}_{(12)}$ and ${\mathcal{O}}_{\tau}={\mathcal{O}}_{(123)}$. (ii) Let $n=4$. The types of $\sigma$ and $\tau$ are $2^{2};4^{1};1^{1}3^{1};1^{2}2^{1}$, respectively. (a). ${\mathcal{O}}_{(12)(34)}$ and ${\mathcal{O}}_{(123)}$ are not square- commutative since $(12)(34)(123)(12)(34)=(214)$, which implies $(12)(34)(123)(12)(34)\notin G^{(123)}.$ (b). ${\mathcal{O}}_{(1234)}$ and ${\mathcal{O}}_{(1234)}$ are not square- commutative since $((1234)(4231))^{2}$ and $((4231)$ $(1234))^{2}$ maps $1$ to $1$ and $2$, respectively. (c). ${\mathcal{O}}_{(1234)}$ and ${\mathcal{O}}_{(123)}$ are not square- commutative since $(1234)(123)(1234)$ maps $1$ to $4$. (d). ${\mathcal{O}}_{(1234)}$ and ${\mathcal{O}}_{(12)}$ are not square- commutative since $(1234)(12)(1234)$ maps $2$ to $4$. (e). ${\mathcal{O}}_{(123)}$ and ${\mathcal{O}}_{(123)}$ are not square- commutative since $(134)(123)(134)$ maps $2$ to $4$. (f). ${\mathcal{O}}_{(123)}$ and ${\mathcal{O}}_{(12)}$ are not square- commutative since $(14)(123)(14)=(423)$. (g). ${\mathcal{O}}_{(12)}$ and ${\mathcal{O}}_{(12)}$ are not square- commutative since $(14)(12)(14)=(42)$. (iii) If $\sigma=(12\cdots r)$ with $n>4$, then $r=n$ or ${\mathcal{O}}_{\sigma}={\mathcal{O}}_{(12)}$ and ${\mathcal{O}}_{\tau}={\mathcal{O}}_{(12)(34)\cdots(n-1\ n)}$ with $n=2k>2$. In fact, obviously, $G^{\sigma}$ is a cycle group generated by $(12\cdots r)$. (a). If $\lambda_{3}^{\prime}\not=0$ and $n-r>1$, set $t=(\cdots,r,r+1,r+2\cdots)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $t\sigma t(r)=r+2$, which implies $t\sigma t\notin G^{\sigma}.$ (b). If $\lambda_{3}^{\prime}\not=0$ and $n-r=1$. Set $t=(1,r,r+1)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $t\sigma t(r+1)=t(2)<r+1$ since $r\not=2$, which implies $t\sigma t\notin G^{\sigma}.$ (c). If there exists $j>3$ such that $\lambda_{j}^{\prime}\not=0$ with $n>r>2$, set $t=(\cdots,r-2,r-1,r,r+1,\cdots)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $t\sigma t(r-2)=r+1$, which implies $t\sigma t\notin G^{\sigma}.$ (d). If there exists $j>3$ such that $\lambda_{j}^{\prime}\not=0$ with $r=2$, set $t=(123\cdots)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $t\sigma t(2)=4$, which implies $t\sigma t\notin G^{\sigma}.$ (e). If the type of $\tau$ is $1^{\lambda_{1}^{\prime}}2^{1}$ with $n>r$, set $t=(r\ r+1)\in{\mathcal{O}}_{\tau}$. See $t\sigma t=(1\cdots r\ r+1)\notin G^{\sigma}.$ (f). If the type of $\tau$ is $1^{\lambda_{1}^{\prime}}2^{\lambda_{2}^{\prime}}$ with $n>r>2$ and $\lambda_{2}^{\prime}>1$, set $t=(r\ r+1)(12)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $t\sigma t=(2\ 1\ \cdots r+1)$ $\notin G^{\sigma}.$ (g). If the type of $\tau$ is $1^{\lambda_{1}^{\prime}}2^{\lambda_{2}^{\prime}}$ with $r=2$, $\lambda_{2}^{\prime}\geq 1$ and $\lambda_{1}^{\prime}\not=0$, set $t=(1)(23)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $t\sigma t=(13)$ $\notin G^{\sigma}.$ From now on assume that both $\sigma$ and $\tau$ are not cycles. (iv) If $n>4$ and $\sigma=(1,2,\cdots,n)$ is a cycle, then it is a contradiction. (a). If $\lambda_{2}^{\prime}\not=0$, set $t=(1,n)(2,3,\cdots)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $t\sigma t\sigma(1)=t(4)\not=1$ and $\sigma t\sigma t(1)=1$. (b). If $\lambda_{3}^{\prime}\not=0$, set $t=(123)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $t\sigma t\sigma(1)=t(4)$ and $\sigma t\sigma t(1)=2$, which implies that $t\sigma t\notin G^{\sigma}$ since $t(4)>3.$ (c). If $\lambda_{4}^{\prime}\not=0$, set $t=(1234)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $t\sigma t\sigma(1)=1$ and $\sigma t\sigma t(1)=5$, which implies $t\sigma t\notin G^{\sigma}.$ (d). If $n>5$ and there exists $j>4$ such that $\lambda_{j}^{\prime}\not=0$, set $t=(12346\cdots)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $\sigma t\sigma t(1)=5$ and $t\sigma t\sigma(1)=6$, which implies that $t\sigma t\notin G^{\sigma}.$ (e). If $n=5$ and there exists $j>4$ such that $\lambda_{j}^{\prime}\not=0$, set $t=(13254)\in{\mathcal{O}}_{\tau}$. See $\sigma t\sigma t(1)=2$ and $t\sigma t\sigma(1)=3$, which implies that $t\sigma t\notin G^{\sigma}.$ (v) If $n>4$ and there exists $r>1$ such that $\lambda_{r}>1$, then $n=\lambda_{r}r$. Let $\sigma=(1,2,\cdots,r)(r+1,\cdots,2r)\cdots((\lambda_{r}-1)r+1,\cdots,\lambda_{r}r)\sigma_{1}$, an independent decomposition of $\sigma$. Assume $n>\lambda_{r}r.$ (a) If $r>2$ and $\lambda_{j}^{\prime}=0$ for any $j>2$, set $t=(1,n)(2,3)t_{1}$, an independent decomposition of $t\in\mathcal{O}_{\tau}$. See $t\sigma t=(n,3,a_{3},\cdots,a_{r})\cdots$, which implies that $t\sigma t\notin G^{\sigma}.$ (b). If $r=2$ and $\lambda_{j}^{\prime}=0$ for any $j>2$, set $t=(1,n)(2,3)t_{1}$, an independent decomposition of $t\in\mathcal{O}_{\tau}$. See $t\sigma t=(n,3)(2,t(4))\cdots$, which implies that $t\sigma t\notin G^{\sigma}$ since $t(4)>2$. (c). If there exists $j>2$ such that $\lambda_{j}^{\prime}\not=0$, set $t=(1,a_{1},\cdots,a_{p},\lambda_{r}r,\lambda_{r}r+1)(2,3,\cdots)t_{1}$, an independent decomposition of $t\in\mathcal{O}_{\tau}$. See $t\sigma t(\lambda_{r}r+1)=t\sigma(1)=3$, which implies that $t\sigma t\notin G^{\sigma}.$ (vi) If $n>4$ and $\lambda_{r}\leq 1$ for any $r>1$, then this is a contradiction. Assume that there exist $r$ and $r^{\prime}$ such that $\lambda_{r^{\prime}}\not=0$ and $\lambda_{r}\not=0$ with $2\leq r^{\prime}<r$. Let $\sigma=(12\cdots r)(r+1\cdots r+r^{\prime})\sigma_{1}$ be an independent decomposition of $\sigma$. (a). If $\lambda_{i}^{\prime}=0$ for any $i>2$, set $t=(1\ n)(23)t_{1}$, an independent decomposition of $t\in\mathcal{O}_{\tau}$. See $t\sigma t=(n\ 3\cdots)\cdots$, which implies that $t\sigma t\notin G^{\sigma}.$ (b). If $r>3$ and there exists $j>2$ such that $\lambda_{j}^{\prime}\not=0$, set $t=(1,a_{1},\cdots,a_{p},r,r+1)(2,3,\cdots)t_{1}$, an independent decomposition of $t\in\mathcal{O}_{\tau}$. See $t\sigma t(r+1)=3$, which implies that $t\sigma t\notin G^{\sigma}.$ (c). If $r=3$ and there exists $j>2$ such that $\lambda_{j}^{\prime}\not=0$, set $t=(123\cdots)(34\cdots)t_{1}$, an independent decomposition of $t\in\mathcal{O}_{\tau}$. See $t\sigma t(1)=4$, which implies that $t\sigma t\notin G^{\sigma}.$ (vii) If $n>4$ and the types of $\sigma$ and $\tau$ are $r^{\lambda_{r}}$, then it is a contradiction. Let $\sigma=(12\cdots r)(r+1\cdots 2r)\cdots.$ Set $t=(r+1,2,\cdots,r)(1,r+2,\cdots,2r)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $t\sigma t\sigma(1)=$ $\left\\{\begin{array}[]{ll}5&\hbox{when }r=3\\\ 5&\hbox{when }r>4\end{array}\right.$ and $\sigma t\sigma t(1)=$ $\left\\{\begin{array}[]{ll}2&\hbox{when }r=3\\\ r+5&\hbox{when }r>4\end{array}\right..$ When $r=4$, set $t=(4231)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. $\sigma t\sigma t(1)=1$ and $t\sigma t\sigma(1)=2$. When $r=2,$ set $t=(14)(25)(36)t_{1}$, an independent decomposition of $t\in{\mathcal{O}}_{\tau}$. See $\sigma t\sigma t(1)=5$ and $t\sigma t\sigma(1)=3$. Then $\sigma t\sigma t\not=t\sigma t\sigma$. $\Box$ In fact, ${\mathcal{O}}_{\sigma}$ and ${\mathcal{O}}_{\tau}$ in cases of Lemma 3.10 (i) (ii) are square- commutative . ###### Lemma 3.11. Let $G=A\rtimes\mathbb{S}_{n}$ with $A\subseteq(C_{2})^{n}$. Then (i) $n=3$, ${\mathcal{O}}_{(a,(123))}$ and ${\mathcal{O}}_{(b,(123))}$ are not square-commutative. (ii) $n=3$, ${\mathcal{O}}_{(a,(12))}$ and ${\mathcal{O}}_{(b,(123))}$ are not square-commutative. (iii) $n=4$, ${\mathcal{O}}_{(a,(12)(34))}$ and ${\mathcal{O}}_{(b,(1234))}$ are not square-commutative. (iv) $n=2k$ with $k>1$, ${\mathcal{O}}_{(a,(12))}$ and ${\mathcal{O}}_{(b,(12)(34))}$ are not square-commutative. (v) If $n=4$, then ${\mathcal{O}}_{(a,(12)(34))}$ and ${\mathcal{O}}_{(b,(12)(34))}$ are square-commutative if and only if the signs of $(a,(12)(34))$ and $(b,(12)(34))$. Proof. For any $(d,\mu)\in G=A\rtimes\mathbb{S}_{n}$, let $(c,\mu\sigma\mu^{-1}):=(d,\mu)(a,\sigma)(d,\mu)^{-1}$, i.e. $c=d(\mu\cdot a)(\mu\sigma\mu^{-1}\cdot d)$. It is clear that $((c,\mu\sigma\mu^{-1})(b,\tau))^{2}=((b,\tau)(c,\mu\sigma\mu^{-1}))^{2}$ if and only if $\displaystyle d(\mu\cdot a)(\mu\sigma\mu^{-1}\cdot d)(\mu\sigma\mu^{-1}\cdot b)(\mu\sigma\mu^{-1}\tau\cdot d)$ $\displaystyle(\mu\sigma\mu^{-1}\tau\mu\cdot a)(\mu\sigma\mu^{-1}\tau\mu\sigma\mu^{-1}\cdot d)(\mu\sigma\mu^{-1}\tau\mu\sigma\mu^{-1}\cdot b)$ $\displaystyle=$ $\displaystyle b(\tau\cdot d)(\tau\mu\cdot a)(\tau\mu\sigma\mu^{-1}\cdot d)(\tau\mu\sigma\mu^{-1}\cdot b)(\tau\mu\sigma\mu^{-1}\tau\cdot d)$ $\displaystyle(\tau\mu\sigma\mu^{-1}\tau\mu\cdot a)(\tau\mu\sigma\mu^{-1}\tau\mu\sigma\mu^{-1}\cdot d),$ which is equivalent to (3.4) $\displaystyle d(\mu\sigma\mu^{-1}\cdot d)(\mu\sigma\mu^{-1}\tau\cdot d)(\mu\sigma\mu^{-1}\tau\mu\sigma\mu^{-1}\cdot d)(\tau\cdot d)(\tau\mu\sigma\mu^{-1}\cdot d)(\tau\mu\sigma\mu^{-1}\tau\cdot d)$ $\displaystyle(\tau\mu\sigma\mu^{-1}\tau\mu\sigma\mu^{-1}\cdot d)=h$ with $h:=(\mu\cdot a)(\mu\sigma\mu^{-1}\cdot b)(\mu\sigma\mu^{-1}\tau\mu\cdot a)(\mu\sigma\mu^{-1}\tau\mu\sigma\mu^{-1}\cdot b)b(\tau\mu\cdot a)(\tau\mu\sigma\mu^{-1}\cdot b)(\tau\mu\sigma\mu^{-1}\tau\mu\cdot a).$ We only need to show that there exists $(d,\mu)\in G$ such that (3.4) does not hold in the four cases above, respectively. Let $d=(g_{2}^{d_{1}},g_{2}^{d_{2}},\cdots,g_{2}^{d_{n}})$ for any $d\in A.$ (i) Let $\sigma=(123)=\tau=\mu$ and $n=3$. (3.4) becomes $d(\sigma\cdot d)=h,$ which implies $d_{1}+d_{3}\equiv h_{1}$ $({\rm mod}\ 2)$. This is a contradiction since $d$ has not this restriction. (ii) Let $\tau=(123)$, $\sigma=(12)=\mu$ and $n=3$. (3.4) becomes $(\tau^{-1}\cdot d)((32)\cdot d)(\tau\cdot d)((13)\cdot d)=h,$ which implies $d_{1}+d_{2}\equiv h_{1}$ $({\rm mod}\ 2)$. This is a contradiction since $d$ has not this restriction. (iii) Let $\sigma=(12)(34)$, $\tau=(1234)$, $\mu=(123)$ and $n=4$. (3.4) becomes $((13)\cdot d)((4321)\cdot d)((1234)\cdot d)((24)\cdot d)=h,$ which implies $d_{1}+d_{2}+d_{3}+d_{4}\equiv h_{1}$ $({\rm mod}\ 2)$. This is a contradiction since $d$ has not this restriction. (iv) Let $\sigma=(12)$, $\lambda=(56)(78)\cdots(n-1\ n)$, $\tau=(12)(34)\lambda$, $\mu=(123)$ . (3.4) becomes (3.5) $\displaystyle d((23)\cdot d)((1342)\lambda\cdot d)((13)(24)\lambda\cdot d)((12)(34)\lambda\cdot d)((1243)\lambda\cdot d)((14)\cdot d)$ $\displaystyle((14)(23)\cdot d)=h,$ which implies $0\equiv h_{i}\ ({\rm mod}\ 2)$ for $i=1,\cdots,n$. By simple computation, we have $(\mu\cdot a)((1342)\mu\cdot a)((12)(34)\mu\cdot a)((14)\mu\cdot a)$ $((23)\cdot b)((13)(24)\cdot b)$ $b((1243)\cdot b)=1,$ which implies $a_{3}+a_{4}\equiv 0\ ({\rm mod}\ 2)$. If $(a,\sigma)$ is a negative cycle, we construct a negative cycle $(a^{\prime},\sigma)$ such that $a^{\prime}_{4}\equiv a_{3}^{\prime}$ does not hold as follows: Let $a^{\prime}_{i}=0$ when $i\not=3,4,$ and $a_{4}^{\prime}=1$, $a_{3}^{\prime}=0.$ If $(a,\sigma)$ is a positive cycle, we construct a positive cycle $(a^{\prime},\sigma)$ such that $a^{\prime}_{4}\equiv a_{3}^{\prime}$ does not hold as follows: Let $a^{\prime}_{i}=0$ when $i\not=1,3,4,$ and $a_{4}^{\prime}=1$, $a_{3}^{\prime}=0$, $a_{1}^{\prime}=1$. Since ${\mathcal{O}}_{(a,\sigma)}={\mathcal{O}}_{(a^{\prime},\sigma)}$, we obtain a contradiction. (v) It is clear that ${\mathcal{O}}_{(12)(34)}^{{\mathbb{S}}_{4}}$ and ${\mathcal{O}}_{(12)(34)}^{{\mathbb{S}}_{4}}$ are commutative. (3.4) becomes $1=h$. That is, $(\mu\cdot a)(\mu\sigma\mu^{-1}\sigma\mu\cdot a)(\sigma\mu\cdot a)(\mu\sigma\mu^{-1}\mu\cdot a)$ $(\mu\sigma\mu^{-1}\cdot b)$ $(\sigma\cdot b)b$ $(\sigma\mu\sigma\mu^{-1}\cdot b)$ $=1$. It is clear that $\sigma(i)$, $\mu\sigma\mu^{-1}(i)$, $(1)(i)$ and $\mu\sigma\mu^{-1}\sigma(i)$ are different each other for any fixed $i$ with $1\leq i\leq 4$ when $\mu\sigma\mu^{-1}\not=\sigma$. Therefore, (3.4) holds for any $(d,\mu)\in A\rtimes\mathbb{S}_{n}$ if and only if $a_{1}+a_{2}+a_{3}+a_{4}$ $\equiv$ $b_{1}+b_{2}+b_{3}+b_{4}$ $({\rm mod}\ 2)$. $\Box$ We give the third main result: ###### Theorem 4. Let $G=A\rtimes\mathbb{S}_{n}$ with $A\subseteq(C_{2})^{n}$ and $n>2$. Assume that there exist different two pairs $(u(C_{1}),i_{1}))$ and $(u(C_{2}),i_{2}))$ with $C_{1},C_{2}$ $\in{\mathcal{K}}_{r}(G)$, $i_{1}\in I_{C_{1}}(r,u)$ and $i_{2}\in I_{C_{2}}(r,u)$ such that $u(C_{1})$, $u(C_{2})$ $\notin A$. If ${\rm dim}\mathfrak{B}(G,r,\overrightarrow{\rho},u)<\infty$, then the following conditions hold: (i) $n=4$. (ii) The type of $u(C)$ is $2^{2}$ for any $u(C)\notin A$ and $C\in{\mathcal{K}}_{r}(G)$. (iii) The signs of $u(C)$ and $u(C^{\prime})$ are the same for any $u(C),u(C^{\prime})\notin A$ and $C,C^{\prime}\in{\mathcal{K}}_{r}(G)$. Proof. If ${\rm dim}\mathfrak{B}(G,r,\overrightarrow{\rho},u)<\infty$, then by [HS, Theorem 8.6], ${\mathcal{O}}_{u(C_{1})}^{G}$ and ${\mathcal{O}}_{u(C_{2})}^{G}$ are square-commutative. Let $(a,\sigma):=u(C_{1})$ and $(b,\tau):=u(C_{2})$. It follows from Lemma 3.9 that ${\mathcal{O}}_{\sigma}^{\mathbb{S}_{n}}$ and ${\mathcal{O}}_{\tau}^{\mathbb{S}_{n}}$ are square-commutative. Considering Lemma 3.10, we have one of the following conditions are satisfied (i) $n=3$, ${\mathcal{O}}_{\sigma}^{\mathbb{S}_{n}}={\mathcal{O}}^{\mathbb{S}_{n}}_{\tau}={\mathcal{O}}^{\mathbb{S}_{n}}_{(123)}$ or ${\mathcal{O}}^{\mathbb{S}_{n}}_{\tau}={\mathcal{O}}_{(12)}$ and ${\mathcal{O}}^{\mathbb{S}_{n}}_{\sigma}={\mathcal{O}}^{\mathbb{S}_{n}}_{(123)}$. (ii) $n=4$, ${\mathcal{O}}^{\mathbb{S}_{n}}_{\sigma}={\mathcal{O}}^{\mathbb{S}_{n}}_{\tau}={\mathcal{O}}^{\mathbb{S}_{n}}_{(12)(34)}$ or ${\mathcal{O}}^{\mathbb{S}_{n}}_{\tau}={\mathcal{O}}_{(1234)}$ and ${\mathcal{O}}^{\mathbb{S}_{n}}_{\sigma}={\mathcal{O}}^{\mathbb{S}_{n}}_{(12)(34)}$. Considering Lemma 3.11, we complete the proof. $\Box$ In other words, we have ###### Remark 3.12. Let $G=A\rtimes\mathbb{S}_{n}$ with $A\subseteq(C_{2})^{n}$ and $n>2$. Let $M=M({\mathcal{O}}_{\sigma_{1}},\rho^{(1)})\oplus M({\mathcal{O}}_{\sigma_{2}},\rho^{(2)})\cdots\oplus M({\mathcal{O}}_{\sigma_{m}},\rho^{(m)})$ be a reducible YD module over $kG$. Assume that there exist $i\not=j$ such that $\sigma_{i}$, $\sigma_{j}$ $\notin A$. If ${\rm dim}\mathfrak{B}(M)<\infty$, then $n=4$, the type of $\sigma_{p}$ is $2^{2}$ and the sign of $\sigma_{p}$ is stable for any $1\leq p\leq m$ with $\sigma_{p}\notin A.$ ## References * [AFZ] Nicol s Andruskiewitsch, Fernando Fantino, Shouchuan Zhang, On pointed Hopf algebras associated with the symmetric groups, Manuscripta Math., 128(2009) 3, 359-371. Also in arXiv:0807.2406. * [AS98] N. Andruskiewitsch and H. J. Schneider, Lifting of quantum linear spaces and pointed Hopf algebras of order $p^{3}$, J. Alg. 209 (1998), 645–691. * [AS00] N. Andruskiewitsch and H. J. Schneider, Finite quantum groups and Cartan matrices, Adv. Math. 154 (2000), 1–45. * [AZ07] N. Andruskiewitsch and Shouchuan Zhang, On pointed Hopf algebras associated to some conjugacy classes in $\mathbb{S}_{n}$, Proc. Amer. Math. Soc. 135 (2007), 2723-2731. * [Ca72] R. W. Carter, Conjugacy classes in the Weyl group, Compositio Mathematica, 25(1972)1, 1–59. * [Gr00] M. Graña, On Nichols algebras of low dimension, Contemp. Math. 267 (2000),111–134. * [He06] I. Heckenberger, Classification of arithmetic root systems, preprint, arXiv:math.QA/0605795. * [HS] I. Heckenberger and H.-J. Schneider, Root systems and Weyl groupoids for Nichols algebras, preprint arXiv:0807.0691. * [Mo93] S. Montgomery, Hopf algebras and their actions on rings. CBMS Number 82, Published by AMS, 1993. * [OZ04] F. Van Oystaeyen and P. Zhang, Quiver Hopf algebras, J. Alg. 280 (2004), 577–589. * [Ra] D. E. Radford, The structure of Hopf algebras with a projection, J. Alg. 92 (1985), 322–347. * [Sa01] Bruce E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Second edition, Graduate Texts in Mathematics 203, Springer-Verlag, 2001. * [Se] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York 1977. * [Su78] Michio Suzuki, Group Theory I, Springer-Verlag, New York, 1978. * [ZZC] Shouchuan Zhang, Y-Z Zhang and H. X. Chen, Classification of PM Quiver Hopf Algebras, J. Alg. and Its Appl. 6 (2007)(6), 919-950. Also see in math.QA/0410150. * [ZCZ] Shouchuan Zhang, H. X. Chen and Y-Z Zhang, Classification of Quiver Hopf Algebras and Pointed Hopf Algebras of Nichols Type, preprint arXiv:0802.3488. * [ZWW] Shouchuan Zhang, Min Wu and Hengtai Wang, Classification of Ramification Systems for Symmetric Groups, Acta Math. Sinica, 51 (2008) 2, 253–264. Also in math.QA/0612508. * [ZZWC] Shouchuan Zhang, Y-Z Zhang, Peng Wang, Jing Cheng, On Pointed Hopf Algebras with Weyl Groups of exceptional type, Preprint arXiv:0804.2602.	arxiv-papers	2009-02-27T02:58:51	2024-09-04T02:49:00.881208	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lingwei Guo, Shouchuan Zhang, Junqin Li", "submitter": "Shouchuan Zhang", "url": "https://arxiv.org/abs/0902.4748" }
0902.4782	††thanks: Corresponding author. flyan@mail.hebtu.edu.cn # Two-step deterministic remote preparation of an arbitrary quantum state in the whole Hilbert space Meiyu Wang, Fengli Yan College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang 050016, China Hebei Advanced Thin Films Laboratory, Shijiazhuang 050016, China ###### Abstract We present a two-step exact remote state preparation protocol of an arbitrary qubit with the aid of a three-particle Greenberger-Horne-Zeilinger state. Generalization of this protocol for higher-dimensional Hilbert space systems among three parties is also given. We show that only single-particle von Neumann measurement, local operation and classical communication are necessary. Moreover, since the overall information of the quantum state can be divided into two different parts, which may be at different locations, this protocol may be useful in the quantum information field. ###### pacs: 03.67.Hk ## I Introduction Quantum information theory has produced many interesting and important developments that are not possible classically in recent years, in which quantum entanglement and classical communication are two elementary resources. Two surprising discoveries in this area are teleportation and remote state preparation (RSP). Quantum teleportation process, originally proposed by Bennett et al s1 , can transmit an unknown quantum state from a sender (called Alice) to a spatially distant receiver (called Bob) via a quantum channel with the help of some classical information. Recently, Lo s2 , Pati s3 and Bennett et al s4 have presented an interesting application of quantum entanglement, i.e., remote state preparation that correlates closely to teleportation. RSP is called ”teleportation of a known quantum state”, which means Alice knows the precise state that she will transmit to Bob. Her task is to help Bob construct a state that is unknown to him by means of a prior shared entanglement and a classical communication channel. So the goal of RSP is the same as that of quantum teleportation. The main difference between RSP and teleportation is that in the former Alice is assumed to know completely the state to be prepared remotely by Bob; in particular, Alice need not own the state, but only know information about the state, while in the latter Alice must own the transmitted state, but neither she nor Bob has knowledge of the transmitted state. So far, RSP has attracted much attention s5 ; s6 ; s7 ; s8 ; s9 ; s10 ; s11 ; s12 ; s13 ; s14 ; s20 . There are many kinds of RSP methods in theory, such as low-entanglement RSP s5 , higher-dimension RSP s6 , optimal RSP s7 , oblivious RSP s8 , RSP without oblivious conditions s9 , RSP for multiparties s10 , and continuous variable RSP in phase spaces11 ; s12 , etc. On the other hand, some RSP schemes have been implemented experimentally with the technique of NMR s15 and spontaneous parametric down-conversion s16 ; s17 . In addition, some authors have also investigated the RSP protocol using different quantum channels such as partial EPR pairs s18 and three-particle Greenberger-Horne- Zeilinger (GHZ) state s19 . To our best knowledge, up to now there is no RSP protocol which determinately generate an arbitrary qubit with unit success probability. They mainly concentrate on RSP of some special ensembles of a quantum state. For example, some schemes discuss how to successfully remotely prepare the state in subspace of the whole real Hilbert space or chosen from equatorial line on Bloch sphere. In this paper, we propose a two-step deterministic RSP protocol via previously shared entanglement, a single-particle von Neumann measurement, local operation and classical communication. Generalization of this protocol for higher-dimensional Hilbert space systems among three parties is also presented. We will see that the overall information of an arbitrary quantum state can be divided into two different parts. They are expressed by $\theta$ and $\varphi$ respectively, which may be at different locations. So this protocol may be useful in the quantum information field, such as quantum state sharing, converging the split information at one point, etc. ## II Deterministic RSP of an arbitrary qubit using a GHZ state as a quantum channel Let us consider a pure state $\|\psi\rangle\in H=C^{2}$ which is the state of a qubit. An arbitrary qubit can be represented as $\|\psi\rangle=\alpha\|0\rangle+\beta\|1\rangle,$ (1) where we can choose $\alpha$ to be real and $\beta$ to be complex number and $\|\alpha\|^{2}+\|\beta\|^{2}=1$. This qubit can be represented by a point on the unit two-dimensional sphere, known as Bloch sphere, with the help of two real parameters $\theta$ and $\varphi$. So we can rewrite Eq.(1) as $\|\psi\rangle=\cos(\theta/2)\|0\rangle+\sin(\theta/2)e^{i\varphi}\|1\rangle.$ (2) Now Alice wants to transmit the above qubit to Bob. The quantum channel shared by Alice and Bob is the three-particle GHZ state $\|\Phi\rangle=\frac{1}{\sqrt{2}}(\|000\rangle+\|111\rangle)_{123}.$ (3) The particles 1 and 2 belong to Alice and the particle 3 is held by Bob. As a matter of fact, the state $\|\Phi\rangle$ can be easily generated from the Bell state $\frac{1}{\sqrt{2}}(\|00\rangle+\|11\rangle)_{23}$, because particles 1 and 2 belong to Alice. A Controlled-Not gate can transform $\frac{1}{\sqrt{2}}\|0\rangle_{1}(\|00\rangle+\|11\rangle)_{23}$ into $\|\Phi\rangle$, when particle 2 and particle 1 are a controlled qubit and a target qubit, respectively. We suppose the qubit $\|\psi\rangle$ is known to Alice, i.e. Alice knows $\theta$ and $\varphi$ completely, but Bob does not know them at all. Since Alice knows the state she can choose to measure the particles 1 and 2 in any basis she wants. First, Alice performs a projective measurement on particle 1. The measurement basis chosen by Alice is a set of mutually orthogonal basis vectors $\\{\|\phi\rangle,\|\phi_{\perp}\rangle\\}$, which is related to the computation basis $\\{\|0\rangle,\|1\rangle\\}$ in the following manner $\displaystyle\|\phi\rangle_{1}=\cos(\theta/2)\|0\rangle_{1}+\sin(\theta/2)\|1\rangle_{1},$ $\displaystyle\|\phi_{\perp}\rangle_{1}=\sin(\theta/2)\|0\rangle_{1}-\cos(\theta/2)\|1\rangle_{1}.$ (4) By this change of basis, the normalization and orthogonality relation between basis vectors are preserved. Using Eq.(4), we can express Eq.(3) as $\|\Phi\rangle=\frac{1}{\sqrt{2}}(\|\phi\rangle_{1}\|\Psi\rangle_{23}+\|\phi_{\perp}\rangle_{1}\|\Psi_{\perp}\rangle_{23}),$ (5) where $\displaystyle\|\Psi\rangle_{23}=\cos(\theta/2)\|00\rangle_{23}+\sin(\theta/2)\|11\rangle_{23},$ $\displaystyle\|\Psi_{\perp}\rangle_{23}=\sin(\theta/2)\|00\rangle_{23}-\cos(\theta/2)\|11\rangle_{23}.$ (6) Now Alice measures the particle 1. For example, if Alice’s von Neumann measurement result is $\|\phi\rangle_{1}$, then the state of particles 2 and 3, as shown by Eq.(5), will collapse into $\|\Psi\rangle_{23}$. Next, Alice performs another projective measurement on particle 2. The measurement basis is also a set of mutually orthogonal basis vectors $\\{\|\eta\rangle,\|\eta_{\perp}\rangle\\}$, the relation between the measurement basis $\\{\|\eta\rangle,\|\eta_{\perp}\rangle\\}$ and the computation basis $\\{\|0\rangle,\|1\rangle\\}$ is given by $\|\eta\rangle_{2}=\frac{1}{\sqrt{2}}(\|0\rangle_{2}+e^{-i\varphi}\|1\rangle_{2}),~{}~{}\|\eta_{\perp}\rangle_{2}=\frac{1}{\sqrt{2}}(\|0\rangle_{2}-e^{-i\varphi}\|1\rangle_{2}).$ (7) Then, we have $\|\Psi\rangle_{23}=\frac{1}{\sqrt{2}}(\|\eta\rangle_{2}\|\psi\rangle_{3}+\|\eta_{\perp}\rangle_{2}\|\psi^{\prime}\rangle_{3}),$ (8) where $\displaystyle\|\psi\rangle=\cos(\theta/2)\|0\rangle+\sin(\theta/2)e^{i\varphi}\|1\rangle,$ $\displaystyle\|\psi^{\prime}\rangle=\cos(\theta/2)\|0\rangle-\sin(\theta/2)e^{i\varphi}\|1\rangle.$ (9) If Alice’s von Neumann measurement result is $\|\eta\rangle_{2}$, the particle 3 can be found in the original state $\|\psi\rangle$, which is nothing but the remote state preparation of the known qubit. If the outcome of Alice’s measurement result is $\|\eta_{\perp}\rangle_{2}$, then the classical communication from Alice will tell Bob that he has obtained a state $\|\psi^{\prime}\rangle$. Bob can carry out the unitary operation $\sigma_{z}=\|0\rangle\langle 0\|-\|1\rangle\langle 1\|$ on his particle 3. That is $\sigma_{z}\|\psi^{\prime}\rangle=\cos(\theta/2)\|0\rangle+\sin(\theta/2)e^{i\varphi}\|1\rangle=\|\psi\rangle.$ (10) This means after Bob’s unitary operation the state $\|\psi\rangle$ has already been prepared in Bob’s qubit. Surely it is possible for Alice to get the state $\|\phi_{\perp}\rangle_{1}$ after her measurement on particle 1. If so, she will choose another measurement basis $\\{\|\xi\rangle,\|\xi_{\perp}\rangle\\}$ on particle 2, which are written as $\|\xi\rangle_{2}=\frac{1}{\sqrt{2}}(\|1\rangle_{2}+e^{-i\varphi}\|0\rangle_{2}),~{}~{}\|\xi_{\perp}\rangle_{2}=\frac{1}{\sqrt{2}}(\|1\rangle_{2}-e^{-i\varphi}\|0\rangle_{2}).$ (11) Obviously, the basis vectors $\\{\|\xi\rangle,\|\xi_{\perp}\rangle\\}$ and $\\{\|\eta\rangle,\|\eta_{\perp}\rangle\\}$ can be mutually converted by a unitary operation $\sigma_{x}=\|0\rangle\langle 1\|+\|1\rangle\langle 0\|$. After Alice’s measurement, for each collapsed state Bob can employ an appropriate unitary operation to convert it to the prepared state $\|\psi\rangle$ except for an overall trivial factor. Here we do not depict them one by one anymore. As a summary, Bob’s corresponding unitary operations to Alice’s measurement results are listed in Table I. One can easily work out that the total probability of RSP is 1 though the classical communication cost is 2 bits. Table 1: Alice’s measurement basis on particle 1 (MB1), Alice’s measurement outcome for particle 1 (AMO1), Alice’s measurement basis on particle 2 (MB2), Alice’s measurement outcome for particle 2 (AMO2), the collapse states for particle 3 (CS3) and Bob’s appropriate unitary operation (BAUO) MB1 \| AMO1 \| MB2 \| AMO2 \| CS3 \| BAUO ---\|---\|---\|---\|---\|--- $\\{\|\phi\rangle,\|\phi_{\perp}\rangle\\}$ \| $\|\phi\rangle_{1}$ \| $\\{\|\eta\rangle,\|\eta_{\perp}\rangle\\}$ \| $\|\eta\rangle_{2}$ \| $\begin{array}[]{c}\cos(\theta/2)\|0\rangle+\\\ \sin(\theta/2)e^{i\varphi}\|1\rangle\end{array}$ \| $I$ $\\{\|\phi\rangle,\|\phi_{\perp}\rangle\\}$ \| $\|\phi\rangle_{1}$ \| $\\{\|\eta\rangle,\|\eta_{\perp}\rangle\\}$ \| $\|\eta_{\perp}\rangle_{2}$ \| $\begin{array}[]{c}\cos(\theta/2)\|0\rangle-\\\ \sin(\theta/2)e^{i\varphi}\|1\rangle\end{array}$ \| $\sigma_{z}$ $\\{\|\phi\rangle,\|\phi_{\perp}\rangle\\}$ \| $\|\phi_{\perp}\rangle_{1}$ \| $\\{\|\xi\rangle,\|\xi_{\perp}\rangle\\}$ \| $\|\xi\rangle_{2}$ \| $\begin{array}[]{c}\sin(\theta/2)e^{i\varphi}\|0\rangle-\\\ \\-\cos(\theta/2)\|1\rangle\end{array}$ \| $\sigma_{z}\sigma_{x}$ $\\{\|\phi\rangle,\|\phi_{\perp}\rangle\\}$ \| $\|\phi_{\perp}\rangle_{1}$ \| $\\{\|\xi\rangle,\|\xi_{\perp}\rangle\\}$ \| $\|\xi_{\perp}\rangle_{2}$ \| $\begin{array}[]{c}\sin(\theta/2)e^{i\varphi}\|0\rangle+\\\ \cos(\theta/2)\|1\rangle\end{array}$ \| $\sigma_{x}$ By the above analysis, one can easily see that unlike the standard teleportation of an unknown qubit, here, we do not require a Bell-state measurement, which is still more difficult according to the present-day technologies. Only single-particle von Neumann measurement and local operation are necessary. On the other hand, the total probability of RSP for an arbitrary qubit is 1 while in the previous schemes only the probability of RSP of some special ensembles of qubit is 1. In addition, what deserves mentioning here is that in this protocol, the overall information of the qubit, which is expressed by $\theta$ and $\varphi$, can be divided into two parts. We must first prepare the part $\theta$ and then prepare the remainder part $\varphi$, which can not be transposed. This indicates that the two parts of information are not equal with each other. As mentioned above, we need only the single-particle measurement and local operation. So, the particle 1 and 2 may be at different locations. In this case, $\|\Phi\rangle$ is a real GHZ state. It is natural to generate it to the three-party RSP. ## III RSP of higher-dimensional quantum state for three parties In this section, we wish to generalize the RSP protocol to systems with larger than two-dimensional Hilbert space among three parties. First we consider the case that two parties (Alice and Bob) collaborate with each other to prepare a 4-dimensional quantum state at Charlie’s location. A quantum state $\displaystyle\|\psi\rangle=$ $\displaystyle\cos\gamma_{1}\|0\rangle+\sin\gamma_{1}\cos\gamma_{2}e^{i\alpha_{1}}\|1\rangle$ (12) $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\cos\gamma_{3}e^{i\alpha_{2}}\|2\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\sin\gamma_{3}e^{i\alpha_{3}}\|3\rangle$ in a four-dimensional Hilbert space can be parameterized by six parameters $\gamma_{1},\gamma_{2},\gamma_{3},\alpha_{1},\alpha_{2}$ and $\alpha_{3}$ such that $0\leq\gamma_{1},\gamma_{2},\gamma_{3}\leq\pi/2$ and $0\leq\alpha_{1},\alpha_{2},\alpha_{3}\leq 2\pi$. Alice and Bob know $\gamma_{1},\gamma_{2},\gamma_{3}$ and $\alpha_{1},\alpha_{2}$ and $\alpha_{3}$ partly respectively, that is, Alice has information of $\gamma_{1},\gamma_{2},\gamma_{3}$, and Bob has information $\alpha_{1},\alpha_{2}$ and $\alpha_{3}$. The quantum channel shared by Alice, Bob and Charlie is a 4-level maximally GHZ state $\|\Phi\rangle_{ABC}=\frac{1}{2}(\|000\rangle+\|111\rangle+\|222\rangle+\|333\rangle)_{ABC},$ (13) where particle $A$, $B$ and $C$ belong to Alice, Bob and Charlie respectively. The method is similar to the case of qubit. First Alice must find a set of orthogonal basis vectors to perform a generalized projective measurement on particle $A$. We shall see below, there exist many sets of orthogonal basis vectors that include the state (12). One such set can be obtained by applying a specific unitary transformation on the computational basis vectors $\displaystyle U(\gamma_{1},\gamma_{2},\gamma_{3})\|0\rangle=\|\phi_{0}\rangle=$ $\displaystyle\cos\gamma_{1}\|0\rangle+\sin\gamma_{1}\cos\gamma_{2}\|1\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\cos\gamma_{3}\|2\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\sin\gamma_{3}\|3\rangle,$ $\displaystyle U(\gamma_{1},\gamma_{2},\gamma_{3})\|1\rangle=\|\phi_{1}\rangle=$ $\displaystyle-\sin\gamma_{1}\cos\gamma_{2}\|0\rangle+\cos\gamma_{1}\|1\rangle$ $\displaystyle-\sin\gamma_{1}\sin\gamma_{2}\sin\gamma_{3}\|2\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\cos\gamma_{3}\|3\rangle,$ $\displaystyle U(\gamma_{1},\gamma_{2},\gamma_{3})\|2\rangle=\|\phi_{2}\rangle=$ $\displaystyle-\sin\gamma_{1}\sin\gamma_{2}\cos\gamma_{3}\|0\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\sin\gamma_{3}\|1\rangle$ $\displaystyle+\cos\gamma_{1}\|2\rangle-\sin\gamma_{1}\cos\gamma_{2}\|3\rangle,$ $\displaystyle U(\gamma_{1},\gamma_{2},\gamma_{3})\|3\rangle=\|\phi_{3}\rangle=$ $\displaystyle\sin\gamma_{1}\sin\gamma_{2}\sin\gamma_{3}\|0\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\cos\gamma_{3}\|1\rangle$ $\displaystyle-\sin\gamma_{1}\cos\gamma_{2}\|2\rangle-\cos\gamma_{1}\|3\rangle.$ Then we have $\displaystyle\|\Phi\rangle_{ABC}=$ $\displaystyle\frac{1}{2}(\|\phi_{0}\rangle_{A}\|\Psi_{0}\rangle_{BC}+\|\phi_{1}\rangle_{A}\|\Psi_{1}\rangle_{BC}$ (15) $\displaystyle+\|\phi_{2}\rangle_{A}\|\Psi_{2}\rangle_{BC}+\|\phi_{3}\rangle_{A}\|\Psi_{3}\rangle_{BC}),$ where $\displaystyle\|\Psi_{0}\rangle_{BC}=$ $\displaystyle\cos\gamma_{1}\|00\rangle+\sin\gamma_{1}\cos\gamma_{2}\|11\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\cos\gamma_{3}\|22\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\sin\gamma_{3}\|33\rangle,$ $\displaystyle\|\Psi_{1}\rangle_{BC}=$ $\displaystyle-\sin\gamma_{1}\cos\gamma_{2}\|00\rangle+\cos\gamma_{1}\|11\rangle$ $\displaystyle-\sin\gamma_{1}\sin\gamma_{2}\sin\gamma_{3}\|22\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\cos\gamma_{3}\|33\rangle,$ $\displaystyle\|\Psi_{2}\rangle_{BC}=$ $\displaystyle-\sin\gamma_{1}\sin\gamma_{2}\cos\gamma_{3}\|00\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\sin\gamma_{3}\|11\rangle$ $\displaystyle+\cos\gamma_{1}\|22\rangle-\sin\gamma_{1}\cos\gamma_{2}\|33\rangle,$ $\displaystyle\|\Psi_{3}\rangle_{BC}=$ $\displaystyle\sin\gamma_{1}\sin\gamma_{2}\sin\gamma_{3}\|00\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\cos\gamma_{3}\|11\rangle$ $\displaystyle-\sin\gamma_{1}\cos\gamma_{2}\|22\rangle-\cos\gamma_{1}\|33\rangle.$ After Alice measures particle $A$, the initial state will be projected onto the measurement basis vectors with the appropriate probability. She has to convey to Bob by classical communication whether to apply the corresponding unitary transformation $\displaystyle U_{1}=\left(\begin{array}[]{cccc}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\\\ \end{array}\right),$ (21) $\displaystyle U_{2}=\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\ -1&0&0&0\\\ 0&1&0&0\\\ \end{array}\right),$ (26) $\displaystyle U_{3}=\left(\begin{array}[]{cccc}0&0&0&-1\\\ 0&0&-1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\ \end{array}\right)$ (31) on his particle $B$ or do nothing. It means that Alice’s measurement outcomes $\|\phi_{0}\rangle$, $\|\phi_{1}\rangle$, $\|\phi_{2}\rangle$, and $\|\phi_{3}\rangle$ correspond to unitary transformations $I$, $U_{1}$, $U_{2}$, and $U_{3}$, respectively. Here $I$ is the identity operator. Next Bob constructs a measurement basis and performs another projective measurement on particle $B$, the relation between the measurement basis $\\{\eta_{0},\eta_{1},\eta_{2},\eta_{3}\\}$ and the computational basis $\\{\|0\rangle,\|1\rangle,\|2\rangle,\|3\rangle$ is given by $\displaystyle\|\eta_{0}\rangle=\frac{1}{2}(\|0\rangle+e^{-i\alpha_{1}}\|1\rangle+e^{-i\alpha_{2}}\|2\rangle+e^{-i\alpha_{3}}\|3\rangle),$ $\displaystyle\|\eta_{1}\rangle=\frac{1}{2}(\|0\rangle+ie^{-i\alpha_{1}}\|1\rangle-e^{-i\alpha_{2}}\|2\rangle- ie^{-i\alpha_{3}}\|3\rangle),$ $\displaystyle\|\eta_{2}\rangle=\frac{1}{2}(\|0\rangle- ie^{-i\alpha_{1}}\|1\rangle-e^{-i\alpha_{2}}\|2\rangle+ie^{-i\alpha_{3}}\|3\rangle),$ $\displaystyle\|\eta_{3}\rangle=\frac{1}{2}(\|0\rangle-e^{-i\alpha_{1}}\|1\rangle+e^{-i\alpha_{2}}\|2\rangle-e^{-i\alpha_{3}}\|3\rangle).$ (32) After Bob measures particle $B$, he will inform Charlie of his measurement result via a classical communication. Charlie can employ an appropriate unitary operation to convert it to the prepared state $\|\psi\rangle$. For example, if Alice’s measurement result is $\|\phi_{1}\rangle_{A}$, the state of particle $B$ and $C$, as shown in Eqs. (15) and (16), will collapse into $\|\Psi_{1}\rangle_{BC}$. After Bob receives Alice’s measurement result $\|\phi_{1}\rangle_{A}$, he first carries out the unitary transformation $U_{1}$ described in Eq.(17) on particle $B$. That is, the unitary operation $U_{1}$ will transform the state $\|\Psi_{1}\rangle_{BC}$ into $\displaystyle\|\Psi^{\prime}_{1}\rangle=$ $\displaystyle\sin\gamma_{1}\cos\gamma_{2}\|10\rangle+\cos\gamma_{1}\|01\rangle$ $\displaystyle+\sin\gamma_{1}\sin\gamma_{2}\sin\gamma_{3}\|32\rangle+\sin\gamma_{1}\sin\gamma_{2}\cos\gamma_{3}\|23\rangle.$ Next, Bob performs the projective measurement on particle $B$ in the basis described in Eq.(18). According to Bob’s different measurement result $\|\eta_{i}\rangle$, Charlie needs to perform the corresponding unitary operation $U_{i}(C)$ on particle $C$, $U_{i}(C)$ may take the form of the following $4\times 4$ matrix $\displaystyle U_{0}(C)=\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\ \end{array}\right),$ (38) $\displaystyle U_{1}(C)=\left(\begin{array}[]{cccc}0&1&0&0\\\ i&0&0&0\\\ 0&0&0&-1\\\ 0&0&-i&0\\\ \end{array}\right),$ (43) $\displaystyle U_{2}(C)=\left(\begin{array}[]{cccc}0&1&0&0\\\ -i&0&0&0\\\ 0&0&0&-1\\\ 0&0&i&0\\\ \end{array}\right),$ (48) $\displaystyle U_{3}(C)=\left(\begin{array}[]{cccc}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\\\ \end{array}\right).$ (53) The RSP is completed. Similarly, for other collapsed state corresponding to Alice’s measurement result, Bob can employ an appropriate unitary operation in Eq.(17) or do nothing and perform the projective measurement on particle $B$ in the basis described in Eq.(18). Here we do not depict them one by one anymore. As a summary, Bob’s measurement outcomes corresponding to Alice’s other measurement results, and Charlie’s corresponding unitary operations to Bob’s measurement results are listed in Table II. Table 2: Alice’s measurement outcome for particle $A$ (AMO), Bob’s measurement outcome for particle $B$ (BMO), and Charlie’s appropriate unitary operation (CAUO) AMO \| BMO \| CAUO ---\|---\|--- $\|\phi_{0}\rangle_{A}$ \| $\|\eta_{0}\rangle_{B}$ \| $I$ $\|\phi_{0}\rangle_{A}$ \| $\|\eta_{1}\rangle_{B}$ \| ${\rm diag}(1,i,-1,-i)$ $\|\phi_{0}\rangle_{A}$ \| $\|\eta_{2}\rangle_{B}$ \| ${\rm diag}(1,-i,-1,i)$ $\|\phi_{0}\rangle_{A}$ \| $\|\eta_{3}\rangle_{B}$ \| ${\rm diag}(1,-1,1,-1)$ $\|\phi_{2}\rangle_{A}$ \| $\|\eta_{0}\rangle_{B}$ \| $\left(\begin{array}[]{cc}0&A_{1}\\\ A_{1}&0\end{array}\right),A_{1}={\rm diag}(1,1)$ $\|\phi_{2}\rangle_{A}$ \| $\|\eta_{1}\rangle_{B}$ \| $\left(\begin{array}[]{cc}0&A_{2}\\\ -A_{2}&0\end{array}\right),A_{2}={\rm diag}(1,i)$ $\|\phi_{2}\rangle_{A}$ \| $\|\eta_{2}\rangle_{B}$ \| $\left(\begin{array}[]{cc}0&A_{3}\\\ -A_{3}&0\end{array}\right),A_{3}={\rm diag}(1,-i)$ $\|\phi_{2}\rangle_{A}$ \| $\|\eta_{3}\rangle_{B}$ \| $\left(\begin{array}[]{cc}0&A_{4}\\\ A_{4}&0\end{array}\right),A_{4}={\rm diag}(1,-1)$ $\|\phi_{3}\rangle_{A}$ \| $\|\eta_{0}\rangle_{B}$ \| $\left(\begin{array}[]{cc}0&A_{5}\\\ A_{5}&0\end{array}\right),A_{5}=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$ $\|\phi_{3}\rangle_{A}$ \| $\|\eta_{1}\rangle_{B}$ \| $\left(\begin{array}[]{cc}0&A_{6}\\\ -A_{6}&0\end{array}\right),A_{6}=\left(\begin{array}[]{cc}0&1\\\ i&0\end{array}\right)$ $\|\phi_{3}\rangle_{A}$ \| $\|\eta_{2}\rangle_{B}$ \| $\left(\begin{array}[]{cc}0&A_{7}\\\ -A_{7}&0\end{array}\right),A_{7}=\left(\begin{array}[]{cc}0&1\\\ -i&0\end{array}\right)$ $\|\phi_{3}\rangle_{A}$ \| $\|\eta_{3}\rangle_{B}$ \| $\left(\begin{array}[]{cc}0&A_{8}\\\ A_{8}&0\end{array}\right),A_{8}=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$ By the above analysis, we may conclude that the essence of our protocol is first preparing a point of the polar circle, and then adding the information of the equatorial state. Now, we suppose that Alice and Bob want to remotely prepare a known $d$-level quantum state at Charlie’s location. However, not all the qudits can be remotely prepared according to Ref. [6], in which the authors have shown that the qudit in real Hilbert space can be remotely prepared when the dimension is 2, 4, or 8. So here we let $d=8$. The state of an eight-dimensional system can be written as $\|\psi\rangle=\sum^{7}_{i=0}\cos\theta_{i}e^{i\varphi_{i}}\|i\rangle,~{}~{}\sum_{i=0}^{7}\|\cos\theta_{i}\|^{2}=1.$ (54) Without loss of generality, we set $\varphi_{0}=0$. According to the analogous procedure described above, the corresponding qudit to be prepared can be remotely prepared exactly onto the particle at Charlie’s location. The measurement basis chosen by Alice can be obtained by $V_{i}\|\psi\rangle$. The unitary operation $V_{i}$ needed is the same as those for eight-dimensional RSP in Ref. [6]. Bob’s measurement basis is written as $\\{\|\eta_{j}\rangle=\sum^{7}_{k=0}e^{(\pi i/4)jk}e^{i\varphi_{j}}\|k\rangle\\}^{7}_{j=0}.$ ## IV Conclusions In summary, we have presented a two-step protocol for the exact remote state preparation of an arbitrary qubit using one three-particle GHZ state as the quantum channel. Only a single-particle von Neumann measurement and local operation are necessary. It has been shown that the overall information of the qubit, can be divided into two different parts, which are expressed by $\theta$ and $\varphi$ respectively. We must first prepare the part $\theta$ and then prepare the remainder part $\varphi$, which can not be transposed. This indicates that the two parts of information are not equal with each other. Generalization of this protocol for higher-dimensional Hilbert space systems among three parties is also presented. Moreover, it should be noticed that in this protocol, the information $\theta$ and $\varphi$ may be at different locations. So this protocol may be useful in the quantum information field, such as quantum state sharing, converging the split information at one point, etc. We hope this will provide new insight for investigating more extensive quantum information processing procedures. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant No: 10671054, Hebei Natural Science Foundation of China under Grant No: 07M006, and the Key Project of Science and Technology Research of Education Ministry of China under Grant No: 207011. ## References * (1) C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). * (2) H.K. Lo, Phys. Rev. A 62, 012313 (2000). * (3) A.K. Pati, Phys. Rev. A 63, 014302 (2001). * (4) C.H. Bennett, D.P. DiVincenzo, P.W. Shor, J.A. Smolin, B.M. Terhal, and W.K. Wootters, Phys. Rev. Lett. 87, 077902 (2001). * (5) I. Devetak and T. Berger, Phys. Rev. Lett. 87, 197901 (2001). * (6) B. Zeng and P. Zhang, Phys. Rev. A 65, 022316 (2002). * (7) D.W. Berry and B.C. Sanders, Phys. Rev. Lett. 90, 057901 (2003). * (8) D.W. Leung and P.W. Shor, Phys. Rev. Lett. 90, 127905 (2003); Z. Kurucz, P. Adam, and J. Janszky, Phys. Rev. A 73, 062301 (2006). * (9) A. Hayashi, T. Hashimoto, and M. Horibe, Phys. Rev. A 67, 052302 (2003). * (10) Y.F. Yu, J. Feng, and M.S. Zhan, Phys. Lett. A 310, 329 (2003); Y.X. Huang and M.S. Zhan, Phys. Lett. A 327, 404 (2004). * (11) M.G.A. Paris, M. Cola, and R. Bonifacio, J. Opt. B 5, s360 (2003). * (12) Z. Kurucz, P. Adam, Z. Kis, and J. Janszky, Phys. Rev. A 72, 052315 (2005). * (13) H.Y. Dai, P.X. Chen, L.M. Liang, and C.Z. Li, Phys. Lett. A 355, 285 (2006). * (14) C.S. Yu, H.S. Song, and Y.H. Wang, Phys. Rev. A 73, 022340 (2006). * (15) F.L. Yan and G.H. Zhang, International Journal of Quantum Information 6, 485 (2008). * (16) X. Peng, X. Zhu, X. Fang, M. Feng, M. Liu, and K. Gao, Phys. Lett. A 306, 271 (2003). * (17) G.Y. Xiang, J. Li, Y. Bo, and G.C. Guo, Phys. Rev. A 72, 012315 (2005). * (18) N.A. Peters, J.T. Barreiro, M.E. Goggin, T.C. Wei, and P.G. Kwiat, Phys. Rev. Lett. 94, 150502 (2005). * (19) B.S. Shi and A. Tomita, J. Opt. B 4, 380 (2002). * (20) J.M. Liu and Y.Z. Wang, Phys. Lett. A 316, 159 (2003).	arxiv-papers	2009-02-27T09:49:42	2024-09-04T02:49:00.888926	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Meiyu Wang, Fengli Yan", "submitter": "Ting Gao", "url": "https://arxiv.org/abs/0902.4782" }
0902.4811	# CP violation, massive neutrinos, and its chiral condensate: new results from Snyder noncommutative geometry Łukasz Andrzej Glinka laglinka@gmail.com _International Institute for Applicable_ _Mathematics & Information Sciences,_ _Hyderabad (India) & Udine (Italy),_ _B.M. Birla Science Centre,_ _Adarsh Nagar, 500 063 Hyderabad, India_ ###### Abstract The Snyder model of a noncommutative geometry due to a minimal scale $\ell$, _e.g._ the Planck or the Compton scale, yields $\ell^{2}$-shift within the Einstein Hamiltonian constraint, and $\gamma^{5}$-term in the free Dirac equation violating CP symmetry manifestly. In this paper the Dirac equation is reconsidered. In fact, there is no any reasonable cause for modification of the Minkowski hyperbolic geometry of a momentum space. It is the consistency – in physics phase space, spacetime (coordinates), and momentum space (dynamics) are independent mathematical structures. It is shown that the modified Dirac equation yields the kinetic mass generation mechanism for the left- and right-handed Weyl chiral fields, and realizes the idea of neutrinos receiving mass due to CP violation. It is shown that the model is equivalent to the gauge field theory of composed two 2-flavor massive fields. The global chiral symmetry spontaneously broken into the isospin group leads to the chiral condensate of massive neutrinos. This result is beyond the Standard Model, but in general can be included into the theory of elementary particles and fundamental interactions. ## 1 Introduction In 1947 the American physicist H. S. Snyder, for elimination of the infrared catastrophe in the Compton effect, proposed employing the model [1] $\dfrac{i}{\hbar}[x,p]=1+\alpha\left(\dfrac{\ell}{\hbar}\right)^{2}p^{2}\quad,\quad\dfrac{i}{\hbar}[x,y]=O(\ell^{2})\quad,$ (1) with $p$ \- a particle’s momentum, $x$, $y$ \- space points, $\ell$ \- a fundamental length scale, $\hbar$ \- the Planck constant, $\alpha\sim 1$ \- a dimensionless constant, $[\cdot,\cdot]$ \- an appropriate Lie bracket. For the Lorentz and Poincaré invariance modified due to $\ell$, Snyder considered a momentum space of constant curvature isometry group, _i.e._ the Poincaré algebra deformation into the De Sitter one. The model (1) is a noncommutative geometry and a deformation (Basics and applications: _e.g._ Ref. [2]). Let us see first it in some general detail. Let $A$ \- an associative Lie algebra, $\tilde{A}=A[[\lambda]]$ \- the module due to the ring of formal series $\mathbb{K}[[\lambda]]$ in a parameter $\lambda$. A deformation of $A$ is a $\mathbb{K}[[\lambda]]$-algebra $\tilde{A}$ such that $\tilde{A}/\lambda\tilde{A}\approx A$. If $A$ is endowed with a locally convex topology with continuous laws, _i.e._ a topological algebra, then $\tilde{A}$ is topologically free. If in $A$ composition law is ordinary product and related bracket is $[\cdot,\cdot]$, then $\tilde{A}$ is associative Lie algebra if for $f,g\in A$ a new product $\star$ and bracket $[\cdot,\cdot]_{\star}$ are $\displaystyle f\star g$ $\displaystyle=$ $\displaystyle fg+\sum_{n=1}^{\infty}\lambda^{n}C_{n}(f,g),$ (2) $\displaystyle\left[f,g\right]_{\star}$ $\displaystyle\equiv$ $\displaystyle f\star g-g\star f=\left[f,g\right]+\sum_{n=1}^{\infty}\lambda^{n}B_{n}(f,g),$ (3) where $C_{n}$, $B_{n}$ are the Hochschild and Chevalley 2-cochains, and for $f,g,h\in A$ hold $(f\star g)\star h=f\star(g\star h)$ and $[[f,g]_{\star},h]_{\star}+[[h,f]_{\star},g]_{\star}+[[g,h]_{\star},f]_{\star}=0$. For each $n$ and $j,k\geqslant 1$, $j+k=n$ the equations are satisfied $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!bC_{n}(f,g,h)\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\sum_{j,k}\left[C_{j}\left(C_{k}(f,g),h\right)-C_{j}\left(f,C_{k}(g,h)\right)\right],$ (4) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\partial B_{n}(f,g,h)\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!\\!\sum_{j,k}\left[B_{j}\left(B_{k}(f,g),h\right)+B_{j}\left(B_{k}(h,f),g\right)+B_{j}\left(B_{k}(g,h),f\right)\right],$ (5) where $b$, $\partial$ are the Hochschild and Chevalley coboundary operators - $b^{2}=0$, $\partial^{2}=0$. Let $C^{\infty}(M)$ \- an algebra of smooth functions on a differentiable manifold $M$. Associativity yields the Hochschild cohomologies. An antisymmetric contravariant 2-tensor $\theta$ trivializing the Schouten–Nijenhuis bracket $[\theta,\theta]_{SN}=0$ on $M$, defines the Poisson bracket $\\{f,g\\}=i\theta df\wedge dg$ with the Jacobi identity and the Leibniz rule; $(M,\\{\cdot,\cdot\\})$ is called a Poisson manifold. In 1997 the Russian mathematician M. L. Kontsevich [3] defined deformation quantization of a general Poisson differentiable manifold. Let $\mathbb{R}^{d}$ endowed with a Poisson bracket $\alpha(f,g)=\sum_{1\leqslant i,j\eqslantless n}\alpha^{ij}\partial_{i}f\partial_{j}g$, $\partial_{k}=\partial/\partial x^{k}$, $1\leqslant k\leqslant d$. For $\star$-product, $n\geqslant 0$, exists a family $G_{n,2}$ of $(n(n+1))^{n}$ oriented graphs $\Gamma$. $V_{\Gamma}$ \- the set of vertices of $\Gamma$; has $n+2$ elements - 1st type $\\{1,\ldots,n\\}$, 2nd type $\\{\bar{1},\bar{2}\\}$. $E_{\Gamma}$ \- the set of oriented edges of $\Gamma$; has $2n$ elements. There is no edge starting at a 2nd type vertex. Star(k) - $E_{\Gamma}$ starting at a 1st type vertex $k$ with cardinality $\sharp k=2$, $\sum_{1\leqslant k\leqslant n}\sharp k=2n$. $\\{e^{1}_{k},\ldots,e^{\sharp k}_{k}\\}$ are the edges of $\Gamma$ starting at vertex $k$. Vortices starting and ending in the edge $v$ are $v=(s(v),e(v))$, $s(v)\in\\{1,\ldots,n\\}$ and $e(v)\in\\{1,\ldots,n;\bar{1},\bar{2}\\}$. $\Gamma$ has no loop and no parallel multiple edges. A bidifferential operator $(f,g)\mapsto B_{\Gamma}(f,g)$, $f,g\in C^{\infty}(\mathbb{R}^{d})$ is associated to $\Gamma$. $\alpha^{e^{1}_{k}e^{2}_{k}}$ are associated to each 1st type vertex $k$ from where the edges $\\{e^{1}_{k},e^{2}_{k}\\}$ start; $f$ is the vertex $1$, $g$ is the vertex $\bar{2}$. Edge $e^{1}_{k}$ acts $\partial/\partial x^{e^{1}_{k}}$ on its ending vertex. $B_{\Gamma}$ is a sum over all maps $I:E_{\Gamma}\rightarrow\\{1,\ldots,d\\}$ $\\!\\!\\!B_{\Gamma}(f,g)=\sum_{I}\left(\prod_{k=1}^{n}\prod_{k^{\prime}=1}^{n}\partial_{I(k^{\prime},k)}\alpha^{I(e^{1}_{k})I(e^{2}_{k})}\right)\\!\\!\left(\prod_{k_{1}=1}^{n}\partial_{I(k_{1},\bar{1})}f\right)\\!\\!\left(\prod_{k_{2}=1}^{n}\partial_{I(k_{2},\bar{2})}g\right).\\!\\!\\!$ (6) Let $\mathcal{H}_{n}$ \- an open submanifold of $\mathbb{C}^{n}$, the configuration space of $n$ distinct points in $\mathcal{H}=\\{x\in\mathbb{C}\|\Im(z)>0\\}$ with the Lobachevsky hyperbolic metric. For the vertex $k$, $1\leqslant k\leqslant n$, $z_{k}\in\mathcal{H}$ \- a variable associated to $\Gamma$. The vertex $1$ associated to $0\in\mathbb{R}$, the vertex $\bar{2}$ to $1\in\mathbb{R}$. If $\tilde{\phi}_{v}=\phi(s(v),e(v))$ \- a function on $\mathcal{H}_{n}$, associated to $v$, with $\phi:\mathcal{H}_{2}\rightarrow\mathbb{R}/2\pi\mathbb{Z}$ \- the angle function $\phi(z_{1},z_{2})=\mathrm{Arg}\dfrac{z_{2}-z_{1}}{z_{2}-\bar{z}_{1}}=\dfrac{1}{2i}\mathrm{Log}\dfrac{\bar{z}_{2}-z_{1}}{z_{2}-\bar{z}_{1}}\dfrac{z_{2}-z_{1}}{\bar{z}_{2}-\bar{z}_{1}},$ (7) then $w(\Gamma)\in\mathbb{R}$, the integral of $2n$-form, is a weight associated to $\Gamma\in G_{n,2}$ $\displaystyle w(\Gamma)=\dfrac{1}{n!(2\pi)^{2n}}\int_{\mathcal{H}_{n}}\bigwedge_{1\leqslant k\leqslant n}\left(d\tilde{\phi}_{e^{1}_{k}}\wedge d\tilde{\phi}_{e^{2}_{k}}\right).$ (8) The weight does not depend on the Poisson structure or the dimension $d$. On $(\mathbb{R}^{d},\alpha)$ the Kontsevich $\star$-product maps $C^{\infty}(\mathbb{R})\times C^{\infty}(\mathbb{R})\rightarrow C^{\infty}(\mathbb{R})[[\lambda]]$ $(f,g)\mapsto f\star g=\sum_{n\geqslant 0}\lambda^{n}C_{n}(f,g)\quad,\quad C_{n}(f,g)=\sum_{\Gamma\in G_{n,2}}w(\Gamma)B_{\Gamma}(f,g),$ (9) with $C_{0}(f,g)=fg$, $C_{1}(f,g)=\\{f,g\\}_{\alpha}=\alpha df\wedge dg$. Equivalence classes of (9) are bijective to the Poisson brackets $\alpha_{\lambda}=\sum_{k\geqslant 0}\lambda^{k}\alpha_{k}$ ones. For linear Poisson structures, _i.e._ on coalgebra $A^{\star}$, (8) of all even wheel graphs vanishes, and (9) coincides with the $\star$-product given by the Duflo isomorphism. This case allows to quantize the class of quadratic Poisson brackets that are in the image of the Drinfeld map which associates a quadratic to a linear bracket. Let us consider the deformations of phase-space and space given by the parameters $\lambda_{ph}$, $\lambda_{s}$ being $\lambda_{ph}=\dfrac{\alpha i\hbar}{2}\quad,\quad\lambda_{s}=\dfrac{i\beta}{2}\quad,\quad\alpha\sim 1,$ (10) and leading to the star products (2), or equivalently the Kontsevich ones (9), on the phase space $(x,p)$ and between two distinct space points $x$ and $y$ $\displaystyle x\star p$ $\displaystyle=$ $\displaystyle px+\sum_{n=1}^{\infty}\left(\dfrac{\alpha i\hbar}{2}\right)^{n}C_{n}(x,p),$ (11) $\displaystyle x\star y$ $\displaystyle=$ $\displaystyle xy+\sum_{n=1}^{\infty}\left(\dfrac{i\beta}{2}\right)^{n}C_{n}(x,y),$ (12) where $C_{n}(x,p)$, $C_{n}(x,y)$ are the appropriate Hochschild cochains in (9). The brackets arising from the star products (11) and (12) are $\displaystyle\left[x,p\right]_{\star}$ $\displaystyle=$ $\displaystyle\left[x,p\right]+\sum_{n=1}^{\infty}\left(\dfrac{\alpha i\hbar}{2}\right)^{n}B_{n}(x,p),$ (13) $\displaystyle\left[x,y\right]_{\star}$ $\displaystyle=$ $\displaystyle\left[x,y\right]+\sum_{n=1}^{\infty}\left(\dfrac{i\beta}{2}\right)^{n}B_{n}(x,y),$ (14) where $B_{n}(x,p)$, $B_{n}(x,y)$ are the Chevalley cochains. By using $[x,p]=-i\hbar$ and $[x,y]=0$, and taking the first approximation of (13) and (14) one obtains $\displaystyle\left[x,p\right]_{\star}=-i\hbar+\dfrac{\alpha i\hbar}{2}B_{1}(x,p)\quad,\quad\left[x,y\right]_{\star}=\dfrac{i\beta}{2}B_{1}(x,y).$ (15) or in the Dirac ”method of classical analogy” form [4] $\displaystyle\dfrac{1}{i\hbar}\left[p,x\right]_{\star}=1-\dfrac{\alpha}{2}B_{1}(x,p)\quad,\quad\dfrac{1}{i\hbar}\left[x,y\right]_{\star}=\dfrac{\beta}{2\hbar}B_{1}(x,y).$ (16) Because, however, for $f,g\in C^{\infty}(M)$: $B_{1}(f,g)=2\theta(df\wedge dg)$, so one has $\displaystyle\dfrac{1}{i\hbar}\left[p,x\right]_{\star}=1-\dfrac{\alpha}{\hbar}(dx\wedge dp)\quad,\quad\dfrac{1}{i\hbar}\left[x,y\right]_{\star}=\dfrac{\beta}{\hbar}dx\wedge dy,$ (17) where $\hbar$ in first relation was introduced for dimensional correctness. Taking into account the simplest space lattice with a fundamental length scale $\ell$ $x=ndx\quad,\quad dx=\ell\quad,\quad n\in\mathbb{Z}\quad\longrightarrow\quad\ell=\dfrac{l_{0}}{n}e^{1/n}\quad,\quad\lim_{n\rightarrow\infty}\ell=0,$ (18) where $l_{0}>0$ is a constant, and the De Broglie coordinate-momentum relation $p=\dfrac{\hbar}{x}$ (19) one receives finally the brackets $\displaystyle\dfrac{i}{\hbar}\left[x,p\right]_{\star}=1+\dfrac{\alpha}{\hbar^{2}}\ell^{2}p^{2}\quad,\quad\dfrac{i}{\hbar}\left[x,y\right]_{\star}=-\dfrac{\beta}{\hbar}\ell^{2},$ (20) that are defining the Snyder model (1). In the 1960s the Soviet physicist M. A. Markov [5] proposed to take a fundamental length scale as the Planck length $\ell=\ell_{Pl}=\sqrt{{\dfrac{\hbar c}{G}}}$, and suppose that a mass $m$ of any elementary particle is $m\leqslant M_{Pl}=\dfrac{\hbar}{c\ell_{Pl}}=\sqrt{{\dfrac{G\hbar}{c^{3}}}}$. Using this idea, since 1978 the Soviet-Russian theoretician V. G. Kadyshevsky and collaborators (See _e.g._ papers in Ref. [6]) have studied widely some aspects of the Snyder noncommutative geometry model. Recently also V. N. Rodionov has developed the Kadyshevsky current independently [7]. The problems discussed in this paper seem to be more related to a general current [8], where the Snyder model (1) is partially employed. Beginning 2000 the Indian scholar B. G. Sidharth [9] showed that in spite of self-evident Lorentz invariance of the structural deformation (1), in general the Snyder modification both breaks the Einstein special equivalence principle as well as violates the Lorentz symmetry so celebrated in relativistic physics. In that case the Einstein Hamiltonian constraint receives an additional term proportional to 4th power of spatial momentum of a relativistic particle and 2nd power of $\ell$ that is a minimal scale, _e.g._ the Planck scale or the Compton one, of a theory (Cf. Ref. [10]) $E^{2}=m^{2}c^{4}+c^{2}p^{2}+\alpha\left(\dfrac{c}{\hbar}\right)^{2}\ell^{2}p^{4}.$ (21) Neglecting negative mass states as nonphysical, Sidharth established a new fact. Namely, as the result of application of the Dirac-like linearization procedure within the modified equivalence principle (21) one concludes the appropriate Dirac Hamiltonian constraint which, however, differs from the standard one by an additional $\gamma^{5}$-term, that is proportional to 2nd power of the spatial momentum of a relativistic particle and to a minimal scale $\ell$ [11] $\gamma^{\mu}p_{\mu}+mc^{2}+\sqrt{\alpha}\dfrac{c}{\hbar}\ell\gamma^{5}p^{2}=0.$ (22) The modified Dirac Hamiltonian constraint (22) formally can be deduced from the equation (21) rewritten in the following compact form $(\gamma^{\mu}p_{\mu})^{2}=m^{2}c^{4}+\alpha\left(\dfrac{c}{\hbar}\right)^{2}\ell^{2}p^{4},$ (23) where $p_{\mu}$ is a relativistic momentum four-vector $p_{\mu}=\left[\begin{array}[]{c}E\\\ -cp\end{array}\right].$ (24) However, in both papers as well as books Sidharth is not noticing that from the Hamiltonian constraint (23) there is arising a one more additional possibility physically nonequivalent to (22), namely, it is given by the Dirac constraint with the correction possessing a negative sign $\gamma^{\mu}p_{\mu}+mc^{2}-\sqrt{\alpha}\dfrac{c}{\hbar}\ell\gamma^{5}p^{2}=0.$ (25) However, the possible physical results following from the Hamiltonian constraint (25) can be deduced by application of the mirror reflection $\ell\rightarrow-\ell$ within the results following from the Dirac Hamiltonian constraint with the positive $\gamma^{5}$-term (22). We are not going to neglect also the negative mass states as nonphysical, because this situation is in strict correspondence with results obtained from the equation (22) by a mirror reflection in mass of a relativistic particle $m\rightarrow-m$. It means that after employing the canonical quantization in the momentum space of a relativistic particle $E\rightarrow\hat{E}=i\hbar\partial_{0}\quad,\quad p\rightarrow\hat{p}=i\hbar\partial_{i}\quad,$ (26) in general one can consider the generalized modification of Dirac equation of the form $\left(\gamma^{\mu}p_{\mu}\pm mc^{2}\pm\sqrt{\alpha}\dfrac{c}{\hbar}\ell\gamma^{5}p^{2}\right)\psi=0,$ (27) which describes 4 physically nonequivalent situations. Here is assumed that in analogy to the conventional Dirac theory, a solution $\psi$ of the equation (38) is four component spinor $\psi=\left[\begin{array}[]{c}\phi_{0}\\\ \phi_{1}\\\ \phi_{2}\\\ \phi_{3}\end{array}\right],$ (28) and that the four-dimensional Clifford algebra of the Dirac $\gamma$-matrices is given in the standard representation $\displaystyle\gamma^{0}$ $\displaystyle=$ $\displaystyle\left[\begin{array}[]{cc}0&\mathbf{1}_{2}\\\ \mathbf{1}_{2}&0\end{array}\right]\quad,\quad\gamma^{i}=\left[\begin{array}[]{cc}0&\sigma^{i}\\\ -\sigma^{i}&0\end{array}\right]\quad,$ (33) $\displaystyle\gamma^{5}$ $\displaystyle=$ $\displaystyle\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=i\left[\begin{array}[]{cc}\mathbf{1}_{2}&0\\\ 0&-\mathbf{1}_{2}\end{array}\right]\quad,\quad\left(\gamma^{5}\right)^{2}=-\mathbf{1}_{4},$ (36) where $\sigma$’s are the Pauli matrices $\sigma^{1}=\left[\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right]\quad,\quad\sigma^{2}=\left[\begin{array}[]{cc}0&-i\\\ i&0\end{array}\right]\quad,\quad\sigma^{3}=\left[\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right].$ (37) A presence of the Dirac’s matrix $\gamma^{5}$ in the Dirac equation (27) causes that it violates parity symmetry manifestly, so in fact there is CP violation and the $\gamma^{5}$-term breaks the full Lorentz symmetry. For simplicity, however, it is useful to consider one of the four situations describing by the equation (27), that is given by the Dirac equation modified due to the Sidharth term $\left(\gamma^{\mu}\hat{p}_{\mu}+mc^{2}+\sqrt{\alpha}\dfrac{c}{\hbar}\ell\gamma^{5}\hat{p}^{2}\right)\psi=0,$ (38) and finally discuss results of application of the mentioned mirror transformations. Recently it was shown [12] that there are some nonequivalent possibilities for establishment of the Hamiltonian from the constraint (21), and it crucially depends on the functional relation between a mass of a relativistic particle and a minimal scale $m(\ell)$. It leads to some nontrivial classical solutions and associated with them nonequivalent quantum theories. This energy-momentum relation is currently under astrophysics’ interesting [13]. Originally the equation (38) was proposed some time ago [11] as an idea for ultra-high energy physics, but any concrete physical predictions arising from this idea still are not well-established. Currently there are only speculations possessing laconic character that the extra term violating the Lorentz symmetry manifestly lies in the new foundations of physics [14]. In fact its meaning is still a great riddle to the same degree as it is an amazing hope. The best test for checking the corrected theory (38) and in general all the theories given by (27) seem to be astrophysical phenomena _i.e._ ultra-high-energy cosmic rays coming from gamma bursts sources, neutrinos coming from supernovas, and others observed in this energy region. This cognitive aspect of the thing is the motivation for reconsidering the equation (38) arising due to the Snyder noncommutative geometry (1), and try pull out extension of well- grounded physical knowledge. ## 2 Massive neutrinos Let us reconsider the modified Dirac equation (38). In fact the Sidharth $\gamma^{5}$-term is the additional effect – the shift of the conventional Dirac theory – arising due to the Snyder noncommutative geometry of phase space $(p,x)$ of a relativistic particle (1). However, it does not mean that Special Relativity will be also modified - the Minkowski hyperbolic geometry of the relativistic momentum space as well as the structure of space-time in fact are preserved. The Einstein theory describes dynamics of a relativistic particle while in the philosophical as well as physical foundations of the algebra deformation we have not any arguments following from dynamics of a particle – strictly speaking the correction is due to finite sizes of a particle. In this manner, the best interpretation of the deformation (21), as well as the appropriate constraint (22), is the energetic constraint corrected by the non-dynamical term. By this reason we propose here to take into account the formalism of the Minkowski geometry of the momentum space independently from a presence of the $\gamma^{5}$-term, and apply it within both the modified Einstein Hamiltonian constraint as well the modified Dirac equation. Application of the standard identity holding in the momentum space of a relativistic particle $p_{\mu}p^{\mu}=\left(\gamma^{\mu}p_{\mu}\right)^{2}=E^{2}-c^{2}p^{2},$ (39) to the modified Dirac equation (38) yields the equation $\left[\gamma^{\mu}\hat{p}_{\mu}+mc^{2}+\dfrac{\sqrt{\alpha}}{\hbar c}\ell\gamma^{5}\left[E^{2}-\left(\gamma^{\mu}\hat{p}_{\mu}\right)^{2}\right]\right]\psi=0,$ (40) which can be rewritten as $\left[-\dfrac{\sqrt{\alpha}}{\hbar c}\ell\gamma^{5}\left(\gamma^{\mu}\hat{p}_{\mu}\right)^{2}+\gamma^{\mu}\hat{p}_{\mu}+mc^{2}+\dfrac{\sqrt{\alpha}}{\hbar c}\ell\gamma^{5}E^{2}\right]\psi=0,$ (41) or equivalently by using of the combination $\gamma^{5}\gamma^{\mu}p_{\mu}$ $\left[\left(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}\right)^{2}-\epsilon\left(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}\right)+E^{2}-\epsilon mc^{2}\gamma^{5}\right]\psi=0,$ (42) where $\epsilon$ is the energy $\epsilon=\dfrac{\hbar c}{\sqrt{\alpha}\ell}.$ (43) Note that for the Planck scale holds $\ell=\ell_{Pl}=\sqrt{{\dfrac{\hbar c}{G}}}$ and the energy (43) coincides with the Planck energy scaled by the factor $\dfrac{1}{\sqrt{\alpha}}$ $\epsilon=\epsilon_{Pl}=\dfrac{1}{\sqrt{\alpha}}\sqrt{{\dfrac{\hbar c^{5}}{G}}}=\dfrac{1}{\sqrt{\alpha}}M_{Pl}c^{2}.$ (44) Similarly for the Compton scale $\ell=\ell_{C}=2\pi\dfrac{\hbar}{m_{p}c}$ is the Compton wavelength of a particle possessing the rest mass $m_{p}$. In this case the energy $\epsilon$ is a particle’s rest energy scaled by the factor $\dfrac{1}{2\pi\sqrt{\alpha}}$ $\epsilon=\epsilon_{C}=\dfrac{1}{2\pi\sqrt{\alpha}}m_{p}c^{2}.$ (45) If the particle has the rest mass that equals the Planck mass $m_{p}\equiv M_{Pl}$ then $\ell_{C}=\dfrac{2\pi G}{c^{2}}M_{Pl}\quad,\quad\epsilon_{C}=\dfrac{\epsilon_{Pl}}{2\pi}.$ (46) In the other words for this case the doubled Compton scale is a circumference of a circle with a radius of the Schwarzschild radius of the Planck mass (Cf. also Ref. [15]) $2\ell_{C}=2\pi r_{S}\left(M_{Pl}\right)\quad,\quad r_{S}(m)=\dfrac{2Gm}{c^{2}}.$ (47) The equation (42) expresses acting of the operator $\left(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}\right)^{2}-\epsilon\left(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}\right)+E^{2}-\epsilon mc^{2}\gamma^{5},$ (48) on the Dirac spinor $\psi$. With using of elementary algebraic manipulations, however, one one can easily deduce that in fact the operator (48) can be rewritten in the reduced form $(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}-\mu_{+})(\gamma^{5}\gamma^{\mu}\hat{p}_{\mu}-\mu_{-}),$ (49) where $M_{pm}$ are the manifestly nonhermitian quantities $\mu_{\pm}=\dfrac{\epsilon}{2}\left(1\pm\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}}}\sqrt{{1+\dfrac{4\epsilon mc^{2}}{\epsilon^{2}-4E^{2}}\gamma^{5}}}\right).$ (50) Principally the quantities (50) are due to the order reduction, and also cause the Dirac-like linearization. Treating energy $E$, mass $m$, and $\epsilon$ (or equivalently the scale $\ell$) in (50) as free parameters one obtains easily that formally the modified Dirac equation (38) and also the generalized equation (27) are equivalent to the following two nonequivalent Dirac equations $\displaystyle\left(\gamma^{\mu}\hat{p}_{\mu}-M_{+}c^{2}\right)\psi=0\qquad,\qquad\left(\gamma^{\mu}\hat{p}_{\mu}-M_{-}c^{2}\right)\psi=0,$ (51) where $M_{\pm}$ are the mass matrices of the Dirac theories generated as the result of the dimensional reduction $M_{\pm}=\dfrac{\epsilon}{2c^{2}}\left(-1\mp\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}+\dfrac{4mc^{2}}{\epsilon}\gamma^{5}}}\right)\gamma^{5}.$ (52) This is nontrivial result – we have obtained usual Dirac theories, where the mass matrices $M_{pm}$ are manifestly nonhermitian $M_{\pm}^{\dagger}\neq M_{\pm}$. However, the total effect from a minimal scale $\ell$ sits within the matrices $M_{pm}$ only, while the four-momentum operator $\hat{p}_{\mu}$ remains exactly the same as in both the conventional Einstein and Dirac theories. Note that this procedure formally is not incorrect - we preserve the Minkowski geometry formalism for the square of spatial momentum that in fact is the fundament of the $\gamma^{5}$-correction, but was not noticed or was omitted in Sidharth’s papers and books. In this manner we have constructed new type mass generation mechanism which deduction within the usual frames of Special Relativity only, _i.e._ for the case of vanishing sizes of the particle $\ell=0$ or equivalently for the maximal energy $\epsilon=\infty$, can not be done. Strictly speaking this mass generation mechanism is due to the order reduction in the operator (48) of the modified Dirac equation. However, both the mass matrices (52) are builded by a square root of the expression containing the matrix $\gamma^{5}$. Let us present now the mass matrices in equivalent way, where the Dirac matrix $\gamma^{5}$ will present in a linear way. Let us see details of the mass matrices $M_{\pm}$. Fist, by application of the Taylor series expansion to the square root present in the defining formula (52) one obtains $\displaystyle\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}+\dfrac{4mc^{2}}{\epsilon}\gamma^{5}}}$ $\displaystyle=$ $\displaystyle\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}}}\sqrt{{1+\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\gamma^{5}}}=$ (53) $\displaystyle=$ $\displaystyle\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}}}\sum_{n=0}^{\infty}\binom{1/2}{n}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\gamma^{5}\right)^{n},$ where the following notation was used $\binom{n}{k}=\dfrac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n+1-k)}$ that is the generalized Newton binomial symbol. Employing now the $\gamma^{5}$-matrix properties – _i.e._ $\left(\gamma^{5}\right)^{2n}=-1$, and $\left(\gamma^{5}\right)^{2n+1}=-\gamma^{5}$ – one decompose the sum present in the last term of (53) onto the two component $\displaystyle\sum_{n=0}^{\infty}\binom{1/2}{n}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\gamma^{5}\right)^{n}=$ $\displaystyle=-\sum_{n=0}^{\infty}\binom{1/2}{2n}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\right)^{2n}-\sum_{n=0}^{\infty}\binom{1/2}{2n+1}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\right)^{2n+1}\gamma^{5}.$ (54) Direct application of standard summation procedure allows to establish the sums presented in the both components in (54) in a compact form $\displaystyle\sum_{n=0}^{\infty}\binom{1/2}{2n}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\right)^{\\!\\!\\!2n}=\sqrt{{1+\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}+\sqrt{{1-\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}},\vspace{10pt}$ (55) $\displaystyle\sum_{n=0}^{\infty}\binom{1/2}{2n+1}\left(\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}\right)^{\\!\\!\\!2n+1}=\sqrt{{1+\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}-\sqrt{{1-\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}.$ (56) In this manner finally one sees easily that both the mass matrices $M_{\pm}$ possess the following formal decomposition $M_{\pm}=\mathfrak{H}(M_{\pm})+\mathfrak{A}(M_{\pm}),$ (57) where $\mathfrak{H}(M_{\pm})$ is hermitian part of $M_{\pm}$ $\mathfrak{H}(M_{\pm})=\pm\dfrac{\epsilon}{2c^{2}}\left[\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}}}\left(\sqrt{{1+\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}-\sqrt{{1-\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}\right)\right],$ (58) and $\mathfrak{A}(M_{\pm})$ is antihermitian part of $M_{\pm}$ $\mathfrak{A}(M_{\pm})=-\dfrac{\epsilon}{2c^{2}}\left[1\pm\sqrt{{1-\dfrac{4E^{2}}{\epsilon^{2}}}}\left(\sqrt{{1+\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}+\sqrt{{1-\dfrac{\dfrac{4mc^{2}}{\epsilon}}{1-\dfrac{4E^{2}}{\epsilon^{2}}}}}\right)\right]\gamma^{5}.$ (59) By application of elementary algebraic manipulations one sees that equivalently the mass matrices $M_{\pm}$ can be decomposed into the basis of the commutating projectors $\left\\{\Pi_{i}:\dfrac{1+\gamma^{5}}{2},\dfrac{1-\gamma^{5}}{2}\right\\}$, $M_{\pm}=\sum_{i}\mu_{i}^{\pm}\Pi_{i}=\mu_{R}^{\pm}\dfrac{1+\gamma^{5}}{2}+\mu_{L}^{\pm}\dfrac{1-\gamma^{5}}{2},$ (60) where $\displaystyle\mu_{R}^{\pm}$ $\displaystyle=$ $\displaystyle-\dfrac{1}{c^{2}}\left(\dfrac{\epsilon}{2}\pm\sqrt{{\epsilon^{2}-4\epsilon mc^{2}-4E^{2}}}\right),$ (61) $\displaystyle\mu_{L}^{\pm}$ $\displaystyle=$ $\displaystyle\dfrac{1}{c^{2}}\left(\dfrac{\epsilon}{2}\pm\sqrt{{\epsilon^{2}+4\epsilon mc^{2}-4E^{2}}}\right),$ (62) are projected masses related to the theories with signs $\pm$ in the matrix mass. By application of the obvious relations for the projectors $\Pi_{i}^{\dagger}\Pi_{i}=\mathbf{1}_{4}$, $\Pi_{1}\Pi_{2}=\dfrac{1}{2}\mathbf{1}_{4}$, $\Pi_{1}^{\dagger}=\Pi_{2}$ and $\Pi_{1}+\Pi_{2}=\mathbf{1}_{4}$ one obtains $M_{\pm}M_{\pm}^{\dagger}=\dfrac{(\mu_{R}^{\pm})^{2}+(\mu_{L}^{\pm})^{2}}{2}\mathbf{1}_{4}.$ (63) Introducing the right- and left-handed chiral Weyl fields $\psi_{R}=\dfrac{1+\gamma^{5}}{2}\psi\quad,\quad\psi_{L}=\dfrac{1-\gamma^{5}}{2}\psi,$ (64) where the Dirac spinor $\psi$ is a solution of the appropriate Dirac equations (51), both the theories (51) can be rewritten as the system of two equations $\left(\gamma^{\mu}\hat{p}_{\mu}+\mu^{+}c^{2}\right)\left[\begin{array}[]{c}\psi_{R}^{+}\\\ \psi_{L}^{+}\end{array}\right]=0\qquad,\qquad\left(\gamma^{\mu}\hat{p}_{\mu}+\mu^{-}c^{2}\right)\left[\begin{array}[]{c}\psi_{R}^{-}\\\ \psi_{L}^{-}\end{array}\right]=0,$ (65) where the mass matrices $\mu^{\pm}$ are hermitian now $\mu^{\pm}=\left[\begin{array}[]{cc}\mu_{R}^{\pm}&0\\\ 0&\mu_{L}^{\pm}\end{array}\right]=\left[\begin{array}[]{cc}\mu_{R}^{\pm}&0\\\ 0&\mu_{L}^{\pm}\end{array}\right]^{\dagger},$ (66) and $\psi_{R,L}^{\pm}$ are the chiral fields related to the mass matrices $\mu_{\pm}$ respectively. Note that the masses (61) and (62) are invariant with respect to choice of the Dirac matrices $\gamma^{\mu}$ representation. By this way they have physical character. It is interesting that for the mirror reflection in a minimal scale $\ell\rightarrow-\ell$ (or equivalently for the change $\epsilon\rightarrow-\epsilon$) we have the exchange $\mu_{R}^{\pm}\leftrightarrow\mu_{L}^{\pm}$ while the chiral Weyl fields are the same. In the case of the mirror reflection in the original mass $m\rightarrow-m$ one has the exchange $\mu_{R}^{\pm}\leftrightarrow-\mu_{L}^{\pm}$. The case of originally massless states $m=0$ is also intriguing from theoretical point of view. From the formulas (61) and (62) one sees easily that in this case $\mu_{R}=-\mu_{L}$. In the case of generic Einstein theory $\ell=0$ one has $\mu_{R}^{\pm}=\left\\{\begin{array}[]{cc}-\infty&\leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\ +\\\ \infty&\leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\ -\end{array}\right.\qquad,\qquad\mu_{L}^{\pm}=\left\\{\begin{array}[]{cc}\infty&\leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\ +\\\ -\infty&\leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\ -\end{array}\right..$ (67) In general, however, for formal correctness of the projection splitting (60) both the neutrinos masses (61) and (62) must be real numbers; strictly speaking when the masses are complex numbers the decomposition (60) does not yield hermitian mass (66, so that the presented construction does not hold and by this reason must be replaced by other one. In the conventional Weyl theory approach neutrinos are massless. In this manner it is evident that employing the Snyder noncommutative geometry generates a new obvious nontriviality – _the kinetic mass generation mechanism that leads to the theory of massive neutrinos_. It must be emphasized that in Sidharth’s books and papers a possibility of neutrino masses was only laconically mentioning as ”due to mass term”, where by the mass term the author understands the $\gamma^{5}$-term in the modified Dirac equation (38). In fact it is not mass term in the common sense of the Standard Model being currently the theory of elementary particles and fundamental interactions. Strictly speaking Sidharth’s statements are incorrect, because we have generated the massive neutrinos due to two-step mechanism - the first was the order reduction of the modified Dirac equation (38), and the second was decomposition of the received mass matrices (52) into the projectors basis and introducing the chiral Weyl fields in the usual way (64). It must be emphasized that a mass generation mechanism is manifestly absent in Sidharth’s contributions and the line of thinking presented there is completely different, omits many interesting physical and mathematical details, and in general does not look like constructive (Cf. _e.g._ Ref. [16]). However, in the result of the procedure proposed above, _i.e._ by application of the Dirac equation with the $\gamma^{5}$-term (22) and direct preservation within this equation the Einstein–Minkowski relativity (39), we have generated the system of equations (65) which describes two left- $\psi_{L}^{\pm}$ and two right- $\psi_{R}^{\pm}$ chiral massive Weyl fields, _i.e._ we have yielded massive neutrinos, related to both cases - any originally massive $m\neq 0$ as well as for originally massless $m=0$ states. By this reason in the proposed approach the notion _neutrino_ takes an essentially new physical meaning; it is a chiral field due to any massive and massless quantum state. Moreover, we have obtained the two massive Weyl theories (65), so that totally with a one quantum state there are associated 4 massive neutrinos. ## 3 The chiral condensate Let us notice that if we want to construct the Lorentz invariant Lagrangian $\mathcal{L}$ of the gauge field theory characterized by the Euler–Lagrange equations of motion (65) for both massive Weyl theories we should put $\displaystyle\mathcal{L}^{\pm}=\bar{\psi}_{R}^{\pm}\gamma^{\mu}\hat{p}_{\mu}\psi_{R}^{\pm}+\bar{\psi}_{L}^{\pm}\gamma^{\mu}\hat{p}_{\mu}\psi_{L}^{\pm}+\mu_{R}^{\pm}c^{2}\bar{\psi}_{R}^{\pm}\psi_{R}^{\pm}+\mu_{L}^{\pm}c^{2}\bar{\psi}_{L}^{\pm}\psi_{L}^{\pm},$ (68) where $\bar{\psi}_{R,L}^{\pm}=\left(\psi_{R,L}^{\pm}\right)^{\dagger}\gamma^{0}$ are the Dirac adjoint of $\psi_{R,L}^{\pm}$, and take into considerations rather the sum of both partial gauge field theories (68) $\mathcal{L}=\mathcal{L}^{+}+\mathcal{L}^{-},$ (69) as the Lagrangian of the appropriate full gauge field theory. One can see straightforwardly that the both partial gauge field theories (68) exhibit the (local) chiral symmetry $SU(2)_{R}^{\pm}\otimes SU(2)_{L}^{\pm}$ $\left\\{\begin{array}[]{c}\psi_{R}^{\pm}\rightarrow\exp\left\\{i\theta_{R}^{\pm}\right\\}\psi_{R}^{\pm}\\\ \psi_{L}^{\pm}\rightarrow\psi_{L}^{\pm}\end{array}\right.\quad\mathrm{or}\quad\left\\{\begin{array}[]{c}\psi_{R}^{\pm}\rightarrow\psi_{R}^{\pm}\\\ \psi_{L}^{\pm}\rightarrow\exp\left\\{i\theta_{L}^{\pm}\right\\}\psi_{L}^{\pm}\end{array}\right.,$ (70) the vector symmetry $U(1)_{V}^{\pm}$ $\left\\{\begin{array}[]{c}\psi_{R}^{\pm}\rightarrow\exp\left\\{i\theta^{\pm}\right\\}\psi_{R}^{\pm}\\\ \psi_{L}^{\pm}\rightarrow\exp\left\\{i\theta^{\pm}\right\\}\psi_{L}^{\pm}\end{array}\right.,$ (71) and the axial symmetry $U(1)_{A}^{\pm}$ $\left\\{\begin{array}[]{c}\psi_{R}^{\pm}\rightarrow\exp\left\\{-i\theta^{\pm}\right\\}\psi_{R}^{\pm}\\\ \psi_{L}^{\pm}\rightarrow\exp\left\\{i\theta^{\pm}\right\\}\psi_{L}^{\pm}\end{array}\right..$ (72) In this manner the total symmetry group is the composed $SU(3)_{C}^{TOT}$ $SU(3)_{C}^{TOT}=SU(3)_{C}^{+}\oplus SU(3)_{C}^{-},$ (73) where $SU(3)_{C}^{\pm}$ are the global (chiral) 3-flavor gauge symmetries related to each of the gauge theories (68), _i.e._ $\displaystyle SU(2)_{R}^{+}\otimes SU(2)_{L}^{+}\otimes U(1)_{V}^{+}\otimes U(1)_{A}^{+}\equiv SU(3)^{+}\otimes SU(3)^{+}=SU(3)_{C}^{+},$ (74) $\displaystyle SU(2)_{R}^{-}\otimes SU(2)_{L}^{-}\otimes U(1)_{V}^{-}\otimes U(1)_{A}^{-}\equiv SU(3)^{-}\otimes SU(3)^{-}=SU(3)_{C}^{-},$ (75) describing 2-flavor massive free quarks – _the neutrinos_ in our proposition. However, by using of the relations for the Weyl fields (64) and applying algebraic manipulations of the Dirac $\gamma$-algebra (as _e.g._ $\left\\{\gamma^{\mu},\gamma^{5}\right\\}=0$) one has $\displaystyle\left(1\mp\gamma^{5}\right)\gamma^{0}\left(1\pm\gamma^{5}\right)$ $\displaystyle=$ $\displaystyle\pm 2\gamma^{0}\gamma^{5},$ (76) $\displaystyle\left(1\mp\gamma^{5}\right)\gamma^{0}\gamma^{\mu}\left(1\pm\gamma^{5}\right)$ $\displaystyle=$ $\displaystyle 2\gamma^{0}\gamma^{5},$ (77) and hence contribution to the right hand side of (68) are $\displaystyle\bar{\psi}_{R,L}^{\pm}\gamma^{\mu}p_{\mu}\psi_{R,L}^{\pm}$ $\displaystyle=$ $\displaystyle\dfrac{1}{2}\bar{\psi^{\pm}}\gamma^{\mu}p_{\mu}\psi^{\pm},$ (78) $\displaystyle\mu_{R,L}^{\pm}c^{2}\bar{\psi}^{\pm}_{R,L}\psi_{R,L}^{\pm}$ $\displaystyle=$ $\displaystyle\pm\dfrac{\mu_{R,L}^{\pm}}{2}c^{2}\bar{\psi^{\pm}}\gamma^{5}\psi^{\pm},$ (79) where $\bar{\psi^{\pm}}=\left(\psi^{\pm}\right)^{\dagger}\gamma^{0}$ is the Dirac adjoint of the Dirac fields $\psi^{\pm}$ related to the Weyl chiral fields by the transformation (64). Both (78) and (79) are the Lorentz invariants. In result the global chiral Lagrangian (69) can be elementary lead to the following form $\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\bar{\psi^{+}}\left(\gamma^{\mu}\hat{p}_{\mu}+\mu_{eff}^{+}c^{2}\right)\psi^{+}+\bar{\psi^{-}}\left(\gamma^{\mu}\hat{p}_{\mu}+\mu_{eff}^{-}c^{2}\right)\psi^{-}=$ (80) $\displaystyle=$ $\displaystyle\bar{\Psi}\left(\gamma^{\mu}\hat{p}_{\mu}+M_{eff}c^{2}\right)\Psi,$ (81) where $\mu_{eff}^{\pm}$ are the effective mass matrices of the gauge fields $\psi^{\pm}$, and $M_{eff}$ is the mass matrix of the effective composed field $\Psi=\left[\begin{array}[]{c}{\psi^{+}}\\\ {\psi^{-}}\end{array}\right]$ $\displaystyle\mu_{eff}^{\pm}$ $\displaystyle=$ $\displaystyle\dfrac{\mu_{R}^{\pm}-\mu_{L}^{\pm}}{2}\gamma^{5},$ (82) $\displaystyle M_{eff}$ $\displaystyle=$ $\displaystyle\left[\begin{array}[]{cc}{\mu^{+}_{eff}}&0\\\ 0&{\mu^{-}_{eff}}\end{array}\right].$ (85) Both the mass matrices $\mu^{\pm}_{eff}$ are hermitian or antihermitian – it depends on a choice of representation, so the same property has the mass matrix $M_{eff}$. Obviously, the full gauge field theory (80), or equivalently (81), is invariant with respect to the composed gauge symmetry $SU(2)_{V}^{TOT}$ transformation $SU(2)_{V}^{TOT}=SU(2)_{V}^{+}\oplus SU(2)_{V}^{-},$ (86) where $SU(2)_{V}^{\pm}$ are the $SU(2)\otimes SU(2)$ transformations used to each of the gauge fields $\psi^{\pm}$ $\left\\{\begin{array}[]{c}\psi^{\pm}\rightarrow\exp\left\\{i\theta^{\pm}\right\\}\psi^{\pm}\\\ \bar{\psi^{\pm}}\rightarrow\bar{\psi^{\pm}}\exp\left\\{-i\theta^{\pm}\right\\}\end{array}\right..$ (87) This means that for the full gauge field theory the composed global chiral symmetry $SU(3)_{C}^{TOT}$ is spontaneously broken to its subgroup – the composed isospin group $SU(2)_{V}^{TOT}$ $SU(3)_{C}^{TOT}\longrightarrow SU(2)_{V}^{TOT}.$ (88) Physically it should be interpreted as the symptom of an existence of the chiral condensate of massive neutrinos being a composition of two chiral condensates, that is the composed effective field theory $SU(2)_{V}^{TOT}=(SU(2)^{+}\otimes SU(2)^{+})\oplus(SU(2)^{-}\otimes SU(2)^{-})$ [17]. However, by the composed global chiral gauge symmetry $SU(3)_{C}^{TOT}$, the gauge theory (68) looks like formally as the theory of free massive quarks which do not interact; this is the situation similar to Quantum Chromodynamics [18], but in the studied case we have formally a composition of two QCDs. For each of the QCDs the space of fields is different then in usual QCD - there are two massive chiral fields only – the left- and right-handed Weyl fields, thats are the massive neutrinos by our proposition. The chiral condensate of massive neutrinos (81) is the result beyond the Standard Model, but essentially it can be included into the theory as the new contribution. ## 4 Discussion It must be emphasized that the energy-momentum relation (21) modified due to the Snyder model of noncommutative geometry (1) differs from the usual Special Relativity’s relation. In particular as is self-evident from the Hamiltonian constraint (21), there is an extra contribution to the Einstein special equivalence principle due to the additional $\ell^{2}$-term. This is brought out very clearly in the manifestly nonhermitian Dirac equations (51), as well as in the hermitian massive Weyl equations (65) describing the neutrinos in our proposition. A massless neutrino in the conventional Weyl theory is now seen to argue as mass, and further, this mass has a two left component and a two right component, as show in (57) and (60). Once this is recognized, the mass matrix which otherwise appears nonhermitian, turs out to be actually hermitian, as seen in (66), but if and only if when the masses (57) and (60) are real. There is no any restrictions, however, for their sign - the masses can be positive as well as negative. In other words the underlying Snyder noncommutative geometry (1) is reflected in the modified energy-momentum relation (22) naturally gives rise to the mass of the neutrino. It was laconically suggested as a possible result in the Ref. [16], however, with no any concrete calculations and proposals for a mass generation mechanism. It must be remembered that in the Standard Model the neutrino is massless, but the Super–Kamiokande experiments in the late nineties showed that the neutrino does indeed have a mass and this is the leading motivation to an exploration of models beyond the Standard Model, as the model presented in this paper. In this connection it is also relevant to mention that currently the Standard Model requires the Higgs Mechanism for the generation of mass in general, though the Higgs particle has been undetected for forty five years and it is hoped will be detected by the Large Hadron Collider, after it is recommissioned. We hope for next development of the both presented massive neutrino model, as well as the chiral condensate. ## Acknowledgements Author benefitted many valuable discussions from Profs. B. M. Barbashov, V. N. Pervushin and A. B. Arbuzov. Interaction with Prof. A. P. Isaev in 2007 was fruitful. Communication with Dr. B. G. Sidharth was enlightening. Remarks of the referees were very helpful for corrections of the primary notes. ## References [1] H. S. Snyder, Phys. Rev. 71, 38-41 (1947); Phys. Rev. 72, 68-71 (1947) * [2] A. Connes, Noncommutative Geometry. Academic Press (1994); N. Seiberg and E. Witten, JHEP 09, 032 (1999) [arXiv:hep-th/9908142]; D. J. Gross, A. Hashimoto, and N. Itzhaki, Adv. Theor. Math. Phys. 4, 893-928 (2000) [arXiv:hep-th/0008075]; M. R. Douglas and N. A. Nekrasov, Rev. Mod. Phys. 73, 977-1029 (2002) [arXiv:hep-th/0106048]; R. J. Szabo, Phys. Rep. 378, 207-299 (2003) [arXiv:hep-th/0109162]; G. Dito and D. Sternheimer, Lect. Math. Theor. Phys. 1, 9-54, (2002) [arXiv:math/0201168]; L. Alvarez-Gaume and M. A. Vazquez-Mozo, Nucl. Phys. B 668, 293-321 (2003) [arXiv:hep-th/0305093]; M. Chaichian, P. P. Kulish, K. Nshijima, and A. Tureanu, Phys. Lett. B 604, 98-102 (2004) [arXiv:hep-th/0408069]; G. Fiore and J. Wess, Phys. Rev. D 75, 105022 (2007) [arXiv:hep-th/0701078]; M. Chaichian, M. N. Mnatsakanova, A. Tureanu, and Yu. S. Vernov, JHEP 0809, 125 (2008) [arXiv:0706.1712 [hep-th]]; M. V. Battisti and S. Meljanac, Phys. Rev. D 79, 067505 (2009) [arXiv:0812.3755 [hep-th]]. * [3] M. Kontsevich, Lett. Math. Phys. 66, 157-216 (2003) [arXiv:q-alg/9709040]. * [4] P. A. M. Dirac, The Principles of Quantum Mechanics. Clarendon Press (1958). * [5] M. A. Markov, Prog. Theor. Phys. Suppl. E65, 85-95 (1965); Sov. Phys. JETP 24, 584 (1967). * [6] V. G. Kadyshevsky, Sov. Phys. JETP 14, 1340-1346 (1962) ; Nucl. Phys. B 141, 477 (1978); in Group Theoretical Methods in Physics: Seventh International Colloquium and Integrative Conference on Group Theory and Mathematical Physics, Held in Austin, Texas, September 11–16, 1978. ed. by W. Beiglböck, A. Böhm, and E. Takasugi, Lect. Notes Phys. 94, 114-124 (1978); Phys. Elem. Chast. Atom. Yadra 11, 5 (1980). V. G. Kadyshevsky and M. D. Mateev, Phys. Lett. B 106, 139 (1981); Nuovo Cim. A 87, 324 (1985). M. V. Chizhov, A. D. Donkov, V. G. Kadyshevsky, and M. D. Mateev, Nuovo Cim. A 87, 350 (1985); Nuovo Cim. A 87, 373 (1985). V. G. Kadyshevsky, Phys. Part. Nucl. 29, 227 (1998). V. G. Kadyshevsky, M. D. Mateev, V. N. Rodionov, and A. S. Sorin, Dokl. Phys. 51, 287 (2006) [arXiv:hep-ph/0512332]; CERN-TH/2007-150, [arXiv:0708.4205 [hep-ph]] * [7] V. N. Rodionov, [arXiv:0903.4420 [hep-ph]] * [8] A. E. Chubykalo, V. V. Dvoeglazov, D. J. Ernst, V. G. Kadyshevsky, and Y. S. Kim, Lorentz Group, CPT and Neutrinos: Proceedings of the International Workshop, Zacatecas, Mexico, 23-26 June 1999. World Scientific (2000). * [9] B. G. Sidharth, The Thermodynamic Universe. World Scientific 2008. * [10] B. G. Sidharth, Found. Phys. 38, 89-95 (2008); _ibid._ , 695-706 (2008). * [11] B. G. Sidharth, Int. J. Mod. Phys. E 14, 1-4 (2005). * [12] L. A. Glinka, Apeiron 16, 147-160 (2009) [arXiv:0812.0551 [hep-th]] * [13] L. Maccione, A. M. Taylor, D. M. Mattingly, and S. Liberati, JCAP 0904, 022 (2009) [arXiv:0902.1756 [astro-ph.HE]] * [14] B. G. Sidharth, _Private communication_ , March-May 2009. * [15] C. Kiefer, Quantum Gravity. 2nd ed., Oxford University Press 2007. * [16] B. G. Sidharth, Int. J. Mod. Phys. E 14, 927-929 (2005); [arXiv:0811.4541 [physics.gen-ph]]; [arXiv:0902.3342 [physics.gen-ph]] * [17] S. Weinberg, The Quantum Theory of Fields. Vol. II Modern Applications, Cambridge University Press 1996. * [18] W. Greiner, S. Schramm, E. Stein, Quantum Chromodynamics. 3rd ed., Springer 2007.	arxiv-papers	2009-02-27T20:52:33	2024-09-04T02:49:00.894067	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "L.A. Glinka", "submitter": "Lukasz Andrzej Glinka", "url": "https://arxiv.org/abs/0902.4811" }
0902.4887	# Quantization of the Maxwell field in curved spacetimes of arbitrary dimension Michael J. Pfenning Michael.Pfenning@usma.edu Department of Physics, United States Military Academy,West Point, New York, 10996-1790, USA (June 10, 2009) ###### Abstract We quantize the massless $p$-form field that obeys the generalized Maxwell field equations in curved spacetimes of dimension $n\geq 2$. We begin by showing that the classical Cauchy problem of the generalized Maxwell field is well posed and that the field possess the expected gauge invariance. Then the classical phase space is developed in terms of gauge equivalent classes, first in terms of the Cauchy data and then reformulated in terms of Maxwell solutions. The latter is employed to quantize the field in the framework of Dimock. Finally, the resulting algebra of observables is shown to satisfy the wave equation with the usual canonical commutation relations. ###### pacs: 04.62.+v, 03.50.-z, 03.70.+k, 11.15.Kc, 11.25.Hf, 14.80.-j ## I Introduction Tensor and/or spinor fields of assorted types in curved spacetime have been studied by various authors, BuchdahlBuchdahl (1958, 1962, 1982a, 1982b, 1984, 1987), GibbonsGibbons (1976), and HiguchiHiguchi (1989) to name a few. It is often found that the straightforward generalization of the flat spacetime field equations to an arbitrary curved spacetime is wrought with internal inconsistencies unless specific conditions are met. For example, in his earliest work Buchdahl Buchdahl (1962) shows that “the natural field equations for particles of spin $\frac{3}{2}$ are consistent if and only if the (pseudo)Riemann space(time) in which they are contemplated is an Einstein space(time),” that is one for which the Ricci tensor satisfies $R_{ab}=\lambda g_{ab}$. Gibbons demonstrates that there is also a breakdown in the Rarita- Schwinger formulation of spin $\frac{3}{2}$ in four-dimensional curved spacetime. Meanwhile, Higuchi derives the constraint condition, $\nabla_{c}R_{ab}=\left[\frac{1}{18}(g_{cb}\partial_{a}+g_{ca}\partial_{b})+\frac{2}{9}g_{ab}\partial_{c}\right]R,$ on the background metric for the generalization of the massive symmetric tensor field equations to curved spacetime. There are two cases where the above condition is met: the metric is a solution to the vacuum Einstein equations or the Ricci tensor $R_{ab}$ is covariantly constant. In this paper, we study the $p$-form field theory (fully antisymmetric rank-$p$ tensors) in curved spacetimes of arbitrary dimension which obey the generalization of the Maxwell equations. Unlike the examples above, the $p$-form field equations generalize to any dimension without inconsistencies or the need for constraints. Thus, $p$-form field theories seem to be a natural model for the study of quantum field theories in higher dimension curved spacetime. For example, the minimally coupled scalar field ($p=0$ in any dimension) and the electromagnetic field ($p=1$ in four dimensions) are two examples of this self-consistent $p$-form theory. Better still, the quantization of the classical theory turns out to be identical for all $p$-form fields, independent of the rank of form. This manuscript proceeds in the following order. In Section II we discuss the classical generalized Maxwell field. This begins with a review of electrodynamics in four dimensions and then proceeds to the generalization of the Maxwell equations into arbitrary dimension. It is at this point that we convert from the traditional notation used in relativistic physics to that of exterior differential calculus thus giving us $p$-form fields. We then discuss fundamental solutions to the resulting wave equation and the initial value problem for the classical field. We shall see that there exists a gauge freedom in the field which complicates the uniqueness of solutions for a given set of initial data. Thus, the Cauchy problem is only well posed if we work with gauge equivalent classes. In Section III we quantize the $p$-form field. From the classical field theory, we obtain a symplectic phase space consisting of a real vector space and a symplectic form. This phase space is quantized by promoting functions in the phase space to operators acting on a Hilbert space while simultaneously requiring the commutator of such operators to be $-i$ times their classical Poisson bracket. In this way, the algebra of observables for the quantized field on a manifold is obtained. Finally, there will be discussion and conclusions. Throughout this paper we will use units where $\hbar=c=1$. The notation $C_{0}^{\infty}(\mathbb{R}^{n})$ denotes the space of smooth, compactly supported 111The support of a function is the closure of the set of points on which it is nonzero., complex-valued functions on $\mathbb{R}^{n}$. We take ${\bm{M}}$ to be a smooth $n$-dimensional manifold (without boundary) which is connected, orientable, Hausdorff, paracompact, and equipped with a smooth metric of index $s$ 222The index is the number of spacelike (i.e., negative norm-squared) basis vectors in any $g$-orthonormal frame.. We denote the space of smooth, complex-valued $p$-forms on ${\bm{M}}$ by $\Omega^{p}({\bm{M}})$; the subspace of compactly supported $p$-forms will be written $\Omega_{0}^{p}({\bm{M}})$. Each $p$-form may be regarded as a fully antisymmetric covariant $p$-tensor field. Our conventions for forms are consistent with that of Abraham, Marsden and Ratiu (AMR) Abraham et al. (1988) and are also summarized in our earlier paper Fewster and Pfenning (2003) to which we refer the reader. Therefore, we only introduce the remaining notational necessities here to make this paper sufficiently self contained. The exterior product between forms will be denoted by $\wedge$, the exterior derivative on forms will be denoted by ${\bf d}$, the Hodge $$–operator by $$ and the co-derivative by $\bm{\delta}$. The Laplace-Beltrami operator, simply called the Laplacian on a Riemannian manifold, is defined $\Box=-\left(\bm{\delta}{\bf d}+{\bf d}\bm{\delta}\right)$. All these are consistent with the previous paper. The only difference between the preceding paper and present one is the definition for the symmetric pairing $\langle\cdot,\cdot\rangle$ of $p$-forms under integration; $\langle\mathcal{U},\mathcal{V}\rangle_{\bm{M}}\equiv\int_{\bm{M}}\mathcal{U}\wedge\mathcal{V}$ (I.1) for any $\mathcal{U},\mathcal{V}\in\Omega^{p}({\bm{M}})$ for which the integral exists. Also, for smooth $\mathcal{U}\in\Omega^{p-1}({\bm{M}})$ and $\mathcal{V}\in\Omega^{p}({\bm{M}})$ we have ${\bf d}(\mathcal{U}\wedge\mathcal{V})={\bf d}\mathcal{U}\wedge\mathcal{V}-\mathcal{U}\wedge\bm{\delta}\mathcal{V}$, therefore by Stokes’ theorem $\langle{\bf d}\mathcal{U},\mathcal{V}\rangle_{\bm{M}}=\langle\mathcal{U},\bm{\delta}\mathcal{V}\rangle_{\bm{M}}$ (I.2) whenever the supports of the forms have compact intersection. In this sense the operators ${\bf d}$ and $\bm{\delta}$ are dual. ## II Classical analysis of the generalized Maxwell field ### II.1 Classical electrodynamics in four dimensions In Minkowski spacetime it is most common to study the electromagnetic field in the abstract index notation Wald (1984) where $F_{ab}$ is the covariant field–strength tensor and the Maxwell equations are $\partial_{[a}F_{bc]}=0\qquad\mbox{and}\qquad\partial^{a}F_{ab}=-4\pi j_{b}.$ (II.1) Here $j_{b}$ is the current density, $\partial_{a}$ is the partial derivative, lowering or raising of indices is done with respect to the metric $\eta_{\mu\nu}=\mbox{diag}(1,-1,-1,-1)$ and its inverse respectively, and $[\ ]$ in the homogeneous Maxwell equation is shorthand for the antisymmetric permutation over the indices. It is known that the generalization of the Maxwell equations to curved four–dimensional spacetimes is internally consistent. This is accomplished by the minimal substitution rule; replace the partial derivatives with $\nabla_{a}$, the unique covariant derivative operator associated with the spacetime metric such that $\nabla_{a}g_{bc}=0$. The Maxwell equations then become $\nabla_{[a}F_{bc]}=0\qquad\mbox{and}\qquad\nabla^{a}F_{ab}=-4\pi j_{b}.$ (II.2) It is also common to introduce the (co)vector potential $A_{a}$ related (at least locally) to the field strength tensor by $F_{ab}=\nabla_{[a}A_{b]}$. Recast in $A_{a}$, the homogeneous equation is trivially satisfied, as a result of the first Bianchi identity, and the inhomogeneous equation becomes $\nabla^{a}\nabla_{a}A_{b}-\nabla_{b}\nabla^{a}A_{a}-{R^{a}}_{b}A_{a}=-4\pi j_{b}.$ (II.3) The Maxwell equations, in both flat and curved spacetime have a gauge freedom in that many different forms of $A_{a}$ give rise to the same $F_{ab}$. This comes from the freedom to add to $A_{a}$ the gradient of any scalar function $\Lambda$. Because the covariant derivatives, like partial derivatives, commute when acting on scalars the addition of the gradient term has no effect on the final outcome of the resulting field strength. Therefore one can choose to work in a particular gauge, say the Lorenz gauge where $\nabla^{a}A_{a}=0$. Then we have considerable simplification to the globally hyperbolic equation $\nabla^{a}\nabla_{a}A_{b}-{R^{a}}_{b}A_{a}=-4\pi j_{b}.$ (II.4) The benefit of doing so is that existence and uniqueness of solutions to globally hyperbolic equations has been well studied and with a little work we can “extract” solutions to our original equation. This will be covered in more detail below. ### II.2 Generalized Maxwell Field The electromagnetic field is a specific example in four dimensions of a much broader theory of fully antisymmetric tensor fields in curved spacetimes of arbitrary dimension. It is mathematically natural to handle such tensors in the language of exterior differential calculus, there being a number of benefits to doing so: (a) All equations are coordinate chart independent, thus we obtain global results directly. (b) Even if a coordinate chart is specified, the differential forms are still independent of the choice of connection, thus substantially simplifying coordinate based calculations. (c) The generalized Maxwell equations do index bookkeeping in a ‘natural’ (albeit hidden) way for forms of different rank or in spacetimes of varying dimension. It is precisely this mechanism by which the generalized Maxwell equations avoid all of the consistency problems discussed above for other types of fields. No subsidiary conditions are needed on the spacetime, and the spacetime itself need not satisfy the Einstein equation. We will elaborate more on this point later. At first glance one could consider the fundamental object of study be the $(p+1)$-form field strength $\mathcal{F}$, which can be thought of as a fully- antisymmetric rank-$(0,p+1)$ tensor. Then, the generalized Maxwell equations are ${\bf d}\mathcal{F}=0\qquad\mbox{ and }\qquad-\bm{\delta}\mathcal{F}=\mathcal{J},$ (II.5) where $\mathcal{J}$ is the $p$-form current density. Electromagnetism happens to be the case where $\mathcal{F}$ is a two-form in a four-dimensional spacetime and given a local coordinate chart, the above equations reduce to the conventional Maxwell equations II.2. However, we have chosen not to take $\mathcal{F}$ to be our fundamental object for three reasons: (a) The action in terms of $\mathcal{F}$, as we will see below, still involves $\mathcal{A}$ in the interaction term with $\mathcal{J}$, thus $\mathcal{A}$ has to be defined from the relation $\mathcal{F}={\bf d}\mathcal{A}$. However, only on spacetimes that have trivial $(p+1)$-th cohomology group, that is $H^{p+1}({\bm{M}})=\\{[0]\\}$, can $\mathcal{F}$ be formulated globally in terms of a $p$-form potential $\mathcal{A}\in\Omega^{p}({\bm{M}})$ such that $\mathcal{F}={\bf d}\mathcal{A}$ is true everywhere. There is a topological restriction to doing this if the cohomology group is nontrivial. Thus starting with $\mathcal{A}$ as the fundamental object avoids cohomological problems early on. (b) Furthermore, even if we were study the free field, dropping all the terms involving $\mathcal{J}$, it is unclear how to then go from the action in terms of $\mathcal{F}$ to field equations without the introduction of $\mathcal{A}$ again. (c) We are specifically working with the massless field here, but from our previous experience with the Proca field we find that $\mathcal{A}$ is the fundamental object of study. Also, when $\mathcal{A}\in\Omega^{0}({\bm{M}})$, the action and field equation are that of the minimally-coupled massless scalar field in curved spacetime. So we have strong reason to treat $\mathcal{A}$ as the fundamental object here which is derivable directly from the action (II.6) below. This last point is rather remarkable. The minimally coupled scalar field and the electromagnetic field are but two examples of a general $p$-form field theory in curved spacetime. In some of our previous work we had indications of this property. Solutions of the massless scalar field theory in four dimensions could be used to construct the gauge photon polarizations of the (co)vector potential and that information could be used to help generate one of the two physically allowed polarization states of the (co)vector potential Fewster and Pfenning (2003). The remaining physical state the comes from Gram- Schmidt orthogonalization. It had also been noted that the quantum inequality for the electromagnetic field was exactly twice that of the minimally-coupled scalar field. ### II.3 The generalized Maxwell field equations and fundamental solutions Let ${\bm{M}}$ be a globally hyperbolic spacetime, that is, a manifold of $\dim({\bm{M}})=n$ with Lorentzian metric of signature $s=n-1$, i.e. the metric is of the form $(+,-,-,\dots)$. On this spacetime we take our fundamental object to be the field $\mathcal{A}\in\Omega^{p}({\bm{M}})$ with $0\leq p<n$. The classical action is given by $\mathcal{S}=(-1)^{p+1}\left[-\frac{1}{2}\langle{\bf d}\mathcal{A},{\bf d}\mathcal{A}\rangle_{\bm{M}}-\langle\mathcal{A},\mathcal{J}\rangle_{\bm{M}}\right]+S(\mathcal{J}),$ (II.6) were $S(\mathcal{J})$ is the remainder of the action for the current density $\mathcal{J}\in\Omega^{p}({\bm{M}})$. The only criteria that we ask of $\mathcal{J}$ is that it be co-closed, i.e. $\bm{\delta}\mathcal{J}=0$ so as to preserve charge/current conservation. Variation with respect to the field yields the generalized Maxwell equation, $-\bm{\delta}{\bf d}\mathcal{A}=\mathcal{J}.$ (II.7) The field strength is then calculated from $\mathcal{A}$ by $\mathcal{F}={\bf d}\mathcal{A}$. It is easily seen from this definition of the field strength that there is a gauge freedom in that to any solution $\mathcal{A}$ of Eq. (II.7) we may add ${\bf d}\Lambda$ where $\Lambda\in\Omega^{p-1}({\bm{M}})$. While this changes the value of the gauge field $\mathcal{A}$ at every point, it leaves the field strength $\mathcal{F}$ unchanged. We will denote any two solutions $\mathcal{A}$ and $\mathcal{A}^{\prime}$ to be gauge equivalent by $\mathcal{A}\sim\mathcal{A}^{\prime}$ if they differ by the exterior derivative of a $(p-1)$-form. Thus when discussing the potential, particularly for the quantum problem, we will often work in gauge equivalent classes denoted by $[\mathcal{A}]=\mathcal{A}+{\bf d}\Omega^{p-1}({\bm{M}})$. In practice one often chooses not to solve the above equation directly but instead work with the constrained Klein-Gordon system, $\Box\mathcal{A}=\mathcal{J}\qquad\mbox{ with }\qquad\bm{\delta}\mathcal{A}=0,$ (II.8) as any solution that satisfies (II.8) is also a solution to (II.7). Typically $\bm{\delta}\mathcal{A}=0$ is called the Lorenz gauge condition. In a given coordinate chart, this constrained Klein-Gordon system can be written in component form as Lichnerowicz (1961) $\nabla^{\beta}\nabla_{\beta}\mathcal{A}_{\alpha_{1}\dots\alpha_{p}}-\sum_{j=1}^{p}{R_{\alpha_{j}}}^{\beta}\mathcal{A}_{\alpha_{1}\dots\beta\dots\alpha_{p}}+\sum_{j,\,k=1,\,j\neq k}^{p}{R_{\alpha_{j}}}^{\beta}{{}_{\alpha_{k}}}^{\gamma}\mathcal{A}_{\alpha_{1}\dots\beta\dots\gamma\dots\alpha_{p}}=\mathcal{J}_{\alpha_{1}\dots\alpha_{p}}$ (II.9) and $\nabla^{\beta}\mathcal{A}_{\beta\alpha_{2}\dots\alpha_{p}}=0.$ (II.10) Here $R_{\alpha\beta}$ and $R_{\alpha\beta\gamma\delta}$ are the Ricci and Riemann tensors, respectively. Also, the notation is such that the index $\beta$ in the first summation occupies the $j^{\rm th}$ place in the tensor component of $\mathcal{A}$ while in the double summation the indices $\beta$ and $\gamma$ occupy the $j^{\rm th}$ and $k^{\rm th}$ spots in the tensor components. The Riemann and Ricci terms are not unexpected; in differential geometry they are a result of the Weitzenböck identity. The advantage of using the constrained Klein-Gordon system of equations is that $\Box$ is a normally hyperbolic operator Bär et al. (2007) with principal part $g^{\mu\nu}\partial_{\mu}\partial_{\nu}$. Thus, there exists a unique advanced and retarded Green’s operator denoted by $E^{\pm}:\Omega^{p}_{0}({\bm{M}})\rightarrow\Omega^{p}({\bm{M}})$, (see Corollary 3.4.3 of Bär, Ginoux and Pfäffle Bär et al. (2007), or Choquet- Bruhat Choquet-Bruhat (1968) and Proposition 3.3 of Sahlmann and Verch Sahlmann and Verch (2001)) which have the properties $\Box E^{\pm}=E^{\pm}\Box=\openone,$ (II.11) and for all $f\in\Omega^{p}_{0}({\bm{M}})$, the ${\rm supp}\,(E^{\pm}f)\subset J^{\pm}({\rm supp}\,(f))$. Furthermore, the map defined by the Green’s operators are sequentially continuous. All of the differential operations on forms commute with the Green’s operator; ###### Proposition II.1. Let $f\in\Omega^{p}_{0}({\bm{M}})$ be a test function, then the following operations involving $E^{\pm}$ commute: (a) ${\bf d}E^{\pm}f=E^{\pm}{\bf d}f$, and (b) $\bm{\delta}E^{\pm}f=E^{\pm}\bm{\delta}f$. ###### Proof. (a) Let $f\in\Omega_{0}^{p}({\bm{M}})$. We know that $\mathcal{A}^{\pm}=E^{\pm}f\in\Omega^{p}({\bm{M}})$ is the unique solution to $\Box\mathcal{A}^{\pm}=f$. Likewise we have $\mathcal{A}^{\prime\pm}=E^{\pm}{\bf d}f\in\Omega^{p+1}({\bm{M}})$ is the unique solution to $\Box\mathcal{A}^{\prime\pm}={\bf d}f$. Consider $\Box{\bf d}\mathcal{A}^{\pm}={\bf d}\Box\mathcal{A}^{\pm}={\bf d}f=\Box\mathcal{A}^{\prime\pm}.$ (II.12) Since $\mathcal{A}^{\prime\pm}$ is unique we deduce ${\bf d}\mathcal{A}^{\pm}=\mathcal{A}^{\prime\pm}$, therefore ${\bf d}E^{\pm}f=E^{\pm}{\bf d}f$. (b) Let $f$ and $\mathcal{A}^{\pm}$ be as defined above. Set $\mathcal{A}^{\prime\pm}=E^{\pm}\bm{\delta}f\in\Omega^{p-1}({\bm{M}})$, which is the unique solution to $\Box\mathcal{A}^{\prime\pm}=\bm{\delta}f$. Then consider $\bm{\delta}\Box\mathcal{A}^{\pm}=\Box\bm{\delta}\mathcal{A}^{\pm}=\bm{\delta}f=\Box\mathcal{A}^{\prime\pm}.$ (II.13) Since $\mathcal{A}^{\prime\pm}$ is unique we deduce $\bm{\delta}\mathcal{A}^{\pm}=\mathcal{A}^{\prime\pm}$, therefore $\bm{\delta}E^{\pm}f=E^{\pm}\bm{\delta}f$. ∎ We also need the advanced minus retarded propagator $E\equiv E^{-}-E^{+}$. Since it is a linear combination of the advanced and retarded propagators, it has all of the same commutation properties above and gives the unique solutions to the homogeneous (source free) Klein-Gordon equation. We are now ready to show that the existence of a fundamental solution for the Klein- Gordon equation also gives us a solution to the generalized Maxwell equations. ### II.4 Initial value formulation The the initial value problem for a gauge field in four dimensional spacetimes has been treated in the initial sections of Dimock and for the Proca field by Furlani. We follow closely the notation and structure of both these papers in this section as we generalize their results to all $p$-form fields with $p<n$ in globally hyperbolic spacetimes of arbitrary dimension. In order to relate initial data to solutions of the wave equation we first need to discuss Green’s theorem for forms. Let ${\bm{M}}$ be a globally hyperbolic spacetime, and $\mathcal{O}\subset{\bm{M}}$ be an open region in the spacetime with boundary $\partial\mathcal{O}$. Define the natural inclusion $i:\partial\mathcal{O}\rightarrow\mathcal{O}$ and $i^{}$ the pullback. On ${\bm{M}}$ let $\mathcal{A}\in\Omega^{p}({\bm{M}})$ and $\mathcal{B}\in\Omega^{p}_{0}({\bm{M}})$, then by Stokes theorem we have $\int_{\mathcal{O}}\left(\mathcal{A}\wedge\Box\mathcal{B}-\mathcal{B}\wedge\Box\mathcal{A}\right)=\int_{\partial\mathcal{O}}i^{}\left(\mathcal{A}\wedge{\bf d}\mathcal{B}+\bm{\delta}\mathcal{A}\wedge\mathcal{B}\right)-\int_{\partial\mathcal{O}}i^{}\left(\mathcal{B}\wedge{\bf d}\mathcal{A}+\bm{\delta}\mathcal{B}\wedge\mathcal{A}\right)$ (II.14) which is called Green’s identity for $\Box$ (Sect. 7.5 of AMR Abraham et al. (1988)). The integrals are all well defined because $\mathcal{B}$ has compact support which does not expand under any of the derivative operations. Next, let $\Sigma\subset{\bm{M}}$ be a Cauchy surface in the spacetime and define $\Sigma^{\pm}\equiv J^{\pm}(\Sigma)\backslash\Sigma$. If we use $\mathcal{O}=\Sigma^{\pm}$ and $\partial\mathcal{O}=\Sigma$ in Green’s identity we have $\int_{\Sigma^{\pm}}\left(\mathcal{A}\wedge\Box\mathcal{B}-\mathcal{B}\wedge\Box\mathcal{A}\right)=\mp\left[\int_{\Sigma}i^{}\left(\mathcal{A}\wedge{\bf d}\mathcal{B}+\bm{\delta}\mathcal{A}\wedge\mathcal{B}\right)-\int_{\Sigma}i^{}\left(\mathcal{B}\wedge{\bf d}\mathcal{A}+\bm{\delta}\mathcal{B}\wedge\mathcal{A}\right)\right]$ (II.15) where the sign difference on the right hand side comes about because of the opposite orientation of the unit normal to the Cauchy surface. For smooth maps, the pullback is natural with respect to both the wedge product and the exterior derivative, thus we may distribute it across the terms above. We define the following operations which act on $p$-forms: $\displaystyle\rho_{(0)}$ $\displaystyle=$ $\displaystyle i^{}$ (II.16) $\displaystyle\rho_{({\bf d})}$ $\displaystyle=$ $\displaystyle(-1)^{p(n-p-1)+(n-1)}\,i^{}{\bf d}$ (II.17) $\displaystyle\rho_{(\bm{\delta})}$ $\displaystyle=$ $\displaystyle i^{}\bm{\delta}$ (II.18) $\displaystyle\rho_{(n)}$ $\displaystyle=$ $\displaystyle(-1)^{(n-p)(p-1)+(n-1)}\,i^{}.$ (II.19) The first operation is the pullback of the form onto the Cauchy Surface, the second is the forward normal derivative, the third is the pullback of the divergence and the last is the forward normalDimock (1992); Furlani (1999). Note, all operations to the right of $i^{}$ act on forms which are defined on the whole manifold ${\bm{M}}$. However, operations to the left of $i^{}$ are defined with respect to the Cauchy surface $\Sigma$, thus in the forward normal derivative and forward normal, the first Hodge star in each expression is with respect to the induced metric on the Cauchy surface. With these operations the above equation reduces to $\int_{\Sigma^{\pm}}\left(\mathcal{A}\wedge\Box\mathcal{B}-\mathcal{B}\wedge\Box\mathcal{A}\right)=\mp\left(\langle\rho_{(0)}\mathcal{A},\rho_{({\bf d})}\mathcal{B}\rangle_{\Sigma}+\langle\rho_{(\bm{\delta})}\mathcal{A},\rho_{(n)}\mathcal{B}\rangle_{\Sigma}-\langle\rho_{(0)}\mathcal{B},\rho_{({\bf d})}\mathcal{A}\rangle_{\Sigma}-\langle\rho_{(\bm{\delta})}\mathcal{B},\rho_{(n)}\mathcal{A}\rangle_{\Sigma}\right).$ (II.20) We begin the discussion of the fundamental solutions to the Klein-Gordon equation. First we look at the mapping of $\Box$ solutions to initial data. ###### Proposition II.2. Let $\mathcal{A}\in\Omega^{p}({\bm{M}})$ be a smooth solution of $\Box\mathcal{A}=\mathcal{J}$ with Cauchy data $\displaystyle A_{(0)}$ $\displaystyle\equiv$ $\displaystyle\rho_{(0)}\mathcal{A}\in\Omega^{p}(\Sigma)$ $\displaystyle A_{({\bf d})}$ $\displaystyle\equiv$ $\displaystyle\rho_{({\bf d})}\mathcal{A}\in\Omega^{p}(\Sigma)$ $\displaystyle A_{(\bm{\delta})}$ $\displaystyle\equiv$ $\displaystyle\rho_{(\bm{\delta})}\mathcal{A}\in\Omega^{p-1}(\Sigma)$ $\displaystyle A_{(n)}$ $\displaystyle\equiv$ $\displaystyle\rho_{(n)}\mathcal{A}\in\Omega^{p-1}(\Sigma).$ Then, for any compactly supported test function $f\in\Omega^{p}_{0}({\bm{M}})$ we have $\int_{\bm{M}}\mathcal{A}\wedgef=\langle\mathcal{J},E^{+}f\rangle_{\Sigma^{+}}+\langle\mathcal{J},E^{-}f\rangle_{\Sigma^{-}}+\langle A_{(0)},\rho_{({\bf d})}Ef\rangle_{\Sigma}+\langle A_{(\bm{\delta})},\rho_{(n)}Ef\rangle_{\Sigma}-\langle A_{({\bf d})},\rho_{(0)}Ef\rangle_{\Sigma}-\langle A_{(n)},\rho_{(\bm{\delta})}Ef\rangle_{\Sigma}.$ (II.21) ###### Proof. Using Eq. II.20, set $\mathcal{A}=\mathcal{A}$ and $\mathcal{B}=E^{\pm}f$ for $\Sigma^{\pm}$ respectively, then $\displaystyle\int_{\Sigma^{\pm}}\left(\mathcal{A}\wedge\Box E^{\pm}f-E^{\pm}f\wedge\Box\mathcal{A}\right)$ $\displaystyle=$ $\displaystyle\mp\left(\langle\rho_{(0)}\mathcal{A},\rho_{({\bf d})}E^{\pm}f\rangle_{\Sigma}+\langle\rho_{(\bm{\delta})}\mathcal{A},\rho_{(n)}E^{\pm}f\rangle_{\Sigma}\right.$ (II.22) $\displaystyle\left.-\langle\rho_{(0)}E^{\pm}f,\rho_{({\bf d})}\mathcal{A}\rangle_{\Sigma}-\langle\rho_{(\bm{\delta})}E^{\pm}f,\rho_{(n)}\mathcal{A}\rangle_{\Sigma}\right).$ Substituting the Cauchy data, noting that $\mathcal{A}$ is a smooth solution to the Klein-Gordon equation with source $\mathcal{J}$ and using $\Box E^{\pm}=\openone$, we can simplify the above to $\int_{\Sigma^{\pm}}\mathcal{A}\wedgef=\langle\mathcal{J},E^{\pm}f\rangle_{\Sigma^{\pm}}\mp\left(\langle A_{(0)},\rho_{({\bf d})}E^{\pm}f\rangle_{\Sigma}+\langle A_{(\bm{\delta})},\rho_{(n)}E^{\pm}f\rangle_{\Sigma}-\langle\rho_{(0)}E^{\pm}f,A_{({\bf d})}\rangle_{\Sigma}-\langle\rho_{(\bm{\delta})}E^{\pm}f,A_{(n)}\rangle_{\Sigma}\right)$ (II.23) When the two above equations are added for $\Sigma^{\pm}$, the result is Eq. II.21. Furthermore, the integrals in this expression are well defined because $f$ has compact support, thus $E^{\pm}f$ and $Ef$ have compact support on all other Cauchy surfaces. ∎ We now address the issues of existence and uniqueness for homogenous solutions to the Klein-Gordon equation. ###### Proposition II.3. (Uniqueness of homogeneous $\Box$ solutions) If $\mathcal{A}$ is a smooth solution to $\Box\mathcal{A}=0$ with Cauchy data $A_{(0)}=0$, $A_{(\bm{\delta})}=0$, $A_{({\bf d})}=0$, and $A_{(n)}=0$ then $\mathcal{A}=0$. ###### Proof. By Proposition II.2 we have $\int_{\bm{M}}\mathcal{A}\wedgef=0$ which is true for all compactly supported $f$ therefore $\mathcal{A}=0$. ∎ ###### Proposition II.4. (Existence of homogenous $\Box$ solutions) Let $A_{(0)},A_{({\bf d})}\in\Omega^{p}_{0}(\Sigma)$ and $A_{(n)},A_{(\bm{\delta})}\in\Omega^{p-1}_{0}(\Sigma)$ specify Cauchy data on $\Sigma$. Then $\mathcal{A}^{\prime}=-E\rho_{({\bf d})}^{\prime}A_{(0)}-E\rho_{(n)}^{\prime}A_{(\bm{\delta})}+E\rho_{(\bm{\delta})}^{\prime}A_{(n)}+E\rho_{(0)}^{\prime}A_{({\bf d})}$ (II.24) is the unique smooth solution of $\Box\mathcal{A}^{\prime}=0$ with these data. ###### Proof. Let $f\in\Omega_{0}^{p}({\bm{M}})$ be any compactly supported test form and consider $\displaystyle\langle\mathcal{A}^{\prime},f\rangle_{\bm{M}}$ $\displaystyle=$ $\displaystyle-\langle E\rho_{({\bf d})}^{\prime}A_{(0)},f\rangle_{\bm{M}}-\langle E\rho_{(n)}^{\prime}A_{(\bm{\delta})},f\rangle_{\bm{M}}+\langle E\rho_{(\bm{\delta})}^{\prime}A_{(n)},f\rangle_{\bm{M}}+\langle E\rho_{(0)}^{\prime}A_{({\bf d})},f\rangle_{\bm{M}}$ (II.25) $\displaystyle=$ $\displaystyle-\langle\rho_{({\bf d})}^{\prime}A_{(0)},E^{\prime}f\rangle_{\bm{M}}-\langle\rho_{(n)}^{\prime}A_{(\bm{\delta})},E^{\prime}f\rangle_{\bm{M}}+\langle\rho_{(\bm{\delta})}^{\prime}A_{(n)},E^{\prime}f\rangle_{\bm{M}}+\langle\rho_{(0)}^{\prime}A_{({\bf d})},E^{\prime}f\rangle_{\bm{M}}.$ The operator $\Box=-(\bm{\delta}{\bf d}-{\bf d}\bm{\delta})$ is self-adjoint therefore the transpose operator $E^{\prime}=-E$ (see Choquet-Bruhat Choquet- Bruhat (1968), corollary to Theorem II). The pullback operators are all linear and continuous, thus there exist transpose operators for the $\rho$’s denoted with the primes, thus we have $\langle\mathcal{A}^{\prime},f\rangle_{\bm{M}}=\langle A_{(0)},\rho_{({\bf d})}Ef\rangle_{\Sigma}+\langle A_{(\bm{\delta})},\rho_{(n)}Ef\rangle_{\Sigma}-\langle A_{(n)},\rho_{(\bm{\delta})}Ef\rangle_{\Sigma}-\langle A_{({\bf d})},\rho_{(0)}Ef\rangle_{\Sigma}.$ (II.26) By Proposition II.2, if $\mathcal{A}$ is a solution to $\Box\mathcal{A}=0$ with the Cauchy data given above, then we have $\langle\mathcal{A}^{\prime},f\rangle_{\bm{M}}=\langle\mathcal{A},f\rangle_{\bm{M}}$ which implies $\mathcal{A}^{\prime}=\mathcal{A}$ in a distributional sense. Thus $\mathcal{A}^{\prime}$ is identified with the unique smooth solution $\mathcal{A}$. ∎ Unless otherwise stated, from this point forward we will be discussing systems whose initial data is smooth and compactly supported on the Cauchy surfaces, i.e., $\left(A_{(0)},A_{({\bf d})},A_{(n)},A_{(\bm{\delta})}\right)\in\Omega^{p}_{0}(\Sigma)\oplus\Omega^{p}_{0}(\Sigma)\oplus\Omega^{p-1}_{0}(\Sigma)\oplus\Omega^{p-1}_{0}(\Sigma)$. We may now discuss the sense in which $\mathcal{A}^{\prime}$, as defined above, varies with respect to the Cauchy data. ###### Proposition II.5. $\mathcal{A}^{\prime}$ is continuously dependent on the Cauchy data $\left(A_{(0)},A_{({\bf d})},A_{(n)},A_{(\bm{\delta})}\right)$. ###### Proof. The proof is a generalization of Theorem 3.2.12 of Bär et. al. Bär et al. (2007). Define $\mathcal{H}^{p}({\bm{M}})\equiv\left\\{\mathcal{A}\in\Omega^{p}({\bm{M}})\|\Box\mathcal{A}=0\right\\}$ as the space of smooth homogeneous Klein-Gordon solutions, then the mapping $\mathcal{P}:\mathcal{H}^{p}({\bm{M}})\rightarrow\Omega^{p}(\Sigma)\oplus\Omega^{p}(\Sigma)\oplus\Omega^{p-1}(\Sigma)\oplus\Omega^{p-1}(\Sigma)$ of a solution to its Cauchy data is by definition both linear and continuous. Next, let $K\subset{\bm{M}}$ be a compact subset of ${\bm{M}}$. On $K$ we have the spaces $\Omega_{0}^{p}(K)\subset\Omega^{p}({\bm{M}})$ and $\Omega_{0}^{p}(K\cap\Sigma)\subset\Omega^{p}(\Sigma)$ for all $p\leq n$. We also define the space $\mathcal{V}_{K}^{p}=\mathcal{P}^{-1}\left[\Omega^{p}_{0}(K\cap\Sigma)\oplus\Omega^{p}_{0}(K\cap\Sigma)\oplus\Omega^{p-1}_{0}(K\cap\Sigma)\oplus\Omega^{p-1}_{0}(K\cap\Sigma)\right]$. Since $\mathcal{P}$ is continuous and $\Omega^{p}_{0}(K\cap\Sigma)\oplus\Omega^{p}_{0}(K\cap\Sigma)\oplus\Omega^{p-1}_{0}(K\cap\Sigma)\oplus\Omega^{p-1}_{0}(K\cap\Sigma)\subset\Omega^{p}(\Sigma)\oplus\Omega^{p}(\Sigma)\oplus\Omega^{p-1}(\Sigma)\oplus\Omega^{p-1}(\Sigma)$ is a closed subset, this implies $\mathcal{V}_{K}^{p}\subset\mathcal{H}^{p}({\bm{M}})$ is also a closed subset. Furthermore, both $\Omega^{p}_{0}(K\cap\Sigma)\oplus\Omega^{p}_{0}(K\cap\Sigma)\oplus\Omega^{p-1}_{0}(K\cap\Sigma)\oplus\Omega^{p-1}_{0}(K\cap\Sigma)$ and $\mathcal{V}_{K}^{p}$ are Fréchet spaces. Also, the map $\mathcal{P}:\mathcal{V}_{K}^{p}\rightarrow\Omega^{p}_{0}(K\cap\Sigma)\oplus\Omega^{p}_{0}(K\cap\Sigma)\oplus\Omega^{p-1}_{0}(K\cap\Sigma)\oplus\Omega^{p-1}_{0}(K\cap\Sigma)$ is linear, continuous, bijective, and by the open mapping theorem for Fréchet spaces (Theorem V.6 of Reed and Simon Reed and Simon (1980)) it is also an open mapping. Since a bijection that is open implies a continuous inverse, we conclude that $\mathcal{P}^{-1}$ is continuous. Finally, if we have a convergent sequence of Cauchy data $\left(A_{(0),i},A_{({\bf d}),i},A_{(n),i},A_{(\bm{\delta}),i}\right)\rightarrow\left(A_{(0)},A_{({\bf d})},A_{(n)},A_{(\bm{\delta})}\right)$ in $\Omega^{p}_{0}(\Sigma)\oplus\Omega^{p}_{0}(\Sigma)\oplus\Omega^{p-1}_{0}(\Sigma)\oplus\Omega^{p-1}_{0}(\Sigma)$, then we can choose a compact subset $K\subset{\bm{M}}$ with the property that $\mbox{supp}\left(A_{(0),i}\right)\cup\mbox{supp}\left(A_{({\bf d}),i}\right)\cup\mbox{supp}\left(A_{(n),i}\right)\cup\mbox{supp}\left(A_{(\bm{\delta}),i}\right)\subset K$ for all $i$ and $\mbox{supp}\left(A_{(0)}\right)\cup\mbox{supp}\left(A_{({\bf d})}\right)\cup\mbox{supp}\left(A_{(n)}\right)\cup\mbox{supp}\left(A_{(\bm{\delta})}\right)\subset K$. Thus, $\left(A_{(0),i},A_{({\bf d}),i},A_{(n),i},A_{(\bm{\delta}),i}\right)\rightarrow\left(A_{(0)},A_{({\bf d})},A_{(n)},A_{(\bm{\delta})}\right)$ in $\Omega^{p}_{0}(K\cap\Sigma)\oplus\Omega^{p}_{0}(K\cap\Sigma)\oplus\Omega^{p-1}_{0}(K\cap\Sigma)\oplus\Omega^{p-1}_{0}(K\cap\Sigma)$ and we conclude that the inverse mapping $\mathcal{P}^{-1}\left(A_{(0),i},A_{({\bf d}),i},A_{(n),i},A_{(\bm{\delta}),i}\right)\rightarrow\mathcal{P}^{-1}\left(A_{(0)},A_{({\bf d})},A_{(n)},A_{(\bm{\delta})}\right)$. ∎ By an appropriate restriction of the Cauchy data for the Klein-Gordon equation we are able to identify useful subspaces of solutions. For example, of all solutions to the Klein-Gordon equation, those that also satisfy the Lorenz gauge condition have the following property; ###### Proposition II.6. (Lorenz solutions) Suppose $\mathcal{A}\in\Omega^{p}({\bm{M}})$ solves $\Box\mathcal{A}=0$ with Cauchy data $\left(A_{(0)},A_{({\bf d})},A_{(n)},A_{(\bm{\delta})}\right)$, then $\bm{\delta}\mathcal{A}=0$ if and only if $\bm{\delta}A_{({\bf d})}=0$, $\bm{\delta}A_{(n)}=0$, and $A_{(\bm{\delta})}=0$. ###### Proof. First suppose that $\mathcal{A}$ solves both $\Box\mathcal{A}=0$ and $\bm{\delta}\mathcal{A}=0$ and let us evaluate the three conditions above. For the forward normal derivative we have $\bm{\delta}A_{({\bf d})}=\bm{\delta}\rho_{({\bf d})}\mathcal{A}=\bm{\delta}\rho_{(n)}{\bf d}\mathcal{A}.$ (II.27) Using the identity for $p$-forms that $\bm{\delta}\rho_{(n)}=(-1)^{np}\rho_{(n)}\bm{\delta}$ we find $\bm{\delta}A_{({\bf d})}=(-1)^{n(p+1)}\rho_{(n)}\bm{\delta}{\bf d}\mathcal{A}=-(-1)^{n(p+1)}\rho_{(n)}{\bf d}\bm{\delta}\mathcal{A}=0,$ (II.28) where we have used the wave equation and the subsidiary condition in the last two steps. For the divergence of the forward normal we find $\bm{\delta}A_{(n)}=\bm{\delta}\rho_{(n)}\mathcal{A}=(-1)^{np}\rho_{(n)}\bm{\delta}\mathcal{A}=0.$ (II.29) Finally, for the pullback of the divergence we have $A_{(\bm{\delta})}=\rho_{(0)}\bm{\delta}\mathcal{A}=0.$ (II.30) On the other hand, assume $\mathcal{A}$ is a Klein-Gordon solution whose Cauchy data satisfies $\bm{\delta}A_{({\bf d})}=0=\bm{\delta}A_{(n)}$ and $A_{(\bm{\delta})}=0$. Let $f\in\Omega_{0}^{p-1}({\bm{M}})$ and evaluate $\displaystyle\langle\bm{\delta}\mathcal{A},f\rangle_{\bm{M}}$ $\displaystyle=$ $\displaystyle\langle\mathcal{A},{\bf d}f\rangle_{\bm{M}}$ (II.31) $\displaystyle=$ $\displaystyle\langle A_{(0)},\rho_{({\bf d})}E{\bf d}f\rangle_{\Sigma}+\langle A_{(\bm{\delta})},\rho_{(n)}E{\bf d}f\rangle_{\Sigma}-\langle A_{({\bf d})},\rho_{(0)}E{\bf d}f\rangle_{\Sigma}-\langle A_{(n)},\rho_{(\bm{\delta})}E{\bf d}f\rangle_{\Sigma}$ $\displaystyle=$ $\displaystyle\langle A_{(0)},\rho_{({\bf d})}{\bf d}Ef\rangle_{\Sigma}+\langle 0,\rho_{(n)}{\bf d}Ef\rangle_{\Sigma}-\langle A_{({\bf d})},\rho_{(0)}{\bf d}Ef\rangle_{\Sigma}-\langle A_{(n)},\rho_{(\bm{\delta})}{\bf d}Ef\rangle_{\Sigma},$ where we have used the fact that ${\bf d}$ commutes with $E$ on forms of compact support. The first term vanishes because $\rho_{({\bf d})}{\bf d}=0$. The second term trivially vanishes. For the third term we have that $\rho_{(0)}$ and ${\bf d}$ commute. For the fourth term we use $\rho_{(\bm{\delta})}=\rho_{(0)}\bm{\delta}$ and the fact that $Ef$ is a solution to the homogeneous Klein-Gordon equation to swap the order of the derivative operators. Therefore $\langle\bm{\delta}\mathcal{A},f\rangle_{\bm{M}}=-\langle A_{({\bf d})},{\bf d}\rho_{(0)}Ef\rangle_{\Sigma}+\langle A_{(n)},\rho_{(0)}{\bf d}\bm{\delta}Ef\rangle_{\Sigma}\ =-\langle\bm{\delta}A_{({\bf d})},\rho_{(0)}Ef\rangle_{\Sigma}+\langle\bm{\delta}A_{(n)},\rho_{(0)}\bm{\delta}Ef\rangle_{\Sigma}=0,$ (II.32) where we have used the conditions on the initial data. Since this is true for all $f$ we deduce that $\bm{\delta}\mathcal{A}=0$. ∎ ###### Corollary II.7. (Coulomb gauge solutions) Solutions in the Coulomb gauge will have Cauchy data $\left(A_{(0)},A_{({\bf d})},0,0\right)$ where $\bm{\delta}A_{({\bf d})}=0$. ###### Proof. Coulomb gauge solutions are Lorenz gauge solutions with the additional constraint $A_{(n)}=0$. ∎ By Proposition II.6 we can infer that the constrained Klein-Gordon system is self consistent in curved spacetimes of arbitrary dimension. A slightly different way to see this is to begin with the evolution equation $\Box\mathcal{A}=\mathcal{J}$ and take $\bm{\delta}$ of it, yielding $0=\bm{\delta}\mathcal{J}=\bm{\delta}\Box\mathcal{A}=\Box\bm{\delta}\mathcal{A}.$ (II.33) We observe that $\bm{\delta}\mathcal{A}$ satisfies the source free Klein- Gordon equation. If the Cauchy data for $\bm{\delta}\mathcal{A}$ vanishes on the initial Cauchy surface then by Proposition II.3 the unique solution is $\bm{\delta}\mathcal{A}=0$. The same holds for Coulomb gauge solutions. What ensures this property are the Riemann and Ricci terms in Eq. (II.9). They commute with the divergence in the proper way as a result of the first and second Bianchi identities. Unlike the results of Buchdahl and Higuchi for spinor and massive symmetric tensor fields, no other conditions are required of the spacetime for the $p$-form fields satisfying the generalized Maxwell equation. In fact, the spacetime does not need to satisfy the Einstein equations in any way. At a deeper level what we have really done is to take the flat space field equations and first make the minimal substitution. Only afterward do we then commute the covariant derivatives. This gives rise to all of the Riemann and Ricci terms. The beauty of using exterior calculus is all this is handled without our having to do it explicitly. Also, note that Lorenz solutions can be gauge related to one and other when considered as solutions to the generalized Maxwell equations. ###### Proposition II.8. (a) Let $\left(A_{(0)},A_{({\bf d})},A_{(n)},0\right)$ with $\bm{\delta}A_{({\bf d})}=0=\bm{\delta}\mathcal{A}_{(n)}$ be Cauchy data on $\Sigma$. If $\mathcal{A},\mathcal{A}^{\prime}\in\Omega^{p}({\bm{M}})$ are Lorenz solutions with this data then $\mathcal{A}\sim\mathcal{A}^{\prime}$. (b) Let $\left(A_{(0)},A_{({\bf d})},A_{(n)},0\right)$ and $(A^{\prime}_{(0)},A^{\prime}_{({\bf d})},A^{\prime}_{(n)},0)$ with $\bm{\delta}A_{({\bf d})}=0=\bm{\delta}A^{\prime}_{({\bf d})}$ and $\bm{\delta}A_{(n)}=0=\bm{\delta}A_{(n)}^{\prime}$ be Cauchy data on a common Cauchy surface $\Sigma$ and $\Lambda\in\Omega^{p-1}({\bm{M}})$ be a $-\bm{\delta}{\bf d}\Lambda=0$ solution. If $\mathcal{A},\mathcal{A}^{\prime}\in\Omega^{p}({\bm{M}})$ are Lorenz solutions with these data, respectively, then $\mathcal{A}^{\prime}\sim\mathcal{A}$ if and only if $A^{\prime}_{(0)}\sim A_{(0)}$, $A^{\prime}_{({\bf d})}=A_{({\bf d})}$, and $A^{\prime}_{(n)}=A_{(n)}+\Lambda_{({\bf d})}$. ###### Proof. (a) We need to show there exists a $\Lambda\in\Omega^{p-1}({\bm{M}})$ such that $\mathcal{A}^{\prime}=\mathcal{A}+{\bf d}\Lambda.$ Applying $\bm{\delta}$ to both sides of the above expression tells us that $\Lambda$ must satisfy $-\bm{\delta}{\bf d}\Lambda=0$. Looking at the four pullbacks that relate a solution to its initial data we find $\Lambda$ must also satisfy ${\bf d}\rho_{(0)}\Lambda=0,\qquad\mbox{ and }\qquad\rho_{({\bf d})}\Lambda=0,$ (II.34) for $\mathcal{A}$ and $\mathcal{A}^{\prime}$ to share common Cauchy data. Choose any set of Cauchy data $\left(\lambda_{(0)},0,\lambda_{(n)},0\right)\in\Omega^{p-1}_{0}(\Sigma)\oplus\Omega^{p-1}_{0}(\Sigma)\oplus\Omega^{p-2}_{0}(\Sigma)\oplus\Omega^{p-2}_{0}(\Sigma)$ with ${\bf d}\lambda_{(0)}=0$ and $\bm{\delta}\lambda_{(n)}=0$. By Proposition II.4 there exists a $\Lambda^{\prime}$ which is the solution to $\Box\Lambda^{\prime}=0$ with this data. Furthermore, the Cauchy data is such that by Proposition II.6 we have $\Lambda^{\prime}$ is a Lorenz solution. It is now trivial to see that $\Lambda=\Lambda^{\prime}$ satisfies $-\bm{\delta}{\bf d}\Lambda=-\bm{\delta}{\bf d}\Lambda^{\prime}=\Box\Lambda^{\prime}=0$ and conditions II.34. Therefore $\Lambda$ exists and we conclude $\mathcal{A}\sim\mathcal{A}^{\prime}$. (b) If $\mathcal{A}^{\prime}=\mathcal{A}+{\bf d}\Lambda$ where $-\bm{\delta}{\bf d}\Lambda=0$ then it immediately follows $\begin{array}[]{rcl}A^{\prime}_{(0)}&=&\rho_{(0)}\mathcal{A}^{\prime}=\rho_{(0)}\mathcal{A}+\rho_{(0)}{\bf d}\Lambda=A_{(0)}+{\bf d}\rho_{(0)}\Lambda=A_{(0)}+{\bf d}\Lambda_{(0)},\\\ A^{\prime}_{({\bf d})}&=&\rho_{({\bf d})}\mathcal{A}^{\prime}=\rho_{({\bf d})}\mathcal{A}+\rho_{({\bf d})}{\bf d}\Lambda=A_{({\bf d})},\\\ A^{\prime}_{(n)}&=&\rho_{(n)}\mathcal{A}^{\prime}=\rho_{(n)}\mathcal{A}+\rho_{(n)}{\bf d}\Lambda=A_{(n)}+\rho_{({\bf d})}\Lambda=A_{(n)}+\Lambda_{({\bf d})},\\\ A^{\prime}_{(\bm{\delta})}&=&\rho_{(\bm{\delta})}\mathcal{A}^{\prime}=\rho_{(\bm{\delta})}\mathcal{A}+\rho_{(\bm{\delta})}{\bf d}\Lambda=\rho_{(0)}\bm{\delta}{\bf d}\Lambda=0.\end{array}$ (II.35) Furthermore, $\bm{\delta}A^{\prime}_{(n)}=\bm{\delta}\rho_{(n)}(\mathcal{A}+{\bf d}\Lambda)=(-1)^{np}\rho_{(n)}\bm{\delta}(\mathcal{A}+{\bf d}\Lambda)=0$. Conversely, let $A^{\prime}_{(0)}=A_{(0)}+{\bf d}\lambda_{(0)}$, $A^{\prime}_{({\bf d})}=A_{({\bf d})}$, $A^{\prime}_{(n)}=A_{(n)}+\lambda_{({\bf d})}$ and $A^{\prime}_{(\bm{\delta})}=A_{(\bm{\delta})}=0$ where $\lambda_{(0)},\lambda_{({\bf d})}\in\Omega^{p-1}_{0}(\Sigma)$ with $\bm{\delta}\lambda_{({\bf d})}=0$. Given $\mathcal{A}$, define $\widetilde{\mathcal{A}}\equiv\mathcal{A}+{\bf d}\Lambda$ where $\Lambda$ solves the homogenous equation $\Box\Lambda=0$ with Cauchy data $\begin{array}[]{rcl}\rho_{(0)}\Lambda&=&\lambda_{(0)},\\\ \rho_{({\bf d})}\Lambda&=&A^{\prime}_{(n)}-A_{(n)},\\\ \rho_{(n)}\Lambda&=&\lambda_{(n)},\\\ \rho_{(\bm{\delta})}\Lambda&=&0,\end{array}$ (II.36) where $\lambda_{(n)}\in\Omega^{p-2}_{0}(\Sigma)$ satisfies $\bm{\delta}\lambda_{(n)}=0$. By Propositions II.4 and II.6 such a Lorenz solution exists and $\Lambda$ is therefore a solution to $-\bm{\delta}{\bf d}\Lambda=0$. Next we evaluate $\displaystyle\Box\widetilde{\mathcal{A}}$ $\displaystyle=$ $\displaystyle\Box\mathcal{A}+\Box{\bf d}\Lambda={\bf d}\Box\Lambda=0,$ (II.37) $\displaystyle\bm{\delta}\widetilde{\mathcal{A}}$ $\displaystyle=$ $\displaystyle\bm{\delta}\mathcal{A}+\bm{\delta}{\bf d}\Lambda=0,$ (II.38) and the Cauchy data $\begin{array}[]{rcl}\rho_{(0)}\widetilde{\mathcal{A}}&=&\rho_{(0)}\left(\mathcal{A}+{\bf d}\Lambda\right)=\rho_{(0)}\mathcal{A}+\rho_{(0)}{\bf d}\Lambda=A_{(0)}+{\bf d}\lambda_{(0)}=A^{\prime}_{(0)},\\\ \rho_{({\bf d})}\widetilde{\mathcal{A}}&=&\rho_{({\bf d})}\left(\mathcal{A}+{\bf d}\Lambda\right)=\rho_{({\bf d})}\mathcal{A}+\rho_{({\bf d})}{\bf d}\Lambda=A_{({\bf d})}=A^{\prime}_{({\bf d})},\\\ \rho_{(n)}\widetilde{\mathcal{A}}&=&\rho_{(n)}\left(\mathcal{A}+{\bf d}\Lambda\right)=\rho_{(n)}\mathcal{A}+\rho_{(n)}{\bf d}\Lambda=A_{(n)}+\lambda_{({\bf d})}=A^{\prime}_{(n)},\\\ \rho_{(\bm{\delta})}\widetilde{\mathcal{A}}&=&\rho_{(\bm{\delta})}\left(\mathcal{A}+{\bf d}\Lambda\right)=\rho_{(\bm{\delta})}{\bf d}\Lambda=\rho_{(n)}\bm{\delta}{\bf d}\Lambda=0.\end{array}$ (II.39) We conclude that $\widetilde{\mathcal{A}}$ is a Maxwell solution with identical Cauchy data to that of $\mathcal{A}^{\prime}$. By part (a) of this proposition we have $\widetilde{\mathcal{A}}\sim\mathcal{A}^{\prime}$ and hence $\mathcal{A}\sim\mathcal{A}^{\prime}$. ∎ ###### Corollary II.9. Let $\mathcal{A}$ be a Lorenz solution with Cauchy data $\left(A_{(0)},A_{({\bf d})},A_{(n)},0\right)$ where $\bm{\delta}A_{({\bf d})}=0=\bm{\delta}A_{(n)}$, then $\mathcal{A}$ is gauge equivalent to the Coulomb solution $\mathcal{A}^{\prime}$ with Cauchy data $\left(A_{(0)},A_{({\bf d})},0,0\right)$. ###### Proof. The Cauchy data for $\mathcal{A}$ and $\mathcal{A}^{\prime}$ given above satisfy the requirements of Proposition II.8, so we conclude $\mathcal{A}^{\prime}\sim\mathcal{A}$. ∎ This completes our discussion of $\Box$ solutions and it is now possible to prove the existence of Maxwell solutions. ###### Proposition II.10. (Existence of homogeneous Maxwell solutions) For any $(A_{(0)},A_{({\bf d})})\in\Omega_{0}^{p}(\Sigma)\times\Omega_{0}^{p}(\Sigma)$ with $\bm{\delta}A_{({\bf d})}=0$, there exists an $\mathcal{A}\in\Omega^{p}({\bm{M}})$ such that $-\bm{\delta}{\bf d}\mathcal{A}=0,\qquad\rho_{(0)}\mathcal{A}=A_{(0)},\qquad\mbox{and}\qquad\rho_{({\bf d})}\mathcal{A}=A_{({\bf d})}.$ (II.40) ###### Proof. The proof is basically a generalization of Dimock’s Proposition 2. To the equations above, we add two additional conditions and still find a solution: (a) First we impose the Lorenz condition $\bm{\delta}\mathcal{A}=0$. Thus, the problem of solving for a solution is equivalent to finding one for $\Box\mathcal{A}=0$. Also, this condition implies $\rho_{(\bm{\delta})}\mathcal{A}=0$. (b) Secondly, we specify that the forward normal $\rho_{(n)}\mathcal{A}=A_{(n)}\in\Omega^{p-1}_{0}(\Sigma)$ satisfies $\bm{\delta}A_{(n)}=0$. Dimock sets $A_{(n)}$ to zero and is therefore solving the problem in the Coulomb gauge. He comments that $A_{(n)}$ could be any other function. That is only true for $A_{(n)}\in\Omega^{0}(\Sigma)$. For higher order forms $A_{(n)}$ must be co-closed, a condition which happens to be trivially satisfied in Dimock’s case. So we are now seeking a solution to $\Box\mathcal{A}=0$ with Cauchy data $\left(A_{(0)},A_{({\bf d})},A_{(n)},0\right)$ where $\bm{\delta}A_{({\bf d})}=0$ and $\bm{\delta}A_{(n)}=0$. By Prop. II.4 such a solution exists and by Prop. II.6 it is a Lorenz solution, thus it satisfies Eqs. II.40. ∎ ###### Theorem II.11. (Existence of inhomogeneous Maxwell solutions) Choose any Cauchy surface $\Sigma$ and suppose $\mathcal{J}\in\Omega^{p}_{0}({\bm{M}})$ obeys $\bm{\delta}\mathcal{J}=0$ on an open globally hyperbolic neighborhood $N$ of $\Sigma$. Set the following Cauchy data on $\Sigma$: $A_{(\bm{\delta})}=A_{(n)}=0,\qquad A_{(0)}\in\Omega^{p}_{0}(\Sigma),\qquad\mbox{ and }\qquad A_{(d)}=\omega,$ (II.41) where $\omega$ is any solution to $\bm{\delta}\omega=(-1)^{n(p+1)+1}\rho_{(n)}\mathcal{J}.$ (II.42) (Solutions to this last equation will certainly exist if $H^{p}(\Sigma)$ is trivial, but might exist in other cases as well; see below.) Then the unique solution $\mathcal{A}$ to $\Box\mathcal{A}=\mathcal{J}$ corresponding to these data satisfies $\bm{\delta}\mathcal{A}=0$ on $N$. In particular, if $\mathcal{J}$ is co-closed on ${\bm{M}}$, $\mathcal{A}$ is a Lorenz gauge solution to $-\bm{\delta}{\bf d}\mathcal{A}=\mathcal{J}$. ###### Proof. We calculate $\displaystyle\rho_{(0)}\bm{\delta}\mathcal{A}$ $\displaystyle=\rho_{(\bm{\delta})}\mathcal{A}=0,$ $\displaystyle\rho_{(\bm{\delta})}\bm{\delta}\mathcal{A}$ $\displaystyle=\rho_{(0)}\bm{\delta}^{2}\mathcal{A}=0,$ $\displaystyle\rho_{(n)}\bm{\delta}\mathcal{A}$ $\displaystyle=(-1)^{np}\bm{\delta}\rho_{(n)}\mathcal{A}=0,$ and also $\rho_{({\bf d})}\bm{\delta}\mathcal{A}=\rho_{(n)}{\bf d}\bm{\delta}\mathcal{A}=-\rho_{(n)}(\mathcal{J}+\bm{\delta}{\bf d}\mathcal{A})=-\rho_{(n)}\mathcal{J}-(-1)^{n(p+1)}\bm{\delta}\rho_{(d)}\mathcal{A}=0$ (II.43) by Eq. (II.42). As $\bm{\delta}\mathcal{A}$ is therefore a solution to $\Box\bm{\delta}\mathcal{A}=\bm{\delta}\Box\mathcal{A}=\bm{\delta}\mathcal{J}=0$ (II.44) with trivial Cauchy data on $\Sigma$, it follows that $\bm{\delta}\mathcal{A}$ vanishes identically in $N$. ∎ If $\mathcal{J}$ is not compactly supported, but has spacelike compact support on Cauchy surfaces (see Bär et. al. Bär et al. (2007)) we can cut it off by multiplying by some smooth function $\chi\in\Omega^{0}({\bm{M}})$ which equals $1$ on $N$ such that $\chi\mathcal{J}\in\Omega_{0}^{p}({\bm{M}})$ is compactly supported. On $N$ we have $\bm{\delta}\chi\mathcal{J}=0$, while on the rest of ${\bm{M}}$ this may not be the case. Then the solution $\mathcal{A}$ given by the theorem is co-closed on $N$; as it is obtained by solving a Cauchy problem within $N$, if we repeat the procedure for a larger globally hyperbolic region $N^{\prime}$ containing $N$, then the new solution (which is co-closed on all of $N^{\prime}$) would agree with the old one on $N$. We may therefore remove the cutoff by expanding $N$ to cover all of ${\bm{M}}$, thereby obtaining a global co-closed solution. This has proved: ###### Theorem II.12. If $\mathcal{J}$ has compact support on Cauchy surfaces and there is a Cauchy surface $\Sigma$ for which $\rho_{(n)}\mathcal{J}$ is co-exact [which always holds if $H^{p}(\Sigma)$ is trivial], then there exists a global Lorenz gauge solution $\mathcal{A}\in\Omega^{p}({\bm{M}})$ to $-\bm{\delta}{\bf d}\mathcal{A}=\mathcal{J}$. This generalizes the current Prop II.2: if $\mathcal{J}$ is compact to the past, take $\Sigma$ to the past of the support of $\mathcal{J}$ and then $\rho_{(n)}\mathcal{J}$ vanishes. Then we may solve (II.42) with $\omega=0$ [regardless of the cohomology of $\Sigma$] and we will of course obtain $\mathcal{A}=E^{+}\mathcal{J}$. The same will hold if $\mathcal{J}$ is compact to the future. ###### Proposition II.13. (a) Let $\left(A_{(0)},A_{({\bf d})},A_{(n)},A_{(\bm{\delta})}\right)$ with $\bm{\delta}A_{({\bf d})}=0$ be Cauchy data on $\Sigma$. If $\mathcal{A},\mathcal{A}^{\prime}\in\Omega^{p}({\bm{M}})$ are solutions to $-\bm{\delta}{\bf d}\mathcal{A}=0$ with this data then $\mathcal{A}\sim\mathcal{A}^{\prime}$. (b) Let $\left(A_{(0)},A_{({\bf d})},A_{(n)},A_{(\bm{\delta})}\right)$ and $\left(A^{\prime}_{(0)},A^{\prime}_{({\bf d})},A^{\prime}_{(n)},A^{\prime}_{(\bm{\delta})}\right)$ with $\bm{\delta}A_{({\bf d})}=0=\bm{\delta}A^{\prime}_{({\bf d})}$ be Cauchy data on a common Cauchy surface $\Sigma$. If $\mathcal{A},\mathcal{A}^{\prime}\in\Omega^{p}({\bm{M}})$ are Maxwell solutions with these data, respectively, then $\mathcal{A}^{\prime}\sim\mathcal{A}$ if and only if $A^{\prime}_{(0)}\sim A_{(0)}$ and $A^{\prime}_{({\bf d})}=A_{({\bf d})}$. ###### Proof. (a) It is easiest to first show that any solution to the generalized Maxwell equation is gauge equivalent to some Coulomb solution. Let $\Lambda\in\Omega^{p-1}({\bm{M}})$ be any solution to $-\bm{\delta}{\bf d}\Lambda=-\bm{\delta}\mathcal{A}\qquad\mbox{ with }\qquad\rho_{(0)}\Lambda=0,\qquad\rho_{({\bf d})}\Lambda=A_{(n)}\quad\mbox{and, }\quad\rho_{(n)}\Lambda=\rho_{(\bm{\delta})}\Lambda=0.$ (II.45) By Propositions II.11 and II.12 we know that such a $\Lambda$ exists. Next, we define $\widetilde{\mathcal{A}}\equiv\mathcal{A}-{\bf d}\Lambda$ and calculate $\begin{array}[]{rcl}\rho_{(0)}\widetilde{\mathcal{A}}&=&\rho_{(0)}\left(\mathcal{A}-{\bf d}\Lambda\right)=A_{(0)}-{\bf d}\rho_{(0)}\Lambda=A_{(0)},\\\ \rho_{({\bf d})}\widetilde{\mathcal{A}}&=&\rho_{({\bf d})}\left(\mathcal{A}-{\bf d}\Lambda\right)=A_{({\bf d})},\\\ \rho_{(n)}\widetilde{\mathcal{A}}&=&\rho_{(n)}\left(\mathcal{A}-{\bf d}\Lambda\right)=A_{(n)}-\rho_{({\bf d})}\Lambda=A_{(n)}-A_{(n)}=0,\\\ \rho_{(\bm{\delta})}\widetilde{\mathcal{A}}&=&\rho_{(\bm{\delta})}\left(\mathcal{A}-{\bf d}\Lambda\right)=A_{(\bm{\delta})}-\rho_{(0)}\bm{\delta}{\bf d}\Lambda=A_{(\bm{\delta})}-\rho_{(0)}\bm{\delta}\mathcal{A}=0.\end{array}$ (II.46) Furthermore, $\bm{\delta}\rho_{({\bf d})}\widetilde{\mathcal{A}}=\bm{\delta}\mathcal{A}_{({\bf d})}=0$. By Proposition II.6 we conclude that $\widetilde{\mathcal{A}}$ is a Lorenz solution. Better still, by Corollary II.7 we know that $\widetilde{\mathcal{A}}$ is in fact a Coulomb gauge solution. Just to emphasize the point, we have shown that any Maxwell solution $\mathcal{A}$ with its associated Cauchy data is gauge equivalent to some Coulomb solution $\widetilde{\mathcal{A}}$ with Cauchy data $(A_{(0)},A_{({\bf d})},0,0)$. By a similar calculation we find $\mathcal{A}^{\prime}\sim\widetilde{\mathcal{A}}^{\prime}$ where $\widetilde{\mathcal{A}}^{\prime}$ has the same Cauchy data as $\widetilde{\mathcal{A}}$. By Proposition II.8(a) we have that $\widetilde{\mathcal{A}}\sim\widetilde{\mathcal{A}}^{\prime}$. Therefore, we conclude $\mathcal{A}\sim\mathcal{A}^{\prime}$. (b) Let $\Lambda\in\Omega^{p-1}({\bm{M}})$ and set $\mathcal{A}^{\prime}=\mathcal{A}+{\bf d}\Lambda$, then it immediately follows $A_{(0)}^{\prime}\sim A_{(0)}$ and $A_{({\bf d})}^{\prime}=A_{({\bf d})}$. Alternatively, in part (a) above, we proved that any Maxwell solution $\mathcal{A}$ with Cauchy data $(A_{(0)},A_{({\bf d})},A_{(n)},A_{(\bm{\delta})})$ is gauge equivalent to some Coulomb solution $\widetilde{\mathcal{A}}$ with Cauchy data $(A_{(0)},A_{({\bf d})},0,0)$. Similarly, $\mathcal{A}^{\prime}$ is gauge equivalent to some Coulomb solution $\widetilde{\mathcal{A}}^{\prime}$ with Cauchy data $\left(A^{\prime}_{(0)},A^{\prime}_{({\bf d})},0,0\right)$. If $A^{\prime}_{(0)}\sim A_{(0)}$ and $A_{({\bf d})}^{\prime}=A_{({\bf d})}$, then by Proposition II.8(b) we find $\widetilde{\mathcal{A}}^{\prime}\sim\widetilde{\mathcal{A}}$ and we therefore conclude $\mathcal{A}^{\prime}\sim\mathcal{A}$. ∎ ###### Proposition II.14. (Fundamental Solutions) (a) Let $\mathcal{J}\in\Omega^{p}({\bm{M}})$ with $\bm{\delta}\mathcal{J}=0$ and $\mbox{supp}(\mathcal{J})$ compact to the past/future, then $\mathcal{A}^{\pm}=E^{\pm}\mathcal{J}$ solves $-\bm{\delta}{\bf d}\mathcal{A}^{\pm}=\mathcal{J}$. (b) If $\mathcal{A}^{\pm}\in\Omega^{p}({\bm{M}})$, $\mbox{supp}(\mathcal{A}^{\pm})$ is compact to the past/future and $-\bm{\delta}{\bf d}\mathcal{A}^{\pm}=\mathcal{J}$ (so $\bm{\delta}\mathcal{J}=0$ and $\mbox{supp}(\mathcal{J})$ compact to the past/future) then $\mathcal{A}^{\pm}\sim E^{\pm}\mathcal{J}$. (c) $\mathcal{A}\in\Omega^{p}({\bm{M}})$ satisfies $-\bm{\delta}{\bf d}\mathcal{A}=0$ on spacetimes with compact spacelike Cauchy surfaces if and only if $\mathcal{A}\sim E\mathcal{J}$ for some $\mathcal{J}\in\Omega^{p}_{0}({\bm{M}})$ with $\bm{\delta}\mathcal{J}=0$. ###### Proof. Part (a) is proven directly in Proposition 4 by Dimock Dimock (1992), to which we refer the reader. Parts (b) and (c) are generalizations of the corresponding parts of proposition 4 in Dimock and we leave the proof to the reader. ∎ ### II.5 Classical Phase Space Finally, as a precursor to quantization we discuss the classical phase space for the $p$-form field which consists of a vector space and a non-degenerate antisymmetric bilinear form. For the most part this section closely follows the discussion of the phase space for the classical electromagnetic field found in Dimock Dimock (1992). All of the propositions in his Section 3 trivially generalize from one forms to p-forms, so we will therefore be very brief. In addition, to simplify further discussion, we will also assume from this point forward that the Cauchy surface is compact, thus $\Omega^{p}(\Sigma)=\Omega^{p}_{0}(\Sigma)$. We conjecture that our analysis can be appropriately extended to spacetimes with noncompact Cauchy surfaces. In a typical Hamiltonian formulation, the Cauchy data for the field is specified on some “constant time” hypersurface. The field’s then evolves according to the flow generated by Hamilton’s equations. We could use the complete set of Cauchy data for our p-form field, but as we have seen above, the Cauchy problem is well posed on gauge equivalent classes. Therefore the initial formulation of the phase space can be accomplished via the vector space where points are $\left(A_{(0)},A_{({\bf d})}\right)\in\mathcal{P}_{0}(\Sigma):=\Omega^{p}_{0}(\Sigma)\times\Omega^{p}_{0}(\Sigma)$, i.e., the part of the Cauchy data (with compact support) which can not be gauge transformed away. Then on $\mathcal{P}_{0}(\Sigma)\times\mathcal{P}_{0}(\Sigma)$ define the antisymmetric bilinear form $\sigma_{\Sigma}\left(A_{(0)},A_{({\bf d})};B_{(0)},B_{({\bf d})}\right)=\langle A_{(0)},B_{({\bf d})}\rangle_{\Sigma}-\langle B_{(0)},A_{({\bf d})}\rangle_{\Sigma}.$ (II.47) Unfortunately this form is degenerate because for all $B_{({\bf d})}$ with $\bm{\delta}B_{({\bf d})}=0$ we have $\langle{\bf d}\chi,B_{({\bf d})}\rangle_{\Sigma}=\langle\chi,\bm{\delta}B_{({\bf d})}\rangle_{\Sigma}=0$ even though ${\bf d}\chi\neq 0$. The way to remove the degeneracy is to pass to gauge equivalent classes of Cauchy data with points being given by the pair $([A_{(0)}],A_{({\bf d})})\in\mathcal{P}:=\Omega^{p}_{0}(\Sigma)/d\Omega^{p-1}_{0}(\Sigma)\times\Omega^{p}_{0}(\Sigma)$ then $\sigma_{\Sigma}\left([A_{(0)}],A_{({\bf d})};[B_{(0)}],B_{({\bf d})}\right)=\langle[A_{(0)}],B_{({\bf d})}\rangle_{\Sigma}-\langle[B_{(0)}],A_{({\bf d})}\rangle_{\Sigma}.$ (II.48) is a suitable weakly non-degenerate bilinear form, as proven in Dimock’s Proposition 5 Dimock (1992) . Given any set of Cauchy data in $\mathcal{P}$, we know from the preceding section that there is a unique equivalence class of solutions to the Maxwell equations with this Cauchy data. Therefore, we can reformulated our phase space to include time evolution without the specific introduction of a Hamiltonian. Define the solution space of all real valued gauge equivalent Maxwell solutions with Cauchy data on $\Sigma$ as $\mathcal{M}^{p}({\bm{M}})\equiv{\left\\{\mathcal{A}\in\Omega^{p}({\bm{M}})\,\|\;-\bm{\delta}{\bf d}\mathcal{A}=0\right\\}}/{d\Omega^{p-1}_{0}({\bm{M}})}.$ (II.49) Since ${\bf d}$ and $\bm{\delta}$ are linear operators on $\Omega^{p}({\bm{M}})$, we have that the numerator of the above expression is a vector space. Furthermore, quotienting by the exact forms is also linear so formally $\mathcal{M}^{p}({\bm{M}})$ is a vector space, elements of which are gauge equivalent classes of Maxwell solutions denoted by $[\mathcal{A}]$. Next, choose any Cauchy surface $\Sigma\subset{\bm{M}}$ with $i:\Sigma\rightarrow{\bm{M}}$ and define the antisymmetric bilinear form $\sigma$ on $\mathcal{M}^{p}({\bm{M}})\times\mathcal{M}^{p}({\bm{M}})$ by $\sigma([\mathcal{A}],[\mathcal{B}])\equiv\int_{\Sigma}i^{}\left([\mathcal{A}]\wedge{\bf d}\mathcal{B}-[\mathcal{B}]\wedge{\bf d}\mathcal{A}\right),$ (II.50) which is by definition gauge invariant. It is also non-degenerate and independent of the choice of Cauchy surface (See Dimock’s proposition 6). Thus $\left(\mathcal{M}^{p}({\bm{M}}),\sigma\right)$ is a suitable symplectic phase space. On this phase space we also want to consider linear functions which map $\mathcal{M}^{p}({\bm{M}})\rightarrow\mathbb{R}$ defined by $[\mathcal{A}]\mapsto\langle[\mathcal{A}],f\rangle_{\bm{M}}$ for all $f\in\Omega^{p}_{0}({\bm{M}})$ with $\bm{\delta}f=0$. Such functions are related to the symplectic form in the following sense… ###### Proposition II.15. For $[\mathcal{A}]\in\mathcal{M}^{p}({\bm{M}})$ and $f\in\Omega^{p}_{0}({\bm{M}})$ where $\bm{\delta}f=0$, we have $\langle[\mathcal{A}],f\rangle_{\bm{M}}=\sigma([\mathcal{A}],[Ef]).$ (II.51) ###### Proof. Choose any Cauchy surface $\Sigma$, then from Eq. II.21 above, we have for all $[\mathcal{A}]$ that are homogeneous solutions of the wave equation $\langle[\mathcal{A}],f\rangle_{\bm{M}}=\int_{\Sigma}i^{}\left([\mathcal{A}]\wedge{\bf d}Ef-Ef\wedge{\bf d}[\mathcal{A}]-\bm{\delta}Ef\wedge[\mathcal{A}]+\bm{\delta}[\mathcal{A}]\wedgeEf\right)=\sigma([\mathcal{A}],Ef).$ (II.52) Also, recall that $Ef$ is a Lorenz solution and thus belongs to some equivalence class, therefore giving us the result. ∎ Furthermore, the symplectic form induces a Poisson bracket operation on the functions over the phase space. (For a detailed description of how this arises, see Wald Wald (1994) and/or Sect. 8.1 of AMR Abraham et al. (1988).) For the linear functions considered above we calculate $\left\\{\sigma([\mathcal{A}],[Ef]),\sigma([A],[Ef^{\prime}])\right\\}=\sigma([Ef],[Ef^{\prime}]).$ (II.53) ## III Quantization of the generalized Maxwell field For electromagnetism in four dimensions, quantization is complicated by gauge freedom. Even in Minkowski space this presents serious problems: as shown by Strocchi Strocchi (1967, 1970) in the Wightman axiomatic approach, the vector potential cannot exist as an operator-valued distribution if it is to transform correctly under the Lorentz group or even display commutativity at spacelike separations in a weak sense. We have already seen that the same gauge freedom exists for the generalized Maxwell p-form field, so we are expecting the same difficulty here. However several researchers have already addressed these issues and quantization of a massless free p-form field in four-dimensional curved spacetime has, to our knowledge, been discussed in three papers: The first is by Folacci Folacci (1991) who quantizes $p$-form fields in a “traditional” manner by adding a gauge-breaking term to the action which then necessitates the introduction of Faddeev-Popov ghost fields to remove spurious degrees of freedom. This is similar in approach to the Gupta–Bleuler formalism for the free electromagnetic field in four dimensional spacetimes Gupta (1950, 1977); Crispino et al. (2001). However, unlike electromagnetism, which requires only a single ghost field, the generic quantized $p$-form field suffers from the phenomenon of having “ghosts for ghosts,” thus there are a multiplicity of fields that need to be handled simultaneously Townsend (1979); Siegel (1980). The second paper, by Furlani Furlani (1995), employs the full Gupta–Bleuler method of quantization for the electromagnetic field in four-dimensional static spacetimes with compact Cauchy surfaces. He constructs a Fock space and a representation of the field operator $\mathcal{A}$ that satisfies the Klein- Gordon equation as an operator identity. This effectively quantizes all four components of the one-form field. To remove the two spurious degrees of freedom requires applying the Lorenz gauge condition as a constraint on the space of states and imposing a sesquilinear form that is only positive on the “physical” Fock space. In a later paper Furlani (1999) Furlani also treats the quantization of the Proca field in four-dimensional globally hyperbolic spacetimes. Within this paper he collects together many of the classical results referenced in the preceding section. The third paper, by Dimock Dimock (1992), uses a more elegant approach to quantize the free electromagnetic field in four-dimensional spacetimes which does not introduce gauge breaking terms and ghost fields. He constructs smeared field operators $\widehat{[\mathcal{A}]}(\mathcal{J})$ which may be smeared only with co-closed (divergence-free) test functions, i.e., $\mathcal{J}\in\Omega^{p}_{0}({\bm{M}})$ must satisfy $\bm{\delta}\mathcal{J}=0$. These objects may be interpreted as smeared gauge- equivalence classes of quantum one-form fields: formally, $\widehat{[\mathcal{A}]}(\mathcal{J})=\langle\mathcal{A},\mathcal{J}\rangle_{\bm{M}},$ where $\mathcal{A}$ is a representative of the equivalence class $[\mathcal{A}]$; since $\bm{\delta}\mathcal{J}=0$, we have $\langle{\bf d}\Lambda,\mathcal{J}\rangle_{\bm{M}}=\langle\Lambda,\bm{\delta}\mathcal{J}\rangle_{\bm{M}}=0$ so this interpretation is indeed gauge independent. The resulting operators satisfy the Maxwell equations in the weak sense and have the correct canonical commutation relation. We adapt this approach to the generalized Maxwell field and quantize in the manner found in Dimock Dimock (1992) and Wald Wald (1994). ### III.1 Quantization via a Fock Space To pass from the classical world into the quantum realm requires replacing our symplectic phase space with a Hilbert space, while simultaneously promoting functions on the classical phase space to self-adjoint operators that act on elements of said Hilbert space. To maintain correspondence with the classical theory, the commutator of such operators must be $-i$ times their classical Poisson bracket. Thus we seek operators on a Hilbert space, indexed by $u\in\mathcal{M}^{p}({\bm{M}})$ and denoted $\widehat{[\mathcal{A}]}(u)\equiv\hat{\sigma}([\mathcal{A}],[u])$, such that $\left[\hat{\sigma}([\mathcal{A}],[u]),\hat{\sigma}([\mathcal{A}],[u^{\prime}])\right]=-i\sigma([u],[u^{\prime}]).$ (III.1) We begin with the construction of our Fock space. Our symplectic phase space is $(\mathcal{M}^{p}({\bm{M}}),\sigma)$ where elements $[\mathcal{A}]$ of $\mathcal{M}^{p}({\bm{M}})$ are gauge equivalent classes of real-valued $p$-form solutions to the Maxwell equation, as defined in the section above. On this space, choose any positive-definite, symmetric, bilinear map $\mu:\mathcal{M}^{p}({\bm{M}})\times\mathcal{M}^{p}({\bm{M}})\rightarrow\mathbb{R}$ such that for all $[\mathcal{A}]\in\mathcal{M}^{p}({\bm{M}})$ we have $\mu([\mathcal{A}],[\mathcal{A}])=\frac{1}{4}\sup_{[\mathcal{B}]\neq 0}\frac{\left[\sigma([\mathcal{A}],[\mathcal{B}])\right]^{2}}{\mu([\mathcal{B}],[\mathcal{B}])}.$ (III.2) Many such $\mu$ of this type exist: For each complex structure $J$ on $\mathcal{M}^{p}({\bm{M}})$ which is compatible with $\sigma$ in the sense that $-\sigma([\mathcal{A}],J[\mathcal{B}])$ is a positive-definite inner product gives rise to such a $\mu$, although this method does not produce all such $\mu$. For further discussion on this point see pp. 41-42 of Wald Wald (1994). We then define the norm $\parallel\cdot\parallel^{2}=2\mu(\cdot,\cdot)$ which is used to form $\mathfrak{m},$ the completion of $\mathcal{M}^{p}({\bm{M}})$ with respect to this norm. Next, define the operator $J:\mathfrak{m}\rightarrow\mathfrak{m}$ by $\sigma([\mathcal{A}],[\mathcal{B}])=2\mu([\mathcal{A}],J[\mathcal{B}])=([\mathcal{A}],J[\mathcal{B}])_{\mathfrak{m}},$ (III.3) where $(\;,\,)_{\mathfrak{m}}$ defined in the equation above is the inner product on $\mathfrak{m}$. As already indicated above, $J$ endows $\mathfrak{m}$ with a complex structure. Furthermore, one can prove straightforwardly that $J$ satisfies $J^{}=-J$ and $J^{}J={\rm id}_{\mathfrak{m}}$. The next step is to complexify $\mathfrak{m}$, i.e. $\mathfrak{m}\rightarrow\mathfrak{m}^{\mathbb{C}}$, and extend $\sigma$, $\mu$, and $J$ by complex linearity. The resulting complex space, endowed with the complex inner product $([\mathcal{A}],[\mathcal{B}])_{\mathfrak{m}^{\mathbb{C}}}=2\mu(\overline{[\mathcal{A}]},[\mathcal{B}])$ (III.4) for $[\mathcal{A}],[\mathcal{B}]\in\mathfrak{m}^{\mathbb{C}}$ is a complex Hilbert space. The operator $J$ can be diagonalized into $\pm i$ eigenspaces, as $iJ$ is a bounded, self-adjoint operator on which we can apply the Spectral Theorem. Therefore, we can decompose $\mathfrak{m}^{\mathbb{C}}$ into two orthogonal subspaces based upon the eigenvalues of $iJ$. Define $\mathcal{H}\subset\mathfrak{m}^{\mathbb{C}}$ to be the subspace with eigenvalue $+i$ for the operator $J$, which satisfies the three properties: (i.) The inner product is positive definite over $\mathcal{H}$, (ii.) $\mathfrak{m}^{\mathbb{C}}$ is equal to the span of $\mathcal{H}$ and its complex conjugate space $\overline{\mathcal{H}}$, and (iii.) all elements of $\mathcal{H}$ are orthogonal to all elements of $\overline{\mathcal{H}}$. We also define the orthogonal projection map $K:\mathfrak{m}^{\mathbb{C}}\rightarrow\mathcal{H}$ with respect to the complex inner product by $K=\frac{1}{2}({\rm id}_{\mathfrak{m}^{\mathbb{C}}}+iJ)$. Restricting this map to $\mathfrak{m}$ defines a real linear map $K:\mathfrak{m}\rightarrow\mathcal{H}$, i.e. a map from the Hilbert space of gauge-equivalent real-valued solutions of the Maxwell equation to the complex Hilbert space $\mathcal{H}$. For any $[\mathcal{A}_{1}],[\mathcal{A}_{2}]\in\mathfrak{m}$ we have $\left(K[\mathcal{A}_{1}],K[A_{2}]\right)_{\mathcal{H}}=-i\sigma\left(\overline{K[\mathcal{A}_{1}]},K[\mathcal{A}_{2}]\right)=\mu([\mathcal{A}_{1}],[\mathcal{A}_{2}])-\frac{i}{2}\sigma([\mathcal{A}_{1}],[\mathcal{A}_{2}]).$ (III.5) Finally, the Hilbert space for our quantum field theory is given by the symmetric Fock space $\mathfrak{F}_{s}(\mathcal{H})$ over $\mathcal{H}$, i.e., $\mathfrak{F}_{s}(\mathcal{H})=\mathbb{C}\oplus\left[\bigoplus_{n=1}^{\infty}\left(\otimes_{s}^{n}\mathcal{H}\right)\right],$ (III.6) where $\otimes_{s}^{n}\mathcal{H}$ represents the $n$-th order symmetric tensor product over $\mathcal{H}$. Our next step is to define the appropriate self-adjoint operators on our Fock space. Let $[f],[g]\in\mathcal{H}$, then for states in $\mathfrak{F}_{s}(\mathcal{H})$ of finite particle number, we define the standard annihilation and creation operators, $\hat{a}(\overline{[f]})$ and $\hat{a}^{}([g])$, respectively, where the annihilation operator is linear in the argument for the complex conjugate space $\overline{\mathcal{H}}$, while the creation operator is linear in the argument for elements of $\mathcal{H}$. (See the appendix of Wald Wald (1994) for more detail.). On a dense domain of the Fock space, the operators satisfy the commutation relation $\left[\hat{a}(\overline{[f]}),\hat{a}^{}([g])\right]=\left([f],[g]\right)_{\mathcal{H}}$ (III.7) for all $[f],[g]$, with all other commutators vanishing. From the analysis of the classical wave solutions in the preceding section, we know that $E$ is a surjective map of all compactly supported test forms $\mathcal{J}\in\Omega^{p}_{0}({\bm{M}})$ which are co-closed into an equivalence class in $\mathcal{M}^{p}({\bm{M}})$, namely $[E\mathcal{J}]$. Furthermore, $\mathcal{M}^{p}({\bm{M}})\subset\mathfrak{m}$, thus, combined with the orthogonal projection $K$, we have that $K[E\mathcal{J}]\in\mathcal{H}$. Therefore, we define the smeared quantum field operator for all co-closed $\mathcal{J}\in\Omega^{p}_{0}({\bm{M}})$) by $\widehat{[{\mathcal{A}}]}(\mathcal{J})=\hat{\sigma}([\mathcal{A}],[E\mathcal{J}])=i\hat{a}(\overline{K[E\mathcal{J}]})-i\hat{a}^{}(K[E\mathcal{J}]).$ (III.8) Note, this is a slight abuse of the notation used earlier where the argument of $[\mathcal{A}]$ was an element of the phase space. We now show that such an operator satisfies the generalized Maxwell equation and canonical commutation relations in the sense of distributions. ###### Proposition III.1. For $\mathcal{J}\in\Omega^{p}_{0}({\bm{M}})$, $\bm{\delta}\mathcal{J}=0$ we have (a) $\widehat{[{\mathcal{A}}]}(\mathcal{J})$ satisfies the generalized Maxwell equation in the weak sense, i.e., $\widehat{[{\mathcal{A}}]}(\bm{\delta}{\bf d}\mathcal{J})=0$. (b) $\left[\widehat{[{\mathcal{A}}]}(\mathcal{J}),\widehat{[{\mathcal{A}}]}(\mathcal{J}^{\prime})\right]=i\langle\mathcal{J},E\mathcal{J}^{\prime}\rangle_{\bm{M}}$. In particular, if $\mbox{supp }\mathcal{J},\mbox{supp }\mathcal{J}^{\prime}$ are spacelike separated then the commutator is zero. ###### Proof. (a) By definition we have $\widehat{[{\mathcal{A}}]}(\bm{\delta}{\bf d}\mathcal{J})=i\hat{a}(\overline{K[E\bm{\delta}{\bf d}\mathcal{J}]})-i\hat{a}^{}(K[E\bm{\delta}{\bf d}\mathcal{J}])$, so we will show $[E\bm{\delta}{\bf d}\mathcal{J}]=0$. For any $\theta\in\Omega^{p}_{0}({\bm{M}})$, we have $\Box E\theta=E\Box\theta=0$, so $[E\bm{\delta}{\bf d}\theta]=-[E{\bf d}\bm{\delta}\theta]=-[{\bf d}E\bm{\delta}\theta]=0$, because all exact forms are in the equivalence class of zero. Since $\mathcal{J}\in\Omega^{p}_{0}({\bm{M}})$, we have $[E\bm{\delta}{\bf d}\mathcal{J}]=0$ and $\widehat{[{\mathcal{A}}]}(\bm{\delta}{\bf d}\mathcal{J})=i\hat{a}(0)-i\hat{a}^{}(0)=0$. Notice, this part of the proposition does not require the co-closed condition $\bm{\delta}\mathcal{J}=0$. (b) Substituting the definition of the field operator into the commutator yields $\displaystyle[\widehat{[{\mathcal{A}}]}(\mathcal{J}),\widehat{[{\mathcal{A}}]}(\mathcal{J}^{\prime})]$ $\displaystyle=$ $\displaystyle\left[\hat{a}(\overline{K[E\mathcal{J}]}),\hat{a}^{}(K[E\mathcal{J}^{\prime}])\right]+\left[\hat{a}^{}(\overline{K[E\mathcal{J}]}),\hat{a}(K[E\mathcal{J}^{\prime}])\right]$ (III.9) $\displaystyle=$ $\displaystyle\left(K[E\mathcal{J}],K[E\mathcal{J}^{\prime}]\right)_{\mathcal{H}}-\overline{\left(K[E\mathcal{J}],K[E\mathcal{J}^{\prime}]\right)}_{\mathcal{H}}$ $\displaystyle=$ $\displaystyle-i\sigma([E\mathcal{J}],[E\mathcal{J}^{\prime}]).$ By Proposition II.15 we obtain the desired result. Finally, if $\mbox{supp }\mathcal{J}$ and $\mbox{supp }\mathcal{J}^{\prime}$ are spacelike separated, then $\mbox{{\rm supp}\,}\mathcal{J}\cap\mbox{{\rm supp}\,}E\mathcal{J}^{\prime}=\emptyset$ and the integral in the inner product vanishes. ∎ ### III.2 Algebraic/Local Quantum Field Theory It is well known that different choices of $\mu$ obviously lead to different constructions of the Fock space $\mathfrak{F}_{s}(\mathcal{H})$ and hence unitarily inequivalent quantum field theories Wald (1994). In Minkowski spacetime, Poincaré invariance picks out a “preferred” $\mu$ which leads to a Hilbert space $\mathcal{H}$ of purely positive frequency solutions to build the Fock space from. There are also purely positive frequency solutions in curved stationary spacetimes where the time translation Killing field generates an isometry similar to that of Poincaré invariance in Minkowski spacetime. In a general curved spacetime there may be no such isometries, so purely positive frequency solutions may no exist and the notion of particles becomes somewhat ambiguous. This situation led to the development of the algebraic approach to quantization, also called local quantum field theory. For a general review of this topic, we recommend the articles by Buchholz Buchholz (2000), Buchholz and Haag Buchholz and Haag (2000), and Wald Wald (2006). For a more thorough discussion see Brunetti et al. (2003); Hollands and Wald (2001, 2002); Dimock (1980). Our notation will closely follow that found in Chapter 4 of Bär et. al. Bär et al. (2007). As our final task, we would like to show that our quantized field theory can be used to generate the Weyl-system commonly used in algebraic quantum field theory. The creation and annihilation operators are unbounded, so in order to work with bounded operators we introduce the unitary operators on our Fock space $W([u])=\exp\left(i\hat{\sigma}([\mathcal{A}],[u])\right)$ (III.10) for all $[u]\in\mathcal{M}^{p}({\bm{M}})$. From the definition of the field operator and its commutation relations, it is relatively straight forward to show that this map satisfies $\displaystyle i.$ $\displaystyle W([0])=id_{\mathfrak{F}_{s}(\mathcal{H})},$ (III.11) $\displaystyle ii.$ $\displaystyle W(-[u])=W([u])^{},$ (III.12) $\displaystyle iii.$ $\displaystyle W([u])\cdot W([v])=e^{-i\sigma([u],[v])/2}W([u]+[v]).$ (III.13) (The last relation follows from the Baker-Campbell-Hausdorff formula.) The canonical commutation relation (CCR) algebra $\mathfrak{A}$ is defined as the $C^{}$-algebra generated by $W([E\mathcal{J}])$ for all $\mathcal{J}\in\Omega^{p}_{0}({\bm{M}})$. The CCR-algebra $\mathfrak{A}$ together with the map $W$ forms a Weyl-system for our symplectic phase space $(\mathcal{M}^{p}({\bm{M}}),\sigma)$ which satisfies the Haag-Kastler axioms as generalized by Dimock Dimock (1980). The elements of the algebra are interpreted as the observables related to the quantum field which satisfy the generalized Maxwell equation. By Theorem 4.2.9 of Bär et. al., this CCR- representation is essentially unique. Lastly, we would like to indicate that two very different constructions of the Weyl-system for a symplectic phase space that could be used are given in Bär et. al. Bär et al. (2007). Unfortunately, both of the Hilbert space representations they construct are not considered physical because the states (vectors) in the Hilbert space are not Hadamard, i.e., the two-point function for these states is not related to a certain $p$-form Klein-Gordon bisolution of Hadamard form. In this manuscript, we have given a framework for the rigorous quantization of the $p$-form field for which the issue of states being Hadamard can be addressed in due course. In the case of the 0-form field, the Maxwell equation and the Klein-Gordon equation are the same, so finding Hadamard state is straightforward. The issue of Hadamard states for the 1-form field in four-dimensional globally hyperbolic spacetimes can be found in Fewster and Pfenning Fewster and Pfenning (2003). We will complete the discussion of Hadamard states for the general $p$-form field and develop the quantum weak energy inequality for these states in our next paper. ## IV Conclusions In this manuscript we quantized the generalized Maxwell field $\mathcal{A}$ on globally hyperbolic spacetimes with compact Cauchy surfaces. We began by taking the Maxwell equations into the language of exterior differential calculus. The resulting field equation II.7 could then be carried to any dimension. Rather remarkably, we found that minimally coupled scalar field and the electromagnetic field are actually two examples of a single $p$-form field theory in arbitrary dimension. We then discussed fundamental solutions and the Cauchy problem for the classical $p$-form field theory where we showed that the Cauchy problem was well posed if we worked in terms of gauge equivalent classes of solutions. This was followed by a discussion of the classical, symplectic phase space consisting of all real valued gauge equivalent Maxwell solutions $\mathcal{M}^{p}({\bm{M}})$ and a non-degenerate antisymetric bilinear form $\sigma([\mathcal{A}],[\mathcal{B}])$ for $\mathcal{A},\mathcal{B}\in\mathcal{M}^{p}({\bm{M}})$. The theory was then quantized by promoting functions on the phase space to operators that act on elements of a Hilbert space. The appropriately selected operators were shown in Proposition III.1 to satisfy the generalized Maxwell equation in the weak sense and have the proper canonical commutation relations. Finally the Weyl- system for our field theory was developed. ###### Acknowledgements. I would like to thank C.J. Fewster and J.C. Loftin for numerous illuminating discussions. I would also like to thank C.J. Fewster for supplying the final version of Theorems II.11 and II.12 that appear in this manuscript as well as commenting on early versions of this manuscript. I also want to thank him for his hospitality and the hospitality of the Department of Mathematics at the University of York where part of this research was carried out. This research was funded by a grant from the US Army Research Office through the USMA Photonics Research Center. ## Appendix A Cauchy Problem for $\mathcal{F}$ ###### Proposition A.1. Let $F_{(0)}\in\Omega_{0}^{p}(\Sigma)$ and $F_{(n)}\in\Omega^{p-1}_{0}(\Sigma)$ with ${\bf d}F_{(0)}=0$ and $\bm{\delta}F_{(n)}=0$ specify Cauchy data for the field strength $\mathcal{F}\in\Omega^{p}({\bm{M}})$, with $0<p\leq n$, such that $\rho_{(0)}\mathcal{F}=F_{(0)}\qquad\mbox{ and}\qquad\rho_{(n)}\mathcal{F}=F_{(n)}.$ (A.1) Given this data, there exists a smooth potential $\mathcal{A}\in\Omega^{p-1}({\bm{M}})$ such that $\mathcal{F}={\bf d}\mathcal{A}$ satisfies the generalized Maxwell equations ${\bf d}\mathcal{F}=0$ and $\bm{\delta}\mathcal{F}=0$, as well as the conditions A.1. ###### Proof. We know that $\mathcal{F}={\bf d}\mathcal{A}$ will satisfy the Maxwell equations if $\bm{\delta}{\bf d}\mathcal{A}=0$. To show that such an $\mathcal{A}$ exists we choose as Cauchy data: ${\bf d}A_{(0)}=F_{(0)},\qquad A_{({\bf d})}=F_{(n)},$ (A.2) while $A_{(n)}$ and $A_{(\bm{\delta})}$ are arbitrary up to ${\bf d}\mathcal{A}_{(\bm{\delta})}=0$. The first thing to address is the existence of $A_{(0)}$. For non-compact Cauchy surface $\Sigma$ we could restrict to only those manifolds that are contractible. By the Poincaré lemma for contractible manifolds (Theorem 6.4.18 of AMR Abraham et al. (1988)) all closed $p$-forms (for $p>0$) are exact. Alternatively, we could require that the compactly supported deRham cohomology group $H^{p}_{c}(\Sigma)$ for $p$-forms on the Cauchy surface be of dimension zero, i.e. $H^{p}_{c}(\Sigma)=\\{[0]\\}$. This is a restriction on the topology of the Cauchy surface. If we do have a trivial deRham cohomology group then all closed $p$-forms $F_{(0)}$ are exact. Either the contractible or cohomology condition is sufficient to allow for the existence of a suitable $A_{(0)}$. From the initial Cauchy data on $\mathcal{F}$ we have $\bm{\delta}A_{({\bf d})}=\bm{\delta}F_{(n)}=0.$ (A.3) Our Cauchy data for $\mathcal{A}$ has the properties necessary to use Proposition II.11. So $\mathcal{A}$ is a solution to $\bm{\delta}{\bf d}\mathcal{A}=0$ and therefore $\mathcal{F}={\bf d}\mathcal{A}$ is a solution to the generalized Maxwell equations. Now we show this also reproduces the Cauchy data. We evaluate $\rho_{(0)}\mathcal{F}=\rho_{(0)}{\bf d}\mathcal{A}={\bf d}\rho_{(0)}\mathcal{A}={\bf d}A_{(0)}=F_{(0)}.$ (A.4) Next we evaluate $\rho_{(n)}\mathcal{F}=\rho_{(n)}{\bf d}\mathcal{A}=\rho_{({\bf d})}\mathcal{A}=A_{({\bf d})}=F_{(n)}.$ (A.5) The remaining two pullbacks are trivially zero since $\rho_{({\bf d})}\mathcal{F}=\rho_{({\bf d})}{\bf d}\mathcal{A}=0$ (A.6) and $\rho_{(\bm{\delta})}\mathcal{F}=\rho_{(\bm{\delta})}{\bf d}\mathcal{A}=\rho_{(0)}\bm{\delta}{\bf d}\mathcal{A}=0.$ (A.7) ∎ ## References Buchdahl (1958) H. A. Buchdahl, Il Nuovo Cimento 10, 3058 (1958). * Buchdahl (1962) H. A. Buchdahl, Il Nuovo Cimento 25, 486 (1962). * Buchdahl (1982a) H. A. Buchdahl, J. Phys. A 15, 1 (1982a). * Buchdahl (1982b) H. A. Buchdahl, J. Phys. A 15, 1057 (1982b). * Buchdahl (1984) H. A. Buchdahl, Class. Quantum Grav. 1, 189 (1984). * Buchdahl (1987) H. A. Buchdahl, Class. Quantum Grav. 4, 1055 (1987). * Gibbons (1976) G. W. Gibbons, J. Phys. A: Math. Gen. 9, 145 (1976). * Higuchi (1989) A. Higuchi, Class. Quantum Grav. 6, 397 (1989). * Abraham et al. (1988) R. Abraham, J. E. Marsden, and T. Ratiu, _Manifolds, Tensor Analysis, and Applications_ (Springer-Verlag, New York, 1988), 2nd ed. * Fewster and Pfenning (2003) C. J. Fewster and M. J. Pfenning, J. Math. Phys. 44, 4480 (2003), gr-qc/0303106. * Wald (1984) R. M. Wald, _General Relativity_ (The University of Chicago Press, Chicago, Illinois, 1984). * Lichnerowicz (1961) A. Lichnerowicz, Publications Mathématiques de l’I.H.É.S. pp. 293–344 (1961). * Bär et al. (2007) C. Bär, N. Ginoux, and F. Pfäffle, _Wave Equations on Lorentzian Manifolds and Quantization_ , ESI Lectures in Mathematics and Physics (European Mathematical Society, Germany, 2007). * Choquet-Bruhat (1968) Y. Choquet-Bruhat, _Batelle Rencontres_ (Benjamin, New York, 1968), chap. IV, pp. 84–106, 1967 Lectures in Mathematics and Physics. * Sahlmann and Verch (2001) H. Sahlmann and R. Verch, Rev. Math. Phys. 13, 1203 (2001), math-ph/0008029. * Dimock (1992) J. Dimock, Rev. Math. Phys. 4, 223 (1992). * Furlani (1999) E. P. Furlani, J. Math. Phys. 40, 2611 (1999). * Reed and Simon (1980) M. Reed and B. Simon, _Functional Analysis_ , vol. I of _Methods of Modern Mathematical Physics_ (Academic Press, San Diego, 1980), revised and enlarged ed. * Wald (1994) R. M. Wald, _Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics_ , Chicago Lectures in Physics (The University of Chicago Press, Chicago, Illinois, 1994). * Strocchi (1967) F. Strocchi, Phys. Rev. 165, 1429 (1967). * Strocchi (1970) F. Strocchi, Phys. Rev. D 2, 2334 (1970). * Folacci (1991) A. Folacci, J. Math. Phys. 32, 2813 (1991). * Gupta (1950) S. N. Gupta, Proc. Phys. Soc. London A63, 681 (1950). * Gupta (1977) S. N. Gupta, _Quantum Electrodynamics_ (Gordon and Breach Science Publishers, Inc., New York, 1977). * Crispino et al. (2001) L. C. B. Crispino, A. Higuchi, and G. E. A. Matsas, Phys. Rev. D 63, 124008 (2001), gr-qc/0011070. * Townsend (1979) P. K. Townsend, Physics Letters 88B, 97 (1979). * Siegel (1980) W. Siegel, Physics Letters 93B, 170 (1980). * Furlani (1995) E. P. Furlani, J. Math. Phys. 36, 1063 (1995). * Buchholz (2000) D. Buchholz (2000), plenary talk given at XIIIth International Congress on Mathematical Physics, London. math-ph/0011044. * Buchholz and Haag (2000) D. Buchholz and R. Haag, J. Math. Phys. 41, 3674 (2000), hep-th/9910243. * Wald (2006) R. M. Wald (2006), contribution to 7th International Conference on the History of General Relativity. gr-qc/0608018. * Brunetti et al. (2003) R. Brunetti, K. Fredenhagen, and R. Verch, Commun. Math. Phys 237, 31 (2003). * Hollands and Wald (2001) S. Hollands and R. M. Wald, Commun. Math. Phys. 223, 289 (2001). * Hollands and Wald (2002) S. Hollands and R. M. Wald, Commun. Math. Phys. 231, 309 (2002). * Dimock (1980) J. Dimock, Comm. Math. Phys. 77, 219 (1980).	arxiv-papers	2009-02-27T18:31:29	2024-09-04T02:49:00.902432	{ "license": "Public Domain", "authors": "Michael J. Pfenning", "submitter": "Michael Pfenning", "url": "https://arxiv.org/abs/0902.4887" }
0903.0006	# The Cycle of Dust in the Milky Way: Clues from the High-Redshift and Local Universe Eli Dwek Frédéric Galliano Anthony Jones ###### Abstract Models for the evolution of dust can be used to predict global evolutionary trends of dust abundances with metallicity and examine the relative importance of dust production and destruction mechanisms. Using these models, we show that the trend of the abundance of polycyclic aromatic hydrocarbons (PAHs) with metallicity is the result of the delayed injection of carbon dust that formed in low mass asymptotic giant branch (AGB) stars into the interstellar medium. The evolution of dust composition with time will have important consequences for determining the opacity of galaxies and their reradiated thermal infrared (IR) emission. Dust evolution models must therefore be an integral part of population synthesis models, providing a self-consistent link between the stellar and dust emission components of the spectral energy distribution (SED) of galaxies. We also use our dust evolution models to examine the origin of dust at redshifts $>6$, when only supernovae and their remnants could have been, respectively, their sources of production and destruction. Our results show that unless an average supernova (or its progenitor) produces between 0.1 and 1 $M_{\odot}$ of dust, alternative sources will need to be invoked to account for the massive amount of dust observed at these redshifts. Observational Cosmology Lab, Code 665, NASA Goddard Space Flight center, Greenbelt, MD 20771 Service d’Astrophysique, CEA/Saclay, L’Orme des Merisiers, 91191 Gif-sur- Yvette, France Institut d’Astrophysique Spatiale (IAS), Bâtiment 121, Université Paris-Sud 11 and CNRS, F-91405 Orsay, France ## 1\. Historical Overview Chemical evolution (CE) models follow the formation, destruction, abundance, and spatial and stellar distribution of elements created during the nucleosynthesis era of the Big Bang, and the different evolutionary stages in stars. Models are pitted against a host of observational test, such as the relative abundances of the various elements and their isotopes in meteorites, stars, and in the interstellar, intergalactic and intracluster media, the G-dwarf metallicity distribution, and the age-metallicity relation of the various systems [e.g. (Matteucci 2001; Pagel 1997)]. CE models provide a natural framework for studying the evolution of dust since the abundance of the elements locked up in dust must be constrained by the availability of refractory elements in the interstellar medium (ISM). CE models need then to be generalized to include processes unique to the evolution of dust: the condensation efficiency of refractory elements in stellar ejecta, the destruction of grains in the ISM by expanding supernova remnants (SNRs), and the growth and coagulation of grains in clouds Dwek (1998). The first dust evolution models (Dwek & Scalo 1979, 1980) (hereafter DS) addressed the origin of the elemental depletion pattern, which was a subject of considerable debate. Field (1974) showed that the depletion pattern correlated with condensation temperature, suggesting that it reflects the condensation efficiency of the elements in their respective sources. Such causal correlation requires that dust undergoes very little interstellar processing that can alter the depletion pattern. An equally good correlation exists between the magnitude of the elements’ depletion and their first ionization potential (Snow 1975), suggesting that the depletion pattern may instead be governed by accretion processes in molecular clouds. Pointing out the intrinsic physical correlation between the condensation temperature, the first ionization potential, and the threshold for grain destruction by sputtering, DS suggested that the depletion pattern could reflect the destruction efficiency of dust in the ISM. It is currently clear that all three processes play some role in establishing the elemental depletion pattern, since it depends on the density of the medium in which it is observed (Savage & Sembach 1996). Globally, their relative importance depends on the prevalence of and the cycling times between the different phases (hot, neutral, and molecular) of the ISM. A more detailed review of dust evolution models is given by Dwek (1998). Since then, several models have been constructed to follow the evolution of dust in dwarf galaxies (Lisenfeld & Ferrara 1998), Damped Ly$\alpha$ systems (Kasimova & Shchekinov 2003), the Milky Way galaxy (Zhukovska et al. 2008), and the origin of isotopic anomalies in meteorites (Clayton & Nittler 2004; Zinner et al. 2006a, b). Early dust evolution models adopted the instantaneous recycling approximation, which assumes that all the elements and dust are promptly injected back into the ISM following the formation of their nascent stars. In contrast Dwek (1998) and Morgan & Edmunds (2003) constructed models that take the finite main sequence (MS) lifetimes of the stars into account. Silicate dust is primarily produced in supernovae (SN) that “instantaneously” recycle their products back to the ISM, whereas carbon dust is mainly produced in low-mass carbon-rich stars which have significantly longer MS lifetimes. Consequently these models predicted that the composition of the dust should evolve as a function of time. Dust evolution models contain many still uncertain parameters such as the dust yields in the various sources and the grain destruction efficiency in the ISM. However, in spite of these uncertainties we will show that they can be very successful in predicting global evolutionary trends, namely the observed correlation of the abundance of polycyclic aromatic hydrocarbon (PAH) molecules with metallicity, and in examining the origin of dust in the high- redshift universe. ## 2\. Main Ingredients of Dust Evolution Models ### 2.1. The Yield of Dust in Stars The dust condensation efficiency, and its composition and size distribution depend very much on the environment in which it is formed. Dust can form in the quiescent outflows of AGB or Wolf-Rayet stars, or in the explosively expelled ejecta of SNe and novae. There is substantial observational evidence for the formation of dust in all these sources, however, their relative importance as sources of interstellar dust, and the composition and size distribution of the condensed dust are still uncertain, especially in SNe. Because of the ease of formation and stability of the CO molecule, dust sources can be divided into two categories: carbon-rich sources, in which the C/O abundance ratio is larger than 1. These sources will produce carbon dust; and oxygen-rich sources with C/O $<$ 1, which will produce silicate-type dust. Such simple arguments assume that CO formation goes to completion, which may not always be the case, as suggested by the dynamical condensation models of Ferrarotti & Gail (2006). SN explosions mark the death of stars more massive than $\sim 8$ $M_{\odot}$. Figure 1 depicts the post-explosive composition of a 25 $M_{\odot}$ star (Woosley 1988). It depicts a typical onion-skin structure in which the composition of the different layers reflect the pre- and post-explosion nuclear burning stages of the star. Globally, the ejecta has a C/O ratio $<$ 1, and should in principle be only producing silicate dust. However, in spite of the mixing between the layers caused by Rayleigh-Taylor instabilities in the ejecta, this mixing is of macroscopic nature and does not occur on an atomic level. So the layers above above $\sim~{}4.2$ $M_{\odot}$, in which C/O $>$ 1, will maintain this ratio and produce carbon dust, whereas the inner layers will produce silicates. If all condensible elements precipitated out of the ejecta and formed dust, the yield of dust in a typical SN would be about 1 $M_{\odot}$ (Kozasa et al. 1989, 1991). Figure 1.: The post-explosive composition of a 25 $M_{\odot}$ star (Woosley 1988). A typical SN can potentially produce 1 $M_{\odot}$ of dust. The yield and composition of dust in lower mass stars depends on the C/O ratio in their atmosphere during the AGB phase of their evolution. Figure 2 depicts the C and O yield in stars. Supernovae yields were taken from Woosley & Weaver (1995) and AGB yields were taken from (Karakas & Lattanzio 2003). Stars with masses above $\sim 8$ $M_{\odot}$ produce carbon and silicate dust, irrespective of the global C/O ratio in their ejecta. The mass range of carbon rich stars depends on the initial stellar metallicity. At zero metallicity (left panel) the mass range of stars producing carbon dust is between 0.8 and 7.8 $M_{\odot}$. This range narrows significantly to masses between 3.0 and 3.6 $M_{\odot}$ at solar metallicity (right panel). Figure 3 presents a qualitative depiction of the method we use to calculate the yield of carbon and silicate dust in AGB stars. A more realistic model calculating the yield of dust in AGB stars was presented by Ferrarotti & Gail (2006) and Hoefner (2009, this conference proceedings). --- Figure 2.: The C and O yields in stars. Stars with masses above $\sim~{}8$ $M_{\odot}$ become SNe and produce both silicate and carbon dust. Stars with masses below $\sim 8$ $M_{\odot}$ produce either carbon or silicate dust, depending on the C/O ratio in their ejecta. The light gray area in the horizontal bar depicts the range of stellar masses in which C/O $>$ 1. --- Figure 3.: Qualitative depiction of the calculated yield of carbon and silicate dust in AGB stars. When $C/O>1$ (left panel), the star produces only carbon dust. The dark shaded area depicts the number of carbon atoms that condense into dust. When $C/O<1$ (right panel) the star produces silicate dust, and the dark shaded area depicts the condensing elements. Figure 4 (left panel) depicts the stellar evolutionary tracks of stars with an initial solar metallicity. Also shown in the figure is their MS lifetime. Similar lifetimes are obtained for different initial metallicities (Portinari et al. 1998). At metallicity of $Z=0$, the first carbon dust producing stars are about 8 $M_{\odot}$ and will evolve of the MS about 50 Myr after their formation. At solar metallicities, the production of carbon by AGB stars will be delayed by about 500 Myr, when $\sim 4$ $M_{\odot}$ stars evolve off the MS. Since most of the interstellar carbon dust is made in AGB stars, this delay in its formation can have important observational consequences. Carbon dust has a significantly higher visual opacity than silicates, so the opacity of galaxies will change with time, with young systems being more transparent than older ones. Figure 4 (right panel) depicts the different evolutionary trends of SN- and AGB-condensed dust calculated for a CE model with by exponential star formation rate characterized by a decay time of 6 Gyr, and a Salpeter initial mass function (Dwek 1998, 2005). The silicate and carbon dust yields were calculated assuming a condensation efficiency of unity in the ejecta, and grain destruction was neglected. The model therefore represents an idealized case, in which grain production is maximized, and grain destruction processes are totally ignored. Also shown in the figure are the separate contributions of AGB stars to the abundance of silicate and carbon dust. The onset of the AGB contribution to the silicate abundance starts at $t\approx 50$ Myr, when $\sim$ 8 M⊙ stars evolve off the main sequence, whereas AGB stars start to contribute to the carbon abundance only at $t\approx 500$ Myr, when 4 M⊙ stars reach the AGB phase. The figure also presents the dust-to-ISM metallicity ratio, which is almost constant at a value of $\sim$ 0.36. --- Figure 4.: Left panel: The H-R diagram of stars and their main sequence lifetime. Right panel: The evolution of silicate (dashed line) and carbon (solid line) dust from SNe (bold curves) and AGB stars (light curves). ## 3\. The Lifetime of Interstellar Dust Following their injection into the ISM, the newly-formed dust particles are subjected to a variety of interstellar processes resulting in the exchange of elements between the solid and gaseous phases of the ISM, including: (a) thermal sputtering in high-velocity ($>$200 km s-1) shocks; (b) evaporation and shattering by grain-grain collisions in lower velocity shocks; and (3) accretion in dense molecular clouds. Detailed description of the various grain destruction mechanisms and grain lifetimes in the ISM were presented by (Jones et al. 1996; Jones 2004). In addition, SN condensates can be destroyed shortly after their formation by reverse shocks that travel through the expanding ejecta (Dwek 2005, Bianchi & Schneider 2007, Nozawa et al. 2008). The most important parameter governing the evolution of the dust is its lifetime, $\tau_{d}$, against destruction by SNRs. In an interstellar medium with a uniform dust-to-gas mass ratio, $Z_{d}$, this lifetime is given by (Dwek & Scalo 1980; McKee 1989): $\tau_{d}={M_{d}(t)\over\left<m_{d}\right>R_{SN}}={M_{g}(t)\over\left<m_{\rm ISM}\right>R_{SN}}$ (1) where $M_{d}$ and $M_{g}$ are, respectively, the total mass of dust and gas in the galaxy, $\left<m_{d}\right>$ is the total mass of elements that are locked up in dust and returned by a single SNR back to the gas phase either by thermal sputtering or evaporative grain-grain collisions. $R_{SN}$ is the SN rate in the galaxy, so that the product $\left<m_{d}\right>\,R_{SN}$ is the destruction rate of dust in the ISM. The parameter $\left<m_{\rm ISM}\right>\equiv\left<m_{d}\right>/Z_{d}$ is the effective ISM mass that is completely cleared of dust by a single SNR, given by (Dwek et al. 2007): $\left<m_{\rm ISM}\right>=\int_{v_{0}}^{v_{f}}\ \zeta_{d}(v_{s})\ \left\|{dM\over dv_{s}}\right\|\ dv_{s}$ (2) where $\zeta_{d}(v_{s})$ is the fraction of the mass of dust that is destroyed in an encounter with a shock wave with a velocity $v_{s}$, $(dM/dv_{s})dv_{s}$ is the ISM mass that is swept up by shocks in the [$v_{s},\ v_{s}+dv_{s}$] velocity range, and $v_{0}$ and $v_{f}$ are the initial and final velocities of the SNR. Figure 5 depicts the mass fraction of carbon and silicate dust that is destroyed after being swept up by a shock of velocity $v_{s}$ as a function of shock velocity. An updated version for the carbon and PAH destruction efficiency was presented by Jones et al. (2009, this conference proceedings). For example, in the Milky Way $M_{g}\approx 5\times 10^{9}$ $M_{\odot}$, $R_{SN}\approx 0.03$ yr-1, and $\left<m_{\rm ISM}\right>\approx 300$ $M_{\odot}$, giving a dust lifetime of $\sim 6\times 10^{8}$ yr. Figure 5.: The mass fraction of carbon and silicate dust that is destroyed as a function of shock velocity (after Jones et al. 1996). In addition, SN condensates can be destroyed shortly after their formation by reverse shocks that travel through their expanding ejecta (Dwek 2004, Nozawa et al. 2008). ## 4\. PAHs and Silicate Dust as Tracers of AGB- and SN-condensed Dust An exciting discovery made by spectral and photometric observations of nearby galaxies with the Infrared Space Observatory (ISO) and Spitzer satellites was the striking correlation between the strength of their mid-infrared (IR) aromatic features, commonly attributed to the emission from PAHs, and their metallicity, depicted in Figure 6 [left panel; see Galliano et al. (2008) for references]. The figure shows the rise of $F_{8/24}$, the 8 $\mu$m-to-24$\mu$m band flux ratio with galaxies’ metallicity, and the existence of a metallicity threshold below which $F_{8/24}$ is equal the flux ratio of the dust continuum emission. The strength of the aromatic feature is a measure of the PAH abundance. Since PAHs are predominantly made in C-rich AGB stars, this correlation provides the first observational evidence for the delayed injection of AGB condensed dust into the ISM, provided the metallicity is a measure of the galaxies’ age. The testing of this hypothesis requires first the determination of the PAH abundance in each galaxy. --- Figure 6.: Left panel: The observed correlation between the 8-to-24 $\mu$m bands flux density and metallicity. Right panel: PAH and dust abundances derived from detailed models of galaxies’ SED versus galaxies’ metallicity. PAHs are very small macromolecules, typically 50 Å in diameter, that are stochastically heated by the ambient radiation field. Consequently, only a fraction of the PAHs are radiating at mid-IR wavelengths at any given time. To determine the total abundance of PAHs, including those too cold to emit in the aromatic features requires the determination of the intensity of the interstellar radiation field (ISRF) to which they are subjected. Figure 7 depicts the steps used by Galliano et al. (2008) in modeling the galaxies’ spectral energy distribution (SED). The galaxy used for this illustrative purpose is the starburst M82. The dust model used in the calculations is the BARE-GR-S model of Zubko et al. (2004), consisting of PAHs, and bare silicate and graphite grains with solar abundances constraints. Figure 7a shows the SED of M82, and its various emission components: stellar emission at optical, and near-IR wavelengths; the PAH spectrum at mid IR wavelengths; dust emission from mid- to far-IR wavelengths; and free-free and synchrotron emission at radio wavelengths. Fig 7b depicts a fit to the dust spectrum using an ISRF characterized by a power-law distribution of radiation field intensities. PAH abundance determined by this method will underestimate the real abundance of PAHs in the galaxy compared to the method outlined below. Our models use the free-free and mid-IR emissions to constrain the gas and dust radiation from the gas and dust from H II regions, and the far-IR and optical emission to constrain the ISRF that heats the dust in photo-dissociation regions (PDRs). The radio emission is uniquely decomposed into free-free (dashed) and synchrotron (dotted) emission components (see Figure 7d). Massive stars are required to produce the ionizing radiation and the expanding SN blast waves that generate, respectively, the observed free-free and synchrotron emission. These stars are produced in an ”instantaneous” burst of star formation, in contrast to the stars that are continuously created over the lifetime of the galaxy and mostly contribute to the optical and near-IR emission (Figure 7c). The ionizing and a fraction of the non-ionizing radiation emitted by the starburst component that is absorbed in the H II region is shown as a shaded area in Figure 7d. This energy is reradiated by the dust and gas, giving rise to the thermal IR and free-free emission components shown in the figure. The non-ionizing radiation from the older stellar population and the radiation escaping the H II regions form the diffuse ISRF that is absorbed by the dust in photodissociation regions (PDR) (Figure 7e). The shaded area depicts the fraction of the radiation from the older stellar population that is absorbed in PDRs. The absorbed radiation is reemitted by the dust, giving rise to the IR emission (figure 7f). Figure 7g depicts the different emission components, and Figure 7h shows the fit of their sum to the SED of M82. The details of the fitting procedure are described in Galliano et al. (2008). Using this physical fitting procedure, Galliano et al. (2008) derived the abundance of PAHs, silicates, and graphite grains in 35 nearby galaxies with metallicities ranging from 1/50 to 3 times solar. The detailed physical modeling of their SEDs gives larger PAH abundances compared to models that employ a template ISRFs to heat the PAHs and the dust. In these models, such as the one depicted in Fig. 7b, PAHs are subjected to the same intense radiation field as that required to produce the mid-IR emission from hot dust. In our model, PAHs are subjected to a weaker radiation field. Since PAHs do not survive in H II regions, all their emission originates from PDRs, which are subjected to lower intensity radiation fields than the H II regions. So, compared to the template ISRF models, a larger amount of PAHs is required to produce the same PAH spectrum with a weaker ISRF. The resulting PAH abundances are plotted versus metallicity in Figure 6 (right panel). The figure shows that the observed trend of increasing 8-to-24 $\mu$m band flux ratio with metallicity indeed reflects a trend of increasing PAH abundance with metallicity. The figure also shows the distinct evolutionary trends of PAHs and the far-IR emitting dust with metallicity. --- --- Figure 7.: Construction of the fit to the SED of the starburst galaxy M82. See text for details. Figure 8 compares the evolution of the PAHs and dust components derived from the dust evolution model to the trend of PAH abundances with metallicity. For the sake of this comparison, the evolution of dust abundances as a function of time was converted to an evolution as a function of metallicity, using the age-metallicity relation derived in the model. We emphasize that the parameters used in the dust evolution model (the star formation rate, the stellar IMF) were identical to those used in the population synthesis model that was used to fit the galaxies’ SED. From the dust evolution model we already derived the two distinct evolutionary trends of SN- and AGB-condensed dust (see Fig. 6, right panel for the idealized example). The current figure compares these results with the derived abundance of the PAHs and of the dust that gives rise to the far-IR emission. The latter is dominated by emission from silicates, and should therefore follow the trend of the SN condensates, since most silicate grains are produced in SNe. The figure shows that the observed far-IR emitting dust falls on the evolutionary track of the calculated SN-condensed dust, and that the observed PAH abundances fall on the evolutionary track of the carbon dust that formed in AGB stars. The shaded regions in the figure represent the range of evolutionary tracks that correspond to different parameters that determine the star formation rate and grain destruction efficiencies in the models. Figure 8.: Comparison between the metallicity trends of the PAH abundance derived from the observed SED and those derived from the chemical evolution model. The shaded area represent the range of model prediction for different grain destruction and star formation rates. Details of the figure are described in Galliano et al. (2008). ## 5\. The presence of massive amounts of dust at high redshift The detection of massive amounts of dust in hyperluminous IR galaxies at redshifts $z>6$ raises challenging questions about the sources capable of producing such large amount of dust during the relatively short lifetime of these galaxies (Maiolino et al. 2006; Beelen et al. 2006; Morgan & Edmunds 2003). For example, the galaxy SDSS J1148+5251 (hereafter J1148+5251) located at $z=6.4$ was observed at far-IR and submillimeter wavelength (Bertoldi et al. 2003; Robson et al. 2004; Beelen et al. 2006). The average IR luminosity of the source is $L_{IR}\sim 2\times 10^{13}$ $L_{\odot}$, and the average dust mass is $M_{d}\sim 2\times 10^{8}$ $M_{\odot}$. Using the Kennicutt (1998) relation, one can derive a star formation rate (SFR) of $\sim 3000$ $M_{\odot}$ yr-1 from the observed far-IR luminosity. For comparison, the Milky Way galaxy is about 10 Gyr old, has an average SFR of $\sim 3$ $M_{\odot}$ yr-1, and contains about $5\times 10^{7}$ $M_{\odot}$ of dust, a significant fraction of which was produced in AGB stars. At $z=6.4$ the universe was only 890 Myr old, using standard $\Lambda$CDM parameters ($\Omega_{m}=0.27$, $\Omega_{\Lambda}=0.73$, and $H_{0}=70$ km s-1 Mpc-1). If J1148+5251 formed at $z=10$ then the galaxy is only 400 Myr old. If the SFR had occured at a constant rate over the lifetime of the galaxy, its initial mass should have been about 1012 $M_{\odot}$, which is significantly larger than the dynamical mass $M_{dyn}\approx 5\times 10^{10}$ $M_{\odot}$ of the galaxy (Walter et al. 2004). The high observed SFR may therefore represent a recent burst of star formation that has lasted for only about 20 Myr. The galaxy J1148+5251 is therefore at most $\sim 400$ Myr old, and probably significantly younger with an age of only $\sim 20$ Myr. Adopting a current gas mass of $M_{g}=3\times 10^{10}$ $M_{\odot}$ for this galaxy we get that the gas mass fraction at 400 Myr is about 0.60. The dust-to-gas mass ratio is given by $Z_{d}\equiv M_{d}/M_{g}=0.0067$. A significant fraction of the dust in the Milky Way was produced in AGB stars. However, these stars are not likely to contribute significantly to the formation of dust in very young galaxies, since the low mass stars ($M\approx 3$ $M_{\odot}$) that produce most of the dust did not have time to evolve off the main sequence (Dwek 1998; Morgan & Edmunds 2003; Dwek 2005). In contrast, core collapse SNe ($M>8$ $M_{\odot}$) and their post-main-sequence progenitors inject their nucleosynthetic products back into the ISM shortly ($t<20$ Myr) after their formation, resulting in the rapid enrichment of the interstellar medium (ISM) with the dust that formed during the mass loss phase prior to the SN event, or in the explosive SN ejecta. We will hereafter attribute both contribution to the SN event, since both are return ”promptly” to the ISM. But can SNe account for the large amount of dust seen in this object? The answer to this question is complicated by the fact that SNe are also the main source of grain destruction during the remnant phase of their evolution (Jones et al. 1996; Jones 2004). The problem can therefore only be quantitatively addressed with CE models for the dust in these systems. The results of the detailed dust evolution models described summarized in Figure 8 show that the contribution of AGB stars to metal and dust abundance can be neglected in galaxies with ages less than about 400 Myr. The equations for the chemical evolution of the galaxy can then be considerably simplified using the instantaneous recycling approximation, which assumes that stars return their ejecta back to the ISM promply after their formation. The evolution of the dust abundance can then be written in analytical form (Morgan & Edmunds 2003; Dwek et al. 2007). In particular, the yield of dust, $Y_{d}$, required to obtain a given dust-to- gas mass ratio, $Z_{d}$, when the galaxy reaches a given gas mass fraction $\mu_{g}$, is given by (Dwek et al. 2007): $Y_{d}=Z_{d}\ \left[{\left<m_{\rm ISM}\right>+R\ m_{\star}\over 1-\mu_{g}^{\nu-1}}\right]$ (3) where, $\nu\equiv{\left<m_{\rm ISM}\right>+m_{\star}\over(1-R)\ m_{\star}}\qquad,$ (4) $R$ is the fraction of the stellar mass that is returned back to the ISM during the stellar lifetime, $\left<m_{\rm ISM}\right>$ is given by eq. (2), and $m_{\star}$ is the mass of all stars born per SN event. For example, $m_{\star}=147$ and $50$ $M_{\odot}$, respectively, for a Salpeter and top- heavy IMF. Figure 9 shows how much dust an average SN must produce in order to give rise to a given dust-to-gas mass ratio, for various grain destruction efficiencies. The value of $Y_{d}$ was calculated when $\mu_{g}$ reaches a value of 0.60, the adopted gas mass fraction of J1148+5251 at 400 Myr. Calculations were performed for two different functional forms of the stellar IMF: a Salpeter IMF in which $\phi(m)\sim m^{-2.35}$ and $0.1<m$($M_{\odot}$) $<100$; and a top heavy IMF characterized by the same mass limits but a flatter slope $\phi(m)\sim m^{-1.50}$. Here, $\phi(m)$ is the number of stars per unit mass interval, normalized to unity between 0.1 and 100 $M_{\odot}$. The figure shows that, for example, to produce a value of $Z_{d}=0.0067$ at $\mu_{g}$=̃ 0.60, a SN must produce about 0.4 (1.2) $M_{\odot}$ of dust for a top-heavy (Salpeter) IMF, provided the dust is not destroyed in the ISM, that is, $\left<m_{\rm ISM}\right>$ = 0. Even with modest amount of grain destruction, $\left<m_{\rm ISM}\right>$ = 100 $M_{\odot}$, the required SN dust yield is dramatically increased to about $1-2$ $M_{\odot}$, depending on the IMF. The horizontal line in the figure corresponds to a value of $Y_{d}$ = 0.054 $M_{\odot}$, the largest mass of SN-condensed dust inferred to be present in a supernova or SNR (Rho et al. 2008; Sugerman et al. 2006). Contrary to the claim by Rho et al. (2008), this yield is not sufficient to account for the large amount of dust observed in high redshift galaxies, since the quoted chemical evolution models of Morgan & Edmunds (2003) do not include the effect of grain destruction. The figure shows that even without grain destruction, the largest observed yield can only give rise to a dust-to-gas mass ratio of $\sim 4\times 10^{-4}$. If the mass of dust in the ejecta of Cas A represents a typical SN yield, then other processes, such as accretion onto preexisting grains in molecular clouds is needed to produce the mass of dust in J1148+5251. Figure 9.: The IMF-averaged yield of dust by type II supernova, $\widehat{Y}_{d}$, that is required to account for a given dust-to-gas mass ratio $Z_{d}$, is presented for different values of $\left<m_{\rm ISM}\right>$ given in units of $M_{\odot}$. Solid and dashed lines correspond to calculations done for a top-heavy and a Salpeter IMF, respectively. The horizontal dashed line near the bottom of the figure corresponds a value of $Y_{d}=0.054$ $M_{\odot}$, the highest inferred yield of dust in supernova ejecta to date Rho et al. (2008). The vertical dotted line represents the value of $Z_{d}$ at $\mu_{g}=0.60$. Curves are labeled by $\left<m_{\rm ISM}\right>$ given in units of $M_{\odot}$. The top two dashed (solid) horizontal lines represent IMF-averaged theoretical dust yields for a Salpeter (top-heavy) IMF, assuming 100% condensation efficiency in the SN ejecta. ## 6\. Summary Dust evolution models have proven to be very successful in predicting global evolutionary trends in dust abundance and composition, and in analyzing the origin of dust in the early universe. An important prediction of these models is that SN- and AGB-condensed dust should follow distinct evolutionary paths because of the different stellar evolutionary tracks of their progenitor stars. By analyzing the UV-to-radio SED of 35 nearby galaxies we have identified silicates and PAHs, respectively, as tracers of SN- and AGB- condensed dust. Our SED fitting procedure used chemical evolution, dust evolution, and population synthesis models in a consistent fashion. The models used the free-free and mid-IR emissions to constrain the gas and dust radiation from the gas and dust from H II regions, and the far-IR and optical emission to constrain the ISRF that heats the dust in PDRs. The observed correlation of the intensity of the mid-IR emission from PAHs with their metallicity can then be interpreted as the result of stellar evolutionary effects which cause the delayed injection of carbon dust into the interstellar medium. The early universe is a unique environment for studying the role of massive stars in the formation and destruction of dust. The equations describing their chemical evolution can be greatly simplified by using the instantaneous recycling approximation, and by neglecting the delayed contribution of low mass stars to the metal and dust abundance of the ISM. Neglecting any accretion of metals onto pre-existing dust in the interstellar medium, the evolution of the dust is then primarily determined by the condensation efficiency of refractory elements in the ejecta of Type II supernovae, and the destruction efficiency of dust by SN blast waves. We applied our general results to J1148+5251, a dusty, hyperluminous quasar at redshift $z=6.4$ and found that the formation of a dust mass fraction of $Z_{d}=0.0067$ in a galaxy with an ISM mass of $3\times 10^{10}$ $M_{\odot}$, requires an average SN to produce between 0.5 and 1 $M_{\odot}$ of dust if there was no grain destruction. Such large amount of dust can be produced if if the condensation efficiency in SNe is about unity. Observationally, the required dust yield is in excess of the largest amount of dust ($\sim 0.054$ $M_{\odot}$) observed so far to have formed in a SN. This suggests that accretion in the ISM may play an important role in the growth of dust mass. For this process to be effective, SNRs must significantly increase, presumably by non-evaporative grain-grain collisions during the late stages of their evolution, the number of nucleation centers onto which refractory elements can condense in molecular clouds. #### Acknowledgments. This work was supported by NASA’s LTSA 03-0000-065. ## References * Beelen et al. (2006) Beelen, A., Cox, P., Benford, D. J., et al. 2006, ApJ, 642, 694 * Bertoldi et al. (2003) Bertoldi, F., Carilli, C. L., Cox, P., et al. 2003, A&A, 406, L55 * Bianchi & Schneider (2007) Bianchi, S. & Schneider, R. 2007, MNRAS, 378, 973 * Clayton & Nittler (2004) Clayton, D. D. & Nittler, L. R. 2004, ARA&A, 42, 39 * Dwek (1998) Dwek, E. 1998, ApJ, 501, 643 * Dwek (2005) Dwek, E. 2005, in American Institute of Physics Conference Series, Vol. 761, The Spectral Energy Distributions of Gas-Rich Galaxies: Confronting Models with Data, ed. C. C. Popescu & R. J. Tuffs, 103–+ * Dwek et al. (2007) Dwek, E., Galliano, F., & Jones, A. P. 2007, ApJ, 662, 927 * Dwek & Scalo (1979) Dwek, E. & Scalo, J. M. 1979, ApJ, 233, L81 * Dwek & Scalo (1980) Dwek, E. & Scalo, J. M. 1980, ApJ, 239, 193 * Ferrarotti & Gail (2006) Ferrarotti, A. S. & Gail, H.-P. 2006, A&A, 447, 553 * Field (1974) Field, G. B. 1974, ApJ, 187, 453 * Galliano et al. (2008) Galliano, F., Dwek, E., & Chanial, P. 2008, ApJ, 672, 214 * Jones (2004) Jones, A. P. 2004, in ASP Conf. Ser. 309: Astrophysics of Dust, ed. A. N. Witt, G. C. Clayton, & B. T. Draine, 347–+ * Jones et al. (1996) Jones, A. P., Tielens, A. G. G. M., & Hollenbach, D. J. 1996, ApJ, 469, 740 * Karakas & Lattanzio (2003) Karakas, A. I. & Lattanzio, J. C. 2003, Publications of the Astronomical Society of Australia, 20, 279 * Kasimova & Shchekinov (2003) Kasimova, E. R. & Shchekinov, Y. A. 2003, Ap&SS, 284, 433 * Kennicutt (1998) Kennicutt, Jr., R. C. 1998, ARA&A, 36, 189 * Kozasa et al. (1989) Kozasa, T., Hasegawa, H., & Nomoto, K. 1989, ApJ, 344, 325 * Kozasa et al. (1991) Kozasa, T., Hasegawa, H., & Nomoto, K. 1991, A&A, 249, 474 * Lisenfeld & Ferrara (1998) Lisenfeld, U. & Ferrara, A. 1998, ApJ, 496, 145 * Maiolino et al. (2006) Maiolino, R., Nagao, T., Marconi, A., et al. 2006, Memorie della Societa Astronomica Italiana, 77, 643 * Matteucci (2001) Matteucci, F. 2001, The chemical evolution of the Galaxy (The chemical evolution of the Galaxy / by Francesca Matteucci, Astrophysics and space science library, Volume 253, Dordrecht: Kluwer Academic Publishers, ISBN 0-7923-6552-6, 2001, XII + 293 pp.) * McKee (1989) McKee, C. 1989, in IAU Symposium, Vol. 135, Interstellar Dust, ed. L. J. Allamandola & A. G. G. M. Tielens, 431–+ * Morgan & Edmunds (2003) Morgan, H. L. & Edmunds, M. G. 2003, MNRAS, 343, 427 * Pagel (1997) Pagel, B. E. J. 1997, Nucleosynthesis and Chemical Evolution of Galaxies (Nucleosynthesis and Chemical Evolution of Galaxies, by Bernard E. J. Pagel, pp. 392. ISBN 0521550610. Cambridge, UK: Cambridge University Press, October 1997.) * Portinari et al. (1998) Portinari, L., Chiosi, C., & Bressan, A. 1998, A&A, 334, 505 * Rho et al. (2008) Rho, J., Kozasa, T., Reach, W. T., et al. 2008, ApJ, 673, 271 * Robson et al. (2004) Robson, I., Priddey, R. S., Isaak, K. G., & McMahon, R. G. 2004, MNRAS, 351, L29 * Savage & Sembach (1996) Savage, B. D. & Sembach, K. R. 1996, ARA&A, 34, 279 * Snow (1975) Snow, Jr., T. P. 1975, ApJ, 202, L87 * Sugerman et al. (2006) Sugerman, B. E. K., Ercolano, B., Barlow, M. J., et al. 2006, Science, 313, 196 * Walter et al. (2004) Walter, F., Carilli, C., Bertoldi, F., et al. 2004, ApJ, 615, L17 * Woosley (1988) Woosley, S. E. 1988, ApJ, 330, 218 * Woosley & Weaver (1995) Woosley, S. E. & Weaver, T. A. 1995, ApJS, 101, 181 * Zhukovska et al. (2008) Zhukovska, S., Gail, H.-P., & Trieloff, M. 2008, A&A, 479, 453 * Zinner et al. (2006a) Zinner, E., Nittler, L. R., Alexander, C. M. O. ., & Gallino, R. 2006a, New Astronomy Review, 50, 574 * Zinner et al. (2006b) Zinner, E., Nittler, L. R., Gallino, R., et al. 2006b, ApJ, 650, 350 * Zubko et al. (2004) Zubko, V., Dwek, E., & Arendt, R. G. 2004, ApJS, 152, 211	arxiv-papers	2009-02-27T21:24:19	2024-09-04T02:49:00.913771	{ "license": "Public Domain", "authors": "Eli Dwek, Frederic Galliano, Anthony Jones", "submitter": "Eli Dwek", "url": "https://arxiv.org/abs/0903.0006" }
0903.0089	# The semilinear Klein-Gordon equation in de Sitter spacetime Karen Yagdjian Correspondence: Department of Mathematics, University of Texas- Pan American 1201 W. University Drive, Edinburg, TX 78541-2999, USA; E-mail:yagdjian@utpa.edu. ###### Abstract In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation $\Box_{g}\phi-m^{2}\phi=-\|\phi\|^{p}$ with the small mass $m\leq n/2$ in de Sitter space-time with the metric $g$. We prove that for every $p>1$ the large energy solution blows up, while for the small energy solutions we give a borderline $p=p(m,n)$ for the global in time existence. The consideration is based on the representation formulas for the solution of the Cauchy problem and on some generalizations of the Kato’s lemma. ## 1 Introduction In this article we study the blow-up phenomena for the solutions of the semilinear Klein-Gordon equation $\Box_{g}\phi-m^{2}\phi=-\|\phi\|^{p}$ with the small mass $m\leq n/2$ in de Sitter space-time. In the model of the universe proposed by de Sitter the line element has the form $ds^{2}=-\left(1-\frac{2M_{bh}}{r}-\frac{\Lambda r^{2}}{3}\right)c^{2}dt^{2}+\left(1-\frac{2M_{bh}}{r}-\frac{\Lambda r^{2}}{3}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}d\phi^{2}).$ The constant $M_{bh}$ may have a meaning of the “mass of the black hole”. The corresponding metric with this line element is called the Schwarzschild - de Sitter metric. The Cauchy problem for the semilinear Klein-Gordon equation in Minkowski spacetime ($M_{bh}=\Lambda=0$) is well investigated. (See, e.g., [7] and references therein.) In particular, Keel and Tao [7] for the semilinear equation $u_{tt}-\Delta u=F(u)$, $u(0,x)=\varepsilon\varphi_{0}(x)$, $u_{t}(0,x)=\varepsilon\varphi_{1}(x)$ proved that if $n=1,2,3$ and $1<p<1+2n$, then there exists a (non-Hamiltonian) nonlinearity $F$ satisfying $\|D^{\alpha}F(u)\|\leq C\|u\|^{p-\|\alpha\|}$ for $0\leq\alpha\leq[p]$ and such that there is no finite energy global solution supported in the forward light cone, for any nontrivial smooth compactly supported $\varphi_{0}$ and $\varphi_{1}$ and for any $\varepsilon>0$. There is an interesting question of instability of the ground state standing solutions $e^{i\omega t}\phi_{\omega}(x)$ for nonlinear Klein-Gordon equation $\partial_{t}^{2}u-\Delta u+u=\|u\|^{p-1}u$. Here $\phi_{\omega}$ is a ground state of the equation $-\Delta\phi+(1-\omega^{2})\phi=\|\phi\|^{p-1}\phi,$ while $0<p-1<4/(N-2)$ and $0\leq\|\omega\|<1$. Ohta and Todorova [9] showed that instability occurs in the very strong sense that an arbitrarily small perturbation of the initial data can make the perturbed solution blow up in finite time. The Cauchy problem for the linear wave equation without source term on the maximally extended Schwarzschild - de Sitter spacetime in the case of non- extremal black-hole corresponding to parameter values $0<M_{bh}<\frac{1}{3\sqrt{\Lambda}}$, is considered by Dafermos and Rodnianski [3]. They proved that in the region bounded by a set of black/white hole horizons and cosmological horizons, solutions converge pointwise to a constant faster than any given polynomial rate, where the decay is measured with respect to natural future-directed advanced and retarded time coordinates. The bounds on decay rates for solutions to the wave equation in the Schwarzschild - de Sitter spacetime is a first step to a mathematical understanding of non- linear stability problems for spacetimes containing black holes. Catania and Georgiev [2] studied the Cauchy problem for the semilinear wave equation $\Box_{g}\phi=\|\phi\|^{p}$ in the Schwarzschild metric $(3+1)$-dimensional space-time, that is the case of $\Lambda=0$ in $0<M_{bh}<\frac{1}{3\sqrt{\Lambda}}$. They established that the problem in the Regge-Wheeler coordinates is locally well-posed in $H^{\sigma}$ for any $\sigma\in[1,p+1)$. Then for the special choice of the initial data they proved the blow-up of the solution in two cases: (a) $p\in(1,1+\sqrt{2})$ and small initial data supported far away from the black hole; (b) $p\in(2,1+\sqrt{2})$ and large data supported near the black hole. In both cases, they also gave an estimate from above for the lifespan of the solution. In the present paper we focus on the another limit case as $M_{bh}\to 0$ in $0<M_{bh}<\frac{1}{3\sqrt{\Lambda}}$, namely, we set $M_{bh}=0$ to ignore completely influence of the black hole. Thus, the line element in de Sitter spacetime has the form $ds^{2}=-\left(1-\frac{r^{2}}{R^{2}}\right)c^{2}\,dt^{2}+\left(1-\frac{r^{2}}{R^{2}}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\,.$ The Lamaître-Robertson transformation [8] $r^{\prime}=\frac{r}{\sqrt{1-r^{2}/R^{2}}}e^{-ct/R}\,,\quad t^{\prime}=t+\frac{R}{2c}\ln\left(1-\frac{r^{2}}{R^{2}}\right)\,,\quad\theta^{\prime}=\theta\,,\quad\phi^{\prime}=\phi,$ leads to the following form for the line element: $ds^{2}=-c^{2}\,d{t^{\prime}}^{2}+e^{2ct^{\prime}/R}(d{r^{\prime}}^{2}+r^{\prime 2}\,d{\theta^{\prime}}^{2}+r^{\prime 2}\sin^{2}\theta^{\prime}\,d{\phi^{\prime}}^{2})$. By defining coordinates $x^{\prime}$, $y^{\prime}$, $z^{\prime}$ connected with $r^{\prime}$, $\theta^{\prime}$, $\phi^{\prime}$ by the usual equations connecting Cartesian coordinates and polar coordinates in a Euclidean space, the line element may be written [8, Sec.134] $ds^{2}=-c^{2}\,d{t^{\prime}}^{2}+e^{2ct^{\prime}/R}(d{x^{\prime}}^{2}+d{y^{\prime}}^{2}+d{z^{\prime}}^{2})\,.$ The new coordinates $r^{\prime}$, $\theta^{\prime}$, $\phi^{\prime}$, $t^{\prime}$ can take all values from $-\infty$ to $\infty$. Here $R$ is the “radius” of the universe. In this paper we study blow-up phenomena for semilinear equation by applying the Lamaître-Robertson transformation and by employing the fundamental solutions for some model linear hyperbolic equation with variable speed of propagation. In [16] the Klein-Gordon operator in Robertson-Walker spacetime, that is ${\mathcal{S}}:=\partial_{t}^{2}-e^{-2t}\bigtriangleup+M^{2}$, is considered. The fundamental solution $E=E(x,t;x_{0},t_{0})$, that is solution of ${\mathcal{S}}E=\delta(x-x_{0},t-t_{0})$, with a support in the forward light cone $D_{+}(x_{0},t_{0})$, $x_{0}\in{\mathbb{R}}^{n}$, $t_{0}\in{\mathbb{R}}$, and the fundamental solution with a support in the backward light cone $D_{-}(x_{0},t_{0})$, $x_{0}\in{\mathbb{R}}^{n}$, $t_{0}\in{\mathbb{R}}$, defined by $D_{\pm}(x_{0},t_{0}):=\big{\\{}(x,t)\in{\mathbb{R}}^{n+1}\,;\,\|x-x_{0}\|$ $\leq\pm(e^{-t_{0}}-e^{-t})\,\big{\\}}$, are constructed. These fundamental solutions have been used to represent solutions of the Cauchy problem and to prove $L^{p}-L^{q}$ estimates for the solutions of the equation with and without a source term that provide with some necessary tools for the studying semilinear equations. In the Robertson-Walker spacetime [5], one can choose coordinates so that the metric has the form $ds^{2}=-dt^{2}+S^{2}(t)d\sigma^{2}\,.$ In particular, the metric in de Sitter and anti-de Sitter spacetime in the Lamaître-Robertson coordinates [8] has this form with $S(t)=e^{t}$ and $S(t)=e^{-t}$, respectively. The matter waves in the de Sitter spacetime are described by the function $\phi$, which satisfies equations of motion. In the de Sitter universe the equation for the scalar field with mass $m$ and potential function $V$ is the covariant Klein-Gordon equation $\square_{g}\phi-m^{2}\phi=V^{\prime}(\phi)\quad\mbox{\rm or}\quad\frac{1}{\sqrt{\|g\|}}\frac{\partial}{\partial x^{i}}\left(\sqrt{\|g\|}g^{ik}\frac{\partial\phi}{\partial x^{k}}\right)-m^{2}\phi=V^{\prime}(\phi)\,,$ with the usual summation convention. Written explicitly in coordinates in the de Sitter spacetime it, in particular, for $V^{\prime}(\phi)=-\|\phi\|^{p}$ has the form $\phi_{tt}+n\phi_{t}-e^{-2t}\Delta\phi+m^{2}\phi=\|\phi\|^{p}\,.$ (1) In this paper we restrict ourselves with consideration of the semilinear equation for particle with small mass $m$, that is $0\leq m\leq n/2$. If we introduce the new unknown function $u=e^{\frac{n}{2}t}\phi$, then it takes the form of the semilinier Klein-Gordon equation for $u$ on de Sitter spacetime $u_{tt}-e^{-2t}\bigtriangleup u-M^{2}u=e^{-\frac{n(p-1)}{2}t}\|u\|^{p},$ (2) where non-negative curved mass $M\geq 0$ is defined as follows: $M^{2}:=\frac{n^{2}}{4}-m^{2}\geq 0\,.$ The equation (2) can be regarded as Klein-Gordon equation with imaginary mass. Equations with imaginary mass appear in several physical models such as $\phi^{4}$ field model, tachion (super-light) fields, Landau-Ginzburg-Higgs equation and others. To solve the Cauchy problem for semilinear equation we use fundamental solution of the corresponding linear operator. We denote by $G$ the resolving operator of the problem $u_{tt}-e^{-2t}\bigtriangleup u-M^{2}u=f,\quad u(x,0)=0,\quad\partial_{t}u(x,0)=0\,.$ (3) Thus, $u=G[f]$. The equation of (3) is strictly hyperbolic. This implies the well-posedness of the Cauchy problem (3) in the different functional spaces. Consequently, the operator is well-defined in those functional spaces. Then, the speed of propagation is variable, namely, it is equal to $e^{-t}$. The second-order strictly hyperbolic equation (3) possesses two fundamental solutions resolving the Cauchy problem without source term $f$. They can be written in terms of the Fourier integral operators, which give complete description of the wave front sets of the solutions. Moreover, the integrability of the characteristic roots, $\int_{0}^{\infty}\|\lambda_{i}(t,\xi)\|dt<\infty$, $i=1,2$, leads to the existence of the so-called “horizon” for that equation. More precisely, any signal emitted from the spatial point $x_{0}\in{\mathbb{R}}^{n}$ at time $t_{0}\in{\mathbb{R}}$ remains inside the ball $B^{n}_{t_{0}}(x_{0}):=\\{x\in{\mathbb{R}}^{n}\,\|\,\|x-x_{0}\|<e^{-t_{0}}\\}$ for all time $t\in(t_{0},\infty)$. In particular, it can cause a nonexistence of the $L^{p}-L^{q}$ decay for the solutions. In [13] this phenomenon is illustrated by a model equation with permanently bounded domain of influence, power decay of characteristic roots, and without $L^{p}-L^{q}$ decay. The above mentioned $L^{p}-L^{q}$ decay estimates are one of the important tools for studying nonlinear problems (see, e.g. [11]). In this paper we show that this phenomenon causes the blow up of the solution. The equation (3) is neither Lorentz invariant nor invariant with respect to usual scaling and that creates additional difficulties. Operator $G$ is constructed in [16] for the case of the large mass $m\geq n/2$. The analytic continuation of this operator in parameter $M$ into ${\mathbb{C}}$ allows us to use $G$ also in the case of small mass $0\leq m\leq n/2$. More precisely, we define the operator $G$ acting on $f(x,t)\in C^{\infty}({\mathbb{R}}\times[0,\infty))$ by $\displaystyle G[f](x,t)$ $\displaystyle:=$ $\displaystyle\int_{0}^{t}db\int_{x-(e^{-b}-e^{-t})}^{x+e^{-b}-e^{-t}}dy\,f(y,b)(4e^{-b-t})^{-M}\Big{(}(e^{-t}+e^{-b})^{2}-(x-y)^{2}\Big{)}^{-\frac{1}{2}+M}$ $\displaystyle\hskip 71.13188pt\times F\Big{(}\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-(x-y)^{2}}{(e^{-b}+e^{-t})^{2}-(x-y)^{2}}\Big{)},$ where $F\big{(}a,b;c;\zeta\big{)}$ is the hypergeometric function.(See, e.g., [1]. For analytic continuation see , e.g., [12, Sec. 1.8] .) If $n$ is odd, $n=2m+1$, $m\in{\mathbb{N}}$, then for $f\in C^{\infty}({\mathbb{R}}^{n}\times[0,\infty))$, we define $\displaystyle G[f](x,t)$ $\displaystyle:=$ $\displaystyle 2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\left(\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{S^{n-1}}f(x+ry,b)\,dS_{y}\right)_{r=r_{1}}$ $\displaystyle\hskip 71.13188pt\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$ $\displaystyle\hskip 71.13188pt\times F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\\!\\!,$ where $c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-2)$. Constant $\omega_{n-1}$ is the area of the unit sphere $S^{n-1}\subset{\mathbb{R}}^{n}$. If $n$ is even, $n=2m$, $m\in{\mathbb{N}}$, then for $f\in C^{\infty}({\mathbb{R}}^{n}\times[0,\infty))$, the operator $G$ is given by the next expression $\displaystyle G[f](x,t)$ $\displaystyle:=$ $\displaystyle 2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\left(\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-2}{2}}\frac{2r^{n-1}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{B_{1}^{n}(0)}\frac{f(x+ry,b)}{\sqrt{1-\|y\|^{2}}}\,dV_{y}\right)_{r=r_{1}}$ $\displaystyle\hskip 71.13188pt\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$ $\displaystyle\hskip 71.13188pt\times F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\\!\\!.$ Here $B_{1}^{n}(0):=\\{\|y\|\leq 1\\}$ is the unit ball in ${\mathbb{R}}^{n}$, while $c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-1)$. Thus, in both cases, of even and odd $n$, one can write $\displaystyle u(x,t)$ $\displaystyle=$ $\displaystyle 2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr\,v(x,r;b)(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r^{2}\right)^{-\frac{1}{2}+M}$ $\displaystyle\hskip 71.13188pt\times F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r^{2}}{(e^{-b}+e^{-t})^{2}-r^{2}}\right),$ where the function $v(x,t;b)$ is a solution to the Cauchy problem for the wave equation $v_{tt}-\bigtriangleup v=0\,,\quad v(x,0;b)=f(x,b)\,,\quad v_{t}(x,0;b)=0\,.$ It can be proved that if $n\big{(}1-\frac{2}{q}\big{)}\leq 1$, then for every given $T>0$ the operator $G$ can be extended to the bounded operator: $G\,:\,C([0,T];L^{q^{\prime}}({\mathbb{R}}^{n}))\longrightarrow C([0,T];L^{q}({\mathbb{R}}^{n}))\,.$ Consequently the operator $G$ maps $G\,:\,C([0,\infty);L^{q^{\prime}}({\mathbb{R}}^{n}))\longrightarrow C([0,\infty);L^{q}({\mathbb{R}}^{n})),$ in the corresponding topologies. Moreover, $G\,:\,C([0,\infty);L^{q^{\prime}}({\mathbb{R}}^{n}))\longrightarrow C^{1}([0,\infty);{\mathcal{D}}^{\prime}({\mathbb{R}}^{n})).$ Let $u_{0}=u_{0}(x,t)$ be a solution of the Cauchy problem $\partial_{t}^{2}u_{0}-e^{-2t}\bigtriangleup u_{0}-M^{2}u_{0}=0,\quad u_{0}(x,0)=\varphi_{0}(x),\quad\partial_{t}u_{0}(x,0)=\varphi_{1}(x)\,.$ (4) Then any solution $u=u(x,t)$ of the equation (2) which takes initial value $u(x,0)=\varphi_{0}(x),\quad\partial_{t}u(x,0)=\varphi_{1}(x)$, solves also integral equation $u(x,t)=u_{0}(x,t)+G[e^{-\frac{n(p-1)}{2}\cdot}\|u\|^{p}](x,t)\,.$ (5) Let $\Gamma\in C([0,\infty))$. For every given function $u_{0}\in C([0,T];L^{q^{\prime}}({\mathbb{R}}^{n}))$ we consider integral equation (5) $u(x,t)=u_{0}(x,t)+G\left[\Gamma(\cdot)\left(\int_{{\mathbb{R}}^{n}}\|u(y,\cdot)\|^{p}dy\right)^{\beta}\|u(y,\cdot)\|^{p}\right](x,t)\,,$ (6) for the function $u\in C([0,T];L^{q}({\mathbb{R}}^{n}))\cap C([0,T];L^{p}({\mathbb{R}}^{n}))$. Here $q^{\prime}\geq q>1$, $p\geq 1$. The last integral equation corresponds to the slightly more general equation than (2), namely, to the nonlocal equation $u_{tt}-e^{-2t}\bigtriangleup u-M^{2}u=\Gamma(t)\left(\int_{{\mathbb{R}}^{n}}\|u(y,t)\|^{p}dy\right)^{\beta}\|u\|^{p}\,.$ (7) If $u_{0}$ is generated by the Cauchy problem (4), then the solution $u=u(x,t)$ of (6) is said to be a weak solution of the Cauchy problem with the initial conditions $u(x,0)=\varphi_{0}(x),\quad\partial_{t}u(x,0)=\varphi_{1}(x)\,,$ for the equation (7). In the present paper we are looking for the conditions on the function $\Gamma$, on constants $M$, $n$, $p$, and $\beta$ that guarantee a non-existence of global in time weak solution, namely, the blow-up phenomena. We are especially interested in the scale of functions $\Gamma(t)=(1+t)^{d_{1}}e^{d_{0}t}$, where $d_{0},d_{1}\in{\mathbb{R}}$. The function $\,e^{-\frac{n(p-1)}{2}t}$ with $d_{0}=-n(p-1)/2$ and $d_{1}=0$ is in that scale and represents equation (2) if $\beta=0$. In particular, we find in the next theorem the upper bound for $d_{0}$ with an existence of the global solution for small initial data. For equation (7) in that scale the bound is given by $d_{0}\geq-M(p(\beta+1)-1)$ and $d_{1}>2$ if $M>0$. ###### Theorem 1.1 Suppose that function $\Gamma\in C^{1}([0,\infty))$ is either non-decreasing or non-increasing, and if $M>0$ then $\displaystyle\Gamma(t)\geq ce^{-M(p(\beta+1)-1)t}t^{2+\varepsilon}\quad\mbox{\rm for all}\quad t\in[0,\infty),$ with the numbers $\varepsilon>0$ and $c>0$, while for $M=0$ it satisfies $\displaystyle\Gamma(t)\geq ct^{-1-p(\beta+1)}\,.$ Then, for every $p>1$, $N$, and $\varepsilon$ there exists $u_{0}\in C^{\infty}({\mathbb{R}}^{n}\times[0,\infty))$ which for any given slice of constant time $t=const\geq 0$ has a compact support in $x$, such that $u_{0}(x,0),\partial_{t}u_{0}(x,0)\in C^{\infty}_{0}({\mathbb{R}}^{n})$, and $\\|u_{0}(x,0)\\|_{C^{N}({\mathbb{R}}^{n})}+\\|\partial_{t}u_{0}(x,0)\\|_{C^{N}({\mathbb{R}}^{n})}<\varepsilon$ but a global in time solution $u\in C([0,\infty);L^{q}({\mathbb{R}}^{n}))$ of the equation (6) with permanently bounded support does not exist for all $q\in[2,\infty)$ and $\beta>1/p-1$. More precisely, there is $T>0$ such that $\lim_{t\nearrow T}\int_{{\mathbb{R}}^{n}}u(x,t)dx=\infty\,.$ This theorem shows that instability of the trivial solution occurs in the very strong sense, that is, an arbitrarily small perturbation of the initial data can make the perturbed solution blowing up in finite time. If we allow large initial data, then according to the next theorem, for every $d_{0}\in{\mathbb{R}}$ and $M>0$ the solution blows up in finite time. ###### Theorem 1.2 Suppose that function $\Gamma(t)=e^{\gamma t}$, where $\gamma\in{\mathbb{R}}$ and that the curved mass is positive, $M>0$. Then, for every $p>1$ and $n$ there exists $u_{0}\in C^{\infty}({\mathbb{R}}^{n}\times[0,\infty))$ which for any given slice of constant time $t=const\geq 0$ has a compact support in $x$, such that $u_{0}(x,0),\partial_{t}u_{0}(x,0)\in C^{\infty}_{0}({\mathbb{R}}^{n})$ but a global in time solution $u\in C([0,\infty);L^{q}({\mathbb{R}}^{n}))$ of the equation (6) with permanently bounded support does not exist for all $q\in[2,\infty)$ and $\beta>1/p-1$. More precisely, there is $T>0$ such that $\lim_{t\nearrow T}\int_{{\mathbb{R}}^{n}}u(x,t)\,dx=\infty\,.$ Thus, for every $p>1$ the large energy classical solution of the Cauchy for equation (1) blows up. We will prove global existence of the small energy solution in a forthcoming paper. The remaining part of this paper is organized as follows. In Section 2 we prove some auxiliary integral representations for the function $\sinh(t)$ and the linear function via Gauss’s hypergeometric function and multidimensional integrals involving also fundamental solution of the Cauchy problem for wave equation in Minkowski spacetime. In Section 3 we suggest two simple generalizations of Kato’s lemma, which allow us to handle the case of differential inequalities with exponentially decreasing kernels. In Section 4 we complete the proofs of both theorems. ## 2 Integral representations of function $M^{-1}\sinh(M(t-b))$ involving hypergeometric function In [1, Sec. 2.4] one can find one-dimensional integrals involving hypergeometric function. In this section we present one more example of such integral and also examples of multidimensional integrals appearing in the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. More examples related to the Tricomi and Gellerstedt equations one can find in [14]. ###### Proposition 2.1 The function $M^{-1}\sinh(M(t-b))$ with $t\geq b\geq 0$, can be represented as follows: (i) The one-dimensional integral $\displaystyle\frac{1}{M}\sinh(M(t-b))$ $\displaystyle=$ $\displaystyle\int_{-(e^{-b}-e^{-t})}^{e^{-b}-e^{-t}}(4e^{-b-t})^{-M}\Big{(}(e^{-t}+e^{-b})^{2}-z^{2}\Big{)}^{-\frac{1}{2}+M}$ $\displaystyle\times F\Big{(}\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-z^{2}}{(e^{-b}+e^{-t})^{2}-z^{2}}\Big{)}\,dz;$ (ii) If $n$ is odd, $n=2m+1$, $m\in{\mathbb{N}}$, then with $c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-2)$, $\displaystyle\frac{1}{M}\sinh(M(t-b))$ $\displaystyle=$ $\displaystyle 2\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\left(\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{S^{n-1}}\,dS_{y}\right)_{r=r_{1}}\\!\\!(4e^{-b-t})^{-M}$ $\displaystyle\times\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right);$ (iii) If $n$ is even, $n=2m$, $m\in{\mathbb{N}}$, then with $c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-1)$, $\displaystyle\frac{1}{M}\sinh(M(t-b))$ $\displaystyle=$ $\displaystyle 2\int_{0}^{e^{-b}-e^{-t}}\\!\\!dr_{1}\\!\\!\left(\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-2}{2}}\frac{2r^{n-1}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{B_{1}^{n}(0)}\frac{1}{\sqrt{1-\|y\|^{2}}}dV_{y}\right)_{r=r_{1}}$ $\displaystyle\hskip 71.13188pt\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$ $\displaystyle\hskip 71.13188pt\times F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\\!\\!.$ Here the constant $\omega_{n-1}$ is the area of the unit sphere $S^{n-1}\subset{\mathbb{R}}^{n}$. Proof. First we consider case (i). According to Theorem 0.3 [16] for every function $f\in$ $C^{\infty}({\mathbb{R}}\times[0,\infty))$, which for any given slice of constant time $t=const\geq 0$ has a compact support in $x$, the function $\displaystyle v(x,t)$ $\displaystyle=$ $\displaystyle\int_{0}^{t}db\int_{x-(e^{-b}-e^{-t})}^{x+e^{-b}-e^{-t}}dy\,(4e^{-b-t})^{-M}\Big{(}(e^{-t}+e^{-b})^{2}-(x-y)^{2}\Big{)}^{-\frac{1}{2}+M}$ $\displaystyle\hskip 65.44142pt\times F\Big{(}\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-(x-y)^{2}}{(e^{-b}+e^{-t})^{2}-(x-y)^{2}}\Big{)}f(y,b)$ is a unique $C^{\infty}$-solution to the Cauchy problem $\partial_{t}^{2}v-e^{-2t}\bigtriangleup v-M^{2}v=f,\quad v(x,0)=0,\quad\partial_{t}v(x,0)=0$ (9) with $n=1$. It follows $\displaystyle\int_{-\infty}^{\infty}v(x,t)dx$ $\displaystyle=$ $\displaystyle\int_{0}^{t}db\Big{(}\int_{-\infty}^{\infty}f(x,b)dx\Big{)}\int_{-(e^{-b}-e^{-t})}^{e^{-b}-e^{-t}}dz(4e^{-b-t})^{-M}\Big{(}(e^{-t}+e^{-b})^{2}-z^{2}\Big{)}^{-\frac{1}{2}+M}$ $\displaystyle\hskip 71.13188pt\times F\Big{(}\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-z^{2}}{(e^{-b}+e^{-t})^{2}-z^{2}}\Big{)}.$ On the other hand, from the linear Klein-Gordon equation (9) and the vanishing initial data, we obtain $\displaystyle\int_{-\infty}^{\infty}v(x,t)dx-M^{2}\int_{0}^{t}d\tau\int_{0}^{\tau}db\int_{-\infty}^{\infty}v(x,b)dx$ $\displaystyle=$ $\displaystyle\int_{0}^{t}d\tau\int_{0}^{\tau}db\int_{-\infty}^{\infty}e^{-2b}\partial_{x}^{2}v(x,b)dx+\int_{0}^{t}d\tau\int_{0}^{\tau}db\int_{-\infty}^{\infty}f(x,b)\,dx\,,$ that is $\displaystyle\int_{-\infty}^{\infty}v(x,t)dx-M^{2}\int_{0}^{t}d\tau\int_{0}^{\tau}db\int_{-\infty}^{\infty}v(x,b)dx$ $\displaystyle=$ $\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}(t-b)db.$ Denote $\displaystyle V(t)=\int_{0}^{t}d\tau\int_{0}^{\tau}db\int_{-\infty}^{\infty}v(x,b)dx,\qquad F(t):=\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}(t-b)db\,,$ (12) then (2) and (12) imply $V_{tt}-M^{2}V=F,\qquad V(0)=V_{t}(0)=0\,.$ We easily find $\displaystyle V(t)$ $\displaystyle=$ $\displaystyle\frac{1}{M}\int_{0}^{t}F(\tau)\sinh(M(t-\tau))d\tau\,.$ Then (2) implies $\displaystyle\int_{-\infty}^{\infty}v(x,t)dx$ $\displaystyle=$ $\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}(t-b)db+M\int_{0}^{t}F(\tau)\sinh(M(t-\tau))d\tau$ $\displaystyle=$ $\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}(t-b)db$ $\displaystyle+M\int_{0}^{t}db\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}\int_{b}^{t}d\tau(\tau-b)\sinh(M(t-\tau)).$ On the other hand $\int_{b}^{t}d\tau(\tau-b)\sinh(M(t-\tau))=-\frac{1}{M}(t-b)+\frac{1}{M^{2}}\sinh(M(t-b))$ implies $\displaystyle\int_{-\infty}^{\infty}v(x,t)dx$ $\displaystyle=$ $\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}(t-b)db$ $\displaystyle+M\int_{0}^{t}db\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}\Big{(}-\frac{1}{M}(t-b)+\frac{1}{M^{2}}\sinh(M(t-b)))\Big{)}$ $\displaystyle=$ $\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}\Big{(}\frac{1}{M}\sinh(M(t-b))\Big{)}db\,.$ Thus, for the arbitrary function $f\in C^{\infty}({\mathbb{R}}\times[0,\infty))$ for all $t$ due to (2) one has $\displaystyle\int_{0}^{t}\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}\Big{(}\frac{1}{M}\sinh(M(t-b))\Big{)}db$ $\displaystyle=$ $\displaystyle\int_{0}^{t}db\Big{(}\int_{-\infty}^{\infty}f(x,b)\,dx\Big{)}\int_{-(e^{-b}-e^{-t})}^{e^{-b}-e^{-t}}dz\,(4e^{-b-t})^{-M}\Big{(}(e^{-t}+e^{-b})^{2}-z^{2}\Big{)}^{-\frac{1}{2}+M}$ $\displaystyle\hskip 71.13188pt\times F\Big{(}\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-z^{2}}{(e^{-b}+e^{-t})^{2}-z^{2}}\Big{)}\,.$ It follows (2.1). Thus (i) is proved. To prove case (ii) with $n$ is odd, $n=2m+1$, $m\in{\mathbb{N}}$, we use the identity $1=\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\int_{S^{n-1}}\,dS_{y}$ and take into consideration that the kernel $(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\\!\\!$ is an even function of $r_{1}$. In the case of (iii) when $n$ is even, $n=2m$, $m\in{\mathbb{N}}$, we apply the identity $1=\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-2}{2}}\frac{2r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\int_{B_{1}^{n}(0)}\frac{1}{\sqrt{1-\|y\|^{2}}}\,dV_{y}\,.$ The proposition is proven. $\Box$ If we set in the above integrals $b=0$ then we get integral representations of the function $\sinh(Mt)$ depending on parameter $M>0$. We also note that both sides of these formulas are translation invariant in $t$. By passing to the limit as $M\to 0$ we arrive at the following corollary. ###### Corollary 2.2 The function $t-b$ with $t\geq b\geq 0$, can be represented as follows: (i) The one-dimensional integral $\displaystyle t-b$ $\displaystyle=$ $\displaystyle\int_{-(e^{-b}-e^{-t})}^{e^{-b}-e^{-t}}\Big{(}(e^{-t}+e^{-b})^{2}-z^{2}\Big{)}^{-\frac{1}{2}}F\Big{(}\frac{1}{2},\frac{1}{2};1;\frac{(e^{-b}-e^{-t})^{2}-z^{2}}{(e^{-b}+e^{-t})^{2}-z^{2}}\Big{)}\,dz;$ (ii) If $n$ is odd, $n=2m+1$, $m\in{\mathbb{N}}$, then with $c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-2)$, $\displaystyle t-b$ $\displaystyle=$ $\displaystyle 2\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\left(\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{S^{n-1}}\,dS_{y}\right)_{r=r_{1}}$ $\displaystyle\times\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}}F\left(\frac{1}{2},\frac{1}{2};1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right);$ (iii) If $n$ is even, $n=2m$, $m\in{\mathbb{N}}$, then with $c_{0}^{(n)}=1\cdot 3\cdot\ldots\cdot(n-1)$, $\displaystyle t-b$ $\displaystyle=$ $\displaystyle 2\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\left(\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-2}{2}}\frac{2r^{n-1}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{B_{1}^{n}(0)}\frac{1}{\sqrt{1-\|y\|^{2}}}\,dV_{y}\right)_{r=r_{1}}$ $\displaystyle\times\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}}F\left(\frac{1}{2},\frac{1}{2};1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\\!\\!.$ ## 3 The second order differential inequalities The second order differential inequality with the power decreasing kernel play key role in proving blow-up of the solutions of the semilinear equations. Kato’s lemma [6] allows us to derive from inequality $\ddot{w}\geq bt^{-1-p}w^{p},\qquad p>1,\,\,b>0,\quad t\,\,\,\mbox{\rm large}$ a boundedness of the life-span of solution with property $w_{t}\geq a>0$. For the equation in de Sitter spacetime the kernel of the corresponding ordinary differential inequality decreases exponentially: $\ddot{w}\geq be^{-Mt}w^{p},\qquad p>1,\,\,b>0,\,M>0,\quad t\,\,\,\mbox{\rm large}.$ There is a global solution to the last inequality. Hence, in order to reach exact conditions on the involving functions we have to generalize Kato’s lemma. It is done in two following lemmas. ###### Lemma 3.1 Suppose $F(t)\in C^{2}([a,b))$, and $\displaystyle F(t)\geq 0\,,\qquad\dot{F}(t)\geq 0\,,\qquad\ddot{F}(t)\geq\Gamma(t)F(t)^{p}\quad\mbox{\rm for all}\quad t\in[a,b)\,,$ where $\Gamma\in C^{1}([a,\infty))$ is non-negative function, $\Gamma(a)>0$, and $p>1$. Assume that for all $t\in[a,b)$ either $\displaystyle\dot{\Gamma}(t)\leq 0\quad\mbox{\rm or }\quad\Gamma(t)\geq const>0\,.$ If there exists $a_{1}\in(a,b)$ such that $\frac{1}{\sqrt{p+1}}\int_{a}^{a_{1}}\Gamma(s)^{1/2}ds>\frac{\sqrt{2}}{p-1}F(a)^{(1-p)/2}\,,\quad\dot{F}(a)^{2}\geq\frac{2}{p+1}\Gamma(a)F(a)^{p+1},$ (13) then $b$ must be finite unless $\lim_{t\to\infty}F(t)$ is finite. Proof. First we consider the case of $\dot{\Gamma}\leq 0$. The conditions of the lemma imply that derivative of the energy density function is non- negative, $\frac{d}{dt}\left(F_{t}(t)^{2}-\frac{2}{p+1}\Gamma(t)F(t)^{p+1}\right)\geq 0\quad\mbox{\rm for all}\quad t\in[a,b)\,.$ We integrate the last inequality and obtain $F_{t}(t)^{2}\geq\frac{2}{p+1}\Gamma(t)F(t)^{p+1}+F_{t}(a)^{2}-\frac{2}{p+1}\Gamma(a)F(a)^{p+1}\quad\mbox{\rm for all}\quad t\in[a,b)\,.$ In fact, according to the second inequality of the condition (13) we have $F_{t}(t)^{2}\geq\frac{2}{p+1}\Gamma(t)F(t)^{p+1}\quad\mbox{\rm for all}\quad t\in[a,b)\,.$ Hence, $F_{t}(t)\geq\sqrt{\frac{2}{p+1}}\Gamma(t)^{1/2}F(t)^{(p+1)/2}\quad\mbox{\rm for all}\quad t\in[a,b)\,.$ It follows $\frac{d}{dt}\left(\frac{2}{1-p}F^{1-(p+1)/2}(t)\right)\geq\sqrt{\frac{2}{p+1}}\Gamma(t)^{1/2}\quad\mbox{\rm for all}\quad t\in[a,b)\,.$ Consequently, $\frac{2}{1-p}F^{(1-p)/2}(t)-\frac{2}{1-p}F^{(1-p)/2}(a)\geq\sqrt{\frac{2}{p+1}}\int_{a}^{t}\Gamma(s)^{1/2}ds\,.$ According to the first inequality of the condition (13) there exists $a_{1}>a$ such that $\displaystyle\frac{2}{1-p}F^{(1-p)/2}(t)$ $\displaystyle\geq$ $\displaystyle\sqrt{\frac{2}{p+1}}\int_{a_{1}}^{t}\Gamma(s)^{1/2}ds+\sqrt{\frac{2}{p+1}}\int_{a}^{a_{1}}\Gamma(s)^{1/2}ds-\frac{2}{p-1}F^{(1-p)/2}(a)$ $\displaystyle\geq$ $\displaystyle\sqrt{\frac{2}{p+1}}\int_{a_{1}}^{t}\Gamma(s)^{1/2}ds$ for all $t\in[a_{1},b)$. Thus, for large $t$ we get contradiction. The case of uniformly positive function $\Gamma$ follows from Kato’s Lemma [6]. Lemma is proven. $\Box$ Next we turn to the case of the small energy and exponentially decreasing $\Gamma(t)$. ###### Lemma 3.2 Suppose $F(t)\in C^{2}([a,b))$, and $F(t)\geq c_{0}A(t),\quad F_{t}(t)\geq 0,\quad F_{tt}(t)\geq\gamma(t)A(t)^{-p}F(t)^{p}\quad\mbox{\rm for all}\,\,t\in[a,b),$ (14) where $A,\gamma\in C^{1}([a,\infty))$ are non-negative functions and $p>1$, $c_{0}>0$. Assume that $\displaystyle\lim_{t\to\infty}A(t)=\infty\,,$ (15) and that $\displaystyle\frac{d}{dt}\left(\gamma(t)A(t)^{-p}\right)\leq 0\quad\mbox{\rm for all}\quad t\in[a,b)\,.$ (16) If there exist $\varepsilon>0$ and $c>0$ such that $\displaystyle\gamma(t)\geq cA(t)(\ln A(t))^{2+\varepsilon}\quad\mbox{\rm for all}\quad t\in[a,b),$ (17) then $b$ must be finite. Proof. There is a point $a_{1}\geq a$ such that ${F}_{t}(a_{1})>0$. Then $F_{t}(t)\geq F_{t}(a_{1})$ for all $t\geq a_{1}$ and consequently $\displaystyle{F}(t)\geq\frac{1}{2}{F}_{t}(a_{1})t\quad\mbox{\rm for all}\quad t\in[a_{2},b)\,,$ for sufficiently large $a_{2}$. Furthermore, according to (16) for the energy density function we have $\frac{d}{dt}\left(F_{t}(t)^{2}-2\frac{1}{p+1}\gamma(t)A(t)^{-p}F(t)^{p+1}\right)\geq 0\quad\mbox{\rm for all}\quad t\in[a_{1},b)\,.$ The last inequality implies $F_{t}(t)^{2}\geq 2\frac{1}{p+1}\gamma(t)A(t)^{-p}F(t)^{p+1}+F_{t}(a_{1})^{2}-2\frac{1}{p+1}\gamma(a_{1})A(a_{1})^{-p}F(a_{1})^{p+1}$ for all $t\in[a_{1},b)$. For sufficiently large $a_{2}\geq a_{1}$ using conditions (14), (15), and (17) we derive $\displaystyle\frac{1}{p+1}\gamma(t)A(t)^{-p}F(t)^{p+1}$ $\displaystyle\geq$ $\displaystyle\frac{1}{p+1}c_{0}^{p+1}\gamma(t)A(t)$ $\displaystyle\geq$ $\displaystyle\frac{1}{p+1}cc_{0}^{p+1}A(t)^{2}(\ln A(t))^{2+\varepsilon}$ $\displaystyle\geq$ $\displaystyle F_{t}(a_{1})^{2}-2\frac{1}{p+1}\gamma(a_{1})A(a_{1})^{-p}F(a_{1})^{p+1}$ for all $t\in[a_{2},b)$. Hence, $F_{t}(t)\geq\sqrt{\frac{1}{p+1}}\gamma(t)^{1/2}A(t)^{-p/2}F(t)^{(p+1)/2}\quad\mbox{\rm for all}\quad t\in[a_{2},b)\,.$ It follows $F_{t}(t)\geq\delta\gamma(t)^{1/2}A(t)^{-p/2}F(t)^{(p-1)/2}(\ln F(t))^{-1-\varepsilon/2}F(t)(\ln F(t))^{1+\varepsilon/2}$ for all $t\in[a_{2},b)$. But with sufficiently large $a_{2}\geq a_{1}$ we obtain $F(t)^{(p-1)/2}(\ln F(t))^{-1-\varepsilon/2}\geq\delta A(t)^{(p-1)/2}(\ln A(t))^{-1-\varepsilon/2}\quad\mbox{\rm for all}\quad t\in[a_{2},b)\,.$ Hence, $F_{t}(t)\geq\delta\left(\gamma(t)A(t)^{-1}(\ln A(t))^{-2-\varepsilon}\right)^{1/2}F(t)(\ln F(t))^{1+\varepsilon/2}\quad\mbox{\rm for all}\quad t\in[a_{2},b)$ implies $F_{t}(t)\geq\delta cF(t)(\ln F(t))^{1+\varepsilon/2}\quad\mbox{\rm for all}\quad t\in[a_{2},b)\,.$ The last nonlinear differential inequality does not have global solution with $F>0$. Lemma is proven. $\Box$ ###### Remark 3.3 We note here that the equation $\displaystyle\ddot{F}(t)=e^{-dt}F(t)^{p}\,,\quad d>0,$ has a global solution $F(t)=c_{F}e^{\frac{d}{p-1}t}$, where $c_{F}=\left({d}/(p-1)\right)^{2/(p-1)}$, while corresponding $A(t)=c_{A}e^{at}$, $a>0$, and $\gamma(t)=c_{\gamma}e^{(pa-d)t}$. The condition (17) implies $a>d/(p-1)$. On the other hand, the first inequality of (14) holds only if $a\leq d/(p-1)$. ## 4 Nonexistence of the global solution for the integral equation associated with the Klein-Gordon equation Since $G$ is a fundamental solution of the strictly hyperbolic operator, for every given function $u_{0}\in C([0,T];$ $L^{q}({\mathbb{R}}^{n}))\cap C^{\infty}([0,T]\times{\mathbb{R}}^{n})$ there exist $T_{0}>0$ and solution $u\in C([0,T_{0}];L^{q}({\mathbb{R}}^{n}))$. Moreover, for every given $T$ one can prove existence of the solution $u\in C([0,T];L^{q}({\mathbb{R}}^{n}))$ provided that $\sup_{t\in[0,T]}\\|u_{0}(\cdot,t)\\|_{L^{q}({\mathbb{R}}^{n})}$ is small enough. Theorem 1.1 shows that the set of such $T$, in general, is bounded. Proof of Theorem 1.1. Let $u_{0}\in C^{\infty}([0,\infty)\times{\mathbb{R}}^{n})$ be a function with the permanently bounded support, that is supp$\,u_{0}(\cdot,t)\subset\\{\,x\in{\mathbb{R}}^{n}\,;\,\|x\|\leq constant\,\\}$ for all $t\geq 0$. We denote $\varphi_{0}(x):=u_{0}(x,0)$ and $\varphi_{1}(x):=\partial_{t}u_{0}(x,0)$. One can find $u_{0}$ such that $\int_{{\mathbb{R}}^{n}}u_{0}(x,t)dx=C_{0}\cosh(Mt)+C_{1}\frac{1}{M}\sinh(Mt)\qquad\mbox{\rm for all}\quad t\geq 0\,,$ (18) where $C_{0}:=\int_{{\mathbb{R}}^{n}}\varphi_{0}(x)dx,\quad C_{1}:=\int_{{\mathbb{R}}^{n}}\varphi_{1}(x)dx\,.$ (19) The solution of the problem (4) with the data $\varphi_{0}(x)$, $\varphi_{1}(x)\in C^{\infty}_{0}({\mathbb{R}}^{n})$ exemplifies such function. Indeed, this unique smooth solution obeys finite propagation speed property that implies supp$\,u_{0}(\cdot,t)\subset\\{\,x\in{\mathbb{R}}^{n}\,;\,\|x\|\leq R_{0}+1-e^{-t}\leq R_{0}+1\,\\}$ if supp$\,\varphi_{0}$, supp$\,\varphi_{1}\subset\\{\,x\in{\mathbb{R}}^{n}\,;\,\|x\|\leq R_{0}\,\\}$. In order to check (18) for that solution $u_{0}$ we integrate (4) with respect to $x$ over ${\mathbb{R}}^{n}$ and then solve the initial problem with data (19) for the obtained ordinary differential equation. Suppose that $u\in C([0,\infty);L^{q}({\mathbb{R}}^{n}))$ with permanently bounded support is a solution to (6) generated by $u_{0}$. According to the definition of the solution, for every given $T>0$ we have $G\left[\Gamma(\cdot)\left(\int_{{\mathbb{R}}^{n}}\|u(y,\cdot)\|^{p}dy\right)^{\beta}\|u\|^{p}\right]\in C([0,T];L^{q}({\mathbb{R}}^{n}))$ and $u(x,0)=\varphi_{0}(x)\,,\quad u_{t}(x,0)=\varphi_{1}(x)\,.$ Then $u\in C([0,\infty);L^{1}({\mathbb{R}}^{n}))$ and we can integrate equation (6): $\int_{{\mathbb{R}}^{n}}u(x,t)\,dx=\int_{{\mathbb{R}}^{n}}u_{0}(x,t)\,dx+\int_{{\mathbb{R}}^{n}}G\left[\Gamma(\cdot)\left(\int_{{\mathbb{R}}^{n}}\|u(y,\cdot)\|^{p}dy\right)^{\beta}\|u\|^{p}\right](x,t)\,dx.$ (20) In particular, $\displaystyle\int_{{\mathbb{R}}^{n}}u(x,0)\,dx=\int_{{\mathbb{R}}^{n}}\varphi_{0}(x)\,dx,\quad\int_{{\mathbb{R}}^{n}}u_{t}(x,0)\,dx=\int_{{\mathbb{R}}^{n}}\varphi_{1}(x)\,dx\,.$ To evaluate the last term of (20) we apply Proposition 2.1. Consider the case of odd $n\geq 3$. Then, for the smooth function $u=u(x,t)$ we obtain $\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)\|u\|^{p}](x,t)\,dx$ $\displaystyle=$ $\displaystyle\int_{{\mathbb{R}}^{n}}\,dx2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr_{1}\Big{(}\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}$ $\displaystyle\times\int_{S^{n-1}}\Big{[}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta}\|u(x+ry,b)\|^{p}\Big{]}\,dS_{y}\Big{)}_{r=r_{1}}$ $\displaystyle\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$ $\displaystyle\times F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\,.$ Therefore, $\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)\|u\|^{p}](x,t)\,dx$ $\displaystyle=$ $\displaystyle 2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr_{1}\Big{(}\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}$ $\displaystyle\times\int_{S^{n-1}}\Big{[}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta}\left(\int_{{\mathbb{R}}^{n}}\|u(x+ry,b)\|^{p}\,dx\right)\Big{]}\,dS_{y}\Big{)}_{r=r_{1}}$ $\displaystyle\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$ $\displaystyle\times F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)$ implies, $\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)\|u\|^{p}](x,t)\,dx$ $\displaystyle=$ $\displaystyle 2\int_{0}^{t}db\int_{0}^{e^{-b}-e^{-t}}dr_{1}\,\Big{(}\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}$ $\displaystyle\times\int_{S^{n-1}}\Big{[}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\Big{]}\,dS_{y}\Big{)}_{r=r_{1}}$ $\displaystyle\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$ $\displaystyle\times F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)\,.$ We obtain, $\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)\|u\|^{p}](x,t)dx$ $\displaystyle=$ $\displaystyle\int_{0}^{t}db\Big{[}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\Big{]}\int_{0}^{e^{-b}-e^{-t}}dr_{1}$ $\displaystyle\times 2\left(\frac{\partial}{\partial r}\Big{(}\frac{1}{r}\frac{\partial}{\partial r}\Big{)}^{\frac{n-3}{2}}\frac{r^{n-2}}{\omega_{n-1}c_{0}^{(n)}}\\!\\!\int_{S^{n-1}}\,dS_{y}\right)_{r=r_{1}}$ $\displaystyle\times(4e^{-b-t})^{-M}\left((e^{-t}+e^{-b})^{2}-r_{1}^{2}\right)^{-\frac{1}{2}+M}$ $\displaystyle\times F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{-b}-e^{-t})^{2}-r_{1}^{2}}{(e^{-b}+e^{-t})^{2}-r_{1}^{2}}\right)$ $\displaystyle=$ $\displaystyle\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db.$ Thus, for the solution $u=u(x,t)$ we have proven $\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)\|u\|^{p}](x,t)\,dx$ $\displaystyle=$ $\displaystyle\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db\,.$ Hence (20) reads $\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx$ $\displaystyle=$ $\displaystyle\int_{{\mathbb{R}}^{n}}u_{0}(x,t)\,dx+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db\,.$ Taking into account (18) and (19) we derive $\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(C_{0}+\frac{C_{1}}{M}\right)e^{Mt}+\frac{1}{2}\left(C_{0}-\frac{C_{1}}{M}\right)e^{-Mt}$ $\displaystyle+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db\,.$ We discuss separately two cases: with positive curved mass, $M>0$, and vanishing curved mass, $M=0$, respectively. In the case of $M>0$ we obtain $\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx$ $\displaystyle=$ $\displaystyle C_{0}\cosh(Mt)+\frac{C_{1}}{M}\sinh(Mt)$ $\displaystyle+$ $\displaystyle\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db\,.$ Denote $\displaystyle F(t)$ $\displaystyle:=$ $\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx\,,$ then the function $F(t)$ is $\displaystyle F(t)$ $\displaystyle=$ $\displaystyle C_{0}\cosh(Mt)+\frac{C_{1}}{M}\sinh(Mt)$ $\displaystyle+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\frac{1}{M}\sinh(M(t-b))\,db\,.$ It follows $F\in C^{2}([0,\infty))$. More precisely, $\displaystyle\dot{F}(t)$ $\displaystyle=$ $\displaystyle C_{1}\cosh(Mt)+MC_{0}\sinh(Mt)$ $\displaystyle+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\cosh(M(t-b))\,db\,,$ $\displaystyle\ddot{F}(t)$ $\displaystyle=$ $\displaystyle M^{2}F(t)+\Gamma(t)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,t)\|^{p}dz\Big{)}^{\beta+1}\,.$ (22) In particular, since $\Gamma(t)\geq 0$, we obtain $F(t)\geq C_{0}\cosh(Mt)+\frac{C_{1}}{M}\sinh(Mt)\,\,\mbox{\rm and}\,\,\ddot{F}(t)\geq\Gamma(t)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,t)\|^{p}dz\Big{)}^{\beta+1}.$ (23) On the other hand, since the solution $u=u(x,t)$ has permanently bounded support, then supp$\,u(\cdot,t)\subset\\{\,x\in{\mathbb{R}}^{n}\,;\,\|x\|\leq R\,\\}$ for some positive number $R$. Using the compact support of $u(\cdot,t)$ and Hölder’s inequality we get with $\tau_{n}$ the volume of the unit ball in ${\mathbb{R}}^{n}$, $\displaystyle\left\|\int_{{\mathbb{R}}^{n}}u(x,t)\,dx\right\|^{p}$ $\displaystyle\leq$ $\displaystyle\left(\int_{\|x\|\leq R}1\,dx\right)^{p-1}\left(\int_{\|x\|\leq R}\|u(x,t)\|^{p}\,dx\right)$ $\displaystyle=$ $\displaystyle\tau_{n}\Gamma(t)^{-1/(\beta+1)}R^{n(p-1)}\left(\Gamma(t)^{1/(\beta+1)}\int_{{\mathbb{R}}^{n}}\|u(x,t)\|^{p}\,dx\right)$ $\displaystyle=$ $\displaystyle\tau_{n}\Gamma(t)^{-1/(\beta+1)}R^{n(p-1)}\Big{(}\ddot{F}(t)-M^{2}F(t)\Big{)}^{1/(\beta+1)}$ $\displaystyle\leq$ $\displaystyle\tau_{n}\Gamma(t)^{-1/(\beta+1)}R^{n(p-1)}\ddot{F}(t)^{1/(\beta+1)}\,.$ Here we assume $\Gamma(t)>0$. Thus $\ddot{F}(t)\geq\tau_{n}^{-(\beta+1)}R^{-n(p-1)(\beta+1)}\Gamma(t)\|F(t)\|^{p(\beta+1)}\quad\mbox{\rm for all}\quad t\in[0,\infty)\,.$ By means of the inequality $MC_{0}+C_{1}>0$ we conclude that $F(t)\geq 0$ and that $\ddot{F}(t)\geq\delta_{0}\Gamma(t)F(t)^{p(\beta+1)}\qquad\mbox{\rm for large}\,\,\,t\quad\mbox{\rm with}\,\,\,\delta_{0}>0\,.$ Hence, for appropriate $C_{0}$ and $C_{1}$ the last inequality together with (4) to (23) implies the following system of the ordinary differential inequalities $\left\\{\begin{array}[]{ccccc}\displaystyle F(t)&\geq&C_{0}\cosh(Mt)+\frac{C_{1}}{M}\sinh(Mt)&\mbox{\rm for all}&\quad t\in[a,b),\\\ \displaystyle\dot{F}(t)&\geq&C_{1}\cosh(Mt)+MC_{0}\sinh(Mt)&\mbox{\rm for all}&\quad t\in[a,b),\\\ \displaystyle\ddot{F}(t)&\geq&\delta_{0}\Gamma(t)F(t)^{p(\beta+1)}&\mbox{\rm for all}&\quad t\in[a,b).\end{array}\right.$ The Lemma 3.1 shows that if $F(t)\in C^{2}([0,b))$ and the energy of particle is large, then $b$ must be finite. The conditions of the Lemma 3.1 are fulfilled on $(0,b)$ for the function $\Gamma(t)=\delta_{0}e^{\gamma t},\quad\gamma\in{\mathbb{R}},$ with $\gamma>0$ without any condition on the energy. They are fulfilled with $\gamma<0$ if the kinetic energy and the potential energy are sufficiently large, that is $C_{0}>0$, $C_{1}>0$, and $C_{1}\geq\sqrt{\frac{2\delta_{0}}{p+1}}C_{0}^{(p+1)/2}\quad\mbox{\rm and}\quad C_{0}^{p-1}>\frac{\gamma^{2}(p+1)}{\delta_{0}(p-1)}\,.$ Next we turn to the case of the small energy and exponentially decreasing $\Gamma(t)$. We apply Lemma 3.2 with $A(t)=e^{Mt}$ and $p$ replaced with $p(\beta+1)$. More precisely, if we set $A(t)=e^{Mt},\qquad\gamma(t)=\Gamma(t)e^{Mp(\beta+1)t},$ then the conditions of the last lemma read: $p(\beta+1)>1\quad\mbox{\rm and }\quad\Gamma_{t}(t)\leq 0\quad\mbox{\rm for all }\quad t\in[0,\infty).$ The last inequality follows from the monotonicity of $\Gamma(t)$. By the condition of the theorem, there exist $\varepsilon>0$ and $c>0$ such that $\displaystyle\Gamma(t)\geq ce^{-M(p(\beta+1)-1)t}t^{2+\varepsilon}\quad\mbox{\rm for all}\quad t\in[a,b),$ that coincides with (17). The case of $M>0$ is proved. Now consider the case of $M=0$. Let $\displaystyle C_{0}:=\int_{{\mathbb{R}}^{n}}\varphi_{0}(x)dx,\quad C_{1}:=\int_{{\mathbb{R}}^{n}}\varphi_{1}(x)dx,\qquad C_{1}>0.$ Then Corollary 2.2 allows us to write $\displaystyle\int_{{\mathbb{R}}^{n}}G[\Gamma(\cdot)\|u\|^{p}](x,t)\,dx$ $\displaystyle=$ $\displaystyle\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}(t-b)\,db.$ Hence (20) reads: $\int_{{\mathbb{R}}^{n}}u(x,t)\,dx=\int_{{\mathbb{R}}^{n}}u_{0}(x,t)\,dx+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\,.$ Now we choose a function $u_{0}\in C^{\infty}([0,\infty)\times{\mathbb{R}}^{n})$ such that $\int_{{\mathbb{R}}^{n}}u_{0}(x,t)dx=C_{0}+C_{1}t\,.$ The solution of the problem (4) with $M=0$ exemplifies such functions. Thus $\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx$ $\displaystyle=$ $\displaystyle C_{0}+C_{1}t+\,\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}(t-b)\,db\,.$ Denote $\displaystyle F(t)$ $\displaystyle:=$ $\displaystyle\int_{{\mathbb{R}}^{n}}u(x,t)\,dx\,,$ then $\displaystyle F(t)$ $\displaystyle=$ $\displaystyle C_{0}+C_{1}t+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}(t-b)\,db\,.$ It follows $F\in C^{2}([0,\infty))$. More precisely, $\displaystyle\dot{F}(t)$ $\displaystyle=$ $\displaystyle C_{1}+\int_{0}^{t}\Gamma(b)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,b)\|^{p}dz\Big{)}^{\beta+1}\,db\,,$ $\displaystyle\ddot{F}(t)$ $\displaystyle=$ $\displaystyle\Gamma(t)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,t)\|^{p}dz\Big{)}^{\beta+1}\,.$ (24) In particular, $\displaystyle F(t)\geq C_{0}+C_{1}t\quad\mbox{\rm and}\quad\ddot{F}(t)=\Gamma(t)\Big{(}\int_{{\mathbb{R}}^{n}}\|u(z,t)\|^{p}dz\Big{)}^{\beta+1}\,.$ (25) On the other hand according to (24) we obtain $\displaystyle\left\|\int_{{\mathbb{R}}^{n}}u(x,t)\,dx\right\|^{p}$ $\displaystyle\leq$ $\displaystyle\left(\int_{\|x\|\leq R}1\,dx\right)^{p-1}\left(\int_{\|x\|\leq R}\|u(x,t)\|^{p}\,dx\right)$ $\displaystyle=$ $\displaystyle\tau_{n}\Gamma(t)^{-1/(\beta+1)}R^{n(p-1)}\left(\Gamma(t)^{1/(\beta+1)}\int_{{\mathbb{R}}^{n}}\|u(x,t)\|^{p}\,dx\right)$ $\displaystyle\leq$ $\displaystyle\tau_{n}\Gamma(t)^{-1/(\beta+1)}R^{n(p-1)}\ddot{F}(t)^{1/(\beta+1)}\,.$ Thus $\ddot{F}(t)\geq\tau_{n}^{-(\beta+1)}R^{-n(p-1)(\beta+1)}\Gamma(t)\|F(t)\|^{p(\beta+1)}$ for all $t$ in $[0,\infty)$. By means of the condition $C_{1}>0$ we conclude $\ddot{F}(t)\geq C\Gamma(t)F(t)^{p(\beta+1)}\qquad\mbox{\rm for large}\,\,t\quad\mbox{\rm with}\,\,C>0\,.$ But for appropriate $C_{0}$ and $C_{1}$ one has $F(t)>0$ and the last inequality together with (25) implies $\displaystyle\left\\{\begin{array}[]{ccccc}\displaystyle F(t)&\geq&C_{0}+C_{1}t&\mbox{\rm for all}&t\in[a,b),\\\ \displaystyle\ddot{F}(t)&\geq&\delta_{0}\Gamma(t)F(t)^{p(\beta+1)}&\mbox{\rm for all}&t\in[a,b).\end{array}\right.$ The Kato’s Lemma 2 [6] shows that if $F(t)\in C^{2}([0,b))$ and $\Gamma(t)\geq t^{-1-p(\beta+1)}$ with $p(\beta+1)>1$, then $b$ must be finite. Theorem is proven. $\square$ ###### Remark 4.1 In fact, we have proved that any solution $u=u(x,t)$ with permanently bounded support blows up if either $MC_{0}+C_{1}>0$ and $M>0$ or $C_{1}>0$ and $M=0$. Proof of Theorem 1.2. The case of $\gamma\geq 0$ is covered by Theorem 1.1 and implies a blow-up even for the small data. Therefore, we restrict ourselves to the case of $\gamma<0$. Then, with a special choice of $C_{0}$ and $C_{1}$ after arguments have been used in the proof of Theorem 1.1 we arrive at the following system of the ordinary differential inequalities $\displaystyle\left\\{\begin{array}[]{ccccc}\displaystyle F(t)&\geq&Ce^{Mt}&\mbox{\rm for all}&t\in[0,b),\\\ \displaystyle\dot{F}(t)&\geq&Ce^{Mt}&\mbox{\rm for all}&t\in[0,b),\\\ \displaystyle\ddot{F}(t)&\geq&\delta_{0}e^{\gamma t}F(t)^{p(\beta+1)}&\mbox{\rm for all}&t\in[0,b),\end{array}\right.$ where $C>0$ and $\delta_{0}>0$. We claim that $b<\infty$. Indeed, we check conditions of Lemma 3.1 with $\Gamma(t)=\delta_{0}e^{\gamma t}\,.$ The condition (13), $\displaystyle\frac{1}{\sqrt{p+1}}\int_{0}^{a_{1}}\Gamma(s)^{1/2}ds>\frac{\sqrt{2}}{p-1}F^{(1-p)/2}(0)\,,\quad\dot{F}^{2}(0)\geq\frac{2}{p+1}\Gamma(0)F(0)^{p+1},$ reads: $\displaystyle\frac{1}{\sqrt{p+1}}\int_{0}^{a_{1}}\delta_{0}^{1/2}e^{\gamma s/2}ds>\frac{\sqrt{2}}{p-1}C_{0}^{(1-p)/2}\,,\quad C_{1}^{2}\geq\frac{2}{p+1}\delta_{0}C_{0}^{p+1}.$ The first inequality is fulfilled if $C_{0}$, that is the initial potential energy, is sufficiently large, while the second one is fulfilled if $C_{1}$, that is the initial kinetic energy, is large enough. Theorem is proven. $\square$ ## References * [1] H. Bateman, A. Erdelyi, “Higher Transcendental Functions”, vol. 1,2, McGraw-Hill, New York, 1953. * [2] D. Catania, V. Georgiev, _Blow-up for the semilinear wave equation in the Schwarzschild metric_ , Differential Integral Equations, 19 (2006), 799–830. MR2235896 (2008c:58021) * [3] M. Dafermos, I. Rodnianski, _The wave equation on Schwarzschild-de Sitter spacetimes_ , preprint, arXiv:0709.2766 * [4] A. Galstian, _$L_{p}$ -$L_{q}$ decay estimates for the wave equations with exponentially growing speed of propagation_, Appl. Anal., 82 (2003), 197–214. MR1970785 (2004b:35231) * [5] S. W. Hawking, G. F. R. Ellis, “The large scale structure of space-time”, Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-New York, 1973. xi+391 pp. * [6] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations. Comm. Pure Appl. Math., 33 (1980), 501–505. * [7] M. Keel, T. Tao, _Small data blow-up for semilinear Klein-Gordon equations_ , Amer. J. Math., 121 (1999), 629–669. * [8] C. M$\o$ller,“The theory of relativity”, Clarendon Press, Oxford, 1952. MR0049685 * [9] M. Ohta, G. Todorova, _Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system_ , SIAM J. Math. Anal., 38 (2007), 1912–1931. * [10] A. Rendall, “Partial differential equations in general relativity”, Oxford Graduate Texts in Mathematics, 16, Oxford University Press, Oxford, 2008. MR2406669 * [11] J. Shatah, M. Struwe, “Geometric wave equations”, Courant Lect. Notes Math., 2. New York Univ., Courant Inst. Math. Sci., New York, 1998. MR1674843 (2000i:35135) * [12] L. J. Slater, “Generalized hypergeometric functions”, Cambridge University Press, Cambridge 1966. * [13] K. Yagdjian, _Global existence in the Cauchy problem for nonlinear wave equations with variable speed of propagation_ , New trends in the theory of hyperbolic equations, 301–385, Oper. Theory Adv. Appl., 159, Birkh$\ddot{\rm a}$user, Basel, 2005. MR2175919 (2007e:35206) * [14] K. Yagdjian, _Global existence for the $n$-dimensional semilinear Tricomi-type equations_, Comm. Partial Diff. Equations, 31 (2006), 907-944. MR2233046 (2007e:35207) * [15] K. Yagdjian, A. Galstian, _Fundamental Solutions of the Wave Equation in Robertson-Walker spaces_ , J. Math. Anal. Appl., 346 (2008), 501–520. MR2433945 * [16] K. Yagdjian, A. Galstian, _Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime_. Comm. Math. Phys., 285 (2009), 293-344.	arxiv-papers	2009-02-28T17:33:18	2024-09-04T02:49:00.919993	{ "license": "Public Domain", "authors": "Karen Yagdjian", "submitter": "Karen Yagdjian", "url": "https://arxiv.org/abs/0903.0089" }
0903.0194	# A Graph Analysis of the Linked Data Cloud Marko A. Rodriguez Semantic Network Research Group Knowledge Reef Systems Inc. Santa Fe, New Mexico 87501 ###### Abstract The Linked Data community is focused on integrating Resource Description Framework (RDF) data sets into a single unified representation known as the Web of Data. The Web of Data can be traversed by both man and machine and shows promise as the de facto standard for integrating data world wide much like the World Wide Web is the de facto standard for integrating documents. On February 27${}^{\text{th}}$ of 2009, an updated Linked Data cloud visualization was made publicly available. This visualization represents the various RDF data sets currently in the Linked Data cloud and their interlinking relationships. For the purposes of this article, this visual representation was manually transformed into a directed graph and analyzed. ††preprint: KRS-2009-01 ## I Introduction The World Wide Web is a distributed document and media repository lee94 . Hyper-Text Markup Language (HTML) documents reference other HTML documents and media (e.g. images, audio, etc.) by means of an href citation. The resulting document citation graph has been the object of scholastic research bowtie:huberman1999 ; bowtie:broder as well as a component utilized in web page ranking anatom:brin1998 . Similarly, the Semantic Web is a distributed resource identifier repository pubsem:lee2001 . The Resource Description Framework (RDF) serves as one of the primary standards of the Semantic Web rdfintro:miller1998 . RDF provides the means by which Uniform Resource Identifiers (URI) uri:berners2005 are interrelated to form a multi-relational or edge labeled graph. If $U$ is the set of all URIs, $L$ is the set of all literals, and $B$ is the set of all blank (or anonymous) nodes, the the Semantic Web RDF graph is defined as the set of triples $G\subseteq(U\cup B)\times U\times(U\cup L\cup B).$ Given that the URI is the foundational standard of both the World Wide Web and the Semantic Web, the Semantic Web serves as an extension to the World Wide Web in that it provides a semantically-rich graph overlay for URIs. Thus, the Semantic Web moves the Web beyond the simplistic href citation into a rich relational structure that can be utilized for numerous end user applications. The Linked Data community is actively focused on integrating RDF data sets into a single connected data set berners:ldata2006 . The Linked Data model allows > “[any man or machine] to start with one data source and then move through a > potentially endless Web of data sources connected by RDF links. Just as the > traditional document Web can be crawled by following hypertext links, the > Web of Data can be crawled by following RDF links. Working on the crawled > data, search engines can provide sophisticated query capabilities, similar > to those provided by conventional relational databases. Because the query > results themselves are structured data, not just links to HTML pages, they > can be immediately processed, thus enabling a new class of applications > based on the Web of Data.” linkeddata:bizer2008 While the Linked Data community has focused on providing a distributed data structure, they have not focused on providing a distributed process infrastructure rodriguez:distributed2008 . Unfortunately, if only a data structure is provided, then processing that data structure will lead to what has occurred with the World Wide Web: a commercial industry focused on downloading, indexing, and providing search capabilities to that data. For the problem space of keyword search, this model suffices. However, the RDF data model is much richer than the World Wide Web citation data model. If data must be downloaded to a remote machine for processing, then only so much of the Web of Data can be processed in a reasonable amount of time. This ultimately limits the sophistication of the algorithms that can be executed on the Web of Data. The RDF data model is rich enough to conveniently support the representation of relational objects activerdf:oren2008 and their computational instructions rodriguez:gpsemnet2007 . Moreover, with respect to searching, the RDF data model requires a new degree of sophistication in graph analysis algorithms semrank:boan2005 . For one, the typical PageRank centrality calculation is nearly meaningless on an edge labeled graph grammar:rodriguez2007 . To leave this algorithmic requirement to a small set of search engines will ultimately yield a limited set of algorithms and not a flourishing democracy of collaborative development. As a remedy to this situation, a distributed process infrastructure (analogous in many ways to the Grid grid:foster2004 ) may be a necessary requirement to ensure the accelerated, grass roots use of the Web of Data, where processes are migrated to the data, not data to the processes. In such a model, computational clock cycles are as open as the data upon which they operate. With respect to the Web of Data as a distributed RDF data structure, this article presents a graph analysis of the March 2009 Linked Data cloud visualization that was published on February 27, 2009 by Chris Bizer.111The March 2009 Linked Data cloud visualization is available at: http://tinyurl.com/b4vfbq. The remainder of this article is organized as follows. §II articulates how the Linked Data cloud graph was constructed from the February 27${}^{\text{th}}$ Linked Data cloud visualization. §III provides a collection of standard graph statistics for the constructed Linked Data cloud graph. Finally §IV provides a more in-depth analysis of the structural properties of the graph. ## II Constructing the Linked Data Cloud Graph The current Linked Data cloud visualization was published by Chris Bizer on February 27, 2009. This visualization is provided in Figure 1. Figure 1: The Linked Data cloud visualization as provided by the Linked Data community. This version is dated February 27, 2009. The author was not responsible for the creation of this visualization. This is only provided in order to better elucidate the means by which the Linked Data cloud graph was created. The Linked Data cloud visualization represents various data sets as vertices (i.e. nodes) and their interlinking relationships as directed unlabeled edges (i.e. links). Moreover, it is assumed that vertex size denotes the number of triples in the data set and edge thickness denotes the extent to which one data set interlinks with another. Data set $A$ links to data set $B$ if data set $B$ has a URI that is maintained (according to namespace) by data set $A$. In this way, by resolving a data set $B$ URI within data set $A$, the man or machine is able to traverse to data set $B$ from $A$. A manual process was undertaken to turn the Linked Data cloud visualization into a Linked Data cloud graph denoted $G=(V,E)$, where $V$ is the set of vertices (i.e. data sets), $E$ is the set of unlabeled edges (i.e data set links), and $E\subseteq(V\times V)$. The link weights and the node sizes in the original visualization were ignored. A new visualization of the manually generated Linked Data cloud graph is represented in Figure 4. The properties of this visualization are discussed throughout the remainder of this article. ## III Standard Graph Statistics Given the constructed Linked Data cloud graph visualized in Figure 4, it is possible to calculate various graph statistics. A collection of standard graph statistics are provided in Table 1. statistic \| statistic value ---\|--- number of vertices \| $86$ number of edges \| $274$ weakly connected \| true strongly connected \| false diameter \| $10$ average path length \| $3.916$ Table 1: A collection of standard graph statistics for the Linked Data cloud graph represented in Figure 4. ### III.1 Strongly Connected Components The Linked Data graph is not strongly connected. This means that there does not exist a path from every data set to every other data set. Therefore, a walk along the graph can lead to an “island” of data sets that can not be returned from. The number of strongly connected components is $31$ with $26$ of those components only maintaining a single data set (that is, they are either the source of a path or the sink of a path). The size of the remaining strongly connected components is $37$, $15$, $4$, $2$, and $2$. The largest component (with size of $37$) is the “DBpedia component”. The second largest (with size of $15$) is the “DBLP RKB Explorer component”. Given the large diameter and average path length, the Linked Data cloud graph can be seen as a two weakly connected components: the larger DBpedia component and the smaller DBLP RKB Explorer component. However, as will be seen later, other communities in the larger DBpedia component exist such as biological and medical communities. ### III.2 Degree Distributions The in- and out-degree distributions of the graph are plotted in Figure 2 and Figure 3 on a log-log plot, respectively. These plots show the number (frequency) of data sets that have a particular in- or out-degree. The top 11 in- and out-degree data sets are presented in Table 2 and Table 3, respectively. It is interesting to note that the two leaders (DBpedia and DBLP RKB Explorer) are also the leaders of the two largest strongly connected components identified previously. Figure 2: The in-degree distribution of the Linked Data cloud graph on a log-log plot. Figure 3: The out-degree distribution of the Linked Data cloud graph on a log-log plot. data set \| in-degree ---\|--- DBpedia \| $14$ DBLP RKB Explorer \| $13$ ACM \| $10$ GeneID \| $10$ GeoNames \| $10$ CiteSeer \| $9$ ePrints \| $9$ UniProt \| $9$ ECS Southampton \| $8$ FOAF Profiles \| $7$ RAE 2001 \| $7$ Table 2: The top $11$ Linked Data data sets with the highest in-degree. data set \| out-degree ---\|--- DBpedia \| $17$ DBLP RKB Explorer \| $14$ ACM \| $10$ CiteSeer \| $9$ EPrints \| $9$ GeneID \| $8$ UniProt \| $8$ DrugBank \| $7$ ECS Southampton \| $7$ FOAF Profiles \| $7$ RAE 2001 \| $7$ Table 3: The top $11$ Linked Data data sets with the highest out-degree. While the number of data points is small, a power-law fit is provided according to a distribution that is defined as $p(x)\sim x^{-\alpha}$, where $p(x)$ is the probability of seeing a data set with a degree of $x$. A power- law fit to the total degree distribution (i.e. ignoring edge directionality) yields an exponent of $\alpha=1.496$. In other words, the larger the degree, the fewer number of data sets. ### III.3 Degree Correlations The correlation between the in- and out-degrees of the vertices yields a Spearman $\rho=0.6753$ with a significant $p<9.85^{-13}$. Similarly, the Kendall $\tau=0.5640$ with a significant $p<7.27^{-12}$. In other words, data sets that frequently link to other data sets tend to get linked to frequently. If a graph is degree assortative then vertices with high degree are connected to other vertices with high degree. Likewise, vertices with low degree connect to vertices with low degree. Assortativity is calculated by creating two vectors of length $\|E\|$. One vector maintains the degree of the vertices at the head of each edge and the other vector maintains the degree of the vertices at the tail of each edge. These two vectors are then correlated. The popular assortative mixing value newman:assort is calculated with a Pearson correlation over the two vectors as $r=\frac{\|E\|\sum_{i}j_{i}k_{i}-\sum_{i}j_{i}\sum_{i}k_{i}}{\sqrt{\left[\|E\|\sum_{i}j^{2}_{i}-\left(\sum_{i}j_{i}\right)^{2}\right]\left[\|E\|\sum_{i}k^{2}_{i}-\left(\sum_{i}k_{i}\right)^{2}\right]}},$ where $j_{i}$ is the degree of the vertex on the tail of edge $i$, and $k_{i}$ is the degree of the vertex on the head of edge $i$. The correlation coefficient $r$ is in $[-1,1]$, where $-1$ represents a fully disassortative graph, $0$ represents an uncorrelated graph, and $1$ represents a fully assortative graph. Given that the degree distribution is non-parametric, a non-parametric assortativity correlation is also provided using both Spearman $\rho$ and Kendall $\tau$. All of these assortativity correlations are presented in Table 4, where the only significant values are from the standard Pearson correlation and all the in-degree correlations. method \| in-degree \| out-degree \| total-degree ---\|---\|---\|--- pearson \| -0.1911 (0.0015) \| -0.1728 (0.0042) \| -0.1868 (0.0019) spearman \| -0.1319 (0.0292) \| -0.0311(0.6089) \| -0.0629 (0.2998) kendall \| -0.0933 (0.0346) \| -0.0193 (0.6626) \| -0.0364 (0.3982) Table 4: Various degree assortativity correlations for the Linked Data cloud graph. The first number is the correlation and the second number in parentheses is the $p$-value. A significant $p$-value is less than $0.05$. These results demonstrate that Linked Data data sets tend to connect to data sets with differing degrees. That is, for instance, high degree data sets connect to low degree data sets. This is made apparent when looking at DBpedia which has a total-degree of $32$. DBpedia’s neighbors in the graph have the following total-degrees: $1$, $1$, $2$, $3$, $3$, $3$, $3$, $3$, $4$, $4$, $4$, $6$, $8$, $9$, $11$, $12$, $12$, and $18$. However, in general, the degree assortativity correlation is weak and for the non-parametric correlations, mostly insignificant. ## IV Structural Analysis This section presents an analysis of the community structures that exist within the Linked Data cloud graph. A community is loosely defined as a set of vertices that have a high number of intra-connections and low number of inter- connections. In other words, vertices in the same community tend to link to vertices in the same community as opposed to vertices in other communities. In order to compare the algorithmically determined structural communities to the metadata properties of the vertices that compose those communities, two metadata properties were gathered: 1. 1. a string label denoting the type of content maintained in the data set 2. 2. an integer value denoting the number of triples contained in the data set. The content labels were determined manually. The set of labels used were: biology, business, computer science, general, government, images, library, location, media, medicine, movie, music, reference, and social. Note that many data sets could have been labeled with more than one label. However, only one label was chosen. Moreover, these labels were determined by reviewing the websites of the data sets and not by looking at the structure of the graph. The data set triple counts were taken from the “Linking Open Data on the Semantic Web” web page.222Linking Open Data on the Semantic Web is available at: http://tinyurl.com/5fcmzm. Of the $86$ data sets in the Linked Data cloud, only $31$ of those data sets have published triple counts. ### IV.1 Labeled Structural Communities The graph analysis method for comparing nominal vertex metadata with structural communities as originally presented in onthe:rodriguez2008 was used to compare the content labels of the data sets to their structural communities. The purpose of this analysis is to determine the semantics of the structural communities. The hypothesis is that structural communities denote shared content. That is, data sets in the same structural community maintain the same type of content data (e.g. biology, medicine, computer science, etc.). A contingency table was created that denotes the number of vertices that have a particular content label and are in a particular structural community. An example contingency table that has community values that were determined using the leading eigenvector community detection algorithm newman-eigen is presented in Table 5. content/community \| 0 \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- biology \| 2 \| 0 \| 4 \| 1 \| 0 \| 0 \| 0 \| 0 \| 3 \| 10 business \| 1 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 computer science \| 1 \| 12 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 0 \| 0 general \| 4 \| 3 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 government \| 3 \| 0 \| 0 \| 0 \| 0 \| 2 \| 0 \| 0 \| 0 \| 0 images \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 library \| 2 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 1 \| 0 \| 0 location \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 1 \| 0 \| 0 media \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 \| 0 medicine \| 0 \| 1 \| 0 \| 4 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 movie \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 music \| 5 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 1 \| 0 \| 0 reference \| 2 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 social \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 \| 5 \| 0 \| 1 Table 5: An example contingency table that denotes how many data sets have a particular content label and structural community. For this example, the structural communities were determined using the leading eigenvector community detection algorithm. The contingency table is subjected to a $\chi^{2}$ analysis in order to determine if the manually generated content labels are statistically related to the algorithmically determined structural communities. Four community detection algorithms (and thus, four individual contingency tables) were used for this analysis and the $\chi^{2}$ $p$-values are presented in Table 6. community algorithm \| $\chi^{2}$ $p$-value ---\|--- Leading Eigenvector \| $6.6^{-12}$ WalkTrap \| $2.2^{-16}$ Edge Betweenness \| $0.0323$ Spinglass \| $2.4^{-16}$ Table 6: The $p$-values for four $\chi^{2}$ tests using four structural community detection algorithms: leading eigenvector newman-eigen , walktrap latapy , edge betweenness girvan-2002 , and spinglass spinglass:reichardt2006 . The analysis demonstrates that data sets that maintain similar content tend to exist in the same structural areas of the graph. This is made salient by a qualitative analysis of various subsets of the graph (see Figure 4 where the vertex colors denote their structural community). Moreover, this makes sense intuitively. Data sets that share the same content labels are more than likely to reference to the same resources. For example, it is true that medical data sets tend to be connected to other medical data sets and not to music data sets. Table 7 provides a review of 15 randomly chosen Linked Data data sets, their structural community values according to the leading eigenvector community detection algorithm, and their manually determined content labels. data set \| community \| content label ---\|---\|--- SurgeRadio \| 0 \| music MusicBrainz \| 0 \| music DBpedia \| 0 \| general Riese \| 5 \| government LinkedCT \| 3 \| medicine World Fact Book \| 5 \| government OpenCyc \| 0 \| general Yago \| 0 \| general DrugBank \| 3 \| medicine DailyMed \| 3 \| medicine UniParc \| 2 \| biology Reactome \| 9 \| biology ACM \| 1 \| computer science CiteSeer \| 1 \| computer science IEEE \| 1 \| computer science Table 7: A sample of 15 Linked Data data sets, their leading eigenvector structual community value, and their manually determined content label. ### IV.2 Data Set Triple Counts Of the $86$ data sets in the Linked Data cloud, only $31$ of those data sets have triple counts that were published on the “Linking Open Data on the Semantic Web” web page. Given the statistically significant, positive correlation between the in-degree and out-degree of the vertices, it is hypothesized that those data sets that are more central in the graph will have a larger triple count. The centrality of all $86$ vertices was determined using the PageRank centrality algorithm with a $\delta=0.85$ page98pagerank . For those $31$ data sets that have triple counts, their triple count value was correlated with their PageRank centrality value. The Spearman $\rho=0.6274$ with a significant $p<0.00016$. Similarly, the Kendall $\tau=0.4566$ with a significant $p<0.00039$. Thus, those data sets that have the most RDF triples tend to be centrally located in the Linked Data cloud. Finally, an assortative mixing calculation over data set triple counts was performed. Given that only $31$ data sets have triple count values, a $31$ vertex subgraph was created. This $31$ vertex graph has $56$ edges. These $56$ edges were used to determine the assortative triple count correlation. Thus, two vectors of length $56$ were created where one vector maintained the triple count of the data sets on the head of each edge and the other vector maintained the triple count of the data sets on the tail of each edge. Table 8 provides three assortativity correlations. Note that the triple count data distribution is non-parametric. From these results, only the non-parametric Kendall correlation is statistically significant with a correlation that demonstrates that the data sets are loosely disassortative according. This means that small data sets tend to connect to large data sets and large data sets tend to connect to small data sets. Again, this correlation is relatively weak. method \| size assortativity ---\|--- pearson \| 0.0682 (0.3230) spearman \| -0.2546 (0.0559) kendall \| -0.2064 (0.0302) Table 8: Data set triple count assortativity correlations for the Linked Data cloud graph. Given that only $31$ data sets have published triple counts, these assortativity values are determined according to this $31$ data set subgraph. The first number is the correlation and the second number in parentheses is the $p$-value. A significant $p$-value is less than $0.05$. ### IV.3 Data Set Centrality The PageRank centrality (with $\delta=0.85$) of each of the $86$ data sets in the Linked Data cloud graph was calculated. Table 9 provides the top $15$ central data sets. From this analysis, and assuming that centrality denotes “importance”, it appears that content in computer science and biology are of major import to the current instantiation of the Linked Data cloud. data set \| page rank \| content label ---\|---\|--- DBLP Berlin \| 0.0484 \| computer science DBLP Hannover \| 0.0464 \| computer science DBpedia \| 0.0384 \| general KEGG \| 0.0370 \| biology UniProt \| 0.0357 \| biology GeneID \| 0.0346 \| biology DBLP RKB Explorer \| 0.0341 \| computer science GeoNames \| 0.0294 \| location ACM \| 0.0257 \| computer science Pfam \| 0.0254 \| biology Prosite \| 0.0233 \| biology ePrints \| 0.0218 \| computer science CiteSeer \| 0.0218 \| computer science PDB \| 0.0209 \| biology Table 9: The top 15 PageRank central data sets in the Linked Data cloud graph. ## V Conclusion The Linked Data initiative is focused on unifying RDF data sets into a single global data set that can be utilized by both man and machine. This initiative is providing a fundamental shift in the way in which data is maintained, exposed, and interrelated. This shift is both technologically and culturally different from the relational database paradigm. For one, the address space of the Web of Data is the URI address space, which is inherently distributed and infinite. Second, the graph data structure is becoming a more accepted, flexible representational medium and as such, may soon displace the linked table data structure of the relational database model. Finally, with respects to culture, the Web of Data maintains publicly available interrelated data. In the relational database world, rarely are database ports made publicly available for harvesting and rarely are relational schemas published for reuse. The Semantic Web, the Linked Data community, and the Web of Data are truly emerging as a radical rethinking of the way in which data is managed and distributed in the modern world. Figure 4: A graph representation of the March 2009 Linked Data Cloud. Each vertex denotes a Linked Data data set. Each edge denotes whether one data set makes reference to another. The size of the vertices are determined by their PageRank centrality according to a $\delta=0.85$ page98pagerank . The vertex colors denote the structural communities as identified by the leading eigenvector community detection algorithm newman-eigen . Finally, the Fruchterman-Reingold layout algorithm was used to visually render this representation layout:fruchter1991 . ## References * (1) T. Berners-Lee, R. Cailliau, A. Luotonen, H. Nielsen, and A. Secret, “The World-Wide Web,” _Communications of the ACM_ , vol. 37, pp. 76–82, 1994\. * (2) A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, and J. Wiener, “Graph structure in the web,” in _Proceedings of the 9th International World Wide Web Conference_ , Amsterdam, Netherlands, May 2000. * (3) B. A. Huberman and L. A. Adamic, “Growth dynamics of the world-wide web,” _Nature_ , vol. 399, 1999. * (4) S. Brin and L. Page, “The anatomy of a large-scale hypertextual web search engine,” _Computer Networks and ISDN Systems_ , vol. 30, no. 1–7, pp. 107–117, 1998. * (5) T. Berners-Lee and J. A. Hendler, “Publishing on the Semantic Web,” _Nature_ , vol. 410, no. 6832, pp. 1023–1024, April 2001. [Online]. Available: http://dx.doi.org/10.1038/35074206 * (6) E. Miller, “An introduction to the Resource Description Framework,” _D-Lib Magazine_ , May 1998. [Online]. Available: http://dx.doi.org/hdl:cnri.dlib/may98-miller * (7) T. Berners-Lee, , R. Fielding, D. Software, L. Masinter, and A. Systems, “Uniform Resource Identifier (URI): Generic Syntax,” January 2005\. * (8) T. Berners-Lee, “Linked data,” World Wide Web Consortium, Tech. Rep., 2006. [Online]. Available: http://www.w3.org/DesignIssues/LinkedData.html * (9) C. Bizer, T. Heath, K. Idehen, and T. Berners-Lee, “Linked data on the web,” in _Proceedings of the International World Wide Web Conference_ , ser. Linked Data Workshop, Beijing, China, April 2008. * (10) M. A. Rodriguez, “A distributed process infrastructure for a distributed data structure,” _Semantic Web and Information Systems Bulletin_ , 2008. [Online]. Available: http://arxiv.org/abs/0807.3908 * (11) E. Oren, B. Heitmann, and S. Decker, “ActiveRDF: Embedding semantic web data into object-oriented languages,” _Web Semantics: Science, Services and Agents on the World Wide Web_ , vol. 6, no. 3, pp. 191–202, 2008. * (12) M. A. Rodriguez, _Emergent Web Intelligence_. Berlin, DE: Springer-Verlag, 2008, ch. General-Purpose Computing on a Semantic Network Substrate. [Online]. Available: http://arxiv.org/abs/0704.3395 * (13) B. Aleman-Meza, C. Halaschek-Wiener, I. B. Arpinar, C. Ramakrishnan, and A. P. Sheth, “Ranking complex relationships on the semantic web,” _IEEE Internet Computing_ , vol. 9, no. 3, pp. 37–44, 2005. * (14) M. A. Rodriguez, “Grammar-based random walkers in semantic networks,” _Knowledge-Based Systems_ , vol. 21, no. 7, pp. 727–739, 2008. [Online]. Available: http://arxiv.org/abs/0803.4355 * (15) I. Foster and C. Kesselman, _The Grid_. Morgan Kaufmann, 2004. * (16) M. Newman, “Assortative mixing in networks,” _Physical Review Letters_ , vol. 89, no. 20, 2002. * (17) M. A. Rodriguez and A. Pepe, “On the relationship between the structural and socioacademic communities of an interdisciplinary coauthorship network,” _Journal of Informetrics_ , vol. 2, no. 3, pp. 195–201, July 2008. [Online]. Available: http://arxiv.org/abs/0801.2345 * (18) M. E. J. Newman, “Finding community structure in networks using the eigenvectors of matrices,” _Physical Review E_ , vol. 74, May 2006. [Online]. Available: http://arxiv.org/abs/physics/0605087 * (19) P. Pons and M. Latapy, “Computing communities in large networks using random walks,” _Journal of Graph Algorithms and Applications_ , vol. 10, no. 2, 2006. * (20) M. Girvan and M. E. J. Newman, “Community structure in social and biological networks,” _Proceedings of the National Academy of Sciences_ , vol. 99, p. 7821, 2002. * (21) J. Reichardt and S. Bornholdt, “Statistical mechanics of community detection,” _Physical Review E_ , vol. 74, no. 016110, 2006. [Online]. Available: http://arxiv.org/abs/cond-mat/0603718 * (22) L. Page, S. Brin, R. Motwani, and T. Winograd, “The PageRank citation ranking: Bringing order to the web,” Stanford Digital Library Technologies Project, Tech. Rep., 1998. * (23) T. Fruchterman and E. Reingold, “Graph drawing by force-directed placement,” _Software Practice and Experience_ , vol. 21, no. 11, pp. 1129–1164, 1991\.	arxiv-papers	2009-03-02T04:47:28	2024-09-04T02:49:00.926679	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Marko A. Rodriguez", "submitter": "Marko A. Rodriguez", "url": "https://arxiv.org/abs/0903.0194" }
0903.0200	# Faith in the Algorithm, Part 1: Beyond the Turing Test Marko A. Rodriguez Theoretical Division – Center for Non-Linear Studies, Los Alamos National Laboratory, email: marko@lanl.govCenter for Embedded Networked Sensing, University of California, Los Angeles, email: apepe@ucla.edu Alberto Pepe Center for Embedded Networked Sensing, University of California, Los Angeles, email: apepe@ucla.edu ###### Abstract Since the Turing test was first proposed by Alan Turing in 1950, the primary goal of artificial intelligence has been predicated on the ability for computers to imitate human behavior. However, the majority of uses for the computer can be said to fall outside the domain of human abilities and it is exactly outside of this domain where computers have demonstrated their greatest contribution to intelligence. Another goal for artificial intelligence is one that is not predicated on human mimicry, but instead, on human amplification. This article surveys various systems that contribute to the advancement of human and social intelligence. > The alleged short-cut to knowledge, which is faith, is only a short-circuit > destroying the mind. > > – Ayn Rand, “For the New Intellectual” ## 1 INTRODUCTION The path towards artificial intelligence, in terms of mimicking human cognitive functionality, has been long, difficult, and painfully incremental. Bottom-up, state of the art vision systems have only accomplished modeling the functional capabilities of the V1, V2, and V4 regions of the visual cortex [36]. Popular, top-down knowledge representation and reasoning system are still primarily monotonic [28], are only beginning to incorporate and understand the ramifications of common sense knowledge [30], and are predicated on logics that do not appear to model the true “rules” of human thought [41]. Moreover, these object recognition and knowledge representation and reasoning developments are but the fringe of a huge landscape of cognitive faculties that must be simulated to achieve human-type artificial intelligence in its fullest form. For example, other less developed agendas are object relation learning in neurally-plausible substrates [23], novel logic acquisition through experience [42], and associative mechanisms for merging the categorizations from different sensory modalities into a single language of thought [15, 19]. The sub-symbolic agenda of artificial intelligence attempts to model the lowest common denominator of the human neural system in order to achieve higher levels of intelligence through experience and learning. Modeling the processing capabilities of individual neurons has been the aim of the connectionist agenda for nearly three decades [35] and beyond various advances in classification, it appears that human type intelligence is still many more decades away. In the area of symbolic artificial intelligence, there have been many developments utilizing computers to solve very specific problems very well, but unfortunately, many of these systems do not have the general, flexible intelligence enjoyed by humans. These statements serve not to criticize the researchers or their methods; rather, they are presented in order to acknowledge the level of difficulty involved in simulating human-type intelligence and the distances that need to be reached if this goal is to be achieved. Is it possible that computers, and their underlying foundation in bivalent logic, centralized processing, and disembodiment, are blinding us as architects and engineers by biasing our approach [9]? Of course, this does not mean that it is impossible to model human intelligence on a computer (assuming that such intelligence can be modeled on a Turing complete system). Instead, it is more a statement that the Turing test [39] – the test for computer intelligence by means of human mimicry – is not a “natural” test of the computer’s abilities in the area of intelligence. Moreover, human mimicry is not a “natural” application of the computer’s abilities. There are many tests that are used to quantify human intelligence. Interestingly, in the mean, a human subject’s scores in all of these tests have a positive correlation. Thus, regardless if a specialist is testing a subject’s ability to manipulate objects in 3D space or the subject’s fluency with language, success in one of these tests is a predictor of success in another. This finding points to a single factor that can account for intelligence. This factor is known as the $g$-factor (or general intelligence factor) [38]. However, any test for intelligence ultimately makes assumptions about the sense modalities through which the test will be administered as well as assumptions about the cultural and common knowledge of the subject. A major trend in intelligence test research is to make intelligence tests devoid of any cultural biases and one day, it may be possible to yield tests that are devoid of any species and modality biases. Species agnostic intelligence tests could be used to measure the intelligence exposed at the level of the human/computer as the autonomous, intelligent entity. Moreover, the degree of intelligence may be greater than what is possible given the human or computer alone [10]. This is because the computer demonstrates unmatched skills in very specific areas such as quickly computing the distance between large vectors of numbers or in maintaining a lossless representation of a presented image in memory. Such skills and their relationship and integration with the skills of the human will continue to yield an advanced degree of real-world intelligence. It is the central thesis of this article that this contribution to intelligence appears to be a more “natural” fit for the computer. This article reviews various systems that, when in combination with humans, yield advanced intelligence – an intelligence that is different than that which can be exposed by the Turing test. ## 2 HUMAN AND SOCIAL AUGMENTATION Computers – the machines and their implemented algorithms – should not simply be interpreted as technological embodiments of solutions to specific problems. There is a larger relationship between the human, their problems and requirements, and designed algorithms and their executing hardware. They are solving larger problems than either the human or the computer could solve alone; in other words, the computer is a contributing component within a larger intelligent system [21]. Sherry Turkle discusses the relationship between humans and computers as not just one in which the computer is a tool used to accomplish human tasks, but one where it is a component that works within the human’s everyday life as a supporting entity [40]. From a “society of minds” perspective [29], the computer, as a cognitive component in human thinking, is very much a well functioning digital information processor much like the hippocampus is a well functioning neural memory device. In other words, the computer has found, not in any affective directed way, an information processing niche that further augments the human much like any other component of the human neural system [37]. To say whether the hippocampus is intelligent or not is to determine whether the results of its processing affect intelligent behavior; that is, does the human know where they are in physical space and do they encode episodic memories correctly? As an autonomous entity, the hippocampus, would appear, to the external human observer, as not being intelligent at all. For one, in isolation, it simply becomes infected and its cells quickly die. However, within the larger schema of the human organism, its role is of great significance to human intelligence. A few minutes interaction with the patient H.M. makes this point obvious [11]. Next, looking at the striate cortex demonstrates a relatively simple system [22] that implements a relatively simple algorithm (albeit on a massive scale) [36]; however, when integrated within the nervous system as a whole, the contribution of the striate cortex to the overall intelligence of the human is immense. Without it, vision, and its associated functionalities, would not be possible. For instance, there would be no notion of a genius painter and the level of intelligence that such a connotation denotes. To this end, how many neural components are required before it is assumed that a human is intelligent? A review of the life and times of Helen Keller should demonstrate how vacuous this question is [26]. Also, like the neural component within the larger system of the human, any other processing component can be utilized in this contribution to intelligence. As such, the measurement of intelligence need not be considered as testing that which is within the confines of the human skin. The relationship between the human and the computer in a technologically- driven society unveils a natural symbiosis which is reminiscent of Hutchins’ theory of distributed cognition [24] and to the notions of collective intelligence found in ant and termite populations [17, 7]. Some of the tasks in which computers are employed in everyday life – from information access to social interaction – make this symbiosis evident. In many respects, traditional, standardized tests of human intelligence test the emergent behavior of the coordinated activity of the individual’s various brain regions. Introducing the computer into this system simply augments or extends the intelligent capabilities of the individual human. It is no accident that this symbiosis has emerged. The computer and its associated algorithms is a needed augmentation to the human given the number of options available in the technologically-rich world and the difficulties in finding one’s global optima within it. Moreover, society, in a collaborative fashion amongst its constituents and its supporting digital infrastructure, is making and will continue to make advances in the area of social intelligence. In this light, the question at hand is: what is the computer’s contribution to intelligence? In order to address this question, the following section explores the emergence of advanced individual and social intelligence within the scope of the technological innovation that has most contributed to this type of augmentation in recent times: the World Wide Web. ## 3 EMERGENT WEB INTELLIGENCE Since the dawn of the World Wide Web, information has been codified and distributed within a shared, universal medium that is accessible by human users world wide. The World Wide Web is unique for two reasons: distribution and standardization. In many respects, the first can not be accomplished without the latter. The Web’s most eminent standard, the Uniform Resource Identifier (URI) has made it possible for the Web to serve as a network of information, from the document to the datum – a shared, global data structure [3]. This distributed data structure is amplifying the intelligence of the individual human and may provide a greater social intelligence. The remainder of this section will address the amplification of intelligence in the context of three general Web system: search engines (index and ranking), recommendation engines (personalized recommendations), and governance engines (collective decision making) [43]. ### 3.1 Search Engines The World Wide Web has emerged as a massive information repository from which humans contribute and consume information. This has not only provided a simple means of retrieving information, but also a simple way to publish and distribute information, thus leading to the increase in human information production. However, information increase inevitably brings about discoverability issues, as the necessity to locate and filter desired information arises. To deal with this problem, algorithms have been developed to augment the individual’s search capabilities. Interestingly, this augmentation is currently predicated on the contribution of many individuals within the stigmergetic environment of the World Wide Web. The early Web maintained rudimentary indexes in the form of Web “yellow pages” that provided short descriptions of web pages. With the explosive growth of the Web, such directory services fell by the wayside as no human operator (or operators) could keep up with the amount of information being published, nor could such rudimentary lists provide the end user a representation of the quality of web pages. By a nearly-Darwinian selection process, these early forms of indexes fell out of use because they were built around a conceptual framework that did not take advantage of the distributed representation of value inherent in every linking webpage made explicit by their authors. As a remedy to this situation, a commercialized Web industry was born and continues to thrive around solving the problem of search. Search engines index massive amounts of data that are gleaned from Web servers world wide. The development of the simple mechanism of ranking web pages by means of their eigenvector component within the web citation graph has proved the most successful to date [8]. It is remarkable that this mechanism is predicated on humans’ decisions to link webpages; that is, the algorithm leverages human interaction with the Web and vice versa in a symbiotic manner. Even more remarkable is the fact that with the approximately 30 billion web pages in existence today, Web users can rest assured that, for the most part, their keyword search will provide the answer to their question within the first few results returned. This level of speed and accuracy of knowledge acquisition was not possible prior to the development of the Web, mainly because the problem of massive-scale indexing and ranking did not make itself apparent until the Web. This problem is solved through the unification of the human’s ability to, in a decentralized fashion, denote the value (or quality) of web pages and the computer’s ability to calculate a global rank over these explicit expressions of value. In this scenario, the Web plays the role of a digital Rolodex providing the human, nearly instantly, a reference to further information on nearly any topic imaginable [14]. Prior to the written document, information was passed from generation to generation in the form of large memorized stories and poems. In the contemporary technologically-rich world, this “algorithm” (cultural process) is no longer necessary. This is not to say that an individual can no longer memorize a long poem if they wish. It is more that a new algorithm has emerged to handle this information indexing requirement and as such, cognitive resources can be appropriated to other tasks. However, the Web is not a large story or poem: it follows no plot, no linear sequence, no poetic meter, no single language – the list of characters is beyond count and no one writing style can be identified. For these reasons, it is posited that no currently existing neural component can memorize, index, and rank the entire Web, and thus, a specialized intelligence is required and, fortunately, has emerged. ### 3.2 Recommendation Engines Large-scale human generated data sets have opened a terrain for numerous algorithms that support individual decision making. Such data sets include the implicit valuation of resources that users leave on the web as they click from web page to web page or from purchased item to purchased item. No individual ever sees the entire Web and for the most part, for the life of the individual, they are confined to a small subset of the greater Web. However, the aggregation of this click-stream information from all individuals provides a collectively generated representation of the inherent relationship between all items on the Web. This collective digital footprint provides not only novel ways to rank resources [5] but also, novel ways to recommend resources [6]. Finally, humans are also developing rich profiles of themselves that include not only identifiable facts such as one’s curriculum vitae, but also the more qualitative aspects of their personality, tastes, and ever changing mood. There are many systems that take advantage of such data sets such as the recommendation engine. A recommendation engine can be defined as any algorithm that provides users with resources (e.g. documents, books, music, movies, life partners, etc.) that are more likely than not to be correlated to the users’ current requirements. The popular collaborative filtering algorithms of document and music services are able to utilize the previous click behavior of an individual to systematically compare it with the click behaviors of others, and from this comparison, recommend a set of resources that will be of interest to the user [20]. For many, the dependency on the librarian and the record shop owner has shifted to a dependence on the community as a whole that is leaving this massive digital footprint. An interesting phenomena to arise in recent years is the development and use of online dating services. In any large city, there are too many individuals for any one human to sift through. Moreover, even if an individual were able to meet everyone, the abilities of the individual may not be keen enough to predict, with any great accuracy, whether or not the person they are meeting will make an optimal partner. For this reason, dating services have emerged to handle, or rather attempt to handle, this common, pervasive problem. Ignoring broader social and cultural considerations for a moment, from a purely statistical perspective, the human’s trial and error methods of sampling small portions of the population through friends or in social, physical environments (bars, restaurants, cafes, etc.) can not compete with the success rates of modern day matchmaking algorithms [2]. Note that matchmaking services are not confined solely to the Web. Newspapers provide “personals” sections, but like the early “yellow pages” of the Web, they can not maintain rich profiles, nor does manually browsing this information compare with the success of a matchmaking algorithm’s recommendation. Again, for those activities for which a human simply does not have the skills to succeed, the human relies on an external augmentation to fulfill the intelligence requirements of the problem at hand. Recommendation services are following a common trend: they are all building more sophisticated models of both humans and resources. The World Wide Web infrastructure has provided the avenues for humans to collectively aggregate in a shared virtual space. Unfortunately, for the most part, the traffic data that is being generated as individuals move from site to site, the profiles that individuals repeatedly create at every online service, and the metadata about the resources that these services index are isolated within the data repositories of the services that utilize this information directly. Fortunately, recent developments in an open data model known as the “web of data” may change this by unifying the information contained in service repositories and exposing, within the shared, global URI address space, every minutia of data [4]. The end benefit of this shift in the perception of ownership and exposure of data will allow for a new generation of algorithms that take advantage of an even richer world model [27, 32]. Such models will include a seamless integration of the individual’s reading, listening, dating, working, etc. behaviors as well as the descriptions of books, songs, movies, people, jobs, etc. At this point, to the algorithms that leverage such data, a human is no longer just a consumer of a particular type of literature or a connoisseur of a particular style of film, but rather, a complex entity that can be subtly oriented, through recommendation, in a direction that ensures that they are experiencing that aspect of the world that is most fitting to who they are. At the extreme of this line of thought, if enough information is gathered and a rich enough world model is generated, then it may be possible to design algorithms that are more fit to determine the life course of an individual human than what the individual, their family, or their community can do for them. This assumes appropriate feedback from the world to the model [16], which may include the perspectives of the individual, their family, and their community. This view suggests that it may be best to rely on a large-scale world model (and algorithms that can efficiently process it) when making decisions about one’s path in life. Such algorithms can take into account the multitude of relations between humans and resources, and improvise a well “thought out” plan of action that ensures that the individual, to the best of the system’s ability, lives a life that is filled with optimal experiences. This is a life in which the others they meet, the restaurants they frequent, the books they read, the classes they attend, and so forth lead to experiences that are completely fulfilling to them as a human. These optimal experiences represent the perfect balance between the psychological states of anxiety and boredom and as such, would increase the individuals’ attentiveness and involvement in such activities – similar to the mental state that is colloquially known as “flow” [12]. Moreover, this state of human experience has been articulated since the times of Aristotle and his notion of the eudaemonic living which arises when one consistently chooses correctly in their life [1]. A large-scale world model has the potential to integrate the collective zeitgeist of a society, the socio-demographic and geographic layouts of cities, the location of its inhabitants, their personal characteristics, their resources and relations. Amazingly, such data currently exist in one form or another, to varying degrees of accuracy, completeness, and levels of access. Further making this information publicly available and integrated would allow for algorithms to evolve, over iterations of development and insight, that are fit to determine the individuals’ global optima. ### 3.3 Governance Engines In many ways, aiding the human in finding global optima is the purpose of a society (within the constraints of taking into account the optima of others) [31]. From high-level governmental decisions to the local cultural rules that determine the way in which humans interact in their environment, the goal of a (benevolent) society is to ensure a life in “the pursuit of happiness” [25]. However, can a society be structured such that the individuals need not pursue, but instead be guaranteed a life full of happiness – or eudaemonia and optimal experiences? The question is then: what are the limits of individual intelligence that can be achieved by the current societal structures alone? And also: are there more efficient and accurate algorithms that can be utilized? Recommendation systems are a step in the direction towards the use of computers to provide the human the right resource at the right time, regardless of what form that resource may take. However, within the grander scheme of society as a whole, the nascent fields of e-governance and computational social choice theory are only beginning to tangentially touch upon the idea that a networked computer infrastructure could be used to foster a new structure for government that is optimized for societal-scale problem- solving. Reflecting on modern voting mechanisms (specifically those within the United States), we find a system that is fragile, inaccurate, and expensive to maintain. Due in part to the outdated infrastructure that citizens use to communicate with their governing body, citizen participation in government decision making is limited. However, these days, with the level of eduction that citizens have, the amount of information that citizens can become aware of, and the sophistication of modern network technologies, it is possible that current government decisions are limited in that they are not leveraging the full potential of an enlightened population (or subset thereof). By making use of both a large-scale and knowledgeable decision making constituency, it is theoretically possible that all rendered decisions are optimal. This statement was validated (under certain simple assumptions) in 1785 by Marquis de Condorcet’s now famous Condorcet jury theorem [13]. With the social networks that are being made explicit on the Web today, and with open standard movements that ensure that this information can be shared across services, it is possible to leverage a relatively simple vote distribution mechanism to remove the representative layers of government and promote full citizen participation in all the decision making affairs of a society. This mechanism, known as dynamically distributed democracy, ensures that any actively participating subset of a population simulates the decision making behavior of the whole [33]. Thus, a simulated, large-scale decision making body can be leveraged in all decisions. A large decision making body is the first requirement of the Condorcet jury theorem. Robin Hanson articulates a vision of government where any individual can participate through a decision system known as a prediction market [18]. The purpose of a prediction market is to provide accurate predictions of objectively determinable states of the world (current or into the future) and its application to governance is noted in the popular phrase “vote on values, but bet on beliefs.” In this form, the self-selecting, monetary mechanisms that determine whether someone participates is based on their degree of knowledge of the problem space. Those that are not knowledgeable, either do not participate or lose money in the process of participating, thus, hampering the individual from participating in matters outside the scope of their abilities into the future. The accuracy of such systems are astounding and have popular uses in election predictions and a short lived run in terrorist predictions (only to be dismantled by the U.S. government because it was considered too morose for market traders to monetarily benefit on the accurate prediction of the death of others). A knowledgeable decision making body is the second requirement of the Condorcet jury theorem and, much like commodity markets, prediction market systems select for knowledgeable individuals. These ideas stress the importance of reflecting on the medium by which society organizes itself, generates its laws, and implements methods in how it will utilize resources most effectively. Like the “yellow pages” of the early Web, it may not be optimal to leave such pressing matters to an operator (or operators). This statement is not a critique of the leaders and doctrines of nations, but instead is a comment on the complexity of the world and the necessity for a new type of intelligence. It is posed as an appeal to rethink government and its role within contemporary networked society [34]. An implementation of a government should not be valued. Instead, what should be valued is the ideals that that implementation is trying to achieve. Moreover, if another implementation would better meet the ideals of the society, then it should be enacted. A distributed value/belief system and algorithmic aggregation mechanism may prove to be the better problem-solving mechanism for societal issues and may prove to be a better mechanism to orchestrate individual lives. It is in this area that computers can greatly contribute to social intelligence, where the unification of the intelligence augmentation gained by the individual human and the society coalesce into a type of intelligence that is novel (beyond human mimicry) and above all beneficial. ## 4 CONCLUSION Humans perceive their world through their sense modalities, create stable representations of the consistent patterns in the world, and utilize those representations to further act and survive to the best of their abilities. Their internal, subjective world is an endless stream of thoughts – a complex, information-rich map of the external world. Manifestations of intelligence inherently depend upon an individual’s internal representation of the external world and their ability to manipulate that representation. By analogy to the field of computer science, this internal map of the world can be regarded as the data structure upon which reasoning mechanisms (i.e. algorithms) function. From an objective perspective, the human mind can only maintain so rich a data structure, process only so many aspects of it, and simulate only so many potential future paths for the individual to choose from. The complexity of the human’s mental calculation grows when considering that many other such simulations are occurring in the minds of their fellow men and women. Like a general-purpose processor, to simulate a machine within a machine reduces the resources available to the original machine to execute other processes. For these reasons, the human is not a perfectly intelligent creature always doing the right thing at the right time. As discussed, with the externalization of the human’s internal world through the explicit expression of themselves, their relation to others, and the resources on which they rely, other processes can utilize this explicit model to aid the human in the process of thought and thus, life. The World Wide Web and the algorithms implemented upon it function like an auxiliary mind, exposed to more information than could be possibly processed by its neural counterpart. While the core specification of these algorithms may be understood, even thoroughly by their designers, ultimately what machines compute are based on such a large-scale model of the world, that to assimilate its results into one’s choices are ultimately based on faith – much like the faith one has in the validity of their episodic memories and their current location in space as provided to them by their hippocampus. ## References * [1] Aristotle, The Nicomachean Ethics, Oxford University Press, 1998. * [2] Aaron Ben-Ze’ev, Love Online: Emotions on the Internet, Cambridge University Press, 2004. * [3] Tim Berners-Lee and James A. Hendler, ‘Publishing on the Semantic Web’, Nature, 410(6832), 1023–1024, (April 2001). * [4] Christian Bizer, Tom Heath, Kingsley Idehen, and Tim Berners-Lee, ‘Linked data on the web’, in Proceedings of the International World Wide Web Conference, Linked Data Workshop, Beijing, China, (April 2008). * [5] Johan Bollen, Herbert Van de Sompel, and Marko A. Rodriguez, ‘Towards usage-based impact metrics: first results from the MESUR project.’, in Proceedings of the Joint Conference on Digital Libraries, pp. 231–240, New York, NY, (2008). ACM Press. * [6] Johan Bollen, Michael L. Nelson, Gary Geisler, and Raquel Araujo, ‘Usage derived recommendations for a video digital library’, Journal of Network and Computer Applications, 30(3), 1059–1083, (2007). * [7] Eric Bonabeau, Marco Dorigo, and Guy Theraulaz, Swarm Intelligence: From Natural to Artificial Systems, Oxford University Press, New York, NY, 1999. * [8] Sergey Brin and Lawrence Page, ‘The anatomy of a large-scale hypertextual web search engine’, Computer Networks and ISDN Systems, 30(1–7), 107–117, (1998). * [9] Andy Clark, Being There: Putting Brain, Body and World Together Again, MIT Press, 1997. * [10] Andy Clark, Supersizing the Mind: Embodiment, Action, and Cognitive Extension, Oxford University Press, 2008. * [11] Neal J. Cohen, Memory, Amnesia, and the Hippocampal System, MIT Press, September 1995. * [12] Mihály Csíkszentmihályi, Flow: The Psychology of Optimal Experience, Harper and Row, New York, NY, 1990. * [13] Marquis de Condorcet. Essai sur l’application de l’analyse á la probabilité des décisions rendues á la pluralité des voix, 1785. * [14] Douglas C. Engelbart, Computer-supported cooperative work: a book of readings, chapter A conceptual framework for the augmentation of man’s intellect, 35–65, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1988\. * [15] Jerry Fodor, The Language of Thought, Harvard University Press, 1975. * [16] Vadas Gintautas and Alfred W. Hübler, ‘Experimental evidence for mixed reality states in an interreality system’, Physical Review E, 75, 057201, (2007). * [17] P. Grasse, ‘La reconstruction du nid et les coordinations inter-individuelles chez bellicositermes natalis et cubitermes sp. la theorie de la stigmergie’, Insectes Sociaux, 6, 41–83, (1959). * [18] Robin Hanson, ‘Shall we vote on values, but bet on beliefs?’, Journal of Political Philosophy, (in press). * [19] Jeff Hawkins and Sandra Blakeslee, On Intelligence, Holt, 2005. * [20] Johnathan L. Herlocker, Joseph A. Konstan, Loren G. Terveen, and John T. Riedl, ‘Evaluating collaborative filtering recommender systems’, ACM Transactions on Information Systems, 22(1), 5–53, (2004). * [21] Francis Heylighen, ‘The global superorganism: an evolutionary-cybernetic model of the emerging network society’, Social Evolution and History, 6(1), 58–119, (2007). * [22] D. H. Hubel and T. N. Wiesel, ‘Receptive fields and functional architecture of monkey striate cortex.’, Journal of Physiology, 195(1), 215–243, (March 1968). * [23] J.E. Hummel and K.J. Holyoak, ‘A symbolic-connectionist theory of relational inference and generalization’, Psychological Review, 110(2), 220–264, (2003). * [24] Edwin Hutchins, Cognition in the Wild, MIT Press, September 1995. * [25] Thomas Jefferson. Declaration of independence, 1776. * [26] Helen Keller, The Story of My Life, Doubleday, Page and Company, New York, NY, 1905. * [27] Lawrence Lessig, Free Culture: The Nature and Future of Creativity, CreateSpace, Paramount, CA, 2008. * [28] Deborah L. McGuinness and Frank van Harmelen. OWL web ontology language overview, February 2004. * [29] Marvin Minsky, The Society of Mind, Simon and Schuster, March 1988. * [30] Erik T. Mueller, Commonsense Reasoning, Morgan Kaufmann, January 2006. * [31] David L. Norton, Democracy and Moral Development: A Politics of Virtue, University of California Press, 1995. * [32] Marko A. Rodriguez, ‘A distributed process infrastructure for a distributed data structure’, Semantic Web and Information Systems Bulletin, (2008). * [33] Marko A. Rodriguez and Daniel J. Steinbock, ‘A social network for societal-scale decision-making systems’, in Proceedingss of the North American Association for Computational Social and Organizational Science Conference, Pittsburgh, PA, (2004). * [34] Marko A. Rodriguez and Jennifer H. Watkins, ‘Revisiting the age of enlightenment from a collective decision making systems perspective’, Technical Report LA-UR-09-00324, Los Alamos National Laboratory, (January 2009). * [35] David E. Rumelhart and James L. McClelland, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, MIT Press, July 1993. * [36] Thomas Serre, Aude Oliva, and Tomaso Poggio, ‘A feedforward architecture accounts for rapid categorization’, Proceedings of the National Academy of Science, 104(15), 6424–6429, (April 2007). * [37] Peter Skagestad, ‘Thinking with machines: Intelligence augmentation, evolutionary epistemology, and semiotic’, Journal of Social and Evolutionary Systems, 16(2), 157–180, (1993). * [38] Charles Spearman, ‘General intelligence objectively determined and measured’, American Journal of Psychology, 15, 201–293, (1904). * [39] Alan M. Turing, ‘Computing machinery and intelligence’, Mind, 58(236), 433–460, (1950). * [40] Sherry Turkle, The Second Self: Computers and the Human Spirit, MIT Press, 1984. * [41] Pei Wang, ‘Cognitive logic versus mathematical logic’, in Proceedings of the Third International Seminar on Logic and Cognition, (May 2004). * [42] Pei Wang, Rigid Flexibility, Springer, 2006. * [43] Jennifer H. Watkins and Marko A. Rodriguez, Evolution of the Web in Artificial Intelligence Environments, chapter A Survey of Web-Based Collective Decision Making Systems, 245–279, Studies in Computational Intelligence, Springer-Verlag, Berlin, DE, 2008.	arxiv-papers	2009-03-02T02:01:40	2024-09-04T02:49:00.931385	{ "license": "Public Domain", "authors": "Marko A. Rodriguez and Alberto Pepe", "submitter": "Marko A. Rodriguez", "url": "https://arxiv.org/abs/0903.0200" }
0903.0250	# Hawking black body spectrum from tunneling mechanism Rabin Banerjee, Bibhas Ranjan Majhi S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata-700098, India E-mail: rabin@bose.res.inE-mail: bibhas@bose.res.in ###### Abstract We obtain, using a reformulation of the tunneling mechanism, the Hawking black body spectrum with the appropriate temperature for a black hole. This is a new result in the tunneling formalism of discussing Hawking effect. Our results are given for a spherically symmetric geometry that is asymptotically flat. Introduction: After Hawking’s observation [1] that black holes radiate, there were several approaches [2, 3, 4, 5, 6, 7] to study this effect. A particularly intuitive and widely used approach is the tunneling mechanism [4, 5]. The essential idea is that a particle-antiparticle pair forms close to the event horizon which is similar to pair formation in an external electric field. The ingoing mode is trapped inside the horizon while the outgoing mode can quantum mechanically tunnel through the event horizon. It is observed at infinity as a Hawking flux. So this effect is totally a quantum phenomenon and the presence of an event horizon is essential. However, in the literature [4, 5, 8, 9, 10, 11, 12], the analysis is confined to obtention of the Hawking temperature only by comparing the tunneling probability of an outgoing particle with the Boltzmann factor. There is no discussion of the spectrum. Hence it is not clear whether this temperature really corresponds to the temperature of a black body spectrum associated with black holes. One has to take recourse to other results to really justify the fact that the temperature found in the tunneling approach is indeed the Hawking black body temperature. In this sense the tunneling method, presented so far, is incomplete. In this paper we rectify this shortcoming. Using density matrix techniques we will directly find the spectrum from a reformulation of the tunneling mechanism. For both bosons and fermions we obtain a black body spectrum with a temperature that corresponds to the familiar semiclassical Hawking expression. Our results are valid for black holes with spherically symmetric geometry. Finally, we show the connection of our formulation with usual tunneling formulations [4, 5] by exploiting the principle of detailed balance. General formulation: Consider a black hole characterised by a spherically symmetric, static space-time and asymptotically flat metric of the form, $\displaystyle ds^{2}=F(r)dt^{2}-\frac{dr^{2}}{F(r)}-r^{2}d\Omega^{2}$ (1) whose event horizon $r=r_{H}$ is defined by $F(r_{H})=0$. For discussing Hawking effect by tunneling, the radial trajectory is relevant [4, 5]. We therefore consider only the $(r-t)$ sector of the metric (1). Now consider the massless Klein-Gordon equation $g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0$ which, in the ($r-t$) sector, reduces to, $\displaystyle-\frac{1}{F(r)}\partial^{2}_{t}\phi+F^{{}^{\prime}}(r)\partial_{r}\phi+F(r)\partial^{2}_{r}\phi=0$ (2) in the black hole space-time (1). Taking the standard WKB ansatz $\displaystyle\phi(r,t)=e^{-\frac{i}{\hbar}S(r,t)}$ (3) and substituting the expansion for $S(r,t)$ $\displaystyle S(r,t)=S_{0}(r,t)+\sum_{i=1}^{\infty}\hbar^{i}S_{i}(r,t)$ (4) in (2) we obtain, in the semiclassical limit (i.e. $\hbar\rightarrow 0$), $\displaystyle\partial_{t}S_{0}(r,t)=\pm F(r)\partial_{r}S_{0}(r,t)$ (5) This is the usual semiclassical Hamilton-Jacobi equation [4, 9] which can also be obtained in a similar way from Dirac [10] or Maxwell equations [11]. Also, this equation is a natural consequence if the chirality (holomorphic) condition on the scalar field with the WKB ansatz (3) is imposed with the $+(-)$ solutions standing for the left (right) movers [14]. Now since the metric (1) is stationary, it has a timelike Killing vector. Therefore we choose an ansatz for $S_{0}(r,t)$ as $\displaystyle S_{0}(r,t)=\omega t+{\tilde{S}}_{0}(r)$ (6) where $\omega$ is the conserved quantity corresponding to the timelike Killing vector. This is identified as the effective energy experienced by the particle at asymptotic infinity. Substituting this in (5) a solution for ${\tilde{S}}_{0}(r)$ is obtained. Inserting this back in (6) yields, $\displaystyle S_{0}(r,t)=\omega(t\pm r_{});\,\,\,\,r_{}=\int\frac{dr}{F(r)}$ (7) For further discussions it is convenient to introduce the sets of null tortoise coordinates which are defined as, $\displaystyle u=t-r_{},\,\,\,v=t+r_{}.$ (8) It is important to note that expressing (7) in these coordinates, defined inside and outside the event horizon, and then substituting in (3) one can obtain the right and left modes for both sectors: $\displaystyle\Big{(}\phi^{(R)}\Big{)}_{\textrm{in}}=e^{-\frac{i}{\hbar}\omega u_{\textrm{in}}};\,\,\,\Big{(}\phi^{(L)}\Big{)}_{\textrm{in}}=e^{-\frac{i}{\hbar}\omega v_{\textrm{in}}}$ $\displaystyle\Big{(}\phi^{(R)}\Big{)}_{\textrm{out}}=e^{-\frac{i}{\hbar}\omega u_{\textrm{out}}};\,\,\,\Big{(}\phi^{(L)}\Big{)}_{\textrm{out}}=e^{-\frac{i}{\hbar}\omega v_{\textrm{out}}}$ (9) Now in the tunneling formalism a virtual pair of particles is produced in the black hole. One member of this pair can quantum mechanically tunnel through the horizon. This particle is observed at infinity while the other goes towards the center of the black hole. While crossing the horizon the nature of the coordinates changes. This can be accounted by working with Kruskal coordinates which are viable on both sides of the horizon. The Kruskal time ($T$) and space ($X$) coordinates inside and outside the horizon are defined as [13], $\displaystyle T_{\textrm{in}}=e^{K(r_{})_{\textrm{in}}}~{}{\textrm{cosh}}(Kt_{\textrm{in}});\,\,\,X_{\textrm{in}}=e^{K(r_{})_{\textrm{in}}}~{}{\textrm{sinh}}(Kt_{\textrm{in}})$ $\displaystyle T_{\textrm{out}}=e^{K(r_{})_{\textrm{out}}}~{}{\textrm{sinh}}(Kt_{\textrm{out}});\,\,\,X_{\textrm{out}}=e^{K(r_{})_{\textrm{out}}}~{}{\textrm{cosh}}(Kt_{\textrm{out}})$ (10) where, as usual, $K=\frac{F^{\prime}(r_{H})}{2}$ is the surface gravity of the black hole. These two sets of coordinates are connected by the relations, $\displaystyle t_{\textrm{in}}\rightarrow t_{\textrm{out}}-i\frac{\pi}{2K};\,\,\,\,(r_{})_{\textrm{in}}\rightarrow(r_{})_{\textrm{out}}+i\frac{\pi}{2K}$ (11) so that, with this mapping, $T_{\textrm{in}}\rightarrow T_{\textrm{out}}$ and $X_{\textrm{in}}\rightarrow X_{\textrm{out}}$. In particular, for the Schwarzschild metric, $K=\frac{1}{4M}$ so that the extra term connecting $t_{\textrm{in}}$ and $t_{\textrm{out}}$ is given by ($-2\pi iM$). Such a result (for the Schwarzschild case) was earlier discussed in [12]. Now, following the definition (8), we obtain the relations connecting the null coordinates defined inside and outside the horizon, $\displaystyle u_{\textrm{in}}=t_{\textrm{in}}-(r_{})_{\textrm{in}}\rightarrow u_{\textrm{out}}-i\frac{\pi}{K}$ $\displaystyle v_{\textrm{in}}=t_{\textrm{in}}+(r_{})_{\textrm{in}}\rightarrow v_{\textrm{out}}$ (12) Under these transformations the inside and outside modes are connected by, $\displaystyle\Big{(}\phi^{(R)}\Big{)}_{\textrm{in}}\rightarrow e^{-\frac{\pi\omega}{\hbar K}}\Big{(}\phi^{(R)}\Big{)}_{\textrm{out}}$ $\displaystyle\Big{(}\phi^{(L)}\Big{)}_{\textrm{in}}\rightarrow\Big{(}\phi^{(L)}\Big{)}_{\textrm{out}}$ (13) Using the above transformations the density matrix operator for an observer outside the event horizon will be constructed in the next section which will lead to the black body spectrum and thermal flux corresponding to the semiclassical Hawking temperature. Black body spectrum and Hawking flux: Now to find the black body spectrum and Hawking flux, we first consider $n$ number of non-interacting virtual pairs that are created inside the black hole. Each of these pairs is represented by the modes defined in the first set of (9). Then the physical state of the system, observed from outside, is given by, $\displaystyle\|\Psi>=N\sum_{n}\|n^{(L)}_{\textrm{in}}>\otimes\|n^{(R)}_{\textrm{in}}>\rightarrow N\sum_{n}e^{-\frac{\pi n\omega}{\hbar K}}\|n^{(L)}_{\textrm{out}}>\otimes\|n^{(R)}_{\textrm{out}}>$ (14) where use has been made of the transformations (13). Here $\|n^{(L)}_{\textrm{out}}>$ corresponds to $n$ number of left going modes and so on while $N$ is a normalization constant which can be determined by using the normalization condition $<\Psi\|\Psi>=1$. This immediately yields, $\displaystyle N=\frac{1}{\Big{(}\displaystyle\sum_{n}e^{-\frac{2\pi n\omega}{\hbar K}}\Big{)}^{\frac{1}{2}}}$ (15) The above sum will be calculated for both bosons and fermions. For bosons $n=0,1,2,3,....$ whereas for fermions $n=0,1$. With these values of $n$ we obtain the normalization constant (15) as $\displaystyle N_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}^{\frac{1}{2}}$ (16) $\displaystyle N_{(\textrm{fermion})}=\Big{(}1+e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}^{-\frac{1}{2}}$ (17) Therefore the normalized physical states of the system for bosons and fermions are respectively, $\displaystyle\|\Psi>_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}^{\frac{1}{2}}\sum_{n}e^{-\frac{\pi n\omega}{\hbar K}}\|n^{(L)}_{\textrm{out}}>\otimes\|n^{(R)}_{\textrm{out}}>$ (18) $\displaystyle\|\Psi>_{(\textrm{fermion})}=\Big{(}1+e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}^{-\frac{1}{2}}\sum_{n}e^{-\frac{\pi n\omega}{\hbar K}}\|n^{(L)}_{\textrm{out}}>\otimes\|n^{(R)}_{\textrm{out}}>$ (19) From here on our analysis will be only for bosons since for fermions the analysis is identical. For bosons the density matrix operator of the system is given by, $\displaystyle{\hat{\rho}}_{(\textrm{boson})}$ $\displaystyle=$ $\displaystyle\|\Psi>_{(\textrm{boson})}<\Psi\|_{(\textrm{boson})}$ (20) $\displaystyle=$ $\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}\sum_{n,m}e^{-\frac{\pi n\omega}{\hbar K}}e^{-\frac{\pi m\omega}{\hbar K}}\|n^{(L)}_{\textrm{out}}>\otimes\|n^{(R)}_{\textrm{out}}><m^{(R)}_{\textrm{out}}\|\otimes<m^{(L)}_{\textrm{out}}\|$ Now tracing out the ingoing (left) modes we obtain the density matrix for the outgoing modes, $\displaystyle{\hat{\rho}}^{(R)}_{(\textrm{boson})}=\Big{(}1-e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}\sum_{n}e^{-\frac{2\pi n\omega}{\hbar K}}\|n^{(R)}_{\textrm{out}}><n^{(R)}_{\textrm{out}}\|$ (21) Therefore the average number of particles detected at asymptotic infinity is given by, $\displaystyle<n>_{(\textrm{boson})}={\textrm{trace}}({\hat{n}}{\hat{\rho}}^{(R)}_{(\textrm{boson})})$ $\displaystyle=$ $\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}\sum_{n}ne^{-\frac{2\pi n\omega}{\hbar K}}$ (22) $\displaystyle=$ $\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}(-\frac{\hbar K}{2\pi})\frac{\partial}{\partial\omega}\Big{(}\sum_{n}e^{-\frac{2\pi n\omega}{\hbar K}}\Big{)}$ $\displaystyle=$ $\displaystyle\Big{(}1-e^{-\frac{2\pi\omega}{\hbar K}}\Big{)}(-\frac{\hbar K}{2\pi})\frac{\partial}{\partial\omega}\Big{(}\frac{1}{1-e^{-\frac{2\pi\omega}{\hbar K}}}\Big{)}$ $\displaystyle=$ $\displaystyle\frac{1}{e^{\frac{2\pi\omega}{\hbar K}}-1}$ where the trace is taken over all $\|n^{(R)}_{\textrm{out}}>$ eigenstates. This is the Bose distribution. Similar analysis for fermions leads to the Fermi distribution: $\displaystyle<n>_{(\textrm{fermion})}=\frac{1}{e^{\frac{2\pi\omega}{\hbar K}}+1}$ (23) Note that both these distributions correspond to a black body spectrum with a temperature given by the Hawking expression, $\displaystyle T_{H}=\frac{\hbar K}{2\pi}$ (24) Correspondingly, the Hawking flux can be obtained by integrating the above distribution functions over all $\omega$’s. For fermions it is given by, $\displaystyle{\textrm{Flux}}=\frac{1}{\pi}\int_{0}^{\infty}\frac{\omega~{}d\omega}{e^{\frac{2\pi\omega}{\hbar K}}+1}=\frac{\hbar^{2}K^{2}}{48\pi}$ (25) Similarly, the Hawking flux for bosons can be calculated, leading to the same answer. Connection with usual approaches: For completeness and for revealing the connection with usual approaches [4, 5, 8] to the tunneling formalism we will show below how one can find only the Hawking temperature using the principle of detailed balance. Since the left moving mode travels towards the center of the black hole, its probability to go inside, as measured by an external observer, is expected to be unity. This is easily seen by computing, $\displaystyle P^{(L)}=\|\phi^{(L)}_{\textrm{in}}\|^{2}\rightarrow\|\phi^{(L)}_{\textrm{out}}\|^{2}=1$ (26) where we have used (13) to recast $\Big{(}\phi^{(L)}\Big{)}_{\textrm{in}}$ in terms of $\Big{(}\phi^{(L)}\Big{)}_{\textrm{out}}$ since measurements are done by an outside observer. This shows that the left moving (ingoing) mode is trapped inside the black hole, as expected. On the other hand the right moving mode ($\phi^{(R)}_{\textrm{in}}$) tunnels through the event horizon. So to calculate the tunneling probability as seen by an external observer one has to use the transformation (13) to recast $\Big{(}\phi^{(R)}\Big{)}_{\textrm{in}}$ in terms of $\Big{(}\phi^{(R)}\Big{)}_{\textrm{out}}$. Then we find, $\displaystyle P^{(R)}=\|\phi^{(R)}_{\textrm{in}}\|^{2}\rightarrow\|e^{-\frac{\pi\omega}{\hbar K}}\Big{(}\phi^{(R)}\Big{)}_{\textrm{out}}\|^{2}=e^{-\frac{2\pi\omega}{\hbar K}}$ (27) Finally, using the principle of “detailed balance” [4, 9], $P^{(R)}=e^{-\frac{\omega}{T_{H}}}P^{(L)}=e^{-\frac{\omega}{T_{H}}}$ and comparison with (27) immediately reproduces the Hawking temperature (24). Conclusions: To conclude, we have provided a novel formulation of the tunneling formalism to highlight the role of coordinate systems. A particular feature of this reformulation is that explicit treatment of the singularity in (7) is not required since we do not carry out the complex path integration. Of course, the singularity at the event horizon is manifested in the transformations (11). In this way our formalism, contrary to the traditional approaches [4, 5, 8], avoids explicit complex path analysis. It is implicit only in the definition (7). Computations were done in terms of the basic modes. From the density matrix constructed from these modes we were able to directly reproduce the black body spectrum, for either bosons or fermions, from a black hole with a temperature corresponding to the standard Hawking expression. We feel that the lack of such an analysis was a gap in the existing tunneling formulations [4, 5, 8, 9, 10, 11, 12, 14] which yield only the temperature rather that the actual black body spectrum. Finally, the connection of our approach with these existing formulations was revealed through the use of the detailed balance principle. ## References * [1] S.W.Hawking, Commun. Math. Phys. 43, 199 (1975). * [2] G.W.Gibbons and S.W.Hawking, Phys. Rev. D 15, 2752 (1977). * [3] S.M.Christensen and S.A.Fulling, Phys. Rev. D 15, 2088 (1977). * [4] K.Srinivasan and T.Padmanabhan, Phys. Rev. D 60, 024007 (1999) [arXiv:gr-qc/9812028]. * [5] M.K.Parikh and F.Wilczek, Phys. Rev. Lett. 85, 5042 (2000) [arXiv:hep-th/9907001]. * [6] S.P.Robinson and F.Wilczek, Phys. Rev. Lett. 95, 011303 (2005) [arXiv:gr-qc/0502074]. * [7] R.Banerjee and S.Kulkarni, Phys. Rev. D 77, 024018 (2008) [arXiv:0707.2449]. R.Banerjee and S.Kulkarni, Phys. Lett. B 659, 827 (2008) [arXiv:0709.3916]. R.Banerjee, Int. J. Mod. Phys. D 17, 2539 (2009) [arXiv:0807.4637]. * [8] M.Arzano, A.J.M.Medved and E.C.Vagenas, JHEP 0509, 037 (2005) [arXiv:hep-th/0505266]. E.T.Akhmedov, V.A.Akhmedova and D.Singleton, Phys. Lett. B 642, 124 (2006) [arXiv:hep-th/0608098]. M.Angheben, M.Nadalini, L.Vanzo and S.Zerbini, JHEP 0505, 014 (2005) [arXiv:hep-th/0503081]. R.Kerner and R.B.Mann, Phys. Rev. D 73, 104010 (2006) [arXiv:gr-qc/0603019]. P.Mitra, Phys. Lett. B 648, 240 (2007) [arXiv:hep-th/0611265]. R.Banerjee and B.R.Majhi, Phys. Lett. B 662, 62 (2008) [arXiv:0801.0200]. R.Banerjee, B.R.Majhi and S.Samanta, Phys. Rev. D 77, 124035 (2008) [arXiv:0801.3583]. S.K.Modak, Phys. Lett. B 671, 167 (2009) [arXiv:0807.0959]. * [9] R.Banerjee and B.R.Majhi, JHEP 0806, 095 (2008) [arXiv:0805.2220]. R.Banerjee and B.R.Majhi, Phys. Lett. B 674, 218 (2009) [arXiv:0808.3688]. * [10] R.Kerner and R.B.Mann, Class. Quant. Grav. 25, 095014,(2008) [arXiv:0710.0612]. R.Kerner and R.B.Mann, Phys. Lett. B 665, 277 (2008) [arXiv:0803.2246]. R.Criscienzo and L.Vanzo, Europhys. Lett. 82, 60001 (2008) [arXiv:0803.0435]. De-You Chen, Q.Q.Jiang and S.Z.Yang, X.Zu, Class. Quant. Grav. 25, 205022 (2008) [arXiv:0803.3248]. B.R.Majhi, Phys. Rev. D 79, 044005 (2009) [arXiv:0809.1508]. * [11] B.R.Majhi and S.Samanta, [arXiv:0901.2258]. * [12] V.Akhmedova, T.Pilling, A.Gill and D.Singleton, Phys. Lett. B 666, 269 (2008) [arXiv:0804.2289]. E.T.Akhmedov, T.Pilling and D.Singleton, Int. J. Mod. Phys. D 17, 2453 (2009) [arXiv:0805.2653]. V.Akhmedova, T.Pilling, A.Gill and D.Singleton, Phys. Lett. B 673, 227 (2009) [arXiv:0808.3413]. * [13] A.K.Raychaudhuri, S.Banerji and A.Banerjee, “General Relativity, Astrophysics, and Cosmology”, Springer, 2003. * [14] R.Banerjee and B.R.Majhi, Phys. Rev. D 79, 064024 (2009) [arXiv:0812.0497].	arxiv-papers	2009-03-02T10:21:52	2024-09-04T02:49:00.937128	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Rabin Banerjee, Bibhas Ranjan Majhi", "submitter": "Bibhas Majhi Ranjan", "url": "https://arxiv.org/abs/0903.0250" }
0903.0301	$\Delta$ contribution in $e^{+}+e^{-}\rightarrow p+\bar{p}$ at small $s$ Hai Qing Zhou 1**E-mail: zhouhq@mail.ihep.ac.cn, Dian Yong Chen 2 and Yu Bing Dong2,3 1 Department of Physics, Southeast University, Nanjing, 211189, P. R. China 2 Institute of High Energy Physics, Chinese Academy of Science, Beijing, 100049, P. R. China 3 Theoretical Physics Center for Science Facilities, CAS, Beijing 100049, China ###### Abstract Two-photon annihilate contributions in the process $e^{+}+e^{-}\rightarrow p+\bar{p}$ including $N$ and $\Delta$ intermediate are discussed in a simple hadronic model. The corrections to the unpolarized cross section and polarized observables $P_{x},P_{z}$ are presented. The results show the two-photon annihilate correction to unpolarized cross section is small and its angle dependence becomes weak at small $s$ after considering the $N$ and $\Delta(1232)$ contributions simultaneously, while the correction to $P_{z}$ is enhanced. PACS numbers: 13.40.Gp, 13.60.-r, 25.30.-c. Key words: Two-Photon Exchange, Delta, Form Factor ## 1 Introduction The Two-Photon-Exchange(TPE) effect has attracted many interests after its success in explaining the un-consistent measurements of $R=\mu_{p}G_{E}/G_{M}$ from $ep\rightarrow ep$ by Rosenbluth technique and polarized methods[1, 2, 3, 4]. It is found that the TPE corrections play an important role in extracting the proton’s form factors due to its explicit angle dependence. Later some other processes[5, 6, 7] are suggested to measure the TPE like effects. The $e^{+}+e^{-}\rightarrow p+\bar{p}$ is one of such processes and the two-photon annihilate corrections in this process have been discussed by [8] where only the $N$ intermediate was included. The estimate by [8] showed the two-photon annihilate corrections are about a few percent in the magnitude but strongly depend on the hadron production angle. On another hand, the calculation in [9] showed the $\Delta(1232)$ intermediates also unneglectable in the TPE corrections in the simple hadronic model [2, 4, 9]. These researches prompt us to extent the estimate of the two-photon annihilate corrections in [8] to include $\Delta$ intermediate state. In this work, we present such results. ## 2 Two-Photon Annihilate Corrections including $N$ and $\Delta(1232)$ as Intermediate State Considering the process $e^{+}(k_{2})+e^{-}(k_{1})\rightarrow p(p_{2})+\bar{p}(p_{1})$, the Born diagram is showed as Fig.1. The differential cross section for this process at the tree level can be written as[10] $\displaystyle(\frac{d\sigma}{d\Omega})_{CM}=\frac{\alpha^{2}\sqrt{1-4M_{N}^{2}/q^{2}}}{4q^{2}}(\|G_{M}\|^{2}(1+cos^{2}\theta)+\frac{1}{\tau}\|G_{E}\|^{2}sin^{2}\theta).$ (1) where $q=k_{1}+k_{2},\tau=q^{2}/4M_{N}^{2}>1$ and $\theta$ is the angle between the momentum of finial antiproton and initial electron in the center of mass frame. The Sachs form factors have been used as $\displaystyle G_{M}(q^{2})=F_{1}(q^{2})+F_{2}(q^{2}),G_{E}(q^{2})=F_{1}(q^{2})+\tau F_{2}(q^{2}).$ (2) In principle, the form factors at certain $s=q^{2}$ can be extracted from the measurement of the unpolarized differential cross section at different angle. To extract the form factors more precisely, the radiative corrections should be considered. Among the one loop radiative corrections, the box and crossed box diagrams play special role due to their strong angle dependence. This leads us restrict our discussions on the two-photon annihilate correction firstly. Figure 1: One photon annihilating diagram for $e^{+}+e^{-}\rightarrow p+\bar{p}$. Figure 2: Two-photon annihilating diagrams (a) with $N$ as intermediate state,(b) with $\Delta(1232)$ as intermediate state. Corresponding cross-box diagrams are implied. Using the simple hadronic model developed in [2, 4, 9] and including $N$ and $\Delta$ as the intermediate state like Fig.2, the unpolarized cross section can be written as $\displaystyle d\sigma=d\sigma_{0}(1+\delta_{N}^{2\gamma}+\delta_{\Delta}^{2\gamma})\propto\sum\limits_{helicity}{\|\mathcal{M}_{0}+\mathcal{M}^{2\gamma}_{N}+\mathcal{M}_{\Delta}^{2\gamma}\|^{2}},$ (3) where $\mathcal{M}_{0}$ is the contribution of one-photon annihilate diagram and $\mathcal{M}^{2\gamma}_{N,\Delta}$ denote the contribution from two-photon annihilate diagrams with $N$ and $\Delta$ as intermediate state. The corrections to the unpolarized cross section can defined as $\displaystyle\delta_{N,\Delta}^{2\gamma}=\frac{\sum\limits_{helicity}{2Re\\{{\mathcal{M}_{N,\Delta}^{2\gamma}\mathcal{M}_{0}^{\dagger}}\\}}}{\sum\limits_{helicity}{\|\mathcal{M}_{0}\|^{2}}}.$ (4) The corrections from $N$ have been discussed in [8]. To discuss the correction from $\Delta$, we take the following matrix elements as [9, 11] $\displaystyle\langle N(p_{2})\|J^{em}_{\mu}\|\Delta(k)\rangle=\frac{-F_{\Delta}(q_{1}^{2})}{M_{N}^{2}}\overline{u}(p_{2})[g_{1}(g^{\alpha}_{\mu}k\\!\\!\\!/q\\!\\!\\!/_{1}-k_{\mu}\gamma^{\alpha}q\\!\\!\\!/_{1}-\gamma_{\mu}\gamma^{\alpha}k\cdot q_{1}+\gamma_{\mu}k\\!\\!\\!/q_{1}^{\alpha})$ $\displaystyle+g_{2}(k_{\mu}q_{1}^{\alpha}-k\cdot q_{1}g^{\alpha}_{\mu})+g_{3}/M_{N}(q_{1}^{2}(k_{\mu}\gamma^{\alpha}-g^{\alpha}_{\mu}k\\!\\!\\!/)+q_{1\mu}(q_{1}^{\alpha}k\\!\\!\\!/-\gamma^{\alpha}k\cdot q_{1}))]\gamma_{5}T_{3}u_{\alpha}^{\Delta}(k),$ $\displaystyle\langle\Delta(k)\overline{N}(p_{1})\|J^{em}_{\nu}\|0\rangle=\frac{-F_{\Delta}(q_{2}^{2})}{M_{N}^{2}}\overline{u}_{\beta}^{\Delta}(k)T_{3}^{+}\gamma_{5}[g_{1}(g^{\beta}_{\nu}q\\!\\!\\!/_{2}k\\!\\!\\!/-k_{\nu}q\\!\\!\\!/_{2}\gamma^{\beta}-\gamma^{\beta}\gamma_{\nu}k\cdot q_{2}+k\\!\\!\\!/\gamma_{\nu}q_{2}^{\beta})$ $\displaystyle+g_{2}(k_{\nu}q_{2}^{\beta}-k\cdot q_{2}g^{\beta}_{\nu})-g_{3}/M_{N}(q_{2}^{2}(k_{\nu}\gamma^{\beta}-g^{\beta}_{\nu}k\\!\\!\\!/)+q_{2\nu}(q_{2}^{\beta}k\\!\\!\\!/-\gamma^{\beta}k\cdot q_{2}))]v(p_{1}),$ (5) where $q_{1}=p_{2}-k,~{}q_{2}=k+p_{1}$ and $T_{3}$ is the third component of the $N\rightarrow\Delta$ isospin transition operator and is $-\sqrt{2/3}$ here. The effective vertexes of $\gamma N\Delta$ are defined as $\overline{u}(p_{2})\Gamma_{\mu}^{\alpha}(\gamma\Delta\rightarrow N)u_{\alpha}^{\Delta}(k)=-ie\langle N(p_{2})\|J^{em}_{\mu}\|\Delta(k)\rangle,~{}\overline{u}_{\beta}^{\Delta}(k)\Gamma_{\nu}^{\beta}(\gamma\rightarrow\overline{N}\Delta)v(p_{1})=-ie\langle\Delta(k)\overline{N}(p_{1})\|J^{em}_{\nu}\|0\rangle$. Both the two vertexes satisfy the conditions $q_{1,2}^{\mu}\Gamma_{\mu}=0$ and $k_{\alpha}\Gamma^{\alpha}=0$, the first condition ensure the gauge invariance of the result and the second condition ensure to select only the physical spin3/2 component [9]. For the propagator of $\Delta$, the same form is employed as [9] $\displaystyle S_{\alpha\beta}^{\Delta}(k)=\frac{-i(k\\!\\!\\!/+M_{\Delta})}{k^{2}-M_{\Delta}^{2}+i\epsilon}P_{\alpha\beta}^{3/2}(k),$ $\displaystyle P_{\alpha\beta}^{3/2}(k)=g_{\alpha\beta}-\gamma_{\alpha}\gamma_{\beta}/3-(k\\!\\!\\!/\gamma_{\alpha}k_{\beta}+k_{\alpha}\gamma_{\beta}k\\!\\!\\!/)/3k^{2}.$ (6) Such propagator is different with the usual R.S one which read as $\displaystyle S_{\alpha\beta}^{RS}(k)=\frac{k\\!\\!\\!/+M_{\Delta}}{k^{2}-M_{\Delta}^{2}+i\epsilon}[-g_{\alpha\beta}+\frac{1}{3}\gamma_{\alpha}\gamma_{\beta}+\frac{1}{3m}(\gamma_{\alpha}k_{\beta}-\gamma_{\beta}k_{\alpha})+\frac{2}{3m^{2}}k_{\alpha}k_{\beta}].$ (7) After using the properties of the vertexes, these two forms result in the same amplitude. By this effective interaction, the amplitude of box diagram Fig.2(b) can be written as $\displaystyle M^{(2b)}=-i\int\frac{d^{4}k}{(2\pi)^{4}}\overline{u}(k_{2})(-ie\gamma_{\mu})\frac{i(p\\!\\!\\!/_{1}+k\\!\\!\\!/-k\\!\\!\\!/_{2}+m_{e})}{(p_{1}+k-k_{2})^{2}-m_{e}^{2}+i\varepsilon}(-ie\gamma_{\nu})v(k_{1})\frac{-i}{(p_{1}+k)^{2}+i\varepsilon}\frac{-i}{(p_{2}-k)^{2}+i\varepsilon}$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overline{u}(p_{2})\Gamma^{\mu\alpha}_{\gamma\Delta\rightarrow N}\frac{-i(k\\!\\!\\!/+M_{\Delta})}{k^{2}-M_{\Delta}^{2}+i\varepsilon}P_{\alpha\beta}^{3/2}(k)\Gamma^{\beta\nu}_{\gamma\rightarrow\overline{N}\Delta}v(p_{1}),$ (8) where Feynamn gauge invariance has been used. Similarly one can get the amplitude of crossed box diagram with $\Delta$ intermediate state. In the practical calculation, we take the form factor $F_{\Delta}$ in the monopole form as $G_{E}$ in $N$ case[8] $\displaystyle F_{\Delta}(q^{2})=G_{E}(q^{2})=G_{M}/\mu_{p}(q^{2})=\frac{-\Lambda_{1}^{2}}{q^{2}-\Lambda_{1}^{2}},$ (9) the coupling parameters and cut-offs are the same as [8, 11] $\displaystyle g_{1}=1.91,g_{2}=2.63,g_{3}=1.58,\Lambda_{1}=0.84GeV.$ (10) ## 3 Numerical Results and Discussion Figure 3: Cosine $\theta$ dependence of two-photon-annihilating corrections to unpolarized cross section. The dashed and dotted lines denote to the correction from $N$ and $\delta_{\Delta}$, respectively, and their sum is given by the solid lines. The left result is for $s=4GeV^{2}$ and the right one for $s=5GeV^{2}$ Figure 4: Cosine $\theta$ dependence of two-photon-annihilating corrections to $P_{x}$ and $P_{z}$. The dashed and dotted lines denote to the correction from $N$ and $\delta_{\Delta}$, respectively, and their sum is given by the solid lines. The left result is for $P_{x}$ and the right one for $P_{z}$, both with $s=4GeV^{2}$. Using the above as input, the two-photon annihilate corrections can be calculated directly. We use the package FeynCalc [12] and LoopTools [13] to carry out the calculation. The IR divergence in the $N$ intermediate case is treated as [8] and there is no divergence in the $\Delta(1232)$ case. The numerical results for $\delta_{N,\Delta}^{2\gamma}$ are showed in Fig.3. The similar calculation can be applied to the polarized quantities $P_{x}$ and $P_{z}$ as [14, 8] with the definitions $\displaystyle\frac{d\sigma}{d\Omega}=\frac{d\sigma_{un}}{d\Omega}[1+P_{y}\xi_{y}+\lambda_{e}P_{x}\xi_{x}+\lambda_{e}P_{z}\xi_{z}].$ (11) The results of the corrections to $P_{x}$ and $P_{z}$ are presented in Fig4. In our previous results[8], when discussing the TPE corrections to polarized observables, only the contributions in term $\frac{d\sigma}{d\Omega}$ are considered, while the corrections in $\frac{d\sigma_{un}}{d\Omega}$ are neglected. Here the calculations are improved to include both corrections. As showed in Fig.3, the correction $\delta^{2\gamma}_{\Delta}$ is found to be always opposite to the corrections $\delta^{2\gamma}_{N}$ in all the angle region. This behavior is similar to the $ep$ scattering case [9]. Detailedly, at $s=4GeV^{2}$ the absolute magnitude of $\delta^{2\gamma}_{\Delta}$ is so close to $\delta^{2\gamma}_{N}$ which results in the large cancelation and small total correction to unpolarized cross section. The small $\delta_{N+\Delta}^{2\gamma}$ and its weak angle dependence suggest the Rosenbluth method will work well in this region. This conclusion is some different with the $ep$ scattering case where the cancelation is much smaller and the total correction still strongly depend on the scattering angle. At $s=5GeV^{2}$, the absolute magnitude of $\delta^{2\gamma}_{\Delta}$ becomes larger than $\delta^{2\gamma}_{N}$ which suggests the important roles played by $\Delta(1232)$ intermediate state in the process of $e^{+}+e^{-}\rightarrow p+\bar{p}$. For the polarized observables, Fig.4 shows the correction to $P_{x}$ from $\Delta$ is much smaller than $N$ and the correction to $P_{z}$ from $\Delta$ is close to $N$. The former property suggests $\Delta(1232)$ gives no new correction than [8] while the latter property increases the two-photon annihilate corrections to $P_{z}$ which enhances our previous suggestion that the nonzero $P_{z}$ at $\theta=\pi/2$ may be a good place to measure the two- photon exchange like effects directly. ## 4 Acknowledgment This work is supported by the National Sciences Foundations of China under Grant No.10747118, No.10805009, No. 10475088, and by CAS Knowledge Innovation Project No. KC2-SW-N02. ## References [1] M.K. Jones et al., Phys. Rev. Lett. 84, (2000)1398 ; O. Gayou et al., Phys. Rev. Lett. 88,(2002) 092301. * [2] P.G. Blunden, W. Melnitchouk, and J.A. Tjon, Phys. Rev. Lett. 91,(2003) 142304. * [3] Y.C. Chen, A.V. Afanasev, S.J. Brodsky, C.E. Carlson, M. Vanderhaeghen, Phys. Rev. Lett 93,(2004) 122301. * [4] P. G. Blunden, W. Melnitchouk and J. A. Tjon, Phys. Rev. C 72, (2005)034612. * [5] M. P. Rekalo, E. Tomasi-Gustafsson and D. Prout, Phys. Rev. C 60, (1999)042202(R). * [6] G. I. Gakh, E. Tomasi-Gustafsson, Nucl. Phys. A 761,(2005) 120. * [7] E. Tomasi-Gustafsson, E. A. Kuraev, S. Bakmaev and S. Pacetti, Phys. Lett. B 659,(2008) 197. * [8] D.Y. Chen, H.Q. Zhou, Y.B. Dong, Phys. Rev. C78,(2008) 045208. * [9] S. Kondratyuk, P.G. Blunden, W. Melnitchouk and J.A. Tjon, Phys. Rev. Lett. 95, (2005) 172503. * [10] N. Cabibbo, Raoul Gatto, Phys.Rev.124,(1961)1577, A.Zichichi et al., Nuovo Cim.24 (1962) 170. * [11] Keitaro Nagata, Hai Qing Zhou, Chung Wen Kao, Shin Nan Yang, arXiv:0811.3539. * [12] R. Mertig, M. Bohm and A. Denner, Comput. Phys. Commun. 64,(1991) 345. * [13] T. Hahn, M. Perez-Victoria, Comput. Phys.Commun, 118, (1999)153. * [14] C. Adamuscin, G. I. Gakh, and E. Tomasi-Gustafsson, arXiv:0704.3375.	arxiv-papers	2009-03-02T14:09:03	2024-09-04T02:49:00.941048	{ "license": "Public Domain", "authors": "Hai Qing Zhou, Dian Yong Chen, Yu Bing Dong", "submitter": "Zhou Haiqing", "url": "https://arxiv.org/abs/0903.0301" }
0903.0494	11institutetext: † ICRA and Centre de Physique Théorique de Luminy, Université de la Méditerranée F-13288, Marseille EU, battisti@icra.it § ICRA, ICRANet, ENEA and Dipartimento di Fisica, Università di Roma “Sapienza” P.le A. Moro 5, 00185 Rome EU, montani@icra.it # Bianchi IX in the GUP approach Marco Valerio Battisti† Giovanni Montani§ ###### Abstract The Bianchi IX cosmological model (through Bianchi I and II) is analyzed in the framework of a generalized uncertainty principle. In particular, the anisotropies of the Universe are described by a deformed Heisenberg algebra. Three main results are in order. (i) The Universe can not isotropize because of the deformed Kasner dynamics. (ii) The triangular allowed domain is asymptotically stationary with respect to the particle (Universe) and its bounces against the walls are not interrupted by the deformed effects. (iii) No reflection law can be in obtained since the Bianchi II model is no longer analytically integrable. The existence of a fundamental scale, by which the continuum space-time picture that we have used from our experience at large scales probably breaks down, may be taken as a general feature of any quantum theory of gravity (for a review see [1]). This claim can be formalized modifying the canonical uncertainty principle as (we adopt units such that $\hbar=c=16\pi G=1$) $\Delta q\Delta p\geq\frac{1}{2}\left(1+\beta(\Delta p)^{2}+\delta\right),$ (1) where $\beta$ and $\delta$ are positive deformation parameters. This is the so-called generalized uncertainty principle (GUP) which appeared in string theory [2], considerations on the proprieties of black holes [3] and de Sitter space [4]. From the string theory point of view, the relation above is a consequence of the fact that strings can not probe distances below the string scale. The GUP (1) implies a finite minimal uncertainty in the position $\Delta q_{0}=\sqrt{\beta}$ and can be recovered by deforming the canonical commutation relations as $[q,p]=i(1+\beta p^{2})$ as soon as $\delta=\beta\langle p\rangle^{2}$. Recently, the GUP framework has received notable interest and a wide work has been made on this field in a large variety of directions (see [5] and references therein). In this paper we describe the dynamics of the Bianchi cosmological models in the GUP framework reviewing the results of [6]. In particular, we analyze the most general homogeneous model (Bianchi IX or Mixmaster) passing through the necessary steps of Bianchi I and II. The GUP approach has been previously implemented to the FRW model filled with a massless scalar field [7] as well as to the Taub Universe [8]. In the first case [7], the big-bang singularity appears to be probabilistically removed but no evidences for a big-bounce, as predicted by the loop approach [9], arise (a cosmological bounce à la loop quantum cosmology has been obtained from a deformed Heisenberg algebra in [10]). The GUP Taub Universe [8] is also singularity-free and this feature is relevant since allows a phenomenological comparison with the polymer (loop) Taub Universe [11]. In fact, in the latter model, the cosmological singularity appears to be not removed. The analysis of the Bianchi models then improve such a research line since the two anisotropies of the Universe are now described by a deformed Heisenberg algebra. The Bianchi Universes are spatially homogeneous cosmological models (for reviews see [12]) and their dynamics is summarized in the scalar constraint which, in the Misner scheme, reads $H=-p_{\alpha}^{2}+p_{+}^{2}+p_{-}^{2}+e^{4\alpha}V(\gamma_{\pm})=0,$ (2) where the lapse function $N=N(t)$ has been fixed by the time gauge $\dot{\alpha}=1$ as $N=-e^{3\alpha}/2p_{\alpha}$. The variable $\alpha=\alpha(t)$ describes the isotropic expansion of the Universe while its shape changes (the anisotropies) are determinated via $\gamma_{\pm}=\gamma_{\pm}(t)$. Homogeneity reduces the phase space of general relativity to six dimensions. In the Hamiltonian framework the cosmological singularity appears for $\alpha\rightarrow-\infty$ and the differences between the Bianchi models are summarized in the potential term $V(\gamma_{\pm})$ which is related to the three-dimensional scalar of curvature. As well-known, to describe the dynamical evolution of a system in general relativity a choice of time has to be performed. This can be basically accomplished in a relational way (with respect to an other field) or with respect to an internal time which is constructed from phase space variables. The ADM reduction of the dynamics relies on the idea to solve the scalar constraint with respect to a suitably chosen momentum. This way, an effective Hamiltonian which depends only on the physical degrees of freedom of the system is naturally recovered. Since the volume $\mathcal{V}$ of the Universe is $\mathcal{V}\propto e^{3\alpha}$, the variable $\alpha$ can be regarded as a good clock for the evolution and therefore the ADM picture arises as soon as the constraint (2) is solved with respect to $p_{\alpha}$. Explicitly, we obtain $-p_{\alpha}=\mathcal{H}=\left(p_{+}^{2}+p_{-}^{2}+e^{4\alpha}V(\gamma_{\pm})\right)^{1/2},$ (3) where $\mathcal{H}$ is a time-dependent Hamiltonian from which is possible to extract, for a given symplectic structure, all the dynamical informations about the homogeneous cosmological models. Let us now analyze the modifications induced on a $2n$-dimensional phase space by the GUP framework. Assuming the translation group as not deformed, i.e. $[p_{i},p_{j}]=0$, and the existence of a new deformation parameter $\beta^{\prime}>0$, the phase space algebra is the one in which the fundamental Poisson brackets are [13] $\displaystyle\\{q_{i},p_{j}\\}$ $\displaystyle=$ $\displaystyle\delta_{ij}(1+\beta p^{2})+\beta^{\prime}p_{i}p_{j},$ (4) $\displaystyle\\{p_{i},p_{j}\\}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle\\{q_{i},q_{j}\\}$ $\displaystyle=$ $\displaystyle\frac{(2\beta-\beta^{\prime})+(2\beta+\beta^{\prime})\beta p^{2}}{1+\beta p^{2}}(p_{i}q_{j}-p_{j}q_{i}).$ These relations are obtained assuming that $\beta$ and $\beta^{\prime}$ are independent constants with respect to $\hbar$. From a string theory point of view, keeping the parameters $\beta$ and $\beta^{\prime}$ fixed as $\hbar\rightarrow 0$ corresponds to keeping the string momentum scale fixed while the string length scale shrinks to zero [2]. The Poisson bracket for any phase space function can be straightforward obtained from (4) and reads $\\{F,G\\}=\left(\frac{\partial F}{\partial q_{i}}\frac{\partial G}{\partial p_{j}}-\frac{\partial F}{\partial p_{i}}\frac{\partial G}{\partial q_{j}}\right)\\{q_{i},p_{j}\\}+\frac{\partial F}{\partial q_{i}}\frac{\partial G}{\partial q_{j}}\\{q_{i},q_{j}\\}.$ (5) It is worth noting that for $\beta^{\prime}=2\beta$ the coordinates $q_{i}$ become commutative up to higher order corrections, i.e. $\\{q_{i},q_{j}\\}=0+\mathcal{O}(\beta^{2})$. This can be then considered a preferred choice of parameters and from now on we analyze this case. However, although we neglect terms like $\mathcal{O}(\beta^{2})$, the case in which $\beta p^{2}\gg 1$ is allowed since in such a framework no restrictions on the $p$-domain arise, i.e. $p\in\mathbb{R}$. The deformed classical dynamics of the Bianchi models can be therefore obtained from the symplectic algebra (4) for $\beta^{\prime}=2\beta$. The time evolution of the anisotropies and momenta, with respect to the ADM Hamiltonian (3), is thus given by ($i,j=\pm$) $\displaystyle\dot{\gamma}_{i}$ $\displaystyle=$ $\displaystyle\\{\gamma_{i},\mathcal{H}\\}=\frac{1}{\mathcal{H}}\left[(1+\beta p^{2})\delta_{ij}+2\beta p_{i}p_{j}\right]p_{j},$ (6) $\displaystyle\dot{p}_{i}$ $\displaystyle=$ $\displaystyle\\{p_{i},\mathcal{H}\\}=-\frac{e^{4\alpha}}{2\mathcal{H}}\left[(1+\beta p^{2})\delta_{ij}+2\beta p_{i}p_{j}\right]\frac{\partial V}{\partial\gamma_{j}},$ where the dot denotes differentiation with respect to the time variable $\alpha$ and $p^{2}=p_{+}^{2}+p_{-}^{2}$. These are the deformed equations of motion for the homogeneous Universes and the ordinary ones are recovered in the $\beta=0$ case. The Bianchi I model is the most simpler homogeneous model and describes a Universe with flat space sections [12]. Its line element is invariant under the group of three-dimensional translations and the spatial Cauchy surfaces can be then identified with $\mathbb{R}^{3}$. This Universe contains as a special case the flat FRW model which is obtained as soon as the isotropy condition is taken into account. Bianchi I corresponds to the case $V(\gamma_{\pm})=0$ and thus, from the Hamiltonian (3), it is described by a two-dimensional massless scalar relativistic particle. The velocity of the particle (Universe) is modified by the deformed symplectic geometry and, from the first equation of (6), it reads $\dot{\gamma}^{2}=\dot{\gamma}_{+}^{2}+\dot{\gamma}_{-}^{2}=\frac{p^{2}}{\mathcal{H}^{2}}\left(1+6\mu+9\mu^{2}\right)=1+6\mu+9\mu^{2},$ (7) where $\mu=\beta p^{2}$. For $\beta\rightarrow 0$ ($\mu\ll 1$), the standard Kasner velocity $\dot{\gamma}^{2}=1$ is recovered. Therefore, the effects of an anisotropies cut-off imply that the point-Universe moves faster than the ordinary case. In the deformed scheme the solution is still Kasner-like ($\dot{\gamma}_{\pm}=C_{\pm}(\beta),\dot{p}_{\pm}=0$), but this behavior is modified by the equation (7). In particular, the second Kasner-relation between the Kasner indices $s_{1},s_{2},s_{3}$ appears to be deformed as $s_{1}^{2}+s_{2}^{2}+s_{3}^{2}=1+4\mu+6\mu^{2},$ (8) while the first one $s_{1}+s_{2}+s_{3}=1$ remains unchanged. As usual, for $\beta=0$, the standard one is recovered. Two remarks are therefore in order. (i) The GUP acts in an opposite way with respect to a massless scalar field in the standard model. In that case the chaotic behavior of the Mixmaster Universe is tamed [14]. On the other hand, in the GUP framework, all the terms on the right hand side of (8) are positive and it means that the Universe cannot isotropize, i.e. it can not reach the stage such that the Kasner indices are equal. (ii) For every non-zero $\mu$, two indices can be negative at the same time. Thus, as the volume of the Universe contracts toward the classical singularity, distances can shrink along one direction and grow along the other two. In the ordinary case the contraction is along two directions. The natural bridge between Bianchi I and the Mixmaster Umiverse is represented by the Bianchi II model. Its dynamics is the one of a two-dimensional particle bouncing against a single wall and it corresponds to the Mixmaster dynamics when only one of the three equivalent potential walls is taken into account [12]. In the Hamiltonian framework, Bianchi II is described by the potential term $V(\gamma_{\pm})=e^{-8\gamma_{+}}$ and this expression can be directly obtained from the one of Bianchi IX in an asymptotic region. The main features of Bianchi IX, as the BKL map, are obtained considering such a simplified model since it is, in the ordinary framework, an integrable system differently from Bianchi IX itself. The BKL map, which is as the basis of the analysis of the stochastic and chaotic proprieties of the Mixmaster Universe, appears to be the reflection law of the $\gamma$-particle (Universe) against the potential wall. A fundamental difference which arises in the deformed framework with respect to the ordinary one, is that $\mathcal{H}$ is no longer a constant of motion near the classical singularity. The wall-velocity $\dot{\gamma}_{w}$ is then modified as [6] $\dot{\gamma}_{w}=\frac{1}{36\mu}\left(-4+22^{1/3}g^{-1/3}+2^{2/3}g^{1/3}\right),$ (9) where $g=2+81\mu\dot{\gamma}^{2}+9\sqrt{\mu\dot{\gamma}^{2}(4+81\mu\dot{\gamma}^{2})}$. From the two velocity equations (7) and (9), it is possible to discuss the details of the bounce. In the standard case the particle (Universe) moves twice as fast as the receding potential wall, independently of its momentum (namely its energy). In the deformed framework the particle-velocity, as well as the potential-velocity, depends on the anisotropy momentum and on the deformation parameter $\beta$. Also in this case the particle moves faster than the wall since the relation $\dot{\gamma}_{w}<\dot{\gamma}$ is always verified (see Fig. 1) and a bounce takes then place also in the deformed picture. Furthermore, in the asymptotic limit $\mu\gg 1$ the maximum angle in order the bounce against the wall to occur is given by $\|\theta_{\max}\|=\pi/2$, differently from the ordinary case ($\dot{\gamma}_{w}/\dot{\gamma}=1/2$) where the maximum incidence angle is given by $\|\theta_{\max}\|=\pi/3$. Figure 1: The potential wall velocity $\dot{\gamma}_{w}$ with respect to the particle one $\dot{\gamma}$ in function of $\mu=\beta p^{2}$. In the $\mu\rightarrow 0$ limit, the ordinary behavior $\dot{\gamma}_{w}/\dot{\gamma}=1/2$ is recovered. The $\gamma$-particle bounce against the wall is thus improved in the sense that no longer maximum limit angle appears. However, the main difference with respect to the ordinary picture is that the deformed Bianchi II model is not analytically integrable. No reflection map can be then in general inferred. In fact, it is no longer possible to find two constants of motion in the GUP picture. For more details see [6]. On the basis of the previous analysis of the GUP Bianchi I and II models we know several features of the deformed Mixmaster Universe. We recall (see [12]) that the Bianchi IX geometry is invariant under the three-dimensional rotation group (this Universe is the generalization of the closed FRW model when the isotropy hypothesis is relaxed) and its potential term is given by $V(\gamma_{\pm})=e^{4(\gamma_{+}+\sqrt{3}\gamma_{-})}+e^{4(\gamma_{+}-\sqrt{3}\gamma_{-})}+e^{-8\gamma_{+}}$. The evolution of the Mixmaster Universe is that of a two-dimensional particle bouncing (the single bounce is described by the Bianchi II model) infinite times against three walls which rise steeply toward the singularity. Between two succeeding bounces the system is described by the Kasner evolution and the permutations of the expanding-contracting directions is given by the BKL map [15] showing the chaotic behavior of such a dynamics [16]. Two conclusions on the deformed Mixmaster Universe can be thus inferred [6]. * • When the ultra-deformed regime is reached ($\mu\gg 1$), i.e. when the $\gamma$-particle (Universe) has the momentum bigger than the cut-off one, the triangular closed domain appears to be stationary with respect to the particle itself. The bounces of the particle are then increased by the presence of deformation terms, i.e. by the non-zero minimal uncertainty in the anisotropies. * • No BKL map (reflection law) can be in general obtained. It arises analyzing the single bounce against a given wall of the equilateral-triangular domain and the Bianchi II model is no longer an integrable system in the deformed picture. The chaotic behavior of the Bianchi IX model is then not tamed by GUP effects, i.e. the deformed Mixmaster Universe is still a chaotic system. As last point we stress the differences between our model and the loop Mixmaster dynamics [17]. In the loop Bianchi IX model the classical reflections of the $\gamma$-particle stop after a finite amount of time and the Mixmaster chaos is therefore suppressed. In this framework, although the analysis is performed through the ADM reduction of the dynamics as we did, all the three scale factors are quantized using the loop techniques. On the other hand, in our approach the time variable (related to the volume of the Universe) is treated in the standard way and only the two physical degrees of freedom of the Universe (the anisotropies) are considered as deformed. Acknowledgments. M. V. B. thanks ”Fondazione Angelo Della Riccia” for financial support. ## References * [1] L. J. Garay, Int.J.Mod.Phys.A 10 (1995) 145. * [2] D. J. Gross and P. F. Mendle, Nucl.Phys.B 303 (1988) 407; D. Amati, M. Ciafaloni and G. Veneziano, Phys.Lett.B 216 (1989) 41. * [3] M. Maggiore, Phys.Lett.B 304 (1993) 65; Phys.Rev.D 49 (1994) 5182. * [4] H. S. Snyder, Phys.Rev. 71 (1947) 38. * [5] I. Dadic, L. Jonke and S. Meljanac, Phys.Rev.D 67 (2003) 087701; B. Vakili, Phys.Rev.D 77 (2008) 044023; B. Vakili and H. R. Sepangi, Phys.Lett.B 651 (2007) 79; N. Khosravi and H. R. Sepangi, Phys.Lett.A 372 (2008) 3356. * [6] M. V. Battisti and G. Montani, arXiv:0808.0831. * [7] M. V. Battisti and G. Montani, Phys.Lett.B 656 (2007) 96. * [8] M. V. Battisti and G. Montani, Phys.Rev.D 77 (2008) 023518. * [9] A. Ashtekar, T. Pawlowski and P. Singh, Phys.Rev.Lett. 96 (2006) 141301. * [10] M. V. Battisti, arXiv:0805.1178. * [11] M. V. Battisti, O. M. Lecian and G. Montani, Phys.Rev.D 78 (2008) 103514. * [12] G. Montani, M. V. Battisti, R. Benini and G. Imponente, Int.J.Mod.Phys.A 23 (2008) 2353; J. M. Heinzle and C. Uggla, arXiv:0901.0776. * [13] S. Benczik et al., Phys.Rev.D 66 (2002) 026003. * [14] V. A. Belinski and I. M. Khalatnikov, Sov.Phys.JETP 36 (1973) 591. * [15] V. A. Belinski, I. M. Khalatnikov and E. M. Lifshitz, Adv.Phys. 19 (1970) 525; Adv.Phys. 31 (1982) 639. * [16] G. P. Imponente and G. Montani, Phys.Rev.D 63 (2001) 103501; T. Damour, M. Henneaux and H. Nicolai, Class.Quant.Grav. 20 (2003) R145; L. Andersson, H. van Elst, W. C. Lim and C. Uggla, Phys.Rev.Lett. 94 (2005) 051101. * [17] M. Bojowald and G. Date, Phys.Rev.Lett. 92 (2004) 071302; M. Bojowald, G. Date and G. M. Hossain, Class.Quant.Grav. 21 (2004) 3541.	arxiv-papers	2009-03-03T11:26:29	2024-09-04T02:49:00.946506	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Marco Valerio Battisti and Giovanni Montani", "submitter": "Marco Valerio Battisti", "url": "https://arxiv.org/abs/0903.0494" }
0903.0506	###### Abstract We consider a charged Brownian particle in an asymmetric bistable electrostatic potential biased by an externally applied or induced time periodic electric field. While the amplitude of the applied field is independent of frequency, that of the one induced by a magnetic field is. Borrowing from protein channel terminology, we define the open probability as the relative time the Brownian particle spends on a prescribed side of the potential barrier. We show that while there is no peak in the open probability as the frequency of the applied field and the bias (depolarization) of the potential are varied, there is a narrow range of low frequencies of the induced field and a narrow range of the low bias of the potential where the open probability peaks. This manifestation of stochastic resonance is consistent with experimental results on the voltage gated $I_{\mbox{\scriptsize Ks}}$ and KCNQ1 potassium channels of biological membranes and on cardiac myocytes. Stochastic resonance with applied and induced fields: the case of voltage- gated ion channels M. Shaked 111Department of Systems, School of Electrical Engineering, The Iby and Aladar Fleischman Faculty of Engineering, Tel-Aviv University, Ramat-Aviv Tel-Aviv 69978, Israel., Z. Schuss 222Department of Mathematics, Tel-Aviv University, Tel-Aviv 69978, Israel. ## 1 Introduction Our recent experimental findings show unusual non-thermal biological effects of a periodic electromagnetic field (EMF) of frequency 16 Hz and amplitude 16 nT (nano Tesla) on the potassium current in human $I_{\mbox{\scriptsize Ks}}$ and KCNQ1 channels [1]. More specifically, we expressed the $I_{\mbox{\scriptsize Ks}}$ channel in Xenopus oocytes and varied the membrane depolarization between -100 mV and +100 mV and measured the membrane potassium current. The current with applied EMF peaked above that without applied EMF at membrane depolarizations between 0 mV and 8 mV to a maximum of about 9% (see Figures 1 and 2). A similar measurement of the potassium current in the KCNQ1 channel protein, expressed in an oocyte, gave a maximal increase of 16% at the same applied EMF and at membrane depolarizations between -10 mV and -3 mV (see Figure 3). Similar experiments with L-type calcium channels showed no response to the electromagnetic field at any frequency between 0.05 and 50 Hz. Figure 1: $I_{\mbox{\scriptsize Ks}}$ current in Xenopus oocytes with applied magnetic field of 16 Hz and 16 nT (red) and without (blue) Figure 2: The quotient of $I_{\mbox{\scriptsize Ks}}$ expressed in Xenopus oocytes with applied magnetic field of 16 Hz and 16 nT and without (red to blue in Figure 1) Figure 3: The quotient of KCNQ1 expressed in Xenopus oocytes with applied magnetic field of 16 Hz and 16 nT and without In a related experiment [2], we applied electromagnetic fields at frequencies 15 Hz, 15.5 Hz, 16 Hz, 16.5 Hz and amplitudes of the magnetic field from below 16 pT and up to 160 nT, to neonatal rat cardiac myocytes in cell culture. In the range 16 pT – 16 nT, we observed that both stimulated and spontaneous activity of the myocytes changed at frequency 16 Hz: the height and duration of cytosolic calcium transients began decreasing significantly about 2 minutes after the magnetic field was applied and kept decreasing for about 30 minutes until it stabilized at about 30% of its initial value and its width decreased to approximately 50%. About 10 minutes following cessation of the magnetic field the myocyte (spontaneous) activity recovered with increased amplitude, duration, and rate of contraction. Outside this range of frequencies and magnetic fields no change in the transients was observed (see Figure 4). When the stereospecific inhibitor of KCNQ1 and $I_{\mbox{\scriptsize Ks}}$ channels chromanol 293B was applied, the phenomenon disappeared, which indicates that the $I_{\mbox{\scriptsize Ks}}$ and KCNQ1 potassium channels in the cardiac myocyte are the targets of the electromagnetic field, in agreement with the former experiment. The effect of changing the outward potassium current in a cardiac myocyte is to change both the height and duration of calcium transients, action potential, sodium current, as indicated by the Luo-Rudy model [3]. Figure 4: Cardiac cells, 4 days in culture, were exposed to magnetic fields of magnitude 160 pT and frequency 16 Hz for 30 min. Characteristic traces of spontaneous cytosolic calcium activity (A,B,C,D) and of electrically stimulated (1 Hz) cytosolic calcium activity (E,F,G,H). Times are measure in seconds from the moment of application of the magnetic field. The specific response at 16 Hz may indicate some form of resonance or stochastic resonance of a gating mechanism of open voltage-gated potassium channels (e.g., a secondary structure or mechanism) with time-periodic induced electric field. Since the induced electric field is too low to interact with any component of the $I_{\mbox{\scriptsize Ks}}$ channel, we conjecture that the induced field may interact with locally stable (metastable) configurations of ions inside the selectivity filter [4]. We propose an underlying scenario for this type of interaction based on the collective motion of three ions in the channel, as represented in the molecular dynamics simulation of [4]. The configurations of three potassium ions in the KcsA channel is represented in [4] in reduced reaction coordinates on a three-dimensional free energy landscape. In our simplified model, we represent the collective motion of the three ions in the channel as diffusion of a higher-dimensional Brownian particle in configuration space. An imitation hypothetical energy landscape with a reaction path (indicated in red) is shown in Figures 5 and 6. Projection onto a reaction path reduces this representation to Brownian motion on one-dimensional landscape of potential barriers (see Figure 7). The stable states represent instantaneous crystallization of the ions into a metastable configuration, in which no current flows through the channel, that is, they represent closed states of the channel. There is also a pathway in the multidimensional energy landscape that corresponds to a steady Figure 5: Hypothetical energy landscape of two ions in the selectivity filter. The reaction path is marked red. The straight segment in the trough may represent the open state in the channel Figure 6: Another view of the hypothetical energy landscape of two ions in the selectivity filter. current flowing in the channel, e.g., an unobstructed trough in the energy landscape. Transitions from the latter into the former represent gating events. In our scenario the motion between closed states is simplified to one- dimensional Brownian motion, e.g., in a trough obstructed with barriers, while the interruptions in the current correspond to exits from the unobstructed trough into the obstructed one. Activated transitions over barriers separating two closed states in the obstructed trough (see Figure 8) affect the probability of transition from closed to open states. Stochastic resonance between two closed states may change the transition rates between them, thus affecting the open (or closed) probability of the channel (see Section 4). We investigate the stochastic resonance (SR) in our mathematical model of a Brownian particle in an asymmetric bistable potential with an induced electric field. The difference between this problem and that of the extensively studied SR with an applied periodic electric field [5], [6] is that according to Faraday’s law (or Maxwell’s equations), the amplitude of Figure 7: Profile of one-dimensional electrostatic potential landscape biased by a constant electric field Figure 8: A simplified version (see eq.(6)) of the wells in Figure 7. The wells at $x_{1}$ and $x_{3}$ and the barrier at $x_{2}$ are now at $x_{1}=-2,x_{3}=0.7$ and $x_{2}=0$. The constant bias in (2) is $c=0$. the induced field is proportional to the frequency of the applied magnetic field. While the traditional manifestation of SR is a peak in the power spectral density of the trajectory of the resonating particle, we consider its manifestation in the probability to be in one of the two meta-stable states. This measure of SR is ineffective for a symmetric potential, because this probability is $1/2$ in the symmetric case and is independent of the applied periodic field. It is effective, however, in asymmetric potentials, for example, when a constant bias field depolarizes the membrane, as is the case in the above mentioned experiments. Note that in the second experiment the depolarization of the myocyte membrane is due to the action potential in the cell. In contrast, asymmetry of the potential can weaken SR with an applied field, as shown in [7], [8]. Our main results concern SR with applied and with induced external periodic forces. In the former case, which we view as a benchmark for our method of analysis, we find that there is no SR as frequency and depolarization are varied, in agreement with known results Figure 9: $P_{o}(\omega,c)/P_{o}(0,c)$ with induced force $A$, $A=0.007,\varepsilon=0.029,\ x_{L}=-2.4,\ x_{R}=1.385$ for $0<\omega<6,-0.1<c<0.1$. Evidently, there is no SR. [5] (see Figure 9). In contrast, the probability to be on one side of the barrier in the case of an induced field peaks at a nearly fixed frequency in a finite window of depolarizations (see Figure 10). We refer to this peak as stochastic resonance, though it may not be the usual SR phenomenon. The folding of the surface in Figure 9 into that in Figure 10 seems to be due to the decrease in the amplitude of the induced field at low frequencies. This observation is consistent with the above mentioned experiments and seems to be new. Figure 10: $P_{o}(\omega,c)/P_{o}(0,c)$ with induced force $Aw$, $A=0.007,\ \varepsilon=0.029,\ x_{L}=-2.4,\ x_{R}=1.385$ To connect the above SR with the cardiac myocyte experiment, we use the Luo- Rudy model [3] of a ventricular cardiac myocyte of a Guinea pig. We express the manifestation of the above SR in the Hodgkin-Huxley equations [9] as a change in the conductance of the $I_{\mbox{\scriptsize Ks}}$ channel in the specific range of depolarizations at the resonant frequency of 16 Hz. We note that the $I_{\mbox{\scriptsize Ks}}$ is one of the delayed rectifier K+ channels that are present in cardiac myocytes [10], in neuron cells [11], [12], and more, that is, it stays open long enough for its (secondary) gating to partially synchronize with the induced field. The SR-increased efflux of potassium (see Figure 11) shortens the action potential, and consequently lowers the peak of the cytosolic calcium concentration (see Figure 12), at the expense of increased sodium concentration (see 13). The shortening of the action potential leads to the shortening of the QT interval (see Figures 14, 15) [10] and was actually observed experimentally [13], [14]. These predictions of the SR modified Luo-Rudy equations are also new. In addition, we obtain from the SR modified Luo-Rudy model an increased conductance during the plateau of the action potential in the cardiac myocyte. This in turn shortens both the action potential and the cytosolic calcium concentration spike durations, lowers their amplitudes, increases cytosolic sodium, and lowers cytosolic potassium concentrations. These theoretical predictions are supported by experimental measurements. Specifically, these effects were communicated in [13], [14], as well as in our own measurements [2]. Figure 11: Cytosolic potassium concentration [mM] vs time [msec] without SR (blue) and with SR (red) in the Luo-Rudy model. SPECIFY UNITS Figure 12: Cytosolic calcium concentration [mM] vs time [msec] without SR (blue) and with SR (red) in the Luo-Rudy model. Figure 13: Cytosolic sodium concentration concentration [mM] vs time [msec] without SR (blue) and with SR (red) in the Luo-Rudy model. Figure 14: Action potential [mV] vs time [msec] without SR (Blue) and with SR (Green) in the Luo-Rudy model Figure 15: Action potential duration [msec] vs time [msec] without SR (Blue) and with SR (Red) in the Luo-Rudy model. ## 2 The mathematical model We consider the dimensionless overdamped dynamics $\displaystyle\dot{x}=-\frac{\partial\phi(x,t)}{\partial x}=-\phi_{x}(x,t)$ (1) in the bistable time-periodic potential $\displaystyle\phi(x,t)=(c-A_{\mbox{\scriptsize Appl,Ind}})\sin\omega t)x+\phi_{0}(x),$ (2) where $A_{\mbox{\scriptsize Appl,Ind}}$ is the amplitude of the applied (induced) electric field and $\phi_{0}(x)$ is a fixed parabolic double well potential that consists of the two parabolas $\displaystyle\phi_{0}(x)=\left\\{\begin{array}[]{lll}\displaystyle\frac{(x-x_{L})^{2}}{x_{L}^{2}}-1&\mbox{for}&x<0\\\ &&\\\ \displaystyle\frac{(x-x_{R})^{2}}{x_{R}^{2}}-1&\mbox{for}&x>0,\end{array}\right.$ (6) where $x_{L}<0<x_{R}$. The amplitude of the electric field induced by the time-periodic magnetic field $B\cos\omega t$ ($B=const$) is $A_{Ind}=A\omega$, where $A=CB$ and $C$ is the proportionality constant in Faraday’s law. The linear term $cx$ represents the membrane depolarization. This model can be considered the limit of the parabolic double well potential that consists of the three parabolas $\displaystyle\phi_{0}(x)=\left\\{\begin{array}[]{lll}\displaystyle\frac{(x-x_{L})^{2}}{x_{L}^{2}}-1+\displaystyle\frac{1}{1+ax_{L}^{2}/2}&\mbox{for}&x<-x_{\delta_{L}}\\\ &&\\\ -\displaystyle\frac{ax^{2}}{2}&\mbox{for}&-x_{\delta_{L}}<x<x_{\delta_{R}}\\\ &&\\\ \displaystyle\frac{(x-x_{R})^{2}}{x_{R}^{2}}-1+\displaystyle\frac{1}{1+ax_{R}^{2}/2}&\mbox{for}&x>x_{\delta_{R}},\end{array}\right.$ (12) where $x_{L}<-x_{\delta_{L}}<0<x_{\delta_{R}}<x_{R}$ and $a>0$. The three parabolas connect smoothly at $-x_{\delta_{L}}$ and $x_{\delta_{R}}$, which implies the relationships $x_{\delta_{L}}=-\displaystyle\frac{x_{L}}{1+ax_{L}^{2}/2}$, $x_{\delta_{R}}=\displaystyle\frac{x_{R}}{1+ax_{R}^{2}/2}$, $\displaystyle\lim_{a\to\infty}ax_{\delta_{L}}=-2/x_{L}$ and $\displaystyle\lim_{a\to\infty}ax_{\delta_{R}}=2/x_{R}$. The potential $\phi(x,t)$ (see Figure 8) has two periodic attractors, $\tilde{x}_{L}(t)$ and $\tilde{x}_{R}(t)$ (see Figure 16) Figure 16: The deterministic trajectories (in dimensionless units) with $\omega=1,\ A=0.5,\ c=-0.05,\ x_{L}=-2.4,\ x_{R}=1.385$ are attracted to the periodic $\tilde{x}_{L}(t)$ (blue lower curve), $\tilde{x}_{M}(t)$ (green middle curve) and $\tilde{x}_{R}(t)$ (red upper curve). and the separatrix333In the model (12) of three parabolas the separatrix is $\tilde{x}_{M}(t)=\frac{c}{a}-\left(\frac{A\omega}{a^{2}+\omega^{2}}\right)[\omega\cos\omega t+a\sin\omega t]$. $\tilde{x}_{M}(t)=0$. The attractors are the stable periodic solutions of (1), given by $\displaystyle\tilde{x}_{i}(\tau)$ $\displaystyle=$ $\displaystyle\alpha_{i}-\tilde{A}_{i}\cos(\tau+\tilde{\varphi}_{i}),\quad i=L,R$ (13) $\displaystyle\tilde{A}_{i}$ $\displaystyle=$ $\displaystyle\frac{A_{Appl,Ind}x_{i}^{2}}{\sqrt{4+x_{i}^{4}\omega^{2}}},\quad\tilde{\varphi}_{i}=\arctan\frac{2}{x_{i}^{2}\omega},\quad\alpha_{i}=\displaystyle\left(x_{i}-\frac{cx_{i}^{2}}{2}\right).$ (14) When small white noise $\sqrt{2\varepsilon}\,\dot{w}(t)$ is added to the dynamics (1), it becomes the stochastic equation $\displaystyle\dot{x}=-\phi_{x}(x,t)+\sqrt{2\varepsilon}\,\dot{w}(t).$ (15) The trajectories of (15) spend relatively long periods of time near the attractors $\tilde{x}_{L}(t)$ and $\tilde{x}_{R}(t)$, crossing $\tilde{x}_{M}(t)$ at random times. The first passage time from $\tilde{x}_{L}(t)$ to $\tilde{x}_{R}(t)$ is defined as $\displaystyle\tau_{L}(t_{0})=\inf\\{t>0\,:\,x(t_{0})={\tilde{x}}_{L}(t_{0}),\,x(t_{0}+t)={\tilde{x}}_{R}(t_{0}+t)\\}$ (16) and the mean first passage time is defined as $\displaystyle\bar{\tau}_{L}=\frac{1}{T}\int_{0}^{T}\hbox{\bb E}\tau_{L}(t_{0})\,dt_{0},$ (17) where 𝔼 denotes ensemble averaging over trajectories of (15) and the period is $\displaystyle T=\displaystyle\frac{2\pi}{\omega}.$ The first passage time $\tau_{R}(t_{0})$ and the mean first passage time $\bar{\tau}_{R}$ are defined in an analogous manner. The fraction of time the random trajectory $x(t)$ spends in the basin of attraction of ${\tilde{x}}_{R}(t)$, that is, the fraction of time that $x(t)>{\tilde{x}}_{M}(t)$, is the right probability $P_{R}(c,\omega,A_{Appl,Ind},\varepsilon)$, given by $\displaystyle P_{R}(c,\omega,A_{Appl,Ind},\varepsilon)$ $\displaystyle=$ $\displaystyle\lim_{n\to\infty}\frac{1}{nT}\int_{0}^{nT}\int_{{\tilde{x}}_{M}(t)}^{\infty}p(x,t)\,dx\,dt$ (18) $\displaystyle=$ $\displaystyle\lim_{n\to\infty}\frac{1}{T}\int_{0}^{T}\int_{{\tilde{x}}_{M}(t+nT)}^{\infty}p(x,t+nT)\,dx\,dt$ $\displaystyle=$ $\displaystyle\frac{1}{T}\int_{0}^{T}\int_{{\tilde{x}}_{M}(t)}^{\infty}p_{\infty}(x,t)\,dx\,dt,$ where $p(x,t)$ is the transition probability density function (pdf) of the random process $x(t)$, generated by the stochastic dynamics (15) and $p_{\infty}(x,t)=\displaystyle\lim_{n\to\infty}p(x,t+nT)$ is the periodic pdf. We obtain in a similar manner $\displaystyle P_{L}(c,\omega,A_{Appl,Ind},\varepsilon)=\frac{1}{T}\int_{0}^{T}\int_{-\infty}^{{\tilde{x}}_{M}(t)}p_{\infty}(x,t)\,dx\,dt,$ (19) For small $\varepsilon$, $\displaystyle P_{L}(c,\omega,A_{Appl,Ind},\varepsilon)\approx\displaystyle\frac{\bar{\tau}_{L}}{\bar{\tau}_{R}+\bar{\tau}_{L}}.$ (20) ## 3 The Fokker-Planck equation The $T$-periodic pdf $p_{\infty}(x,t)$ is the $T$-periodic solution of the Fokker-Planck equation $\displaystyle\frac{\partial p(x,t)}{\partial t}$ $\displaystyle=$ $\displaystyle\varepsilon\frac{\partial^{2}p(x,t)}{\partial x^{2}}+\frac{\partial[\phi_{x}(x,t)p(x,t)]}{\partial x}\quad\mbox{for}\quad-\infty<x<\infty,\ 0<t<\infty.$ (21) We construct a WKB approximation to $p_{\infty}(x,t)$ for small $\varepsilon$, $\displaystyle p_{\infty}(x,t)\sim\frac{\exp\left\\{-\displaystyle\frac{\psi(x,t,\varepsilon)}{\varepsilon}\right\\}}{\displaystyle\int_{-\infty}^{\infty}\exp\left\\{-\displaystyle\frac{\psi(x,t,\varepsilon)}{\varepsilon}\right\\}\,dx},$ (22) where $\psi(x,t,\varepsilon)$ is a $T$-periodic regular function of $\varepsilon$. Expanding $\displaystyle\psi(x,t,\varepsilon)=\psi(x,t,0)+\varepsilon\psi_{1}(x,t)+\ldots,$ (23) we find from large deviations theory that $\psi(x,t,0)$ is the+ minimum of the integral $\displaystyle I(x(\cdot))(x,t)=\int_{0}^{t}\left[\dot{x}(s)+\phi_{x}(x(s),s)\right]^{2}\,ds$ (24) over all continuous trajectories $x(\cdot)$ such that $x(0)=x$. Setting $\tau=\omega t$, we write the Hamilton-Jacobi (eikonal) equation for the minimal values of $I(x,\tau)$ in the domains $x>0$ and $x<0$ as $\displaystyle-\psi_{\tau}^{i}(x,\tau,0)=\frac{1}{\omega}\displaystyle\left(\psi_{x}^{i}\right)^{2}(x,\tau,0)-\frac{1}{\omega}\displaystyle\left[c-A_{Appl,Ind}\sin\tau+\frac{2(x-x_{i})}{x_{i}^{2}}\right]\psi_{x}^{i}(x,\tau,0),$ (25) for $i=L,R$. The solution can be constructed in the quadratic form [17] $\displaystyle\psi^{i}(x,\tau,0)=\displaystyle\frac{[x-\tilde{x}_{i}(\tau)]^{2}}{x_{i}^{2}}+a_{i}$ (26) and the constants $a_{i}$ are determined from the Freidlin-Wentzell extremum principle [16]. According to this principle the local minima $\psi^{L}(x,\tau,0)$ and $\psi^{R}(x,\tau,0)$ are joined into a global minimum function $\psi(x,\tau,0)$ by the requirement that the steady state probability current across the separatrix $\tilde{x}_{M}$ vanishes [17], $\displaystyle J(0,\tau)=\int_{0}^{2\pi}[J_{L}(0,\tau)+J_{R}(0,\tau)]\,d\tau=0,$ (27) where the probability flux density is $\displaystyle J_{i}(0,\tau)=-\varepsilon\frac{\partial p^{i}(0,\tau)}{\partial x}+\phi_{x}(0,\tau)p^{i}(0,\tau)=-\varepsilon\frac{\partial p^{i}(0,\tau)}{\partial x}.$ (28) Using the WKB approximation (22) for $p(x,\tau)$, we find that the minimum condition is $\displaystyle J_{i}(0,\tau)=\frac{-2\tilde{x}_{i}(\tau)}{x_{i}^{2}}\exp\left\\{-\frac{1}{\varepsilon}\left(\displaystyle\frac{\tilde{x}_{i}^{2}(\tau)}{x_{i}^{2}}+a_{i}\right)\right\\}.$ (29) Using (29) in (27), we find that $\displaystyle e^{(a_{L}-a_{R})/\varepsilon}=-\displaystyle\frac{x_{R}^{2}\displaystyle\int_{0}^{2\pi}\tilde{x}_{L}(\tau)\exp\left\\{-\frac{\tilde{x}_{L}^{2}(\tau)}{\varepsilon x_{L}^{2}}\right\\}\,d\tau}{x_{L}^{2}\displaystyle\int_{0}^{2\pi}\tilde{x}_{R}(\tau)\exp\left\\{-\frac{\tilde{x}_{R}^{2}(\tau)}{\varepsilon x_{R}^{2}}\right\\}\,d\tau}.$ (30) It should be noted that the phases $\tilde{\varphi}_{L}$ and $\tilde{\varphi}_{R}$ may be disregarded in the integrals of equation (30), therefore we set them to zero. Expanding the integrals in (30) by the Laplace method for small $\varepsilon$ about the maxima of the integrands, at $\displaystyle x_{1}=\tilde{x}_{R}(0)=\alpha_{R}-\tilde{A}_{R},\quad x_{2}=\tilde{x}_{L}(\pi)=\alpha_{L}-\tilde{A}_{L},$ (31) we get $\displaystyle e^{(a_{L}-a_{R})/\varepsilon}$ $\displaystyle=$ $\displaystyle-\displaystyle\frac{x_{R}^{2}\sqrt{\displaystyle\frac{2\pi\varepsilon x_{L}^{2}}{[\tilde{x}_{L}^{2}]^{{}^{\prime\prime}}(\pi)}}\tilde{x}_{L}(\pi)\exp\left\\{-\displaystyle\frac{\tilde{x}_{L}^{2}(\pi)}{\varepsilon x_{L}^{2}}\right\\}}{x_{L}^{2}\sqrt{\displaystyle\frac{2\pi\varepsilon x_{R}^{2}}{[\tilde{x}_{R}^{2}]^{{}^{\prime\prime}}(0)}}\tilde{x}_{R}(0)\exp\left\\{-\displaystyle\frac{\tilde{x}_{R}^{2}(0)}{\varepsilon x_{R}^{2}}\right\\}}$ (32) $\displaystyle=$ $\displaystyle-\displaystyle\frac{x_{R}^{2}\sqrt{\displaystyle\frac{x_{L}^{2}}{-2\tilde{A}_{L}x_{2}}}x_{2}\exp\left\\{-\displaystyle\frac{x_{2}^{2}}{\varepsilon x_{L}^{2}}\right\\}}{x_{L}^{2}\sqrt{\displaystyle\frac{x_{R}^{2}}{2\tilde{A}_{R}x_{1}}}x_{1}\exp\left\\{-\displaystyle\frac{x_{1}^{2}}{\varepsilon x_{R}^{2}}\right\\}},$ where, according to (13) and $\tilde{\varphi}_{i}=0$ ### 3.1 The left probability $P_{L}(c,\omega,A_{Appl,Ind},\varepsilon)$ To calculate the probability $P_{L}(c,\omega,A_{Appl,Ind},\varepsilon)$, we use the WKB approximation (22) in (19) and evaluate the integrals by the Laplace method, as in (29), to get $\displaystyle P_{L}(c,\omega,A_{Appl,Ind},\varepsilon)=\displaystyle\frac{1}{1+\displaystyle\frac{x_{R}}{-x_{L}}e^{(a_{L}-a_{R})/\varepsilon}}.$ (33) Using the result from (32) in (33), we find that $\displaystyle P_{L}(c,\omega,A_{Appl,Ind},\varepsilon)=\displaystyle\frac{1}{1+\displaystyle\frac{x_{R}^{3}}{\|x_{L}\|^{3}}\sqrt{\displaystyle\frac{\sqrt{4+x_{L}^{4}\omega^{2}}}{\sqrt{4+x_{R}^{4}\omega^{2}}}}\sqrt{\displaystyle\frac{-x_{2}}{x_{1}}}\exp\left\\{\displaystyle\frac{x_{1}^{2}}{\varepsilon x_{R}^{2}}-\displaystyle\frac{x_{2}^{2}}{\varepsilon x_{L}^{2}}\right\\}}.$ (34) We normalize the frequency-dependent left probability $P_{L}(c,\omega,A_{Appl,Ind},\varepsilon)$ by the left probability of the unforced dynamics $P_{L}^{0}(c,\omega,A_{Appl,Ind}=0,\varepsilon)$. To calculate $P_{L}^{0}(c,\omega,A_{Appl,Ind}=0,\varepsilon)$, we set $A=0$ in (33) and obtain $\displaystyle P_{L}^{0}(c,\omega,A_{Appl,Ind}=0,\varepsilon)$ $\displaystyle=$ $\displaystyle\left[1+\frac{x_{R}^{2}}{x_{L}^{2}}\displaystyle\left\|\displaystyle\frac{2-cx_{L}}{2-cx_{R}}\right\|\,\exp\left\\{\frac{c(x_{R}-x_{L})}{\varepsilon}\left(\frac{c(x_{R}+x_{L})}{4}-1\right)\right\\}\right]^{-1}.$ Note that (LABEL:Popen_A0) is not other than (20), where $\displaystyle\bar{\tau}_{i}=\displaystyle\sqrt{\frac{2\pi\varepsilon}{\phi_{xx}^{i}(\tilde{x}_{i})\|\phi_{x}^{i}(\tilde{x}_{M})\,\|^{2}}}\displaystyle\exp\left(\displaystyle\frac{\phi^{i}(\tilde{x}_{M})-\phi^{i}(\tilde{x}_{i})}{\varepsilon}\right).$ (36) The MFPT $\bar{\tau}_{i}$ in (36) is the Kramers escape rate of a Brownian particle over a high sharp barrier $\tilde{x}_{M}$ [19]. ## 4 Coarse-grained Markov model of secondary gating We consider the movement of a Brownian particle over two unequal barriers of heights $\Delta\phi_{21}=\phi(x_{2})-\phi(x_{1})$ and $\Delta\phi_{43}=\phi(x_{4})-\phi(x_{3})$, respectively, such that $\Delta\phi_{21}\ll\Delta\phi_{43}$ (see Figure 7). Both $x_{1}$ and $x_{3}$ are closed states of the channel whereas $x_{5}$ represents the open state (see Figure 5). Our goal is to elucidate the influence of SR between the periodic force and the activation over the local small barrier $\Delta\phi_{21}$ at $x_{2}$, within the closed state, on the open probability of the channel. Specifically, when SR increases the time spent in the well at $x_{3}$ relative to that at $x_{1}$, the attempt frequency to cross the barrier at $x_{4}$ into the open state $x_{5}$ increases, thus increasing the open probability of the channel. More specifically, we evaluate the influence of SR on the mean closed time, that is, on the mean time spent in the wells at $x_{1}$ and $x_{3}$ prior to passage into $x_{5}$ (which we denote $\bar{\tau}_{1,3}^{c}$). For that purpose, we can assume that $x_{5}$ is an absorbing boundary. First, we note that steady state considerations can be applied in describing SR in the wells at $x_{1}$ and $x_{3}$. Indeed, we assume that $\displaystyle\phi(x_{4})-\phi(x_{5}),\phi(x_{4})-\phi(x_{3})>\phi(x_{2})-\phi(x_{1})>\phi(x_{2})-\phi(x_{3})\gg\varepsilon,$ (37) which means that a transition over the barrier at $x_{4}$ between the open and closed states occurs at a much lower rate than those over the barrier at $x_{2}$, between the two closed substates. In particular, the first inequality in (37) means that there will be many transitions over the barrier at $x_{2}$ before a transition occurs from $x_{3}$ to $x_{5}$ over the barrier at $x_{4}$. Thus we confine our attention to transitions over the former and consider $x_{5}$ to be an absorbing state, as mentioned above. The assumption of high barriers (the last inequality in (37) means that a quasi steady state is reached in each of the wells before a transition over $x_{4}$ occurs. Therefore the pdf of the quasi-steady state in each well can be represented by the principal eigenfunction and eigenvalue in that well, with absorbing boundary conditions. We coarse-grain the trajectory of the diffusion process $x(t)$ into that of a continuous-time three state Markov jump process $\tilde{x}(t)$, that jumps between $x_{1}$ and $x_{3}$ and is absorbed in $x_{5}$, $\displaystyle x_{1}\rightleftarrows x_{3}\rightarrow x_{5}.$ (38) The three state continuous-time Markov chain $\tilde{x}(t)$ is not stationary due to the passage to the open state $x_{5}$. There are two time scales of passages: a short scale corresponding to the transitions between $x_{1}$ and $x_{3}$, and a long one for the transitions between $x_{3}$ and $x_{5}$. Due to the long time scale, the dynamics between the closed states ($x_{1}$ and $x_{3}$)is quasi-stationary. We assume it as a stationary dynamics in our analysis. The jump of the Markov process from $x_{1}$ to $x_{3}$ occurs when $x(t)$ reaches $x_{3}$ for the first time after it was at $x_{1}$, and so on. The Chapman-Kolmogorov equation for the transition probability matrix $\mbox{\boldmath$P$}_{t}$ of the Markov process is [15] $\dot{\mbox{\boldmath$P$}_{t}}=\mbox{\boldmath$P$}_{t}\mbox{\boldmath$R$},$ (39) where $\displaystyle P_{t}(i,j)=\Pr\\{\tilde{x}(t)=j\,\|\,\tilde{x}_{0}=i\\},\quad\mbox{for}\quad 1\leq i,j\leq 3,$ (40) and the elements of the instantaneous jump rate matrix $R$ are $\displaystyle\mbox{\boldmath$R$}=\left[\begin{array}[]{ccc}-r_{13}&r_{13}&0\\\ &&\\\ r_{31}&-(r_{31}+r_{35})&r_{35}\\\ &&\\\ 0&0&0\end{array}\right].$ (46) The stationary distribution $\pi$ of the process is $\mbox{\boldmath$\pi$}=\left[\begin{array}[]{lll}0,&0,&1\end{array}\right],$ because $x_{5}$ is an absorbing state. Next, we calculate the time-dependent probability distribution $\displaystyle\mbox{\boldmath$p$}_{t}=\left[\begin{array}[]{lll}\Pr\\{\tilde{x}(t)=x_{1}\\},&\Pr\\{\tilde{x}(t)=x_{3}\\},&\Pr\\{\tilde{x}(t)=x_{3}\\}\end{array}\right]^{T}.$ (48) According to (39) and (40), $\mbox{\boldmath$p$}_{t}$ the sum of elements in each column of the matrix $\mbox{\boldmath$P$}_{t}$ $Pr\\{\tilde{x}(t)=x_{j}\\}=\sum_{i=1}^{3}\Pr\\{\tilde{x}(t)=j\,\|\,\tilde{x}_{0}=i\\}\quad\mbox{for}\quad 1\leq j\leq 3$ and therefor satisfies the Chapman-Kolmogorov equation $\displaystyle\dot{\mbox{\boldmath$p$}}_{t}=\mbox{\boldmath$R$}^{T}\mbox{\boldmath$p$}_{t},$ (49) given by $\displaystyle\mbox{\boldmath$p$}_{t}=e^{\mbox{\boldmath$R$}^{T}t}\mbox{\boldmath$p$}_{0},$ (50) where we assume that $\mbox{\boldmath$p$}_{0}$ is the stationary distribution of the chain $x_{1}\rightleftarrows x_{3}$, namely, $\displaystyle\mbox{\boldmath$p$}_{0}=\left[\begin{array}[]{lll}\displaystyle{\frac{r_{31}}{r_{13}+r_{31}}},&\displaystyle{\frac{r_{13}}{r_{13}+r_{31}}},&0\\\ \end{array}\right]^{T}.$ (52) We further express the vector $\mbox{\boldmath$p$}_{t}$ as a linear combination of the eigenvectors $\\{\mbox{\boldmath$v$}_{1},\mbox{\boldmath$v$}_{2},\mbox{\boldmath$v$}_{3}\\}$ of the matrix $\mbox{\boldmath$R$}^{T}$, corresponding to the eigenvalues $\lambda_{1},\lambda_{2},\lambda_{3}$, $\displaystyle\mbox{\boldmath$p$}_{t}=\alpha_{1}e^{\lambda_{1}t}\mbox{\boldmath$v$}_{1}+\alpha_{2}e^{\lambda_{2}t}\mbox{\boldmath$v$}_{2}+\alpha_{3}e^{\lambda_{3}t}\mbox{\boldmath$v$}_{3},$ (53) where $\displaystyle\lambda_{1}$ $\displaystyle=$ $\displaystyle\displaystyle{\frac{-r_{13}r_{35}}{S}}$ (54) $\displaystyle\lambda_{2}$ $\displaystyle=$ $\displaystyle-S+\displaystyle{\frac{r_{13}r_{35}}{S}}$ (55) $\displaystyle\lambda_{3}$ $\displaystyle=$ $\displaystyle 0.$ (56) and $S=r_{13}+r_{31}+r_{35}$. Using the fact that $\displaystyle\lim_{t\rightarrow\infty}\mbox{\boldmath$p$}_{t}=\mbox{\boldmath$\pi$}$, we get that $\alpha_{3}=1$. The MFPTs $\bar{\tau}_{3}$ and $\bar{\tau}_{1}$ are related to the exit rates $r_{31}$ and $r_{13}$ over a non-sharp boundary [18] according to $\displaystyle\bar{\tau}_{3}=\displaystyle\frac{1}{2r_{31}}$ $\displaystyle\bar{\tau}_{1}=\displaystyle\frac{1}{2r_{13}}.$ (57) In order to find mean closed time $\bar{\tau}_{1,3}^{c}$ prior to the first arrival to $x_{5}$, it is enough to consider the matrix $\displaystyle\tilde{\mbox{\boldmath$R$}}^{T}=\left[\begin{array}[]{lc}-r_{13}&r_{31}\\\ r_{13}&-(r_{13}+r_{35})\end{array}\right],$ (60) because this time is determined by the first two elements of the vector $\mbox{\boldmath$p$}_{t}$, $\displaystyle\mbox{\boldmath$p$}_{t}^{1,3}=\left[\begin{array}[]{ll}\Pr\\{\tilde{x}(t)=x_{1}\\},&\Pr\\{\tilde{x}(t)=x_{3}\\}\end{array}\right]^{T}.$ (62) An eigenvector expansion $\mbox{\boldmath$p$}_{t}^{1,3}$, similar to that in (53), is $\displaystyle\mbox{\boldmath$p$}_{t}^{1,3}=\alpha_{1}e^{\lambda_{1}t}\tilde{\mbox{\boldmath$v$}_{1}}+\alpha_{2}e^{\lambda_{2}t}\tilde{\mbox{\boldmath$v$}_{2}},$ (63) where $\lambda_{1}$ and $\lambda_{2}$ are given in (54) and (55), respectively, and $\tilde{\mbox{\boldmath$v$}_{1}}$ and $\tilde{\mbox{\boldmath$v$}_{2}}$ are the eigenvectors of the matrix $\tilde{\mbox{\boldmath$R$}}^{T}$ $\displaystyle\tilde{\mbox{\boldmath$v$}_{1}}$ $\displaystyle=$ $\displaystyle\displaystyle{\left[r_{31}S,r_{13}S-r_{13}r_{35}\right]^{T}}$ $\displaystyle\tilde{\mbox{\boldmath$v$}_{2}}$ $\displaystyle=$ $\displaystyle\displaystyle{\left[r_{31}S,r_{13}S-S^{2}+r_{13}r_{35}\right]^{T}}.$ Using the initial condition $\displaystyle\mbox{\boldmath$p$}_{0}^{1,3}=\left[\begin{array}[]{ll}\displaystyle{\frac{r_{31}}{r_{13}+r_{31}}},&\displaystyle{\frac{r_{13}}{r_{13}+r_{31}}}\\\ \end{array}\right]^{T},$ we obtain $\displaystyle\alpha_{1}=\displaystyle{\frac{S^{2}+r_{13}r_{35}}{S^{3}(r_{13}+r_{35})}}\approx\displaystyle{\frac{1}{(r_{13}+r_{31})^{2}}},\quad\alpha_{2}=\displaystyle{\frac{r_{13}r_{35}}{-S^{3}(r_{13}+r_{31})}}.$ Setting $P(t)=\mbox{\boldmath$p$}_{t}^{1,3}(1,1)+\mbox{\boldmath$p$}_{t}^{1,3}(2,1)$ and substituting the values of $\alpha_{1}$, $\alpha_{2}$, $\tilde{\mbox{\boldmath$v$}_{1}}$ and $\tilde{\mbox{\boldmath$v$}_{2}}$ into (63), we obtain $\displaystyle P(t)=e^{\lambda_{1}t}-e^{\lambda_{2}t}\displaystyle{\frac{(r_{13}r_{35})^{2}}{(r_{13}+r_{31})^{4}}}.$ (65) Hence $\displaystyle\bar{\tau}_{1,3}^{c}=E[\tau_{1,3}^{c}]=\int_{0}^{\infty}P(t)\,dt\sim\displaystyle{\frac{1}{\|\lambda_{1}\|}}=\displaystyle{\frac{S}{r_{13}r_{35}}}\simeq\displaystyle{\frac{1}{r_{35}}}\displaystyle{\left(1+\displaystyle{\frac{r_{31}}{r_{13}}}\right)}.$ (66) Using (57) in (66) and setting $P_{3}^{R}=\displaystyle\frac{\bar{\tau}_{3}}{\bar{\tau}_{3}+\bar{\tau}_{1}}$, we obtain that $\displaystyle\bar{\tau}_{1,3}^{c}=\displaystyle{\frac{1}{r_{35}}}\displaystyle{\left(1+\displaystyle{\frac{\bar{\tau}_{1}}{\bar{\tau}_{3}}}\right)}=\displaystyle{\frac{1}{r_{35}}}\frac{1}{P_{3}^{R}},$ (67) We further coarse-grain the trajectories of the process $\tilde{x}(t)$ into that of a telegraph process with two states, closed state (corresponding to $x_{1}$ and $x_{3}$) and open state (corresponding to $x_{5}$) $c\rightarrow o.$ Denoting $\bar{\tau}_{o}$ as the mean first passage time from $x_{5}$, using equation (67) we get $\displaystyle P_{open}=\displaystyle\frac{\bar{\tau}_{o}}{\bar{\tau}_{o}+\bar{\tau}_{1,3}^{c}}=\displaystyle\frac{\bar{\tau}_{o}}{\bar{\tau}_{o}+\displaystyle{\frac{1}{r_{35}}}\frac{1}{P_{3}^{R}}}.$ (68) Applying the theory proposed in 3.1, to the dynamics between the closed states $x_{1}$ and $x_{3}$, the SR effect increases $P_{3}^{R}$, by using a negative depolarization, $c$ . According to (68) an increase of $P_{3}^{R}$ causes to an increase of $P_{open}$. ### 4.1 High barrier approximation to $r_{35}$, $r_{31}$, $r_{13}$ We consider the autonomous stochastic differential equation $\displaystyle dx$ $\displaystyle=$ $\displaystyle-\phi^{\prime}(x)\,dt+\sqrt{2\varepsilon}\,dw$ (69) $\displaystyle x\left(0\right)$ $\displaystyle=$ $\displaystyle x.$ The transition rates between the wells are the probability fluxes in the direction of the transition at the top of the barrier. Thus, denoting by $\Phi_{i}(x)$ and $\lambda_{i}$ the principal eigenfunction and eigenvalue in well $i\ (i=1,3)$, we have $\displaystyle r_{13}=-\varepsilon{\Phi^{\prime}_{1}}(x_{2}),\quad r_{31}=\varepsilon{\Phi^{\prime}_{3}}(x_{2}),\quad r_{35}=-\varepsilon{\Phi^{\prime}_{3}}(x_{4}).$ (70) To calculate the fluxes, we have to construct the eigenfunctions $\Phi_{i}(x)$, which are the solutions of $\displaystyle\varepsilon\Phi_{1}^{\prime\prime}(x)+\left[\phi^{\prime}(x)\Phi_{1}(x)\right]^{\prime}$ $\displaystyle=$ $\displaystyle-\lambda_{1}\Phi_{1}(x)\quad\mbox{for}\quad-\infty<x<x_{2}$ (71) $\displaystyle\Phi_{1}(x_{2})$ $\displaystyle=$ $\displaystyle 0,\quad\Phi_{1}(x)\to 0\quad\mbox{for}\quad x\to-\infty$ (72) $\displaystyle\varepsilon\Phi_{3}^{\prime\prime}(x)+\left[\phi^{\prime}(x)\Phi_{3}(x)\right]^{\prime}$ $\displaystyle=$ $\displaystyle-\lambda_{3}\Phi_{3}(x)\quad\mbox{for}\quad- x_{2}<x<x_{4}$ (73) $\displaystyle\Phi_{3}(x_{2})$ $\displaystyle=$ $\displaystyle 0,\quad\Phi_{3}(x_{4})=0.$ (74) The asymptotic structure of the eigenfunctions is given in [19] as $\displaystyle\Phi_{1}(x)$ $\displaystyle\sim$ $\displaystyle-{\cal N}_{1}^{-1}e^{-\phi(x)/\varepsilon}\sqrt{\frac{2}{\pi}}\int_{0}^{\omega_{2}(x-x_{2})/\sqrt{\varepsilon}}e^{-z^{2}/2}dz\quad\mbox{for}\quad x<x_{2}$ (75) $\displaystyle\Phi_{3}(x)$ $\displaystyle\sim$ $\displaystyle{\cal N}_{3}^{-1}e^{-\phi(x)/\varepsilon}\sqrt{\frac{2}{\pi}}\left[\int_{\omega_{4}(x-x_{3})/\sqrt{\varepsilon}}^{\omega_{2}(x-x_{2})/\sqrt{\varepsilon}}e^{-z^{2}/2}dz-1\right]\quad\mbox{for}\quad x_{2}<x<x_{4},$ (76) where $\omega_{i}=\sqrt{\|\phi^{\prime\prime}(x_{i})\|},\ (i=1,2,3,4)$ and $\displaystyle{\cal N}_{1}=\int_{-\infty}^{x_{2}}\Phi_{1}(x)\,dx\sim\frac{\sqrt{2\pi}}{\omega_{1}}e^{-\phi(x_{1})/\varepsilon},\quad{\cal N}_{3}=\int_{x_{2}}^{x_{4}}\Phi_{3}(x)\,dx\sim\frac{\sqrt{2\pi}}{\omega_{3}}e^{-\phi(x_{3})/\varepsilon}.$ (77) According to (70) and (75)-(77), $\displaystyle r_{13}\sim\frac{\omega_{1}\omega_{2}}{\pi}e^{-[\phi(x_{2})-\phi(x_{1})]/\varepsilon},\quad r_{31}\sim\frac{\omega_{3}\omega_{2}}{\pi}e^{-[\phi(x_{2})-\phi(x_{3})]/\varepsilon}\quad r_{35}\sim\frac{\omega_{3}\omega_{4}}{\pi}e^{-[\phi(x_{4})-\phi(x_{3})]/\varepsilon},$ (78) which are Kramers’ rates for the corresponding barriers [19]. ## 5 Effect of SR in the Luo-Rudy model of cardiac myocytes The Luo-Rudy model [3] describes ionic concentrations and cardiac ventricular action potential by a system of $21$ ordinary differential equations. It reflects the guinea-pig electrophysiology by detailed Hodgkin-Huxley models of ionic currents. The most significant currents are the slow $I_{\mbox{\scriptsize Ks}}$ and rapid $I_{\mbox{\scriptsize Kr}}$ delayed rectifier potassium currents, a time-independent potassium current, a plateau potassium current (ultra-rapid $I_{\mbox{\scriptsize Kur}}$), a transient outward current, fast and background sodium currents, L- and T-type calcium currents, a background calcium current, calcium pumps, sodium-potassium pumps, and sodium-calcium exchangers. In addition, the model describes $Ca^{2+}$ handling processes, that is, calcium dynamic release from the sarcoplasmic- reticulum and from the calcium buffers troponin, calmodulin, and calsequestrin. The stochastic resonance described above changes the open probability of the $I_{\mbox{\scriptsize Ks}}$ channel, and therefore it affects its conductance. To incorporate this effect into the Luo-Rudy model, we modify the Hodgkin- Huxley equation for the $I_{\mbox{\scriptsize Ks}}$ current-voltage relation by shifting the stationary open probability of the channel in the Luo-Rudy model [3], $\displaystyle P_{O}(V)=\displaystyle\frac{1}{1+\displaystyle\exp\left\\{-\displaystyle\frac{V-1.5}{16.7}\right\\}},$ (79) to $\displaystyle\tilde{P}_{O}(V)=\displaystyle\frac{1}{1+\displaystyle\exp\left\\{-\displaystyle\frac{V+4.12}{16.7}\right\\}},$ (80) which imitates the experimentally observed shift (see figure 1). Figure 17: Resonant increase (red) of $20\%$ in the open probability of the $I_{\mbox{\scriptsize Ks}}$ channel, and normal regime (blue) This changes the channel conductance $\bar{G}_{\mbox{\scriptsize Ks}}P_{O}^{2}(V)$ in the Luo-Rudy model ($\bar{G}_{\mbox{\scriptsize Ks}}$ is the open channel conductance) to $\bar{G}_{\mbox{\scriptsize Ks}}\tilde{P}_{O}^{2}(V)$, which changes, in turn, the membrane potassium current $\langle I_{\mbox{\scriptsize Ks}}\rangle$, averaged over many channels, to [3] $\displaystyle\langle I_{\mbox{\scriptsize Ks}}\rangle\to\bar{G}_{\mbox{\scriptsize Ks}}\tilde{P}_{O}^{2}(V)(V-E_{\mbox{\scriptsize Ks}})\quad\mbox{as}\quad t\to\infty$ (81) ($E_{Ks}$ is the reversal potential of the channel). The effect of this modification of the Luo-Rudy model is shown in Figure 11. The duration of the action potential is reduced and accordingly, the peak of the cytosolic calcium concentration is lowered (see Figures 12), as in the experiment described in the Introduction. On the other hand, sodium concentration is increased (see Figure 13). The shortening of the action potential duration in the ventricular cardiac myocytes affects the QT interval in the electrocardiogram, which consists of a sum of several different action potentials created in the myocardium [10] (see Figures 14, 15). These theoretical predictions are supported by experimental measurements. Specifically, these effects in vivo were communicated in [13], [14], as well as in our own in vitro measurements [2]. ## 6 Conclusion and Discussion This paper tries to explain the results of the experiment of exposing human potassium $I_{\mbox{\scriptsize Ks}}$ channels and cardiac myocytes, which contain these channels, to weak and slow electromagnetic fields. We offer a scenario of a new kind of stochastic resonance between the induced periodic field and the thermally activated transitions between locally stable configurations of the mobile ions in the selectivity filter. More specifically, since the induced electric field is too weak to interact with any component of the $I_{\mbox{\scriptsize Ks}}$ channel protein, our model cannot describe the primary gating mechanism of a voltage gated channel. We therefore resort to a mathematical model, which postulates interaction of the induced field with configurations of the mobile ions inside the selectivity filter. These configurations may be much more susceptible to the weak induced field than any components of the surrounding protein, because the potential barriers separating the metastable configurations of the mobile ions can be of any height. According to our scenario, the observed resonance is due to the dependence of the induced electric field amplitude on frequency, in contrast to an applied external electric field with fixed frequency, which is known not to exhibit stochastic resonance with changing frequency and depolarization. In our theory the observed SR between two closed (or inactivated) states affects the open probability of the channel. Our model describes the dynamics of a Brownian particle in an asymmetric bistable potential forced by a periodic induced electric field. The analysis of this model is based on the construction of an asymptotic solution to the time-periodic Fokker-Planck equation in the WKB form. We evaluate the dependence of the steady state probability to be on one side of the potential barrier on the frequency, amplitude, depolarization, and noise intensity. Our main results are shown in Figure 10, which indicates that there is a peak in the open probability in a relatively narrow range of depolarizations and frequencies. We refer to this peak as stochastic resonance, though it is not be the usual SR phenomenon. This observation is consistent with the results of the $I_{\mbox{\scriptsize Ks}}$ channel experiment mentioned in the Introduction. Another result is the incorporation of the SR result into the Luo-Rudy model of cardiac myocytes. We found that the increased conductance of the $I_{\mbox{\scriptsize Ks}}$ channel reduces the duration of the action potential, the peak height of the cytosolic calcium concentration, in good agreement with the experimental results. The shortening of the action potential duration in the ventricular cardiac myocytes affects the QT interval in the electrocardiogram. Acknowledgment: We wish to thank S. Laniado, T. Kamil and M. Scheinowitz for introducing us to the in vivo resonance experiments, T. Zinman, A. Shainberg and S. Barzilai for the in vitro cardiac myocytes experiments, G. Gibor and B. Attali for the oocyte experiments, and N. Dascal and A. Moran for experiments on L-type channels. We thank Y. Rudy, F. Bezanilla, and G. Deutscher for useful discussions. ## References * [1] M. Shaked, G. Gibor, B. Attali and Z. Schuss, ”Weak EMF at 16 Hz increases conductance of $I_{\mbox{\scriptsize Ks}}$ and KCNQ1 channels in a narrow window of depolarizations”, (preprint 2009). * [2] M. Shaked, T. Zinman, A. Shainberg and Z. Schuss, ”The effect of extremely low frequency and amplitude electromagnetic fields in cytosolic calcium of cardiac myocytes”, (preprint 2009). * [3] J. Zeng, K.R. Laurita, D.S. Rosenbaum, Y. Rudy, “Two components of the delayed rectifier K+ current in ventricular myocyctes of the Guinea pig type”, Circ. Res. 77, pp.140–152 (1995). * [4] S. Bernèche and B. Roux, ”Energetics of ion conduction through the K+ channel”, Nature, 414, pp.73-77 (2001). * [5] L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, ”Stochastic Resonance”, Rev. Mod. Phys. 70, pp.223–288 (1998). * [6] A. Nikitin, N. G. Stocks and A. R. Bulsara, ”Asymmetric bistable systems subject to periodic and stochastic forcing in the strongly nonlinear regime: Switching time distributions”, Phys. Rev E 68, 016103 (2003). * [7] H.S. Wio and S. Bouzat, ”Stochastic Resonance: The role of Potential Asymmetry and Non Gaussian Noises”, Brazilian Journal of Physics 29 (1), pp.1-8 (1999). * [8] J.H. Li, ”Effect of asymmetry on stochastic resonance and stochastic resonance induced by multiplicative noise and by mean-field coupling”, Phys. Rev. E 66, pp.0311041-0311047 (2002). * [9] A. L. Hodgkin and A. F. Huxley, ”A Quantitative Description of Membrane Current and its Application to Conduction and Excitation in Nerve”, J. Physiology 117, pp.500–544 (1952). * [10] L.H. Opie, Heart Physiology: From Cell to Circulation, Lippincott, Williams & Wilkins; 4th edition 2003. * [11] C. Koch, Biophysics of Computation, Oxford University Press, NY 1999. * [12] D. Johnston and S.M. Wu, Foundations of Cellular Neurophysiology, MIT Press, Cambridge, MA 1995. * [13] R. Mazhari, H.B. Nuss, A.A. Armoundas, R.L. Winslow, E. Marban, ”Ectopic expression of KCNE3 accelerates cardiac repolarization and abbreviates the QT interval”, J. Clin. Invest. 109, pp.1083-1090 (2002). * [14] J.H. Jeong, J.S. Kim, B.C. Lee, Y.S. Min, D.S. Kim, J.S. Ryu, K.S. Soh, K.M. Seo, U.D. Sohn, ”Influence of exposure to electromagnetic field on the cardiovascular system”, Autonomic & Autacoid Pharmacology 25 (1), pp.17-23 (7) (2005). * [15] S.M. Ross, Stochastic Processes, John Wiley & Sons, Inc. NY 1983. * [16] M.A. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, NY 1984. * [17] R. Graham and T. Tél, ”Weak-noise limit of Fokker-Planck models and nondifferentiable potentials for dissipative dynamical systems”, Phys.Rev A 31 (2), pp.1109–1122 (1985). * [18] B.J. Matkowsky,Z. Schuss and C.Tier “Uniform expansion of the transition rate in Kramers’ problem”, J. Stat. Phys. 35(3/4), pp. 443–456 (1984). * [19] Z. Schuss, Theory and Applications of Stochastic Differential Equations, Wiley, NY 1980.	arxiv-papers	2009-03-03T11:55:03	2024-09-04T02:49:00.950665	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Meir Shaked and Zeev Schuss", "submitter": "Zeev Schuss", "url": "https://arxiv.org/abs/0903.0506" }
0903.0569	# Factorization of low-energy gluons in exclusive processes Geoffrey T. Bodwin High Energy Physics Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA Xavier Garcia i Tormo High Energy Physics Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2G7 111Current address Jungil Lee Department of Physics, Korea University, Seoul 136-701, Korea ###### Abstract We outline a proof of factorization in exclusive processes, taking into account the presence of soft and collinear modes of arbitrarily low energy, which arise when the external lines of the process are taken on shell. Specifically, we examine the process of $e^{+}e^{-}$ annihilation through a virtual photon into two light mesons. In an intermediate step, we establish a factorized form that contains a soft function that is free of collinear divergences. In contrast, in soft-collinear effective theory, the low-energy collinear modes factor most straightforwardly into the soft function. We point out that the cancellation of the soft function, which relies on the color- singlet nature of the external hadrons, fails when the soft function contains low-energy collinear modes. ###### pacs: 12.38.-t ††preprint: ANL-HEP-PR-09-6 Alberta Thy 08-10 ## I Introduction Factorization theorems are fundamental to modern calculations in QCD of the amplitudes for hard-scattering exclusive hadronic processes. They allow one to separate contributions to the amplitudes that involve states of high virtuality from those that involve states of low virtuality. The former, short-distance contributions can, by virtue of asymptotic freedom, be calculated in perturbation theory, while the latter, long-distance contributions are parametrized in terms of inherently nonperturbative matrix elements of QCD operators in hadronic states. States of low virtuality can arise from the emission of a soft gluon, whose four-momentum components are all small, or from the emission of a collinear gluon, whose four-momentum is nearly parallel to the four-momentum of a gluon or light quark. In some discussions of factorization that employ soft- collinear effective theory (SCET) Bauer:2000yr ; Bauer:2001yt ; Bauer:2002nz or diagrammatic methods Bodwin:2008nf , it is assumed that gluons can have no transverse momentum components that are smaller than the QCD scale $\Lambda_{\rm QCD}$. That is, gluons can have hard momentum, in which all components are of order the hard-scattering scale $Q$, soft momentum, in which all components are of order $\Lambda_{\rm QCD}$, or collinear momentum, in which the transverse components are of order $\Lambda_{\rm QCD}$ and the energy and longitudinal spatial component are much larger than $\Lambda_{\rm QCD}$ (usually taken to be of order $Q$). This assumption is appropriate to the discussion of physical hadrons, in which confinement provides a nonperturbative IR cutoff of order $\Lambda_{\rm QCD}$. However, in perturbative matching calculations of short-distance coefficients, one usually takes the external quark and gluon states to be on their mass shells, and, in this situation, soft and collinear gluons of arbitrarily low energy can be emitted. In order to establish the consistency of such calculations, one must prove, to all orders in perturbation theory, that these soft and collinear gluons factor from the hard-scattering process and that the factorized form is identical to the conventional one that is obtained in the presence of an infrared cutoff of order $\Lambda_{\rm QCD}$.222In Ref. Manohar:2005az , it was asserted that, if a factorized form exists when one considers only modes with scales of order $\Lambda_{\rm QCD}$ or greater, then the short-distance coefficients are independent of all infrared modes, even in perturbative calculations in which modes with scales below $\Lambda_{\rm QCD}$ are present. This was shown to be the case in a one-loop example. However, no general, all- orders proof of that assertion was given. In the absence of such a proof, one would have no guarantee in the matching calculation that low-virtuality soft and collinear contributions either cancel or can be absorbed entirely into the standard nonperturbative functions (distribution functions in inclusive processes and distribution amplitudes in exclusive processes). At the one-loop level, gluons of arbitrarily low energy can be treated along with higher-energy soft and collinear gluons, and the conventional proofs of factorization apply. However, as we shall see, the proof of factorization of low-energy gluons becomes more complicated beyond one loop. In multiloop integrals, in the on-shell case, one finds contributions at leading power in the hard-scattering momentum in which collinear gluons of low energy couple to soft gluons. Our goal is to construct a proof of factorization that takes this possibility into account. To our knowledge, the existing discussions of factorization, either in the context of SCET or diagrammatic methods, have not addressed this possibility. In on-shell perturbative calculations in SCET, gluon transverse momenta extend to zero. Hence, the possibility of low-energy gluons with momenta collinear to one of the external particles arises. At one-loop level, the soft and collinear contributions can be separated through the use of an additional cutoff Manohar:2006nz . However, as we have already mentioned, at two-loop level and higher, a low-energy collinear gluon can attach to a soft gluon. The SCET action is formulated so that soft gluons can be decoupled from collinear gluons through a field redefinition, but there is no corresponding provision to decouple collinear gluons from soft gluons. Therefore, it seems that, in SCET, low-energy collinear gluons would be treated most straightforwardly as part of the soft (or ultrasoft) contribution. This results in a factorized form in which the soft function contains gluons with both soft and collinear momenta and, hence, contains both soft and collinear divergences. Alternatively, one can consider a factorized form in which gluons with collinear momenta are factored completely from the soft function, so that they reside only in jet functions that are associated with the initial- or final- state hadrons. Such an alternative factorized form, in which the soft function is free of collinear divergences, has been discussed in the context of factorization for the Drell-Yan process in Refs. Bodwin:1984hc ; Collins:1985ue ; Collins:1989gx , although the details of the factorization of gluons with collinear momenta from the soft function were not given. This alternative factorized form has also been discussed in connection with resummation of logarithms in, for example, Refs. Collins:1984ik ; Sterman:1986aj ; Contopanagos:1996nh ; Kidonakis:1998bk ; Kidonakis:1998nf ; Sterman:2002qn . Furthermore, it has been discussed in an axial gauge in the context of on-shell quark scattering Sen:1982bt . Axial gauges are somewhat problematic, in that they introduce unphysical singularities into gluon and ghost propagators. Such singularities could potentially spoil contour- deformation arguments that are used to ascertain the leading regions of integration in Feynman diagrams Collins:1982wa .333For a discussion of a class of gauges that may ameliorate some of these difficulties, see Ref. Sterman:1978bi . For this reason, we believe that it is important to construct a proof of factorization in a covariant gauge, such as the Feynman gauge, which we employ in the present paper. A factorized form in which the soft function contains no gluons with collinear momenta has several useful features. One is that contributions in which there are two logarithms per loop (one collinear and one soft) reside entirely in the jet functions, which have a diagonal color structure, rather than in the soft function, which has a more complicated color structure. Here we focus on a feature that is crucial for factorization proofs: A factorized form in which the soft function contains no collinear modes allows one to establish a cancellation of the soft function when it connects to a color-singlet hadron. As we shall explain below, if the soft function contains gluons with collinear momenta, then the cancellation of the soft function fails at leading order in the large momentum scale. In this paper, we outline the proof of factorization at leading order in the hard-scattering momentum for the case of on-shell external partons. For concreteness, we discuss the example of the exclusive production of two light mesons in $e^{+}e^{-}$ annihilation. In an intermediate step, the factorized form that we obtain contains a soft function that is free of collinear divergences. This allows us to demonstrate the cancellation of the soft function at leading order in the large momentum scale. Our proof makes use of standard all-orders diagrammatic methods for proving factorization Bodwin:1984hc ; Collins:1985ue ; Collins:1989gx . We find that the factorization of gluons of arbitrarily low energy can be dealt with conveniently by focusing on the factorization of contributions to loop integrals from singular regions, i.e., regions that contain the soft and collinear singularities. Such singular regions are discussed in Refs. Collins:1985ue ; Collins:1989gx . However, the coupling of low-energy collinear gluons to soft gluons is not discussed in those papers. The remainder of this paper is organized as follows. In Sec. II we describe the model that we use for the production amplitude. Section III contains a heuristic discussion of the regions of loop momenta that give leading contributions. This discussion is aimed at making contact with previous work on factorization and also sets the stage for a more precise discussion of the singular regions of loop momenta. In Sec. IV, we discuss the diagrammatic topology of the leading contributions and also the topology of the soft and collinear singular contributions. We treat the collinear and soft contributions by making use of collinear and soft approximations that are valid in the singular regions. These are discussed in Sec. V, along with the decoupling relations for the collinear and soft singular contributions. In Sec. VI, we outline the factorization of the collinear and soft singularities and describe how one arrives at the standard factorized form for the production amplitude. We also outline the proof of factorization in the case in which the relative momentum between the quark and the antiquark in a meson is taken to be nonzero. Here, we discuss the difficulty that arises in the cancellation of the soft function if the soft function contains gluons with collinear momenta. Finally, in Sec. VII, we summarize our results. ## II Model for the amplitude Let us consider the exclusive production of two light mesons in $e^{+}e^{-}$ annihilation through a single virtual photon. We work in the $e^{+}e^{-}$ center-of-momentum frame and in the Feynman gauge, and we write four-vectors in terms of light-cone components: $k=(k^{+},k^{-},\bm{k}_{\perp})$, with $k^{\pm}=(1/\sqrt{2})(k^{0}\pm k^{3})$. We take each meson to be moving in the plus (minus) direction and to consist of an on-shell quark with momentum $p_{1q}$ ($p_{2q}$) and an on-shell antiquark with momentum $p_{1\bar{q}}$ ($p_{2\bar{q}}$): $\displaystyle p_{1q}$ $\displaystyle=$ $\displaystyle\left[\frac{z_{1}Q}{\sqrt{2}},\frac{\bm{p}_{1\perp}^{2}}{\sqrt{2}z_{1}Q},\bm{p}_{1\perp}\right],$ (1a) $\displaystyle p_{1\bar{q}}$ $\displaystyle=$ $\displaystyle\left[\frac{(1-z_{1})Q}{\sqrt{2}},\frac{\bm{p}_{1\perp}^{2}}{\sqrt{2}(1-z_{1})Q},-\bm{p}_{1\perp}\right],$ (1b) $\displaystyle p_{2q}$ $\displaystyle=$ $\displaystyle\left[\frac{\bm{p}_{2\perp}^{2}}{\sqrt{2}z_{2}Q},\frac{z_{2}Q}{\sqrt{2}},\bm{p}_{2\perp}\right],$ (1c) $\displaystyle p_{2\bar{q}}$ $\displaystyle=$ $\displaystyle\left[\frac{\bm{p}_{2\perp}^{2}}{\sqrt{2}(1-z_{2})Q},\frac{(1-z_{2})Q}{\sqrt{2}},-\bm{p}_{2\perp}\right],$ (1d) where $0<z_{i}<1$ and $z_{i}$ does not lie near the endpoints of its range. The momentum $P_{i}$ of the meson $M_{i}$ is given by $P_{i}=p_{iq}+p_{i\bar{q}}.$ (2) The large scale $Q$ is equal to the invariant mass of the virtual photon, up to corrections of relative order $\bm{p}_{i\perp}/Q$. We assume that the components of $\bm{p}_{i\perp}$ are all of order $\Lambda_{\rm QCD}$. It is useful for subsequent discussions to introduce a dimensionless parameter $\lambda\equiv\Lambda_{\rm QCD}/Q.$ (3) In order to simplify the initial discussion, we set $\bm{p}_{i\perp}=0$. We will discuss at the end of the factorization argument the effect of keeping the $\bm{p}_{i\perp}$ nonzero. ## III Leading regions of loop momenta Let us now discuss the regions of loop momenta that are leading in powers of the large scale $Q$. Our analysis will be somewhat heuristic, in that, as we will see, the boundaries between the various momentum types are indistinct. We carry out this analysis in order to make contact with previous discussions of factorization and to set the stage for our proof of factorization. That proof focuses on the soft and collinear singular regions of loop momenta, which are distinct.444Power counting in the neighborhoods of pinch singularities has been discussed in Refs. Sterman:1978bi ; Libby:1978bx . Suppose that a virtual gluon with momentum $k$ attaches to external $q$ or $\bar{q}$ lines with momentum $p_{i}$ and $p_{j}$. (In the remainder of this paper, we call lines that originate in an external $q$ or $\bar{q}$ “outgoing fermion lines.”) In the limit in which the components of $k$ are all small compared to the largest components of $p_{i}$ and $p_{j}$, the amplitude associated with this process is proportional to $\int d^{4}k\frac{4p_{i}\cdot p_{j}}{(2p_{i}\cdot k+i\varepsilon)(-2p_{j}\cdot k+i\varepsilon)}\frac{1}{k^{2}+i\varepsilon}.$ (4) Because the integral is independent of the scale of $k$, leading contributions arise from arbitrarily small momentum $k$. One can emit an additional virtual gluon of momentum $k^{\prime}$ from an outgoing fermion line at a point to the interior of the emission of a gluon with momentum $k$, provided that $k^{\prime}\cdot p_{i}\gtrsim k\cdot p_{i}$. Such emissions are arranged in a hierarchy along the outgoing fermion lines, according to the virtualities that the emissions produce on the outgoing fermion lines. Now let us establish some nomenclature to describe the regions of loop momenta that yield contributions that are leading in powers of the large scale $Q$. We call such momentum regions “leading regions.” We outline below the construction of an argument to prove that these are the only possible leading regions. We consider hard ($H$), soft ($S$), collinear-to-plus ($C^{+}$), and collinear-to-minus ($C^{-}$) momenta, whose components have the following orders of magnitude: $\displaystyle H\,\,\,$ : $\displaystyle Q(1,1,\bm{1}_{\perp}),$ (5a) $\displaystyle S\,\,\,\,$ : $\displaystyle Q\epsilon_{S}(1,1,\bm{1}_{\perp}),$ (5b) $\displaystyle C^{+}$ : $\displaystyle Q\epsilon^{+}[1,(\eta^{+})^{2},\bm{\eta}^{+}_{\perp}],$ (5c) $\displaystyle C^{-}$ : $\displaystyle Q\epsilon^{-}[(\eta^{-})^{2},1,\bm{\eta}^{-}_{\perp}].$ (5d) We call a line in a Feynman diagram that carries momentum of type $X$ an “$X$ line.” The parameters $\epsilon_{S}$, $\epsilon^{+}$, and $\epsilon^{-}$ set the energy scales of the momenta. We define the soft region of momentum space by the condition $\epsilon_{S}\ll 1.$ (6) We define the collinear region of momentum space by the conditions $\displaystyle\epsilon^{\pm}$ $\displaystyle\lesssim$ $\displaystyle 1,$ $\displaystyle\eta^{\pm}$ $\displaystyle\ll$ $\displaystyle 1.$ (7) In our definitions of momentum regions, the positions of the boundaries between regions are somewhat vague. That is because there is no clear distinction between the $H$, $S$, and $C^{\pm}$ regions near the boundaries between regions: When $\epsilon_{S}\sim 1$, an $S$ momentum is essentially an $H$ momentum; when $\eta^{\pm}\sim 1$, a $C^{\pm}$ momentum is essentially an $S$ momentum. Soft singularities occur in the limit $\epsilon_{S}\to 0$, and $C^{\pm}$ singularities occur in the limits $\eta^{\pm}\to 0$. Hence, we see that, unlike the soft and collinear momentum regions, the soft and collinear singularities are distinct. There are also singularities that are associated with the scales of the collinear momenta. These appear in the limit $\epsilon^{\pm}\to 0$. If $\eta^{\pm}$ is finite, these are essentially soft singularities, but they can occur in conjunction with a collinear singularity if $\eta^{\pm}\to 0$. We do not consider gluon loop momenta of the “Glauber” type Bodwin:1981fv , in which $k^{+},k^{-}\ll\|\bm{k}_{\perp}\|$. The reason for this is that, for exclusive processes, the $k^{+}$ and $k^{-}$ contours of integration are not pinched in the Glauber region, and, hence, one can always deform them out of that region Collins:1982wa . If we take $\epsilon^{\pm}$ to be of order one and $\epsilon_{S}$ and $\eta^{\pm}$ to be of order $\lambda$ [Eq. (3)], then the resulting momenta are those that are treated in SCETII Bauer:2002aj . Soft momenta with $\epsilon_{S}$ of order $\lambda^{2}$ have been considered in Ref. Beneke:2000ry in the context of two-loop-order contributions to $B$-meson decays, and the possibility of leading momentum regions involving momenta of arbitrarily small energy is mentioned in Ref. Smirnov:1999bza for the case of massive particles. \| $k$ $\backslash$ $p$ \| $S$ \| $C^{\pm}$ \| $\tilde{C}^{\pm}$ ---\|---\|---\|--- $S$ \| $\epsilon_{S_{k}}\sim\epsilon_{S_{p}}$ \| $\epsilon_{p}^{\pm}(\eta^{\pm}_{p})^{2}\lesssim\epsilon_{S_{k}}\ll\epsilon_{p}^{\pm}$ \| $\epsilon_{p}^{\pm}\tilde{\eta}^{\pm}_{p}\lesssim\epsilon_{S_{k}}\ll\epsilon_{p}^{\pm}$ \| $k$ $\backslash$ $p$ \| $S$ \| $C^{\mp}$ \| $\tilde{C}^{\mp}$ \| $CC$ ---\|---\|---\|---\|--- $C^{\pm}$ \| $\epsilon^{\pm}_{k}\sim\epsilon_{S_{p}}$ \| $\epsilon_{p}^{\mp}(\eta^{\mp}_{p})^{2}\lesssim\epsilon^{\pm}_{k}\lesssim\epsilon^{\mp}_{p}$ \| $\epsilon^{\mp}_{p}\tilde{\eta}^{\mp}_{p}\lesssim\epsilon^{\pm}_{k}\lesssim\epsilon^{\mp}_{p}$ \| $\epsilon^{\pm}_{k}\sim\epsilon_{{CC}_{p}}$ $CC$ \| $\epsilon_{CC_{k}}\sim\epsilon_{S_{p}}$ \| $\epsilon_{p}^{\mp}(\eta^{\mp}_{p})^{2}\lesssim\epsilon_{CC_{k}}\ll\epsilon^{\mp}_{p}$ \| $\epsilon^{\mp}_{p}\tilde{\eta}^{\mp}_{p}\lesssim\epsilon_{CC_{k}}\ll\epsilon^{\mp}_{p}$ \| $\epsilon_{CC_{k}}\sim\epsilon_{CC_{p}}$ Table 1: Conditions under which a gluon with momentum $k$ can attach to a line with momentum $p$. In each table, the left-hand column gives the momentum type of the gluon with momentum $k$, and the top row gives the momentum type of the line with momentum $p$. Each entry gives the conditions that must be fulfilled if the attachment is to satisfy our conventions for attachments, as described in the text, and also yield a contribution that is not suppressed by powers of ratios of momentum components. For purposes of power counting, an $H$ line behaves as a soft line with $\epsilon_{S}\sim 1$. The rules for the attachment if $k$ is a $\tilde{C}^{\pm}$ momentum are the same as the rules of attachment if $k$ is a $C^{\pm}$ momentum. As is explained in the text, if $k$ is $S$, and the lines to which it attaches have momentum $p_{i}$ and $p_{j}$, then $p_{i}$ and $p_{j}$ cannot both be $C^{+}$ or $C^{-}$. Furthermore, if $k$ is $C^{\pm}$, then at least one of $p_{i}$ and $p_{j}$ is $C^{\pm}$. We wish to determine the configurations of the various momentum types in a Feynman diagram that are leading, in the sense that they are not suppressed by powers of the ratios of momentum components. In our analysis, we begin with the hard subdiagram plus the bare external $q$ and $\bar{q}$ for each meson. Then we add one gluon at a time to the diagram. (Each added gluon possibly contains quark, gluon, and ghost vacuum polarization loops.) There are many redundant procedures for adding gluons to obtain a diagram with a given momentum configuration. We adopt the following convention: We say that a gluon with momentum $l$ can attach to a line with momentum $p$ only if the momentum $p+l$ is predominantly of the same type as momentum $p$. For example, an $S$ gluon with momentum $l$ can attach to a $C^{\pm}$ line with momentum $p$ only if $\epsilon_{S}$ is of order $\epsilon^{\pm}\eta^{\pm}$ or smaller, so that the plus (minus) component of $p+l$ is the dominant component. Similarly, a $C^{\pm}$ gluon with momentum $l$ can attach to an $S$ gluon with momentum $p$ only if $\epsilon^{\pm}$ is of order $\epsilon_{S}$ or smaller, so that all components of $p+l$ are approximately equal. We call the sum of a $C^{\pm}$ momentum and an $S$ momentum with $\epsilon_{S}\sim\epsilon^{\pm}\eta^{\pm}$ a $\tilde{C}^{\pm}$ momentum. The sum of a $C^{\pm}$ momentum and a $C^{\mp}$ momenta with $\epsilon^{\pm}(\eta^{\pm})^{2}\ll\epsilon^{\mp}\ll\epsilon^{\pm}$ is also a $\tilde{C}^{\pm}$ momentum. We also allow the attachment of a $C^{\pm}$ momentum to a $C^{\mp}$ momentum with $\epsilon^{+}\sim\epsilon^{-}$, and, in this case, we call the sum of the $C^{+}$ momentum and $C^{-}$ momentum a $CC$ momentum. These combination momenta have the following orders of magnitude: $\displaystyle\tilde{C}^{+}$ : $\displaystyle Q\epsilon^{+}(1,\tilde{\eta}^{+},\bm{\eta}^{+}_{\perp}),$ (8a) $\displaystyle\tilde{C}^{-}$ : $\displaystyle Q\epsilon^{-}(\tilde{\eta}^{-},1,\bm{\eta}^{-}_{\perp}),$ (8b) $\displaystyle CC$ : $\displaystyle Q\epsilon_{CC}(1,1,\bm{\eta}_{CC\perp}),$ (8c) where $1\gg\tilde{\eta}^{\pm}\gg(\eta^{\pm})^{2}.$ (9) In order to determine the momenta of attached gluons that can result in a leading power count, it is useful to consider the expression (4). In the first two factors in the denominator of the expression (4), terms of the form $p_{i}^{2}$ and $k^{2}$ have been dropped. Thus, the denominator of the expression (4) gives a lower bound on the order of magnitude of the exact denominator. Because of our convention for the allowed momentum types for $k$, the numerator $p_{i}\cdot p_{j}$ in the expression (4) gives the leading behavior unless $p_{i}$ and $p_{j}$ are both either $C^{+}$ or $C^{-}$. For such cases, we need to consider numerator factors $k^{2}$, $k\cdot p_{i}$, and $k\cdot p_{j}$, in addition to $p_{i}\cdot p_{j}$. Otherwise, we can use the expression (4) as it stands to obtain an upper bound on the magnitude of the factors that appear when one adds a gluon. The expression (4) has the useful property that it is independent of the scales of the momenta $k$, $p_{i}$, and $p_{j}$, and so it can be used to determine rules for the leading momentum configurations that are independent of the scales of the momenta. From these considerations, it is easy to see that $k$ must be $S$, $C^{+}$, or $C^{-}$ in order to obtain a leading power count. We regard these momentum types as primary, in the sense that the loop-integration variables correspond to these momenta. Other momentum types can arise when we add these primary types, following our convention above for allowed attachments. It follows that, if $k$ is $S$, then $p_{i}$ and $p_{j}$ cannot both be $C^{+}$ or $C^{-}$. It also follows that, if $k$ is $C^{\pm}$, then at least one of $p_{i}$ and $p_{j}$ is $C^{\pm}$. If we restore the terms of the form $p_{i}^{2}$ and $k^{2}$ in the denominators of the expression (4), then there can be an additional suppression of the amplitude.555 In counting powers in this case, we assume that a $C^{\pm}$ line is off shell by an amount of order $Q^{2}(\epsilon^{\pm})^{2}(\eta^{\pm})^{2}$ and that an $S$ line is off shell by an amount of order $Q^{2}(\epsilon_{S})^{2}$. In the integrations over the momenta that are associated with the virtual particles, there are contributions from the neighborhoods of the mass-shell poles. However, because the poles in the $k^{+}$ and $k^{-}$ complex planes are well separated, one can always deform the $k^{+}$ and $k^{-}$ contours of integration into the complex plane such that a gluon never has virtuality smaller than of order the square of its transverse momentum. In order to obtain a leading contribution, we must have $\displaystyle k\cdot p$ $\displaystyle\gtrsim$ $\displaystyle k^{2},$ $\displaystyle k\cdot p$ $\displaystyle\gtrsim$ $\displaystyle p^{2}.$ (10) Taking into account the additional conditions in Eq. (10), we obtain the rules for the leading contributions that are given in Table 1. In Table 1, the symbol “$\sim$” means that quantities have the same order of magnitude. In each expression in Table I, if the quantity with subscript $k$ is much greater than the quantity with subscript $p$, then the attachment is not allowed because $p+k$ is not essentially of the same momentum type as $p$. If the quantity with subscript $k$ is much less than the quantity with subscript $p$, then the contribution is suppressed by a power of the ratio of those quantities.666Suppose that we add an $S$ gluon to a $C^{\pm}$ gluon with $\epsilon_{S}\sim\eta^{\pm}\epsilon^{\pm}$ or that we add a $C^{\mp}$ gluon to a $C^{\pm}$ gluon with $\epsilon^{\mp}\sim\eta^{\pm}\epsilon^{\pm}$. Then, the sum of the momenta is no longer of the $C^{\pm}$ type. Because this change in momentum can propagate through the diagram, such additions of gluons can affect vertices other than those of the added gluon and propagators other than those adjacent to a vertex of the added gluon. In these cases, one must check that the rules in Table 1 still allow the attachments at the affected vertices. The rules in Table 1 also apply when the added gluon attaches to one of the outgoing fermion lines. In that case, one sets $\eta^{+}=0$ or $\eta^{-}=0$ on the outgoing fermion line. In Table 1, we have not given the rules for the attachments of gluons with $C^{\pm}$ or $\tilde{C}^{\pm}$ momenta to lines with $C^{\pm}$ or $\tilde{C}^{\pm}$ momenta. The rules for such attachments are complicated and cannot be characterized simply in terms of the magnitudes of the momentum components, as is the case for the attachments listed in Table 1. For our purposes, it suffices to note that necessary conditions for such attachments are given in Eq. (10). The constraints in Eq. (10) imply that an attachment of a gluon to a given line is allowed only if the virtuality that it produces on that line is of order or greater than the virtuality that is produced by the gluons that attach to that line to the outside of the attachment in question. Here, and throughout this paper, “outside” means toward the on-shell ends of the external quark and antiquark lines. If a gluon with momentum $k$ of type $C^{\pm}$, $\tilde{C}^{\pm}$, $S$, $C^{\mp}$, or $CC$ attaches to a $C^{\pm}$ line from an on-shell outgoing quark or antiquark, it adds virtuality $Q^{2}\epsilon_{k}^{\pm}(\eta_{k}^{\pm})^{2}$, $Q^{2}\epsilon_{k}^{\pm}\tilde{\eta}_{k}^{\pm}$, $Q^{2}\epsilon_{S_{k}}$, $Q^{2}\epsilon_{k}^{\mp}$, or $Q^{2}\epsilon_{CC_{k}}$, respectively. ## IV Topology of the leading contributions ### IV.1 Topology of the leading momentum regions By taking into account the allowed gluon attachments in Table 1, one arrives at the topology of Feynman diagrams that is shown in Fig. 1. This topology is similar in appearance to topologies that have been discussed previously in connection with the identification of IR (pinch) singularities in Feynman diagrams Sterman:1978bi ; Sterman:1978bj ; Libby:1978bx ; Collins:1989gx . However, as we will explain, the subdiagrams in Fig. 1 contain finite ranges of momenta, whereas those in Refs. Sterman:1978bi ; Sterman:1978bj ; Collins:1989gx contain only infinitesimal neighborhoods of the soft and collinear singularities. (We will discuss the topology of the soft and collinear singularities in Sec. IV.3.) Figure 1: Leading regions for double light-meson production in $e^{+}e^{-}$ annihilation. The wavy line represents the virtual photon. In the topology of Fig. 1, there is a jet subdiagram for each of the collinear regions (corresponding to each light meson), there is a hard subdiagram that includes the production process at lowest order in $\alpha_{s}$, and there is a soft subdiagram. We include in the hard subdiagram all propagators that are off shell by order $Q^{2}$. That is, we include lines carrying both momentum $H$ and momentum $CC$ with $\epsilon_{CC}\sim 1$. (The propagators in the Born process carry momenta $CC$ with $\epsilon_{CC}\sim 1$.) The soft subdiagram includes gluons with $S$ momenta, which may contain quark, gluon, and ghost loops. The soft subdiagram attaches to the jet subdiagrams through any number of $S$-gluon lines, according to the rules in Table 1. Note that a gluon carrying momentum $S_{i}$ cannot attach to a line carrying momentum $S_{j}$ unless $\epsilon_{S_{i}}\sim\epsilon_{S_{j}}$, and so various part of the soft subdiagram cannot attach to each other. The $C^{\pm}$-jet subdiagram $J^{\pm}$ contains the external quark lines for the meson with $C^{\pm}$ momentum, as well as gluons with $C^{\pm}$ momenta, which may contain quark, gluon and ghost loops. We also include in $J^{\pm}$ lines carrying $CC$ momentum with $\epsilon_{CC}\ll 1$ that occur when a gluon carrying momentum $C^{\mp}$ from a $J^{\mp}$ jet attaches to a line carrying $C^{\pm}$ momentum in $J^{\pm}$. Each jet subdiagram attaches to the hard subdiagram through the external quark and antiquark lines and through any number of $C^{\pm}$ gluons with $\epsilon^{\pm}\sim 1$. A gluon carrying $C^{\pm}$ or $\tilde{C}^{\pm}$ momentum can connect the $J^{\pm}$ subdiagram to the $J^{\mp}$ subdiagram, but only with the attachments in Table 1. Of particular note is the fact that a gluon carrying momentum $C^{\pm}$ or $\tilde{C}^{\pm}$ can connect a $C^{\pm}$ jet to an $S$ line in the soft subdiagram, provided that $\epsilon^{\pm}\sim\epsilon_{S}$. This is a feature of scattering processes in the on-shell case that does not appear when one has an infrared cutoff of order $\Lambda_{\rm QCD}$. The factorization of gluons carrying collinear momenta from the soft subdiagram is one of the principal technical issues that we address in this paper. In order to prove factorization, we need to show that the nonperturbative contributions to Feynman diagrams (those with virtualities of order $\Lambda_{\rm QCD}^{2}$ or less) either cancel or can be factored into the meson distribution amplitudes. Specifically, we will argue that the nonperturbative contributions associated with the soft divergences factor from the $J^{\pm}$ subdiagrams and cancel and that the nonperturbative contributions associated with the $C^{\pm}$ divergences factor from the $J^{\mp}$, hard, and soft subdiagrams and can be absorbed into the $J^{\pm}$ meson distribution amplitude. These factorizations and cancellations establish that the production amplitude depends only on the properties of the individual mesons, and not on correlations between the two mesons, except through the hard subprocess. ### IV.2 Two-loop example In Fig. 2 we show a two-loop example in which a $C^{+}$ gluon attaches to an $S$ gluon. Figure 2: A two-loop example in which a $C^{+}$ gluon attaches to an $S$ gluon. The $V_{i}$ are the vertex factors, and the $D_{i}$ are the propagator factors. We take the $C^{+}$ momentum to be $l_{1}=Q\epsilon^{+}(1,\eta^{2},\bm{\eta}^{+}_{\perp})$ and the $S$ momentum to be $l_{2}=Q\epsilon_{S}(1,1,\bm{1}_{\perp})$. We assume that $\epsilon^{+}\lesssim\epsilon_{S}$, and we route the $l_{1}$ momentum through the $D_{5}$ propagator. Then, we find the factors for the diagram that are shown on the right side of Fig. 2. Combining these factors, we obtain the following order of magnitude for the two-loop correction: $\epsilon_{S}\epsilon^{+}/(\epsilon_{S}^{2}+\epsilon_{S}\epsilon^{+})$. We see that this result is independent of $Q$, as expected, and is also independent of $\eta$. This contribution is leading if $\epsilon^{+}\sim\epsilon_{S}$, but it vanishes in the limit $\epsilon^{+}/\epsilon_{S}\to 0$, in accordance with the rule in Table 1. ### IV.3 Topology of the singular momentum regions In the preceding discussion, as we have noted, the soft and collinear momentum regions are not well distinguished. If $\eta^{\pm}\sim 1$, then a collinear momentum is virtually identical to a soft momentum. Similarly, if the components of a soft momentum have significantly different sizes, then a soft momentum can be virtually identical to a collinear momentum. In discussions of factorization, we rely on collinear approximations that are accurate only for $\eta^{\pm}\ll 1$. In order to apply such approximations, we must avoid the problems in distinguishing soft and collinear momenta that arise near the boundaries between these regions. Furthermore, the soft approximation for the attachment of a soft gluon to a $C^{\pm}$ line becomes inaccurate as the soft momentum becomes more nearly a $C^{\pm}$ momentum. Again, we encounter a problem that occurs near the boundary between momentum regions. In the discussion that follows, we avoid such boundary issues by focusing on infinitesimal neighborhoods of the soft and collinear singularities (singular regions).777It has been suggested that problems that arise near boundaries between momentum regions can be avoided by implementing a subtraction scheme that is akin to the Bogoliubov-Parasiuk-Hepp-Zimmerman formalism for subtraction of ultraviolet divergences Collins:1985ue ; Collins:1989gx . Such a subtraction scheme has not yet been constructed, although one-loop examples have been given in the context of the zero-bin-subtraction method of SCET Manohar:2006nz . As a first step in proving factorization, we will demonstrate the factorization of these singular regions. The topologies of soft and collinear singular regions have been discussed in the context of factorization theorems for inclusive processes in Refs. Collins:1985ue ; Collins:1989gx . These topologies follow from the rules for power counting that we have given in Sec. III. Let us describe the relationship of the topologies of the singular regions to the topologies in Fig. 1. The $C^{\pm}$ singularities reside in the outermost part of the $J^{\pm}$ subdiagram, which we call the $\tilde{J}^{\pm}$ subdiagram. (We consider the $\tilde{J}^{\pm}$ subdiagram to be part of the $J^{\pm}$ subdiagram, and we call the part of the $J^{\pm}$ subdiagram that excludes the $\tilde{J}^{\pm}$ subdiagram the $J^{\pm}-\tilde{J}^{\pm}$ subdiagram.) The soft singularities reside in the outermost part of the $S$ subdiagram, which we call the $\tilde{S}$ subdiagram. (We consider the $\tilde{S}$ subdiagram to be part of the $S$ subdiagram, and we call the part of the $S$ subdiagram that excludes the $\tilde{S}$ subdiagram the $S-\tilde{S}$ subdiagram.) $S$ singular gluons connect the $\tilde{S}$ subdiagram only to the $\tilde{J}^{\pm}$ subdiagrams. The $\tilde{J}^{\pm}$ subdiagrams connect to the $J^{\pm}$, $J^{\mp}$, $S$, and $H$ subdiagrams via $C^{\pm}$ gluons. We emphasize that the $\tilde{J}^{\pm}$ subdiagrams connect, via $C^{\pm}$ gluons to the $\tilde{S}$ subdiagram. This last type of connection is a feature that was not included in the discussion of leading (pinch) singularities in Refs. Collins:1985ue ; Collins:1989gx . Otherwise, the topologies that we find are the same as in Refs. Collins:1985ue ; Collins:1989gx , provided that we identify the hard subdiagram in those references with the union of all of the subdiagrams in our topology except for $\tilde{S}$, $\tilde{J}^{+}$, and $\tilde{J}^{-}$. We call this union $\tilde{H}$. ## V Collinear and soft approximations and decoupling relations Our strategy is to show that contributions from the $\tilde{J}^{\pm}$ subdiagrams factor from the $J^{\mp}$, hard, and $S$ subdiagrams and can be absorbed into the $J^{\pm}$ meson distribution amplitude and that contributions from the $\tilde{S}$ subdiagram factor from the $\tilde{J}^{\pm}$ subdiagrams and cancel. We treat the contributions from the $C^{\pm}$ singular regions by making use of a collinear-to-plus (minus) approximation Bodwin:1984hc ; Collins:1985ue ; Collins:1989gx for the $C^{\pm}$ gluons that attach the $\tilde{J}^{\pm}$ subdiagram to the $J^{\mp}$, $H$, and $\tilde{S}$ subdiagrams. The $C^{\pm}$ approximations capture all of the collinear-to-plus (minus) singularities, but become increasingly inaccurate as one moves away from the singularities. Similarly, we treat the contributions from the $S$ singular regions by using a soft approximation for the gluons with $S$ momentum that attach the $\tilde{S}$ subdiagram to the $\tilde{J}^{\pm}$ subdiagrams. The soft approximation captures all of the soft singularities, but becomes increasingly inaccurate as one moves away from the singularities. ### V.1 Collinear approximation Let us now describe the collinear approximation explicitly. Suppose that a gluon carrying momentum in the $C^{\pm}$ singular regions attaches to a line carrying $H$, $C^{\mp}$, $\tilde{C}^{\mp}$, $S$, or $CC$ momentum. Then, we can apply a collinear approximation to that gluon Bodwin:1984hc ; Collins:1985ue ; Collins:1989gx with no loss of accuracy. The collinear-to- plus ($C^{+}$) and collinear-to-minus ($C^{-}$) approximations consist of the following replacements in the gluon-propagator numerator: $g_{\mu\nu}\,\,\Longrightarrow\,\,\left\\{\begin{array}[]{ll}\displaystyle\frac{k_{\mu}\bar{n}_{1\nu}}{k\cdot\bar{n}_{1}-i\varepsilon}&(C^{+}),\\\\[8.61108pt] \displaystyle\frac{k_{\mu}\bar{n}_{2\nu}}{k\cdot\bar{n}_{2}+i\varepsilon}&(C^{-}).\end{array}\right.$ (11) The index $\mu$ corresponds to the attachment of the gluon to the hard, soft, or $J^{\mp}$ subdiagram, and the index $\nu$ corresponds to the attachment of the gluon to the $J^{\pm}$ subdiagram. Our convention is that $k$ flows out of a $C^{+}$ line and into a $C^{-}$ line. There is considerable freedom in choosing the auxiliary vectors $\bar{n}_{1}$ and $\bar{n}_{2}$. In order to reproduce the amplitude in the collinear singular region, it is only necessary to have $\bar{n}_{1}\cdot p_{1q}>0$ (or $\bar{n}_{1}\cdot p_{1\bar{q}}>0$) and $\bar{n}_{2}\cdot p_{2q}>0$ (or $\bar{n}_{2}\cdot p_{2\bar{q}}>0$). We choose $\bar{n}_{1}$ and $\bar{n}_{2}$ to be lightlike vectors in the minus and plus directions such that, for any vector $q$, $q\cdot\bar{n}_{1}=q^{+}$ and $q\cdot\bar{n}_{2}=q^{-}$. The $C^{\pm}$ approximation relies on the fact that the $\pm$ component of $k$ dominates in the collinear limit, provided that the $\mu$ index connects to a current in which the $\mp$ component is nonzero. Because of this last stipulation, we cannot apply the collinear approximations to a gluon carrying momentum in the $C^{\pm}$ singular region when it attaches to a line that is also carrying momentum in the $C^{\pm}$ singular region. In the $C^{\pm}$ approximation, the gluon’s polarization is longitudinal, i.e., proportional to the gluon’s momentum, which is essential to the application of graphical Ward identities to derive decoupling relations. ### V.2 Soft approximation Suppose that a gluon that carries momentum $k$ in the $S$ singular region attaches to a line carrying momentum $p$ that lies outside the $S$ singular region. Then we can apply the soft approximation without loss of accuracy. The soft approximation Grammer:1973db ; Collins:1981uk consists of replacing $g_{\mu\nu}$ in the gluon-propagator numerator with $k_{\mu}p_{\nu}/k\cdot p$, where the index $\mu$ corresponds to the attachment of the gluon to the line with momentum $p$. Unlike the collinear approximation, the soft approximation depends on the momentum of the line to which the gluon attaches. For the attachment of the gluon with momentum $k$ to any line with momentum in the $C^{+}$ ($C^{-}$) singular region, the soft approximation consists of the following replacements in the gluon-propagator numerator: $g_{\mu\nu}\,\,\Longrightarrow\,\,\left\\{\begin{array}[]{ll}\displaystyle\frac{k_{\mu}n_{1\nu}}{k\cdot n_{1}+i\varepsilon}&(C^{+}),\\\\[8.61108pt] \displaystyle\frac{k_{\mu}n_{2\nu}}{k\cdot n_{2}-i\varepsilon}&(C^{-}),\end{array}\right.$ (12) where $n_{1}$ and $n_{2}$ are lightlike vectors that are proportional to $p_{1q}$ (or $p_{1\bar{q}}$) and $p_{2q}$ (or $p_{2\bar{q}}$), respectively, and are normalized such that, for any vector $q$, $n_{1}\cdot q=q^{-}$ and $n_{2}\cdot q=q^{+}$. The index $\mu$ contracts into the line carrying the momentum of type $C^{+}$ ($C^{-}$). ### V.3 Decoupling relations Once we have implemented a collinear or soft approximation, we can make use of decoupling relations to factor contributions to the amplitude. The decoupling relations for collinear and soft gluons have the same graphical form, which is shown in Fig. 3. Figure 3: Graphical form of the decoupling relations for collinear and soft gluons. The relations show the decoupling of longitudinally polarized gluons, which are represented by curly lines. The $C^{+}$ ($C^{-}$) decoupling relation applies when the longitudinally polarized gluons all have momenta in the $C^{+}$ ($C^{-}$) singular region. The $S^{+}$ ($S^{-}$) decoupling relation applies when the longitudinally polarized gluons all have momenta in the $S$ singular region and the subdiagram that is represented by an oval contains only lines with momenta in the $C^{+}$ ($C^{-}$) singular region. The longitudinally polarized gluons are to be attached in all possible ways to the oval. The arrows on the gluon lines represent the factors $k^{\mu}\bar{n}^{\nu}/(k\cdot\bar{n})$ [$k^{\mu}n^{\nu}/(k\cdot n)$] that appear in the collinear (soft) approximation. The external lines with hash marks are truncated. In addition, the subdiagram can include any number of untruncated on-shell external legs (not shown), provided that the polarizations of the on-shell gluons are orthogonal to their momenta. $p_{i}$ are momenta, and the $a_{i}$ are color indices. The double lines are $C^{+}$, $C^{-}$, $S^{+}$, or $S^{-}$ eikonal lines, which are described in the text. If any number of longitudinally polarized gluons carrying momenta in the $C^{+}$ ($C^{-}$) singular region attach in all possible ways to a subdiagram, then the $C^{+}$ ($C^{-}$) decoupling relation applies. The subdiagram can have any number of truncated external legs and any number of untruncated on- shell external legs, provided that the polarization of each on-shell gluon is orthogonal to its momentum. In the $C^{+}$ ($C^{-}$) case, the eikonal (double) lines shown in Fig. 3 have the Feynman rules that a vertex is $\mp igT_{a}\bar{n}_{1\mu}$ ($\pm igT_{a}\bar{n}_{2\mu}$) and a propagator is $i/(k\cdot\bar{n}_{1}-i\varepsilon)$ [$i/(k\cdot\bar{n}_{2}+i\varepsilon)$], where the upper (lower) sign in the vertex is for eikonal lines that attach to quark (antiquark) lines. Here, $T_{a}$ is an $SU(3)$ color matrix in the fundamental representation. (Our convention is that a QCD gluon-quark vertex is $igT_{a}\gamma_{\mu}$.) We call these eikonal lines $C^{+}$ and $C^{-}$ eikonal lines, respectively. An analogous decoupling relation holds when any number of longitudinally polarized gluons with momenta in the soft singular region attach in all possible ways to a subdiagram that contains only lines with momenta in the $C^{+}$ ($C^{-}$) singular regions. Again, the subdiagram can have any number of truncated external legs and any number of untruncated on-shell external legs, provided that the polarization of each untruncated on-shell gluon is orthogonal to its momentum. In this case, the eikonal lines have the Feynman rules that a vertex is $\pm igT_{a}n_{1\mu}$ ($\mp igT_{a}n_{2\mu}$) and a propagator is $i/(k\cdot n_{1}+i\varepsilon)$ [($i/(k\cdot n_{2}-i\varepsilon)$] when the subdiagram is $C^{+}$ ($C^{-}$). We call these eikonal lines $S^{+}$ and $S^{-}$ eikonal lines, respectively.888The decoupling relations rely on the fact that, in the collinear and soft singular regions, the momenta of the attached gluons are effectively parallel to each other. This fact is obvious in the case of the collinear singular regions. In the case of the soft singular region, this is also the case because the currents to which the soft gluons attach are all in the plus (minus) direction when the soft gluons attach to a $C^{+}$ ($C^{-}$) subdiagram. Hence, only the minus (plus) components of the gluons’ momenta appear in invariants. In Refs. Collins:1985ue ; Collins:1989gx , an alternative definition of the soft approximation is given in which this fact is made manifest. In this definition, if the soft gluon attaches to the $\tilde{J}^{+}$ ($\tilde{J}^{-}$) subdiagram, then the momentum $k$ is replaced, in the subdiagram and in the soft approximation, with a collinear momentum $\tilde{k}=\bar{n}_{1}k\cdot n_{1}$ ($\tilde{k}=\bar{n}_{1}k\cdot n_{1}$). This alternative definition of the soft approximation is equivalent to the one that is implied by the Feynman rules for SCET. It has the property that the decoupling relation (field redefinition in SCET) holds even outside the soft singular region. The Feynman rules for the eikonal lines in the collinear and soft decoupling relations are summarized in Table 2. Type \| Vertex \| Propagator ---\|---\|--- $C^{+}$ \| $\mp igT_{a}\bar{n}_{1\mu}$ \| $\displaystyle\frac{i}{k\cdot\bar{n}_{1}-i\varepsilon}$ $C^{-}$ \| $\pm igT_{a}\bar{n}_{2\mu}$ \| $\displaystyle\frac{i}{k\cdot\bar{n}_{2}+i\varepsilon}$ $S^{+}$ \| $\pm igT_{a}n_{1\mu}$ \| $\displaystyle\frac{i}{k\cdot{n}_{1}+i\varepsilon}$ $S^{-}$ \| $\mp igT_{a}n_{2\mu}$ \| $\displaystyle\frac{i}{k\cdot{n}_{2}-i\varepsilon}$ Table 2: Feynman rules for the collinear ($C^{\pm}$) and soft ($S^{\pm}$) eikonal lines. The upper (lower) sign is for the eikonal line that attaches to a quark (antiquark) line. ## VI Factorization Now let us describe the factorization of the contributions from the $C^{+}$, $C^{-}$, and $S$ singular regions. We can determine the momentum assignments that give singular contributions by making use of the power-counting rules that we have outlined in Sec. III. When we apply these rules to the attachments of gluons with momenta in the singular regions, the symbol $\sim$ and the phrase “of the same order” mean that quantities do not differ by an infinite factor, while the phrases “much less than” and “much greater than” mean that quantities do differ by an infinite factor. Hence, for gluons with momenta in the singular regions, our convention that an allowed attachment of a gluon cannot change the essential nature of the momentum of the line to which it attaches has the following meaning: The attaching gluon cannot have an energy that is greater by an infinite factor than the energy of the line to which it attaches. The rules in Sec. III lead to complicated relationships between the allowed momenta of gluons in a given diagrammatic topology. However, there is a general principle, which we have already mentioned, that allows us to organize the discussion: The attachments of gluons to a given line must be ordered so that a given attachment produces a virtuality along the line that is of order or greater than the virtualities that are produced by the attachments that lie to the outside of it. In particular, the virtuality that a $C^{\pm}$, $\tilde{C}^{\pm}$, or $S$ singular gluon produces on a $C^{\mp}$, $\tilde{C}^{\mp}$, or $S$ line is of order its energy times the energy of the line to which it attaches. ### VI.1 Characterization of the singular contributions The relationships between allowed momenta lead to a hierarchy of scales as the singular limits are approached. Consider, for example, the contribution in which an additional soft gluon is attached to the diagram of Fig. 2 to the same outgoing fermion lines as the other gluons, but to the outside of them. In order for this contribution to be leading, the additional soft gluon must produce a virtuality on the outgoing fermion lines that is of order or less than the virtuality of $D_{1}$ or $D_{3}$. The former condition implies that the energy scale of the additional soft gluon $\epsilon_{S}^{\prime}$ must be of order or less than $\epsilon^{+}(\eta^{+})^{2}$. Since $\epsilon_{S}\sim\epsilon^{+}$, this implies that $\epsilon_{S}^{\prime}\sim\epsilon_{S}(\eta^{+})^{2}$. That is, in the collinear singular limit, $\epsilon_{S}^{\prime}$ is infinitesimal with respect to $\epsilon_{S}$. From such arguments it is clear that an infinite hierarchy of virtualities of various infinitesimal orders appears. However, these orders of virtuality are well separated in the singular limits. That is, the various gluon energy scales differ by infinite factors, as in our example. This property allows us to organize the singular contributions in such a way that we can apply the soft and $C^{\pm}$ approximations to obtain the factorized form. In order to carry out the factorization, we need to distinguish two cases for the ordering of the energy scale of a collinear momentum relative to the energy scale of a soft momentum. Both of these orderings can yield contributions that are nonvanishing in the limits $\epsilon_{S}\to 0$, $\eta^{\pm}\to 0$. Case 1: As $\epsilon_{S}\to 0$, $\epsilon^{\pm}/\epsilon_{S}$ is finite. (It is easy to see that the contribution in which $\epsilon^{\pm}/\epsilon_{S}$ goes to zero vanishes. See for example, Sec. IV.2.) In this case, we say that the collinear singular momentum and the soft singular momentum have energies that are of the same order. Case 2: As $\epsilon_{S}\to 0$, $\epsilon_{S}/\epsilon^{\pm}\to 0$.999This is the situation that was discussed in Refs. Collins:1985ue ; Collins:1989gx . In this case we say that the soft singular momentum has energy that is infinitesimal in comparison with the energy of the collinear singular momentum. We will use an iterative procedure to factor gluons at the different levels of the hierarchy of energy scales. It is useful to establish first a general nomenclature to characterize this hierarchy of energy scales. We characterize each level in the hierarchy by the energy scale of the soft singular gluons in that level. We call that energy scale the “nominal scale”. We call soft singular and collinear singular gluons that have energies of order this scale nominal-scale gluons. We call collinear singular gluons that have energies that are infinitely larger than the nominal energy scale but infinitely smaller than the next-larger soft-gluon scale “large-scale” collinear gluons. The nominal-scale collinear gluons are of the type in case 1 above with respect to the nominal-scale soft gluons. The large-scale collinear gluons are of the type in case 2 above with respect to the nominal-scale soft gluons. ### VI.2 Factorization of the singular contributions Let us now describe the factorization of the singular contributions. We make use of an iterative procedure in which gluons of higher energies are factored before gluons of lower energies. As we shall see, this ordering of the factorization procedure is convenient because it allows us to apply the decoupling relations rather straightforwardly to decouple gluons whose attachments lie toward the inside of the Feynman diagrams before we decouple gluons whose attachments lie to the outside of the Feynman diagrams. We will illustrate the factorization of the large-scale collinear gluons and the nominal-scale soft and collinear gluons for double light-meson production in $e^{+}e^{-}$ annihilation by referring to the diagram that is shown in Fig. 4. In this diagram, we have suppressed gluons with energies that are much less than the nominal scale. These gluons have attachments that lie to the outside of the attachments of the gluons that are shown explicitly. In the diagram in Fig. 4, each gluon represents any finite number of gluons, including zero gluons. For clarity, we have suppressed the antiquark lines in each meson and we have shown explicitly only the attachments of the gluons to the quark line in each meson and only a particular ordering of those attachments. However, we take the diagram in Fig. 4 to represent a sum of many diagrams, which includes all of the attachments that we specify in the arguments below of the singular gluons to the quark and antiquark in each meson, to other singular gluons, and to the $\tilde{H}$ subdiagram. Figure 4: Diagram to illustrate the factorization of large-scale collinear gluons and nominal-scale soft and collinear gluons for double light-meson production in $e^{+}e^{-}$ annihilation. $C^{i}_{\textrm{LS}}$ denotes a large-scale $C^{i}$ singular gluon, $C^{i}_{\textrm{NS}}$ denotes a nominal- scale $C^{i}$ singular gluon, and $S_{\textrm{NS}}$ denotes a nominal-scale $S$ singular gluon. #### VI.2.1 Factorization of the large-scale $C^{\pm}$ gluons We begin with the large-scale $C^{\pm}$ gluons that have the largest energy scale, and proceed iteratively through all of the scales of the large-scale $C^{\pm}$ gluons. In the first step of the iteration, those are gluons with finite-energy collinear singular momenta. In the subsequent steps, only gluons with infinitesimal collinear singular momenta are present. Gluons with relatively infinitesimal energies may attach to a gluon that carries a $C^{\pm}$ singular momentum. We still consider that gluon to carry $C^{\pm}$ singular momentum. First, we wish to apply the $C^{+}$ approximation and the $C^{+}$ decoupling relation (Fig. 3) to decouple the large-scale $C^{+}$ gluons that originate in the $\tilde{J}^{+}$ subdiagram from the $\tilde{H}$ and $\tilde{J}^{-}$ subdiagrams. In applying the decoupling relation, we need to know the extent of the subdiagram in Fig. 3: Eikonal lines appear at the points at which the subdiagram is truncated. We include the attachments of gluons with large-scale $C^{+}$ momenta to the $\tilde{J}^{-}$ subdiagram that are allowed by our conventions and by power counting. Here, and in the discussions to follow, we consider a $C^{\pm}$ gluon to be attached to the $\tilde{J}^{\mp}$ subdiagram if and only if its momentum routes through $\tilde{H}$. We include all of the attachments of gluons with large-scale $C^{+}$ momenta to $\tilde{H}$. We include the attachments that are allowed by our conventions and by power counting. However, we also include formally attachments to $\tilde{H}$ that yield vanishing contributions in the singular limits. (In subsequent iterations, we include formally, as well, the vanishing attachments of large-scale $C^{+}$ gluons to points on $C^{-}$ eikonal lines that lie to the interior of the outermost attachment of a $C^{-}$ singular gluon.) In applying the $C^{+}$ decoupling relation, we do not include attachments of gluons with large-scale $C^{+}$ momentum to a gluon with nominal-scale $S$ momentum: Such attachments violate our convention for allowed attachments because they alter the nature of the $S$ singular momentum. However, as we mentioned above, gluons with nominal-scale $S$ singular momenta can attach to a gluon with large-scale $C^{+}$ singular momentum without altering the nature of the $C^{+}$ singular momentum. We carry these attachments along as we attach the gluon with finite $C^{+}$ singular momentum to other lines in the diagram. We follow this same procedure in discussions below in treating gluons whose energies are infinitesimal with respect to the energy of an $S$ or a $C^{\pm}$ singular gluon to which they attach. The allowed attachments of gluons with large-scale $C^{+}$ momenta to a $C^{-}$ singular line lie to the inside of the attachments of gluons with nominal-scale $S$ or $C^{\pm}$ momenta. Therefore, one might expect that, when the $C^{+}$ decoupling relation is applied, a $C^{+}$ eikonal-line contribution would appear at the vertex immediately to the outside of the outermost allowed attachment of a large-scale $C^{+}$ gluon. In fact, such an eikonal-line contribution vanishes because the propagator on the $C^{-}$ singular line just to the outside of the outermost allowed attachment of a gluon with large-scale $C^{+}$ singular momentum is on shell, and, in the case of a gluon line, has physical polarization (polarization orthogonal to its momentum), up to relative corrections of infinitesimal size. Therefore, we omit such eikonal-line contributions in applying the decoupling relation. Then, the result of applying the decoupling relation is that the gluons with large-scale $C^{+}$ momenta attach to $C^{+}$ eikonal lines that attach to the outgoing fermion lines in $\tilde{J}^{+}$ just to the outside of the $\tilde{H}$ subdiagram. Next we decouple the gluons that originate in the $\tilde{J}^{-}$ subdiagram and have large-scale $C^{-}$ momenta from the $\tilde{H}$ and $J^{+}$ subdiagrams. The procedure follows the same argument as for the gluons with large-scale $C^{+}$ singular momenta, except for one new ingredient: We must include formally the vanishing attachments of the gluons with large-scale $C^{-}$ momenta to the $C^{+}$ eikonal lines from the previous step. Note that we need to include only the attachments that lie to the interior of the attachment of the outermost gluon with $C^{+}$ singular momentum in order to apply the $C^{-}$ decoupling relation. The result of applying the $C^{-}$ decoupling relation is that gluons with large-scale $C^{-}$ momenta attach to $C^{-}$ eikonal lines that attach to $C^{-}$ outgoing fermion lines just to the outside of the $\tilde{H}$ subdiagram. Now, we iterate this procedure for the large-scale $C^{\pm}$ gluons at the next-lower energy scale. The result of applying the $C^{\pm}$ decoupling relations is that the large-scale $C^{\pm}$ gluons attach to $C^{\pm}$ eikonal lines that attach to the outgoing fermion lines just to the inside of the $C^{\pm}$ eikonal lines from the previous iteration. It is easy to see that, on each outgoing fermion line, the $C^{\pm}$ eikonal line from the current iteration can be combined with the $C^{\pm}$ eikonal line from the previous iteration into a single $C^{\pm}$ eikonal line. On the combined $C^{\pm}$ eikonal line, the attachments of $C^{\pm}$ gluons with the smaller energy scale lie to the outside of the attachments of gluons with the larger energy scale. (Other orderings yield vanishing contributions.) We continue iteratively in this fashion until we have factored all of the large-scale $C^{\pm}$ gluons. After this decoupling step, the sum of diagrams represented by Fig. 4 becomes a sum of diagrams represented by Fig. 5. Figure 5: Diagram representing the sum of diagrams that occurs after one applies the decoupling of the large-scale collinear gluons that is described in Sec. VI.2.1. #### VI.2.2 Factorization of the nominal-scale $C^{\pm}$ gluons Next we factor the nominal-scale $C^{\pm}$ gluons. In applying the $C^{+}$ decoupling relation, we include the allowed attachments of these gluons to the $\tilde{J}^{-}$ subdiagram and the attachments to the nominal-scale soft gluons. We also include formally the vanishing contributions from the attachments of the nominal-scale gluons to the $\tilde{H}$ subdiagram and to the $C^{-}$ eikonal lines. Because of the ordering of virtualities along a line with $C^{+}$ singular momentum, the outermost attachment to such a line of a gluon with nominal-scale $C^{+}$ momentum must lie to the outside of the outermost attachment of a gluon with nominal-scale $S$ momentum. It is then easy to see that, for every attachment described above of a $C^{+}$ line to a line with momentum that is not $C^{+}$ singular, the $C^{+}$ approximation holds exactly. The $C^{-}$ propagator that lies to the outside of the outermost allowed attachment of a gluon with nominal-scale $C^{+}$ momentum to a line with $C^{-}$ singular momentum is on-shell, and, in the case of a gluon line, has physical polarization, up to relative corrections of infinitesimal size. Therefore, when we apply the $C^{+}$ decoupling relation, no eikonal line appears at the vertex immediately to the outside of this outermost attachment of a gluon with nominal-scale $C^{+}$ momenta. The result of applying the $C^{+}$ decoupling relation is that the nominal-scale $C^{+}$ gluons attach to several $C^{+}$ eikonal lines. These eikonal lines attach in the following locations: to the outgoing $C^{+}$ fermion lines just to the outside of $\tilde{H}$, but to the inside of the large-scale $C^{+}$ eikonal lines; just to the soft-gluon side of each vertex involving a nominal-scale soft gluon and a $C^{+}$ singular gluon of the large scale or a larger scale. In a similar fashion, we factor the nominal-scale $C^{-}$ gluons. The result of applying the $C^{-}$ decoupling relation is that the $C^{-}$ singular gluons attach to several $C^{-}$ eikonal lines. These eikonal lines attach to the following locations: to the outgoing $C^{-}$ fermion lines just to the outside of $\tilde{H}$, but to the inside of the large-scale $C^{-}$ eikonal lines; just to the soft-gluon side of each vertex involving a nominal-scale soft gluon and a $C^{-}$ singular gluon of the large scale or a larger scale. After this decoupling step, the sum of diagrams represented by Fig. 5 becomes the sum of diagrams represented by Fig. 6. Figure 6: Diagram representing the sum of diagrams that occurs after one applies the initial decoupling of the nominal-scale collinear gluons that is described in Sec. VI.2.2. #### VI.2.3 Factorization of the nominal-scale $S$ gluons We now wish to apply the soft decoupling relations to factor the nominal-scale soft gluons. In order to do this, we implement the $S^{\pm}$ approximations for the attachments of the soft gluons to the $C^{\pm}$ singular lines of the large scale or a larger scale. However, we make a slight modification to the soft approximation by combining the momentum of the nominal-scale soft gluon with the total momentum of the associated nominal-scale $C^{\pm}$ eikonal line. Then, when we implement the $S^{\pm}$ decoupling relations, the nominal- scale $C^{\pm}$ eikonal lines are carried along with the nominal-scale soft- gluon attachments. In applying the $S^{+}$ decoupling relation, we include attachments of nominal-scale soft gluons to the $\tilde{J}^{+}$ subdiagram, and in applying the $S^{-}$ decoupling relation, we include attachments of nominal-scale soft gluons to the $\tilde{J}^{-}$ subdiagram. Because we have already factored the attachments of nominal-scale $C^{\pm}$ gluons, the $S^{\pm}$ approximations hold, up to relative corrections of infinitesimal size. We also include vanishing attachments of the nominal-scale soft gluons to the large-scale eikonal lines Collins:1985ue , including only those soft- gluon attachments that lie to the inside of the outermost $C^{+}$-gluon attachments. The $C^{\pm}$ propagator that lies to the outside of the outermost allowed attachment of a nominal-scale soft gluon to a $C^{\pm}$ line is on shell, up to relative corrections of infinitesimal size. Therefore, when we apply the $S^{\pm}$ decoupling relations, no $S^{\pm}$ eikonal lines appear at the vertices just to the outside of the outermost allowed attachments. The result of applying the $S^{\pm}$ decoupling relations is that soft gluons attach to $S^{\pm}$ eikonal lines. These eikonal lines attach to the outgoing $C^{\pm}$ fermion lines just to the outside of the nominal-scale $C^{\pm}$ eikonal lines and just to the inside of the large-scale $C^{\pm}$ eikonal lines. Associated with each attachment of a nominal-scale soft gluon to an $S^{\pm}$ eikonal line is a $C^{\pm}$ eikonal line to which nominal-scale $C^{\pm}$ gluons attach. After this decoupling step, the sum of diagrams represented by Fig. 6 becomes a sum of diagrams represented by Fig. 7. Figure 7: Diagram representing the sum of diagrams that occurs after one applies the decoupling of the nominal-scale soft gluons that is described in Sec. VI.2.3. #### VI.2.4 Further factorization of the nominal-scale $C^{\pm}$ gluons We next factor the nominal-scale $C^{\pm}$ gluons from the $S^{\pm}$ eikonal lines. In order do this, we include formally the vanishing contributions that arise when one attaches the nominal-scale $C^{\pm}$ gluons to all points on the $S^{\pm}$ eikonal lines that lie to the inside of the outermost attachment of a nominal-scale soft gluon. We also make use of the following facts: Each nominal-scale $C^{\pm}$ eikonal line that attaches to an outgoing $C^{\pm}$ fermion line is identical to the eikonal line that one would obtain by applying the $C^{\pm}$ decoupling relation to the attachments of the nominal- scale $C^{\pm}$ gluons to an on-shell fermion line; each nominal-scale $C^{\pm}$ eikonal line that attaches to a nominal-scale gluon is identical to the eikonal line that one would obtain by applying the $C^{\pm}$ decoupling relation to the attachments of nominal-scale $C^{\pm}$ gluon to an on-shell gluon line. Then, applying the $C^{+}$ decoupling relation, we find that the nominal-scale $C^{+}$ gluons attach to $C^{+}$ eikonal lines that attach to the outgoing fermion lines just to the inside of the large-scale $C^{+}$ eikonal lines. Similarly, applying the $C^{-}$ decoupling relation, we find that the nominal-scale $C^{-}$ gluons attach to $C^{-}$ eikonal lines that attach to the outgoing fermion lines just to the inside of the large-scale $C^{-}$ eikonal lines. This result is represented by the diagram that is shown in Fig. 8. The nominal-scale $C^{\pm}$ eikonal lines can then be combined with the large-scale $C^{\pm}$ eikonal lines. After performing those steps, we arrive at the final factorized form, which is represented by the diagram in Fig. 9. Figure 8: Diagram representing the sum of diagrams that occurs after one applies the further decoupling of the nominal-scale collinear gluons that is described in Sec. VI.2.4. Figure 9: Diagram representing the sum of diagrams that occurs after one completely decouples the large-scale collinear gluons and the nominal-scale soft and collinear gluons. #### VI.2.5 Completion of the factorization Now we can iterate the procedure that we have given in Secs. VI.2.1–VI.2.4, taking the nominal scale to be the next-smaller soft-gluon scale. In these subsequent iterations, we include formally, in the steps of Secs. VI.2.1 and VI.2.2, the vanishing contributions from the attachments of the large-scale and nominal-scale $C^{+}$ and $C^{-}$ gluons to the soft gluons of higher levels and to the $S^{+}$ and $S^{-}$ eikonal lines that are associated with those soft gluons. (We also include formally the vanishing contributions from the attachments of the large-scale and nominal-scale $C^{+}$ and $C^{-}$ gluons to $\tilde{H}$, as in the first iteration.) Proceeding iteratively through all of the soft-gluon scales, we produce new nominal-scale $S^{\pm}$ eikonal lines at each step that attach to the outgoing fermion lines just to the outside of the existing $S^{\pm}$ eikonal lines. The nominal-scale $C^{\pm}$ eikonal lines that attach to the outgoing $C^{\pm}$ fermion lines after the steps of Sec. VI.2.2 are situated just to the inside of these nominal-scale $S^{\pm}$ eikonal lines. After the further factorization of the nominal-scale $C^{\pm}$ gluons that is described in Sec. VI.2.4, the $S^{\pm}$ eikonal lines that attach to a given outgoing fermion can be combined into a single $S^{\pm}$ eikonal line. The soft gluons of a lower energy scale attach to the outside of the soft gluons of a higher energy scale. This is the only ordering that produces a nonvanishing contribution. Following this procedure, we arrive at the standard factorized form for the singular contributions. The $\tilde{S}$ subdiagram now attaches only to $S^{+}$ eikonal lines that attach to the outgoing fermion lines from $\tilde{J}^{+}$ just outside of $\tilde{H}$ and to $S^{-}$ eikonal lines that attach to the outgoing fermion lines from $\tilde{J}^{-}$ just outside of $\tilde{H}$. The attachments involve only gluons with $S$ singular momenta. All of the $C^{\pm}$ singular contributions are contained in the $J^{\pm}$ subdiagram, which attaches via $C^{\pm}$ singular gluons to $C^{\pm}$ eikonal lines that attach to the outgoing fermion lines from $\tilde{J}^{\pm}$ just outside of the $S^{\pm}$ eikonal lines. This factorized form is illustrated in Fig. 10. Figure 10: Illustration of the factorized form for double light-meson production in $e^{+}e^{-}$ annihilation. After the use of the decoupling relations, gluons with momenta in the $S$ singular region attach to $S^{\pm}$ eikonal lines and gluons with momenta in the $C^{\pm}$ singular regions attach to $C^{\pm}$ eikonal lines. ### VI.3 Cancellation of the eikonal lines At this point the $\tilde{S}$ subdiagram and associated soft eikonal lines, which we call $\bar{S}$, have the form of the vacuum-expectation value of a time-ordered product of four eikonal lines: $\displaystyle\bar{S}(x_{1q},x_{1\bar{q}},x_{2q},x_{2\bar{q}})$ $\displaystyle=$ $\displaystyle\langle 0\|T\\{[x_{1\bar{q}},\infty^{+}][\infty^{+},x_{1q}]\otimes[x_{2\bar{q}},\infty^{-}][\infty^{-},x_{2q}]\\}\|0\rangle_{S},$ where $[x,y]=\exp\left[\int_{x}^{y}igT_{a}A_{\mu}^{a}dx^{\mu}\right]$ (14) is the exponentiated line integral (eikonal line) running between $x$ and $y$, $\infty^{+}=(\infty,0,\bm{0}_{\perp})$, and $\infty^{-}=(0,\infty,\bm{0}_{\perp})$. The symbol $\otimes$ indicates a direct product of the color factors that are associated with the soft-gluon attachments to meson 1 and the soft-gluon attachments to meson 2. We note that eikonal-line self-energy subdiagrams, which were absent in our derivation of $\bar{S}$, vanish for lightlike eikonal lines in the Feynman gauge. The subscript on the matrix element indicates that only contributions from the soft singular region are kept. Because the $H$ and $J^{+}-\tilde{J}^{+}$ subdiagrams are insensitive to a momentum in the $S$ singular region flowing through them, we can ignore the difference between $x_{1q}$ and $x_{1\bar{q}}$ in Eq. (LABEL:S-tilde-me). Then the $S^{+}$ eikonal lines cancel. Note that this cancellation relies on the color-singlet nature of the external meson. In a similar fashion, we can ignore the difference between $x_{2q}$ and $x_{2\bar{q}}$ in Eq. (LABEL:S-tilde-me), and the $S^{-}$ quark and antiquark eikonal lines cancel. We can make a Fierz rearrangement to decouple the color factors of the $\tilde{J}^{+}$ and $\tilde{J}^{-}$ subdiagrams from $\tilde{H}$. Then, we can write the $\tilde{J}^{\pm}$ subdiagrams and their associated eikonal lines, which we call $\bar{J}^{+}$ and $\bar{J}^{-}$, as follows: $\displaystyle\bar{J}^{\pm}_{\alpha\beta}(z_{i})$ $\displaystyle=$ $\displaystyle\frac{P_{i}^{\pm}}{\pi}\int_{-\infty}^{+\infty}dx^{\mp}\exp[-i(2z_{i}-1)P_{i}^{\pm}x^{\mp}]\langle M_{i}(P_{i})\|\bar{\Psi}_{\alpha}(x^{\mp})T\\{[x^{\mp},\infty^{\mp}][\infty^{\mp},-x^{\mp}]\\}\Psi_{\beta}(-x^{\mp})\|0\rangle_{C^{\pm}}.$ (15) Here, $z_{i}$ is the fraction of $P_{i}^{\pm}$ that is carried by the quark in meson $i$, $\alpha$ and $\beta$ are Dirac indices, and the upper (lower) sign in Eq. (15) corresponds to $i=1$ ($i=2$). It is understood that the fields $\Psi$ and $\bar{\Psi}$ in the matrix element are in a color-singlet state. The subscripts on the matrix elements indicate that only the contributions from the collinear singular regions are kept. There is a partial cancellation of the eikonal lines in $\bar{J}^{+}$ and $\bar{J}^{-}$, with the result that the residual eikonal lines run directly from $-x^{\mp}$ to $x^{\mp}$: $\displaystyle\bar{J}^{\pm}_{\alpha\beta}(z_{i})=\frac{P_{i}^{\pm}}{\pi}\int_{-\infty}^{+\infty}dx^{\mp}\exp[-i(2z_{i}-1)P_{i}^{\pm}x^{\mp}]\langle M_{i}(P_{i})\|\bar{\Psi}_{\alpha}(x^{\mp})P[x^{\mp},-x^{\mp}]\Psi_{\beta}(-x^{\mp})\|0\rangle_{C^{\pm}}.$ (16) Here, we have written the time-ordered product of the exponentiated line integral as a path-ordered product. Because the integrations over $z_{1}$ and $z_{2}$ have nonvanishing ranges of support in $\tilde{H}$, $x^{\mp}$ and $-x^{\mp}$ in Eq. (16) are typically separated by a distance of order $1/Q$. This shows that the $C^{\pm}$ singular contributions that have energies much less than $Q$ cancel, once they have been factored. ### VI.4 Factorized form We have shown that the contributions from $C^{\pm}$ singular regions factor from the $\tilde{J}^{\mp}$, $S$, and $H$ subdiagrams and are contained entirely in the $\bar{J}^{\pm}$ subdiagrams and that the contributions from the $S$ singular region factor from the $\tilde{J}^{\pm}$ subdiagrams and cancel. The $\bar{J}^{\pm}$ subdiagrams each have precisely the form of a meson distribution amplitude. Hence, we have arrived at the conventional factorized form, except for the following facts: the $\bar{J}^{\pm}$ subdiagrams contain only the infinitesimal $C^{\pm}$ singular regions, whereas they are conventionally defined to contain finite regions of integration; the $\tilde{H}$ subdiagram is not yet free of nonperturbative contributions from collinear momenta with transverse components of order $\Lambda_{\rm QCD}$ or less.101010At this stage, we have shown that, if one uses dimensional regularization for the soft and collinear divergences in the production amplitudes, then the soft poles in $\epsilon=(4-d)/2$ cancel and the collinear poles can be factored into $\bar{J}^{\pm}$. Next we extend the ranges of integration in the logarithmically ultraviolet divergent integrals in $\bar{J}^{\pm}$ from infinitesimal neighborhoods of the collinear singularities to finite neighborhoods that are defined by an ultraviolet cutoff $\mu_{F}\sim Q$, which is the factorization scale. In making such an extension, we do not encounter any new singularities in $\bar{J}^{\pm}$. The soft singularities that do not involve the eikonal lines have already been shown to cancel. There is the possibility that $S$ or $C^{\mp}$ singularities could arise from the eikonal lines in $\bar{J}^{\pm}$. However, as we have mentioned, after the cancellation of the quark and antiquark eikonal lines, the remaining segment of eikonal line is finite in length, with length of order $1/Q$. Hence, $S$ or $C^{\mp}$ modes with virtualities much less than $Q$ cannot propagate on these eikonal lines. Finally, having extended the momentum ranges in $\bar{J}^{\pm}$, we redefine $\tilde{H}$ to be the factor that, when convolved with $\bar{J}^{\pm}$, produces the complete production amplitude. This is precisely the conventional definition of the hard subdiagram. Since the soft divergences have canceled and the collinear divergences are contained in $\bar{J}^{\pm}$, $\tilde{H}$ is a finite function (after ultraviolet renormalization), and depends only on the scales $Q$ and $\mu_{F}$ and the renormalization scale. Therefore, $\tilde{H}$ contains only contributions from momenta of order $Q$ or $\mu_{F}$, i.e., from momenta in the perturbative regime. We have now established the conventional factorized form for the production amplitude $\mathcal{A}$, which reads $\mathcal{A}=\bar{J}^{-}\otimes\tilde{H}\otimes\bar{J}^{+},$ (17) where the symbol $\otimes$ denotes a convolution over the longitudinal momentum fraction $z_{i}$ of the corresponding meson and we have suppressed the Dirac indices on $\bar{J}^{\pm}$ and $\tilde{H}$. The $\bar{J}^{\pm}$ are now given by Eq. (16), but without the subscript $C^{\pm}$ on the matrix element. They can be decomposed into a sum of products of Dirac-matrix and kinematic factors and the standard light-cone distributions for the mesons. ### VI.5 Nonzero relative momentum between the quark and antiquark Now let us return to the situation in which the $\bm{p}_{i\perp}$ in Eq. (1) are nonzero and of order $\Lambda_{\rm QCD}$. In this case the outgoing quark and antiquark in each meson are moving in slightly different light-cone directions. Therefore, we must define separate singular regions $C_{1q}$ ($C_{2q}$) for the quark direction and $C_{1\bar{q}}$ ($C_{2\bar{q}}$) for the antiquark direction in meson 1 (2). As we have mentioned, in defining the collinear approximations, we can choose any auxiliary vectors $\bar{n}_{i}$ that, for the collinear singular region associated with $p_{i}$, satisfy the relation $\bar{n}_{i}\cdot p_{i}>0$. We choose the lightlike auxiliary vector $\bar{n}_{1}$, which is in the minus direction, for both the $C_{1q}$ and $C_{1\bar{q}}$ singular regions and the lightlike auxiliary vector $\bar{n}_{2}$, which is in the plus direction, for both the $C_{2q}$ and $C_{2\bar{q}}$ singular regions. That is, we take the same collinear approximation for the collinear regions associated with the quark and the antiquark in a meson. Then the factorization of the collinear singular regions goes through exactly as in the case $p_{\perp}=0$. For gluons with momenta in the soft singular region, one can still define soft approximations, but the approximations are different for the couplings to lines in the quark and antiquark collinear singular regions. When gluons with momenta in the soft singular region attach to lines with momenta in the $C_{iq}$ ($C_{i\bar{q}}$) singular region, one can use a unit lightlike vector $n_{iq}$ ($n_{i\bar{q}}$) that is proportional to $p_{iq}$ ($p_{i\bar{q}}$) to define the soft approximation. Then, the gluons with momenta in the soft singular region still factor. However, in the factored form, the soft eikonal line that attaches to the quark (antiquark) line in meson $i$ is parametrized by the auxiliary vector $n_{iq}$ ($n_{i\bar{q}}$). Because $n_{iq}$ and $n_{i\bar{q}}$ differ by an amount of relative order $\lambda$ [Eq. (3)], the quark and antiquark soft eikonal lines in each meson fail to cancel completely. These noncancelling soft contributions violate factorization because they couple one meson to the other in the production amplitude. If we take the approximation $n_{iq}=n_{i\bar{q}}$, but keep $n_{jq}\neq n_{j\bar{q}}$, then the quark and antiquark eikonal lines cancel in meson $i$, but not in meson $j$. However, the remaining soft subdiagram, which attaches only to the quark and antiquark eikonal lines in $\bar{J}$, can be absorbed into the definition of the $\bar{J}$ subdiagram for meson $j$.111111It can be shown, by making use of the methods in Sec. VI.2, that the configuration in which the soft subdiagram attaches only to the quark and antiquark soft eikonal lines that are associated with $\bar{J}^{+}$ ($\bar{J}^{-}$) is precisely the configuration that one would obtain by using the soft decoupling relation to factor a soft subdiagram that attaches only to $\bar{J}^{+}$ ($\bar{J}^{-}$). Here one must use the fact that the contributions in which $C^{+}$ ($C^{-}$) collinear gluons with infinitesimal energy attach to the collinear eikonal line in $\bar{J}^{+}$ ($\bar{J}^{-}$) vanish, owing to the finite length of the eikonal line. Therefore, we see that we obtain a violation of factorization only if the quark and antiquark eikonal lines fail to cancel in both mesons. Hence, the violations of factorization that arise from the soft function are of relative order $\lambda^{2}$. In order to express the amplitude in terms of the light-cone distributions $\bar{J}^{\pm}$ in Eq. (15), it is necessary to neglect in $\tilde{H}$ the minus and transverse components of $p_{1q}$ and $p_{1\bar{q}}$ and the plus and transverse components of $p_{2q}$ and $p_{2\bar{q}}$. In doing so, we make an error of relative order $p_{i\perp}/Q\sim\lambda$. ### VI.6 Failure of the soft cancellation for low-energy collinear gluons Now let us discuss the cancellation of the soft diagram for the factorized form in which the soft subdiagram contains collinear gluons. As we have mentioned, such a factorized form is the one that would seem to follow most straightforwardly from SCET Bauer:2001yt ; Bauer:2002nz . Suppose that a gluon with momentum $k$ attaches to the soft eikonal line that attaches to the quark line in meson 1. That contribution contains a factor $1/k\cdot n_{1q}$, which is singular in the limit in which $k$ becomes collinear to $p_{1q}$ ($n_{1q}$). On the other hand, for the contribution in which the gluon with momenta $k$ attaches to the soft eikonal line that attaches to the antiquark line in meson 1, there is a factor $1/k\cdot n_{1\bar{q}}$, which is not singular in the limit in which $k$ becomes collinear to $p_{1q}$. Hence, the attachments of the gluon with momentum $k$ to the quark and antiquark lines fail to cancel when $k$ is in the $C_{1q}$ (or $C_{1\bar{q}}$) singular region. Furthermore, the uncanceled contribution is not suppressed by a power of $Q$ and is, in fact, divergent. Thus, we see that, in the factorized form in which the soft subdiagram contains collinear gluons, the soft subdiagram fails to cancel, and one cannot establish the conventional factorized form.121212There is also a potential difficulty in apply the soft approximation to low-energy collinear gluons. For example, as we have mentioned, if a soft gluon attaches to the $J^{+}$ subdiagram, then soft approximations in SCET and Refs. Collins:1985ue ; Collins:1989gx and the soft decoupling relation involve the replacement of the soft momentum $k$ with a collinear momentum $\tilde{k}=\bar{n}_{1}k\cdot n_{1}$ in the $\tilde{J}^{+}$ subdiagram. Hence, $\tilde{k}$ vanishes when $k$ becomes $C^{+}$.. It might seem that one could recover the cancellation of the soft subdiagram by setting $p_{i\perp}$ exactly to zero. However, at $p_{\perp}=0$, the cancellation of the quark and antiquark eikonal lines becomes ill-defined for $k$ collinear to the quark and antiquark because of the infinite factors that arise from the eikonal denominators $k\cdot n_{1q}$ and $k\cdot n_{1\bar{q}}$. 131313One might also consider the possibility of defining a single soft-approximation auxiliary vector $n_{i}$ for each meson, where $n_{i}$ lies between $n_{iq}$ and $n_{i\bar{q}}$. However, the resulting soft approximation fails to reproduce the collinear divergences that occur if the soft subdiagram contains low-energy gluons that are parallel to the quark or the antiquark. We note that this issue also arises in inclusive processes, for example the Drell-Yan process, in the decoupling of the soft subdiagram from color-singlet hadrons. ## VII Summary We have established, to all orders in perturbation theory, factorization of the amplitude for the exclusive production of two light mesons in $e^{+}e^{-}$ annihilation through a single virtual photon for the case in which the external mesons are represented by an on-shell quark and an on-shell antiquark. The case of on-shell external particles is important for perturbative matching calculations. The presence of on-shell external particles opens the possibility of soft and collinear momentum modes of arbitrarily low energy. In this situation, low- energy collinear gluons can couple to soft gluons. That coupling leads to additional complications in the factorization proof. Nevertheless, we have shown that one can derive the standard factorized form, in which the production amplitude is written as a hard factor convolved with a distribution amplitude for each meson. The hard factor is free of soft and collinear divergences and depends only on the hard-scattering scale $Q$, the collinear factorization scale $\mu_{C}$, and an ultraviolet renormalization scale. The meson distribution amplitudes contain all of the collinear divergences and all of the nonperturbative contributions that involve virtualities of order $\Lambda_{\rm QCD}$ or less. We find that the factorization formula holds up to corrections of relative order $\Lambda_{\rm QCD}/Q$. As an intermediate step in the factorization proof, we obtain a form in which the soft subdiagram does not contain gluons with momenta in the collinear singular regions. This form of factorization may be useful in the resummation of soft logarithms, as the contributions with two logarithms per loop are contained entirely in the jet functions, which are diagonal in color. It is essential in establishing the standard factorized form for exclusive processes with on-shell external partons because, as we have shown, the cancellation of the attachment of the soft diagram to a color-singlet hadron fails at leading order in $Q$ if the soft subdiagram would contain gluons with momenta that are collinear to the constituents of the hadron. This issue also arises in inclusive processes in the decoupling of the soft subdiagram from color- singlet hadrons. In on-shell perturbative calculations in SCET, low-energy gluons with momenta collinear to the external particles can appear. At two-loop level and higher, these low-energy collinear gluons can couple to soft gluons. Since SCET has no provision to decouple the collinear gluons from the soft gluons, it seems that it would be most straightforward in SCET to treat the low-energy gluons as part of the soft contribution. In such an approach, the soft subdiagram contains gluons with momenta in both the soft and collinear singular regions. As we have said, the soft subdiagram would fail to cancel in this case, and one would not achieve the standard factorized form. Therefore, in the absence of a further factorization argument, there would be no assurance in a matching calculation that the low-virtuality contributions could all be absorbed into the meson distribution amplitudes: Some low-virtuality contributions might be associated with a soft function that could not be factored from the meson distribution amplitudes. Alternatively, one could abandon the notion that SCET should reproduce the contributions of full QCD on a diagram-by-diagram basis and assume that SCET is valid only after one sums over all Feynman diagrams. Furthermore, one could consider the collinear action in SCET to apply to all collinear momenta of arbitrarily low energy. Then, as is asserted in Ref. Bauer:2002nz , the production amplitude in SCET would take the form of a hard-scattering diagram, a $\bar{J}^{+}$ light-cone distribution and a $\bar{J}^{-}$ light-cone distribution that are convolved with the hard subdiagram, and a soft subdiagram that is free of collinear momenta and that connects to the $\bar{J}^{+}$ and $\bar{J}^{-}$ light-cone distributions with interactions that are given by the collinear action. That factorized form is the one that we would obtain after the decoupling of the collinear gluons from the soft gluons if we were to extend the ranges of integration in the $\tilde{S}$, $\bar{J}^{+}$, and $\bar{J}^{-}$ subdiagrams from the singular regions to finite regions of $S$, $C^{+}$ and $C^{-}$ momenta. Issues of double counting arise when one extends the ranges of integration. They could be dealt with, for example, by making use of the method of zero-bin subtractions Manohar:2006nz . Once the double-counting issues are resolved, our proof shows that such a form for the amplitude is correct. However, this result does not follow obviously from QCD or from SCET. It requires a derivation, such as the one that we have given in this paper. The low-energy contributions that we have discussed involve integrands that are homogeneous in the integration momenta. Therefore, one might argue that, if one applies the method of regions Beneke:1997zp , then such contributions lead to scaleless integrals and vanish. The difficulty in making use of such an argument to prove factorization is that the method of regions extends the range of integration for each region to infinity. There is no proof of the validity of such an extension, and, hence, there is the possibility of double counting. Double counting between the soft and collinear subdiagrams is dealt with in SCET through the use of zero-bin subtractions Manohar:2006nz . However, the zero-bin subtractions are formulated rigorously in terms of a hard cutoff. In Ref. Manohar:2006nz , examples of the zero-bin subtraction in dimensional regularization in one-loop perturbation theory are given. To our knowledge, no proof of an all-orders zero-bin subtraction scheme in dimensional regularization has been given. In physical hadrons, gluon momenta are cut off by confinement at a scale of order $\Lambda_{\rm QCD}$. In that situation, one does not need to consider the possibility that collinear gluons can attach to soft gluons in order to demonstrate the factorization of nonperturbative contributions, i.e., those contributions that involve momentum components of order $\Lambda_{\rm QCD}$. However, if one wishes to factor logarithmic contributions up to a scale of order $Q$, for example, for the purpose of resummation, then it is again necessary to treat the attachments of collinear gluons to soft gluons along the lines that we have described in this paper. In this paper, we have focused on a specific exclusive process. However, we expect that our method can be generalized straightforwardly to the other exclusive processes and, possibly, to inclusive processes. In the latter case, one must consider Glauber-type momenta explicitly, as contributions that arise from such momenta cancel only once one has summed over all possible final- state cuts Bodwin:1984hc ; Collins:1983ju ; Collins:1985ue ; Collins:1988ig ; Collins:1989gx . However, it seems plausible that one can implement this cancellation, using standard techniques, independently of the factorization arguments that we have presented here. ###### Acknowledgements. We thank John Collins and George Sterman for many useful comments and suggestions. We also thank Thomas Becher, Dave Soper, and Iain Stewart for helpful discussions. We thank In-Chol Kim for his assistance in preparing the figures in this paper. The work of G.T.B. and X.G.T. was supported in part by the U.S. Department of Energy, Division of High Energy Physics, under Contract No. DE-AC02-06CH11357. The research of X.G.T. was also supported by Science and Engineering Research Canada. The work of J.L. was supported by the Korea Ministry of Education, Science, and Technology through the National Research Foundation under Contract No. 2010-0000144. ## References * (1) C. W. Bauer, S. Fleming, D. Pirjol, and I. W. Stewart, Phys. Rev. D 63, 114020 (2001) [arXiv:hep-ph/0011336]. * (2) C. W. Bauer, D. Pirjol, and I. W. Stewart, Phys. Rev. D 65, 054022 (2002). [arXiv:hep-ph/0109045] * (3) C. W. Bauer et al., Phys. Rev. D 66, 014017 (2002). [arXiv:hep-ph/0202088] * (4) G. T. Bodwin, X. Garcia i Tormo, and J. Lee, Phys. Rev. Lett. 101, 102002 (2008) [arXiv:0805.3876 [hep-ph]]. * (5) A. V. Manohar, Phys. Lett. B 633, 729 (2006) [arXiv:hep-ph/0512173]. * (6) A. V. Manohar and I. W. Stewart, Phys. Rev. D 76, 074002 (2007) [arXiv:hep-ph/0605001]. * (7) G. T. Bodwin, Phys. Rev. D 31, 2616 (1985) [Erratum-ibid. D 34, 3932 (1986)]. * (8) J. C. Collins, D. E. Soper, and G. Sterman, Nucl. Phys. B 261, 104 (1985). * (9) J. C. Collins, D. E. Soper, and G. Sterman, Adv. Ser. Direct. High Energy Phys. 5, 1 (1988). [arXiv:hep-ph/0409313] * (10) J. C. Collins, ANL-HEP-PR-84-36 (1984). * (11) G. Sterman, Nucl. Phys. B 281, 310 (1987). * (12) H. Contopanagos, E. Laenen, and G. Sterman, Nucl. Phys. B 484, 303 (1997) [arXiv:hep-ph/9604313]. * (13) N. Kidonakis, G. Oderda, and G. Sterman, Nucl. Phys. B 525, 299 (1998) [arXiv:hep-ph/9801268]. * (14) N. Kidonakis, G. Oderda, and G. Sterman, Nucl. Phys. B 531, 365 (1998) [arXiv:hep-ph/9803241]. * (15) G. Sterman and M. E. Tejeda-Yeomans, Phys. Lett. B 552, 48 (2003) [arXiv:hep-ph/0210130]. * (16) A. Sen, Phys. Rev. D 28, 860 (1983). * (17) J. C. Collins, D. E. Soper, and G. Sterman, Nucl. Phys. B 223, 381 (1983). * (18) G. Sterman, Phys. Rev. D 17, 2773 (1978). * (19) S. B. Libby and G. Sterman, Phys. Rev. D 18, 4737 (1978). * (20) G. T. Bodwin, S. J. Brodsky, and G. P. Lepage, Phys. Rev. Lett. 47, 1799 (1981). * (21) C. W. Bauer, D. Pirjol, and I. W. Stewart, Phys. Rev. D 67, 071502 (2003). [arXiv:hep-ph/0211069]. * (22) M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Nucl. Phys. B 591, 313 (2000) [arXiv:hep-ph/0006124]. * (23) V. A. Smirnov, Phys. Lett. B 465, 226 (1999). [arXiv:hep-ph/9907471] * (24) G. Sterman, Phys. Rev. D 17, 2789 (1978). * (25) G. Grammer, Jr. and D. R. Yennie, Phys. Rev. D 8, 4332 (1973). * (26) J. C. Collins and D. E. Soper, Nucl. Phys. B 193, 381 (1981) [Erratum-ibid. B 213, 545 (1983)]. * (27) M. Beneke and V. A. Smirnov, Nucl. Phys. B 522, 321 (1998) [arXiv:hep-ph/9711391]. * (28) J. C. Collins, D. E. Soper, and G. Sterman, Phys. Lett. B 134, 263 (1984). * (29) J. C. Collins, D. E. Soper, and G. Sterman, Nucl. Phys. B 308, 833 (1988).	arxiv-papers	2009-03-03T17:03:11	2024-09-04T02:49:00.957832	{ "license": "Public Domain", "authors": "Geoffrey T. Bodwin (Argonne), Xavier Garcia i Tormo (Argonne & Alberta\n U.), Jungil Lee (Korea U.)", "submitter": "Xavier Garcia i Tormo", "url": "https://arxiv.org/abs/0903.0569" }
0903.0618	# Thermal Geo-axions Hooman Davoudiasl 111email: hooman@bnl.gov Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA Patrick Huber 222email: pahuber@vt.edu Department of Physics, IPNAS, Virginia Tech, Blacksburg, VA 24061, USA ###### Abstract We estimate the production rate of axion-type particles in the core of the Earth, at a temperature $T\approx 5000$ K. We constrain thermal geo-axion emission by demanding a core-cooling rate less than $\mathcal{O}{(100)}$ K/Gyr, as suggested by geophysics. This yields a “quasi-vacuum” (unaffected by extreme stellar conditions) bound on the axion-electron fine structure constant $\alpha_{ae}^{QV}\lesssim 10^{-18}$, stronger than the existing accelerator (vacuum) bound by 4 orders of magnitude. We consider the prospects for measuring the geo-axion flux through conversion into photons in a geoscope; such measurements can further constrain $\alpha_{ae}^{QV}$. A variety of scenarios for physics beyond the Standard Model (SM) give rise to light pseudo-scalar particles, generically referred to as axions. The Peccei- Quinn (PQ) solution to the SM strong CP problem provided the initial context for axions Peccei:1977hh . Axion-type particles are ubiquitous in string theory constructs and have also been considered in cosmological model building Preskill:1982cy . There are stringent astrophysical and cosmological constraints on the couplings of axions, as a result of which they are largely assumed to be very weakly interacting. Some of the strongest bounds on axion- SM couplings come from astrophysics, where stellar evolution and cooling arguments imply that the axion (PQ) scale $f\gtrsim 10^{9}$ GeV. Such analyzes are based on the requirement that new exotic processes should not significantly perturb a standard picture of the energetics that govern the evolution of various astrophysical objects. Since axions (or other light weakly interacting particles) can directly drain energy out of such objects, one can obtain bounds on the coupling of axions to matter. For a concise summary of various astrophysical bounds, see Ref. Amsler:2008zzb . More recent astrophysical bounds on axion-type particles have been presented in Ref. Gondolo:2008dd . In this work, we consider the possibility that the hot core of the Earth can convert some of its thermal energy into a flux of axions of $\mathcal{O}{({\rm eV})}$ energy Sivaram 333Non-thermal geo-axions, produced in radioactive decays within the Earth, have been examined in Ref. Liolios:2007gu . This work does not find a currently detectable signal even in the most favorable case considered therein.. Then, it would be interesting to find out what bounds can be obtained from geological considerations and also to determine the prospects for discovering the geo-axions emanating from the terrestrial core. The Earth’s core is at a temperature of around $5000\,\mathrm{K}$ corresponding to $0.4\,\mathrm{eV}$. Although this is a much lower temperature than those of stellar interiors, which have temperatures of order $\mathrm{keV}$, there are a number of considerations that motivate our analysis. First of all, the core of the Earth is only a short distance away, compared to any astronomical object. This greatly enhances the prospects for measuring a geo-axion flux and can potentially compensate for the low core temperature. Secondly, the Earth’s core is quite different from other axion emitting environments, being mainly made up of hot molten or crystallized iron. Hence, in principle, the intuition and calculations that apply to stellar plasmas may not be adequate to estimate geo-axion emission and new effects may need to be considered. Finally, the Earth’s center is a far less extreme medium compared to stellar media. The possibility of the dependence of axion properties on the environment has been proposed medium1 in the context of reports of large vacuum birefringence by PVLAS Zavattini:2005tm . This result if it were confirmed would have implied an axion like particle with an axion-photon coupling in stark violation of astrophysical bounds. As a consequence, a number of models were developed to reconcile the laboratory result with the astrophysical bounds medium2 . Although the initial PVLAS result could not be reproduced Cantatore:2008zz , it highlighted the necessity for obtaining complementary bounds on axions in a wide variety of production environments. If axion couplings are temperature and/or density dependent, the geo-axion bounds could be viewed as independent new data on axion physics in “quasi- vacuum” conditions. Thus, in this letter we do not attempt to supersede existing, stringent astrophysical bounds, but to supplement them by examining axion production in a novel environment. Motivated by the above discussion, we will next derive an estimate for the thermal geo-axion flux. We will use geodynamical considerations to constrain this flux and hence the axion-electron coupling $\alpha_{ae}$ in the core. This bound is not competitive with its astrophysical counterparts, but, as mentioned before, is derived in a very different regime. Note that collider bounds on $\alpha_{ae}$ that are derived in a similar regime are much weaker than our geo-axion bound. We will then consider detection of the geo-axion flux, via magnetic conversion into photons, using a “geoscope,” in analogy with the helioscope concept Sikivie:1983ip ; vanBibber:1988ge . A discussion and a summary of our results are presented at the end of this work. The core of the Earth is mainly made of iron (Fe). The inner core, which extends to a radius of $R_{ic}\approx 1200$ km, is thought to be in solid crystalline form at a temperature $T\sim 6000$ K. The outer core, which extends to $R_{c}\approx 3500$ km, is made up of molten iron at $T\sim 4000$ K core . Since Fe is a transition metal, with the electronic configuration $[{\rm Ar}]\;3d^{6}\;4s^{2}$, both $3d$ and $4s$ electrons are important in determining its properties. However, for a simplified treatment, we only consider the $4s$ electrons as nearly free. The effective nuclear charge seen by the $4s$ electrons is $Z_{\rm eff}\simeq 5.4$ Zeff . Given that the solid iron core makes up a negligible mass of the total core, we will ignore its contributions to our estimate. This is partly done to avoid a complicated treatment of the interactions of electrons and phonons inside a hot crystal, far from the plasma regime. However, we note that a more complete analysis should take these effects into account. We adopt $T_{c}\approx 5000$ K$\approx 0.4$ eV core as the mean temperature of the molten iron core. We are also ignoring the contribution of other trace elements, such as nickel, which have more or less the same properties as iron, for our purposes. Given the metallic nature of the core, we will treat it as a plasma composed of a degenerate gas of free electrons, with a Fermi energy $E_{F}\approx 10.3$ eV FerroFe . The resulting Fermi momentum is given by $p_{F}=\sqrt{2m_{e}\,E_{F}}\approx 3.3$ keV, where $m_{e}\simeq 0.5$ MeV is the mass of the electron. These free electrons move in the background of Fe ions with effective charge $Z_{\rm eff}\simeq 5.4$. The free electron density in the core is given by GR $n_{e}=\frac{p_{F}^{3}}{3\pi^{2}}$ (1) and hence we get $n_{e}\approx 2\times 10^{23}$ cm-3. Let us define the radius $a_{e}=\left(\frac{3}{4\pi n_{e}}\right)^{1/3},$ (2) for the mobile charged particles in the plasma, which we take to be electrons here. The quantity $\Gamma\equiv\frac{Z_{\rm eff}^{2}\alpha}{a_{e}T_{c}},$ (3) with $\alpha=1/137$, is a measure of the relative strength of Coulomb interactions and the kinetic energy of the electrons. For the core parameters, we get $a_{e}\approx 10^{-8}$ cm and $\Gamma\sim 10^{3}$. We take $\Gamma\gg 1$ as indicative of a strongly coupled plasma GR . Since the iron core of the Earth is in a molten state and not yet a crystal, this interpretation is reasonable, despite the large value of $\Gamma$. The effect of the geomagnetic field in the core on the density of states close to the Fermi surface can be neglected since the thermal energy is large compared to the energy difference between successive Landau levels. In any case, we note that a more detailed numerical treatment may reveal important corrections to the estimates that follow. Interestingly enough, there is an astrophysical environment that is described by the above key features. This is the interior of White Dwarfs (WD’s) which is a strongly coupled plasma of Carbon and Oxygen, supported by a degenerate gas of electrons, similar to the iron core of the Earth. Hence, we adopt the formalism used for WD cooling by axion emission in the bremsstrahlung process $e\,N(Z,A)\to e\,N(Z,A)\,a$ Raffelt:1985nj , in order to estimate the geo- axion flux; $Z$ is the ionic charge and $A$ is the atomic mass. We will ignore Primakoff Primakoff contributions to this flux, resulting from the interactions of thermal photons in the plasma. This is justified, since the density of such photons is roughly given by $({\rm eV})^{3}\sim 10^{15}$ cm-3, which is much smaller than $n_{e}$ in the core. For a plasma with only one species of nuclei, the energy emission rate, in axions, per unit mass is given by Raffelt:1985nj $\varepsilon_{a}=(Z^{2}\alpha^{2}\alpha_{ae})/(Am_{e}^{2}m_{u})\,T^{4}\xi(p_{F}),$ (4) where $m_{u}\simeq 1.7\times 10^{-24}$ g is the atomic mass unit and $\xi(p_{F})$ is a numerical factor which only depends on $p_{F}$. Numerical calculations relevant for WD’s indicate that $\xi\simeq 1$ to a good approximation, over a wide range of parameters in the strongly coupled regime GR . We thus take $\xi\sim 1$ in our calculations. For geo-axion emission, we then obtain $\varepsilon_{a}\sim 10^{7}\alpha_{ae}\,T_{3}^{4}\;\;{\rm erg}\,{\rm g}^{-1}{\rm s}^{-1},$ (5) where we have set $Z=Z_{\rm eff}\simeq 5.4$, $A=56$, and $T_{3}\equiv T/10^{3}$ K. Given a core mass density of $\rho_{c}\simeq 10$ g cm-3 and $T_{c}\approx 5T_{3}$, we get $\Phi_{a}\sim 10^{37}\alpha_{ae}\;\;{\rm erg}\,{\rm s}^{-1},$ (6) for the flux of geo-axions. It is interesting to inquire how geological considerations can constrain the estimate in Eq. (6). As a simple criterion, and in the spirit of analogous considerations for stellar objects, we will demand that the rate of core- cooling $\Phi_{a}$ be less than that inferred from geodynamical considerations. This rate has been estimated to be in the range of 100 K/Gyr$=10^{-7}$ K/yr CB2006 . Given that the heat capacity of the Earth’s core is estimated to be $C_{\oplus}\sim 10^{34}$ erg/K CB2006 , we get for the geological rate of core-cooling $\Phi_{\oplus}\sim 10^{27}{\rm erg/yr}\sim 10^{19}\;\;{\rm erg}\,{\rm s}^{-1},$ (7) in agreement with Ref. core . Requiring $\Phi_{a}<\Phi_{\oplus}$ yields $\alpha_{ae}^{\oplus}\lesssim 10^{-18}\quad({\rm core}{\rm-}{\rm cooling}).$ (8) This bound is not strong compared to those from astrophysics. For example, the bound from solar age is $\alpha_{ae}\lesssim 10^{-22}$ and the one from red giant constraints is $\alpha_{ae}\lesssim 10^{-26}$ GR . However, the bound in (8) is within a few orders of magnitude of the solar. Again, we note that the bound in (8) is valid for a quasi-vacuum regime and not the extreme stellar environments. The closest such bounds for quasi-vacuum environments are from $e^{+}e^{-}$ collider experiments, and correspond to $\alpha_{ae}\lesssim 10^{-14}$ ($f\gtrsim 1000$ GeV) Amsler:2008zzb , weaker than our geo-axion bound (8) by 4 orders of magnitude. Next, we will examine the prospects for detecting a geo-axion flux consistent with this bound. The geo-axion flux $F_{a}$, corresponding to $\Phi_{a}$ in Eq. (6), at $R_{\oplus}\simeq 6.4\times 10^{3}$ km (surface of the Earth) is given by $F_{a}=\Phi_{a}/(4\pi R_{\oplus}^{2})\sim\alpha_{ae}10^{30}\;\;{\rm eV}\,{\rm cm}^{-2}{\rm s}^{-1}.$ (9) Assuming average axion energy $\langle E_{a}\rangle\simeq 1$ eV, we get $\frac{dN_{a}}{d{\cal A}\,dt}\sim\alpha_{ae}10^{30}\;\;{\rm cm}^{-2}{\rm s}^{-1}$ (10) for the flux of eV-axions at the surface of the Earth444For axions, the non- radial flux also contributes, thus in principle Gauß’ law can not be used. In this case, the difference compared to an exact treatment is about 5%.. In principle, there are two ways to detect these axions: The obvious choice is to exploit their coupling to electrons $\alpha_{ea}$ via the axio-electric effect, in analogy with the photo-electric effect. The cross section for the axio-electric effect is reduced by a factor of Avignone:2008uk $\frac{\alpha_{ea}}{2\alpha}\left(\frac{E_{a}}{m_{e}}\right)^{2}\sim 10^{-10}\alpha_{ea}$ (11) compared to the ordinary photo-electric cross section. Using our bound derived in Eq. (8) and the typical size of photo-electric cross sections of $\mathcal{O}{(10^{-20})}\,\mathrm{cm}^{2}$, the resulting axio-electric cross section is $\mathcal{O}{(10^{-48})}\,\mathrm{cm}^{2}$, which is quite small. For comparison typical neutrino cross sections are $\mathcal{O}{(10^{-42})}\,\mathrm{cm}^{2}$. Therefore, we next consider the possibility to use magnetic axion-photon mixing, to convert axions into photons. We use the simple formula GR $P_{a\gamma}\simeq\left(\frac{Bg_{a\gamma}L}{2}\right)^{2},$ (12) where $B$ is the strength of the transverse magnetic field along the axion path, $g_{a\gamma}$ is the axion-photon coupling, and $L$ is the length of the magnetic region. The above equation is valid when the $q\ll 1/L$ with $q=\|m_{a}^{2}-m_{\gamma}^{2}\|/(2E_{a}),$ (13) where $m_{a}$ is the axion mass and $m_{\gamma}$ is the photon effective mass. Note that for $m_{a}\lesssim 10^{-3}$ eV, the geo-axion oscillation length $q^{-1}\gtrsim 0.6$ m in vacuum with $m_{\gamma}=0$. We will assume this mass range for the purposes of our discussion. An obvious choice would be to consider an LHC-class magnet, such as the one used by the CAST experiment Andriamonje:2004hi , with $B=10$ T and $L=10$ m. However, this magnet has a cross sectional area of order 14 cm2. Given that the core of the Earth subtends an angle of order $30^{\circ}$ as viewed from its surface, we see that the CAST magnet will capture a very small portion of the relevant “field of view.” Thus, we have to consider other magnets of similar strength, but larger field of view. Fortunately, such magnets are used in Magnetic Resonance Imaging (MRI), to carry out medical research. For example, the MRI machine at the University of Illinois at Chicago has a magnetic field of 9.4 T, over length scales of order 1 m MRI . Hence, we will use $B=10$ T and $L=1$ m, as presently accessible values, for our estimates. The current best laboratory bound on $g_{a\gamma}$ for nearly massless axions, derived under vacuum conditions, was recently obtained by the GammeV collaboration Chou:2007zzc : $g_{a\gamma}<3.5\times 10^{-7}\,\mathrm{GeV}^{-1}$. With this value the upper bound for the axion photon conversion probability is $P_{a\gamma}\simeq 3\times 10^{-12}\,.$ (14) Thus, we conclude that magnetic detection is more promising than axio-electric detection. Using Eq. (10), we then get $\frac{dN_{\gamma}}{d{\cal A}\,dt}\sim\alpha_{ae}10^{18}\;\;{\rm cm}^{-2}{\rm s}^{-1}$ (15) for the flux of converted photons in the signal. The bound in (8) then suggests that a sensitivity to a photon flux of order $1\;{\rm cm}^{-2}{\rm s}^{-1}$, using our reference geoscope parameters, is required to go beyond the geodynamical constraint and look for a signal. Modern superconducting transition edge bolometers have demonstrated single photon counting in the near infrared with background rates as low as $10^{-3}\,\mathrm{Hz}$ noise and quantum efficiencies close to unity qe . With such a detector photon fluxes as small as $10^{-5}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$ can be detected with an integration time of $10^{7}\,\mathrm{s}$. Our results on the prospects of direct search for geo-axions are summarized in figure 1. Figure 1: The black lines show the resulting photon fluxes $\phi$ in units of $\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$ for a geoscope with $L=1\,\mathrm{m}$ and $B=10\,\mathrm{T}$ as a function of $\alpha_{ae}$ and $g_{a\gamma}$. The gray lines show the current astro-physical bounds on $\alpha_{ae}$ from the cooling of white dwarfs Amsler:2008zzb and on $g_{a\gamma}$ from the non-observation of solar axions by CAST Andriamonje:2004hi . The colored/shaded regions indicate the parameter space excluded by photon regeneration Chou:2007zzc and the cooling of the Earth (this work). Before closing, we would like to point out a few directions for improving our estimates. First of all, our picture of the iron core is quite simplified. A more detailed treatment of electron-ion interactions in the molten core, as well as the inner core contribution, which was ignored here, could reveal extra enhancements or suppressions that were left out in our analysis. This could, in principle, require a numerical simulation of the strongly coupled plasma (molten Fe) and the crystalline solid core. Another issue is the possible role that the Fe $3d$-orbital electrons play, given that they are delocalized over a few nuclei and may contribute to pseudo-scattering processes inside the hot Fe medium. Also, there could be important bound-bound and free-bound processes that result in the emission of axions from Fe atoms at high temperatures. These processes have been ignored here, but could provide contributions comparable to those we have estimated. In principle, more detailed geodynamical analyzes may yield stronger bounds on non- convective energy transfer out of the Earth’s core. This can result in tighter bounds on axion-electron coupling $\alpha_{ae}$, in the regime we considered here. Finally, our estimate of a geoscope signal assumed an axion flux transverse to the magnetic field. Given the angular size of the core, as viewed through the geoscope, we expect that effective transverse field is, on average, suppressed by roughly $\cos^{2}30^{\circ}$, which does not affect our conclusions, given the approximate nature of our estimates. In summary, we have derived estimates on possible emission of axions from the hot core of the Earth. Our analysis allows for possible dependence of axion properties on non-vacuum production media, such as astrophysical environments. We approximated the molten core as a strongly coupled plasma of free degenerate electrons in the background of Fe nuclei. We adapted the existing axion emission estimate from a White Dwarf interior, which is a strongly coupled plasma supported by a degenerate electron gas. We obtained the bound $\alpha_{ae}^{\oplus}\lesssim 10^{-18}$ on the axion-electron coupling, by considering geodynamical constraints on core-cooling rates. Given that geo- axions would originate from a far less extreme environment than stellar cores, our bound is nearly a vacuum bound. Hence, our result improves existing accelerator constraints on $\alpha_{ae}$, in vacuo, by 4 orders of magnitude. We also estimated the signal strength to be expected in a dedicated search for geo-axions using a geoscope, based on magnetic axion-photon conversion. ###### Acknowledgements. We would like to thank G. Khodaparast, S. King, and Y. Semertzidis for useful discussions. The work of H.D. is supported in part by the United States Department of Energy under Grant Contract DE-AC02-98CH10886. ## References * (1) R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977); Phys. Rev. D 16, 1791 (1977). * (2) J. Preskill, M. B. Wise and F. Wilczek, Phys. Lett. B 120, 127 (1983); L. F. Abbott and P. Sikivie, Phys. Lett. B 120, 133 (1983); M. Dine and W. Fischler, Phys. Lett. B 120, 137 (1983); M. S. Turner, Phys. Rev. D 33, 889 (1986). * (3) C. Amsler et al. [Particle Data Group], Phys. Lett. B 667, 1 (2008). * (4) P. Gondolo and G. Raffelt, arXiv:0807.2926 [astro-ph]. * (5) Emission of axions from the core of Jupiter has been considered in: C. Sivaram, Earth, Moon, and Planets 37, 155 (1987). * (6) A. Liolios, Phys. Lett. B 645, 113 (2007). * (7) E. Masso and J. Redondo, Phys. Rev. Lett. 97, 151802 (2006) [arXiv:hep-ph/0606163]. * (8) E. Zavattini et al. [PVLAS Collaboration], Phys. Rev. Lett. 96, 110406 (2006) [Erratum-ibid. 99, 129901 (2007)] [arXiv:hep-ex/0507107]. * (9) See, J. Redondo, arXiv:0807.4329 [hep-ph], and references therein. * (10) G. Cantatore [PVLAS Collaboration], Lect. Notes Phys. 741, 157 (2008). * (11) P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983) [Erratum-ibid. 52, 695 (1984)]. * (12) K. van Bibber, P. M. McIntyre, D. E. Morris and G. G. Raffelt, Phys. Rev. D 39, 2089 (1989). * (13) R. D. van der Hilst, et al., Science 315, 1813 (2007). * (14) E. Clementi and D. L. Raimondi, J. Chem. Phys. 1963, 38, 2686. * (15) T. Nautiyal and S. Auluck, Phys. Rev. B 32, 6424 (1985). * (16) G. G. Raffelt, Stars as Laboratories for Fundamental Physics (The University of Chicago Press, 1996). * (17) G. G. Raffelt, Phys. Lett. B 166, 402 (1986). * (18) H. Primakoff, Phys. Rev. 81, 899 (1951). * (19) S. O. Costin and S. L. Butler, Phys. Earth Plan. Int. 157, 55 (2006). * (20) F. T. . Avignone, arXiv:0810.4917 [nucl-ex]. * (21) K. Zioutas et al. [CAST Collaboration], Phys. Rev. Lett. 94, 121301 (2005) [arXiv:hep-ex/0411033]. S. Andriamonje et al. [CAST Collaboration], JCAP 0704, 010 (2007) [arXiv:hep-ex/0702006]. * (22) See, for example: I. C. Atkinson et al., J. Magn. Reson. Imaging 26: 1222 (2007). * (23) A. S. Chou et al. [GammeV (T-969) Collaboration], Phys. Rev. Lett. 100, 080402 (2008) [arXiv:0710.3783 [hep-ex]]. * (24) A. J. Miller, et al., Appl. Phys. Lett. 83, 791 (2003). * (25) A. E. Lita, A. J. Miller, and S. W. Nam, Optics Express 16 3032-3040, (2008).	arxiv-papers	2009-03-03T21:00:30	2024-09-04T02:49:00.966528	{ "license": "Public Domain", "authors": "Hooman Davoudiasl, Patrick Huber", "submitter": "Patrick Huber", "url": "https://arxiv.org/abs/0903.0618" }
0903.0643	# Cones and convex bodies with modular face lattices. Daniel Labardini-Fragoso , Max Neumann-Coto and Martha Takane Graduate Program in Mathematics, Northeastern University, Boston, Ma. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Cuernavaca, México. Dedicated to Claus M. Ringel on the occasion of his 60th birthday. ###### Abstract. If a convex body $C$ has modular and irreducible face lattice (and is not strictly convex), there is a face-preserving homeomorphism from $C$ to a section of a cone of hermitian matrices over $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, or $C$ has dimension 8, 14 or 26. ###### Key words and phrases: convex, face lattice, modular, hermitian matrix, projective space Research partially supported by PAPIIT-UNAM grant IN103508 and a fellowship from PASPA ###### 1991 Mathematics Subject Classification: 52A20, 06C05, 51A05, 15A48 ## 1\. Introduction. Let $C$ be a convex body in $\mathbb{R}^{n}$. A subset $F$ of $C$ is a face of $C$ if every open interval in $C$ that contains a point of $F$ is contained in $F$. An extreme point is a 1-point face. If $S$ is any subset of $C$, the face generated by $S$ is the minimal face of $C$ containing $S$. The set $\mathcal{F}(C)$ of all faces of $C$ ordered by inclusion is a lattice, where $F\wedge G$ is the intersection of $F$ and $G$, and $F\vee G$ is the face generated by $F\cup G$. The lattice $\mathcal{L}(C)$ is always algebraic (the chains of faces are finite), atomic (faces are generated by extreme points) and complemented (for every face $F$ there exists a face $G$ such that $F\wedge G=\emptyset$ and $F\vee G=C$). We want to consider convex bodies for which $\mathcal{F}(C)$ is modular, i.e. $F\vee(G\wedge H)=(F\vee G)\wedge H$ whenever $F\leq H$. Modularity is a ‘weak distributivity’ property satisfied by the lattice of normal subgroups of a group and by the lattice of subspaces of a vector space. For algebraic, atomic lattices, modularity is equivalent to the existence of a rank function such that $rk(F)+rk(G)=rk(F\vee G)+rk(F\wedge G)$ for all $F$ and $G$ [4]. Strictly convex bodies and simplices clearly have modular face lattices. No other polytopes have this property [3], but there are beautiful examples of non-polytopal convex bodies in which every pair of extreme points is contained in a proper face and every pair of faces with more than one point meet. If $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are lattices, their direct product is given by $(\mathcal{L}_{1}\times\mathcal{L}_{2},\leq),$ where $(a,b)\leq(c,d)$ if and only if $a\leq c$ and $b\leq d$. It follows that the direct product of two lattices is modular if and only if the factors are modular. A lattice is called irreducible if it is not isomorphic to a direct product of two nontrivial lattices. If $C_{1}\subset\mathbb{R}^{m}$ and $C_{2}\subset\mathbb{R}^{n}$ are convex bodies, define $C_{1}\ast C_{2}\subset\mathbb{R}^{m+n+1}$ as the convex hull of a copy of $C_{1}$ and a copy of $C_{2}$ placed in general position in the sense that their linear spans are disjoint and have no common directions. So $C_{1}\ast C_{2}$ is well defined up to a linear transformation: it is the convex join of $C_{1}$ and $C_{2}$ of largest dimension. For example $C\ast\left\\{pt\right\\}$ is a pyramid with base $C$. Let’s say that a convex body $C$ is $\ast$-decomposable if $C=C_{1}\ast C_{2}$ for two convex bodies $C_{1}$ and $C_{2}$. The natural correspondence (up to linear transformation) between convex bodies in $\mathbb{R}^{n}$ and closed cones in $\mathbb{R}^{n+1}$ gives an isomorphism of face lattices in which $C_{1}\ast C_{2}$ corresponds to the direct product of the cones, so the results of this paper apply to cones. This project started with the undergraduate thesis of D. Labardini-Fragoso [9], who showed that in dimension less than 6 any cone with modular face lattice is strictly convex or is decomposable (this was conjectured by Barker in [3]). ###### Lemma 1. A convex body $C$ is $\ast$-decomposable if and only if its lattice of faces $\mathcal{L}(C)$ is reducible. ###### Proof. Let $C=C_{1}\ast C_{2}$. Observe that each point $p$ of $C_{1}\ast C_{2}$ with $p\notin C_{i}$, lies in a unique segment joining a point $p_{1}$ of $C_{1}$ and a point $p_{2}$ of $C_{2}$. For, if a point lies in two segments $p_{1}p_{2}$ and $p_{1}^{\prime}p_{2}^{\prime}$ then the lines $p_{1}p_{1}^{\prime}$ and $p_{2}p_{2}^{\prime}$ are parallel or they intersect, contradicting the assumptions on the spans of $C_{1}$ and $C_{2}$. Moreover, if a point $p\ $moves along a straight line in $C_{1}\ast C_{2}$ then the corresponding points $p_{1}$ and $p_{2}$ move along straight lines in $C_{1}$ and $C_{2}$: If $p$ and $q$ are points in $C$ and $x\in$ $pq$ then $x=tp+(1-t)q=t\lambda p_{1}+t(1-\lambda)p_{2}+(1-t)\mu q_{1}+(1-t)(1-\mu)q_{2}$ which can be rewritten as a linear combination of a point in $p_{1}q_{1}$ and a point in $p_{2}q_{2}$ with coefficients adding up to 1 so $x_{1}\in p_{1}q_{1}$ and $x_{2}\in p_{2}q_{2}$. Now if $C_{i}^{\prime}$ is a face of $C_{i}$ then $C_{1}^{\prime}\ast C_{2}^{\prime}$ is a face of $C_{1}\ast C_{2}$. For, if $x\in C_{1}^{\prime}\ast C_{2}^{\prime}$ and $x=\lambda p+(1-\lambda)q$ with $p,q\in C_{1}\ast C_{2}$, then $x_{1}$ lies in $p_{1}q_{1}$ and $x_{2}$ lies in $p_{2}q_{2}$ so as $C_{i}^{\prime}$ is a face of $C_{i}$, $p_{i}$ and $q_{i}$ lie in $C_{i}^{\prime}$ so $p$ and $q$ lie in $C_{1}^{\prime}\ast C_{2}^{\prime}$. Conversely, if $C^{\prime}$ is a face of $C_{1}\ast C_{2}$ and $p\in C^{\prime}$ then $p_{1}$ and $p_{2}$ lie in $C^{\prime}$ so $C^{\prime}=(C^{\prime}\cap C_{1})\ast(C^{\prime}\cap C_{2})$. It remains to show that $C^{\prime}\cap C_{i}$ is a face of $C_{i}$. If $x\in C^{\prime}\cap C_{1}$ and $x=\lambda p+(1-\lambda)q$ with $p,q\in C_{1}\ast C_{2}$ then as $C^{\prime}$, and $C_{1}=C_{1}\ast\emptyset$ are faces of $C_{1}\ast C_{2}$, $p$ and $q$ lie in $C^{\prime}$ and also in $C_{1}$, so $C^{\prime}\cap C_{1}$ is a face of $C_{1}$. $C^{\prime}\cap C_{2}$ is a face of $C_{2}$. So $\mathcal{L}(C_{1}\ast C_{2})\simeq\mathcal{L}(C_{1})\times\mathcal{L}(C_{2}).$ If $\mathcal{L}(C)\approx\mathcal{L}_{1}\ast\mathcal{L}_{2}$ then $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are isomorphic to sublattices of $\mathcal{L}(C)$, so $\mathcal{L}_{i}\approx\mathcal{L}(C_{i})$ for two faces of $C$ with $C_{1}\wedge C_{2}=\varnothing$ and $C_{1}\vee C_{2}=C$. To show that $C=C_{1}\ast C_{2}$ we need to prove that $span(C_{1})$ and $span(C_{2})$ are disjoint and have no directions in common. Suppose that $x\in span(C_{1})\cap span(C_{2})$. Take $x_{i}\in Int(C_{i})$ then the line through $x$ and $x_{i}$ meets $\partial C_{i}$ at two points $a_{i}$ and $b_{i}.$ As $a_{2}$ lies in a proper subface $C_{2}^{\prime}$ of $C_{2}$, the face generated by $C_{1}$ and $a_{2}$ lies in $C_{1}\vee C_{2}^{\prime}$ which is a proper subface of $C_{1}\vee C_{2}$. But the points $a_{1}$, $b_{1}$, $a_{2}$, $b_{2}$ determine a plane quadrilateral whose side $a_{i}b_{i}$ lies in the interior of $C_{i}$ so its diagonals intersect at an interior point $c$ of $C_{1}\vee C_{2}$ so the face generated by $C_{1}$ and $a_{2}$ (which contains $c$) must be $C_{1}\vee C_{2}$, a contradiction. Now suppose that $span(C_{1})$ and $span(C_{2})$ have a common direction $v$. Take $x_{i}\in Int(C_{i})$ then the line through $x_{i}$ in the direction $v$ meets $\partial C_{i}$ at two points $a_{i}$ and $b$. As before $a_{1}$, $b_{1}$, $a_{2}$, $b_{2}$ determine a plane quadrilateral whose diagonals intersect at an interior point $c$ of $C_{1}\vee C_{2}$, but $c$ lies in the face generated by $C_{1}$ and $a_{2}$ which is a proper face of $C_{1}\vee C_{2}$. ∎ Recall that a projective space consists of a set $P$ (the points) and a set $L$ (the lines) so that (a) Each pair of points is contained in a unique line, (b) If $a,b,c,d$ are distinct points and the lines $ab$ and $cd$ intersect, then the lines $ac$ and $bd$ intersect (c) Each line contains at least 3 points and there are at least 2 lines (d) Every chain of subspaces (also called flats) has finite length. The maximum length of a chain starting with a point is the projective dimension of the space. The flats of a projective space form an algebraic, atomic, irreducible, modular lattice. Conversely, any lattice with these properties is the lattice of flats of a projective space, whose points are atoms and whose lines are joins of two atoms [6]. It is a classic result of Hilbert [8] that a projective space in which Desargues theorem holds is isomorphic to the projective space $\mathbb{AP}^{n}$ determined by the linear subspaces of $\mathbb{A}^{n+1}$, for some division ring $\mathbb{A}$, and that $\mathbb{AP}^{n}$ and $\mathbb{BP}^{m}$ are isomorphic if and only if $\mathbb{A}$ and $\mathbb{B}$ are isomorphic and $m=n$. All projective spaces of dimension larger than $2$ are desarguesian, but there are many non- desarguesian projective planes. Examples of convex bodies whose face lattices determine the projective spaces $\mathbb{RP}^{n}$, $\mathbb{CP}^{n}$ and $\mathbb{HP}^{n}$, and the octonionic projective plane arise as sections of some classical cones. ###### Example 1. Let $\mathbb{F\in\\{R}$,$\mathbb{C}$,$\mathbb{H}\\}$, let $H_{n}(\mathbb{F})$ be the set of Hermitian (self-adjoint) $n\times n$ matrices with coefficients in $\mathbb{F}$, and let $C_{n}(\mathbb{F})$ be the subset of positive- semidefinite matrices ($A$ is positive-semidefinite if $\overline{v}Av^{T}\geq 0,$ for all $v\in$ $\mathbb{F}^{n}$). Then $C_{n}(\mathbb{F})$ is a real cone whose face lattice is isomorphic to the lattice of subspaces of $\mathbb{F}^{n}$. To see this, let $A,B\in C_{n}(\mathbb{K})$, and let $\varphi(B)$ denote the face generated by $B$. Then $A\in\varphi(B)$ if and only if $\ker A\supseteq\ker B$. For, $A\in\varphi(B)$ $\Leftrightarrow\exists\lambda>0$ such that $B-\lambda A\in$ $C_{n}(\mathbb{K})\Leftrightarrow\exists\lambda>0$ such that $\overline{w}Bw^{T}\geq\lambda\overline{w}Aw^{T}\geq 0$ for all $w\in$ $\mathbb{F}^{n}$ $\Longleftrightarrow\overline{w}Bw^{T}=0$ implies $\overline{w}Aw^{T}=0$ for all $w\in$ $\mathbb{F}^{n}\Longleftrightarrow\ker A\supseteq\ker B$ (since for $A\in C_{n}(\mathbb{K})$, $\overline{w}Aw^{T}=0$ if and only if $Aw^{T}=0$). Therefore $\varphi(A)\rightarrow\left(\ker A\right)^{\perp}$ defines a bijection $\nu$ from the set of faces of $C_{n}(\mathbb{K})$ to the set of linear subspaces of $\mathbb{K}^{n}$. To prove that $\nu$ is an isomorphism of lattices observe that $\nu(\varphi(A)\vee\varphi(B))=\nu(\varphi(A+B))=\left(\ker\left(A+B\right)\right)^{\perp}=\left(\ker A\cap\ker B\right)^{\perp}=\left(\ker A\right)^{\perp}\cup\left(\ker B\right)^{\perp}$, and on the other hand, if $\varphi(A)\wedge\varphi(B)$ is a non-empty face, then it is generated by a matrix $C$ with $\ker C=span(\ker A\cup\ker B)$, so $\nu(\varphi(A)\wedge\varphi(B))=\left(\ker C\right)^{\perp}=\left(\ker A\right)^{\perp}\cap\left(\ker B\right)^{\perp}$. ###### Example 2. Let $H_{3}(\mathbb{O)}$ be the set of Hermitian $3\times 3$ matrices over $\mathbb{O}$. Then the subset $C_{3}(\mathbb{O)}$ of all sums of squares of elements in $H_{3}(\mathbb{O)}$ is a real cone whose face lattice determines an octonionic projective plane. This can be shown using the nontrivial fact that each matrix in $H_{3}(\mathbb{O})$ is diagonalizable by an automorphism of $H_{3}(\mathbb{O})$ that leaves the trace invariant [2], so: (a) A matrix $A$ in $H_{3}(\mathbb{O)}$ lies in $C_{3}(\mathbb{O)}$ if and only if it can be diagonalized to a matrix $A^{\prime}$ with non-negative entries, because if $A$ lies in $C_{3}(\mathbb{O)}$ then $A^{\prime}$ is a sum of squares of matrices in $H_{3}(\mathbb{O)}$, which have non-negative diagonal entries. (b) All the idempotent matrices in $H_{3}(\mathbb{O)}$ lie in $C_{3}(\mathbb{O)}$ as they are squares $(A=A^{2})$. The idempotent matrices of trace 1 correspond to the extreme rays of $C_{3}(\mathbb{O)}$ since they can’t be written as non-negative combinations of other idempotent matrices. (c) Each face of $C_{3}(\mathbb{O)}$ is generated by an idempotent matrix, because in any cone all the positive linear combinations of the same set of vectors generate the same face, so a diagonal matrix with non-negative entries generates the same face as a matrix with only zeros and ones. (d) Any two idempotent matrices of trace 1 lie in a face generated by an idempotent matrix of trace 2, because they can be put simultaneously in the form $\left[\begin{array}[]{ccc}a&x&0\\\ \overline{x}&b&0\\\ 0&0&0\end{array}\right],$ and these lie in the face generated by $\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&0\end{array}\right]$. (e) $A\in C_{3}(\mathbb{O)}$ is an idempotent of trace 1 if and only if $I-A$ is an idempotent with trace 2. If $A$ and $B$ are idempotents of trace 1, then $A$ lies in the face generated by $I-B$ if and only if $B$ lies in the face generated by $I-A$. This duality and (d) show that any two faces generated by idempotent matrices of trace 2 meet in a face generated by an idempotent matrix of trace 1. ## 2\. Face lattices defining projective spaces. If $C$ is a convex body whose face lattice is modular and irreducible and $C$ is not strictly convex, the set of extreme points of $C$ is a projective space with flats determined by the faces of $C$. We would like to know which projective spaces arise in this way, and what convex bodies give rise to them. By Blaschke selection theorem [7], the space of all compact, convex subsets of a convex body in $\mathbb{R}^{n}$, with the Hausdorff metric, is compact. So the subspace formed by the compact convex subsets of the boundary is closed, but the subspace formed by the faces is not closed in general. ###### Lemma 2. If $C$ is a convex body whose face lattice $\mathcal{F}$ is modular, then the set $\mathcal{F}_{h}$ of faces of rank $h$, with the Hausdorff metric, is compact for each $h$. ###### Proof. Let $F_{i}$ be a sequence of faces of rank $h$. Then $F_{i}$ has a subsequence $F_{i_{j}}$ that is convergent in $\mathcal{C}$, and its limit is a compact convex set $K$ contained in $\partial C$, so $K$ generates a proper face $F$ of $C$ of some rank $h^{\prime}$. We claim that $h^{\prime}=h$ and $K=F$. If the rank of $F$ was less than $h$, there would be a face $F^{c}$ of $C$ of rank $n-h$ with $F^{c}\cap F=\phi$. As $F$ and $F^{c}$ are two disjoint compact sets in $\mathbb{R}^{n}$, there exists $\epsilon>0$ such that the $\varepsilon$-neighborhoods of $F$ and $F^{c}$ in $\mathbb{R}^{n}$ are disjoint. But as $F_{i_{j}}\rightarrow K\subset F$ in the Hausdorff metric, then for sufficiently large $j$, $F_{i_{j}}$ is contained in the $\varepsilon$-neighborhood of $F$, therefore $F_{i_{j}}\cap F^{c}=\phi$, but these 2 faces have ranks that add up to $n$, so they should meet, a contradiction. If the rank of $F$ is $h$ and $K\neq F$, there is an extreme point $p\in F-K$, and there is a face $F^{\prime}$ of rank $n-h$ that meets $F$ only at $p$, so $F^{\prime}\cap K=\phi$ and the previous argument gives a contradiction. To show that the rank of $F$ cannot be larger than $h$, proceed inductively on $n-h$. As a limit of proper faces is contained in a proper face, the claim holds if $n-h=1$. Given a sequence $F_{i}$ of faces of rank $h$, let $F$ be a face generated by the limit of a convergent subsequence $F_{i_{j}}$. If $p$ is an extreme point of $C$ not in $F$, then for sufficiently large $j$, $p\notin F_{i_{j}}$ (otherwise $p$ would be in $F$). Let $G_{i_{j}}$ be a face of rank $h+1$ containing $F_{i_{j}}$ and $p$. Now we can assume inductively that the limit of a convergent subsequence of $G_{i_{j}}$ generates a face of rank $h+1$. This face contains $F$ properly (because it contains $p$) so $h^{\prime}<h+1$. ∎ Now recall that a topological projective space is a projective space in which the sets of flats of each rank are given nontrivial topologies that make the join and meet operations $\vee$ and $\wedge$ continuous, when restricted to pairs of flats of fixed ranks whose join or meet have a fixed rank [5]. ###### Lemma 3. If $C$ is a convex body whose face lattice is modular and irreducible then $C$ is strictly convex or the set of extreme points $\mathcal{E(}C\mathcal{)}$ is a topological projective space which is compact and connected. ###### Proof. A natural topology for the set of flats is given by the Hausdorff distance between the faces. By lemma 2, $\mathcal{E=F}_{0}$ is compact. As the lattice is irreducible and has more than 2 points, the 1-flats have more than 2 points, so (as they are topological spheres) they are connected. Now every pair of points in $\mathcal{E}$ is contained in one of these spheres, so $\mathcal{E}$ is connected (one can actually show that each.$\mathcal{F}_{h}$ is connected). It remains to show that $\vee$ and $\wedge$ are continuous on the preimages of each $\mathcal{F}_{h}$. Suppose $A_{i}\rightarrow A$ , $B_{i}\rightarrow B$ where $A_{i}\wedge B_{i}$ and $A\wedge B$ are faces corresponding to $h$ flats. We need to show that $A_{i}\wedge B_{i}\rightarrow A\wedge B$. By lemma 2 $C_{i}=A_{i}\wedge B_{i}$ has convergent subsequences and the limit of a convergent subsequence $C_{i_{\alpha}}$ is a face $C_{\alpha}$ corresponding to an $h$ flat. As $C_{i_{\alpha}}$ is contained in $A_{i_{\alpha}}$ and $B_{i_{\alpha}}$, $C_{\alpha}$ is contained in $A\wedge B$. But $C_{\alpha}$ and $A\wedge B$ are both faces corresponding to $h$ flats, so $C_{\alpha}=A\wedge B$. Similarly, if $A_{i}\rightarrow A$ , $B_{i}\rightarrow B$ and $A_{i}\vee B_{i}$ , $A\vee B$ are faces corresponding to $h$ flats, the limit of each convergent subsequence of $D_{i}=A_{i}\vee B_{i}$ is a face $D$ corresponding to an $h$ flat. As $D_{i}$ contains $A_{i}$ and $B_{i}$, $D$ contains $A\vee B$, and as both faces correspond to $h$ flats they must be equal. ∎ Let $C$ be any convex body. Denote by $\mathcal{B}(C\mathcal{)}\subset C$ the set of baricenters of faces of $C$ and let $b:\mathcal{F}(C)\rightarrow\mathcal{B}(C)$ the function that assigns to each face its baricenter. ###### Lemma 4. (a) If $\mathcal{F}(C)$ is compact then $b$ is a homeomorphism. (b) If $\mathcal{E}(C)$ is compact then a sequence of faces $F_{i}$ converges to a face $F$ if and only if $\mathcal{E}(F_{i})$ converges to $\mathcal{E}(F)$. ###### Proof. (a) The function that assigns to each compact convex set in $\mathbb{R}^{n}$ its baricenter is continuous, so $b:\mathcal{F}(C)\rightarrow\mathcal{B}(C)$ is a continuous bijective map from a compact Hausdorff space to a metric space. (b) The Hausdorff distance between two compact convex sets is bounded above by the Hausdorff distance between their sets of extreme points. If $F_{i}$ converges to $F$ but $\mathcal{E}(F_{i})$ doesn´t converge to $\mathcal{E}(F)$ then there is a subsequence $\mathcal{E}(F_{i_{j}})$ that stays at distance at least $\varepsilon>0$ from $\mathcal{E}(F)$. For each $i_{j}$ there is an extreme point $p_{i_{j}}\in F_{i_{j}}$ whose distance from $\mathcal{E}(F)$ is larger than $\varepsilon$, or an extreme point $q_{i}\in F$ whose distance from $\mathcal{E}(F_{i_{j}})$ is larger than $\varepsilon$. If there is a convergent subsequence $p_{i_{k}}\rightarrow p\in F$ then $p$ is at distance at least $\varepsilon$ from $\mathcal{E}(F)$, so $p$ can’t be an extreme point of $C.$ If there is a convergent subsequence $q_{i_{k}}\rightarrow q\in F$, take $p_{i_{k}}^{\prime}\in F_{i_{k}}$ with $p_{i_{k}}^{\prime}\rightarrow q$. Eventually $\left\|p_{i_{k}}^{\prime}-q_{i_{k}}\right\|<\frac{\varepsilon}{2}$ so the distance from $p_{i_{k}}^{\prime}$ to $\mathcal{E}(F_{i_{k}})$ is at least $\frac{\varepsilon}{2}$, so $p_{i_{k}}^{\prime}$ is the center of a straight interval $I_{i_{k}}$ of length $\varepsilon$ contained in $F_{i_{k}}$. A convergent subsequence of these intervals yields a straight interval centered at $q$ and contained in $F$, so $q$ can’t be an extreme point of $C$, contradicting the compacity of $\mathcal{E}(C)$. ∎ ###### Lemma 5. If $C$ and $C^{\prime}$ are convex bodies with $\mathcal{F}(C)$ and $\mathcal{F}(C^{\prime})$ compact, then any continuous ”face preserving” map $\varphi:\mathcal{E}(C)\rightarrow\mathcal{E}(C^{\prime})$ extends naturally to a continuous map $\varphi:C\rightarrow C^{\prime}$. ###### Proof. $\varphi$ determines a function $\Psi:\mathcal{F}(C)\rightarrow\mathcal{F}(C^{\prime})$. $\Psi$ is continuous because by lemma 4, $F_{i}\rightarrow F$ implies $\mathcal{E}(F_{i})\rightarrow\mathcal{E}(F)$, then uniform continuity of $\varphi$ on $\mathcal{E}(C)$ implies that $\varphi(\mathcal{E}(F_{i}))\rightarrow\varphi(\mathcal{E}(F))$ so by definition $\mathcal{E}(\Psi(F_{i}))\rightarrow\mathcal{E}(\Psi(F))$ and so $\Psi(F_{i})\rightarrow\Psi(F)$. So $\varphi$ can be extended to a continuous function $\varphi:\mathcal{B}(C)\rightarrow\mathcal{B}(C^{\prime})$ as $b\circ\Psi\circ b^{-1}$ (recall that $\mathcal{E}(C)\subset\mathcal{B}(C)$). Now we can extend $\varphi$ to the interiors of the faces of $C$ defining it linearly on rays, as follows. For each point $a\in C,$ let $F(a)$ be the unique face of $C$ containing $a$ in its interior and let $b(a)$ be the baricenter of $F(a)$. Although $F(a)$ and $b(a)$ are not continuous functions of $a$ on all of $C$, they are continuous on the union of the interiors of the faces corresponding to $h$-flats for each $h$. If $a\neq b(a)$ let $p(a)$ be the projection of $a$ to $\partial F(a)$ from $b(a)$ and let $\lambda(a)=\frac{\left\|a-b(a)\right\|}{\left\|p(a)-b(a)\right\|}$ (or $0$ if $a=b(a)$) so $a=(1-\lambda(a))b(a)+\lambda(a)p(a)$. Define $\varphi(a)=(1-\lambda(a))\varphi(b(a))+\lambda(a)\varphi(p(a)).$ Assume inductively that $\varphi$ is continuous on the union of $\mathcal{B}(C)$ and the faces of $C$ of dimension less than $d$ (this set is closed because the limit of faces of dimension less than $d$ has dimension less than $d$). and let’s show that for each sequence of points $a_{i}$ in the interiors of faces of dimension $d$, $a_{i}\rightarrow a$ implies $\varphi(a_{i})\rightarrow\varphi(a)$. We may assume that the $a_{i}$ are not baricenters, so $p(a_{i})$ is well defined. Case 1. $F(a_{i})\rightarrow F(a)$ then $b(a_{i})\rightarrow b(a)$ by the continuity of $b$ on faces. If $b(a)\neq a$ then $p(a_{i})\rightarrow p(a)$ and $\lambda(a_{i})\rightarrow\lambda(a)$ so $\varphi(a_{i})=(1-\lambda(a_{i}))\varphi(b(a_{i}))+\lambda(a_{i})\varphi(p(a_{i}))\rightarrow(1-\lambda(a))\varphi(b(a))+\lambda(a)\varphi(p(a))=\varphi(a)$. If $b(a)=a$ then $\lim b(a_{i})=\lim a_{i}$ but $p(a_{i})$ may not converge, so consider a convergent subsequence $p(a_{i_{j}})$: If $\lim p(a_{i_{j}})\neq$ $\lim b(a_{i_{j}})=\lim a_{i_{j}}$ then $\lim\lambda(a_{i_{j}})=0$ so $\varphi(a_{i_{j}})=\varphi(b(a_{i_{j}}))+\lambda(a_{i_{j}})\left[\varphi(p(a_{i_{j}}))-\varphi(b(a_{i_{j}}))\right]\rightarrow\varphi(b(a))+0=\varphi(a)$. If $\lim p(a_{i_{j}})=$ $\lim b(a_{i_{j}})$ then $\lim\varphi(p(a_{i_{j}}))=\lim\varphi(b(a_{i_{j}}))$ (by continuity of $\varphi$ in the baricenters and faces of lower dimension) and as $\varphi(a_{i_{j}})$ lies between them, $\lim\varphi(a_{i_{j}})=\lim\varphi(b\circ F(a_{i_{j}}))=\varphi(b(a))=\varphi(a)$. Case 2. $F(a_{i})\nrightarrow F(a)$, then for any convergent subsequence $F(a_{i_{j}})$ with limit a face $F\neq F(a)$, $a$ lies in $F$ and so $a$ must lie in $\partial F$, so $\left\|a_{i_{j}}-p(a_{i_{j}})\right\|\rightarrow 0$ and $\lambda(a_{i_{j}})\rightarrow 1$, so $\lim\varphi(a_{i_{j}})=\lim(1-\lambda(a_{i_{j}}))\varphi(b(a_{i_{j}}))+\lambda(a_{i_{j}})\varphi(p(a_{i_{j}}))=\lim\varphi(p(a_{i_{j}}))=\varphi(a)$ (by continuity of $\varphi$ on the faces of lower dimension). ∎ ###### Theorem 1. Let $C$ be a convex body whose face lattice defines a n-dimensional projective space. If $n=2$, then $C$ has dimension $5,8,14$ or $26$. If $n>2$ (or the space is desarguesian) there is a face-preserving homeomorphism from $C$ to a section of a cone of Hermitian matrices over $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$. ###### Proof. First consider the case $n=2$. All the lines of a topological projective plane $\mathcal{P}$ are homeomorphic because if $l$ is a line and $p$ is a point not in $l$ then the projection $\phi:\mathcal{P}-p\rightarrow l$, $\phi(x)=(x\vee p)\wedge l$ is continuous and its restriction to each projective line not containing $p$ is one to one. If the projective lines are topological spheres then a famous result of Adams [5, p.1278], shows that their dimension must be $d=0,1,2,4$ or $8$. To compute the dimension of $C$ take 3 faces of rank 1, $F_{0}$, $F_{1}$ and $F_{2}$ so that $F_{1}$ and $F_{2}$ meet at a point $p$ not in $F_{0}$. The projection $\phi:\mathcal{E}(C)-\\{p\\}\rightarrow\partial F_{0}$ extends to a continuous map $\phi:\cup\left\\{F\text{ }\|\text{ }F\text{ face of }C\text{, }p\notin F\right\\}\rightarrow F_{0}$ whose restriction to each face is one to one (see proof of lemma 5). Now $U=\cup\left\\{IntF\text{ }/\text{ }F\text{ face of }C\text{, }p\notin F\right\\}$ is an open subset of $\partial C$ and the function $\Phi:U\rightarrow F_{0}\times\left(\partial F_{1}-\\{p\\}\right)\times\left(\partial F_{2}-\\{p\\}\right)$ defined as $\Phi(x)=(\phi(x),\partial F(x)\wedge\partial F_{1},\partial F(x)\wedge\partial F_{2})$ is continuous and bijective, so $U$ has the same dimension as $F\times\partial F\times\partial F$, which is $3d+1$, therefore $C$ has dimension $3d+2$. Note that the discrepancy between the dimensions of the union of the boundaries of the faces ($2d$) and the union of the faces ($3d+1$) arises because the boundaries of the faces overlap (as the lines in a projective plane do) but the interiors of the faces are disjoint. When $n>2$, there is a similar homeomorphism from an open subset of $\partial C$ and a product $F_{0}\times\left(\partial F_{1}-\\{p\\}\right)\times\left(\partial F_{2}-\\{p\\}\right)\times...x\left(\partial F_{n}-\\{p\\}\right)$ where $F_{0}$ is a face of rank $r-1$ and $F_{1},F_{2},...,F_{n}$ are faces of rank 1. So $\dim(C)=\dim(F_{0})+rd+1$ and it follows by induction that $\dim C=\frac{n(n-1)}{2}d+n-1$. Now assume that the projective space determined by$\ \mathcal{E}(C)$ is desarguesian. Every topological desarguesian projective space is isomorphic to a projective space over a topological division ring $A$ (defined on a line minus a point) and the isomorphism is a homeomorphism [5, p.1261]. By a classic result of Pontragin [5, p.1263] the only locally compact, connected division rings are $\mathbb{R},$ $\mathbb{C}$ and $\mathbb{H}$ . So the projective space determined by $\mathcal{E}(C)$ is isomorphic to $\mathbb{RP}^{r},\mathbb{CP}^{r}$ or $\mathbb{HP}^{r}$. Therefore $\mathcal{E}(C)$ is isomorphic to $\mathcal{E}(C^{\prime})$ where $C^{\prime}$ is a section of a cone of Hermitian matrices, and so by 5 there is a face- preserving homeomorphism from $C$ to $C^{\prime}$. ∎ ## 3\. Face lattices defining affine spaces. Now let us consider closed (but not necessarily compact) convex sets in $\mathbb{R}^{n}$ whose faces meet as the subspaces of an affine space. An abstract affine plane consists of a set of points and a set of lines so that 1) there are at least 2 points and 2 lines 2) every pair of points is contained in a line and 3) given a line and a point not contained in it, there is a unique line containing the point and parallel to the line. The axioms of an abstract affine space are not so simple, but it is enough to know that if $P$ is a projective space then the complement of a maximal flat of $P$ is an affine space, and any affine space $A$ can be embedded in a projective space in this fashion, by attaching to $A$ a point at infinity for each parallelism class of affine lines. Observe that if a closed convex set $C$ in $\mathbb{R}^{n}$ is non-compact, it contains a ray (half of a euclidean line) and if $C$ contains a ray then it contains all the parallel rays starting at points of $C$ (we say that $C$ contains an infinite direction). So if $C$ contains a line, $C$ is the product of that line and a closed convex set $C^{\prime}$ of lower dimension and the face lattice of $C$ and $C^{\prime}$ are isomorphic. So from now on we will assume that $C$ doesn’t contain lines. It is easy to see that the faces of a polytope cannot determine an affine space (the faces of rank $i$ would have dimension $i$, two parallel faces of rank 1 generate a face of rank 2 with at least 4 vertices, but the sides of a polygon don’t define an affine plane). ###### Example 3. Let $C$ be a convex body in $\mathbb{R}^{n}$ whose faces determine a projective space. Take a cone over $C$ and slice it with a hyperplane parallel to a support hyperplane containing a maximal face. The result is a closed non- compact convex set $C^{\prime}$ in $\mathbb{R}^{n}$ whose faces determine an affine space. In particular, the cones of Hermitian matrices have non-compact sections whose face lattice determines a real, complex or quaternionic affine space or an octonionic affine plane. $\mathbb{RP}^{n}$ can be seen as the space of lines through the origin in $\mathbb{R}^{n+1}$ or as the quotient of the unit sphere $\mathbb{S}^{n}$ (or the sphere at infinity of $\mathbb{R}^{n+1}$) by the action of the antipodal map. Identifying $\mathbb{R}^{n}$ with a hyperplane of $\mathbb{R}^{n+1}$ that doesn’t contain the origin gives an embedding of $\mathbb{R}^{n}$ as a dense open subset of $\mathbb{RP}^{n}$. The remaining points of $\mathbb{RP}^{n}$ correspond to lines through the origin in $\mathbb{R}^{n+1}$ that don’t meet the hyperplane, i.e., parallelism classes of lines in $\mathbb{R}^{n}$ (or pairs of antipodal points in the sphere at infinity). Define a set in $\mathbb{RP}^{n}$ to be convex if it is the image of a convex set in $\mathbb{R}^{n}$ under one of these embeddings. As convex sets in $\mathbb{RP}^{n}$ correspond to convex cones based at the origin of $\mathbb{R}^{n+1}$, convexity in $\mathbb{RP}^{n}$ doesn’t depend on the particular embedding, and a convex set in $\mathbb{RP}^{n}$ has the usual properties of a convex set in $\mathbb{R}^{n}$. Now if $C$ is a closed convex set in $\mathbb{R}^{n}$ that doesn’t contain lines, its closure $\overline{C}$ is a convex set in $\mathbb{RP}^{n}$. The faces of $\overline{C}$ are the closures of faces of $C$ and their intersections with the sphere at infinity modulo the antipodal map. ###### Lemma 6. In a closed convex set in $\mathbb{R}^{n}$ that has semi-modular face lattice, two faces of rank 1 can share at most one direction, and it corresponds to a ray. ###### Proof. We are considering convex bodies without lines. Suppose that two rank 1 faces $F_{1}$ and $F_{2}$ have a common direction, i.e., there are segments of parallel euclidean lines $l_{1}$ and $l_{2}$ lying in $F_{1}$ and $F_{2}$. We may assume that $l_{i}$ goes through an interior point of $F_{i},$ so $l_{i}$ meets $\partial F_{i}$ in one or two extreme points. If $l_{1}$ or $l_{2}$ has two extreme points then there is a convex quadrilateral with sides in $l_{1}$ and $l_{2}$ with 3 extreme points as vertices. The interior of the quadrilateral lies in the interior of the rank 2 face generated by the 3 extreme points, but the intersection of its diagonals lies in the rank 1 face generated by 2 extreme points, a contradiction. So $l_{i}$ meets $\partial F_{i}$ in only one point and so $F_{i}$ contains a ray $l_{i}^{+}$. If $F_{1}$ and $F_{2}$ have two common directions, there is a common direction which meets $F_{1}$ in 2 points, giving the same contradiction. ∎ ###### Lemma 7. If the faces of a closed convex set $C$ in $\mathbb{R}^{n}$ define an affine space, then each face representing a line contains a unique ray, and faces representing parallel lines contain parallel rays. ###### Proof. We are assuming again that $C$ doesn’t contain lines, so the points of the affine space correspond to the extreme points of $C$. Each affine lines is represented by the boundary of a convex set of dimension at least 2, which is connected, so set of extreme points in $C$ is connected. Let’s first show that the faces representing affine lines cannot be compact. Suppose that $C$ has a compact face $F$. Let $p$ and $q$ be two extreme points in $F$ and let $q_{i}$ be a sequence of extreme points not in $F$ that converge to $q$. Let $F_{i}$ be the face generated by $p$ and $q_{i}$. If $F_{i}$ is non-compact, it contains a ray $r_{i}$ through $p$. So $F_{i}$ contains the ”parallelogram” determined by the interval $pq_{i}$ and the ray $r_{i}.$An infinite sequence of $l_{i}$’s would have a subsequence converging to a ray $l$ through $q$, so the parallelogram determined by the interval $pq$ and the ray $l$ would be contained in $\partial C$, so it would have to be contained in a face of $C$, which would have to be $F$ because it contains $p$ and $q$. This contradicts the assumption that $F$ is compact and shows that if $q_{i}$ is sufficiently close to $q$, the face generated by $p$ and $q_{i}$ is compact. Now take a face $F^{\prime}$ that doesn’t meet $F$ (i.e., $F$ and $F^{\prime}$ represent parallel affine lines). As $F$ is compact and $F^{\prime}$ is closed in $\mathbb{R}^{n}$, there is an $\varepsilon$ neighborhood of $F$ that doesn’t intersect $F^{\prime}$. By the previous argument there is a point $q_{i}$ not in $F$ so that the face $F_{i}$ generated by $p$ and $q_{i}$ is contained in the $\varepsilon$ neighborhood of $F$. So $F_{i}$ doesn’t meet $F^{\prime}$, but $F$ was supposed to be the only face containing $p$ and disjoint from $F^{\prime}$. This proves that $F$ is non-compact, so it contains rays. Let’s show that two faces representing parallel affine lines contain parallel rays. Let $p$ be an extreme point outside $F$, so $F$ and $p$ generate a face $H$ representing an affine plane. There are extreme points $p_{0},p_{1},p_{2},...$ in $F$ so that the sequence of intervals $p_{0}p_{i}$ converges to a ray $l_{+}$ contained in $F$ (because $F$ is closed). The sequence of intervals $pp_{i}$ lie in $\partial H$ and converge to a ray $m_{+}$ parallel to $l_{+}$ and containing $p$ so (as $H$ is closed) $m_{+}$ is contained in a face $G$ of $\partial H$ representing an affine line. As two faces that contain parallel rays cannot meet at a single point, $G$ doesn’t meet $F$ so (as $F$ and $G$ are contained in $H$) $G$ represents the affine line parallel to $F$ through $p$. If $F$ has nonparallel rays, one can construct as before two nonparallel rays $l_{+}$ and $l_{+}^{\prime}$ in $F$ and faces $G$ and $G^{\prime}$ through $p$ and containing rays $m_{+}$ and $m_{+}^{\prime}$. The uniqueness of parallel affine lines implies that $G=G^{\prime}$, so $F$ and $G$ have more than one common direction, contradicting the previous lemma. ∎ This shows that if the faces of a closed convex set $C$ define an affine space, $C$ is non-compact. One can show that if the faces of a closed convex set $C$ (containing no lines) define a projective space, $C$ must be compact. For this, one has to give a topology to the space of closed convex sets in $\mathbb{R}^{n}$ that makes it locally compact, show that this makes $\mathcal{E}(C)$ into a locally compact projective space, and observe that these spaces are necessarily compact. ###### Theorem 2. If $C$ is a closed convex set in $\mathbb{R}^{n}$ whose faces determine an affine space, there is a projective transformation in $\mathbb{RP}^{n}$ taking $\overline{C}$ to a compact convex set in $\mathbb{R}^{n}$ whose faces determine a projective space. If the space is desarguesian, there is a face- preserving homeomorphism from $C$ to a non-compact slice of a cone of Hermitian matrices. ###### Proof. We need to show that if the face lattice of $C$ determines an affine space, the face lattice of $\overline{C}\subset\mathbb{RP}^{n}$ determines its projective completion. The faces of $C$ representing affine lines are non- compact, and two of them share an infinite direction in $\mathbb{R}^{n}$ if and only if they represent parallel affine lines. The closure $\overline{C}\subset\mathbb{RP}^{n}$ contains one point at infinity for each infinite direction in $C$, so $\overline{C}$ contains an extreme point at infinity for each class of faces of $C$ representing parallel lines. This corresponds precisely with the definition of the projective completion of the affine space. Now the result for $C$ follows by applying theorem 1 to $\overline{C}$. ∎ ## 4\. Projective planes and the case $d=1$. The face lattice of a convex body $C$ (not a triangle) determines a projective plane if every pair of extreme points is contained in a proper face and every pair of faces with more than one point meet. By theorem 1 this projective plane is compact and connected, so for some $d\in\left\\{1,2,4,8\right\\}$, all the faces of $C$ have dimension $d+1$ and $C$ has dimension $n=3d+2$. ###### Lemma 8. A $d+1$ dimensional subspace of $\mathbb{R}^{n}$ is the span of a face of $C$ if and only if it meets all the spans of faces of $C$ . ###### Proof. Let $S$ be an affine subspace that intersects $span(F)$ for every $F\in\mathcal{F}_{1}$, the set of faces of rank 1. Then $\left\\{F\in\mathcal{F}_{1}\text{ }\|\text{ }\dim(S\cap span(F))\geq i\right\\}$ is closed in $\mathcal{F}_{1}$ for each $i$. Case 1. $d=1.$ We claim that if $S$ is not the span of a face then $S$ cannot intersect $span(F)$ in more than one point. For, if $S\cap span(F)$ contains a line, then $span(S\cup F)$ is 3 dimensional. Take an extreme point $p\notin span(S\cup F)$ and let $F_{1}$ and $F_{2}$ be 2 faces containing $p$, and meeting $F$ at points $p_{1}$ and $p_{2}$ not in $S$. If $p_{1}^{\prime}$ and $p_{2}^{\prime}$ are points in $S\cap span(F_{1})\ $and $S\cap span(F_{2})\ $respectively, then $p$, $p_{i}$ and $p_{i}^{\prime}$ are not aligned (otherwise $p$ would be in the span of $S\cup span(F)$) and so the span of $p$, $p_{i}$ and $p_{i}^{\prime}$, which is $span(F_{i})$, is contained in $span(p$ $\cup S\cup F)$. But $span(p$ $\cup S\cup F)$ is 4 dimensional, so it cannot contain the 3 faces $F$, $F_{1}$ and $F_{2}$ because if it did, it would contain each face that meets $F$, $F_{1}$ and $F_{2}$ at $3$ different points, but every face is a limit of such faces, so it would contain all the faces of $C$, but $C$ has dimension 5. This shows that $S$ intersects each $span(F)$ at exactly one point, and so $S$ contains at most one extreme point of $C$. The function $I:\mathcal{F}_{1}\rightarrow S$ that maps each face $F_{i}$ to the point of intersection of $span(F)$ with $S$ is continuous, and as the spans of faces meet only at extreme points, $I$ only fails to be injective on the faces containing the extreme point in $S$ (if any). But $\mathcal{F}_{1}$ is a 2-dimensional closed surface in which the faces that contain an extreme point form a closed curve, and there are no continuous maps from a closed surface to the plane that fail to be injective only along a curve. Case 2. $S$ doesn’t contain extreme points of some face $F$. Choose $F$ that minimizes the dimension of the subspace $S\cap span(F)$. Then for every $F^{\prime}$ in a neighborhood of $F$, $S\cap span(F)$ is a subspace of minimal dimension and with no extreme points. If $S^{\prime}$ is the orthogonal complement of $S\cap span(F)$ in $S$ then $S^{\prime}$ intersects $span(F)$ in one point for all $F^{\prime}$ in a smaller neighborhood $V$ of $F.$ Then the function $I:V\rightarrow S^{\prime}$ that maps $F^{\prime}$ to $S^{\prime}\cap span(F)$ is continuous, and it is injective as the spans of faces only meet at extreme points. But an injective map between manifolds can only exist when the domain has dimension no larger than the target so $2d\leq\dim S^{\prime}\leq\dim S\leq d+1$, so $d=1$ and we are in case 1. Case 3. $S$ contains extreme points of each face $F.$ As $C$ is convex, either $S\cap C\subset\partial C$ or $\partial_{S}(S\cap C)=S\cap\partial C$. In the first case $S\cap C$ is contained in a face $F_{1}$ of $C$ and so either $S\cap C=F_{1}$ (so $F_{1}\subset S$) or there is an extreme point $p$ of $F_{1}$ not contained in $S$, but then a face $F_{2}$ that meets $F_{1}$ at $p$ doesn’t meet $F_{1}\cap S\supset S\cap C$ so $S$ doesn’t contain extreme points of $F_{2}$. Let $p$ be an extreme point not in $S$, and consider the set $\mathcal{F}_{1}^{p}$ of faces of rank 1 containing $p$. If $F$ and $F^{\prime}$ are distinct faces in $\mathcal{F}_{1}^{p}$, $S\cap F$ and $S\cap F^{\prime}$ are disjoint. Choose $F$ so that $S\cap F$ has minimal dimension, then for all $F^{\prime}$ in some neighborhood $V$ of $F$, $S\cap F^{\prime}$ has the same dimension and the map $I_{B}:\mathcal{F}_{1}^{p}\cap V$ $\rightarrow S\cap\partial C$ that sends $F^{\prime}$ to the baricenter of $S\cap F^{\prime}$ is continuous and injective. As $I_{B}$ is a map between manifolds, $d=\dim\mathcal{F}_{1}^{p}\leq\dim S\cap\partial C\leq d$ and so by domain invariance the image of $I_{B}$ is an open subset of $S\cap\partial C=\partial_{S}(S\cap C)$. This implies that for each $F^{\prime}\in\mathcal{F}_{1}^{p}\cap V$, $S\cap F^{\prime}$ consists of one point (if a face of a convex set has more than 1 point, its baricenter is arbitrarily close to points in the boundary that are not baricenters of other faces, namely, the points in the face) and so, by hypothesis, $S\cap F^{\prime}$ is an extreme point of $C$. So part of the boundary of $S\cap C$ in $S$ is strictly convex, therefore the line segment joining two extreme points in it lies in $Int_{S}(S\cap C)$, but that line segment lies in the face of $C$ containing the 2 extreme points, so it must lie in $S\cap\partial C=\partial_{S}(S\cap C)$, a contradiction. ∎ ###### Lemma 9. The boundaries of the faces of rank 1 of $C$ are semi-algebraic sets. If $d=1$, they are conic sections. ###### Proof. By lemma 8, the set $\mathcal{S}$ of spans of faces of $C$ is the same as the set of $d+1$-dimensional subspaces of $\mathbb{R}^{n}$ that intersect every element of $\mathcal{S}$. The set of all $d+1$-dimensional affine subspaces of $\mathbb{R}^{n}$ forms a real algebraic variety and the condition that the subspaces meet a fixed subspace is algebraic, so (by the finite descending chain condition) there is a finite family of spans $S_{1},S_{2},...,S_{m}\in\mathcal{S}$ such that $S\in\mathcal{S}$ if it intersects these $S_{i}$’s (see [1]). Now for $(x_{1},x_{2},...,x_{m})\in S_{1}\times S_{2}...\times S_{m}$, the subspace $span(x_{1},...,x_{m})$ has dimension at least $d+1$ (otherwise it would be contained in two subspaces of dimension $d+1$ that meet each $S_{i}$, so they would both be in $\mathcal{S}$, but two spans can only meet in 1 point). So $span(x_{1},...,x_{m})$ lies in $\mathcal{S}$ if and only if its dimension is $d+1$, and this happens if and only if some determinants (given by polynomials on $x_{1},...,x_{m}$ ) vanish. Therefore the set $X=\left\\{(x_{1},x_{2},...,x_{m})\in S_{1}\times S_{2}...\times S_{m}\text{ }\|\text{ }span(x_{1},...,x_{m})\in\mathcal{S}\right\\}$ is real algebraic, as is the set $X^{p}$ formed by the elements of $X$ that contain a fixed point $p$. If $F_{1}$ is the face in $S_{1}$ and $p$ is an extreme point of $C$ outside $F_{1}$ then $\partial F_{1}$ consists of the intersections of $S_{1}$ with the elements of $\mathcal{S}$ containing $p$. So $\partial F_{1}$ is the one to one projection of the algebraic set $X^{p}$ to $S_{1}$, so $\partial F_{1}$ is at least semi-algebraic. Figure 1 Now assume $d=1$ so $n=5$. Every projective plane has 7 points and 6 lines so that each line contains 3 points as in figure 1, so $C$ has 7 extreme points and 6 faces intersecting in that way. The 7 points are in general position in $\mathbb{R}^{5}$ because as each face of $C$ is spanned by 3 points, the span of any 6 of those points contains the span of 3 faces, which is all of $\mathbb{R}^{5}$. Therefore we may assume (by applying a projective transformation) that the 7 points are $p_{0}=(0,0,0,0,0),p_{1}=(1,0,0,0,0),...,p_{5}=(0,0,0,0,1),p_{6}=(1,1,1,1,1)$. Let $S_{i}$ be the plane spanned by the face $F_{i}$. A plane $S$ that intersects $S_{1},S_{2}$ and $S_{3}$ has a parametrization $(x,y,z,v,w)=r(a,b,0,0,0)+s(0,0,c,d,0)+t(e,e,e,e,f)$ with $r+s+t=1$. $S$ intersects $S_{4},S_{5}$ and $S_{6}$ only if three systems of linear equations in $r,s,t$ represented by the following matrices have nontrivial solutions: $\left\|\begin{array}[]{ccc}a&0&e\\\ 0&d&e\\\ b-1&c-1&2e+f-1\end{array}\right\|\left\|\begin{array}[]{ccc}b&0&e\\\ 0&c&e\\\ a-1&d-1&2e+f-1\end{array}\right\|\left\|\begin{array}[]{ccc}b&0&e-f\\\ 0&d&e-f\\\ a-1&c-1&2e-f-1\end{array}\right\|$ As the determinants of these matrices are linear functions on the variables $e$ and $f$, they vanish simultaneously if and only if the matrix of this new system has determinant 0: $\det\left\|\begin{array}[]{ccc}-ac+2ad-bd+a+d&ad&-ad\\\ -ac+2bc- bd+b+c&bc&-bc\\\ -ad-bc+2bd+b+d&ad+bc-bd-b-d&-bd\end{array}\right\|=0$ This determinant factors as the product of a linear and a quadratic function of $a$ and $b$ (with coefficients in $c$ and $d$). Since the boundary of the face $F_{1}$ is formed by the intersections of $S_{1}$ with the planes that meet all $S_{i}$’s and go through a fixed point in the boundary of $F_{2}$ (this corresponds to fixing $c$ and $d$), the boundary of $F_{1}$ is contained in the union of a line and a conic. As the boundary of $F_{1}$ is strictly convex, it must be the conic. ∎ ###### Theorem 3. All convex bodies in $\mathbb{R}^{5}$ with modular and irreducible face lattice are projectively equivalent. ###### Proof. Let $C$ and $C^{\prime}$ be two such bodies. Take extreme points $p_{0},p_{1},...,p_{6}$ and faces $F_{1},...F_{6}$ of $C$ as in figure 1. Pick an extreme point $p_{0}^{\prime}$ in $C^{\prime}$ and two faces $F_{1}^{\prime}$ and $F_{2}^{\prime}$ of $C^{\prime}$ intersecting at $p_{0}^{\prime}$. Let $S_{i}$ be the span of $F_{i}$. As the faces of $C$ and $C^{\prime}$ are conics, there are linear transformations from $S_{1}$ to $S_{1}^{\prime}$ taking $F_{1}$ to $F_{1}^{\prime}$ and from $P_{2}$ to $P_{2}^{\prime}$ taking $F_{2}$ to $F_{2}^{\prime}$. Together, they define a linear transformation $l$ from $span(F_{1}\cup F_{2})$ to $span(F_{1}^{\prime}\cup F_{2}^{\prime})$. Let $p_{i}^{\prime}=l(p_{i})$ for $i=1,...,4$. The faces $F_{4},F_{5},F_{6}$ are generated by unique pairs of $p_{i}$’s with $i\leq 4$. Let $F_{4}^{\prime},F_{5}^{\prime},F_{6}^{\prime}$ be the faces generated by the corresponding pairs of $p_{i}^{\prime}$s. Finally, let $p_{5}^{\prime}=S_{4}^{\prime}\cap S_{5}^{\prime}$, let $F_{3}^{\prime}$ be the face generated by $p_{0}^{\prime}$ and $p_{5}^{\prime}$ and let $p_{6}^{\prime}=S_{3}^{\prime}\cap S_{6}^{\prime}$. The linear transformation $l$ can be extended to a projective transformation $\rho$ in $\mathbb{RP}^{5}$ that takes $p_{5}$ to $p_{5}^{\prime}$ and $p_{6}$ to $p_{6}^{\prime}$. As $\rho$ sends each $p_{i}$ to $p_{i}^{\prime}$, it sends each $S_{i}$ to $S_{i}^{\prime}$, so it sends each plane in $\mathbb{R}^{5}$ intersecting every $S_{i}$ to a plane intersecting every $S_{i}^{\prime}$. Since by construction $\rho$ takes those planes that meet $\partial F_{1}$ and $\partial F_{2}$ to planes that meet $\partial F_{1}^{\prime}$ and $\partial F_{2}^{\prime}$, lemma 8 implies that $\rho$ maps spans of faces of $C$ to spans of faces of $C^{\prime}$ and therefore it maps faces to faces. ∎ Question 1: Are all the convex bodies whose face lattices determine classical projective spaces projectively equivalent to sections of cones of hermitian matrices? Question 2: Can two convex bodies of the same dimension define non isomorphic projective planes (so they are not related by a face-preserving homeomorphism)? In dimensions $8$ and $14$ this is equivalent to ask if the projective planes are always desarguesian. In dimension $26$ there might be enough space for non-equivalent non-desarguesian examples. ## References * [1] Bochnak,J., Coste,M., Roy,M., Real Algebraic geometry. Springer, 1998. * [2] Baez,J., The Octonions, Bull. Amer. Math. Soc. 39 (2001), 145-205. * [3] Barker,G.P., Theory of cones. Linear Algebra Appl 39 (1981), 263–291. * [4] Birkhoff,G., Lattice Theory, 3rd edition, Amer. Math. Soc. Colloq. Publ., AMS 1967. * [5] Grundhofer,T., Lowen,R., Linear topological geometries, in Handbook of Incidence Geometry. Buildings and Foundations. North-Holland. 1995. * [6] Crawley,P., Dilworth R.P., Algebraic Theory of Lattices, Prentice-Hall, Inc., 1973. * [7] Gruber,P.M., Willis,J.M., Handbook of Convex Geometry. Volume A. North-Holland. 1993. * [8] Hartshorne,R., Foundations of Projective Geometry. W.A. Benjamin, Inc. 1967\. * [9] Labardini-Fragoso,D., The face lattice of a finite dimensional cone, Bachelor Thesis UNAM, 2004. * [10] Salzmann,H., Betten,D., Grundhoefer,T., Haehl,H., Loewn,R., Stroppel, M., Compact Projective Planes. De Gruyter Expositions in Mathematics. 1995.	arxiv-papers	2009-03-04T20:48:08	2024-09-04T02:49:00.971723	{ "license": "Public Domain", "authors": "D. Labardini-Fragoso, M. Neumann-Coto and M. Takane", "submitter": "Max Neumann-Coto", "url": "https://arxiv.org/abs/0903.0643" }
0903.0651	††thanks: Supported in part by a grant from Prince of Songkla University††thanks: Supported in part by NSF Grant DMS-0555862 # Toeplitz operators on generalized Bergman spaces Kamthorn Chailuek Department of Mathematics Prince of Songkla University Hatyai, Songkhla, Thailand 90112 kamthorn.c@psu.ac.th Brian C. Hall Department of Mathematics University of Notre Dame 255 Hurley Building Notre Dame IN 46556-4618 USA bhall@nd.edu (Date: July 10, 2009) ###### Abstract. We consider the weighted Bergman spaces $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}),$ where we set $d\mu_{\lambda}(z)=c_{\lambda}(1-\left\|z\right\|^{2})^{\lambda}~{}d\tau(z),$ with $\tau$ being the hyperbolic volume measure. These spaces are nonzero if and only if $\lambda>d.$ For $0<\lambda\leq d,$ spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces. ###### Key words and phrases: Bergman space; Toeplitz operator; quantization; holomorphic Sobolev space; Berezin transform ###### 1991 Mathematics Subject Classification: Primary 47B35; Secondary 32A36, 81S10 ## 1\. Introduction ### 1.1. Generalized Bergman spaces Let $\mathbb{B}^{d}$ denote the (open) unit ball in $\mathbb{C}^{d}$ and let $\tau$ denote the hyperbolic volume measure on $\mathbb{B}^{d},$ given by $d\tau(z)=(1-\|z\|^{2})^{-(d+1)}~{}dz,$ (1.1) where $dz$ denotes the $2d$-dimensional Lebesgue measure. The measure $\tau$ is natural because it is invariant under all of the automorphisms (biholomorphic mappings) of $\mathbb{B}^{d}.$ For $\lambda>0,$ let $\mu_{\lambda}$ denote the measure $d\mu_{\lambda}(z)=c_{\lambda}(1-\|z\|^{2})^{\lambda}~{}d\tau(z),$ where $c_{\lambda}$ is a positive constant whose value will be specified shortly. Finally, let $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ denote the (weighted) Bergman space, consisting of those holomorphic functions on $\mathbb{B}^{d}$ that are square-integrable with respect to $\mu_{\lambda}.$ (Often these are defined using the Lebesgue measure as the reference measure, but all the formulas look nicer if we use the hyperbolic volume measure instead.) These spaces carry a projective unitary representation of the group $SU(d,1).$ If $\lambda>d,$ then the measure $\mu_{\lambda}$ is finite, so that all bounded holomorphic functions are square-integrable. For $\lambda>d,$ we choose $c_{\lambda}$ so that $\mu_{\lambda}$ is a probability measure. Calculation shows that $c_{\lambda}=\frac{\Gamma(\lambda)}{\pi^{d}\Gamma(\lambda-d)},\quad\lambda>d.$ (1.2) (This differs from the value in Zhu’s book [Z2] by a factor of $\pi^{d}/d!,$ because Zhu uses normalized Lebesgue whereas we use un-normalized Lebesgue measure in (1.1).) On the other hand, if $\lambda\leq d,$ then $\mu_{\lambda}$ is an infinite measure. In this case, it is not hard to show that there are no nonzero holomorphic functions that are square-integrable with respect to $\mu_{\lambda}$ (no matter which nonzero value for $c_{\lambda}$ we choose). Although the holomorphic $L^{2}$ space with respect to $\mu_{\lambda}$ is trivial (zero dimensional) when $\lambda\leq d,$ there are indications that life does not end at $\lambda=d.$ First, the reproducing kernel for $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ is given by $K_{\lambda}(z,w)=\frac{1}{(1-z\cdot\bar{w})^{\lambda}}$ for $\lambda>d$. The reproducing kernel is defined by the property that it is anti-holomorphic in $w$ and satisfies $\int_{\mathbb{B}^{d}}K_{\lambda}(z,w)f(w)~{}d\mu_{\lambda}(w)=f(z)$ for all $f\in\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}).$ Nothing unusual happens to $K_{\lambda}$ as $\lambda$ approaches $d.$ In fact, $K_{\lambda}(z,w):=(1-z\cdot\bar{w})^{-\lambda}$ is a “positive definite reproducing kernel” for all $\lambda>0.$ Thus, it is possible to define a reproducing kernel Hilbert space for all $\lambda>0$ that agrees with $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ for $\lambda>d.$ Second, in representation theory, one is sometimes led to consider spaces like $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ but with $\lambda<d.$ Consider, for example, the much-studied metaplectic representation of the connected double cover of $SU(1,1)\cong Sp(1,\mathbb{R}).$ This representation is a direct sum of two irreducible representations, one of which can be realized in the Bergman space $\mathcal{H}L^{2}(\mathbb{B}^{1},\mu_{3/2})$ and the other of which can be realized in (a suitably defined version of) the Bergman space $\mathcal{H}L^{2}(\mathbb{B}^{1},\mu_{1/2}).$ To be precise, we can say that the second summand of the metaplectic representation is realized in a Hilbert space of holomorphic functions having $K_{\lambda},$ $\lambda=1/2,$ as its reproducing kernel. See [14, Sect. 4.6]. Last, one often wants to consider the infinite-dimensional ($d\rightarrow\infty$) limit of the spaces $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}).$ (See, for example, [25] and [23].) To do this, one wishes to embed each space $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ isometrically into a space of functions on $\mathbb{B}^{d+1},$ as functions that are independent of $z_{n+1}.$ It turns out that if one uses (as we do) hyperbolic volume measure as the reference measure, then the desired isometric embedding is achieved by embedding $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ into $\mathcal{H}L^{2}(\mathbb{B}^{d+1},\mu_{\lambda}).$ That is, if we use the same value of $\lambda$ on $\mathbb{B}^{d+1}$ as on $\mathbb{B}^{d},$ then the norm of a function $f(z_{1},\ldots,z_{d})$ is the same whether we view it as a function on $\mathbb{B}^{d}$ or as a function on $\mathbb{B}^{d+1}$ that is independent of $z_{d+1}.$ (See, for example, Theorem 4, where the inner product of $z^{m}$ with $z^{n}$ is independent of $d$.) If, however, we keep $\lambda$ constant as $d$ tends to infinity, then we will eventually violate the condition $\lambda>d.$ Although it is possible to describe the Bergman spaces for $\lambda\leq d$ as reproducing kernel Hilbert spaces, this is not the most convenient description for calculation. Instead, drawing on several inter-related results in the literature, we describe these spaces as “holomorphic Sobolev spaces,” also called Besov spaces. The inner product on these spaces, which we denote as $H(\mathbb{B}^{d},\lambda),$ is an $L^{2}$ inner product involving both the functions and derivatives of the functions. For $\lambda>d,$ $H(\mathbb{B}^{d},\lambda)$ is identical to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ (the same space of functions with the same inner product), but $H(\mathbb{B}^{d},\lambda)$ is defined for all $\lambda>0.$ It is worth mentioning that in the borderline case $\lambda=d,$ the space $H(\mathbb{B}^{d},\lambda)$ can be identified with the Hardy space of holomorphic functions that are square-integrable over the boundary. To see this, note that the normalization constant $c_{\lambda}$ tends to zero as $\lambda$ approaches $d$ from above. Thus, the measure of any compact subset of $\mathbb{B}^{d}$ tends to zero as $\lambda\rightarrow d^{+},$ meaning that most of the mass of $\mu_{\lambda}$ is concentrated near the boundary. As $\lambda\rightarrow d^{+},$ $\mu_{\lambda}$ converges, in the weak-$\ast$ topology on $\overline{\mathbb{B}^{d}},$ to the unique rotationally invariant probability measure on the boundary. Alternatively, we may observe that the formula for the inner product of monomials in $H(\mathbb{B}^{d},d)$ (Theorem 4 with $\lambda=d$) is the same as in the Hardy space. ### 1.2. Toeplitz operators One important aspect of Bergman spaces is the theory of Toeplitz operators on them. If $\phi$ is a bounded measurable function, the we can define the Toeplitz operator $T_{\phi}$ on $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ by $T_{\phi}f=P_{\lambda}(\phi f),$ where $P_{\lambda}$ is the orthogonal projection from $L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ onto the holomorphic subspace. That is, $T_{\phi}$ consists of multiplying a holomorphic function by $\phi,$ followed by projection back into the holomorphic subspace. Of course, $T_{\phi}$ depends on $\lambda,$ but we suppress this dependence in the notation. The function $\phi$ is called the (Toeplitz) symbol of the operator $T_{\phi}.$ The map sending $\phi$ to $T_{\phi}$ is known as the Berezin–Toeplitz quantization map and it (and various generalizations) have been much studied. See, for example, the early work of Berezin [5, 6], which was put into a general framework in [26, 27], along with [22, 8, 7, 10], to mention just a few works. The Berezin–Toeplitz quantization may be thought of as a generalization of the anti-Wick-ordered quantization on $\mathbb{C}^{d}$ (see [15]). When $\lambda<d,$ the inner product on $H(\mathbb{B}^{d},\lambda)$ is not an $L^{2}$ inner product, and so the “multiply and project” definition of $T_{\phi}$ no longer makes sense. Our strategy is to find alternative formulas for computing $T_{\phi}$ in the case $\lambda>d,$ with the hope that these formulas will continue to make sense (for certain classes of symbols $\phi$) for $\lambda\leq d.$ Specifically, we will identify classes of symbols $\phi$ for which $T_{\phi}$ can be defined as: * • A bounded operator on $H(\mathbb{B}^{d},\lambda)$ (Section 4) * • A Hilbert–Schmidt operator on $H(\mathbb{B}^{d},\lambda)$ (Section 5). We also consider in Section 3 Toeplitz operators whose symbols are polynomials in $z$ and $\bar{z}$ and observe some unusual properties of such operators in the case $\lambda<d.$ ### 1.3. Acknowledgments The authors thank M. Engliš for pointing out to them several useful references and B. Driver for useful suggestions regarding the results in Section 4. This article is an expansion of the Ph.D. thesis of the first author, written under the supervision of the second author. We also thank the referee for helpful comments and corrections. ## 2\. $H(\mathbb{B}^{d},\lambda)$ as a holomorphic Sobolev space In this section, we construct a Hilbert space of holomorphic functions on $\mathbb{B}^{d}$ with reproducing kernel $(1-z\cdot\bar{w})^{-\lambda},$ for an arbitrary $\lambda>0.$ We denote this space as $H(\mathbb{B}^{d},\lambda).$ The inner product on this space is an $L^{2}$ inner product with respect to the measure $\mu_{\lambda+2n},$ where $n$ is chosen so that $\lambda+2n>d.$ The inner product, however, involves not only the holomorphic functions but also their derivatives. That is, $H(\mathbb{B}^{d},\lambda)$ is a sort of holomorphic Sobolev space (or Besov space) with respect to the measure $\mu_{\lambda+2n}.$ When $\lambda>d,$ our space is identical to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$—not just the same space of functions, but also the same inner product. When $\lambda\leq d,$ the Hilbert space $H(\mathbb{B}^{d},\lambda),$ with the associated projective unitary action of $SU(d,1),$ is sometimes referred to as the analytic continuation (with respect to $\lambda$) of the holomorphic discrete series. Results in the same spirit as—and in some cases almost identical to—the results of this section have appeared in several earlier works, some of which treat arbitrary bounded symmetric domains and not just the ball in $\mathbb{C}^{d}.$ For example, in the case of the unit ball in $\mathbb{C}^{d},$ Theorem 3.13 of [30] would presumably reduce to almost the same expression as in our Theorem 4, except that Yan has all the derivatives on one side, in which case the inner product has to be interpreted as a limit of integrals over a ball of radius $1-\varepsilon.$ (Compare the formula for $\mathcal{D}_{\lambda}^{k}$ on p. 13 of [30] to the formula for $A$ and $B$ in Theorem 4.) See also [2, 4, 21, 31, 32]. Note, however, a number of these references give a construction that yields, for $\lambda>d,$ the same space of functions as $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ with a different but equivalent norm. Such an approach is not sufficient for our needs; we require the same inner product as well as the same space of functions. Although our results in this section are not really new, we include proofs to make the paper self-contained and to get the precise form of the results that we want. The integration-by-parts argument we use also serves to prepare for our definition of Toeplitz operators on $H(\mathbb{B}^{d},\lambda)$ in Section 4. We ourselves were introduced to this sort of reasoning by the treatment in Folland’s book [14] of the disk model for the metaplectic representation. The paper [16] obtains results in the same spirit as those of this section, but in the context of a complex semisimple Lie group. We begin by showing that for $\lambda>d,$ the space $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ can be expressed as a subspace of $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})$, with a Sobolev-type norm, for any positive integer $n.$ Let $N$ denote the “number operator,” defined by $N=\sum_{j=1}^{d}z_{j}\frac{\partial}{\partial z_{j}}.$ This operator satisfies $Nz^{m}=\|m\|z^{m}$ for all multi-indices $m.$ If $f$ is holomorphic, then $Nf$ coincides with the “radial derivative” $\left.df(rz)/dr\right\|_{r=1}.$ We use also the operator $\bar{N}=\sum_{j=1}^{d}\bar{z}_{j}\partial/\partial\bar{z}_{j}.$ A simple computation shows that $(1-\|z\|^{2})^{\alpha}=\left(I-\frac{N}{\alpha+1}\right)(1-\|z\|^{2})^{\alpha+1}=\left(I-\frac{\bar{N}}{\alpha+1}\right)(1-\|z\|^{2})^{\alpha+1}.$ (2.1) We will use (2.1) and the following integration by parts result, which will also be used in Section 4. ###### Lemma 1. If $\lambda>d$ and $\psi$ is a continuously differentiable function for which $\psi$ and $N\psi$ are bounded, then $\displaystyle c_{\lambda}\int_{\mathbb{B}^{d}}\psi(z)(1-\|z\|^{2})^{\lambda-d-1}dz$ $\displaystyle=c_{\lambda+1}\int_{\mathbb{B}^{d}}\left[\left(I+\frac{N}{\lambda}\right)\psi\right](z)(1-\|z\|^{2})^{\lambda-d}~{}dz$ $\displaystyle=c_{\lambda+1}\int_{\mathbb{B}^{d}}\left[\left(I+\frac{\bar{N}}{\lambda}\right)\psi\right](z)(1-\|z\|^{2})^{\lambda-d}~{}dz.$ Here $dz$ is the $2d$-dimensional Lebesgue measure on $\mathbb{B}^{d}.$ ###### Proof. We start by applying (2.1) and then think of the integral over $\mathbb{B}^{d}$ as the limit as $r$ approaches 1 of the integral over a ball of radius $r<1.$ On the ball of radius $r,$ we write out $\partial/\partial z_{j}$ in terms of $\partial/\partial x_{j}$ and $\partial/\partial y_{j}.$ For, say, the $\partial/\partial x_{j}$ term we express the integral as a one- dimensional integral with respect to $x_{j}$ (with limits of integration depending on the other variables) followed by an integral with respect to the other variables. We then use ordinary integration by parts in the $x_{j}$ integral, and similarly for the $\partial/\partial y_{j}$ term. The integration by parts will yield a boundary term involving $z_{j}\psi(z)(1-\|z\|^{2})^{\lambda-d}$; this boundary term will vanish as $r$ tends to 1, because we assume $\lambda>d.$ In the nonboundary term, the operator $N$ applied to $(1-\|z\|^{2})^{\lambda-d}$ will turn into the operator $-\sum_{j=1}^{d}\partial/\partial z_{j}\circ z_{j}=-(dI+N)$ applied to $\psi.$ Computing from (1.2) that $c_{\lambda}/c_{\lambda+1}=(\lambda-d)/\lambda$, we may simplify and let $r$ tend to 1 to obtain the desired result involving $N.$ The same reasoning gives the result involving $\bar{N}$ as well. ∎ We now state the key result, obtained from (2.1) and Lemma 1, relating the inner product in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ to the inner product in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+1})$ (compare [14, p. 215] in the case $d=1$). ###### Proposition 2. Suppose that $\lambda>d$ and $f$ and $g$ are holomorphic functions on $\mathbb{B}^{d}$ for which $f,$ $g,$ $Nf,$ and $Ng$ are all bounded. Then $\left\langle f,g\right\rangle_{L^{2}(\mathbb{B}^{d},\mu_{\lambda})}=\left\langle f,\left(I+\frac{N}{\lambda}\right)g\right\rangle_{L^{2}(\mathbb{B}^{d},\mu_{\lambda+1})}=\left\langle\left(I+\frac{N}{\lambda}\right)f,g\right\rangle_{L^{2}(\mathbb{B}^{d},\mu_{\lambda+1})}.$ (2.2) ###### Proof. Recalling the formula (1.1) for the measure $\tau,$ we apply Lemma 1 with $\psi(z)=\overline{f(z)}g(z)$ with $f$ and $g$ holomorphic. Observing that $N(\bar{f}g)=\bar{f}Ng$ gives the first equality and observing that $\bar{N}(\bar{f}g)=\overline{(Nf)}g$ gives the second equality. ∎ Now, a general function in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ is not bounded. Indeed, the pointwise bounds on elements of $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}),$ coming from the reproducing kernel, are not sufficient to give a direct proof of the vanishing of the boundary terms in the integration by parts in Proposition 2. Nevertheless, (2.2) does hold for all $f$ and $g$ in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}),$ provided that one interprets the inner product as the limit as $r$ approaches 1 of integration over a ball of radius $r.$ (See [14, p. 215] or [30, Thm. 3.13].) We are going to iterate (2.2) to obtain an expression for the inner product on $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ involving equal numbers of derivatives on $f$ and $g.$ This leads to the following result. ###### Theorem 3. Fix $\lambda>d$ and a non-negative integer $n.$ Then a holomorphic function $f$ on $\mathbb{B}^{d}$ belongs to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ if and only if $N^{l}f$ belongs to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})$ for $0\leq l\leq n.$ Furthermore, $\left\langle f,g\right\rangle_{\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})}=\left\langle Af,Bg\right\rangle_{\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})}$ (2.3) for all $f,g\in\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}),$ where $\displaystyle A$ $\displaystyle=\left(I+\frac{N}{\lambda+n}\right)\left(I+\frac{N}{\lambda+n+1}\right)\cdots\left(I+\frac{N}{\lambda+2n-1}\right)$ $\displaystyle B$ $\displaystyle=\left(I+\frac{N}{\lambda}\right)\left(I+\frac{N}{\lambda+1}\right)\cdots\left(I+\frac{N}{\lambda+n-1}\right).$ Let us make a few remarks about this result before turning to the proof. Let $\sigma=\lambda+2n.$ It is not hard to see that $N^{k}f$ belongs to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\sigma})$ for $0\leq k\leq n$ if and only if all the partial derivatives of $f$ up to order $n$ belong to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\mu}),$ so we may describe this condition as “$f$ has $n$ derivatives in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\sigma}).$” This condition then implies that $f$ belongs to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\sigma-2n}),$ which in turn means that $f(z)/(1-\left\|z\right\|^{2})^{n}$ belongs to $L^{2}(\mathbb{B}^{d},\mu_{\sigma}).$ Since $1/(1-\left\|z\right\|^{2})^{n}$ blows up at the boundary of $\mathbb{B}^{d},$ saying that $f(z)/(1-\left\|z\right\|^{2})^{n}$ belongs to $L^{2}(\mathbb{B}^{d},\mu_{\sigma})$ says that $f(z)$ has better behavior at the boundary than a typical element of $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\sigma}).$ We may summarize this discussion by saying that each derivative that $f\in\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\sigma})$ has in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\sigma})$ results, roughly speaking, in an improvement by a factor of $(1-\left\|z\right\|^{2})$ in the behavior of $f$ near the boundary. This improvement is also reflected in the pointwise bounds on $f$ coming from the reproducing kernel. If $f$ has $n$ derivatives in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\sigma}),$ then $f$ belongs to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\sigma-2n}),$ which means that $f$ satisfies the pointwise bounds $\displaystyle\left\|f(z)\right\|$ $\displaystyle\leq\left\\|f\right\\|_{L^{2}(\mathbb{B}^{d},\mu_{\sigma-2n})}\left(K_{\sigma-2n}(z,z)\right)^{1/2}$ $\displaystyle=\left\\|f\right\\|_{L^{2}(\mathbb{B}^{d},\mu_{\sigma-2n})}\left(\frac{1}{1-\left\|z\right\|^{2}}\right)^{\frac{\sigma}{2}-n}.$ (2.4) These bounds are better by a factor of $(1-\left\|z\right\|^{2})^{n}$ than the bounds on a typical element of $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\sigma}).$ See also [16] for another setting in which the existence of derivatives in a holomorphic $L^{2}$ space can be related in a precise way to improved pointwise behavior of the functions. The results of the two previous paragraphs were derived under the assumption that $\lambda=\sigma-2n>d.$ However, Theorem 4 will show that (2.4) still holds under the assumption $\lambda=\sigma-2n>0.$ ###### Proof. If $f$ and $g$ are polynomials, then (2.3) follows from iteration of Proposition 2. Note that $N$ is a non-negative operator on polynomials, because the monomials form an orthogonal basis of eigenvectors with non- negative eigenvalues. It is well known and easily verified that for any $f$ in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}),$ the partial sums of the Taylor series of $f$ converge to $f$ in norm. We can therefore choose polynomials $f_{j}$ converging in norm to $f$. If we apply (2.3) with $f=g=(f_{j}-f_{k})$ and expand out the expressions for $A$ and $B,$ then the positivity of $N$ will force each of the terms on the right-hand side to tend to zero. In particular, $N^{l}f_{j}$ is a Cauchy sequences in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n}),$ for all $0\leq l\leq n.$ It is easily seen that the limit of this sequence is $N^{l}f$; for holomorphic functions, $L^{2}$ convergence implies locally uniform convergence of the derivatives to the corresponding derivatives of the limit function. This shows that $N^{l}f$ is in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n}).$ For any $f,g\in\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}),$ choose sequences $f_{j}$ and $g_{k}$ of polynomials converging to $f,g.$ Since $N^{l}f_{j}$ and $N^{l}g_{j}$ converge to $N^{l}f$ and $N^{l}g,$ respectively, plugging $f_{j}$ and $g_{j}$ into (2.3) and taking a limit gives (2.3) in general. In the other direction, suppose that $N^{l}f$ belongs to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})$ for all $0\leq l\leq n.$ Let $f_{j}$ denote the $j$th partial sum of the Taylor series of $f$. Then since $Nz^{m}=\|m\|z^{m}$ for all multi-indices $m,$ the functions $N^{l}f_{j}$ form the partial sums of a Taylor series converging to $N^{l}f_{j},$ and so these must be the partial sums of the Taylor series of $N^{l}f.$ Thus, for each $l,$ we have that $N^{l}f_{j}$ converges to $N^{l}f$ in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n}).$ If we then apply (2.3) with $f=g=f_{j}-f_{k},$ convergence of each $N^{l}f_{j}$ implies that all the terms on the right-hand side tend to zero. We conclude that $f_{j}$ is a Cauchy sequence in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}),$ which converges to some $\hat{f}.$ But $L^{2}$ convergence of holomorphic functions implies pointwise convergence, so the limit in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ (i.e., $\hat{f}$) coincides with the limit in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})$ (i.e., $f$). This shows that $f$ is in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}).$ ∎ Now, when $\lambda\leq d,$ Proposition 2.2 no longer holds. This is because the boundary terms, which involve $(1-\|z\|^{2})^{\lambda-d},$ no longer vanish. This failure of equality is actually a good thing, because if we take $f=g$, then $c_{\lambda}\int_{\mathbb{B}^{d}}\left\|f\left(z\right)\right\|^{2}(1-\|z\|^{2})^{\lambda}~{}d\tau(z)=+\infty$ for all nonzero holomorphic functions, no matter what positive value we assign to $c_{\lambda}.$ (Recall that when $\lambda>d,$ $c_{\lambda}$ is chosen to make $\mu_{\lambda}$ a probability measure, but this prescription does not make sense for $\lambda\leq d.$) Although the left-hand side of (2.2) is infinite when $f=g$ and $\lambda\leq d,$ the right-hand side is finite if $\lambda+1>d$ and, say, $f$ is a polynomial. More generally, for any $\lambda\leq d,$ we can choose $n$ big enough that $\lambda+2n>d.$ We then take the right-hand side of (2.3) as a definition. ###### Theorem 4. For all $\lambda>0,$ choose a non-negative integer $n$ so that $\lambda+2n>d$ and define $H(\mathbb{B}^{d},\lambda)=\left\\{f\in\mathcal{H}(\mathbb{B}^{d})\left\|N^{k}f\in\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n}),~{}0\leq k\leq n\right.\right\\}.$ Then the formula $\left\langle f,g\right\rangle_{\lambda}=\left\langle Af,Bg\right\rangle_{\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})}$ where $\displaystyle A$ $\displaystyle=\left(I+\frac{N}{\lambda+n}\right)\left(I+\frac{N}{\lambda+n+1}\right)\cdots\left(I+\frac{N}{\lambda+2n-1}\right)$ $\displaystyle B$ $\displaystyle=\left(I+\frac{N}{\lambda}\right)\left(I+\frac{N}{\lambda+1}\right)\cdots\left(I+\frac{N}{\lambda+n-1}\right)$ defines an inner product on $H(\mathbb{B}^{d},\lambda)$ and $H(\mathbb{B}^{d},\lambda)$ is complete with respect to this inner product. The monomials $z^{m}$ form an orthogonal basis for $H(\mathbb{B}^{d},\lambda)$ and for all multi-indices $l$ and $m$ we have $\left\langle z^{l},z^{m}\right\rangle_{\lambda}=\delta_{l,m}\frac{m!\Gamma(\lambda)}{\Gamma(\lambda+\|m\|)}.$ Furthermore, $H(\mathbb{B}^{d},\lambda)$ has a reproducing kernel given by $K_{\lambda}(z,w)=\frac{1}{(1-z\cdot\bar{w})^{\lambda}}.$ Using power series, it is easily seen that for any holomorphic function $f,$ if $N^{n}f$ belongs to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n}),$ then $N^{k}f$ belongs to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})$ for $0\leq k\,<n.$ Note that the reproducing kernel and the inner product of the monomials are independent of $n.$ Thus, we obtain the same space of functions with the same inner product, no matter which $n$ we use, so long as $\lambda+2n>d.$ From the reproducing kernel we obtain the pointwise bounds given by $\left\|f(z)\right\|^{2}\leq\left\\|f\right\\|_{\lambda}^{2}(1-\left\|z\right\|^{2})^{-\lambda}.$ ###### Proof. Using a power series argument, it is easily seen that if $f$ and $N^{k}f$ belong to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})$, then $\left\langle f,N^{k}f\right\rangle_{L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})}\geq 0.$ From this, we obtain positivity of the inner product $\left\langle\cdot,\cdot\right\rangle_{\lambda}.$ If $f_{j}$ is a Cauchy sequence in $H(\mathbb{B}^{d},\lambda),$ then positivity of the coefficients in the expressions for $A$ and $B$ imply that for $0\leq k\leq n,$ $N^{k}f_{j}$ is a Cauchy sequence in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n}),$ which converges (as in the proof of Theorem 3) to $N^{k}f.$ This shows that $N^{k}f$ is in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})$ for each $0\leq k\leq n,$ and so $f\in H(\mathbb{B}^{d},\lambda).$ Further, convergence of each $N^{k}f_{j}$ to $N^{k}f$ implies that $f_{j}$ converges to $f$ in $H(\mathbb{B}^{d},\lambda).$ To compute the inner product of two monomials in $H(\mathbb{B}^{d},\lambda),$ we apply the definition. Since $Nz^{m}=\|m\|z^{m},$ we obtain $\displaystyle\left\langle z^{l},z^{m}\right\rangle_{\lambda}$ $\displaystyle=\delta_{l,m}\left(\frac{\lambda+\|m\|}{\lambda}\right)\left(\frac{\lambda+1+\|m\|}{\lambda+1}\right)\cdots\left(\frac{\lambda+2n-1+\|m\|}{\lambda+2n-1}\right)\frac{m!\Gamma(\lambda+2n)}{\Gamma(\lambda+2n+\|m\|)}$ $\displaystyle=\delta_{l,m}\frac{m!\Gamma(\lambda)}{\Gamma(\lambda+\|m\|)},$ where we have used the known formula for the inner product of monomials in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})$ (e.g., [Z2]). Completeness of the monomials holds in $H(\mathbb{B}^{d},\lambda)$ for essentially the same reason it holds in the ordinary Bergman spaces. For $f\in\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$, expand $f$ in a Taylor series and then consider $\left\langle z^{m},f\right\rangle_{\lambda}$. Each term in the inner product is an integral over $\mathbb{B}^{d}$ with respect to $\mu_{\lambda+2n}$, and each of these integrals can be computed as the limit as $r$ tends to 1 of integrals over a ball of radius $r<1.$ On the ball of radius $r,$ we may interchange the integral with the sum in the Taylor series. But distinct monomials are orthogonal not just over $\mathbb{B}^{d}$ but also over the ball of radius $r,$ as is easily verified. The upshot of all of this is that $\left\langle z^{m},f\right\rangle_{\lambda}$ is a nonzero multiple of the $m$th Taylor coefficient of $f.$ Thus if $\left\langle z^{m},f\right\rangle_{\lambda}=0$ for all $m,$ $f$ is identically zero. Finally, we address the reproducing kernel. Although one can use essentially the same argument as in the case $\lambda>d,$ using the orthogonal basis of monomials and a binomial expansion (see the proof of Theorem 12), it is more enlightening to relate the reproducing kernel in $H(\mathbb{B}^{d},\lambda)$ to that in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n}).$ We require some elementary properties of the operators $A$ and $B$; since the monomials form an orthogonal basis of eigenvectors for these operators, these properties are easily obtained. We need that $A$ is self-adjoint on its natural domain and that $A$ and $B$ have bounded inverses. Let $\chi_{z}^{\lambda+2n}$ be the unique element of $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})$ for which $\left\langle\chi_{z}^{\lambda+2n},f\right\rangle_{L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})}=f(z)$ for all $f$ in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n}).$ Explicitly, $\chi_{z}^{\lambda+2n}(w)=(1-\bar{z}\cdot w)^{-(\lambda+2n)}.$ (This is Theorem 2.2 of [Z2] with our $\lambda$ corresponding to $n+\alpha+1$ in [Z2].) Now, a simple calculation shows that $(I+N/a)(1-\bar{z}\cdot w)^{-a}=(1-\bar{z}\cdot w)^{-(a+1)},$ (2.5) where $N$ acts on the $w$ variable with $z$ fixed. From this, we see that $N^{k}\chi_{z}^{\lambda+2n}$ is a bounded function for each fixed $z\in\mathbb{B}^{d}$ and $k\in\mathbb{N},$ so that $\chi_{z}^{\lambda+2n}$ is in $H(\mathbb{B}^{d},\lambda).$ For any $f\in H(\mathbb{B}^{d},\lambda)$ we compute that $\displaystyle\left\langle f,(AB)^{-1}\chi_{z}^{\lambda+2n}\right\rangle_{\lambda}$ $\displaystyle=\left\langle Af,B(AB)^{-1}\chi_{z}^{\lambda+2n}\right\rangle_{L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n)}}$ $\displaystyle=\left\langle f,\chi_{z}^{\lambda+2n}\right\rangle_{L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n)}}=f(z).$ This shows that the reproducing kernel for $H(\mathbb{B}^{d},\lambda)$ is given by $K_{\lambda}(z,w)=\overline{[(AB)^{-1}\chi_{z}^{\lambda+2n}](w)}.$ Using (2.5) repeatedly gives the desired result. ∎ We conclude this section with a simple lemma that will be useful in Section 5. ###### Lemma 5. For all $\lambda_{1},\lambda_{2}>0,$ if $f$ is in $H(\mathbb{B}^{d},\lambda_{1})$ and $g$ is in $H(\mathbb{B}^{d},\lambda_{2})$ then $fg$ is in $H(\mathbb{B}^{d},\lambda_{1}+\lambda_{2}).$ ###### Proof. If, say, $\lambda_{1}>d,$ then we have the following simple argument: $\displaystyle\left\\|fg\right\\|_{\lambda_{1}+\lambda_{2}}^{2}$ $\displaystyle=c_{\lambda_{1}+\lambda_{2}}\int_{\mathbb{B}^{d}}\left\|f(z)\right\|^{2}\left\|g(z)\right\|^{2}(1-\left\|z\right\|^{2})^{\lambda_{1}+\lambda_{2}}~{}d\tau(z)$ $\displaystyle\leq c_{\lambda_{1}+\lambda_{2}}\left\\|g\right\\|_{\lambda_{2}}^{2}\int_{\mathbb{B}^{d}}\left\|f(z)\right\|^{2}(1-\left\|z\right\|^{2})^{-\lambda_{2}}(1-\left\|z\right\|^{2})^{\lambda_{1}+\lambda_{2}}~{}d\tau(z)$ $\displaystyle=\frac{c_{\lambda_{1}+\lambda_{2}}}{c_{\lambda_{1}}}\left\\|f\right\\|_{\lambda_{1}}^{2}\left\\|g\right\\|_{\lambda_{2}}^{2}.$ Unfortunately, $c_{\lambda_{1}+\lambda_{2}}/c_{\lambda_{1}}$ tends to infinity as $\lambda_{1}$ approaches $d$ from above, so we cannot expect this simple inequality to hold for $\lambda_{1}<d.$ For any $\lambda_{1},\lambda_{2}>0,$ choose $n$ so that $\lambda_{1}+n>d$ and $\lambda_{2}+n>d.$ Then $fg$ belongs to $H(\mathbb{B}^{d},\lambda_{1}+\lambda_{2})$ provided that $N^{n}(fg)$ belongs to $\mathcal{H}L^{2}(\mathbb{B}^{d},\lambda_{1}+\lambda_{2}+2n).$ But $N^{n}(fg)=\sum_{k=0}^{n}\binom{n}{k}N^{k}f~{}N^{n-k}g.$ (2.6) Using Theorem 4, it is easy to see that if $f$ belongs to $H(\mathbb{B}^{d},\lambda_{1})$ then $N^{k}f$ belongs to $H(\mathbb{B}^{d},\lambda_{1}+2k)$. Thus, $\left\|N^{k}f(z)\right\|^{2}\leq a_{k}(1-\left\|z\right\|^{2})^{-(\lambda_{1}+2k)}.$ Now, for each term in (2.6) with $k\leq n/2$, we then obtain the following norm estimate: $\displaystyle c_{\lambda_{1}+\lambda_{2}+2n}\int_{\mathbb{B}^{d}}\left\|N^{k}f(z)N^{n-k}g(z)\right\|^{2}~{}(1-\left\|z\right\|^{2})^{\lambda_{1}+\lambda_{2}+2n}~{}d\tau(z)$ $\displaystyle\leq c_{\lambda_{1}+\lambda_{2}+2n}a_{k}\int_{\mathbb{B}^{d}}\left\|N^{n-k}g(z)\right\|^{2}(1-\left\|z\right\|^{2})^{\lambda_{2}+2n-2k}~{}d\tau(z).$ (2.7) Since $k\leq n/2,$ we have $\lambda_{2}+2n-2k\geq\lambda_{2}+n>d.$ We are assuming that $g$ is in $H(\mathbb{B}^{d},\lambda_{2}),$ so that $N^{n-k}g$ is in $H(\mathbb{B}^{d},\lambda_{2}+2n-2k),$ which coincides with $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda_{2}+2n-2k}).$ Thus, under our assumptions on $f$ and $g,$ each term in (2.6) with $k\leq n/2$ belongs to $\mathcal{H}L^{2}(\mathbb{B}^{d},\lambda_{1}+\lambda_{2}+2n).$ A similar argument with the roles of $f$ and $g$ reversed takes care of the terms with $k\geq n/2.$ ∎ ## 3\. Toeplitz operators with polynomial symbols In this section, we will consider our first examples of Toeplitz operators on generalized Bergman spaces, those whose symbols are (not necessarily holomorphic) polynomials. Such examples are sufficient to see some interesting new phenomena, that is, properties of ordinary Toeplitz operator that fail when extended to these generalized Bergman spaces. The definition of Toeplitz operators for the case of polynomial symbols is consistent with the definition we use in Section 4 for a larger class of symbols. For $\lambda>d,$ we define the Toeplitz operator $T_{\phi}$ by $T_{\phi}f=P_{\lambda}(\phi f)$ for all $f$ in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ and all bounded measurable functions $\phi.$ Recall that $P_{\lambda}$ is the orthogonal projection from $L^{2}(\mathbb{B}^{d},\tau)$ onto the holomorphic subspace. Because $P_{\lambda}$ is a self-adjoint operator on $L^{2}(\mathbb{B}^{d},\mu_{\lambda}),$ the matrix entries of $T_{\phi}$ may be calculated as $\left\langle f_{1},T_{\phi}f_{2}\right\rangle_{\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})}=\left\langle f_{1},\phi f_{2}\right\rangle_{L^{2}(\mathbb{B}^{d},\mu_{\lambda})},\quad\lambda>d,$ (3.1) for all $f_{1},f_{2}\in\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}).$ From this formula, it is easy to see that $T_{\bar{\phi}}=(T_{\phi})^{\ast}.$ If $\psi$ is a bounded holomorphic function and $\phi$ is any bounded measurable function, then it is easy to see that $T_{\phi\psi}=T_{\phi}M_{\psi}.$ Thus, for any two multi-indices $m$ and $n,$ we have $T_{\bar{z}^{m}z^{n}}=(M_{z^{m}})^{\ast}(M_{z^{n}}).$ (3.2) We will take (3.2) as a definition for $0<\lambda\leq d.$ Our first task, then, is to show that $M_{z^{n}}$ is a bounded operator on $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ for all $\lambda>0.$ ###### Proposition 6. For all $\lambda>0$ and all multi-indices $n,$ the multiplication operator $M_{z^{n}}$ is a bounded operator on $H(\mathbb{B}^{d},\lambda).$ Thus, for any polynomial $\phi,$ the Toeplitz operator $T_{\phi}$ defined in (3.2) is a bounded operator on $H(\mathbb{B}^{d},\lambda).$ ###### Proof. The result is a is a special case of a result of Arazy and Zhang [3] and also of the results of Section 4, but it is easy to give a direct proof. It suffices to show that $M_{z_{j}}$ is bounded for each $j.$ Since $M_{z_{j}}$ preserves the orthogonality of the monomials, we obtain $\left\\|M_{z_{j}}\right\\|=\sup_{m}\frac{\left\\|z_{j}z^{m}\right\\|_{\lambda}}{\left\\|z^{m}\right\\|_{\lambda}}=\sup_{m}\frac{m_{j}+1}{\|m\|+\lambda}.$ Note that $m_{j}\leq\left\|m\right\|$ with equality when $m_{k}=0$ for $k\neq j.$ Thus the supremum is finite and is easily seen to have the value of 1 if $\lambda\geq 1$ and $1/\lambda$ if $\lambda<1.$ ∎ We now record some standard properties of Toeplitz operators on (ordinary) Bergman spaces. These properties hold for Toeplitz operators (defined by the “multiply and project” recipe) on any holomorphic $L^{2}$ space. We will show that these properties do not hold for Toeplitz operators with polynomial symbols on the generalized Bergman spaces $H(\mathbb{B}^{d},\lambda),$ $\lambda<d.$ ###### Proposition 7. For $\lambda>d$ and $\phi(z)$ bounded, the Toeplitz operator $T_{\phi}$ on the space $\mathcal{H}L^{2}(\mathbb{B}^{d},d\mu_{\lambda}),$ which is defined by $T_{\phi}f=P_{\lambda}(\phi f),$ has the following properties. 1. (1) $\\|T_{\phi}\\|\leq\sup_{z}\|\phi(z)\|$ 2. (2) If $\phi(z)\geq 0$ for all $z,$ then $T_{\phi}$ is a positive operator. Both of these properties fail when $\lambda<d.$ In fact, for $\lambda<d,$ there is no constant $C$ such that $\\|T_{\phi}\\|\leq C\sup_{z}\|\phi(z)\|$ for all polynomials $\phi.$ As we remarked in the introduction, when $\lambda=d,$ the space $H(\mathbb{B}^{d},\lambda)$ may be identified with the Hardy space. Thus Properties 1 and 2 in the proposition still hold when $\lambda=d,$ if, say, $\phi$ is continuous up to the boundary of $\mathbb{B}^{d}$ (or otherwise has a reasonable extension to the closure of $\mathbb{B}^{d}$). ###### Proof. When $\lambda>d,$ the projection operator $P_{\lambda}$ has norm 1 and the multiplication operator $M_{\phi}$ has norm equal to $\sup_{z}\|\phi(z)\|$ as an operator on $L^{2}(\mathbb{B}^{d},\mu_{\lambda})$. Thus, the restriction to $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ of $P_{\lambda}M_{\phi}$ has norm at most $\sup_{z}\|\phi(z)\|.$ Meanwhile, if $\phi$ is non-negative, then from (3.1) we see that $\left\langle f,T_{\phi}f\right\rangle\geq 0$ for all $f\in\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}).$ Let us now assume that $0<\lambda<d.$ Computing on the orthogonal basis in Theorem 4, it is a simple exercise to show that $T_{\bar{z}_{j}z_{j}}(z^{m})=\frac{\Gamma(\lambda+\|m\|)}{m!}\frac{(m+e_{j})!}{\Gamma(\lambda+\|m\|+1)}z^{m}=\frac{1+m_{j}}{\lambda+\|m\|}z^{m}.$ (3.3) If we take $\phi(z)=\|z\|^{2},$ then summing (3.3) on $j$ gives $T_{\phi}z^{m}=\frac{d+\|m\|}{\lambda+\|m\|}z^{m}.$ Since $\lambda<d,$ this shows that $\left\\|T_{\phi}\right\\|>1,$ even though $\left\|\phi(z)\right\|\,<1$ for all $z\in\mathbb{B}^{d}.$ Thus, Property 1 fails for $\lambda<d.$ (From this calculation it easily follows that if $\phi(z)=(1-\left\|z\right\|^{2})/(\lambda-d),$ then $T_{\phi}$ is the bounded operator $(\lambda I+N)^{-1},$ for all $\lambda\neq d.$) For the second property, we let $\psi(z)=1-\phi(z)=1-\|z\|^{2}$ which is positive. From the above calculation we obtain $\langle T_{\psi}z^{m},z^{m}\rangle_{H_{\lambda}}=\\|z^{m}\\|_{H_{\lambda}}^{2}-\left(\frac{d+\|m\|}{\lambda+\|m\|}\right)\\|z^{m}\\|_{H(\mathbb{B}^{d},\lambda)}^{2},$ which is negative if $0<\lambda<d$. We now show that there is no constant $C$ such that $\\|T_{\phi}\\|\leq C\sup_{z}\|\phi(z)\|$. Consider $\displaystyle\phi_{k}(z)$ $\displaystyle:=(\|z\|^{2})^{k}=\left(\sum_{i=1}^{d}\|z_{i}\|^{2}\right)^{k}$ $\displaystyle=\sum_{\|i\|=k}\frac{k!}{i!}(\|z_{1}\|^{2})^{i_{1}}(\|z_{2}\|^{2})^{i_{2}}\cdots(\|z_{d}\|^{2})^{i_{d}}=\sum_{\|i\|=k}\frac{k!}{i!}\overline{z}^{i}z^{i}.$ Computing on the orthogonal basis in Theorem 4 we obtain $T_{\phi_{k}}\mathbf{1}=\sum_{\|i\|=k}\frac{k!}{i!}(T_{\overline{z}^{i}z^{i}}\mathbf{1})=\sum_{\|i\|=k}\frac{k!}{i!}\frac{i!\Gamma(\lambda)}{\Gamma(\lambda+k)}\mathbf{1}=\mathcal{I}\frac{k!\Gamma(\lambda)}{\Gamma(\lambda+k)}\mathbf{1,}$ where $\mathbf{1}$ is the constant function. Here, $\mathcal{I}$ is the number of multi-indices $i$ of length $d$ such that $\|i\|=k,$ which is equal to ${\binom{k+d-1}{d-1}}$. Thus $T_{\phi_{k}}\mathbf{1}=\frac{(k+d-1)!}{(d-1)!}\frac{\Gamma(\lambda)}{\Gamma(\lambda+k)}\mathbf{1}=\frac{(d+k-1)\cdots(d)}{(\lambda+k-1)\cdots(\lambda)}\mathbf{1}=\prod_{j=0}^{k-1}\frac{d+j}{\lambda+j}\mathbf{1}.$ Consider $\prod_{j=0}^{k-1}\frac{d+j}{\lambda+j}=\prod_{j=0}^{k-1}\left(1+\frac{d-\lambda}{\lambda+j}\right)$. Since $d>\lambda,$ the terms $\frac{d-\lambda}{\lambda+j}$ are positive and $\sum_{j=0}^{\infty}\frac{d-\lambda}{\lambda+j}$ diverges. This implies $\prod_{j=0}^{\infty}\frac{d+j}{\lambda+j}=\infty$. Since $\sup_{z}\|\phi_{k}(z)\|=1$ for all $k$, there is no a constant $C$ such that $\\|T_{\phi}\\|\leq C\sup_{z}\|\phi(z)\|$. ∎ ###### Remark 8. For $\lambda<d,$ there does not exist any positive measure $\nu$ on $\mathbb{B}^{d}$ such that $\left\\|f\right\\|_{\lambda}=\left\\|f\right\\|_{L^{2}(\mathbb{B}^{d},\nu)}$ for all $f$ in $H(\mathbb{B}^{d},\lambda).$ If such a $\nu$ did exist, then the argument in the first part of the proof of Proposition 7 would show that Properties 1 and 2 in the proposition hold. ## 4\. Bounded Toeplitz operators In this section, we will consider a class of symbols $\phi$ for which we will be able to define a Toeplitz operator $T_{\phi}$ as a bounded operator on $H(\mathbb{B}^{d},\lambda)$ for all $\lambda>0.$ Our definition of $T_{\phi}$ will agree (for the relevant class of symbols) with the usual “multiply and project” definition for $\lambda>d.$ In light of the examples in the previous section, we cannot expect boundedness of $\phi$ to be sufficient to define $T_{\phi}$ as a bounded operator. Instead, we will consider functions $\phi$ for which $\phi$ and a certain number of derivatives of $\phi$ are bounded. Our strategy is to use integration by parts to give an alternative expression for the matrix entries of a Toeplitz operator with sufficiently regular symbol, in the case $\lambda>d.$ We then take this expression as our definition of Toeplitz operator in the case $0<\lambda\leq d.$ ###### Theorem 9. Assume $\lambda>d$ and fix a positive integer $n.$ Let $\phi$ be a function that is $2n$ times continuously differentiable and for which $\bar{N}^{k}N^{l}\phi$ is bounded for all $0\leq k,l\leq n.$ Then $\left\langle f,T_{\phi}g\right\rangle_{\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})}=c_{\lambda+2n}\int_{\mathbb{B}^{d}}C\left[\left(\overline{f(z)}\phi(z)g(z)\right)\right]\left(1-\|z\|^{2}\right)^{\lambda+2n}d\tau(z)$ for all $f,g\in\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}),$ where $C$ is the operator given by $C=\left(I+\frac{\bar{N}}{\lambda+2n-1}\right)\cdots\left(I+\frac{\bar{N}}{\lambda+n}\right)\left(I+\frac{N}{\lambda+n-1}\right)\cdots\left(I+\frac{N}{\lambda}\right).$ (4.1) Thus, there exist constants $A_{jklm}$ (depending on $n$ and $\lambda$) such that $\left\langle f,T_{\phi}g\right\rangle_{\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})}=\sum_{j,k,l,m=1}^{n}A_{jklm}\left\langle N^{j}f,\left(\bar{N}^{k}N^{l}\phi\right)N^{m}g\right\rangle_{L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})}.$ (4.2) ###### Proof. Assume at first that $f$ and $g$ are polynomials, so that $f$ and $g$ and all of their derivatives are bounded. We use (3.1) and apply the first equality in Lemma 1 with $\psi=\bar{f}\phi g.$ We then apply the first equality in the lemma again with $\psi=(I+N/\lambda)[\bar{f}\phi g].$ We continue on in this fashion until we have applied the first equality in Lemma 1 $n$ times and the second equality $n$ times. This establishes the desired equality in the case that $f$ and $g$ are polynomials. For general $f$ and $g$ in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}),$ we approximate by sequences $f_{a}$ and $g_{a}$ of polynomials. From Theorem 3 we can see that convergence of $f_{a}$ and $g_{a}$ in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ implies convergence of $N^{j}f_{a}$ and $N^{k}g_{a}$ to $N^{j}f$ and $N^{k}g,$ so that applying (4.2) to $f_{a}$ and $g_{a}$ and taking a limit establishes the desired result for $f$ and $g.$ ∎ ###### Definition 10. Assume $0<\lambda\leq d$ and fix a positive integer $n$ such that $\lambda+2n>d.$ Let $\phi$ be a function that is $2n$ times continuously differentiable and for which $\bar{N}^{k}N^{l}\phi$ is bounded for all $0\leq k,l\leq n.$ Then we define the Toeplitz operator $T_{\phi}$ to be the unique bounded operator on $H(\mathbb{B}^{d},\lambda)$ whose matrix entries are given by $\left\langle f,T_{\phi}g\right\rangle_{H(\mathbb{B}^{d},\lambda)}=c_{\lambda+2n}\int_{\mathbb{B}^{d}}C\left[\left(\overline{f(z)}\phi(z)g(z)\right)\right]\left(1-\|z\|^{2}\right)^{\lambda+2n}dz,$ (4.3) where $C$ is given by (4.1). Note that from Theorem 4, $N^{j}f$ and $N^{m}g$ belong to $L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})$ for all $0\leq j,m\leq n,$ for all $f$ and $g$ in $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda}).$ Furthermore, $\left\\|N^{j}f\right\\|_{L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})}$ and $\left\\|N^{m}g\right\\|_{L^{2}(\mathbb{B}^{d},\mu_{\lambda+2n})}$ are bounded by constants times $\left\\|f\right\\|_{\lambda}$ and $\left\\|g\right\\|_{\lambda}$, respectively. Thus, the right-hand side of (4.3) is a continuous sesquilinear form on $H(\mathbb{B}^{d},\lambda),$ which means that there is a unique bounded operator $T_{\phi}$ whose matrix entries are given by (4.3). If $\lambda=d,$ then (as discussed in the introduction) the Hilbert space $H(\mathbb{B}^{d},\lambda)$ is the Hardy space of holomorphic functions that are square-integrable over the boundary. In that case, the Toeplitz operator $T_{\phi}$ will be the zero operator whenever $\phi$ is identically zero on the boundary of $\mathbb{B}^{d}.$ If $\lambda=d-1,$ $d-2,$ $\ldots,$ then the inner product on $H(\mathbb{B}^{d},\lambda)$ can be related to the inner product on the Hardy space. It is not hard to see that in these cases, $T_{\phi}$ will be the zero operator if $\phi$ and enough of its derivatives vanish on the boundary of $\mathbb{B}^{d}.$ Let us consider the case in which $\phi(z)=\overline{\psi_{1}(z)}\psi_{2}(z),$ where $\psi_{1}$ and $\psi_{2}$ are holomorphic functions such that the function and the first $n$ derivatives are bounded. Then when applying $C$ to $\overline{f(z)}\phi(z)g(z),$ all the $N$-factors go onto the expression $\psi_{2}(z)g(z)$ and all the $\bar{N}$-factors go onto $\overline{f(z)}\overline{\psi_{1}(z)}.$ Recalling from Theorem 4 the formula for the inner product on $H(\mathbb{B}^{d},\lambda)$, we see that $\left\langle f,T_{\phi}g\right\rangle_{\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})}=\left\langle\psi_{1}f,\psi_{2}g\right\rangle_{H(\mathbb{B}^{d},\lambda)},$ as expected. This means that in this case, $T_{\bar{\psi}_{1}\psi_{2}}=(M_{\psi_{1}})^{\ast}(M_{\psi_{2}}),$ as in the case $\lambda>d.$ In particular, Definition 10 agrees with the definition we used in Section 3 in the case that $\phi$ is a polynomial in $z$ and $\bar{z}.$ ## 5\. Hilbert–Schmidt Toeplitz operators ### 5.1. Statement of results In this section, we will give sufficient conditions under which a Toeplitz operator $T_{\phi}$ can be defined as a Hilbert–Schmidt operator on $H(\mathbb{B}^{d},\lambda).$ Specifically, if $\phi$ belongs to $L^{2}(\mathbb{B}^{d},\tau)$ then $T_{\phi}$ can be defined as a Hilbert–Schmidt operator, provided that $\lambda>d/2.$ Meanwhile, if $\phi$ belongs to $L^{1}(\mathbb{B}^{d},\tau),$ then $T_{\phi}$ can be defined as a Hilbert–Schmidt operator for all $\lambda>0.$ In both cases, we define $T_{\phi}$ in such a way that for all bounded functions $f$ and $g$ in $H(\mathbb{B}^{d},\lambda),$ we have $\left\langle f,T_{\phi}g\right\rangle_{\lambda}=c_{\lambda}\int_{\mathbb{B}^{d}}\overline{f(z)}\phi(z)g(z)(1-\|z\|^{2})^{\lambda}~{}d\tau(z),$ (5.1) where $c_{\lambda}$ is defined by $c_{\lambda}=\Gamma(\lambda)/(\pi^{d}\Gamma(\lambda-d)).$ This expression is identical to (3.1) in the case $\lambda>d.$ The value of $c_{\lambda}$ should be interpreted as 0 when $\lambda-d=0,-1,-2,\ldots$. This means that for $\phi$ in $L^{2}(\mathbb{B}^{d},\tau)$ or $L^{1}(\mathbb{B}^{d},\tau)$ (but not for other classes of symbols!), $T_{\phi}$ is the zero operator when $\lambda=d,d-1,\ldots.$ This strange phenomenon is discussed in the next subsection. Note that we are not claiming $T_{\phi}=0$ for arbitrary symbols when $\lambda=d,d-1,\ldots,$ but only for symbols that are integrable or square-integrable with respect to the hyperbolic volume measure $\tau.$ Such functions must have reasonable rapid decay (in an average sense) near the boundary of $\mathbb{B}^{d}.$ In the case $\phi\in L^{2}(\mathbb{B}^{d},\tau),$ the restriction $\lambda>d/2$ is easy to explain: the function $(1-\|z\|^{2})^{\lambda}$ belongs to $L^{2}(\mathbb{B}^{d},\tau)$ if and only if $\lambda>d/2.$ Thus, if $f$ and $g$ are bounded and $\phi$ is in $L^{2}(\mathbb{B}^{d},\tau),$ then (5.1) is absolutely convergent for $\lambda>d/2.$ In this subsection, we state our results; in the next subsection, we discuss some unusual properties of $T_{\phi}$ for $\lambda<d$; and in the last subsection of this section we give the proofs. We begin by considering symbols $\phi$ in $L^{2}(\mathbb{B}^{d},\tau).$ ###### Theorem 11. Fix $\lambda>d/2$ and let $c_{\lambda}=\Gamma(\lambda)/(\pi^{d}\Gamma(\lambda-d)).$ (We interpret $c_{\lambda}$ to be zero if $\lambda$ is an integer and $\lambda\leq d$.) Then the operator $A_{\lambda}$ given by $A_{\lambda}\phi(z)=c_{\lambda}^{2}\int_{\mathbb{B}^{d}}\left[\frac{(1-\|z\|^{2})(1-\|w\|^{2})}{(1-w\cdot\bar{z})(1-\bar{w}\cdot z)}\right]^{\lambda}\phi(w)\,d\tau(w)$ is a bounded operator from $L^{2}(\mathbb{B}^{d},\tau)$ to itself. ###### Theorem 12. Fix $\lambda>d/2.$ Then for each $\phi\in L^{2}(\mathbb{B}^{d},\tau)$, there is a unique Hilbert–Schmidt operator on $H(\mathbb{B}^{d},\lambda),$ denoted $T_{\phi},$ with the property that $\left\langle f,T_{\phi}g\right\rangle_{\lambda}=c_{\lambda}\int_{\mathbb{B}^{d}}\overline{f(z)}\phi(z)g(z)(1-\|z\|^{2})^{\lambda}~{}d\tau(z)$ (5.2) for all bounded holomorphic functions $f$ and $g$ in $H(\mathbb{B}^{d},\lambda).$ The Hilbert–Schmidt norm of $T_{\phi}$ is given by $\left\\|T_{\phi}\right\\|_{HS}^{2}=\left\langle\phi,A_{\lambda}\phi\right\rangle_{L^{2}(\mathbb{B}^{d},\tau)}.$ If $\lambda>d$ and $\phi\in L^{2}(\mathbb{B}^{d},\tau)\cap L^{\infty}(\mathbb{B}^{d},\tau)$, then the definition of $T_{\phi}$ in Theorem 12 agrees with the “multiply and project” definition; compare (3.1). Applying Lemma 5 with $\lambda_{1}=\lambda_{2}=\lambda$ and $\lambda>d/2,$ we see that for all $f$ and $g$ in $H(\mathbb{B}^{d},\lambda),$ the function $z\rightarrow\overline{f(z)}g(z)(1-\left\|z\right\|^{2})^{\lambda}$ is in $L^{2}(\mathbb{B}^{d},\tau).$ This means that the integral on the right-hand side of (5.2) is absolutely convergent for all $f,g\in H(\mathbb{B}^{d},\lambda).$ It is then not hard to show that (5.2) holds for all $f,g\in H(\mathbb{B}^{d},\lambda).$ The operator $A_{\lambda}$ coincides, up to a constant, with the Berezin transform. Let $\chi_{z}^{\lambda}(w):=K_{\lambda}(z,w)$ be the coherent state at the point $z,$ which satisfies $f(z)=\left\langle\chi_{z}^{\lambda},f\right\rangle_{\lambda}$ for all $f\in H(\mathbb{B}^{d},\lambda).$ Then one standard definition of the Berezin transform $B_{\lambda}$ is $B_{\lambda}\phi=\frac{\left\langle\chi_{z}^{\lambda},T_{\phi}\chi_{z}^{\lambda}\right\rangle_{\lambda}}{\left\langle\chi_{z}^{\lambda},\chi_{z}^{\lambda}\right\rangle_{\lambda}}.$ The function $B_{\lambda}\phi$ may be thought of as the Wick-ordered symbol of $T_{\phi},$ where $T_{\phi}$ is thought of as the anti-Wick-ordered quantization of $\phi.$ Using the formula (Theorem 4) for the reproducing kernel along with (5.2), we see that $A_{\lambda}=c_{\lambda}B_{\lambda}.$ (Note that $\chi_{z}^{\lambda}(w)$ is a bounded function of $w$ for each fixed $z\in\mathbb{B}^{d}$ and that $\left\langle\chi_{z}^{\lambda},\chi_{z}^{\lambda}\right\rangle_{\lambda}=K_{\lambda}(z,z).$) Note that $\tau$ is an infinite measure, which means that if $\phi$ is in $L^{2}(\mathbb{B}^{d},\tau)$ or $L^{1}(\mathbb{B}^{d},\tau),$ then $\phi$ must tend to zero at the boundary of $\mathbb{B}^{d},$ at least in an average sense. This decay of $\phi$ is what allows (5.2) to be a convergent integral. If, for example, we want to take $\phi(z)\equiv 1,$ then we cannot use (5.2) to define $T_{\phi},$ but must instead use the definition from Section 3 or Section 4. Note also that the space of Hilbert–Schmidt operators on $H(\mathbb{B}^{d},\lambda)$ may be viewed as the quantum counterpart of $L^{2}(\mathbb{B}^{d},\tau).$ It is thus natural to investigate the question of when the Berezin–Toeplitz quantization maps $L^{2}(\mathbb{B}^{d},\tau)$ into the Hilbert–Schmidt operators. We now show that if one considers a symbol $\phi$ in $L^{1}(\mathbb{B}^{d},\tau),$ then one obtains a Hilbert–Schmidt Toeplitz operator $T_{\phi}$ for all $\lambda>0.$ ###### Theorem 13. Fix $\lambda>0$ and let $c_{\lambda}$ be as in Theorem 12. Then for each $\phi\in L^{1}(\mathbb{B}^{d},\tau),$ there exists a unique Hilbert–Schmidt operator on $H(\mathbb{B}^{d},\lambda),$ denoted $T_{\phi},$ with the property that $\left\langle f,T_{\phi}g\right\rangle_{\lambda}=c_{\lambda}\int_{\mathbb{B}^{d}}\overline{f(z)}\phi(z)g(z)(1-\|z\|^{2})^{\lambda}~{}d\tau(z)$ (5.3) for all bounded holomorphic functions $f$ and $g$ in $H(\mathbb{B}^{d},\lambda).$ The Hilbert–Schmidt norm of $T_{\phi}$ satisfies $\left\\|T_{\phi}\right\\|_{HS}\leq c_{\lambda}\left\\|\phi\right\\|_{L^{1}(\mathbb{B}^{d},\tau)}.$ Using the pointwise bounds on elements of $H(\mathbb{B}^{d},\lambda)$ coming from the reproducing kernel, we see immediately that for all $f,g\in H(\mathbb{B}^{d},\lambda),$ the function $z\rightarrow\overline{f(z)}g(z)(1-\|z\|^{2})^{\lambda}$ is bounded. It is then not hard to show that (5.3) holds for all $f,g\in H(\mathbb{B}^{d},\lambda).$ We have already remarked that the definition of $T_{\phi}$ given in this section agrees with the “multiply and project” definition when $\lambda>d$ (and $\phi$ is bounded). It is also easy to see that the definition of $T_{\phi}$ given in this section agrees with the one in Section 4, when $\phi$ falls under the hypotheses of both Definition 10 and either Theorem 12 or Theorem 13. For some positive integer $n,$ consider the set of $\lambda$’s for which $\lambda+2n>d$ and $\lambda>d/2,$ i.e., $\lambda>\max(d-2n,d/2).$ Now suppose that $\phi$ belongs to $L^{2}(\mathbb{B}^{d},\tau)$ and that $N^{k}\bar{N}^{l}\phi$ is bounded for all $0\leq k,l\leq n.$ It is easy to see that the matrix entries $\left\langle f,T_{\phi}g\right\rangle_{\lambda}$ depend real-analytically on $\lambda$ for fixed polynomials $f$ and $g,$ whether $T_{\phi}$ is defined by Definition 10 or by Theorem 12. For $\lambda>d,$ the two matrix entries agree because both definitions of $T_{\phi}$ agree with the “multiply and project” definition. The matrix entries therefore must agree for all $\lambda>\max(d-2n,d/2).$ Since polynomials are dense in $H(\mathbb{B}^{d},\lambda)$ and both definitions of $T_{\phi}$ give bounded operators, the two definitions of $T_{\phi}$ agree. The same reasoning shows agreement of Definition 10 and Theorem 13. ### 5.2. Discussion Before proceeding on with the proof, let us make a few remarks about the way we are defining Toeplitz operators in this section. For $\lambda>d,$ $c_{\lambda}$ is the normalization constant that makes the measure $\mu_{\lambda}$ a probability measure, which can be computed to have the value $\Gamma(\lambda)/(\pi^{d}\Gamma(\lambda-d)).$ For $\lambda\leq d,$ although the measure $(1-\|z\|^{2})^{\lambda}~{}d\tau(z)$ is an infinite measure, we simply use the same formula for $c_{\lambda}$ in terms of the gamma function. We understand this to mean that $c_{\lambda}=0$ whenever $\lambda$ is an integer in the range $(0,d].$ It also means that $c_{\lambda}$ is negative when $d-1<\lambda<d$ and when $d-3<\lambda<d-2,$ etc. In the cases where $c_{\lambda}=0$, we have that $T_{\phi}=0$ for all $\phi$ in $L^{1}(\mathbb{B}^{d},\tau)$ or $L^{2}(\mathbb{B}^{d},\tau).$ This first occurs when $\lambda=d.$ Recall that for $\lambda=d,$ the space $H(\mathbb{B}^{d},\lambda)$ can be identified with the Hardy space of holomorphic functions square-integrable over the boundary. Meanwhile, having $\phi$ being integrable or square-integrable with respect to $\tau$ means that $\phi$ tends to zero (in an average sense) at the boundary, in which case it is reasonable that $T_{\phi}$ should be zero as an operator on the Hardy space. For other integer values of $\lambda\leq d,$ the inner product on $H(\mathbb{B}^{d},\lambda)$ can be expressed using the methods of Section 2 in terms of integration over the boundary, but involving the functions and their derivatives. In that case, we expect $T_{\phi}$ to be zero if $\phi$ has sufficiently rapid decay at the boundary, and it is reasonable to think that having $\phi$ in $L^{1}$ or $L^{2}$ with respect to $\tau$ constitutes sufficiently rapid decay. Note, however, that the conclusion that $T_{\phi}=0$ when $c_{\lambda}=0$ applies only when $\phi$ is in $L^{1}$ or $L^{2}$; for other classes of symbols, such as polynomials, $T_{\phi}$ is not necessarily zero. For example, $T_{z^{m}}$ is equal to $M_{z^{m}},$ which is certainly a nonzero operator on $H(\mathbb{B}^{d},\lambda),$ for all $\lambda>0.$ Meanwhile, if $c_{\lambda}<0,$ then we have the curious situation that if $\phi$ is positive and in $L^{1}$ or $L^{2}$ with respect to $\tau,$ then the operator $T_{\phi}$ is actually a negative operator. This is merely a dramatic example of a phenomenon we have already noted: for $\lambda<d,$ non-negative symbols do not necessarily give rise to non-negative Toeplitz operators. Again, though, the conclusion that $T_{\phi}$ is negative for $\phi$ positive applies only when $\phi$ belongs to $L^{1}$ or $L^{2}.$ For example, the constant function $\mathbf{1}$ always maps to the (positive!) identity operator, regardless of the value of $\lambda.$ ### 5.3. Proofs As motivation, we begin by computing the Hilbert–Schmidt norm of Toeplitz operators in the case $\lambda>d.$ For any bounded measurable $\phi,$ we extend the Toeplitz operator $T_{\phi}$ to all of $L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ by making it zero on the orthogonal complement of the holomorphic subspace. This extension is given by the formula $P_{\lambda}M_{\phi}P_{\lambda}.$ Then the Hilbert–Schmidt norm of the operator $T_{\phi}$ on $\mathcal{H}L^{2}(\mathbb{B}^{d},\mu_{\lambda})$ is the same as the Hilbert–Schmidt norm of the operator $P_{\lambda}M_{\phi}P_{\lambda}$ on $L^{2}(\mathbb{B}^{d},\mu_{\lambda}).$ Since $P_{\lambda}$ is computed as integration against the reproducing kernel, we may compute that $P_{\lambda}M_{\phi}P_{\lambda}f(z)=\int_{\mathbb{B}^{d}}\mathcal{K}_{\phi}(z,w)f(w)\,d\mu_{\lambda}(w),$ where $\mathcal{K}_{\phi}(z,w)=\int_{\mathbb{B}^{d}}K(z,u)\phi(u)K(u,w)\,d\mu_{\lambda}(u).$ If we can show that $\mathcal{K}_{\phi}$ is in $L^{2}(\mathbb{B}^{d}\times\mathbb{B}^{d},\mu_{\lambda}\times\mu_{\lambda})$, then it will follow by a standard result that $T_{\phi}$ is Hilbert–Schmidt, with Hilbert–Schmidt norm equal to the $L^{2}$ norm of $\mathcal{K}_{\phi}.$ For sufficiently nice $\phi,$ we can compute the $L^{2}$ norm of $\mathcal{K}_{\phi}$ by rearranging the order of integration and using twice the reproducing identity $\int K(z,w)K(w,u)~{}d\mu_{\lambda}(w)=K(z,u).$ (This identity reflects that $P_{\lambda}^{2}=P_{\lambda}.$) This yields $\int_{\mathbb{B}^{d}\times\mathbb{B}^{d}}\left\|\mathcal{K}_{\phi}(z,w)\right\|^{2}~{}d\mu_{\lambda}(z)~{}d\mu_{\lambda}=\left\langle\phi,A\phi\right\rangle_{L^{2}(\mathbb{B}^{d},\tau)},$ where $A_{\lambda}$ is the integral operator given by $\displaystyle A_{\lambda}\phi(z)$ $\displaystyle=c_{\lambda}^{2}\int_{\mathbb{B}^{d}}\left\|K(z,w)\right\|^{2}(1-\|z\|^{2})^{\lambda}(1-\|w\|^{2})^{\lambda}\phi(w)~{}d\tau(w)$ $\displaystyle=c_{\lambda}^{2}\int_{\mathbb{B}^{d}}\left[\frac{(1-\|z\|^{2})(1-\|w\|^{2})}{(1-\bar{w}\cdot z)(1-\bar{z}\cdot w)}\right]^{\lambda}\phi(w)\,d\tau(w).$ (5.4) In the case $d/2<\lambda\leq d,$ it no longer makes sense to express $T_{\phi}$ as $P_{\lambda}M_{\phi}P_{\lambda}.$ Nevertheless, we can consider an operator $A_{\lambda}$ defined by (5.4). Our goal is to show that for all $\lambda>d/2,$ (1) $A_{\lambda}$ is a bounded operator on $L^{2}(\mathbb{B}^{d},\tau)$ and (2) if we define $T_{\phi}$ by (5.1), then the Hilbert–Schmidt norm of $T_{\phi}$ is given by $\left\langle\phi,A_{\lambda}\phi\right\rangle_{L^{2}(\mathbb{B}^{d},\tau)}.$ We will obtain similar results for all $\lambda>0$ if $\phi\in L^{1}(\mathbb{B}^{d},\tau).$ ###### Proof of Theorem 11. We give two proofs of this result; the first generalizes more easily to other bounded symmetric domains, whereas the second relates $A_{\lambda}$ to the Laplacian for $\mathbb{B}^{d}$ (compare [13]). First Proof. We let $F_{\lambda}(z,w)=c_{\lambda}^{2}\left[\frac{(1-\|z\|^{2})(1-\|w\|^{2})}{(1-\bar{w}\cdot z)(1-\bar{z}\cdot w)}\right]^{\lambda}\text{;}$ i.e., $F_{\lambda}$ is the integral kernel of the operator $A_{\lambda}.$ A key property of $F_{\lambda}$ is its invariance under automorphisms: $F_{\lambda}(\psi(z),\psi(w))=F_{\lambda}(z,w)$ for each automorphism (biholomorphism) $\psi$ of $\mathbb{B}^{d}$ and all $z,w\in\mathbb{B}^{d}.$ To establish the invariance of $F_{\lambda},$ let $f_{\lambda}(z)=c_{\lambda}^{2}(1-\|z\|^{2})^{\lambda}.$ (5.5) According to Lemma 1.2 of [Z2], $F_{\lambda}(z,w)=f_{\lambda}(\phi_{w}(z)),$ where $\phi_{w}$ is an automorphism of $\mathbb{B}^{d}$ taking $0$ to $w$ and satisfying $\phi_{w}^{2}=I.$ Now, if $\psi$ is any automorphism, the classification of automorphisms (Theorem 1.4 of [Z2]) implies that $\psi\circ\phi_{w}=\phi_{\psi(w)}\circ U$ for some unitary matrix $U.$ From this we can obtain $\phi_{\psi(w)}=U\circ\phi_{w}\circ\psi^{-1},$ and so $f_{\lambda}(\phi_{\psi(w)}(\psi(z)))=f_{\lambda}(U(\phi_{w}(\psi^{-1}(\psi(z))))=f_{\lambda}(\phi_{w}(z)),$ i.e., $F_{\lambda}(\psi(z),\psi(w))=F_{\lambda}(z,w).$ The invariance of $F_{\lambda}$ under automorphisms means that $A_{\lambda}\phi$ can be thought of as a convolution (over the automorphism group $PSU(d,1)$) of $\phi$ with the function $f_{\lambda}.$ What this means is that $A_{\lambda}\phi(z)=\int_{G}f_{\lambda}(gh^{-1}\cdot 0)\phi(h\cdot 0)~{}dh,$ where $g\in G$ is chosen so that $g\cdot 0=z.$ Here $G=PSU(d,1)$ is the group of automorphisms of $\mathbb{B}^{d}$ (given by fractional linear transformations) and $dh$ is an appropriately normalized Haar measure on $G.$ Furthermore, $L^{2}(\mathbb{B}^{d},\tau)$ can be identified with the right-$K$-invariant subspace of $L^{2}(G,dg)$, where $K:=U(d)$ is the stabilizer of $0.$ If $\lambda>d,$ then $f_{\lambda}$ is in $L^{1}(\mathbb{B}^{d},\tau),$ in which case it is easy to prove that $A_{\lambda}$ is bounded; see, for example, Theorem 2.4 in [5]. This argument does not work if $\lambda\leq d.$ Nevertheless, if $\lambda>d/2,$ an easy computation shows that $f_{\lambda}$ belongs to $L^{2}(\mathbb{B}^{d},\tau)$ and also to $L^{p}(\mathbb{B}^{d},\tau)$ for some $p<2.$ We could at this point appeal to a general result known as the Kunze–Stein phenomenon [24]. The result states that on connected semisimple Lie groups $G$ with finite center (including $PSU(d,1)$), convolution with a function in $L^{p}(G,dg),$ $p<2,$ is a bounded operator from $L^{2}(G,dg)$ to itself. (See [11] for a proof in this generality.) However, the proof of this result is simpler in the case we are considering, where the function in $L^{p}(G,dg)$ is bi-$K$-invariant and the other function is right-$K$-invariant. (In our case, the function in $L^{p}(G,dg)$ is the function $g\rightarrow f_{\lambda}(g\cdot 0)$ and the function in $L^{2}(G,dg)$ is $g\rightarrow\phi(g\cdot 0).$) Using the Helgason Fourier transform along with its behavior under convolution with a bi-$K$-invariant function ([19, Lemma III.1.4]), we need only show that the spherical Fourier transform of $f_{\lambda}$ is bounded. (Helgason proves Lemma III.1.4 under the assumption that the functions are continuous and of compact support, but the proof also applies more generally.) Meanwhile, standard estimates show that for every $\varepsilon>0,$ the spherical functions are in $L^{2+\varepsilon}(G/K),$ with $L^{2+\varepsilon}(G/K)$ norm bounded independent of the spherical function. (Specifically, in the notation of [18, Sect. IV.4], for all $\lambda\in\mathfrak{a}^{\ast},$ we have $\left\|\phi_{\lambda}(g)\right\|\leq\phi_{0}(g),$ and estimates on $\phi_{0}$ (e.g., [1, Prop. 2.2.12]) show that $\phi_{0}$ is in $L^{2+\varepsilon}$ for all $\varepsilon>0.$) Choosing $\varepsilon$ so that $1/p+1/(2+\varepsilon)=1$ establishes the desired boundedness. Second proof. If $c_{\lambda}=0$ (i.e., if $\lambda\in\mathbb{Z}$ and $\lambda\leq d$), then there is nothing to prove. Thus we assume $c_{\lambda}$ is nonzero, in which case $c_{\lambda+1}$ is also nonzero. The invariance of $F_{\lambda}$ under automorphisms together with the square-integrability of the function $(1-\|z\|^{2})^{\lambda}$ for $\lambda>d/2$ show that the integral defining $A_{\lambda}f(z)$ is absolutely convergent for all $z.$ We introduce the (hyperbolic) Laplacian $\Delta$ for $\mathbb{B}^{d},$ given by $\Delta=(1-\|z\|^{2})\sum_{j,k=1}^{d}(\delta_{jk}-\bar{z}_{j}z_{k})\frac{\partial^{2}}{\partial\bar{z}_{j}\partial z_{k}}.$ (5.6) (This is a negative operator.) This operator commutes with the automorphisms of $\mathbb{B}^{d}.$ It is known (e.g., [28]) that $\Delta$ is an unbounded self-adjoint operator on $L^{2}(\mathbb{B}^{d},\tau),$ on the domain consisting of those $f$’s in $L^{2}(\mathbb{B}^{d},\tau)$ for which $\Delta f$ in the distribution sense belongs to $L^{2}(\mathbb{B}^{d},\tau).$ In particular, if $f\in L^{2}(\mathbb{B}^{d},\tau)$ is $C^{2}$ and $\Delta f$ in the ordinary sense belongs to $L^{2}(\mathbb{B}^{d},\tau),$ then $f\in Dom(\Delta).$ We now claim that $\Delta_{z}F_{\lambda}(z,w)=\lambda(\lambda-d)(F_{\lambda}(z,w)-F_{\lambda+1}(z,w)),$ (5.7) where $\Delta_{z}$ indicates that $\Delta$ is acting on the variable $z$ with $w$ fixed. Since $\Delta$ commutes with automorphisms, it again suffices to check this when $w=0,$ in which case it is a straightforward algebraic calculation. Suppose, then, that $\phi$ is a $C^{\infty}$ function of compact support. In that case, we are free to differentiate under the integral to obtain $\Delta A_{\lambda}\phi=\lambda(\lambda-d)A_{\lambda}\phi-\lambda(\lambda-d)A_{\lambda+1}\phi.$ (5.8) Now, the invariance of $F_{\lambda}$ tells us that $L^{2}(\mathbb{B}^{d},\tau)$ norm of $F_{\lambda}(z,w)$ as a function of $z$ is finite for all $w$ and independent of $w.$ Putting the $L^{2}$ norm inside the integral then shows that $A_{\lambda}\phi$ and $A_{\lambda+1}\phi$ are in $L^{2}(\mathbb{B}^{d},\tau).$ This shows that $A_{\lambda}\phi$ is in $Dom(\Delta).$ Furthermore, the condition $\lambda>d/2$ implies that $\lambda(\lambda-d/2)>-d^{2}/4.$ It is known that the $L^{2}$ spectrum of $\Delta$ is $(-\infty,-d^{2}/4].$ For general symmetric space of the noncompact type, the $L^{2}$ spectrum of the Laplacian is $(-\infty,-\left\\|\rho\right\\|^{2}],$ where $\rho$ is half the sum of the positive (restricted) roots for $G/K,$ counted with their multiplicity. In our case, there is one positive root $\alpha$ with multiplicity $(2d-2)$ and another positive root $2\alpha$ with multiplicity 1. (See the entry for “A IV” in Table VI of Chapter X of [17].) Thus, $\rho=d\alpha.$ It remains only to check that if the metric is normalized so that the Laplacian comes out as in (5.6), then $\left\\|\alpha\right\\|^{2}=1/4.$ This is a straightforward but unilluminating computation, which we omit. Since $\lambda(\lambda-d)$ is in the resolvent set of $\Delta,$ we may rewrite (5.8) as $A_{\lambda}\phi=-\lambda(\lambda-d)[\Delta-\lambda(\lambda-d)I]^{-1}A_{\lambda+1}\phi.$ Suppose now that $\lambda+1>d,$ so that (as remarked above) $A_{\lambda+1}$ is bounded. Since $[\Delta-cI]^{-1}$ is a bounded operator for all $c$ in the resolvent of $\Delta,$ we see that $A_{\lambda}$ has a bounded extension from $C_{c}^{\infty}(\mathbb{B}^{d})$ to $L^{2}(\mathbb{B}^{d},\tau).$ Since the integral computing $A_{\lambda}\phi(z)$ is a continuous linear functional on $L^{2}(\mathbb{B}^{d},\tau)$ (integration against an element of $L^{2}(\mathbb{B}^{d},\tau)$), it is easily seen that this bounded extension coincides with the original definition of $A_{\lambda}.$ The above argument shows that $A_{\lambda}$ is bounded if $\lambda>d/2$ and $\lambda+1>d.$ Iteration of the argument then shows boundedness for all $\lambda>d/2.$ ∎ ###### Proof of Theorem 12. We wish to show that for all $\lambda>d/2,$ if $\phi$ is in $L^{2}(\mathbb{B}^{d},\tau),$ then there is a unique Hilbert–Schmidt operator $T_{\phi}$ with matrix entries given in (5.1) for all polynomials, and furthermore, $\left\\|T_{\phi}\right\\|_{HS}^{2}=\left\langle\phi,A_{\lambda}\phi\right\rangle_{\lambda}.$ At the beginning of this section, we had an calculation of $\left\\|T_{\phi}\right\\|$ in terms of $A_{\lambda},$ but this argument relied on writing $T_{\phi}$ as $P_{\lambda}M_{\phi}P_{\lambda},$ which does not make sense for $\lambda\leq d.$ We work with an orthonormal basis for $H(\mathbb{B}^{d},\lambda)$ consisting of normalized monomials, namely, $e_{m}(z)=z^{m}\sqrt{\frac{\Gamma(\lambda+\|m\|)}{m!\Gamma(\lambda)}},$ for each multi-index $m.$ Then we want to establish the existence of a Hilbert–Schmidt operator whose matrix entries in this basis are given by $a_{lm}:=c_{\lambda}\int_{\mathbb{B}^{d}}\overline{e_{l}(z)}\phi(z)e_{m}(z)(1-\|z\|^{2})^{\lambda}~{}d\tau(z).$ (5.9) There will exist a unique such operator provided that $\sum_{l,m}\|a_{lm}\|^{2}<\infty.$ If we assume, for the moment, that Fubini’s Theorem applies, we obtain $\displaystyle\sum_{l,m}\left\|a_{lm}\right\|^{2}$ $\displaystyle=c_{\lambda}^{2}\int_{\mathbb{B}^{d}}\int_{\mathbb{B}^{d}}\sum_{l,m}\frac{\Gamma(\lambda+\|l\|)}{l!\Gamma(\lambda)}\frac{\Gamma(\lambda+\|m\|)}{m!\Gamma(\lambda)}\bar{z}^{l}w^{l}z^{m}\bar{w}^{m}$ $\displaystyle\times\phi(z)\overline{\phi(w)}(1-\|z\|^{2})^{\lambda}(1-\|w\|^{2})^{\lambda}~{}d\tau(z)~{}d\tau(w),$ (5.10) where $l$ and $m$ range over all multi-indices of length $d.$ We now apply the binomial series $\frac{1}{(1-r)^{\lambda}}=\sum_{k=0}^{\infty}\binom{\lambda+k-1}{k}r^{k}$ for $r\in\mathbb{C}$ with $\left\|r\right\|<1,$ where $\binom{\lambda+k-1}{k}=\frac{\Gamma(\lambda+k)}{k!\Gamma(\lambda)}.$ (This is the so-called negative binomial series.) We apply this with $r=\sum_{j}\bar{z}_{j}w_{j},$ and we then apply the (finite) multinomial series to the computation of $(\bar{z}\cdot w)^{k}.$ The result is that $\sum_{l}\frac{\Gamma(\lambda+\|l\|)}{l!\Gamma(\lambda)}\bar{z}^{l}w^{l}=\frac{1}{(1-\bar{z}\cdot w)^{\lambda}},$ (5.11) where the sum is over all multi-indices $l.$ Applying this result, (5.10) becomes $\sum_{l,m}\left\|a_{lm}\right\|^{2}=\left\langle\phi,A_{\lambda}\phi\right\rangle_{\lambda},$ (5.12) which is what we want to show. Assume at first that $\phi$ is “nice,” say, continuous and supported in a ball of radius $r<1.$ This ball has finite measure and $\phi$ is bounded on it. Thus, if we put absolute values inside the sum and integral on the right-hand side of (5.10), finiteness of the result follows from the absolute convergence of the series (5.11). Thus, Fubini’s Theorem applies in this case. Now for a general $\phi\in L^{2}(\mathbb{B}^{d},\tau),$ choose $\phi_{j}$ converging to $\phi$ with $\phi_{j}$ “nice.” Then (5.12) tells us that $T_{\phi_{j}}$ is a Cauchy sequence in the space of Hilbert–Schmidt operators, which therefore converges in the Hilbert–Schmidt norm to some operator $T.$ The matrix entries of $T_{\phi_{j}}$ in the basis $\\{e_{m}\\}$ are by construction given by the integral in (5.9). The matrix entries of $T$ are the limit of the matrix entries of $T_{\phi_{j}},$ hence also given by (5.9), because $e_{l}$ and $e_{m}$ are bounded and $(1-\|z\|^{2})^{\lambda}$ belongs to $L^{2}(\mathbb{B}^{d},\tau)$ for $\lambda>d/2.$ We can now establish that (5.2) in Theorem 12 holds for all bounded holomorphic functions $f$ and $g$ in $H(\mathbb{B}^{d},\lambda)$ by approximating these functions by polynomials. ∎ ###### Proof of Theorem 13. In the proof of Theorem 12, we did not use the assumption $\lambda>d/2$ until the step in which we approximated arbitrary functions in $L^{2}(\mathbb{B}^{d},\tau)$ by “nice” functions. In particular, if $\phi$ is nice, then (5.9) makes sense for all $\lambda>0,$ and (5.12) still holds. Now, since $F_{\lambda}(z,w)=f_{\lambda}(\phi_{w}(z)),$ where $f_{\lambda}$ is given by (5.5), we see that $\left\|F_{\lambda}(z,w)\right\|\leq c_{\lambda}^{2}$ for all $z,w\in\mathbb{B}^{d}.$ Thus, $\left\langle\phi,A_{\lambda}\phi\right\rangle_{\lambda}\leq c_{\lambda}^{2}\left\\|\phi\right\\|_{L^{1}(\mathbb{B}^{d},\tau)}^{2}$ for all nice $\phi.$ An easy approximation argument then establishes the existence of a Hilbert–Schmidt operator with the desired matrix entries for all $\phi\in L^{1}(\mathbb{B}^{d},\tau),$ with the desired estimate on the Hilbert–Schmidt norm. ∎ ## References * [1] J.-P. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces. Geom. Funct. Anal. 9 (1999), 1035-1091. * [2] J. Arazy, Integral formulas for the invariant inner products in spaces of analytic functions on the unit ball. Function spaces (Edwardsville, IL, 1990), 9–23, Lecture Notes in Pure and Appl. Math., 136, Dekker, New York, 1992. * [3] J. Arazy and G. Zhang, Homogeneous multiplication operators on bounded symmetric domains. J. Funct. Anal. 202 (2003), 44–66. * [4] F. Beatrous Jr. and J. Burbea, Holomorphic Sobolev spaces on the ball. Dissertationes Math. (Rozprawy Mat.) 276 (1989), 60 pp. * [5] F. A. Berezin, Quantization. Math. USSR Izvestija, 8 (1974), 1109-1165. * [6] F. A. Berezin, Quantization in complex symmetric spaces. Math. USSR Izvestija 9 (1976), 341-379. * [7] M. Bordemann, E. Meinrenken, and M. Schlichenmaier, Toeplitz quantization of Kähler manifolds and $gl(N),$ $N\rightarrow\infty$ limits. Comm. Math. Phys. 165 (1994), 281–296. * [8] D. Borthwick, A. Lesniewski, and H. Upmeier, Nonperturbative deformation quantization of Cartan domains. J. Funct. Anal. 113 (1993), 153–176. * [9] D. Borthwick, T. Paul, and A. Uribe, Legendrian distributions with applications to relative Poincaré series. Invent. Math. 122 (1995), 359–402. * [10] L. A. Coburn, Deformation estimates for the Berezin-Toeplitz quantization. Comm. Math. Phys. 149 (1992), 415–424. * [11] M. Cowling, The Kunze–Stein phenomenon. Ann. Math. 107 (1978), 209-234. * [12] P. Duren and A. Schuster, Bergman spaces. Mathematical surveys and monographs; no.100, American Mathematical Society, 2004. * [13] M. Engliš, Berezin transform and the Laplace-Beltrami operator. Algebra i Analiz 7 (1995), 176–195; translation in St. Petersburg Math. J. 7 (1996), 633–647. * [14] G. Folland, Harmonic analysis in phase space. Princeton University Press, 1989. * [15] B. C. Hall, Holomorphic methods in analysis and mathematical physics. In: First Summer School in Analysis and Mathematical Physics (S. Pérez-Esteva and C. Villegas-Blas, Eds.), 1–59, Contemp. Math., 260, Amer. Math. Soc., 2000. * [16] B. C. Hall and W. Lewkeeratiyutkul, Holomorphic Sobolev spaces and the generalized Segal–Bargmann transform. J. Funct. Anal. 217 (2004), 192–220. * [17] S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Academic Press, 1978. * [18] S. Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, corrected reprint of the 1984 edition. Amer. Math. Soc., 2000. * [19] S. Helgason, Geometric analysis on symmetric spaces. Amer. Math. Soc., 1994. * [20] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces. Springer-Verlag, 2000. * [21] H. T. Kaptanoğlu, Besov spaces and Bergman projections on the ball. C. R. Math. Acad. Sci. Paris 335 (2002), 729–732. * [22] S. Klimek and A. Lesniewski, Quantum Riemann surfaces I. The unit disc. Commun. Math. Phys., 46 (1976) 103-122. * [23] A. Konechny, S. G. Rajeev, and O. T. Turgut, Classical mechanics on Grassmannian and disc. In: Geometry, integrability and quantization (Varna, 2000), 181–207, Coral Press Sci. Publ., Sofia, 2001. * [24] R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the $2\times 2$ unimodular group. Amer. J. Math. 82 (1960), 1-62. * [25] S. G. Rajeev and O. T. Turgut, Geometric quantization and two-dimensional QCD. Comm. Math. Phys. 192 (1998), 493–517. * [26] J. H. Rawnsley, Coherent states and Kähler manifolds. Quart. J. Math. Oxford Ser. (2) 28 (1977), 403–415. * [27] J. H. Rawnsley, M. Cahen, S. Gutt, Quantization of Kähler manifolds. I. Geometric interpretation of Berezin’s quantization. J. Geom. Phys. 7 (1990), 45–62. * [28] R. Strichartz, Harmonic analysis as spectral theory of Laplacians. J. Funct. Anal. 87 (1989), 51–148. * [29] A. Unterberger and H. Upmeier The Berezin transform and invariant differential operators. Commun. Math. Phys, 164 (1994), 563-597. * [30] Z. Yan, Invariant differential operators and holomorphic function spaces. J. Lie Theory 10 (2000), 31 pp. * [31] R. Zhao and K. H. Zhu, Theory of Bergman spaces in the unit ball of $\mathbb{C}^{n}$. Preprint, arxiv.org/abs/math/0611093. * [32] K. Zhu, Holomorphic Besov spaces on bounded symmetric domains. Quart. J. Math. Oxford Ser. (2) 46 (1995), 239–256. * [Z2] K. Zhu, Spaces of holomorphic functions in the unit ball. Springer-Verlag, 2004.	arxiv-papers	2009-03-03T23:06:34	2024-09-04T02:49:00.978093	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Kamthorn Chailuek, Brian C. Hall", "submitter": "Brian C. Hall", "url": "https://arxiv.org/abs/0903.0651" }
0903.0672	# Spatial Variations in Galactic H I Structure on AU-Scales Toward 3C 147 Observed with the Very Long Baseline Array T. Joseph W. Lazio Naval Research Laboratory, 4555 Overlook Avenue SW, Washington, DC 20375-5351 Joseph.Lazio@nrl.navy.mil C. L. Brogan National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903-2475 W. M. Goss National Radio Astronomy Observatory, P. O. Box O, 1003 Lopezville Road, Socorro, NM 87801 Snežana Stanimirović Department of Astronomy, University of Wisconsin, Madison, WI 53706 (Received 2008 June 13; Revised 2009 February 24; Accepted 2009 March 1) ###### Abstract This paper reports dual-epoch, Very Long Baseline Array observations of H I absorption toward 3C 147 (catalog 3C). One of these epochs (2005) represents new observations while one (1998) represents the reprocessing of previous observations to obtain higher signal-to-noise results. Significant H I opacity and column density variations, both spatially and temporally, are observed with typical variations at the level of $\Delta\tau\approx 0.20$ and in some cases as large as $\Delta\tau\approx 0.70$, corresponding to column density fluctuations of order $5\times 10^{19}$ cm-2 for an assumed 50 K spin temperature. The typical angular scale is 15 mas; while the distance to the absorbing gas is highly uncertain, the equivalent linear scale is likely to be about 10 AU. Approximately 10% of the face of the source is covered by these opacity variations, probably implying a volume filling factor for the small- scale absorbing gas of no more than about 1%. Comparing our results with earlier results toward 3C 138 (catalog 3C) (Brogan et al.), we find numerous similarities, and we conclude that small-scale absorbing gas is a ubiquitious phenomenon, albeit with a low probability of intercept on any given line of sight. Further, we compare the volumes sampled by the line of sight through the Galaxy between our two epochs and conclude that, on the basis of the motion of the Sun alone, these two volumes are likely to be substantially different. In order to place more significant constraints on the various models for the origin of these small-scale structures, more frequent sampling is required in any future observations. galaxies: individual (3C 147 (catalog 3C)) — ISM: general — ISM: structure — radio lines: ISM — techniques: interferometric ## 1 Introduction Beginning with a two-antenna very long baseline interferometric (VLBI) observation of 3C 147 (catalog 3C) by Dieter et al. (1976), a variety of H I absorption studies over the past three decades have found AU-scale optical depth variations in the Galactic interstellar medium (ISM). The initial detections were confirmed by Diamond et al. (1989), and the first images of the small-scale H I in the direction of 3C 138 (catalog 3C) and 3C 147 (catalog 3C) were made by Davis et al. (1996) using MERLIN. Faison et al. (1998) and Faison & Goss (2001) used the Very Long Baseline Array (VLBA) to improve the resolution toward a number of sources to approximately 20 mas ($\sim 10$ AU). Significant variations were detected in the direction of 3C 138 (catalog 3C) and 3C 147 (catalog 3C), while no significant variations in H I opacity were found in the direction of five other compact radio sources. An independent means of probing small-scale neutral structures is multi-epoch H I absorption measurements of high proper motion pulsars (Frail et al., 1994; Johnston et al., 2003; Stanimirović et al., 2003). While early pulsar observations suggested that small-scale structure might be ubiquitous, more recent observations suggest that it could be more sporadic. A significant advantage of VLBI observations is that they provide 2-D images of the opacity variations, rather than 1-D samples as in the case of pulsars observations. Brogan et al. (2005) revisited the observations of 3C 138 (catalog 3C), by re- analyzing the 1995 VLBA observations (Faison et al., 1998) and by obtaining two new epochs of observations (1999 and 2002). They confirmed the initial results of Faison & Goss (2001), that there are small-scale opacity changes along the line of sight to 3C 138 (catalog 3C) at the level of $\Delta\tau_{\mathrm{max}}=0.50\pm 0.05$, with typical sizes of roughly 50 mas ($\sim 25$ AU). However, with multiple epochs and improvements in data analysis techniques (yielding an increase of a factor of 5 in the sensitivity of the 1995 epoch), they reached a number of additional significant conclusions: 1. 1. They found clear evidence for temporal variations in the H I opacity over the seven-year time span of the three epochs, consistent with structures moving across the line of sight at velocities of a few tens of kilometers per second, though the infrequent sampling in time means that they could not determine whether these structures were persistent. 2. 2. They found no evidence for a drop in the H I spin temperature, as would be evidenced by a narrowing of line widths at small scales compared to single dish measurements. In turn, a constant H I spin temperature implies that the small-scale opacity variations are due to density enhancements, although these enhancements would necessarily be extremely over-pressured relative to the mean interstellar pressure, far from equilibrium, and likely of relatively short duration. 3. 3. For the first time they determined that the plane of sky covering fraction of the small-scale H I gas is roughly 10%. In turn, this small covering fraction suggests that the volume filling factor of such gas, within the cold neutral medium, is quite low ($\lesssim 1$%), in agreement with HST observations of high-pressure gas in the ISM (Jenkins & Tripp, 2001; Jenkins, 2004). 4. 4. They simulated pulsar observations that have been used to search for H I opacity variations and showed that the existing pulsar observations have generally been too sparsely sampled (in time) to be useful in studying the details of small-scale H I opacity variations. While the multi-epoch study of Brogan et al. (2005) represented a substantial improvement, nonetheless their conclusions rested on observations of only one line of sight. In light of this sample of one, their conclusions might seem rather audacious, particularly given the larger sample observed by Faison et al. (1998) and Faison & Goss (2001), in which most of the objects did not show variations in the H I absorption. The 3C 138 (catalog 3C) study has shown that the key to a successful small scale H I study is a background source with both high surface brightness ($\gtrsim 60$ mJy beam-1) and large angular extent ($>100$ mas). The quasar 3C 147 (catalog 3C) is one of the few sources that shares these characteristics with 3C 138 (catalog 3C). This paper presents dual-epoch observations of 3C 147 (catalog 3C) that were designed specifically to confront the conclusions of Brogan et al. (2005) with a second line of sight. Section 2 of this paper describes the observations, focussing on the new observations acquired for the second epoch, §3 discusses the results, and §4 presents our conclusions and recommendations for future work. ## 2 Observations We have observed the Galactic H I absorption (near $-10\,\mbox{km~{}s${}^{-1}$}$) toward the quasar 3C 147 (catalog 3C) at two epochs. Epoch I was 1998 October 22, and the results from those observations have been published previously by Faison & Goss (2001). Epoch II consists of new data observed on 2005 August 21. Table 1 summarizes the basic observing parameters for the two epochs. Table 1: Observational Log Parameter \| Faison & Goss (2001) \| Epoch I \| Epoch II ---\|---\|---\|--- Date \| 1998 October 22 \| 1998 October 22 \| 2005 August 21 Number of IFs \| 4 \| 4 \| 4 Bandwidth per IF (MHz) \| 0.5 \| 0.5 \| 0.5 Spectral channels \| 256 \| 256 \| 512 Channel separation (km s-1) \| 0.41 \| 0.41 \| 0.21 Velocity resolution (km s-1)aaAfter Hanning smoothing during imaging process. \| 0.4 \| 0.41 \| 0.21 clean beam (mas)bbThe clean beam before convolution to 10 mas. All subsequent values are for the convolved 10 mas resolution images. \| 5 $\times$ 4 \| 8.2 $\times$ 5.6 \| 7.6 $\times$ 7.1 Continuum Peak (Jy beam-1)ccFor the image after it has been convolved to 10 mas resolution. \| 2.31 \| 1.681 \| 1.801 Continuum rms noise (mJy beam-1)ccFor the image after it has been convolved to 10 mas resolution. \| 6.5 \| 1.7 \| 1.1 Spectral line rms noise (mJy beam-1)ccFor the image after it has been convolved to 10 mas resolution. \| 7.6 \| 5.0 \| 3.5 General Parameters for 3C 147 (catalog 3C) Position (equatorial, J2000) \| $05^{\rm h}42^{\rm m}36\fs 13788$ \| $+49\arcdeg 07\arcmin 51\farcs 2335$ \| Position (Galactic, longitude & latitude) \| $+161.69\arcdeg$ \| $+10.30\arcdeg$ \| Redshift \| 0.545 \| \| Note. — The values listed under the Faison & Goss (2001) column are for the original analysis. The values listed under the Epoch I column are for this analysis, after the reprocessing of the data as described in the text. For both epochs the data were obtained using the 10 antennas of the Very Long Baseline Array combined with the Very Large Array with its 27 antennas operating in a phased-array mode. For the 2005 epoch, the Green Bank Telescope was also used. The observing duration was 12 hr for the 1998 epoch and 16 hr for the 2005 epoch, including time spent on calibration sources. The proximity of the VLA to the VLBA antenna at Pie Town, New Mexico, significantly increased our sensitivity to large-scale structures. Four separate spectral windows or intermediate frequency bands (IFs) were used, with one IF centered on the absorption line (at an approximate LSR velocity of $-10\,\mbox{km~{}s${}^{-1}$}$) and three IFs separated by at least 100 km s-1 in velocity in order to sample the 21 cm continuum emission. For the 1998 epoch, the data were correlated with velocity channels of 0.4 km s-1, with a bandwidth of 500 kHz per IF over 256 spectral channels; for the 2005 epoch, improvements in the correlator allowed the number of spectral channels per IF to be increased to 512, with a concomitant improvement in the velocity resolution to 0.2 km s-1. Broadly similar data reduction procedures were used for the two epochs. For the 2005 epoch, the data were calibrated for the frequency dependence of the bandpass using observations of 3C 48 (catalog 3C) and amplitude calibrated using system temperatures measured at the individual antennas. The most significant difference in the calibration is that for the 2005 epoch, we attempted to phase-reference the observations to the compact source IVS B0532$+$506 (catalog IVS), separated by 1$\fdg$3 from 3C 147 (catalog 3C). Our initial motivation for this change in procedure is that 3C 147 (catalog 3C) has a sufficiently complex structure that fringe-fitting assuming a point- source model could yield erroneous residual delay and rate solutions. In practice, phase referencing did not prove useful. The phase-referencing cycle time was short enough that latency in the VLA system often resulted in the VLA acquiring no data. The most significant difficulty, however, was that only one of the epochs was phase-referenced. There was an apparent offset in the core position between the two epochs (with a magnitude of a fraction of the synthesized beam width or a few milliarcseconds) that biased any attempt to compare results from the two epochs (e.g., comparing the integrated line profiles). Consequently, we did not make use of the phase-referenced data for constructing the H I line profile or opacity images. One obvious impact on our results is that the sensitivity of the 2005 epoch H I line data is less than it could have otherwise been due to the phase-referencing cycling between 3C 147 (catalog 3C) and IVS B0532+506 (catalog IVS). Two of the three continuum IFs were then averaged together and several iterations of hybrid imaging (iterative imaging and self-calibration) were performed. After the final iteration of self-calibration, the phase and amplitude solutions were applied to the IF containing the H I line. The line- free velocity channels in this IF were averaged together to produce a continuum data set, which underwent a final round of hybrid imaging, the solutions from which were applied to the velocity channels containing the line. Finally, the continuum emission was subtracted from the velocity channels containing the H I line and the resulting line data set was imaged. We also reprocessed the observations of Faison & Goss (2001) in a similar fashion. A significant difference from the original analysis of Faison & Goss (2001) is that we used the continuum image from the 2005 epoch as an initial model for fringe fitting the 1998 epoch data (for which no phase referencing was performed). The combination of a better initial model and improvements in the imaging software and analysis procedures led to a substantial improvement in the reprocessed 1998 epoch data. The noise in the 1998 epoch continuum image has improved by nearly an order of magnitude, and the improvement in the spectral line images is a factor of a few. As was the case for 3C 138 (catalog 3C), the original analysis found a significantly higher peak brightness than we do, by a similar factor ($\approx 30$%). Like Brogan et al. (2005), we attribute this difference to the use of a point source model by Faison & Goss (2001) in the original fringe fitting along with other details of the subsequent imaging and self-calibration. Following the procedure of Faison & Goss (2001), both the continuum images and continuum-subtracted line cubes were convolved to 10 mas resolution. The $u$-$v$ coverages for the visibility data from the two epochs were similar, producing images with angular resolutions of approximately 7 mas (Table 1). The convolution of the continuum images and continuum-subtracted line cubes is an attempt to minimize any effects of modest differences in the $u$-$v$ coverage between the epochs. The second epoch data were also smoothed in velocity so that their velocity resolution matched that of the first epoch. An optical depth cube, calculated as $\tau_{H\,I}(\alpha,\delta,v)=-\ln[1-I_{\mathrm{line}}(\alpha,\delta,v)/I_{\mathrm{cont}}(\alpha,\delta)]$, where $I_{\mathrm{line}}$ and $I_{\mathrm{cont}}$ are, respectively, the images from formed from line and line-free channels. Because the signal-to- noise ratio in the optical depth images is low where the continuum emission is weak, the optical depth images were blanked where the continuum emission was less than 5% of the peak emission. ## 3 Results ### 3.1 21 cm Continuum Figure 1 presents the 21 cm continuum image of 3C 147 (catalog 3C) from the new observations of the 2005 epoch. There is good qualitative agreement between our image and previously published images at comparable wavelengths (18–20 cm, Readhead & Wilkinson, 1980; Zhang et al., 1991; Polatidis et al., 1995; Faison & Goss, 2001). The source displays its well-known core-jet structure, with the jet extending some 200 mas to the southwest before bending to the north. Also prominent is diffuse emission to the east of the core, extending to the north, first noticed by Zhang et al. (1991). The resolution of our observations is not high enough to resolve the northeast extension from the core found by Readhead & Wilkinson (1980). Figure 1: The Epoch II (2005) 21 cm continuum image of 3C 147 (catalog 3C) obtained with the VLBA and phased VLA. The clean beam is 7.6 mas $\times$ 7.1 mas at a position angle of $-15\arcdeg$. The rms noise level is 0.5 mJy beam-1, and the contours are set at 0.5 mJy beam-1 $\times$ $-3$, 5, 7.07, 10, 14.1, 20, $\ldots$. The gray scale is linear between 1.5 and 500 mJy beam-1. This image shows the source at the full resolution; for subsequent analysis, the image was convolved to 10 mas resolution. The origin is at (J2000) $05^{\mathrm{h}}\,42^{\mathrm{m}}\,36\fs 1379$ $+49\arcdeg\,51\arcmin\,07\farcs 234$. We assessed the continuum images from the two epochs for variability. Creating a difference image between the two epochs, we find that any variability in the source is below the 20 mJy beam-1 level. Even if the source is variable at this level, as for the Brogan et al. (2005) analysis, variability will not impact our optical depth calculations, because (1) the continuum appropriate for each epoch was used in the optical depth calculations, (2) amplitude self- calibration solutions were never transferred between the epochs, and (3) the intrinsic continuum morphology at 10 mas resolution does not appear to change from epoch to epoch. The flux density in our image is 18 Jy. The VLA Calibrator Manual lists of flux density of 22.5 Jy, indicating that our observations have recovered 80% of the source’s total flux density. ### 3.2 H I Line Profile Figure 2 shows the average optical depth profile from the new observation of 2005. Even with its relatively high latitude ($b=+10\arcdeg$), the profile is complex, making it similar to 3C 138 (catalog 3C) (Brogan et al., 2005). Our profile is in good agreement with previously published profiles (Kalberla et al., 1985; Faison & Goss, 2001), and the difference between the line profiles from the two epochs shows only modest variations. Like Faison & Goss (2001), we shall restrict our attention to the three most prominent velocity components, those with approximate central velocities of $-10.4$, $-8.0$, and 0.4 km s-1 (see below). Figure 2: The solid line shows the average H I optical depth profile toward 3C 147 (catalog 3C) from Epoch II (2005). The dotted line shows the difference between the line profiles from the two epochs. Also marked are the three significant velocity components at which further optical depth analysis is performed (viz. Table 2). Faison & Goss (2001) have discussed the difficulties with assessing the distance to the absorbing gas. Under the simple assumption that all H I gas is confined to a 100 pc thick layer, the absorbing gas must be within 0.6 kpc. However, the kinematic distance to the gas causing the $-8\,\mbox{km~{}s${}^{-1}$}$ absorption is uncertain, with distances as large as 1.1 kpc allowed. As a nominal value, we adopt the conversion that our resolution of 10 mas corresponds to a linear distance of 7.5 AU ($1\,\mathrm{mas}=0.75\,\mathrm{AU}$), implying a distance of 750 pc to the gas, though differences of as much 50% are possible. For subsequent analysis, we fit the 2005 epoch optical depth line cube by gaussian components, using the profile of Figure 2 as an guide to initial values for the component parameters. For the fitting, we focussed on the three significant components identified. The fitting was done on a pixel-by-pixel basis, with an independent three-component fit for each pixel. Table 2 summarizes results of the fits, _averaged_ over the face of the the source. Guided by the results of the fitting from the 2005 epoch, a similar fitting was performed for the 1998 epoch. Table 2: Optical Depth Profile Gaussian Component Fit Results Component \| Epoch \| Central Velocity \| Velocity Width \| Maximum Optical Depth ---\|---\|---\|---\|--- \| \| (km s-1) \| (km s-1) \| 1 \| 1998 \| 0.4 \| 5.1 \| 0.5 \| 2005 \| 0.3 \| 4.9 \| 0.5 2 \| 1998 \| $-8.0$ \| 1.6 \| 0.8 \| 2005 \| $-8.0$ \| 1.5 \| 0.8 3 \| 1998 \| $-10.4$ \| 6.1 \| 0.3 \| 2005 \| $-10.4$ \| 6.2 \| 0.3 ### 3.3 Small-Scale Structure Figures 3 and 4 show _column density fluctuation_ images at the two epochs for the three different velocity components. From the gaussian fits, column density images were constructed from the fitting results by multiplying the maximum optical depth by the velocity width ($N_{\mathrm{H\,I}}/T_{s}\propto\tau\sigma_{v}$, see below regarding the spin temperature $T_{s}$). In order to highlight fluctuations, the average column density from the 2005 epoch across the face of the source was subtracted from these column density images to produce the _column density fluctuation_ images. The signal-to-noise is not uniform across the face of the source, and tends to decrease near the edges. In order to aid in assessing the reality of features, Figures 3 and 4 also show the column density fluctuation signal-to- noise ratio images. Figure 3: Column density _fluctuation_ images, for the 1998 epoch (Epoch I), with the average value of the 2005 epoch column density subtracted, $\Delta N_{\mathrm{H\,I}}/T_{s}\propto\int dv\,(\tau-\langle\tau_{2005}\rangle)$. Horizontal white lines are pixels where the gaussian fitting failed to converge. _Left_ panels show the signal-to-noise ratio, on a linear scale with the dynamic range restricted to 0–15 and with a single contour showing a signal-to-noise ratio of 5. _Right_ panels show the column density fluctuations, on a linear scale. The gray scale bar shows the column density fluctuation, in units of $10^{19}$ cm-2. (Top) Column density fluctuations for $-10.4\,\mbox{km~{}s${}^{-1}$}$ velocity component in 1998, the 2005 average _optical depth_ that was subtracted is $\langle\tau_{2005}\rangle=0.31$; (Middle) Column density fluctuations for $-8.0\,\mbox{km~{}s${}^{-1}$}$ velocity component in 1998, $\langle\tau_{2005}\rangle=0.93$; and (Bottom) Column density fluctuations for $0.4\,\mbox{km~{}s${}^{-1}$}$ velocity component in 1998, $\langle\tau_{2005}\rangle=0.54$. Figure 4: Column density fluctuation images, as for Figure 3 but for the 2005 epoch (Epoch II). We show column density fluctuation images, in contrast to Brogan et al. (2005) who showed optical depth channel images. For 3C 147 (catalog 3C), analysis of the column density fluctuations is required because the velocity field of the absorping gas appears to change somewhat within each velocity component. For 3C 138 (catalog 3C), Brogan et al. (2005) compared the optical depth channel images at different velocities and concluded that any velocity field fluctuations were negligible, in contrast to the situation for 3C 147 (catalog 3C). Also, in converting to column density, we assume a uniform spin temperature of the gas of $T_{s}=50$ K (Heiles, 1997). Unlike 3C 138 (catalog 3C) which was part of the Millennium Arecibo 21 cm Survey (Heiles & Troland, 2003), 3C 147 (catalog 3C) is outside the Arecibo declination range so that there is less direct information about the absorbing gas along this line of sight. Figures 5–8 show cuts through the column density fluctuation images. In all cases, spatially significant changes in the column density between the two epochs are clearly apparent for all of the H I components, at significance levels exceeding $5\sigma$ over most of the face of the source. We have also conducted an analysis in which we consider a constant column density cross-cut to be the null hypothesis. In a $\chi^{2}$ sense, the null hypothesis can be firmly rejected as typical values, for both epochs and all velocity components, are $\chi^{2}\sim 10$ (reduced $\chi^{2}$). Typical angular scales for column density variations are approximately 15 mas, corresponding to a linear scale of approximately 10 AU. Figure 5: (Top Left) Cross-cuts taken approximately along the major axis showing the column density fluctuations for the 1998 epoch. (Top Right) Cross- cuts taken approximately along the major axis showing the column density fluctuations for the 2005 epoch. (Bottom) Illustration showing where the cross cuts were taken taken, with the column density fluctuations from the 0.4 km s-1, 2005 epoch shown for reference. Figure 6: As for Figure 5, but for the minor axis. Figure 7: Comparison of a cross-cut along the major axis at the two epochs for the three velocity components. For clarity, the 1998 epoch is shifted (by $15\times 10^{19}$ cm-2) relative to the 2005 epoch. For both epochs, a spin temperature $T_{s}=50$ K is assumed. Also shown are uncertainties ($\pm 2\sigma$), although in many cases they are only slightly larger than the symbol size. Further, because the restoring beam can induce correlations, we plot only every fifth datum. The major axis illustrated is the central of the three in Figure 5. Figure 8: As for Figure 7 (and Figure 6), but for a minor axis slice. Brogan et al. (2005) discussed the possible systematic effects that might affect the extraction of reliable optical depth or column density variations from dual-epoch imaging such as presented here. We do not repeat their discussion, but consider many of the same issues and conclusions to hold. Namely, while small differences in the images from epoch to epoch may be due to the details of the observations and data reduction (e.g., spatial frequency or $u$-$v$ coverage, slight differences in the imaging), a number of steps were taken during the analysis in an effort to minimize the differences between the epochs, and the column density variations are significant. The column density fluctuations in Figures 5–7 correpond to peak-to-peak optical depth variations as large as $\Delta\tau\approx 0.7$, and typical optical depth variations on scales of approximately 15 mas ranges from 0.1–0.3. The associated uncertainties in the optical depth are $\sigma_{\tau}\approx 0.07$, implying significant variations at the 3$\sigma$ level, and the column density cross-cuts indicate variations at even higher significance are present. Further, the magnitude of the variations is approximately correlated with the strength of the average H I absorption toward 3C 147 (catalog 3C), in that the largest variations are observed at $-8.0$ km s-1, followed by 0.4 km s-1, with the smallest variations at $-10.4$ km s-1 (cf. Figures 2 and 5–8). The opacity variations toward 3C 138 (catalog 3C) show changes that are consistent with motion of structures across the line of sight (Brogan et al., 2005), though there is considerable uncertainty with making these identifications (as they discuss). Possible motions of structures across the line of sight toward 3C 147 (catalog 3C) are also visible in the cross-cuts. Examples of such motions include the features at distances between 50 and 100 mas (Figures 7). Caution in interpreting these features as arising from motions is clearly warranted, however, given that we have only two epochs. Nonetheless, typical position shifts appear to be of the order of 5 mas. Over the 7 yr interval between observations, the implied proper motion is just under 1 mas yr-1, equivalent to a velocity of order 3 km s-1, at a distance of 750 pc. For comparison, and recognizing that there is considerable uncertainty in these comparisons, Brogan et al. (2005) find larger values for the apparent velocities of structures toward 3C 138 (catalog 3C) ($\approx 20\,\mbox{km~{}s${}^{-1}$}$). One of the motivations for undertaking these observations was to assess whether the optical depth variations, both spatial and temporal, found by Brogan et al. (2005) toward 3C 138 (catalog 3C) indicated that the line of sight to that source was in some sense “special” or anomalous. Comparison of Figures 3 and 4 and Figures 5–8 with the corresponding ones from Brogan et al. (2005) show that they are qualitatively similar, with clearly significant opacity or column density variations occurring both in space and time. Quantitatively, there are modest differences in the opacity/column density variations between 3C 147 (catalog 3C) and 3C 138 (catalog 3C). We estimate that the typical angular scale of opacity variations is 15 mas ($\approx 10$ AU), as opposed to about 50 mas ($\approx 25$ AU) toward 3C 138 (catalog 3C), though the linear scales are comparable. The magnitude of the variations toward 3C 147 (catalog 3C) also seems somewhat smaller. Optical depth changes (both in space and time), and corresponding column density fluctuations, are a factor of a few to several larger for 3C 138 (catalog 3C)—optical depth changes of 0.4 and larger ($>10^{20}$ cm-2) for 3C 138 (catalog 3C) vs. optical depths typically not exceeding 0.3 ($\sim 5\times 10^{19}$ cm-2) for 3C 147 (catalog 3C). ### 3.4 Small-Scale H I Covering and Filling Factors Brogan et al. (2005) used their observations of 3C 138 (catalog 3C) to conclude that the (two-dimensional) _covering factor_ of small-scale H I opacity variations was about 10%, from which they inferred a three-dimensional filling factor of probably less than 1%. Although the optical depth variations appear qualitatively similar for lines of sight toward 3C 138 (catalog 3C) and 3C 147 (catalog 3C), we have repeated their analysis in order to determine the covering and filling factors for the line of sight to 3C 147 (catalog 3C). Figure 9 shows the fractional number of pixels in a optical depth channel image for the three different velocities at both epochs. Most of the opacity variations are at a level less than about 0.2 in optical depth, and we do not consider them significant. Restricting to optical depth variations larger than approximately 0.2 ($\approx 3\sigma$), most of the covering fractions are about 10%, ranging from a low value of a few percent to a high exceeding 25%. Figure 9: Fractional number of pixels in an optical depth channel image as a function of the optical depth variation $\Delta\tau\equiv\|\tau-\langle\tau\rangle\|$. Left panels show the first epoch (1998), and right panels show the second epoch (2005). (Top) $-10.4\,\mbox{km~{}s${}^{-1}$}$; (Middle) $-8.0\,\mbox{km~{}s${}^{-1}$}$; and (Bottom) $0.4\,\mbox{km~{}s${}^{-1}$}$. While we have decomposed the optical depth profile into gaussian components, our decomposition is likely not unique and we fit only for three components, whereas Figure 2 clearly shows that there are could be more components. Thus, a plausible upper limit to the volume filling factor of the small-scale absorbing gas is obtained by assuming that there are multiple components, each of which contributes equally. We obtain an upper limit of 1%, a value which, as Brogan et al. (2005) emphasize, is not directly measurable. ## 4 Discussion and Conclusions We have presented two epochs of observations of the H I optical depth across the face of the source 3C 147 (catalog 3C) on scales of approximately 10 mas. The motivation for these observations was assessing whether spatial and temporal H I opacity variations found in multi-epoch observations of 3C 138 (catalog 3C) by Brogan et al. (2005) were in some sense “special” or anomalous. We find qualitatively similar opacity and column density variations toward 3C 147 (catalog 3C) as were found toward 3C 138 (catalog 3C). Quantitatively, the variations toward 3C 147 (catalog 3C) appear to be somewhat smaller in angular scale (15 mas vs. 50 mas) and smaller in magnitude (by a factor $\sim 5$). While the typical angular scale of the variations toward 3C 147 (catalog 3C) appears smaller, the absorbing gas may be more distant than that causing the absorption toward 3C 138 (catalog 3C) (§2). If so, the resulting linear scales are comparable (10 AU for 3C 147 (catalog 3C) vs. 25 AU for 3C 138 (catalog 3C)), though the uncertainties are large. Further similarities are observed in the covering and filling factors of the small-scale absorbing gas toward both sources. For both lines of sight, the covering factor appears to be approximately 10%, and the volume filling factor, while not measured directly, has a plausible upper limit of 1% (and potentially much less). Both 3C 138 (catalog ) and 3C 147 (catalog 3C) display significant H I opacity variations across their faces, implying variations within the ISM on scales of about 10 to 50 AU over path lengths ranging from 100 to 1000 pc. Using MERLIN, Goss et al. (2008) also have resolved significant H I opacity variations across the faces of 3C 111 (catalog ), 3C 123 (catalog ), and 3C 161 (catalog ), implying structure on scales of 50 to 500 AU. We conclude that the conditions that cause such small-scale variations are fairly widespread within the Galactic ISM. The reason that so few other sources display such small- scale opacity variations is likely to be, as Brogan et al. (2005) discuss, that few sources other than 3C 138 (catalog 3C) and 3C 147 (catalog 3C) have the combination of angular extent and surface brightness required to conduct these observations. One unfortunate implication of this conclusion is that milliarcsecond-scale H I observations of other sources will largely not be useful for probing the small-scale structure without a significant increase in sensitivity. One possible target, particularly with the existing High Sensitivity Array (HSA),111 The VLBA combined with other large aperture telescopes such as the phased VLA, the Green Bank Telescope, Arecibo, or the 100-m Efflesberg telescope. may be 3C 380 (catalog 3C). In one aspect, however, the lines of sight to 3C 147 (catalog 3C) and 3C 138 (catalog 3C) do differ. For the 3C 138 (catalog 3C) analysis, Brogan et al. (2005) used optical depth velocity channel images whereas, for 3C 147 (catalog 3C), we fit the optical depth line cube with gaussian components and used the resulting column density images. This difference in approach was motivated by the velocity structure that was apparent within an H I component in the 3C 147 (catalog 3C) optical depth line cube. We have not been able to find a ready explanation for this difference. Both sources are seen toward the Galactic anticenter, with Galactic coordinates (longitude, latitude) of (161$\fdg$7, 10$\fdg$3) for 3C 147 (catalog 3C) and (187$\fdg$4, $-11\fdg 3$) for 3C 138 (catalog 3C), respectively. To first order, the lines of sight to both cut almost perpendicular to the Perseus spiral arm. Further, were velocity crowding the explanation, it would seem that that should be more of an issue for 3C 138 (catalog 3C) than for 3C 147 (catalog 3C). We have also consulted the WHAM H$\alpha$ survey and the Green supernova remnant (SNR) catalog (Green, 2006), reasoning that H$\alpha$ and SNRs might serve as a tracers of turbulence injected by winds or explosions from massive stars. There are no obvious indications that the line of sight to 3C 147 (catalog 3C) should be affected any such turbulence—indeed a comparison of the 3C 138 (catalog 3C) and 3C 147 (catalog 3C) lines of sight suggest that the line of sight to 3C 138 (catalog 3C) would be _more_ likely to be the one that would display any such evidence of turbulence. An alternate possibility is that the kinematic differences between these two lines of sight reflect small-scale features, and possibly the past history of the gas. Kalberla et al. (1985) imaged the H I emission around the line of sight toward 3C 147 (catalog 3C) at 1′ resolution ($\approx 1$ pc linear scale). They found a series of filaments and small clumps of H I emission, and they were able to associate at least some of the absorption features with small emission clumps. The amount of small-scale structure (in H I emission) toward 3C 147 (catalog 3C) is not generally observed in the on-going GALFA H I survey at Arecibo. Further, Kalberla et al. (1985) find that a large fraction ($\sim 80$%) of the H I in emission in the direction of 3C 147 (catalog 3C) has a temperature of 500–2000 K. At this temperature, the gas would be thermally unstable. While the warm H I is not responsible for the absorption, the possibility that this line of sight contains thermally unstable H I is consistent with a scenario in which the microphysics, and potentially the past history of the gas, leads to kinematic variations within an H I absorption component. Kalberla et al. (1985) also suggested a relative distance ordering of the gas. Comparison of the opacity variations (Figures 5–8) suggests that the opacity variations are smallest in amplitude for the $-10.4$ km s-1 velocity component and increase in magnitude for the 0.4 km s-1 and the $-8.0$ km s-1 velocity components. One interpretation is that the opacity variations result from structures of essentially constant size, which are comparable to or smaller than the equivalent linear size of our beam ($\sim 10$ AU). If this were the case, we could obtain a relative distance ordering of the gas, with the $-8.0$ km s-1 material being the nearest, followed by the 0.4 km s-1 material, and the $-10.4$ km s-1 material being the most distant. An alternate interpretation (see above) would attribute these differences to the history of exposure of the gas to shocks or other interstellar disturbances. Whichever is the case, there is clearly significant structure on large scales ($\sim 1$ pc), suggesting that such structure could persist to smaller scales. The combination of VLBA and VLA data (as well as potentially MERLIN data) might be able to explore the connection between the small- and large-scale opacity variations. Bregman et al. (1983) have set an upper limit (3$\sigma$) on the magnetic field toward 3C 147 (catalog 3C) of $B_{\parallel}<50\,\mu\mathrm{G}$, based on Zeeman effect measurements in H I spectra. Under the standard assumption that discrete H I structures require densities $n\sim 10^{5}$ cm-3 (Heiles, 1997), with a typical velocity width of $v\approx 3\,\mbox{km~{}s${}^{-1}$}$ (Figure 2 and Table 2), one concludes that magnetic and turbulent equipartition requires a magnetic field strength of order 400 $\mu$G, well above the observed upper limit. As for the optical depth variations toward 3C 138 (Brogan et al., 2005), the optical depth variations cannot be in magnetic and turbulent equilibrium, unless there is significant blending and dilution of the magnetic field on the angular scales over which the Zeeman effect measurements were made. Our observations of 3C 147 (catalog 3C) do not produce any new constraints on the nature of the small-scale structures vis-a-vis whether they represent “statistical” fluctuations (e.g., Deshpande, 2000), “non-equilibrium” physical structures (e.g., Jenkins & Tripp, 2001; Hennebelle & Audit, 2007), or discrete “tiny scale atomic structures” (Heiles, 1997). While the level of opacity variations toward 3C 147 (catalog 3C) are lower than those toward 3C 138 (catalog 3C), we believe that this lower level can be accommodated easily within any of these scenarios. A wide range of opacity variations might be expected if these result from non-equilibrium processes, particularly because the level of opacity variations could depend upon the history of the gas. Also, as Brogan et al. (2005) note, the predicted level of opacity variations within a statistical description depends sensitively upon the assumed spectral index of the underlying power law; within the current uncertainties for this spectral index, a large range of opacity variations is allowed. What would be required in order to place significant constraints on these small-scale opacity variations? Ideally, one should monitor the same volume of gas and determine how the structures evolve. In the most simple comparison, discrete structures should show only linear motion across the line of sight, while fluctuations or non-equilibrium physical conditions might also cause the appearance of the opacity variations to change significantly. The elapsed times between observations presented here and those in Brogan et al. (2005) range from 3 to 7 yr. These intervals are only a few percent of the estimated time for discrete structures to change substantially ($\approx 500$ yr, Heiles, 2007). However, on milliarcsecond scales, the line of sight to 3C 147 (catalog 3C) effectively samples a volume through the Galaxy (Marscher et al., 1993; Dieter-Conklin, 2009). The Sun’s velocity through space causes this volume to move between our observing epochs (Figure 10), in addition to any motion that the gas itself might have. A simple estimate of the Sun’s motion suggests that an entirely new volume through the Galaxy could be sampled on time scales of approximately 3 yr. That is, in the typical interval between VLBI observations, essentially an entirely new volume of the Galaxy is sampled by the line of sight. Figure 10: An illustration of the different volumes of the Galaxy sampled by multi-epoch VLBI observations. The distance to the absorbing gas is assumed to be 750 pc, the space velocity of the Sun is assumed to be 30 km s-1, and the gas is assumed to be stationary. The resulting effective proper motion is 8.4 mas yr-1; a smaller assumed distance for the gas would result in a larger proper motion while a smaller space velocity for the Sun would result in a smaller proper motion. Shown is the apparent position of the source at three hypothetical epochs, each separated by 3 yr, comparable to the typical separation in epochs for the existing multi-epoch VLBI H I absorption observations. (Epoch I is red, Epoch II is green, and Epoch III is blue.) The white areas indicate the _only_ sampled volumes of gas common to all three epochs. Consequently, we also conclude that the time sampling of the existing multi- epoch VLBI observations has been too coarse to distinguish between the various models for the small-scale opacity variations. Ideally, one would like to monitor the same volume of gas, to determine if the opacity variations appear to be simply in motion or also changing in appearance. We estimate that, for either 3C 138 (catalog ) and 3C 147 (catalog 3C), an appropriate sampling interval is no longer than about 9 months, with even more rapid sampling desirable. We thank N. Dieter-Conklin for helpful discussions on the motion of the lines of sight through the Galaxy. We thank J. Dickey, the referee, who made insightful comments that we believe improved the analysis presented here. The Wisconsin H-Alpha Mapper is funded by the National Science Foundation. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This research has made use of NASA’s Astrophysics Data System. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. Basic research in radio astronomy at the NRL is supported by 6.1 NRL Base funding. ## References * Bregman et al. (1983) Bregman, J. D., Forster, J. R., Troland, T. H., Schwarz, U. J., Goss, W. M., & Heiles, C. 1983, A&A, 118, 157 * Brogan et al. (2005) Brogan, C. L., Zauderer, B. A., Lazio, T. J., Goss, W. M., DePree, C. G., & Faison, M. D. 2005, AJ, 130, 698 * Dieter-Conklin (2009) Dieter-Conklin, N. 2009, AJ, submitted * Davis et al. (1996) Davis, R. J., Diamond, P. J., & Goss, W. M. 1996, MNRAS, 283, 1105 * Deshpande (2000) Deshpande, A. A. 2000, MNRAS, 317, 199 * Deshpande et al. (2000) Deshpande, A. A., Dwarakanath, K. S., & Goss, W. M. 2000, ApJ, 543, 227 * Diamond et al. (1989) Diamond, P. J., Goss, W. M., Romney, J. D., Booth, R. S., Kalberla, P. M. W., & Mebold, U. 1989, ApJ, 347, 302 * Dieter et al. (1976) Dieter, N. H., Welch, W. J., & Romney, J. D. 1976, ApJ, 206, L113 * Faison & Goss (2001) Faison, M. D., & Goss, W. M. 2001, AJ, 121, 2706 * Faison et al. (1998) Faison, M. D., Goss, W. M., Diamond, P. J., & Taylor, G. B. 1998, AJ, 116, 2916 * Frail et al. (1994) Frail, D. A., Weisberg, J. M., Cordes, J. M., & Mathers, C. 1994, ApJ, 436, 144 * Goss et al. (2008) Goss, W. M., Richards, A. M. S., Muxlow, T. W. B., & Thomasson, P. 2008, MNRAS, in press * Green (2006) Green D. A., 2006, “A Catalogue of Galactic Supernova Remnants (2006 April version),” http://www.mrao.cam.ac.uk/surveys/snrs/ * Heiles (2007) Heiles, C. 2007, in SINS—Small Ionized and Neutral Structures in the Diffuse Interstellar Medium, eds. M. Haverkorn & W. M. Goss (ASP: San Francisco) p. 3 * Heiles (1997) Heiles, C. 1997, ApJ, 481, 193 * Heiles & Troland (2003) Heiles, C. & Troland, T. H. 2003, ApJS, 145, 329 * Hennebelle & Audit (2007) Hennebelle, P., & Audit, E. 2007, A&A, 465, 431 * Jenkins (2004) Jenkins, E. B. 2004, Ap&SS, 289, 215 * Jenkins & Tripp (2001) Jenkins, E. B. & Tripp, T. M. 2001, ApJS, 137, 297 * Johnston et al. (2003) Johnston, S., Koribalski, B., Wilson, W., & Walker, M. 2003, MNRAS, 341, 941 * Kalberla et al. (1985) Kalberla, P. M. W., Schwarz, U. J., & Goss, W. M. 1985, A&A, 144, 27 * Marscher et al. (1993) Marscher, A., Moore, E., & Bania, T. 1993, ApJ, 419, L101 * Polatidis et al. (1995) Polatidis, A. G., Wilkinson, P. N., Xu, W., Readhead, A. C. S., Pearson, T. J., Taylor, G. B., & Vermeulen, R. C. 1995, ApJS, 98, 1 * Readhead & Wilkinson (1980) Readhead, A. C. S., & Wilkinson, P. N. 1980, ApJ, 235, 11 * Stanimirović et al. (2003) Stanimirović, S., Weisberg, J. M., Hedden, A., Devine, K. E., & Green, J. T. 2003, ApJ, 598, L23 * Zhang et al. (1991) Zhang, F. J., Akujor, C. E., Chu, H. S., Mutel, R. L., Spencer, R. E., Wilkinson, P. N., Alef, W., Matveyenko, L. I., & Preuss, E. 1991, MNRAS, 250, 650	arxiv-papers	2009-03-04T01:32:34	2024-09-04T02:49:00.985913	{ "license": "Public Domain", "authors": "T. Joseph W. Lazio (1), C. L. Brogan (2), W. M. Goss (2), S.\n Stanimirovic (3) ((1) NRL; (2) NRAO; (3) Wisconsin)", "submitter": "Joseph Lazio", "url": "https://arxiv.org/abs/0903.0672" }
0903.0774	# Classical membrane in a time dependent orbifold Przemysław Małkiewicz† and Włodzimierz Piechocki‡ Theoretical Physics Department Institute for Nuclear Studies Hoża 69, 00-681 Warszawa, Poland; †pmalk@fuw.edu.pl, ‡piech@fuw.edu.pl ###### Abstract We analyze classical theory of a membrane propagating in a singular background spacetime. The algebra of the first-class constraints of the system defines the membrane dynamics. A membrane winding uniformly around compact dimension of embedding spacetime is described by two constraints, which are interpreted in terms of world-sheet diffeomorhisms. The system is equivalent to a closed bosonic string propagating in a curved spacetime. Our results may be used for finding a quantum theory of a membrane in the compactified Milne space. ###### pacs: 46.70.Hg, 11.25.-w, 02.20.Sv ## I Introduction In our previous papers we have examined the evolution of a particle Malkiewicz:2005ii ; Malkiewicz:2006wq and a string Malkiewicz:2006bw ; Malkiewicz:2008dw across the singularity of the compactified Milne (CM) space. The case of a membrane is technically more complicated because functions describing membrane dynamics depend on three variables. The Hamilton equations for these functions constitute a system of coupled non-linear equations in higher dimensional phase space. Owing to this complexity, we only try to identify some non-trivial membrane states which propagate through the cosmological singularity. An action integral of a membrane winding uniformly around compact dimension of CM space (equivalently, a closed string in curved spacetime) is reparametrization invariant. The first-class constraints describing membrane dynamics are generators of gauge transformations in the phase space of the system. We present the relationship between these symmetries. Our results constitute prerequisite for quantization of membrane dynamics in CM space. The paper is organized as follows: In Sec II we recall a general formalism for propagation of a $p$-brane in a fixed spacetime and we indicate that the Hamiltonian for a membrane winding uniformly around compact dimension of CM space reduces to the Hamiltonian of a string. Sec III concerns the algebra of Hamiltonian constraints of a membrane. The constraint satisfy the Poisson algebra, but may be turned into a Lie algebra by some reinterpretation of constraints. In Sec IV we analyze the algebra of conformal transformations connected with the symmetry of the Polyakov action integral of a string in a fixed gauge and we present a homomorphism between this algebra and the constraints algebra. Some insight into this relationship is given in Sec V. We conclude in Sec VI. Appendix consists of useful details clarifying the content of our paper. ## II General formalism The Polyakov action for a test $p$-brane embedded in a background spacetime with metric $g_{\tilde{\mu}\tilde{\nu}}$ has the form $S_{p}=-\frac{1}{2}\mu_{p}\int d^{p+1}\sigma\sqrt{-\gamma}\;\big{(}\gamma^{ab}\partial_{a}X^{\tilde{\mu}}\partial_{b}X^{\tilde{\nu}}g_{\tilde{\mu}\tilde{\nu}}-(p-1)\big{)},$ (1) where $\mu_{p}$ is a mass per unit $p$-volume, $(\sigma^{a})\equiv(\sigma^{0},\sigma^{1},\ldots,\sigma^{p})$ are $p$-brane worldvolume coordinates, $\gamma_{ab}$ is the $p$-brane worldvolume metric, $\gamma:=det[\gamma_{ab}]$, $~{}(X^{\tilde{\mu}})\equiv(X^{\mu},\Theta)\equiv(T,X^{k},\Theta)\equiv(T,X^{1},\ldots,X^{d-1},\Theta)$ are the embedding functions of a $p$-brane, i.e. $X^{\tilde{\mu}}=X^{\tilde{\mu}}(\sigma^{0},\ldots,\sigma^{p}$), in $d+1$ dimensional background spacetime. It has been found Turok:2004gb that the total Hamiltonian, $H_{T}$, corresponding to the action (1) is the following $H_{T}=\int d^{p}\sigma\mathcal{H}_{T},~{}~{}~{}~{}\mathcal{H}_{T}:=AC+A^{i}C_{i},~{}~{}~{}~{}~{}i=1,\ldots,p$ (2) where $A=A(\sigma^{a})$ and $A^{i}=A^{i}(\sigma^{a})$ are any functions of $p$-volume coordinates, $C:=\Pi_{\tilde{\mu}}\Pi_{\tilde{\nu}}g^{\tilde{\mu}\tilde{\nu}}+\mu_{p}^{2}\;det[\partial_{a}X^{\tilde{\mu}}\partial_{b}X^{\tilde{\nu}}g_{\tilde{\mu}\tilde{\nu}}]\approx 0,$ (3) $C_{i}:=\partial_{i}X^{\tilde{\mu}}\Pi_{\tilde{\mu}}\approx 0,$ (4) and where $\Pi_{\tilde{\mu}}$ are the canonical momenta corresponding to $X^{\tilde{\mu}}$. Equations (3) and (4) define the first-class constraints of the system. The Hamilton equations are $\dot{X}^{\tilde{\mu}}\equiv\frac{\partial{X}^{\tilde{\mu}}}{\partial\tau}=\\{X^{\tilde{\mu}},H_{T}\\},~{}~{}~{}~{}~{}~{}\dot{\Pi}_{\tilde{\mu}}\equiv\frac{\partial{\Pi}_{\tilde{\mu}}}{\partial\tau}=\\{\Pi_{\tilde{\mu}},H_{T}\\},~{}~{}~{}~{}~{}~{}\tau\equiv\sigma^{0},$ (5) where the Poisson bracket is defined by $\\{\cdot,\cdot\\}:=\int d^{p}\sigma\Big{(}\frac{\partial\cdot}{\partial X^{\tilde{\mu}}}\frac{\partial\cdot}{\partial\Pi_{\tilde{\mu}}}-\frac{\partial\cdot}{\partial\Pi_{\tilde{\mu}}}\frac{\partial\cdot}{\partial X^{\tilde{\mu}}}\Big{)}.$ (6) In what follows we restrict our considerations to the compactified Milne, CM, space. The CM space is one of the simplest models of spacetime implied by string/M theory Khoury:2001bz . Its metric is defined by the line element $ds^{2}=-dt^{2}+dx^{k}dx_{k}+t^{2}d\theta^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}+t^{2}d\theta^{2}=g_{\tilde{\mu}\tilde{\nu}}dx^{\tilde{\mu}}dx^{\tilde{\nu}},$ (7) where $\eta_{\mu\nu}$ is the Minkowski metric, and $\theta$ parameterizes a circle. Orbifolding $\mathbb{S}^{1}$ to a segment $~{}\mathbb{S}^{1}/\mathbb{Z}_{2}~{}$ gives the model of spacetime in the form of two planes which collide and re-emerge at $t=0$. Such model of spacetime has been used in Steinhardt:2001vw ; Steinhardt:2001st . Our results do not depend on the choice of topology of the compact dimension. In our previous papers Malkiewicz:2006bw ; Malkiewicz:2008dw and the present one we analyze the dynamics of a $p$-brane which is winding uniformly around the $\theta$-dimension. The $p$-brane in such a state is defined by the conditions $\sigma^{p}=\theta=\Theta~{}~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}\partial_{\theta}X^{\mu}=0=\partial_{\theta}\Pi_{\mu},$ (8) which lead to $\frac{\partial}{\partial\theta}(X^{\tilde{\mu}})=(0,\ldots,0,1)~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}\frac{\partial}{\partial\tau}(X^{\tilde{\mu}})=(\dot{T},\dot{X}^{k},0).$ (9) The conditions (8) reduce (3)-(6) to the form in which the canonical pair $(\theta,\Pi_{\theta})$ does not occur Turok:2004gb . Thus, a $p$-brane in the winding zero-mode state is described by (3)-(6) with $\tilde{\mu},\tilde{\nu}$ replaced by $\mu,\nu$. The propagation of a $p$-brane reduces effectively to the evolution of $(p-1)$-brane in the spacetime with dimension $d$ (while $d+1$ was the original one). ## III Algebra of constraints of a membrane In the case of a membrane in the winding zero-mode state the constraints are $C=\Pi_{\mu}(\tau,\sigma)\;\Pi_{\nu}(\tau,\sigma)\;\eta^{\mu\nu}+\kappa^{2}\;T^{2}(\tau,\sigma)\acute{X}^{\mu}(\tau,\sigma)\acute{X}^{\nu}(\tau,\sigma)\;\eta_{\mu\nu}\approx 0,$ (10) $C_{1}=\acute{X}^{\mu}(\tau,\sigma)\;\Pi_{\mu}(\tau,\sigma)\approx 0,~{}~{}~{}~{}~{}C_{2}=0,$ (11) where $\acute{X}^{\mu}:=\partial X^{\mu}/\partial\sigma~{},~{}~{}\sigma:=\sigma^{1},$ $\kappa:=\theta_{0}\mu_{2}$, and where $\theta_{0}:=\int d\theta$. To examine the algebra of constraints we ‘smear’ the constraints as follows $\check{A}:=\int_{-\pi}^{\pi}d\sigma\;f(\sigma)A(X^{\mu},\Pi_{\mu}),~{}~{}~{}~{}f\in\\{C^{\infty}[-\pi,\pi]\,\|\,f^{(n)}(-\pi)=f^{(n)}(\pi)\\}.$ (12) The Lie bracket is defined as $\\{\check{A},\check{B}\\}:=\int_{-\pi}^{\pi}d\sigma\;\Big{(}\frac{\partial\check{A}}{\partial X^{\mu}}\frac{\partial\check{B}}{\partial\Pi_{\mu}}-\frac{\partial\check{A}}{\partial\Pi_{\mu}}\frac{\partial\check{B}}{\partial X^{\mu}}\Big{)}.$ (13) The constraints in an integral form satisfy the algebra $\\{\check{C}(f_{1}),\check{C}(f_{2})\\}=\check{C}_{1}\big{(}4\kappa^{2}T^{2}(f_{1}\acute{f}_{2}-\acute{f}_{1}f_{2})\big{)},$ (14) $\\{\check{C}_{1}(f_{1}),\check{C}_{1}(f_{2})\\}=\check{C}_{1}(f_{1}\acute{f}_{2}-\acute{f}_{1}f_{2}),$ (15) $\\{\check{C}(f_{1}),\check{C}_{1}(f_{2})\\}=\check{C}(f_{1}\acute{f}_{2}-\acute{f}_{1}f_{2}).$ (16) Equations (14)-(16) demonstrate that $C$ and $C_{1}$ are first-class constraints because the Poisson algebra closes. However, it is not a Lie algebra because the factor $T^{2}$ is not a constant, but a function on phase space. Little is known about representations of such type of an algebra. Similar mathematical problem occurs in general relativity (see, e.g. TT ). The smearing (12) of constraints helps to get the closure of the algebra in an explicit form. A local form of the algebra includes the Dirac delta so the algebra makes sense but in the space of distributions (see Appendix A for more details). It seems that such an arena is inconvenient for finding a representation of the algebra which is required in the quantization process. The original algebra of constraints may be rewritten in a tractable form by making use of the redefinitions $C_{\pm}:=\frac{C\pm C_{1}}{2}$ (17) where $C:=\frac{\mbox{\scriptsize{original}}~{}C}{2\kappa T},~{}~{}~{}~{}C_{1}:=\mbox{\scriptsize{original}}~{}C_{1},$ (18) where ‘original’ means defined by (10) and (11). The new algebra reads $\\{\check{C}_{+}(f),\check{C}_{+}(g)\\}=\check{C}_{+}(f\acute{g}-g\acute{f}),$ (19) $\\{\check{C}_{-}(f),\check{C}_{-}(g)\\}=\check{C}_{-}(f\acute{g}-g\acute{f}),$ (20) $\\{\check{C}_{+}(f),\check{C}_{-}(g)\\}=0.$ (21) The redefined algebra is a Lie algebra. The redefinition (18) seems to be a technical trick without a physical interpretation. In what follows we show that it corresponds to the specification of the winding zero-mode state of a membrane not at the level of the constraints (10) and (11), but at the level of an action integral. The Nambu-Goto action for a membrane in the CM space reads $\displaystyle S_{NG}$ $\displaystyle=$ $\displaystyle-\mu_{2}\int d^{3}\sigma\sqrt{-det(\partial_{a}X^{\mu}\partial_{b}X^{\nu}g_{\mu\nu})}$ (22) $\displaystyle=$ $\displaystyle-\mu_{2}\int d^{3}\sigma\sqrt{-det(-\partial_{a}T\partial_{b}T+T^{2}\partial_{a}\Theta\partial_{b}\Theta+\partial_{a}X^{k}\partial_{b}X_{k})}$ (23) where $(T,\Theta,X^{k})$ are embedding functions of the membrane corresponding to the spacetime coordinates $(t,\theta,x^{k})$ respectively. An action $S_{NG}$ in the lowest energy winding mode, defined by (8), has the form $\displaystyle S_{NG}$ $\displaystyle=$ $\displaystyle-\mu_{2}\theta_{0}\int d^{2}\sigma\sqrt{-T^{2}det(-\partial_{a}T\partial_{b}T+\partial_{a}X^{k}\partial_{b}X_{k})}$ (24) $\displaystyle=$ $\displaystyle-\mu_{2}\theta_{0}\int d^{2}\sigma\sqrt{-det(\partial_{a}X^{\alpha}\partial_{b}X^{\beta}\widetilde{g}_{\alpha\beta})}.$ (25) where $a,b\in\\{0,1\\}$ and $\widetilde{g}_{\alpha\beta}=T\eta_{\alpha\beta}$. It is clear that the dynamics of a membrane in the state (8) is equivalent to the dynamics of a string with tension $\mu_{2}\theta_{0}$ in the spacetime with the metric $\widetilde{g}_{\alpha\beta}$. One can verify that the Hamiltonian corresponding to the string action (25) has the form $H_{T}=\int d\sigma\mathcal{H}_{T},~{}~{}~{}~{}\mathcal{H}_{T}:=AC+A^{1}C_{1},$ (26) where $C:=\frac{1}{2\mu_{2}\theta_{0}T}\Pi_{\alpha}\Pi_{\beta}\eta^{\alpha\beta}+\frac{\mu_{2}\theta_{0}}{2}\;T\;\partial_{a}X^{\alpha}\partial_{b}X^{\beta}\eta_{\alpha\beta}\approx 0,~{}~{}~{}~{}C_{1}:=\partial_{\sigma}X^{\alpha}\Pi_{\alpha}\approx 0,$ (27) and $A=A(\tau,\sigma)$ and $A^{1}=A^{1}(\tau,\sigma)$ are any regular functions. Therefore (27) and (18) coincide, which gives an interpretation for the redefinition of the constraints. ## IV Algebra of conformal transformations The Nambu-Goto action (25) is equivalent to the Polyakov action $S_{p}=-\frac{1}{2}\mu_{2}\theta_{0}\int d^{2}\sigma\sqrt{\gamma}(\gamma^{ab}\partial_{a}X^{\alpha}\partial_{b}X^{\beta}~{}T\eta_{\alpha\beta})$ (28) because variation with respect to $\gamma^{ab}$ (and using $\delta\gamma=\gamma\gamma^{ab}\delta\gamma_{ab}$) gives $\partial_{a}X^{\alpha}\partial_{b}X^{\beta}~{}T\eta_{\alpha\beta}-\frac{1}{2}\gamma_{ab}\gamma^{cd}\partial_{c}X^{\alpha}\partial_{d}X^{\beta}~{}T\eta_{\alpha\beta}=0.$ (29) The insertion of (29) into the Polyakov action (28) reproduces the Nambu-Goto action (25). In the gauge $\sqrt{-\gamma}\gamma^{ab}=1-\delta_{ab}$ the action (28) reads $S_{p}=-\mu_{2}\theta_{0}\int d^{2}\sigma(\partial_{+}X^{\alpha}\partial_{-}X^{\beta}~{}T\eta_{\alpha\beta})$ (30) where $\partial_{\pm}=\frac{\partial}{\partial{\sigma_{\pm}}}$, and where $\sigma_{\pm}:=\sigma_{0}\pm\sigma_{1}$. The least action principle applied to (30) gives the following equations of motion $\displaystyle\partial_{-}(T\partial_{+}X^{k})+\partial_{+}(T\partial_{-}X^{k})=0$ (31) $\displaystyle\partial_{-}(T\partial_{+}T)+\partial_{+}(T\partial_{-}T)+\partial_{+}X^{\alpha}\partial_{-}X^{\beta}~{}\eta_{\alpha\beta}=0,$ (32) where (29), due to the gauge $\sqrt{-\gamma}\gamma^{ab}=1-\delta_{ab}$, reads $\partial_{+}X^{\alpha}\partial_{+}X^{\beta}~{}\eta_{\alpha\beta}=0=\partial_{-}X^{\alpha}\partial_{-}X^{\beta}~{}\eta_{\alpha\beta}.$ (33) On the other hand, the action (30) is invariant under the conformal transformations, i.e. $\sigma_{\pm}\longrightarrow\sigma_{\pm}+{\epsilon}_{\pm}(\sigma_{\pm})$. It is so because for such transformations we have $\delta X^{\alpha}=-{\epsilon}_{-}\partial_{-}X^{\alpha}-{\epsilon}_{+}\partial_{+}X^{\alpha}$ and hence $\delta S_{p}=-\mu_{2}\theta_{0}\int d^{2}\sigma\big{(}\partial_{-}(-{\epsilon}_{-}\partial_{+}X^{\alpha}\partial_{-}X^{\beta}~{}T\eta_{\alpha\beta})+\partial_{+}(-{\epsilon}_{+}\partial_{+}X^{\alpha}\partial_{-}X^{\beta}~{}T\eta_{\alpha\beta})\big{)},$ (34) which is equal to zero since the fields $X^{\alpha}$ either vanish at infinity or are periodic. Now let assume that the fields $X^{\alpha}$ satisfy (31) and (32). Then (34) can be rewritten as $\displaystyle\delta S_{p}=-\mu_{2}\theta_{0}\int d^{2}\sigma\big{(}\partial_{-}(-{\epsilon}_{-}\partial_{+}X^{\alpha}\partial_{-}X^{\beta}~{}T\eta_{\alpha\beta})+\partial_{+}(-{\epsilon}_{-}\partial_{-}X^{\alpha}\partial_{-}X^{\beta}~{}T\eta_{\alpha\beta})$ $\displaystyle+~{}\partial_{+}(-{\epsilon}_{+}\partial_{+}X^{\alpha}\partial_{-}X^{\beta}~{}T\eta_{\alpha\beta})+\partial_{-}(-{\epsilon}_{+}\partial_{+}X^{\alpha}\partial_{+}X^{\beta}~{}T\eta_{\alpha\beta})\big{)}$ (35) which leads to $\partial_{-}T_{++}=0,~{}~{}~{}~{}~{}~{}\partial_{+}T_{--}=0$ (36) where $T_{++}={\epsilon}_{+}\partial_{+}X^{\alpha}\partial_{+}X^{\beta}~{}T\eta_{\alpha\beta},~{}~{}~{}~{}T_{--}={\epsilon}_{-}\partial_{-}X^{\alpha}\partial_{-}X^{\beta}~{}T\eta_{\alpha\beta}~{}.$ (37) One can verify that the vector fields ${\epsilon}_{-}\partial_{-}$ and ${\epsilon}_{+}\partial_{+}$ satisfy the following Lie algebra $[f_{+}\partial_{+},g_{+}\partial_{+}]=(f_{+}\acute{g}_{+}-g_{+}\acute{f}_{+})\partial_{+},$ (38) $[f_{-}\partial_{-},g_{-}\partial_{-}]=(f_{-}\acute{g}_{-}-g_{-}\acute{f}_{-})\partial_{-},$ (39) $[f_{+}\partial_{+},g_{-}\partial_{-}]=0.$ (40) The constraints algebra (19)-(21) defined on the phase space is the representation of the algebra of the conformal transformations (38)-(40) defined on the constraints surface (33). The Lie algebra homomorphism is defined by $\check{C}_{+}(f(\sigma))\longrightarrow f(\sigma_{+})\,\partial_{+},~{}~{}~{}~{}~{}\check{C}_{-}(f(\sigma))\longrightarrow f(\sigma_{-})\,\partial_{-},$ (41) where $\sigma_{\pm}\in\mathbb{R}$ and $\sigma\in\mathbb{S}$. ## V Transformations generated by constraints An action integral of a string is invariant with respect to smooth and invertible maps of worldsheet coordinates $(\tau,\sigma)\rightarrow(\tau^{\prime},\sigma^{\prime}).$ (42) These diffeomorphisms considered infinitesimally form an algebra of local fields $-\epsilon(\tau,\sigma)\partial_{\tau}$ and $-\eta(\tau,\sigma)\partial_{\sigma}$ (we refer to their actions on the fields as $\dot{\delta}_{\epsilon}$ and $\delta^{\prime}_{\eta}$, respectively). Mapping (42) leads to the infinitesimal changes of the fields $X^{\mu}(\tau,\sigma)$ and $\Pi_{\mu}(\tau,\sigma)=\partial L/\partial\dot{X}^{\mu}=\mu(\frac{1}{A}g_{\mu\nu}\dot{X}^{\nu}-\frac{A^{1}}{A}g_{\mu\nu}\acute{X}^{\nu})$ as follows $\delta X^{\mu}=\dot{\delta}_{\epsilon}X^{\mu}+\delta^{\prime}_{\eta}X^{\mu}=\epsilon\dot{X}^{\mu}+\eta\acute{X}^{\mu},~{}~{}~{}~{}~{}\delta\Pi_{\mu}=\epsilon\dot{\Pi}_{\mu}+\acute{\epsilon}(A^{1}{\Pi}_{\mu}+\mu Ag_{\mu\nu}\acute{X}^{\nu})+(\eta{\Pi}_{\mu})^{\prime}.$ (43) The transformations (43) are defined along curves in the phase space with coordinates $(X^{\mu},\Pi_{\mu})$ and are expected to be generated by the first-class constraints $\check{C}$ and $\check{C}_{1}$ according to the theory of gauge systems PAM ; HT . One verifies that $\\{X^{\mu},\check{C}(\varphi)\\}=\frac{\varphi}{\mu}\Pi_{\mu},~{}~{}~{}~{}\\{\Pi_{\mu},\check{C}(\varphi)\\}=-\frac{\varphi}{2\mu}(\Pi_{\alpha}\Pi_{\beta}g^{\alpha\beta}_{,X^{\mu}}+\acute{X}^{\alpha}\acute{X}^{\beta}g_{\alpha\beta,X^{\mu}})+\mu(\varphi g_{\mu\nu}\acute{X}^{\nu})^{\prime},$ (44) $\\{X^{\mu},\check{C}_{1}(\phi)\\}=\phi\acute{X}^{\mu},~{}~{}~{}~{}\\{\Pi_{\mu},\check{C}_{1}(\phi)\\}=(\phi\Pi_{\mu})^{\prime},$ (45) where $\phi(\sigma,\tau)$ and $\varphi(\sigma,\tau)$ are smearing functions depending on two variables, and the integration defining the smearing of the constraints $C$ and $C_{1}$ does not include the integration with respect to $\tau$ variable (see (12)). The comparison of (43) with (44)-(45) gives specific relations between these two transformations. For the action of the constraints along curves in the phase space, which are solutions to the equations of motion, we get $\\{X^{\mu},\check{C}(\varphi)\\}=\dot{\delta}_{\frac{\varphi}{A}}X^{\mu}-\delta^{\prime}_{\frac{A^{1}\varphi}{A}}X^{\mu},~{}~{}~{}~{}\\{\Pi_{\mu},\check{C}(\varphi)\\}=\dot{\delta}_{\frac{\varphi}{A}}\Pi_{\mu}-\delta^{\prime}_{\frac{A^{1}\varphi}{A}}\Pi_{\mu},$ (46) $\\{X^{\mu},\check{C}_{1}(\phi)\\}=\delta^{\prime}_{\phi}X^{\mu},~{}~{}~{}~{}\\{\Pi_{\mu},\check{C}_{1}(\phi)\\}=\delta^{\prime}_{\phi}\Pi_{\mu}.$ (47) Since $A$ and $A^{1}$ (see (63)) are invariant with respect to conformal isometries with respect to the worldsheet metric, the solutions to the equations of motion with fixed $A$ and $A^{1}$ have still some gauge freedom. The reduction of transformations (46)-(47) to the conformal transformations $\sigma_{\pm}\longrightarrow\sigma_{\pm}+\rho_{\pm}(\sigma_{\pm})$ for the curves in the orthonormal gauge $A=1$ and $A^{1}=0$, leads to $\frac{1}{2}(\dot{\delta}_{\rho_{\pm}}\pm\delta^{\prime}_{\rho_{\pm}})F\big{(}X^{\mu}(\sigma,\tau),\Pi_{\mu}(\sigma,\tau)\big{)}=\\{F\big{(}X^{\mu}(\sigma,\tau),\Pi_{\mu}(\sigma,\tau)\big{)},\check{C}_{\pm}(\rho_{\pm})\\},$ (48) where $F$ is a smooth function on phase space. One may show that (48) corresponds to the transformations defined by the algebra (38)-(40) but limited to the solutions of the equations of motion. On the other hand, the transformations (48) and (46)-(47) coincide with the algebra (19)-(21), for fixed $\tau$ . Now, we can see that the homomorphism (41) represents the reduction of the algebra of general conformal transformations (for fields not necessarily satisfying the equations of motion) to the algebra of generators of conformal transformations acting on curves $(X^{\mu},\Pi_{\mu})$ for fixed $\tau$. The latter algebra is equivalent to the algebra of generators $\check{C}$ and $\check{C}_{1}$ acting on the phase space $(X^{\mu},\Pi_{\mu})$. ## VI Conclusions In this paper we have considered states of membrane winding uniformly around compact dimension of the background space. Dynamics of a membrane in such special states is equivalent to the dynamics of a closed string in curved target space. However, the problem of quantization of a string in curved spacetime has not been solved yet (see, e.g. Thiemann:2004qu ). The construction of satisfactory quantum theory of membrane presents a challenge . The first-class constraints specifying the dynamics of a membrane propagating in the compactified Milne space satisfy the algebra which is a Poisson algebra. Methods for finding a self-adjoint representation of such type of an algebra are very complicated TT ; Thiemann:2004qu . We overcome this problem by the reduction and redefinition of the constraints algebra. Resulting algebra is a Lie algebra which simplifies the problem of quantization of the membrane dynamics. We have found a homomorphism between the algebra of conformal transformations and the algebra of transformations generated by the first-class constraints of the system. This may enable the construction of quantum dynamics of a membrane by making use of representations of conformal algebra. Details concerning quantization procedure will be presented elsewhere PMWP . ## Appendix A Local form of the constraints algebra One can verify that the constraints (10) and (11) satisfy the algebra $\\{C(\sigma),C(\sigma^{\prime})\\}=8\kappa^{2}T^{2}(\sigma)\;C_{1}(\sigma)\frac{\partial}{\partial\sigma}\delta(\sigma^{\prime}-\sigma)+4\kappa^{2}\delta(\sigma^{\prime}-\sigma)\frac{\partial}{\partial\sigma}\big{(}T^{2}(\sigma)C_{1}(\sigma)\big{)},$ (49) $\\{C(\sigma),C_{1}(\sigma^{\prime})\\}=2\;C(\sigma)\frac{\partial}{\partial\sigma}\delta(\sigma^{\prime}-\sigma)+\delta(\sigma^{\prime}-\sigma)\frac{\partial}{\partial\sigma}C(\sigma),$ (50) $\\{C_{1}(\sigma),C_{1}(\sigma^{\prime})\\}=2\;C_{1}(\sigma)\frac{\partial}{\partial\sigma}\delta(\sigma^{\prime}-\sigma)+\delta(\sigma^{\prime}-\sigma)\frac{\partial}{\partial\sigma}C_{1}(\sigma),$ (51) where $\partial X^{\mu}(\sigma^{\prime})/\partial X^{\nu}(\sigma)=\delta^{\mu}_{\nu}\delta(\sigma^{\prime}-\sigma)=\partial\Pi_{\nu}(\sigma^{\prime})/\partial\Pi_{\mu}(\sigma)$ (with other partial derivatives being zero), and where the Poisson bracket is defined to be $\\{\cdot,\cdot\\}:=\int_{-\pi}^{\pi}d\sigma\;\Big{(}\frac{\partial\cdot}{\partial X^{\mu}}\frac{\partial\cdot}{\partial\Pi_{\mu}}-\frac{\partial\cdot}{\partial\Pi_{\mu}}\frac{\partial\cdot}{\partial X^{\mu}}\Big{)}.$ (52) ## Appendix B Relation between gauges The least action principle applied to the Nambu-Goto action, $\delta S_{NG}=0$, gives $\displaystyle\partial_{a}(\frac{\partial_{b}X^{\alpha}\partial_{b}X^{\beta}g_{\alpha\beta}}{\sqrt{-det(\partial_{a}X^{\alpha}\partial_{b}X^{\beta}{g}_{\alpha\beta})}}\partial_{a}X_{\mu}-\frac{\partial_{a}X^{\alpha}\partial_{b}X^{\beta}g_{\alpha\beta}}{\sqrt{-det(\partial_{a}X^{\alpha}\partial_{b}X^{\beta}{g}_{\alpha\beta})}}\partial_{b}X_{\mu})$ $\displaystyle-\frac{(\partial_{a}X^{\alpha}\partial_{a}X^{\beta}g_{\alpha\beta})\partial_{b}X^{\alpha}\partial_{b}X^{\beta}-(\partial_{a}X^{\alpha}\partial_{b}X^{\beta}g_{\alpha\beta})\partial_{a}X^{\alpha}\partial_{b}X^{\beta}}{2\sqrt{-det(\partial_{a}X^{\alpha}\partial_{b}X^{\beta}{g}_{\alpha\beta})}}g_{\alpha\beta,\mu}=0.$ (53) In the case of the Polyakov action the least action principle, $\delta S_{p}=0$, gives $\partial_{a}(\sqrt{-\gamma}\gamma^{ab}\partial_{b}X_{\mu})=\frac{1}{2}\sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{\alpha}\partial_{b}X^{\beta}g_{\alpha\beta,\mu},$ (54) $\partial_{a}X^{\alpha}\partial_{b}X^{\beta}~{}g_{\alpha\beta}-\frac{1}{2}\gamma_{ab}\gamma^{cd}\partial_{c}X^{\alpha}\partial_{d}X^{\beta}~{}g_{\alpha\beta}=0.$ (55) On the other hand, the Hamilton equations read $\displaystyle\dot{X}^{\mu}$ $\displaystyle=$ $\displaystyle\\{{X}^{\mu},H_{T}\\}\approx A\frac{1}{\mu}\Pi_{\nu}g^{\nu\mu}+A^{1}\partial_{\sigma}X^{\mu},$ (56) $\displaystyle\dot{\Pi}_{\mu}$ $\displaystyle=$ $\displaystyle\\{\Pi_{\mu},H_{T}\\}\approx-A\frac{1}{2\mu}(\Pi_{\alpha}\Pi_{\beta}\frac{\partial g^{\alpha\beta}}{\partial X^{\mu}}+\mu^{2}\partial_{\sigma}X^{\alpha}\partial_{\sigma}X^{\beta}\frac{\partial g_{\alpha\beta}}{\partial X^{\mu}})+\mu\partial_{\sigma}(Ag_{\nu\mu}\partial_{\sigma}X^{\nu})$ (57) $\displaystyle+$ $\displaystyle\partial_{\sigma}(A^{1}\Pi_{\mu}),$ which in the case $g_{\alpha\beta}=T\eta_{\alpha\beta}$ give $\displaystyle\dot{X}^{\mu}$ $\displaystyle=$ $\displaystyle\\{{X}^{\mu},H_{T}\\}\approx A\frac{1}{\mu T}\Pi_{\nu}\eta^{\nu\mu}+A^{1}\partial_{\sigma}X^{\mu},$ (58) $\displaystyle\dot{\Pi}_{\mu}$ $\displaystyle=$ $\displaystyle\\{\Pi_{\mu},H_{T}\\}\approx-A\frac{\delta_{\mu 0}}{2\mu}(-\Pi_{\alpha}\Pi_{\beta}\frac{\eta^{\alpha\beta}}{T^{2}}+\mu^{2}\partial_{\sigma}X^{\alpha}\partial_{\sigma}X^{\beta}\eta_{\alpha\beta})+\mu\partial_{\sigma}(AT\eta_{\nu\mu}\partial_{\sigma}X^{\nu})$ (59) $\displaystyle+$ $\displaystyle\partial_{\sigma}(A^{1}\Pi_{\mu}).$ Now, we are ready to find the relations among $\gamma_{ab}$, $A$, $A^{1}$ and the induced metric. It is not difficult to see that $\frac{1}{\sqrt{-det(\partial_{a}X^{\mu}\partial_{b}X^{\nu}{g}_{\mu\nu})}}\left(\begin{array}[]{cc}-\partial_{\sigma}X^{\mu}\partial_{\sigma}X^{\nu}g_{\mu\nu}&\partial_{\sigma}X^{\mu}\partial_{\tau}X^{\nu}g_{\mu\nu}\\\ \partial_{\tau}X^{\mu}\partial_{\sigma}X^{\nu}g_{\mu\nu}&-\partial_{\tau}X^{\mu}\partial_{\tau}X^{\nu}g_{\mu\nu}\\\ \end{array}\right)=-\sqrt{-\gamma}\gamma^{ab},$ (60) $\left(\begin{array}[]{cc}\frac{1}{A}&-\frac{A^{1}}{A}\\\ -\frac{A^{1}}{A}&-A+\frac{(A^{1})^{2}}{A}\\\ \end{array}\right)=-\sqrt{-\gamma}\gamma^{ab}.$ (61) For instance, $\sqrt{-\gamma}\gamma^{ab}=(-1)^{a}~{}\delta_{ab}$ translates into $A=1$ and $A^{1}=0$. There exists an interesting discussion of the ADM like gauges in the context of a constrained Hamiltonian approach to the bosonic p-branes in the Minkowski space Banerjee:2004un ; Banerjee:2005bb . We postpone finding the relation between our choice of gauges and the ADM type and its usefulness in the context of the singularity problem to our next papers. ## Appendix C Position-velocity and phase spaces The position-velocity space is a space of pairs of fields $(X^{\mu}(\sigma),\dot{X}^{\mu}(\sigma))$, whereas the space of pairs $(X^{\mu}(\sigma),\Pi_{\mu}(\sigma))$ defines a phase space. The transformation $\\{X^{\mu},\dot{X}^{\mu}\\}\rightarrow\\{X^{\mu},\Pi_{\mu}=\frac{\mu}{\sqrt{-g}}(-(\acute{X})^{2}g_{\mu\nu}\dot{X}^{\nu}+(\acute{X}\dot{X})g_{\mu\nu}\acute{X}^{\nu})\\}$ (62) is a surjection onto the surface $C=0=C^{1}$. It becomes a bijection for fixed $A:=-\frac{\sqrt{-g}}{(\acute{X})^{2}},~{}~{}~{}~{}~{}A^{1}:=\frac{(\dot{X}\acute{X})}{(\acute{X})^{2}},$ (63) where $\acute{X}^{\mu}\acute{X}^{\nu}g_{\mu\nu}>0$ and $\dot{X}^{\mu}\dot{X}^{\nu}g_{\mu\nu}<0$, and $g<0$. We say that such choice of $A,A^{1}$ defines the $(A,A^{1})$-sector. Thus, the mapping $\\{X^{\mu}(\sigma),\Pi_{\mu}(\sigma)\\}\rightarrow\\{X^{\mu}(\sigma),\dot{X}^{\mu}(\sigma)=\frac{A}{\mu}\Pi^{\mu}+A^{1}\acute{X}^{\mu}\\}$ (64) presents the one-to-one correspondence between the phase space surface $C=0=C^{1}$ and the $(A,A^{1})$-sector. If $A$ and $A^{1}$ depend on $\tau$, then the $(A,A^{1})$-sector and the correspondence depend on $\tau$ as well. All $(A,A^{1})$-sectors are equivalent ($A\neq 0$) in the sense that all solutions to dynamics are mapped from one sector to another by a diffeomorphism (42). ###### Acknowledgements. This work has been supported by the Polish Ministry of Science and Higher Education Grant NN 202 0542 33. ## References * (1) P. Małkiewicz and W. Piechocki, “The simple model of big-crunch/big-bang transition”, Class. Quant. Grav., 23 (2006) 2963 [arXiv:gr-qc/0507077]. * (2) P. Małkiewicz and W. Piechocki, “Probing the cosmic singularity with a particle”, Class. Quant. Grav., 23 (2006), to appear arXiv:gr-qc/0606091. * (3) P. Malkiewicz and W. Piechocki, “Propagation of a string across the cosmic singularity”, Class. Quant. Grav. 24 (2007) 915 [arXiv:gr-qc/0608059]. * (4) P. Malkiewicz and W. Piechocki, “Excited states of a string in a time dependent orbifold”, Class. Quant. Grav. (2007), in print [arXiv:0807.2990 [gr-qc]]. * (5) N. Turok, M. Perry and P. J. Steinhardt, “M theory model of a big crunch / big bang transition”, Phys. Rev. D 70 (2004) 106004 [arXiv:hep-th/0408083]. * (6) J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Steinhardt and N. Turok, “From big crunch to big bang”, Phys. Rev. D 65 (2002) 086007 [arXiv:hep-th/0108187]. * (7) P. J. Steinhardt and N. Turok, “A cyclic model of the universe”, Science 296 (2002) 1436 [arXiv:hep-th/0111030]. * (8) P. J. Steinhardt and N. Turok, “Cosmic evolution in a cyclic universe”, Phys. Rev. D 65 (2002) 126003 [arXiv:hep-th/0111098]. * (9) T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge: Cambridge University Press, 2007). * (10) P. A. M. Dirac, Lectures on Quantum Mechanics (New York: Belfer Graduate School of Science Monographs Series, 1964). * (11) M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton: Princeton University Press, 1992). * (12) L. Smolin, “Covariant quantization of membrane dynamics,” Phys. Rev. D 57 (1998) 6216 [arXiv:hep-th/9710191]. * (13) P. Horava, “Membranes at Quantum Criticality”, arXiv:0812.4287 [hep-th]. * (14) T. Thiemann, “The LQG string: Loop quantum gravity quantization of string theory. I: Flat target space,” Class. Quant. Grav. 23 (2006) 1923 [arXiv:hep-th/0401172]. * (15) P. Malkiewicz and W. Piechocki, “Quantum membrane in a time dependent orbifold”, in preparation. * (16) R. Banerjee, P. Mukherjee and A. Saha, “Interpolating action for strings and membranes: A study of symmetries in the constrained Hamiltonian approach”, Phys. Rev. D 70 (2004) 026006 [arXiv:hep-th/0403065]. * (17) R. Banerjee, P. Mukherjee and A. Saha, “Genesis of ADM decomposition: A brane-gravity correspondence”, Phys. Rev. D 72 (2005) 066015 [arXiv:hep-th/0501030].	arxiv-papers	2009-03-04T14:37:02	2024-09-04T02:49:00.992223	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Przemyslaw Malkiewicz and Wlodzimierz Piechocki", "submitter": "Wlodzimierz Piechocki", "url": "https://arxiv.org/abs/0903.0774" }
0903.0843	# Algorithms for Weighted Boolean Optimization Vasco Manquinho1, Joao Marques-Silva2, Jordi Planes3 1 IST/UTL - INESC-ID, vasco.manquinho@inesc-id.pt 2 University College Dublin, jpms@ucd.ie 3 Universitat de Lleida, jplanes@diei.udl.cat ###### Abstract The Pseudo-Boolean Optimization (PBO) and Maximum Satisfiability (MaxSAT) problems are natural optimization extensions of Boolean Satisfiability (SAT). In the recent past, different algorithms have been proposed for PBO and for MaxSAT, despite the existence of straightforward mappings from PBO to MaxSAT and vice-versa. This papers proposes Weighted Boolean Optimization (WBO), a new unified framework that aggregates and extends PBO and MaxSAT. In addition, the paper proposes a new unsatisfiability-based algorithm for WBO, based on recent unsatisfiability-based algorithms for MaxSAT. Besides standard MaxSAT, the new algorithm can also be used to solve weighted MaxSAT and PBO, handling pseudo-Boolean constraints either natively or by translation to clausal form. Experimental results illustrate that unsatisfiability-based algorithms for MaxSAT can be orders of magnitude more efficient than existing dedicated algorithms. Finally, the paper illustrates how other algorithms for either PBO or MaxSAT can be extended to WBO. ## 1 Introduction In the area of Boolean-based decision and optimization procedures, natural extensions of Boolean Satisfiability (SAT) include Maximum Satisfiability (MaxSAT) [10] and Pseudo-Boolean Optimization (PBO) [6]. Algorithms for MaxSAT and PBO have been the subject of significant improvements over the last few years. This in turn, motivated the use of both PBO and, more recently, of MaxSAT in a number of practical applications. Interestingly, albeit there are simple translations from any MaxSAT variant to PBO and vice-versa (by encoding to CNF) [1, 18], algorithms for MaxSAT and PBO have evolved separately, and often use fairly different algorithmic organizations. Nevertheless, there exists work that acknowledges this relationship and algorithms that can solve instances of MaxSAT and of PBO have already been proposed [1, 18]. Recent work has provided more alternatives for solving either MaxSAT or PBO, by using SAT solvers and the identification of unsatisfiable sub-formulas [16, 27]. However, the proposed algorithms were restricted to the plain and partial variants of MaxSAT and to a restricted form of Binate Covering for PBO. This paper extends this recent work in a number of directions. First, the paper proposes a simple algorithm for (Partial) Weighted MaxSAT, using unsatisfiable sub-formula identification. Second, the paper generalizes MaxSAT and PBO by introducing Weighted Boolean Optimization (WBO), a new modeling framework for solving linear optimization problems over Boolean domains. Third, the paper shows how to extend the unsatisfiability-based algorithm for MaxSAT for solving WBO problems. Finally, the paper suggests how other algorithms can be used for solving WBO. Besides the proposed contributions, the paper also provides empirical evidence that unsatisfiability-based MaxSAT and WBO solvers can outperform state-of-the-art solvers on problem instances from practical problems. The paper is organized as follows. Section 2 provides a brief overview of the topics addressed in the paper, namely MaxSAT, PBO, translations from MaxSAT to PBO and vice-versa, and unsatisfiability-based algorithms for MaxSAT. Section 3 details an algorithm for (Partial) Weighted MaxSAT based on unsatisfiable sub-formula identification. Next, Section 4 introduces Weighted Boolean Optimization (WBO), and shows how to extend the algorithm of Section 3 to WBO. Section 5 analyzes the experimental results, obtained on representative classes of problem instances. Section 6 overviews related work, and Section 7 concludes the paper. ## 2 Preliminaries This section briefly introduces the Maximum Satisfiability (MaxSAT) problem and its variants, as well as the Pseudo-Boolean Optimization (PBO) problem. The main approaches used by state-of-the-art solvers are summarized. Moreover, translation procedures from MaxSAT to PBO and vice-versa are overviewed. Finally, unsatisfiability-based MaxSAT algorithms are surveyed, all of which the paper uses in later sections. ### 2.1 Maximum Satisfiability Given a CNF formula $\varphi$, the Maximum Satisfiability (MaxSAT) problem can be defined as finding an assignment that maximizes the number of satisfied clauses (which implies that the assignment minimizes the number of unsatisfied clauses). Besides the classical MaxSAT problem, there are also three well- known variants of MaxSAT: weighted MaxSAT, partial MaxSAT and weighted partial MaxSAT. All these formulations have been used in a wide range of practical applications, namely scheduling, FPGA routing [34], design automation [31], among others. A partial CNF formula is described as the conjunction of two CNF formulas $\varphi_{h}$ and $\varphi_{s}$, where $\varphi_{h}$ represents the _hard_ clauses and $\varphi_{s}$ represents the _soft_ clauses. The _partial_ MaxSAT problem consists in finding an assignment to the problem variables such that all hard clauses ($\varphi_{h}$) are satisfied, and the number of satisfied soft clauses ($\varphi_{s}$) is maximixed. A weighted CNF formula is a set of weighted clauses. A weighted clause is a pair $(\omega,c)$, where $\omega$ is a classical clause and $c$ is a natural number corresponding to the cost of unsatisfying $\omega$. Given a weighted CNF formula, the _weighted_ MaxSAT problem consists in finding an assignment to the problem variables such that the total weight of the satified clauses is maximized (which implies that the total weight of the unsatisfied clauses is minimized). A weighted partial CNF formula is the conjunction of a weighted CNF formula (soft clauses) and a classical CNF formula (hard clauses). The _weighted partial_ MaxSAT problem consists in finding an assignment to the variables such that all hard clauses are satisfied and the total weight of satisfied soft clauses is maximized. Observe that, for both partial MaxSAT and weighted partial MaxSAT, hard clauses can also be represented as weighted clauses: one can consider that the weight is greater than the sum of the weights of the soft clauses. Starting with the seminal work of Borchers and Furman [10], there has been an increasing interest in developing efficient MaxSAT solvers. Following such work, two branch and bound based solvers have been developed: (i) MaxSatz [20], the first solver to implement a unit propagation based lower bound and a failed literal based lower bound, both closely linked with a set of inference rules; (This solver has been extended into several solvers: IncMaxSatz [22], WMaxSatz [3], WMaxSatz_icss [13].) (ii) MiniMaxSAT [18], a solver created on top of MiniSAT with MaxSAT resolution [9] applied over an unsatisfiable sub- formula detected by the unit propagation based lower bound. A different approach has been the conversion of MaxSAT into a different formalism. The most notable works using this approach have been: Toolbar [19], a weighted CSP solver which converts MaxSAT instances into a weighted constraint network; SAT4J MAXSAT [7], a solver which iteratively converts a MaxSAT instance into a PBO instance; Clone [29] and sr(w) [30], solvers which convert a MaxSAT instance into a deterministic decomposable negation normal form (d-DNNF) instance; and MSUnCore [27], a solver which solves MaxSAT using the unsatisfiable cores detected by iteratively encoding the problem instance into SAT. In the Max-SAT Evaluations [4], this latter approach has been shown to be effective for industrial problems. ### 2.2 Pseudo-Boolean Optimization The Pseudo-Boolean Optimization (PBO) problem is another extension of SAT where constraints can be any linear inequality with integer coefficients (also known as pseudo-Boolean constraints) defined over the set of problem variables. The objective in PBO is to find an assignment to problem variables such that all problem constraints are satisfied and the value of a linear objective function is optimized. Any pseudo-Boolean formulation can be easily translated into a normal form [6] such that all integer coefficients are non- negative. $\begin{array}[]{lll}\mbox{minimize }&\sum\limits_{j\in N}c_{j}\cdot x_{j}\\\ \mbox{subject to }&\sum\limits_{j\in N}a_{ij}l_{j}\geq b_{i},\\\ \mbox{ }&l_{j}\in\\{x_{j},\bar{x}_{j}\\},x_{j}\in\\{0,1\\},a_{ij},b_{i},c_{j}\in\mathbf{N}_{0}^{+}\\\ \end{array}$ (1) Almost all algorithms to solve PBO rely on the generalization of the most effective techniques already used in SAT solvers, namely Boolean Constraint Propagation, conflict-based learning and conflict-directed backtracking [24, 11]. Nevertheless, there are several approaches to solve PBO formulations. The most common using SAT solvers is to make a linear search on the value of the objective function. The idea is to generalize SAT algorithms to deal natively with pseudo-Boolean constraints [6] and whenever a solution for the problem constraints is found, a new constraint is added such that only solutions with a lower value for the objective function can be accepted. The algorithm finishes when the solver cannot improve on the last solution found, therefore proving its optimality. Another common approach is branch and bound, where lower bounding procedures to estimate the value of the objective function are used. Several lower bounding procedures have been proposed, namely Maximum Independent Set of constraints [12], Linear Programming Relaxation [21, 23], among others [23]. There are also algorithms that encode pseudo-Boolean constraints into propositional clauses [33, 5, 15] and solve the problem by subsequently using a SAT solver. This approach has been proved to be very effective for several problem sets, in particular when the clause encoding is not much larger than the original pseudo-Boolean formulation. ### 2.3 Translations between MaxSAT and PBO Although MaxSAT and PBO are different formalisms, it is possible to encode any MaxSAT instance into a PBO instance and vice-versa [2, 1, 17]. This section focus solely on weighted partial MaxSAT, since the encodings of the other variants easily follow. The encoding of hard clauses from weighted partial MaxSAT to PBO is straightforward, since propositional clauses are a particular case of pseudo- Boolean constraints. However, for each soft clause $\omega_{i}=(l_{1}\vee l_{2}\vee\ldots\vee l_{k})$ with weight $c_{i}$, the encoding to PBO involves the use of an additional selection variable $s_{i}$, such that the corresponding constraint in PBO to $\omega_{i}$ would be $s_{i}+\sum_{j=1}^{k}l_{j}\geq 1$. This ensures that variable $s_{i}$ is assigned to true whenever $\omega_{i}$ is not satisfied. The objective function of the corresponding PBO instance is to minimize the weighted sum of the selection variables. For each selection variable $s_{i}$ in the objective function, its coefficient is the weight $c_{i}$ of the corresponding soft clause $\omega_{i}$. Example. Consider the following weighted partial MaxSAT instance. $\begin{array}[]{rrl}\varphi_{h}&=\\{&(x_{1}\vee x_{2}\vee\bar{x}_{3}),(\bar{x}_{2}\vee x_{3}),(\bar{x}_{1}\vee x_{3})\\}\\\ \varphi_{s}&=\\{&(\bar{x}_{3},6),(x_{1}\vee x_{2},3),(x_{1}\vee x_{3},2)\\}\\\ \end{array}$ (2) According to the described encoding, the corresponding PBO instance would be: $\begin{array}[]{rrl}\mbox{minimize }&6s_{1}+3s_{2}+2s_{3}\\\ \mbox{subject to }&x_{1}+x_{2}+\bar{x}_{3}\geq 1\\\ &\bar{x}_{2}+x_{3}\geq 1\\\ &\bar{x}_{1}+x_{3}\geq 1\\\ &s_{1}+\bar{x}_{3}\geq 1\\\ &s_{2}+x_{1}+x_{2}\geq 1\\\ &s_{3}+x_{1}+x_{3}\geq 1\\\ \end{array}$ (3) $\Box$ The encoding of PBO constraints into MaxSAT can be done using any of the proposed encodings from pseudo-Boolean constraints to clauses [33, 5, 15]. Hence, for each pseudo-Boolean constraint there will be a set of hard clauses encoding it in the respective MaxSAT instance. The number of clauses and additional variables, depends on the translation process used. The encoding is trivial when the original constraint in the PBO instance is already a clause. The objective function of PBO instances can be encoded into MaxSAT with the use of weighted soft clauses. The idea is that for each variable $x_{j}$ with coefficient $c_{j}$ in the objective function, a corresponding soft clause $(\bar{x}_{j})$ with weight $c_{j}$ is added to the MaxSAT instance. Therefore, the solution of the MaxSAT formulation minimizes the weighted sum of problem variables, as required in the PBO instance. Example. For illustration purposes, consider the following PBO instance: $\begin{array}[]{rrl}\mbox{minimize }&4x_{1}+2x_{2}+x_{3}\\\ \mbox{subject to }&2x_{1}+3x_{2}+5x_{3}\geq 5\\\ &\bar{x}_{1}+\bar{x}_{2}\geq 1\\\ &x_{1}+x_{2}+x_{3}\geq 2\\\ \end{array}$ (4) Note that the first and third constraint must be encoded into CNF, but the second constraint is already a clause and so it can be represented directly as a hard clause. The corresponding MaxSAT instance would be: $\begin{array}[]{rrl}\varphi_{h}&=\\{&\mbox{CNF}(2x_{1}+3x_{2}+5x_{3}\geq 5),(\bar{x}_{1}\vee\bar{x}_{2}),\mbox{CNF}(x_{1}+x_{2}+x_{3}\geq 2)\\}\\\ \varphi_{s}&=\\{&(\bar{x}_{1},4),(\bar{x}_{2},2),(\bar{x}_{3},1)\\}\\\ \end{array}$ (5) $\Box$ ### 2.4 Unsatisfiability-Based MaxSAT Recent work proposed the use of SAT solvers to solve (partial) MaxSAT, by iteratively identifying and relaxing unsatisfiable sub-formulas [16, 27, 26, 25]. In this paper we refer to these algorithms generically as MSU (Maximum Satisfiability with Unsatisfiability) algorithms. The original algorithm of Fu&Malik (referred to as MSU1.0) iteratively identifies unsatisfiable sub-formulas. For each computed unsatisfiable sub- formula, all original (soft) clauses are relaxed with fresh relaxation variables. Moreover, a new Equals1 (or AtMost1) constraint relates the relaxation variables of each iteration, i.e. exactly 1 of these relaxation variables can be assigned value 1. The MSU1.0 algorithm can use more than one relaxation variables for each clause. In the original algorithm [16], a quadratic pairwise encoding of the Equals1 constraint was used. Finally, observe that the Equals1 constraint in line 1 of Algorithm 1 can be replaced by an AtMost1 constraint, without affecting the correctness of the algorithm. More recently, several new MSU algorithms were proposed [26, 27]. The differences of the MSU algorithms include the number of cardinality constraints used, the encoding of cardinality constraints (of which the AtMost1 and Equals1 constraints are a special case), the number of relaxation variables considered for each clause, and how the MSU algorithm proceeds. Extensive experimentation (from [25] but also from the MaxSAT Evaluation [4]) suggests that an optimized variation of Fu&Malik’s algorithm[25] is currently the best performing MSU algorithm. * $\textnormal{msu1}(\varphi)$ 1$\varphi_{W}\leftarrow\varphi$ $\rhd$ Working formula, initially set to $\varphi$ 2while true 3 do$(\textnormal{st},\varphi_{C})\leftarrow\textsc{SAT}(\varphi_{W})$ 4 $\rhd$ $\varphi_{C}$ is an unsatisfiable sub-formula if $\varphi_{W}$ is unsat 5 if $\textnormal{st}=\textbf{UNSAT}$ 6 then $V_{R}\leftarrow\emptyset$ 7 for each $\omega\in\varphi_{C}$ 8 doif not hard($\omega$) 9 then $r$ is a new relaxation variable 10 $\omega_{R}\leftarrow\omega\cup\\{r\\}$ $\rhd$ $\omega_{R}$ is tagged non-auxiliary 11 $\varphi_{W}\leftarrow\varphi_{W}-\\{\omega\\}\cup\\{\omega_{R}\\}$ 12 $V_{R}\leftarrow V_{R}\cup\\{r\\}$ 13 $\varphi_{R}\leftarrow\textnormal{CNF}(\sum_{r\in V_{R}}r=1)$ $\rhd$ Equals1 constraint 14 Set all clauses in $\varphi_{R}$ as hard clauses 15 $\varphi_{W}\leftarrow\varphi_{W}\cup\varphi_{R}$ $\rhd$ Clauses in $\varphi_{R}$ are declared hard 16 else $\rhd$ Solution to MaxSAT problem 17 $\nu\leftarrow\|\,\textnormal{blocking variables w/ value 1}\,\|$ 18 return $\|\varphi\|-\nu$ Algorithm 1 The (Partial) MaxSAT algorithm of Fu&Malik [16] ## 3 Unsatisfiability-Based Weighted MaxSAT This section describes extensions of MSU1.X, described in Algorithm 1, for solving (Partial) Weighted MaxSAT problems. One simple solution is to create $c_{j}$ replicas of clause $\omega_{j}$, where $c_{j}$ is the weight of clause $\omega_{j}$. The resulting extended CNF formula can then be solved by MSU1.X. The proof of Fu&Malik’s paper would also apply in this case, and so correctness follows. The operation of this solution for (Partial) Weighted MaxSAT justifies a few observations. Consider an unsatisfiable sub-formula $\varphi_{C}$ where the smallest weight is ${\mathit{min}}_{c}$. Each clause would be replaced by a number of replicas. Hence, this unsatisfiable sub- formula would be identified ${\mathit{min}}_{c}$ times. Clearly, this solution is unlikely to scale for clauses with very large weights. Hence, a more effective solution is needed, which is detailed below. * $\textnormal{wmsu1}(\varphi)$ 1$\varphi_{W}\leftarrow\varphi$ $\rhd$ Working formula, initially set to $\varphi$ 2$\mathit{cost}_{\mathit{lb}}\leftarrow 0$ 3while true 4 do$(\textnormal{st},\varphi_{C})\leftarrow\textsc{SAT}(\varphi_{W})$ 5 $\rhd$ $\varphi_{C}$ is an unsatisfiable sub-formula if $\varphi_{W}$ is unsat 6 if $\textnormal{st}=\textbf{UNSAT}$ 7 then $\mathit{min}_{c}\leftarrow\infty$ 8 for each $\omega\in\varphi_{C}$ 9 doif not hard($\omega$) and $\mathit{cost}(\omega)<\mathit{min}_{c}$ 10 then $\mathit{min}_{c}\leftarrow\mathit{cost}(\omega)$ 11 $\mathit{cost}_{\mathit{lb}}\leftarrow\mathit{cost}_{\mathit{lb}}+\mathit{min}_{c}$ 12 $V_{R}\leftarrow\emptyset$ 13 for each $\omega\in\varphi_{C}$ 14 doif not hard($\omega$) 15 then $r$ is a new relaxation variable 16 $V_{R}\leftarrow V_{R}\cup\\{r\\}$ 17 $\omega_{R}\leftarrow\omega\cup\\{r\\}$ $\rhd$ $\omega_{R}$ is tagged non-auxiliary 18 $\mathit{cost}(\omega_{R})\leftarrow\mathit{min}_{c}$ 19 if $\mathit{cost}(\omega)>\mathit{min}_{c}$ 20 then $\varphi_{W}\leftarrow\varphi_{W}\cup\\{\omega_{R}\\}$ 21 $\mathit{cost}(\omega)\leftarrow\mathit{cost}(\omega)-\mathit{min}_{c}$ 22 else $\varphi_{W}\leftarrow\varphi_{W}-\\{\omega\\}\cup\\{\omega_{R}\\}$ 23 $\varphi_{R}\leftarrow\textnormal{CNF}(\sum_{r\in V_{R}}r=1)$ $\rhd$ Equals1 constraint 24 Set all clauses in $\varphi_{R}$ as hard clauses 25 $\varphi_{W}\leftarrow\varphi_{W}\cup\varphi_{R}$ $\rhd$ Clauses in $\varphi_{R}$ are declared hard 26 else $\rhd$ Solution to Weighted MaxSAT problem 27 return $\mathit{cost}_{\mathit{lb}}$ Algorithm 2 Unsatisfiability-based (Partial) Weighted MaxSAT algorithm An alternative solution is to split a clause only when the clause is included in an unsatisfiable sub-formula. The way the clause is split depends on its weight. An algorithm implementing this solution is shown in Algorithm 2. For each unsatisfiable sub-formula, the smallest weight $\mathit{min}_{c}$ of the clauses in the sub-formula is computed. This smallest weight is then used to update a lower bound on minimum cost of unsatisfiable clauses. Clauses in the unsatisfiable sub-formula are relaxed. However, if the weight of a clause is larger than $\mathit{min}_{c}$, then the clause is split: a new relaxed clause with weight $\mathit{min}_{c}$ is created, and the weight of the original clause is decreased by $\mathit{min}_{c}$. Example. Consider the partial MaxSAT instance in (2). Assume that the unsatisfiable sub-formula detected in line 4 of Algorithm 2 is: $\displaystyle\varphi_{C}$ $\displaystyle=$ $\displaystyle\\{\,(\bar{x}_{2}\vee x_{3}),(\bar{x}_{1}\vee x_{3}),\quad(\bar{x}_{3},6),(x_{1}\vee x_{2},3)\,\\}.$ (6) Then, the smallest weight $min_{c}$ is 3, and the new formula becomes $\varphi_{W}=\varphi_{h}\cup\varphi_{s}$, where $\begin{array}[]{rll}\varphi_{h}&=\\{&(x_{1}\vee x_{2}\vee\bar{x}_{3}),(\bar{x}_{2}\vee x_{3}),(\bar{x}_{1}\vee x_{3}),\mbox{CNF}(s_{1}+s_{2}=1)\,\\}\\\ \varphi_{s}&=\\{&(\bar{x}_{3},3),(x_{1}\vee x_{3},2),(s_{1}\vee\bar{x}_{3},3),(s_{2}\vee x_{1}\vee x_{2},3)\,\\}.\end{array}$ (7) $\Box$ Observe that the new algorithm can be viewed as a direct optimization of the naive algorithm outlined earlier. The main difference is that each iteration of the algorithm collapses $\mathit{min}_{c}$ iterations of the naive algorithm. For clauses with large weights the difference can be significant. Theorem. [Correctness of WMSU1] The value returned by Algorithm 2 is minimum cost of non-satisfied clauses in $\varphi$. $\Box$ Proof. The previous discussion and the proof in [16]. $\Box$ ## 4 Weighted Boolean Optimization This section introduces Weighted Boolean Optimization (WBO), a new framework for modeling with hard and soft pseudo-Boolean constraints, that extends both MaxSAT and its variants and PBO. Furthermore, a new algorithm based on identifying unsatisfiable sub-formulas is also proposed for solving WBO. An Weighted Boolean Optimization (WBO) formula $\varphi$ is composed of two sets of pseudo-Boolean constraints, $\varphi_{s}$ and $\varphi_{h}$, where $\varphi_{s}$ contains the soft constraints and $\varphi_{h}$ contains the hard constraints. For each soft constraint $\omega_{i}\in\varphi_{s}$ there is an associated integer weight $c_{i}>0$. The WBO problem consists in finding an assignment to the problem variables such that all hard constraints are satisfied and the total weight of the unsatisfied soft constraints is minimized (i.e. the total weight of satisfied soft constraints is maximized). It should be noted that WBO represents a generalization of weighted partial MaxSAT by introducing the use of pseudo-Boolean constraints instead of just using propositional clauses. Hence, more compact formulations can be obtained with WBO than with MaxSAT. Moreover, PBO formulations can also be linearly encoded into WBO. Constraints in PBO can be directly encoded as hard constraints in WBO and the objective function can also be encoded as described in section 2.3. Therefore, WBO is a generalization of MaxSAT and its variants, as well as of PBO, allowing a unified modeling framework to integrate both of these Boolean optimization problems. ### 4.1 Unsatisfiability-Based WBO This section describes how Algorithm 2 (introduced in Section 3) for weighted partial MaxSAT can be modified for solving WBO formulas. First of all, in a WBO formula, constraints are not restricted to be propositional clauses. Both soft and hard constraints can be pseudo-Boolean constraints. Hence, $\varphi$ is a pseudo-Boolean formula, instead of a CNF formula. Moreover, the use of a SAT solver in line 6 is replaced with a pseudo-Boolean solver extended with the ability to generate an unsatisfiable sub-formula from the original pseudo- Boolean formula. Next, if the formula is unsatisfiable, the weight associated with the unsatisfiable sub-formula is computed in the same way (lines 9-13) and the soft constraints in the core must also be relaxed using new relaxation variables (lines 15-24). Consider that $\omega=\sum a_{j}l_{j}\geq b$ denotes the pseudo-Boolean constraint to be relaxed using variable $r$. The resulting relaxed constraint in line 19 will be $\omega_{R}=b\cdot r+\sum a_{j}l_{j}\geq b$. Finally, the constraint on the new relaxation variables in line 25 does not need to be encoded into CNF. The pseudo-Boolean constraint $\sum_{r\in V_{R}}r=1$ can be directly added to $\varphi_{W}$, resulting in a more compact formulation, in particular if the number of soft constraints in the core is large. In some cases, for an unsatisfiable sub-formula with $k$ soft constraints, it is possible to use less than $k$ additional variables. Consider the following soft constraints $\omega_{1}=\sum_{l_{j}\in L_{1}}a_{1j}l_{j}\geq b_{1}$ and $\omega_{2}=\sum_{l_{j}\in L_{2}}a_{2j}l_{j}\geq b_{2}$ in a given unsatisfiable sub-formula, where $L_{1}$ and $L_{2}$ denote respectively the set of literals in constraints $\omega_{1}$ and $\omega_{2}$. Additionally, let $x_{k}\in L_{1}$, $\bar{x}_{k}\in L_{2}$, $a_{1k}\geq b_{1}$ and $a_{2k}\geq b_{2}$, i.e. assigning $x_{k}$ to true satisfies $\omega_{1}$ and assigning $x_{k}$ to false satisfies $\omega_{2}$.111This is a generalization to pseudo-Boolean constraints. Note that if the WBO instance corresponds to a MaxSAT instance, this is very common to occur, since $\omega_{1}$ and $\omega_{2}$ are clauses. In this case, these constraints can share the same relaxing variable. This is due to the fact that it is impossible for both $\omega_{1}$ and $\omega_{2}$ to be unsatisfied by the same assignment, since either $x_{k}$ satisfies $\omega_{1}$ or $\bar{x}_{k}$ satisfies $\omega_{2}$. Therefore, by using the same relaxing variable on both constraints, it is maintained the restriction that at most one soft constraint in the core can be relaxed. Example. Suppose that the following set of soft constraints defines an unsatisfiable sub-formula in a WBO instance: $\begin{array}[]{rrl}\omega_{1}=&2x_{1}+3x_{2}+5x_{3}&\geq 5\\\ \omega_{2}=&\bar{x}_{1}+\bar{x}_{2}&\geq 1\\\ \omega_{3}=&x_{2}+\bar{x}_{3}&\geq 1\\\ \omega_{4}=&x_{1}+\bar{x}_{3}&\geq 1\\\ \end{array}$ (8) In this case, constraints $\omega_{1}$ and $\omega_{3}$ can share the same relaxation variable, since the assignment of a value to $x_{3}$ implies that either $\omega_{1}$ or $\omega_{3}$ is satisfied. The same occurs with $\omega_{2}$ and $\omega_{4}$, given that the assignment to $x_{1}$ either satisfies $\omega_{2}$ or $\omega_{4}$. Therefore, after the relaxation, the resulting formula can include just two relaxation variables, instead of four. The resulting formula would be: $\begin{array}[]{rl}5s_{1}+2x_{1}+3x_{2}+5x_{3}&\geq 5\\\ s_{2}+\bar{x}_{1}+\bar{x}_{2}&\geq 1\\\ s_{1}+x_{2}+\bar{x}_{3}&\geq 1\\\ s_{2}+x_{1}+\bar{x}_{3}&\geq 1\\\ s_{1}+s_{2}&\leq 1\\\ \end{array}$ (9) $\Box$ The application of this reduction rule of relaxing variables raises the problem of finding the smallest number of relaxation variables to be used. This problem can be mapped into finding a matching of maximum cardinality in an undirected graph. In such a graph, there is a vertex for each constraint in the unsatisfiable sub-formula, while edges connect vertexes corresponding to constraints that can share a relaxation variable. The problem of finding a matching of maximum cardinality in an undirected graph can be solved in polynomial time [14]. Nevertheless, our prototype implementation of WBO solver uses a greedy algorithmic approach. ### 4.2 Other Algorithms for WBO An alternative solution for solving WBO is to extend existing PBO algorithms. For example, soft pseudo-Boolean constraints can be represented in a PBO instance as relaxable constraints, and the overall cost function becomes the weighted sum of the relaxation variables of all soft pseudo-Boolean constraints of the original WBO formulation. This solution resembles the existing approach for solving MaxSAT with PBO [2, 1], and has the same potential drawbacks. One additional alternative solution is to generalize branch and bound weighted partial MaxSAT solvers to deal with soft and hard pseudo-Boolean constraints. However, note that these approaches focus on a search process that uses successive refinements on the upper bound of the WBO solution, while the algorithm proposed in section 4.1 works by refining lower bounds on the optimum solution value. ## 5 Results With the objective of evaluating the new (partial) weighted MaxSAT algorithm and the new WBO solver, a set of industrially-motivated problem instances was selected. The characteristics of the classes of instances considered are shown in Table 1. For each class of instances, the table provides the class name, the number of instances (#I), the type of MaxSAT variant, and the source for the class of instances. Table 1: Classes of problem instances Class \| #I \| MaxSAT Variant \| Source ---\|---\|---\|--- IND \| 110 \| Partial Weighted \| To Appear in MaxSAT Evaluation 2009 FIR \| 59 \| Partial \| Pseudo-Boolean Evaluation 2007 SYN \| 74 \| Partial \| Pseudo-Boolean Evaluation 2005 Moreover, a wide range of MaxSAT and PBO solvers were considered, all among the best performing in either the MaxSAT or the Pseudo-Boolean evaluations. The weighted MaxSAT solvers considered were WMaxSatz [3], MiniMaxSat [18], IncWMaxSatz [22], Clone [29], and SAT4J (MaxSAT) [7]. In addition, a new version of MSUnCore [26, 27, 25], integrating the weighted MaxSAT algorithm proposed in Section 3, was also evaluated. The PBO solvers considered were BSOLO [23], PBS [1], Pueblo [32], Minisat+[15], and SAT4J (PB) [7]. Finally, results for the new WBO solver, implementing the WBO organization described in Section 4 is also shown. All experiments were run on a cluster of Linux AMD Opteron 2GHz servers with 1GB of RAM. The CPU time limit was set to 1800 seconds, and the RAM limit was set to 1 GB. All algorithms were run on all problem instances considered. The original representations were used, in order to avoid introducing any bias towards any of the problem representations. Tables 2 and 3 summarize the number of instances aborted by each solver for each class of instances. As can be concluded, for practical problem instances, only a small number of MaxSAT solvers is effective. The results are somewhat different for the PBO solvers, where several can be competitive for different classes of instances. It should be noted that the IND benchmarks can be considered challenging for pseudo- Boolean solvers due to the large clause weights used. For class IND and for the MaxSAT solvers, the results are somewhat surprising. Some of the solvers perform extremely well, whereas the others cannot solve most of the problem instances. IncWMaxSatz, MSUnCore and WBO are capable of solving all problem instances, but other MaxSAT solvers abort the vast majority of the problem instances. One additional observation is the very good performance of IncWMaxSatz when compared to WMaxSatz. This clearly indicates that the lower bound computation used in IncWMaxSatz can be very effective, even for industrial problem instances. For the PBO solvers, given the set of benchmark instances considered, SAT4J (PB) and BSOLO come out as the best performing. Clearly, this conclusion is based on the class of instances considered, which nevertheless derive from practical applications. Moreover, SAT4J (PB) performs significantly better than SAT4J (MaxSAT). This may be the result of a less effective encoding internally to SAT4J. Table 2: Solved Instances for MaxSAT Solvers Class \| WMaxSatz \| MiniMaxSat \| IncWMaxSatz \| Clone \| SAT4J (MS) \| MSUncore ---\|---\|---\|---\|---\|---\|--- IND \| 11 \| 0 \| 110 \| 0 \| 10 \| 110 FIR \| 7 \| 14 \| 33 \| 5 \| 10 \| 45 SYN \| 22 \| 29 \| 19 \| 13 \| 21 \| 34 Total (Out of 243) \| 40 \| 43 \| 162 \| 18 \| 41 \| 189 Table 3: Solved Instances for PBO & WBO Solvers Class \| BSOLO \| PBS \| Pueblo \| Minisat+ \| SAT4J (PB) \| WBO ---\|---\|---\|---\|---\|---\|--- IND \| 17 \| 0 \| 0 \| 0 \| 60 \| 110 FIR \| 20 \| 11 \| 14 \| 22 \| 7 \| 39 SYN \| 51 \| 19 \| 30 \| 30 \| 22 \| 33 Total (Out of 243) \| 88 \| 30 \| 44 \| 52 \| 89 \| 172 Motivated by the overall results, the best MaxSAT, PBO and the WBO solver were analyzed in more detail. Given the experimental results, IncWMaxSatz, MSUnCore, and WBO were selected. Figure 1 shows the results for the selected solvers by increasing run times. 020040060080010001200140016001800050100150200250 CPU time instancesMSUnCoreWBOIncWMaxSatz Figure 1: Run times for IncWMaxSatz, MSUnCore, and WBO for all instances As can be concluded, the plot confirms the trends in the tables of results. MSUnCore is the best performing, followed by WBO and IncWMaxSatz. For smaller run times (instances from class IND), IncWMaxSatz can be more efficient than WBO. Moreover, these results indicate that, for the classes of instances considered, encoding cardinality constraints into CNF (as done in MSUnCore) may be a better solution than natively handling cardinality and pseudo-Boolean constraints (as done in WBO). It should be noted that all the instances considered can be encoded with cardinality constraints, for which existing polynomial encodings guarantee arc-consistency. This is not true for problem instances that use other pseudo-Boolean constraints, and for which encodings that ensure arc-consistency are exponential in the worst-case [15]. Finally, another source of difference in the experimental results is that whereas MSUnCore is built on top of PicoSAT [8], WBO is built on top of Minisat2. The different underlying SAT solvers may also contribute to explain some of the differences observed. ## 6 Related Work A brief account of MaxSAT and PBO solvers is provided in Section 2. The use of unsatisfiability for solving MaxSAT was first proposed in 2006 [16]. This work was later extended [26, 27, 25], to accommodate several alternative algorithms and a number of optimizations to the first algorithm. To the best of our knowledge, MSUnCore is the first algorithm for solving (Partial) Weighted MaxSAT with unsatisfiable sub-formula identification. Also, to the best of our knowledge, WBO represents a new modeling framework, and the associate algorithm is new. The use of optimization variants of decision procedures has also been proposed in the area of SMT [28], and a few SMT solvers now offer the ability for solving optimization problems. The approaches used for solving optimization problems in SMT are based on the use of relaxation variables, similarly to the PBO approach for solving MaxSAT [1]. ## 7 Conclusions and Future Work This paper proposes a new algorithm for (Partial) Weighted MaxSAT, based on unsatisfiable sub-formula identification. In addition, the paper introduces Weighted Boolean Optimization (WBO), that aggregates and generalizes PBO and MaxSAT. The paper then shows how unsatisfiability-based algorithms for (Partial) Weighted MaxSAT can be extended to WBO. Finally, the paper illustrates how to extend other algorithms for PBO and MaxSAT to solve WBO. Experimental results, obtained on a representative set of benchmark instances shows that the new algorithm for weighted MaxSAT can outperform other existing algorithms by orders of magnitude. The experimental results also provide a preliminary (albeit possibly biased) study on the performance differences between handling pseudo-Boolean constraints natively and encoding to CNF. Finally, the paper shows that a general algorithm for WBO can be as efficient as other dedicated algorithms. The integration of MaxSAT and PBO into a unique optimization extension of SAT increases the range of problems that can be solved. It also allows developing other general purpose algorithms, integrating the best techniques from both domains. Future research work will address adapting other algorithms for WBO. One concrete example is the use of PBO solvers. The other is extending the existing family of MSU algorithms for WBO. Acknowledgement. This work is partially supported by EU grant ICT/217069 and FCT grant PTDC/EIA/76572/2006. ## References * [1] F. Aloul, A. Ramani, I. Markov, and K. A. Sakallah. Generic ILP versus specialized 0-1 ILP: An update. In International Conference on Computer-Aided Design, pages 450–457, 2002. * [2] L. Amgoud, C. Cayrol, and D. L. Berre. Comparing arguments using preference ordering for argument-based reasoning. In International Conference on Tools with Artificial Intelligence, pages 400–403, 1996. * [3] J. Argelich, C. M. Li, and F. Manà. An improved exact solver for partial max-sat. In Proceedings of the International Conference on Nonconvex Programming: Local and Global Approaches (NCP-2007), pages 230–231, 2007. * [4] J. Argelich, C. M. Li, F. Manyà, and J. Planes. Third Max-SAT evaluation. www.maxsat.udl.cat/08/, 2008. * [5] O. Bailleux, Y. Boufkhad, and O. Roussel. A translation of pseudo Boolean constraints to SAT. Journal on Satisfiability, Boolean Modeling and Computation, 2:191–200, 2006. * [6] P. Barth. A Davis-Putnam Enumeration Algorithm for Linear Pseudo-Boolean Optimization. Technical Report MPI-I-95-2-003, Max Plank Institute for Computer Science, 1995. * [7] D. L. Berre. SAT4J library. www.sat4j.org. * [8] A. Biere. PicoSAT essentials. Journal on Satisfiability, Boolean Modeling and Computation, 2:75–97, 2008. * [9] M. L. Bonet, J. Levy, and F. Manyà. Resolution for Max-SAT. Artificial Intelligence, 171(8–9):606–618, 2007. * [10] B. Borchers and J. Furman. A two-phase exact algorithm for MAX-SAT and weighted MAX-S AT problems. Journal of Combinatorial Optimization, 2:299–306, 1999. * [11] D. Chai and A. Kuehlmann. A fast pseudo-Boolean constraint solver. In Design Automation Conference, pages 830–835, 2003. * [12] O. Coudert. On Solving Covering Problems. In Proceedings of the Design Automation Conference, pages 197–202, 1996. * [13] S. Darras, G. Dequen, L. Devendevill, and C. M. Li. On inconsistent clause-subsets for Max-SAT solving. In CP-07, volume 4741 of LNCS, pages 225–240. Springer, 2007\. * [14] J. Edmonds. Paths, trees and flowers. Canadian Journal of Mathematics, 17:449–467, 1965. * [15] N. Een and N. Sörensson. Translating pseudo-Boolean constraints into SAT. Journal on Satisfiability, Boolean Modeling and Computation, 2, 2006\. * [16] Z. Fu and S. Malik. On solving the partial MAX-SAT problem. In International Conference on Theory and Applications of Satisfiability Testing, pages 252–265, August 2006. * [17] F. Heras, J. Larrosa, and A. Oliveras. MiniMaxSat: a new weighted Max-SAT solver. In International Conference on Theory and Applications of Satisfiability Testing, pages 41–55, May 2007. * [18] F. Heras, J. Larrosa, and A. Oliveras. MiniMaxSAT: An efficient weighted Max-SAT solver. Journal of Artificial Intelligence Research, 31:1–32, 2008. * [19] J. Larrosa, F. Heras, and S. de Givry. A logical approach to efficient Max-SAT solving. Artificial Intelligence, 172(2-3):204–233, 2008. * [20] C. M. Li, F. Manyà, and J. Planes. New inference rules for Max-SAT. Journal of Artificial Intelligence Research, 30:321–359, 2007. * [21] S. Liao and S. Devadas. Solving Covering Problems Using LPR-Based Lower Bounds. In Proceedings of the Design Automation Conference, pages 117–120, 1997. * [22] H. Lin and K. Su. Exploiting inference rules to compute lower bounds for MAX-SAT solving. In IJCAI-07, pages 2334–2339, 2007. * [23] V. Manquinho and J. Marques-Silva. Effective lower bounding techniques for pseudo-boolean optimization. In Design, Automation and Test in Europe Conference, pages 660–665. ACM Press, 2005. * [24] V. M. Manquinho and J. Marques-Silva. Search pruning techniques in SAT-based branch-and-bound algorithms for the binate covering problem. IEEE Transactions on Computer-Aided Design, 21(5):505–516, 2002\. * [25] J. Marques-Silva and V. Manquinho. Towards more effective unsatisfiability-based maximum satisfiability algorithms. In International Conference on Theory and Applications of Satisfiability Testing, pages 225–230, March 2008. * [26] J. Marques-Silva and J. Planes. On using unsatisfiability for solving maximum satisfiability. Computing Research Repository, abs/0712.0097, December 2007. * [27] J. Marques-Silva and J. Planes. Algorithms for maximum satisfiability using unsatisfiable cores. In Design, Automation and Testing in Europe Conference, pages 408–413, March 2008. * [28] R. Nieuwenhuis and A. Oliveras. On SAT modulo theories and optimization problems. In International Conference on Theory and Applications of Satisfiability Testing, pages 156–169, 2006. * [29] K. Pipatsrisawat, A. Palyan, M. Chavira, A. Choi, and A. Darwiche. Solving weighted Max-SAT problems in a reduced search space: A performance analysis. Journal on Satisfiability Boolean Modeling and Computation (JSAT), 4:191–217, 2008. * [30] M. Ramírez and H. Geffner. Structural relaxations by variable renaming and their compilation for solving MinCostSAT. In C. Bessiere, editor, The 13th International Conference on Principles and Practice of Constraint Programming, volume 4741 of LNCS, pages 605–619. Springer, 2007. * [31] S. Safarpour, H. Mangassarian, A. Veneris, M. H. Liffiton, and K. A. Sakallah. Improved design debugging using maximum satisfiability. In Formal Methods in Computer-Aided Design, 2007. * [32] H. Sheini and K. A. Sakallah. Pueblo: A hybrid pseudo-Boolean SAT solver. Journal on Satisfiability, Boolean Modeling and Computation, 2:165–189, 2006. * [33] J. P. Warners. A linear-time transformation of linear inequalities into conjunctive normal form. Information Processing Letters, 68(2):63–69, 1998. * [34] H. Xu, R. A. Rutenbar, and K. A. Sakallah. sub-SAT: a formulation for relaxed boolean satisfiability with applications in routing. IEEE Trans. on CAD of Integrated Circuits and Systems, 22(6):814–820, 2003.	arxiv-papers	2009-03-04T20:21:56	2024-09-04T02:49:00.997469	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Vasco Manquinho and Joao Marques-Silva and Jordi Planes", "submitter": "Jordi Planes", "url": "https://arxiv.org/abs/0903.0843" }
0903.0915	# Centrality bin size dependence of multiplicity correlation in central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV Yu-Liang Yan1, Dai-Mei Zhou2, Bao-Guo Dong1,3, Xiao-Mei Li1, Hai-Liang Ma1, Ben-Hao Sa1,2,4111Corresponding author: sabh@ciae.ac.cn 1 China Institute of Atomic Energy, P.O. Box 275(18), Beijing 102413, China 2 Institute of Particle Physics, Huazhong Normal University, Wuhan 430079, China 3 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Collisions, Lanzhou 730000,China 4 CCAST (World Laboratory), P. O. Box 8730 Beijing 100080, China ###### Abstract We have studied the centrality bin size dependence of charged particle forward-backward multiplicity correlation strength in 5%, 0-5%, and 0-10% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV with a parton and hadron cascade model, PACIAE based on PYTHIA. The real (total), statistical, and NBD (Negative Binomial Distribution) correlation strengths are calculated by the real events, the mixed events, and fitting the charged particle multiplicity distribution to the NBD, respectively. It is turned out that the correlation strength increases with increasing centrality bin size monotonously. If the discrepancy between real (total) and statistical correlation strengths is identified as dynamical one, the dynamical correlation may just be a few percent of the total (real) correlation. ###### pacs: 24.10.Lx, 24.60.Ky, 25.75.Gz ## I INTRODUCTION The study of fluctuations and correlations has been suggested as a useful means for revealing the mechanism of particle production and Quark-Gluon- Plasma (QGP) formation in Relativistic Heavy Ion Collisions hwa2 ; naya . Correlations and fluctuations of the thermodynamic quantities and/or the produced particle distributions may be significantly altered when the system undergoes phase transition from hadronic matter to quark-gluon matter because the degrees of freedom in two matters is very different. The experimental study of fluctuations and correlations becomes a hot topic in relativistic heavy ion collisions with the availability of high multiplicity event-by-event measurements at the CERN-SPS and BNL-RHIC experiments. An abundant experimental data have been reported star2 ; phen2 ; phob where a lot of new physics arise and are urgent to be studied. A lot of theoretical investigations have been reported as well paja ; hwa1 ; konc ; brog ; fu ; yan ; bzda . Recently STAR collaboration have measured the charged particle forward- backward multiplicity correlation strength $b$ in Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV star3 ; star4 . The outstanding features of STAR data are: * • In most central collisions, the correlation strength $b$ is approximately flat across a wide range in $\Delta\eta$ which is the distance between the centers of forward and backward (pseudo)rapidity bins. * • This trend disappears slowly with decreasing centrality and approaches a exponential function of $\Delta\eta$ at the peripheral collisions. That has stimulated a lot of theoretical interests konc ; brog ; fu ; bzda . In Ref. fu , a statistical model was proposed to calculate the charged particle forward-backward multiplicity correlation strength $b$ in 0-10% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV. One outstanding feature of STAR data, the $b$ as a function of $\Delta\eta$ is approximately flat, was well reproduced. The calculated value of $b\approx 0.44$ was compared with STAR data of $b\approx 0.60$ star3 ; star4 . However, in this statistical model fu the Negative Binomial Distribution (NBD) is assumed for the charged multiplicity distribution and the NBD parameters of $\mu$ and $k$ (see later) are extracted from fit in with PHENIX charged particle multiplicity distribution phen3 . It is turned out in Ref. yan that the experimental $\eta$ and $p_{T}$ acceptances have large influences on the correlation strength $b$. The STAR experimental acceptances are quite different from PHENIX, thus the inconsistency, using PHENIX multiplicity data to explain STAR correlation data, involved in fu have to be studies further. Meanwhile, what is the discrepancy between $b\approx 0.60$ (STAR datum) and $b\approx 0.44$ (NBD) also needs to be answered. In this paper we use a parton and hadron cascade model PACIAE sa , to investigate the centrality bin size dependence of charged particle multiplicity correlation in 5, 0-5, and 0-10% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV. Following Ref. yan we generate the real events (6000) by the PACIAE model, construct the mixed events according to real events one by one, and extract the NBD parameters ($\mu$ and $k$) from fitting the real events charged particle multiplicity distribution to the NBD. Then the charged particle forward-backward multiplicity correlation strength $b$ is calculated for the real events (real correlation strength), the mixed events (statistical correlation strength), and the NBD (NBD correlation strength), respectively. They are all nearly flat across a widw range in $\Delta\eta$. Their magnitude in 0-10% most central Au+Au collisions are about 0.63, 0.59, and 0.52, respectively. So the corresponding STAR data is well reproduced. It is turned out that the real (total), statistical, and NBD correlation strengths increase with increasing centrality bin size monotonously. If the discrepancy between real (total) and statistical correlation strengths is identified as dynamical one, then the dynamical correlation strength may just be a few percent of the total (real) correlation strength. ## II PACIAE MODEL The parton and hadron cascade model, PACIAE sa , is based on PYTHIA soj2 which is a model for hadron-hadron ($hh$) collisions. The PACIAE model is composed of four stages: parton initialization, parton evolution (rescattering), hadronization, and hadron evolution (rescattering). ### II.1 Parton initialization In the PACIAE model a nucleon-nucleon ($NN$) collision is described with PYTHIA model, where a $NN$ ($hh$) collision is decomposed into the parton- parton collisions. The hard parton-parton collision is described by the lowest-leading-order (LO) pQCD parton-parton cross section comb with modification of parton distribution function in the nucleon. And the soft parton-parton interaction is considered empirically. The semihard, between hard and soft, QCD $2\rightarrow 2$ processes are also involved in PYTHIA (PACIAE) model. Because of the initial- and final-state QCD radiation added to the above processes, the PYTHIA (PACIAE) model generates a multijet event for a $NN$ ($hh$) collision. That is followed, in the PYTHIA model, by the string- based fragmentation scheme (Lund model and/or Independent Fragmentation model), thus a hadronic state is reached for a $NN$ ($hh$) collision. However, in the PACIAE model above string fragmentation is switched off temporarily, so the result is a multijet event (composed of quark pairs, diquark pairs and gluons) instead of a hadronic state. If the diquarks (anti-diquarks) are split forcibly into quarks (anti-quarks) randomly, the consequence of a $NN$ ($hh$) collision is its initial partonic state composed of quarks, anti-quarks, and gluons. A nucleus-nucleus collision, in the PACIAE model, is decomposed into the nucleon-nucleon collisions based on the collision geometry. A nucleon in the colliding nucleus is randomly distributed in the spatial coordinate space according to the Woods-Saxon distribution ($r$) and the 4$\pi$ uniform distribution ($\theta$ and $\phi$). The beam momentum is given to $p_{z}$ and $p_{x}=p_{y}=0$ is assumed for each nucleon in the colliding nucleus. A closest approaching distance of two assumed straight line trajectories is calculated for each $NN$ pair. If this distance is less than or equal to $\displaystyle{\sqrt{\sigma_{\rm{tot}}/\pi}}$, then it is considered as a collision pair. Here $\sigma_{\rm{tot}}$ refers to the total cross section of $NN$ collision assumed to be 40 mb. The corresponding collision time of this collision pair is then calculated. So the particle list and the $NN$ collision (time) list can be constructed. A $NN$ collision pair with smallest collision time is selected from the $NN$ collision (time ) list and performed by the method in former paragraph. After upgrading the particle list and collision (time) list we select and perform a new $NN$ collision pair again. Repeat these processes until the collision (time) list is empty we obtain a initial partonic state for a nucleus-nucleus collision. ### II.2 Parton evolution (rescattering) The next step, in the PACIAE model, is parton evolution (partonic rescattering). Here the $2\rightarrow 2$ LO-pQCD differential cross sections comb are employed. The differential cross section for a subprocess $ij\rightarrow kl$ reads $\frac{d\sigma_{ij\rightarrow kl}}{d\hat{t}}=K\frac{\pi\alpha_{s}^{2}}{\hat{s}}\sum_{ij\rightarrow kl},$ (1) where the $K$ factor is introduced for higher order corrections and the non- perturbative QCD correction as usual. Take the process $q_{1}q_{2}\rightarrow q_{1}q_{2}$ as an example, one has $\sum_{q_{1}q_{2}\rightarrow q_{1}q_{2}}=\frac{4}{9}\frac{\hat{s}^{2}+\hat{u}^{2}}{\hat{t}^{2}},$ (2) which can be regularized as $\sum_{q_{1}q_{2}\rightarrow q_{1}q_{2}}=\frac{4}{9}\frac{\hat{s}^{2}+\hat{u}^{2}}{(\hat{t}-m^{2})^{2}}$ (3) by introducing the parton colour screen mass, $m$=0.63 GeV. In above equation $\hat{s}$, $\hat{t}$, and $\hat{u}$ are the Mandelstam variables and $\alpha_{s}$= 0.47 stands for the running coupling constant. The total cross section of the parton collision $i+j$ is $\sigma_{ij}(\hat{s})=\sum_{k,l}\int_{-\hat{s}}^{0}d\hat{t}\frac{d\sigma_{ij\to kl}}{d\hat{t}}.$ (4) With these total and differential cross sections the parton evolution (rescattering) can be simulated by the Monte Carlo method until the parton- parton collision is ceased (partonic freeze-out). ### II.3 Hadronization In the PACIAE model the partons can be hadronized with the string-based fragmentation scheme or by the coalescence (recombination) models biro ; csiz ; grec ; hwo ; frie . The Lund string fragmentation regime, involved in the PYTHIA model, is adopted for hadronization in this paper, see soj2 for the details. Meanwhile, we have proposed a simulant coalescence (recombination) model which can be briefly explained as follows: 1. 1. The Field-Feynman parton generation mechanism ff1 is first applied to deexcite the energetic parton and thus to increase the parton multiplicity. This deexcitation of an energetic parton plays a similar role as string multiple fragmentation in the Lund model and1 . 2. 2. The gluons are forcibly split into $q\bar{q}$ pair randomly. 3. 3. In the program there is a hadron table composed of mesons and baryons. The pseudoscalar and vector mesons made of u, d, s, and c quarks, as well as $B^{+}$, $B^{0}$, $B^{0}$, and $\Upsilon$ are considered. The SU(4) multiplets of baryons made of u, d, s, and c quarks (except those with double c quarks) as well as $\Lambda^{0}_{b}$ are considered. 4. 4. Two partons can coalesce into a meson and three partons into a baryon (antibaryon) according to the flavor, momentum, and spatial coordinates of partons and the valence quark structure of hadron. 5. 5. When the coalescing partons can form either a pseudoscalar meson or a vector meson (e. g. $u\bar{d}$ can form either a $\pi^{+}$ or a $\rho^{+}$) a judgment of less discrepancy between the invariant mass of coalescing partons and the mass of coalesced hadron is invoked to select one from two mesons above. In the case of baryon, e. g. both $p$ and $\Delta^{+}$ are composed of $uud$, the same judgment is invoked to select one baryon from both of $\frac{1}{2}^{+}$ and $\frac{3}{2}^{+}$ baryons. 6. 6. Four momentum conservation is required. 7. 7. There is a phase space condition $\frac{16\pi^{2}}{9}\Delta r^{3}\Delta p^{3}=\frac{h^{3}}{d},$ (5) where $h^{3}/d$ is the volume occupied by a single hadron in the phase space, $d$=4 refers to the spin and parity degeneracies, $\Delta r$ and $\Delta p$ stand for the spatial and momentum distances between coalescing partons, respectively. ### II.4 Hadron evolution (rescattering) We obtain a configuration of hadrons in spatial and momentum coordinate spaces for a nucleus-nucleus collision after the hadronization. If one only considers the rescattering among $\pi,k,p,n,\rho(\omega),\Delta,\Lambda,\Sigma,\Xi,\Omega,J/\Psi$ and their antiparticles, the particle list is then constructed by the above hadrons. A closest approaching distance of two assumed straight line trajectories is calculated for each $hh$ pair. If this distance is less than or equal to $\displaystyle{\sqrt{\sigma_{\rm{tot}}^{hh}/\pi}}$ sa1 , then it is considered as a collision pair. Here $\sigma_{\rm{tot}}^{hh}$ refers to the total cross section of $hh$ collision. The corresponding collision time of this collision pair is then calculated. So the $hh$ collision (time) list can be constructed. A $hh$ collision pair with smallest collision time is selected from the collision (time) list and performed by the usual two-body collision method sa1 . After upgrading the particle list and collision (time) list we select and perform a new $hh$ collision pair again. Repeat these processes until the collision (time) list is empty (hadronic freeze-out). A isospin averaged parametrization formula is used for the $hh$ cross section koch ; bald . However, we also provide a option of constant total, elastic, and inelastic cross sections sa1 : $\sigma_{\rm{tot}}^{NN}=40$ mb, $\sigma_{\rm{tot}}^{\pi N}=25$ mb, $\sigma_{\rm{tot}}^{kN}=35$ mb, $\sigma_{\rm{tot}}^{\pi\pi}=10$ mb, and the assumed ratio of inelastic to total cross section equals 0.85. We also assume $\sigma_{pp}=\sigma_{pn}=\sigma_{nn}=\sigma{\Delta N}=\sigma{\Delta\Delta}.$ (6) The cross section of $\pi\bar{N}$ and $k\bar{N}$, for instance, is assumed to be equal to the cross section of $\pi N$ and $kN$, respectively. The momentum of scattered particles in a $hh$ elastic collision is simulated according to that the $hh$ differential cross section, $d\sigma_{\rm{tot}}^{hh}/dt$, is assumed to be an exponential function of $t$ which is squared momentum transfer sa1 . As it is impossible to include all inelastic channels, we consider only a part of them ($\approx$ 600) which have noticeable effects on the hadronic final state, and the rest is attributed to the elastic scattering. Take incident channel $\pi N$ as an example, if there are possible final channels of $\pi N\rightarrow\pi\Delta$, $\pi N\rightarrow\rho N$, and $\pi N\rightarrow k\Lambda$, their relative probabilities are then used to select one among above three channels. The momentum of scattered particles in a $hh$ inelastic collision is simulated according to the usual two-body kinematics sa1 ; pdg . ## III CALCULATION AND RESULT Following cape the charged particle forward-backward multiplicity correlation strength $b$ is defined as $\ b=\frac{\langle n_{f}n_{b}\rangle-\langle n_{f}\rangle\langle n_{b}\rangle}{\langle n_{f}^{2}\rangle-\langle n_{f}\rangle^{2}}=\frac{{\rm{cov}}(n_{f},n_{b})}{{\rm{var}}(n_{f})},$ (7) where $n_{f}$ and $n_{b}$ are, respectively, the number of charged particles in forward and backward pseudorapidity bins defined relatively and symmetrically to a given pseudorapidity $\eta$. $\langle n_{f}\rangle$ refers to the mean value of $n_{f}$ for instance. cov($n_{f}$,$n_{b}$) and var($n_{f}$) are the forward-backward multiplicity covariance and forward multiplicity variance, respectively. Table 1: Total charged particle multiplicity in three $\eta$ fiducial ranges in 0-6% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV. \| $N_{\rm{ch}}(\|\eta\|<4.7)$ \| $N_{\rm{ch}}(\|\eta\|<5.4)$ \| $N_{\rm{ch}}$(total) ---\|---\|---\|--- PHOBOSa \| 4810 $\pm$ 240 \| 4960 $\pm$ 250 \| 5060 $\pm$ 250 PACIAE \| 4819 \| 4983 \| 5100 a The experimental data are taken from phob2 . In the calculations the default values given in the PYTHIA model are adopted for all model parameters except the parameters $K$ and $b_{s}$ (in the Lund string fragmentation function). The $K$=3 is assumed and the $b_{s}$=6 is fixed by fitting the charged particle multiplicity to the corresponding PHOBOS data in 0-6% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV phob2 as shown in Tab. 1. Therefore in the generation of real events there is no free parameters. The mapping relation sa2 between the centrality definition in theory and experiment $b_{i}=\sqrt{g}b_{i}^{\rm{max}},\qquad b_{i}^{\rm{max}}=R_{A}+R_{B},$ (8) is employed. In the above equation $b_{i}$ (in fm) refers to the theoretical impact parameter and $g$ stands for the percentage of geometrical (total) cross section used in experiment to define the centrality. $R_{A}=1.12A^{1/3}+0.45$ fm, for instance, is the radius of nucleus $A$. Thus the 0-10, 0-6, 0-5, and 5% most central collisions, for instance, are mapped to $0<b_{i}<4.46$, $0<b_{i}<3.53$, $0<b_{i}<3.20$, and $b_{i}=3.20$ fm, respectively. In this paper we propose a mixed event method where the mixed events are generated according to real events one by one. We first assume the charged particle multiplicity $n$ in a mixed event is the same as one corresponding real event. However, $n$ particles of this mixed event are sampled randomly from the particle reservoir composed of all particles in all real events. Therefore, there is no dynamical relevance among the particles in a mixed event. So the correlation calculated by mixed events is reasonably to be identified as the statistical correlation yan . It is known that the statistical correlation can also be studied by the NBD method, because the charged particle multiplicity distribution in high energy heavy-ion collisions is close to NBD phen3 . For an integer $n$ the NBD reads $\ P(n;\mu,k)=\begin{pmatrix}n+k-1\\\ k-1\end{pmatrix}\frac{(\mu/k)^{n}}{(1+\mu/k)^{n+k}},$ (9) where $\mu\equiv\langle n\rangle$ is a parameter, $P(n;\mu,k)$ is normalized in $0\leq n\leq\infty$, and $k$ is another parameter responsible for the shape of the distribution. As proved in yan the correlation strength can be expressed as $\ b=\frac{\langle n_{f}\rangle}{\langle n_{f}\rangle+k},$ (10) where the parameter $k$ is fixed by fitting the charged particle multiplicity to the NBD usually. Figure 1: Charged particle pseudorapidity distribution in Au+Au collision at $\sqrt{s_{\rm{NN}}}$=200 GeV: (a) 0-6% most central collision and (b) 0-10, 0-5, and 5% most central collision. The experimental data are taken from phob2 . We compare the theoretical charged particle pseudorapidity distribution (open circles) in 0-6% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV with the corresponding PHOBOS data (solid squares) phob2 in Fig. 1 (a). One sees here that the PHOBOS data are well reproduced. In Fig. 1 (b), we compare the charged particle pseudorapidity distributions in 0-5% (open circles) and 5% (open triangles) most central Au+Au collisions with the 0-10% one (open squares). We see in Fig. 1 (b) that the pseudorapidity distribution in 5% most central collision is quite close to the 0-10% one, because the 5% centrality is nearly equal to the average centrality of 0-10% centrality bin. Figure 2: Charged particle forward-backward multiplicity correlation strength $b$ in 0-10, 0-5, and 5% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV. The experimental data are taken from star3 . In Fig. 2 we compare the calculated real (total) correlation strength $b$ (open squares) as a function of $\Delta\eta$ in 0-10% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV with the corresponding STAR data (solid squares) star3 . The STAR data feature of correlation strength $b$ is approximately flat across a wide range in $\Delta\eta$ are well reproduced. For comparison we also give the real (total) correlation strength in 0-5 and 5% most central collisions by open circles and triangles, respectively. One sees here that the real (total) correlation strength decreases with decreasing centrality bin size monotonously, because the charged particle multiplicity fluctuation decreases from 0-10 to 0-5 and to 5% monotonously, as one will see in Fig. 3. This first result of the correlation strength increases with increasing centrality bin size monotonously given in the transport model remains to be proved experimentally. Figure 3: Charged particle multiplicity distributions in 0-10 (open squares), 0-5 (open circles), and 5% (open triangles) most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV. The dotted, dashed, and solid lines are the corresponding NBD fits, respectively. The calculated charged particle multiplicity distributions in 0-10, 0-5, and 5% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV are given in Fig. 3, respectively, by the open squares, circles and triangles. The corresponding NBD fits are shown by dotted, dashed, and solid lines, respectively. One sees in Fig. 3 that the charged particle multiplicity fluctuation is increased and the NBD fit is worsened with increasing centrality bin size monotonously. Figure 4: The calculated charged particle total (real), statistical, and NBD correlation strengths in 0-10, 0-5 and, 5% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV. In Fig. 4 we compare the calculated charged particle real (solid symbols), statistical (open symbols), and NBD (lines) correlation strengths as a function of $\Delta\eta$ in 0-10, 0-5, and 5% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV. The solid squares, open squares, and dotted line are for 0-10% most central collisions, solid circles, open circles, and dashed line for 0-5%, and solid triangles, open triangles, and solid line for 5%, respectively. We see in Fig. 4 that the behavior of correlation strength increases with increasing centrality bin size monotonously is not only existed in the real correlation strength but also in the statistical and NBD ones. If the discrepancy between real (total) and statistical correlation strengths is identified as the dynamical correlation strength, one then sees in Fig. 4 that the dynamical correlation strength may just be a few percent of the total (real) correlation strength. The dynamical correlation strength in 0-10% most central collision is close to the one in 5% most central collision globally speaking. That is because the later centrality is nearly the average of the former one. The dynamical correlation strength in 0-10% most central collisions is globally less than 0-5% most central collision. That is because the interactions (represented by the collision number for instance) in the former collisions is weaker than the later one. We also see in Fig. 4 that the statistical correlation strength is nearly the same as the NBD one in the 5% most central collision, that is consistent with the results in $p+p$ collisions at the same energy yan . However the discrepancy between statistical and NBD correlation strengths seems to be increased with increasing centrality bin size monotonously. That is mainly because the NBD fitting to the charged particle multiplicity distribution becomes worse with increasing centrality bin size monotonously. ## IV CONCLUSION In summary, we have used a parton and hadron cascade model, PACIAE, to study the centrality bin size dependence of charged particle forward-backward multiplicity correlation strength in 5, 0-5, and 0-10% most central Au+Au collisions at $\sqrt{s_{\rm{NN}}}$=200 GeV. The real (total), statistical, and NBD correlation strengths are calculated by real events, mixed events, and NBD method, respectively. The corresponding STAR data feature of the correlation strength $b$ is approximately flat across a wide range in $\Delta\eta$ in most central Au+Au collisions is well reproduced. It is turned out that the correlation strength increases with increasing centrality bin size monotonously. This first result, given in the transport model, remains to be proved experimentally. If the discrepancy between real (total) and statistical correlation strengths is identified as dynamical one yan , then the dynamical correlation may be just a few percent of the total (real) correlation. As a next step, we will investigate the relation between correlation strength $b$ and the centrality bin size in the mid-central and peripheral collisions, and the STAR data feature of $b$ approaches an exponential function of $\Delta\eta$ at the peripheral collisions. ACKNOWLEDGMENT Finally, the financial support from NSFC (10635020, 10605040, and 10705012) in China is acknowledged ## References (1) R. C. Hwa, Int. J. Mod. Phys. E 16, 3395 (2008). * (2) T. K. Nayak, J. of Phys. G 32, S187 (2006). * (3) J. Adams, et al., STAR Collaboration, Phys. Rev. C 75, 034901 (2007). * (4) A. Adare, et al., PHENIX Collaboration, Phys. Rev. Lett. 98, 232302 (2007). * (5) Zheng-Wei Chai , et al., PHOBOS Collaboration, J. of Phys.: Conference Series 27, 128 (2005). * (6) N. S. Amelin, N. Armesto, M. A. Braun, E. G. Ferreiro, and C. Pajares, Phys. Rev. Lett. 73, 2813 (1994); N. Armesto, M. A. Braun, and C. Pajares, Phys. Rev. C 75, 054902 (2007). * (7) R. C. Hwa and C. B. Yang , nucl-th/0705.3073. * (8) V. P. Konchakovski, M. I. Gorenstein, and E. L. Bratkovskaya, Phys. Rev. C 76, 031901(R) (2007); V. P. Konchakovski, M. Hauer, G. Torrieri, M. I. Gorenstein, and E. L. Bratkovskaya, nucl-th/0812.3967 . * (9) P.Brogueira, J. Dias de Deus, and J. G. Milhano, Phys. Rev. C 76, 064901 (2007). * (10) Jinghua Fu, Phys. Rev. C 77, 027902 (2008). * (11) Yu-Liang Yan, Bao-Guo Dong, Dai-Mei Zhou, Xiao-Mei Li, and Ben-Hao Sa, Phys. Lett. B 660, 478 (2008). * (12) A. Bzdak, hep-ph/0902.2639. * (13) T. Tarnowsky, STAR Collaboration, arXiv:0711.1175v1 (PoS CP0D07 (2007) 019); Int. J. Mod. Phys. E 16, 3363 (2008). * (14) B. K. Srivastavs, STAR Collaboration, Int. J. Mod. Phys. E 16, 3371 (2008). * (15) S. S. Adler et al., PHENIX Collaboration, Phys. Rev. C, 76, 034903 (2007). * (16) Dai-Mei Zhou, Xiao-Mei Li, Bao-Guo Dong, and Ben-Hao Sa, Phys. Lett. B 638, 461 (2006); Ben-Hao Sa, Xiao-Mei Li, Shou-Yang Hu, Shou-Ping Li, Jing Feng, and Dai-Mei Zhou, Phys. Rev. C 75, 054912 (2007). * (17) T. Sj$\ddot{o}$strand, Comput. Phys. Commun. 82, 74 (1994). * (18) B. L. Combridge, J. Kripfgang, and J. Ranft, Phys. Lett. B 70, 234 (1977). * (19) T. S. Biró, P. Lévai, and J. Zimányi, Phys. Rev. C 59, 1547 (1999). * (20) P. Csizmadia and P. Lévai, Phys. Rev. C 61, 031903(R) (2000). * (21) V. Greco, C. M. Ko, and P. Lévai, Phys. Rev. Lett. 90, 202302 (2003). * (22) R. C. Hwa and C. B. Yang, Phys. Rev. C 67, 034902 (2003). * (23) R. J. Fries, B. Müller, and C.Nonaka, Phys. Rev. Lett. 90, 202303 (2003). * (24) R. D. Field and R. P. Feynman, Phys. Rev. D 15, 2590 (1977); Nucl. Phys. B 138, 1(1978); R. P. Feynman, R. D. Field, and G. C. Fox, Phys. Rev. D 18, 3320 (1978). * (25) B. Andersson, G. Gustafson, G. Ingelman, T. Sj$\ddot{o}$strand, Phys. Rep. 97, 33 (1983); B. Andersson, G. Gustafson, B. S$\ddot{o}$derberg, Nucl. Phys. B 264, 29 (1986). * (26) Ben-Hao Sa and Tai An, Comput. Phys. Commun. 90, 121 (1995); Tai An and Ben-Hao Sa, Comput. Phys. Commun. 116, 353 (1999). * (27) P. Koch, B. Müller, and J. Rafelski Phys. Rep. 142, 167 (1986). * (28) A. Baldini, et al., “Total cross sections for reactions of high energy particles”, Springer-Verlag, Berlin, 1988. * (29) PDG, “Particle Physics Booklet”, Extracted from C. Amsler, et al., Phys. Lett. B 667, 1 (2008). * (30) A. Capella, U. Sukhatme, C.-I. Tan, and J. Tran Thanh Van, Phys. Rep. 236, 225 (1994). * (31) B. B. Back, et al., PHOBOS Collaboration, Phys. Rev. Lett. 91, 052303 (2003). * (32) Ben-Hao Sa, A. Bonasera, An Tai and Dai-Mei Zhou, Phys. Lett. B 537, 268 (2002).	arxiv-papers	2009-03-05T06:13:21	2024-09-04T02:49:01.004515	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yu-Liang Yan, Dai-Mei Zhou, Bao-Guo Dong, Xiao-Mei Li, Hai-Liang Ma,\n Ben-Hao Sa", "submitter": "Yuliang Yan", "url": "https://arxiv.org/abs/0903.0915" }
0903.0985	# Nonequilibrium Extension of Onsager Relations for Thermoelectric Effects of Mesoscopic Conductors Eiki Iyoda1 E-mail address: iyoda@issp.u-tokyo.ac.jp Yasuhiro Utsumi1,2 and Takeo Kato1 1 Institute for Solid State Physics1 Institute for Solid State Physics the University of Tokyo the University of Tokyo Chiba 277-8581 Chiba 277-8581 Japan 2 CREST Japan 2 CREST Japan Science and Technology (JST) Japan Science and Technology (JST) Saitama Saitama 332-0012 332-0012 Japan Japan fluctuation theorem, Onsager relation, nonequilibrium quantum transport It is well known that the Onsager reciprocal relations give important identities between transport coefficients for heat and charge conduction in the linear response regime. [1, 2] For example, the Peltier coefficient $\Pi$ is related to the Seebeck coefficient $S$ in a simple form as $S=\Pi/T$, where $T$ is the temperature. The Onsager-Casimir symmetry, which is the extension of the Onsager relation in the presence of an external magnetic field, is also derived from the microscopic reversibility. [3] Recently, a general relation called the steady-state fluctuation theorem has been derived by the same concept of the microscopic reversibility. [4, 5] It gives an identity equation on the probability of the entropy production $\Delta S$ during time $\tau$ as $\ln\left[\frac{P(\Delta S)}{P(-\Delta S)}\right]=\Delta S,$ (1) in the asymptotic limit $\tau\rightarrow\infty$. While this expression reproduces the ordinal Onsager relations in the linear response regime, it provides useful information even in the far-from-equilibrium regime. The importance of the Onsager-Casimir relation on coherent quantum transport of mesoscopic devices is known for long time. [6, 7] Recently, Saito and Utsumi have derived a quantum version of the fluctuation theorem, and applied it for derivation of universal relations among nonlinear transport coefficients [8] by means of the full-counting statistics in the Keldysh formalism. [9, 10, 11] In Ref. Saito08, however, its physical meaning on thermoelectric effects has not been addressed in detail. In this note, we discuss extension of the Onsager relation between the Peltier effect and the Seebeck effect to nonequilibrium steady states in mesoscopic devices. For simplicity, we consider a two-terminal setup, though extension to a multi-terminal setup is straightforward. We consider the Hamiltonian $H=\sum_{r={\rm L},{\rm R}}H_{r}+H_{d}+H_{T}$, where $\displaystyle H_{r}=$ $\displaystyle\sum_{k}\epsilon_{rk}a^{\dagger}_{rk}a_{rk},$ (2) $\displaystyle H_{d}=$ $\displaystyle\sum_{i=1}^{m}\epsilon_{i}d^{\dagger}_{i}d_{i}+\sum_{\langle i,j\rangle}^{m}t_{ij}(c_{i}^{\dagger}c_{j}+{\rm h.c.})+H_{I},$ (3) $\displaystyle H_{T}=$ $\displaystyle\sum_{rki}t_{rki}(d^{\dagger}_{i}a_{rk}+{\rm h.c.}).$ (4) Each of two leads is described by $H_{r}$ ($r={\rm L},{\rm R}$), where $a_{rk}$ is an annihilation operator of an electron with a wave vector $k$. A mesoscopic device is modeled in a general form by $H_{d}$ consisting of an arbitrary number of local energy levels labeled by $\epsilon_{i}$ ($1\leq i\leq m$), where $d_{i}$ is an annihilation operator at the $i$th site. In this model, electron hopping $t_{ij}$ and electron-electron interaction (denoted by $H_{I}$) between arbitrary pairs of sites are assumed. [12] A coupling between the leads and the mesoscopic device is described by $H_{T}$ with electron hopping $t_{rki}$. An external magnetic field $B$ is introduced by the Peierls phase on the hopping elements as $t_{rki}=\left\|t_{rki}\right\|\exp({\rm i}\phi_{rki})$ and $t_{ij}=\left\|t_{ij}\right\|\exp({\rm i}\phi_{ij})$ where $\phi_{rki}$ and $\phi_{ij}$ are odd functions of the magnetic field: $\phi(-B)=-\phi(B)$. In this note, we use a unit $\hbar=k_{B}=e=1$. We introduce a cumulant generating function (CGF) ${\cal F}(\chi_{c},\chi_{e};B)$ for the steady state, where $\chi_{c}$ and $\chi_{e}$ are counting fields for charge and heat current. Current operators are defined as $I_{c}=\dot{N}_{\rm L}={\rm i}[N_{L},H_{T}]$ and $I_{e}=\dot{H}_{\rm L}={\rm i}[H_{L},H_{T}]$ [13], where $N_{L}$ is a number of particle of the left lead. The CGF generates cumulants by the derivatives with respect to the counting fields. For example, the charge current and noise are generated as $\displaystyle\langle\\!\langle I_{c}\rangle\\!\rangle$ $\displaystyle=\langle I_{c}\rangle=\left.\frac{\partial{\cal F}}{\partial i\chi_{c}}\right\|_{\chi_{c}=\chi_{e}=0},$ (5) $\displaystyle\langle\\!\langle I_{c}^{2}\rangle\\!\rangle$ $\displaystyle=\langle I_{c}^{2}\rangle-\langle I_{c}\rangle^{2}=\left.\frac{\partial^{2}{\cal F}}{(\partial i\chi_{c})^{2}}\right\|_{\chi_{c}=\chi_{e}=0},$ (6) respectively. General relations among nonlinear transport coefficients are derived from a symmetry relation $\displaystyle{\cal F}(\chi_{c},\chi_{e};B)={\cal F}(-\chi_{c}+i{\cal A}_{c},-\chi_{e}+i{\cal A}_{e};-B),$ (7) which is a consequence of the microscopic reversibility [8]. Here, ${\cal A}_{c}=\beta_{L}\mu_{L}-\beta_{R}\mu_{R}$ and ${\cal A}_{e}=-(\beta_{L}-\beta_{R})$ are affinities. All the cumulants are expanded with respect to ${\cal A}_{c}$ and ${\cal A}_{e}$ as $\displaystyle\langle\\!\langle I_{c}^{k_{1}}I_{e}^{k_{2}}\rangle\\!\rangle$ $\displaystyle=\sum_{l_{1}=0}^{\infty}\sum_{l_{2}=0}^{\infty}\frac{L_{l_{1},l_{2}}^{k_{1},k_{2}}}{l_{1}!l_{2}!}{\cal A}_{c}^{l_{1}}{\cal A}_{e}^{l_{2}},$ (8) $\displaystyle L^{k_{1},k_{2}}_{l_{1},l_{2}}\left(B\right)$ $\displaystyle=\left.\frac{\partial^{l_{1}+l_{2}}\langle\\!\langle I^{k_{1}}_{c}I^{k_{2}}_{e}\rangle\\!\rangle}{\partial{\cal A}^{l_{1}}_{c}\partial{\cal A}^{l_{2}}_{e}}\right\|_{{\cal A}_{c}={\cal A}_{e}=0}.$ (9) By symmetrizing and anti-symmetrizing the transport coefficients with respect to $B$ as $\displaystyle L^{k_{1},k_{2}}_{l_{1},l_{2},\pm}\left(B\right)$ $\displaystyle=L^{k_{1},k_{2}}_{l_{1},l_{2}}\left(B\right)\pm L^{k_{1},k_{2}}_{l_{1},l_{2}}\left(-B\right),$ (10) general relations are derived from Eq. (7) as [8] $\displaystyle L^{k_{1},k_{2}}_{l_{1},l_{2},\pm}$ $\displaystyle=\pm\sum_{n_{1}=0}^{l_{1}}\sum_{n_{2}=0}^{l_{2}}\binom{l_{1}}{n_{1}}\binom{l_{2}}{n_{2}}$ $\displaystyle\times\left(-1\right)^{n_{1}+n_{2}+k_{1}+k_{2}}L^{k_{1}+n_{1},k_{2}+n_{2}}_{l_{1}-n_{1},l_{2}-n_{2},\pm}.$ (11) We can show immediately that Eq. (11) includes the Onsager relation in the linear response region as follows. First, $L_{l_{1},l_{2}}^{k_{1},k_{2}}(B)$ are related to ordinary observables by changing affinities to the external- bias variables, ${\cal A}_{c}$ and ${\cal A}_{e}$ to the bias voltage $V=\mu_{L}-\mu_{R}$ and the temperature difference $\Delta T=T_{L}-T_{R}$. (For example, the linear conductance of charge current, the charge current noise at equilibrium and the Seebeck coefficient are written as $G=L_{1,0}^{1,0}/T$, $S_{I,I}=L_{0,0}^{2,0}$ and $S=L_{0,1}^{1,0}/TL_{1,0}^{1,0}$, respectively.) We note that the linear response of charge and heat currents are expressed by $I_{c}=L_{1,0}^{1,0}(B)(V/T)+L_{0,1}^{1,0}(B)(\Delta T/T^{2})$ and $I_{e}=L_{1,0}^{0,1}(B)(V/T)+L_{0,1}^{0,1}(B)(\Delta T/T^{2})$. [2] Then, specific equations between coefficients satisfying $N\equiv k_{1}+k_{2}+l_{1}+l_{2}=2$ lead to the Onsager relation $L_{0,1}^{1,0}(B)=L_{1,0}^{0,1}(-B)$. Next, we prove that the magnitude of the nonlinear Peltier effect can be determined only by information of charge current measurements without heat current measurement; we show that all the coefficients for heat current appearing in the expansion $\displaystyle I_{e}=\sum_{l_{1}=0}^{\infty}\frac{L_{l_{1},0}^{0,1}}{l_{1}!}{\cal A}_{c}^{l_{1}}=L^{0,1}_{1,0}{\cal A}_{c}+L^{0,1}_{2,0}{\cal A}_{c}^{2}/2+{\cal O}({\cal A}_{c}^{3}),$ (12) under the isothermal condition (${\cal A}_{e}=0$) can always be rewritten by the transport coefficients of the higher-order charge current cumulant $L_{l_{1},1}^{k_{1},0}$. By substituting $k_{2}=0$ and $l_{2}=1$ into the general equation (11), we obtain $\displaystyle L_{N-i,1,\pm}^{i,0}=\pm\sum_{j=0}^{N}M_{ij}\left(L_{N-j,1,\pm}^{j,0}-L_{N-j,0,\pm}^{j,1}\right),$ (13) where $M$ is a matrix whose matrix elements are given as $\displaystyle M_{ij}=\left\\{\begin{array}[]{cc}\displaystyle\binom{N\\!-\\!i}{j\\!-\\!i}(-1)^{j}&(0\leq i\leq j\leq N)\\\ \displaystyle 0&(0\leq j<i\leq N)\end{array}\right..$ (16) By utilizing $\sum_{j}M_{ij}M_{jk}\\!=\\!\delta_{ik}$, we obtain $\displaystyle L_{N,0,\pm}^{0,1}=\mp\sum_{k=1}^{N}M_{0k}L_{N-k,1,\pm}^{k,0}.$ (17) We notice that Eq. (17) relates the nonlinear response of the heat current under ${\cal A}_{e}=0$ to that of the charge current, current noise, and higher cumulants up to the linear response in ${\cal A}_{e}$. This means that the magnitude of nonlinear Peltier effect can be evaluated without direct measurement of the heat current. This result can be regarded as a nonequilibrium extension of the Kelvin-Onsager relation. Finally, we discuss the lowest-order nonlinear correction of the Peltier coefficient defined by $\displaystyle\Pi({\cal A}_{c})=\frac{I_{e}}{I_{c}}=\Pi^{(0)}+\Pi^{(1)}I_{c}+{\cal O}(I_{c}^{2}).$ (18) under the isothermal condition (${\cal A}_{e}=0$). The linear-response term satisfies the Kelvin-Onsager relation $\Pi^{(0)}=TS^{(0)}(-B)$ as already mentioned, where $S^{(0)}(B)$ is the ordinal Seebeck coefficient defined in the linear-response regime. The first-order correction $\Pi^{(1)}$ is formally expressed by using the definition of $L_{l_{1},l_{2}}^{k_{1},k_{2}}$ as $\displaystyle\Pi^{(1)}=\frac{L^{0,1}_{1,0}(B)}{2L^{1,0}_{1,0}(B)^{2}}\left(\frac{L^{0,1}_{2,0}(B)}{L^{0,1}_{1,0}(B)}-\frac{L^{1,0}_{2,0}(B)}{L^{1,0}_{1,0}(B)}\right).$ By utilizing the general relation in Eq. (11) for $N=2$, the correction is rewritten as $\displaystyle\frac{\Pi^{(1)}}{T}=$ $\displaystyle-\frac{{S}(-B)}{2}\biggl{(}\frac{L^{1,0}_{2,0}(B)}{L^{1,0}_{1,0}(B)^{2}}$ $\displaystyle+$ $\displaystyle\frac{{L}^{2,0}_{0,1}(-B)-2{L}^{1,0}_{1,1}(-B)}{L^{1,0}_{1,0}(B){L}^{1,0}_{0,1}(-B)}\biggl{)}\,.$ (19) Thus, though the expression is a little complicated, one can find that the nonlinear correction of the Peltier coefficient can be calculated only by coefficients $L_{l_{1},l_{2}}^{k_{1},0}$, which needs no heat-current measurements. In a similar way, higher-order correction of the Peltier coefficient $($e.g. $\Pi^{(2)})$ can be expressed by higher-order cumulants of the charge current (e.g. skewness). We proved that the magnitude of the nonlinear Peltier effect under the isothermal condition can be evaluated from transport coefficients for charge current cumulants (current, current noise, and higher cumulants) up to the linear response to a thermal bias. Inversely, if precise measurement of both charge and heat currents is possible, thermoelectric effects in the nonlinear regime can be used for an experimental proof of the fluctuation theorem in quantum systems, as performed in a recent experiment by means of simultaneous measurement of conductance and shot noise. [14] By combining recent progress in measurement techniques of thermoelectric effects, [15] the present result will provide an important key to understand nonequilibrium thermoelectric effects in mesoscpic systems. We would like to thank Keiji Saito for drawing our attention to the FT and the thermal transport. Y. U. acknowledges financial support by Strategic International Cooperative Program JST. E. I. and T. K. acknowledge financial support by JSPS and MAE under the Japan-France Integrated Action Program (SAKURA). This work was supported by Grant-in-Aid for Young Scientists (B) (21740220). ## References * [1] L. Onsager: Phys. Rev. 37 (1931) 405. * [2] Thermodynamics and an Introduction to Thermostatistics, 2nd edition, H. B. Callen (Wiley, USA, 1985). * [3] H. B. G. Casimir: Rev. Mod. Phys. 17 (1945) 343. * [4] D. J. Evans et al.: Phys. Rev. Lett. 71 (1993) 2401. * [5] For a recent review, M. Esposito et al: Rev. Mod. Phys. 81 (2009) 1665. * [6] Electronic Transport in Mesoscopic Systems, S. Datta, (Cambridge University Press, Cambridge, 1995); Mesoscopic Electron Transport (NATO Science Series E), eds. L. L. Sohn et al., (Springer-Verlag, Berlin, 1997); * [7] M. Büttiker: Phys. Rev. Lett. 57 (1986) 1761. * [8] K. Saito and A. Dhar: Phys. Rev. Lett. 99 (2007) 180601; K. Saito and Y. Utsumi: Phys. Rev. B 78 (2008) 115429. * [9] L. V. Keldysh: Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [Sov. Phys. JETP 20 (1965) 1018]. * [10] L. S. Levitov and G. B. Lesovik: JETP Lett. 58 (1993) 230. * [11] Quantum Noise in Mesoscopic Physics, in NATO Science Series II: eds. Yu. V. Nazarov (Kluwer, Dordrecht/Academic, London, 2003) * [12] We note that the model considered here can cover a large number of mesoscopic systems, e.g., including Coulomb interaction $H_{I}=\sum V_{ij}n_{i}n_{j}$ ($n_{i}=d^{\dagger}_{i}d_{i}$) and spin degrees of freedom (not explicitly written in this note). * [13] Because the leads consists of noninteracting electron systems, the energy current coincides with the heat current in the present definition. * [14] S. Nakamura et al.: arXiv:0911.3470. * [15] R. Scheibner et al.: Phys. Rev. B 75 (2007) 041301.	arxiv-papers	2009-03-05T13:33:28	2024-09-04T02:49:01.009440	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Eiki Iyoda, Yasuhiro Utsumi and Takeo Kato", "submitter": "Eiki Iyoda", "url": "https://arxiv.org/abs/0903.0985" }
0903.0997	# New application of Dirac’s representation: N-mode squeezing enhanced operator and squeezed state ††thanks: Work was supported by the National Natural Science Foundation of China under grants 10775097 and 10874174. Xue-xiang Xu1, Li-yun Hu1,2 and Hong-yi Fan1 1Department of Physics, Shanghai Jiao Tong University, Shanghai, 200240, China 2College of Physics & Communication Electronics, Jiangxi Normal University, Nanchang 330022, China Corresponding author. E-mail addresses: hlyun2008@126.com. ###### Abstract It is known that $\exp\left[\mathtt{i}\lambda\left(Q_{1}P_{1}-\mathtt{i}/2\right)\right]$ is a unitary single-mode squeezing operator, where $Q_{1}$,$P_{1}$ are the coordinate and momentum operators, respectively. In this paper we employ Dirac’s coordinate representation to prove that the exponential operator $S_{n}\equiv\exp[\mathtt{i}\lambda\sum\limits_{i=1}^{n}(Q_{i}P_{i+1}+Q_{i+1}P_{i}))],$ ($Q_{n+1}=Q_{1}$, $P_{n+1}=P_{1}$), is a n-mode squeezing operator which enhances the standard squeezing. By virtue of the technique of integration within an ordered product of operators we derive $S_{n}$’s normally ordered expansion and obtain new n-mode squeezed vacuum states, its Wigner function is calculated by using the Weyl ordering invariance under similar transformations. PACS: 03.65.-w; 03.65.Ud Keywords: Dirac’s representation; The IWOP technique; Squeezing enhanced operator; Squeezed sate ## 1 Introduction Squeezed state has been a hot topic in quantum optics since Stoler [1] put forward the concept of the optical squeezing in 1970’s. $S_{1}=$ $\exp\left[\mathtt{i}\lambda\left(Q_{1}P_{1}-\mathtt{i}/2\right)\right]$ is a unitary single-mode squeezing operator, where $Q_{1}$, $P_{1}$ are the coordinate and momentum operators, respectively, $\lambda$ is a squeezing parameter. Their variances in the squeezed state $S_{1}\left\|0\right\rangle=$sech${}^{1/2}\lambda\exp\left[-\frac{1}{2}a_{1}^{\dagger 2}\tanh\lambda\right]\left\|0\right\rangle$ are $\Delta Q_{1}=\frac{1}{4}e^{2\lambda},\text{ }\Delta P_{1}=\frac{1}{4}e^{-2\lambda},\text{ }(\Delta Q_{1})(\Delta P_{1})=\frac{1}{4}.$ Some generalized squeezed state have been proposed since then. Among them the two-mode squeezed state not only exhibits squeezing, but also quantum entanglement between the idle-mode and the signal-mode in frequency domain, therefore is a typical entangled states of continuous variable. In recent years, various entangled states have attracted considerable attention and interests of physists because of their potential uses in quantum communication [2]. Theoretically, the two-mode squeezed state is constructed by acting the two-mode squeezing operator $S_{2}=\exp[\lambda(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})]$ on the two- mode vacuum state $\left\|00\right\rangle$[3, 4, 5], $S_{2}\left\|00\right\rangle=\text{sech}\lambda\exp\left[-a_{1}^{\dagger}a_{2}^{\dagger}\tanh\lambda\right]\left\|00\right\rangle.$ (1) We also have $S_{2}=\exp\left[\mathtt{i}\lambda\left(Q_{1}P_{2}+Q_{2}P_{1}\right)\right],$ where $Q_{i}$ and $P_{i}$ are the coordinate and momentum operators related to Bose operators ($a_{i},a_{i}^{\dagger}$) by $Q_{i}=(a_{i}+a_{i}^{\dagger})/\sqrt{2},\ P_{i}=(a_{i}-a_{i}^{\dagger})/(\sqrt{2}\mathtt{i})$ (2) In the state $S_{2}\left\|00\right\rangle$, the variances of the two-mode quadrature operators of light field, $\mathfrak{X}=(Q_{1}+Q_{2})/2,\text{ }\mathfrak{P}=(P_{1}+P_{2})/2,\text{ \ }[\mathfrak{X},\mathfrak{P}]=\frac{\mathtt{i}}{2},$ (3) take the standard form, i.e., $\left\langle 00\right\|S_{2}^{\dagger}\mathfrak{X}^{2}S_{2}\left\|00\right\rangle=\frac{1}{4}e^{-2\lambda},\text{ \ }\left\langle 00\right\|S_{2}^{\dagger}\mathfrak{P}^{2}S_{2}\left\|00\right\rangle=\frac{1}{4}e^{2\lambda},\text{ \ and }(\Delta\mathfrak{X})(\Delta\mathfrak{P})=\frac{1}{4}.$ (4) On the other hand, the two-mode squeezing operator has a neat and natural representation in the entangled state $\left\|\eta\right\rangle$ representation [6], $S_{2}=\int\frac{d^{2}\eta}{\pi\mu}\left\|\frac{\eta}{\mu}\right\rangle\left\langle\eta\right\|,$ (5) where $\left\|\eta\right\rangle=\exp(-\frac{1}{2}\left\|\eta\right\|^{2}+\eta a_{1}^{\dagger}-\eta^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger})\left\|00\right\rangle,$ (6) makes up a complete set $\int\frac{d^{2}\eta}{\pi}\left\|\eta\right\rangle\left\langle\eta\right\|=1.$ $\left\|\eta\right\rangle$ was constructed according to the idea of quantum entanglement innitiated by Einstein, Podolsky and Rosen in their argument that quantum mechanics is incomplete [7]. An interesting question naturally arises: is the $n$-mode exponential operator $S_{n}\equiv\exp\left[\mathtt{i}\lambda\sum_{i=1}^{n}(Q_{i}P_{i+1}+Q_{i+1}P_{i})\right],\text{ \ }(Q_{n+1}=Q_{1},\ P_{n+1}=P_{1}),\ n\geqslant 2,$ (7) a squeezing operator? If yes, what kind of squeezing for $n$-mode quadratures of field it can engenders? To answer these questions we must know what is the normally ordered expansion of $S_{n}$ and what is the state $S_{n}\left\|\mathbf{0}\right\rangle$ ($\left\|\mathbf{0}\right\rangle$ is the n-mode vacuum state)? In this work we shall analyse $S_{n}$ in detail. But how to disentangle the exponential of $S_{n}?$ Since the terms in the set $Q_{i}P_{i+1}\ $and $Q_{i+1}P_{i}$ ($i=1,2,\cdots,n$) do not make up a closed Lie algebra, the problem of what is $S_{n}$’s normally ordered form seems difficult. Thus we appeal to Dirac’s coordinate representation and the technique of integration within an ordered product (IWOP) of operators [8, 9] to solve this problem. Our work is arranged as follows: firstly we use the IWOP technique to derive the normally ordered expansion of $S_{n}$ and obtain the explicit form of$\ S_{n}\left\|\mathbf{0}\right\rangle$; then we examine the variances of the $n$-mode quadrature operators in the state $S_{n}\left\|\mathbf{0}\right\rangle$, we find that $S_{n}$ causes squeezing which is stronger than the standard squeezing. Thus $S_{n}$ is an $n$-mode squeezing-enhanced operator. The Wigner function of $S_{n}\left\|\mathbf{0}\right\rangle$ is calculated by using the Weyl ordering invariance under similar transformations. Some examples are discussed in the last section. ## 2 Normal Product Form of $S_{n}$ derived by Dirac’s coordinate representation In order to disentangle operator $S_{n}$, let $A$ be $A=\left(\begin{array}[]{ccccc}0&1&0&\cdots&1\\\ 1&0&1&\cdots&0\\\ 0&1&0&\ddots&0\\\ \vdots&\vdots&\ddots&\ddots&\vdots\\\ 1&0&\cdots&1&0\end{array}\right),$ (8) then $S_{n}$ in (7) is compactly expressed as $S_{n}=\exp[\mathtt{i}\lambda Q_{i}A_{ij}P_{j}],$ (9) here and henceforth the repeated indices represent Einstein’s summation notation. Using the Baker-Hausdorff formula, $e^{A}Be^{-A}=B+\left[A,B\right]+\frac{1}{2!}\left[A,\left[A,B\right]\right]+\frac{1}{3!}\left[A,\left[A,\left[A,B\right]\right]\right]+\cdots,$ we have $\displaystyle S_{n}^{-1}Q_{k}S_{n}$ $\displaystyle=$ $\displaystyle Q_{k}-\lambda Q_{i}A_{ik}+\frac{1}{2!}\mathtt{i}\lambda^{2}\left[Q_{i}A_{ij}P_{j},Q_{l}A_{lk}\right]+\cdots$ (10) $\displaystyle=$ $\displaystyle Q_{i}(e^{-\lambda A})_{ik}=(e^{-\lambda\tilde{A}})_{ki}Q_{i},$ $\displaystyle S_{n}^{-1}P_{k}S_{n}$ $\displaystyle=$ $\displaystyle P_{k}+\lambda A_{ki}P_{i}+\frac{1}{2!}\mathtt{i}\lambda^{2}\left[A_{ki}P_{j},Q_{l}A_{lm}P_{m}\right]+\cdots$ (11) $\displaystyle=$ $\displaystyle(e^{\lambda A})_{ki}P_{i}.$ From Eq.(10) we see that when $S_{n}$ acts on the n-mode coordinate eigenstate $\left\|\vec{q}\right\rangle,$ where $\widetilde{\vec{q}}=(q_{1},q_{2},\cdots,q_{n})$, it squeezes $\left\|\vec{q}\right\rangle$ in this way: $S_{n}\left\|\vec{q}\right\rangle=\left\|\Lambda\right\|^{1/2}\left\|\Lambda\vec{q}\right\rangle,\text{ }\Lambda=e^{-\lambda\tilde{A}},\text{ }\left\|\Lambda\right\|\equiv\det\Lambda.$ (12) Thus $S_{n}$ has the representation on the Dirac’s coordinate basis $\left\langle\vec{q}\right\|$[10] $S_{n}=\int d^{n}qS_{n}\left\|\vec{q}\right\rangle\left\langle\vec{q}\right\|=\left\|\Lambda\right\|^{1/2}\int d^{n}q\left\|\Lambda\vec{q}\right\rangle\left\langle\vec{q}\right\|,\text{ \ \ }S_{n}^{\dagger}=S_{n}^{-1},$ (13) since $\int d^{n}q\left\|\vec{q}\right\rangle\left\langle\vec{q}\right\|=1.$ Using the expression of $\left\|\vec{q}\right\rangle$ in Fock space $\displaystyle\left\|\vec{q}\right\rangle=\pi^{-n/4}\colon\exp\left[-\frac{1}{2}\widetilde{\vec{q}}\vec{q}+\sqrt{2}\widetilde{\vec{q}}a^{{\dagger}}-\frac{1}{2}\tilde{a}^{{\dagger}}a^{{\dagger}}\right]\left\|\mathbf{0}\right\rangle,\text{ }$ $\displaystyle\tilde{a}^{{\dagger}}=(a_{1}^{{\dagger}},a_{2}^{{\dagger}},\cdots,a_{n}^{{\dagger}})\text{,}$ (14) and the normally ordered form of n-mode vacuum projector $\left\|\mathbf{0}\right\rangle\left\langle\mathbf{0}\right\|=\colon\exp[-\tilde{a}^{{\dagger}}a^{{\dagger}}]\colon$, we can put $S_{n}$ into the normal ordering form, $\displaystyle S_{n}$ $\displaystyle=$ $\displaystyle\pi^{-n/2}\left\|\Lambda\right\|^{1/2}\int d^{n}q\colon\exp[-\frac{1}{2}\widetilde{\vec{q}}(1+\widetilde{\Lambda}\Lambda)\vec{q}+\sqrt{2}\widetilde{\vec{q}}(\widetilde{\Lambda}a^{{\dagger}}+a)$ (15) $\displaystyle-\frac{1}{2}(\widetilde{a}a+\tilde{a}^{{\dagger}}a^{{\dagger}})-\tilde{a}^{{\dagger}}a]\colon.$ To perform the integration in Eq.(15) by virtue of the IWOP technique, using the mathematical formula $\int d^{n}x\exp[-\widetilde{x}Fx+\widetilde{x}v]=\pi^{n/2}(\det F)^{-1/2}\exp\left[\frac{1}{4}\widetilde{v}F^{-1}v\right],$ (16) then we derive $\displaystyle S_{n}$ $\displaystyle=$ $\displaystyle\left(\frac{\det\Lambda}{\det N}\right)^{1/2}\exp\left[\frac{1}{2}\tilde{a}^{{\dagger}}\left(\Lambda N^{-1}\widetilde{\Lambda}-I\right)a^{{\dagger}}\right]$ (17) $\displaystyle\times\colon\exp\left[\tilde{a}^{{\dagger}}\left(\Lambda N^{-1}-I\right)a\right]\colon\exp\left[\frac{1}{2}\widetilde{a}\left(N^{-1}-I\right)a\right],$ where $N=(1+\widetilde{\Lambda}\Lambda)/2$. Eq.(17) is just the normal product form of $S_{n}.$ ## 3 Squeezing property of $S_{n}\left\|\mathbf{0}\right\rangle$ Operating $S_{n}$ on the n-mode vacuum state $\left\|\mathbf{0}\right\rangle,$ we obtain the squeezed vacuum state $S_{n}\left\|\mathbf{0}\right\rangle=\left(\frac{\det\Lambda}{\det N}\right)^{1/2}\exp\left[\frac{1}{2}\tilde{a}^{{\dagger}}\left(\Lambda N^{-1}\widetilde{\Lambda}-I\right)a^{{\dagger}}\right]\left\|\mathbf{0}\right\rangle.$ (18) Now we evaluate the variances of the n-mode quadratures. The quadratures in the n-mode case are defined as $X_{1}=\frac{1}{\sqrt{2n}}\sum_{i=1}^{n}Q_{i},\text{ }X_{2}=\frac{1}{\sqrt{2n}}\sum_{i=1}^{n}P_{i},$ (19) obeying $[X_{1},X_{2}]=\frac{\mathtt{i}}{2}.$ Their variances are $\left(\Delta X_{i}\right)^{2}=\left\langle X_{i}^{2}\right\rangle-\left\langle X_{i}\right\rangle^{2}$, $i=1,2.$ Noting the expectation values of $X_{1}$ and $X_{2}$ in the state $S_{n}\left\|\mathbf{0}\right\rangle$, $\left\langle X_{1}\right\rangle=\left\langle X_{2}\right\rangle=0$, then using Eqs. (10) and (11) we see that the variances are $\displaystyle\left(\triangle X_{1}\right)^{2}$ $\displaystyle=$ $\displaystyle\left\langle\mathbf{0}\right\|S_{n}^{-1}X_{1}^{2}S_{n}\left\|\mathbf{0}\right\rangle=\frac{1}{2n}\left\langle\mathbf{0}\right\|S_{n}^{-1}\sum_{i=1}^{n}Q_{i}\sum_{j=1}^{n}Q_{j}S_{n}\left\|\mathbf{0}\right\rangle$ (20) $\displaystyle=$ $\displaystyle\frac{1}{2n}\left\langle\mathbf{0}\right\|\sum_{i=1}^{n}Q_{k}(e^{-\lambda A})_{ki}\sum_{j=1}^{n}(e^{-\lambda\tilde{A}})_{jl}Q_{l}\left\|\mathbf{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{2n}\underset{i,j}{\sum^{n}}(e^{-\lambda A})_{ki}(e^{-\lambda\tilde{A}})_{jl}\left\langle\mathbf{0}\right\|Q_{k}Q_{l}\left\|\mathbf{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{4n}\underset{i,j}{\sum^{n}}(e^{-\lambda A})_{ki}(e^{-\lambda\tilde{A}})_{jl}\left\langle\mathbf{0}\right\|a_{k}a_{l}^{\dagger}\left\|\mathbf{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{4n}\underset{i,j}{\sum^{n}}(e^{-\lambda A})_{ki}(e^{-\lambda\tilde{A}})_{jl}\delta_{kl}=\frac{1}{4n}\underset{i,j}{\sum^{n}}(\widetilde{\Lambda}\Lambda)_{ij},$ similarly we have $\left(\triangle X_{2}\right)^{2}=\left\langle\mathbf{0}\right\|S_{n}^{-1}X_{2}^{2}S_{n}\left\|\mathbf{0}\right\rangle=\frac{1}{4n}\underset{i,j}{\sum^{n}}\left[(\widetilde{\Lambda}\Lambda)^{-1}\right]_{ij}.$ (21) Eqs. (20) -(21) are the quadrature variance formula in the transformed vacuum state acted by the operator $\exp[\mathtt{i}\lambda Q_{i}A_{ij}P_{j}].$ By observing that $A$ in (9) is a symmetric matrix, we see $\underset{i,j}{\sum^{n}}\left[(A+\tilde{A})^{l}\right]_{i\text{ }j}=2^{2l}n,$ (22) then using $A\tilde{A}=\tilde{A}A,$ so $\widetilde{\Lambda}\Lambda=e^{-\lambda(A+\tilde{A})}$, a symmetric matrix, we have $\underset{i,j=1}{\sum^{n}}(\widetilde{\Lambda}\Lambda)_{i\text{ }j}=\sum_{l=0}^{\infty}\frac{(-\lambda)^{l}}{l!}\underset{i,j}{\sum^{n}}\left[(A+\tilde{A})^{l}\right]_{i\text{ }j}=n\sum_{l=0}^{\infty}\frac{(-\lambda)^{l}}{l!}2^{2l}=ne^{-4\lambda},$ (23) and $\underset{i,j=1}{\sum^{n}}(\widetilde{\Lambda}\Lambda)_{i\text{ }j}^{-1}=ne^{4\lambda}.$ (24) It then follows $\displaystyle\left(\triangle X_{1}\right)^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{4n}\underset{i,j}{\sum^{n}}(\widetilde{\Lambda}\Lambda)_{ij}=\frac{e^{-4\lambda}}{4},$ (25) $\displaystyle\left(\triangle X_{2}\right)^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{4n}\underset{i,j}{\sum^{n}}\left[(\widetilde{\Lambda}\Lambda)^{-1}\right]_{ij}=\frac{e^{4\lambda}}{4}.$ (26) This leads to $(\triangle X_{1})(\triangle X_{2})=\frac{1}{4},$ which shows that $S_{n}$ is a correct n-mode squeezing operator for the n-mode quadratures in Eq.(19). Furthermore, Eqs.(25) and (26) clearly indicate that the squeezed vacuum state $S_{n}\left\|\mathbf{0}\right\rangle$ may exhibit stronger squeezing ($e^{-4\lambda}$) in one quadrature than that ($e^{-2\lambda}$) of the usual two-mode squeezed vacuum state. This is a way of enhancing squeezing. ## 4 The Wigner function of $S_{n}\left\|\mathbf{0}\right\rangle$ Wigner distribution functions [12] of quantum states are widely studied in quantum statistics and quantum optics. Now we derive the expression of the Wigner function of $S_{n}\left\|\mathbf{0}\right\rangle.$ Here we take a new method to do it. Recalling that in Ref. [13] we have introduced the Weyl ordering form of single-mode Wigner operator $\Delta_{1}\left(q_{1},p_{1}\right)$, $\Delta_{1}\left(q_{1},p_{1}\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{1}-Q_{1}\right)\delta\left(p_{1}-P_{1}\right)\genfrac{}{}{0.0pt}{}{:}{:},$ (27) its normal ordering form is $\Delta_{1}\left(q_{1},p_{1}\right)=\frac{1}{\pi}\colon\exp\left[-\left(q_{1}-Q_{1}\right)^{2}-\left(p_{1}-P_{1}\right)^{2}\right]\colon$ (28) where the symbols $\colon\colon$ and $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$ denote the normal ordering and the Weyl ordering, respectively. Note that the order of Bose operators $a_{1}$ and $a_{1}^{\dagger}$ within a normally ordered product and a Weyl ordered product can be permuted. That is to say, even though $[a_{1},a_{1}^{\dagger}]=1$, we can have $\colon a_{1}a_{1}^{\dagger}\colon=\colon a_{1}^{\dagger}a_{1}\colon$ and$\genfrac{}{}{0.0pt}{}{:}{:}a_{1}a_{1}^{\dagger}\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}a_{1}^{\dagger}a_{1}\genfrac{}{}{0.0pt}{}{:}{:}.$ The Weyl ordering has a remarkable property, i.e., the order-invariance of Weyl ordered operators under similar transformations, which means $U\genfrac{}{}{0.0pt}{}{:}{:}\left(\circ\circ\circ\right)\genfrac{}{}{0.0pt}{}{:}{:}U^{-1}=\genfrac{}{}{0.0pt}{}{:}{:}U\left(\circ\circ\circ\right)U^{-1}\genfrac{}{}{0.0pt}{}{:}{:},$ (29) as if the “fence” $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$did not exist. For n-mode case, the Weyl ordering form of the Wigner operator is $\Delta_{n}\left(\vec{q},\vec{p}\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\vec{q}-\vec{Q}\right)\delta\left(\vec{p}-\vec{P}\right)\genfrac{}{}{0.0pt}{}{:}{:},$ (30) where $\widetilde{\vec{Q}}=(Q_{1},Q_{2},\cdots,Q_{n})$ and $\widetilde{\vec{P}}=(P_{1},P_{2},\cdots,P_{n})$. Then according to the Weyl ordering invariance under similar transformations and Eqs.(10) and (11) we have $\displaystyle S_{n}^{-1}\Delta_{n}\left(\vec{q},\vec{p}\right)S_{n}$ $\displaystyle=$ $\displaystyle S_{n}^{-1}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\vec{q}-\vec{Q}\right)\delta\left(\vec{p}-\vec{P}\right)\genfrac{}{}{0.0pt}{}{:}{:}S_{n}$ (31) $\displaystyle=$ $\displaystyle\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q_{k}-(e^{-\lambda\tilde{A}})_{ki}Q_{i}\right)\delta\left(p_{k}-(e^{\lambda A})_{ki}P_{i}\right)\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=$ $\displaystyle\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(e^{\lambda\tilde{A}}\vec{q}-\vec{Q}\right)\delta\left(e^{-\lambda A}\vec{p}-\vec{P}\right)\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=$ $\displaystyle\Delta\left(e^{\lambda\tilde{A}}\vec{q},e^{-\lambda A}\vec{p}\right).$ Thus using Eqs.(27) and (31) the Wigner function of $S_{n}\left\|\mathbf{0}\right\rangle$ is $\displaystyle\left\langle\mathbf{0}\right\|S_{n}^{-1}\Delta_{n}\left(\vec{q},\vec{p}\right)S_{n}\left\|\mathbf{0}\right\rangle$ (32) $\displaystyle=$ $\displaystyle\frac{1}{\pi^{n}}\left\langle\mathbf{0}\right\|\colon\exp[-(e^{\lambda\tilde{A}}\vec{q}-\vec{Q})^{2}-(e^{-\lambda A}\vec{p}-\vec{P})^{2}]\colon\left\|\mathbf{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{\pi^{n}}\exp[-(e^{\lambda\tilde{A}}\vec{q})^{2}-\left(e^{-\lambda A}\vec{p}\right)^{2}]$ $\displaystyle=$ $\displaystyle\frac{1}{\pi^{n}}\exp\left[-\widetilde{\vec{q}}e^{\lambda A}e^{\lambda\tilde{A}}\vec{q}-\widetilde{\vec{p}}e^{-\lambda\tilde{A}}e^{-\lambda A}\vec{p}\right]$ $\displaystyle=$ $\displaystyle\frac{1}{\pi^{n}}\exp\left[-\widetilde{\vec{q}}\left(\Lambda\widetilde{\Lambda}\right)^{-1}\vec{q}-\widetilde{\vec{p}}\Lambda\widetilde{\Lambda}\vec{p}\right],$ From Eq.(32) we see that once the explicit expression of $\Lambda\tilde{\Lambda}=\exp[-\lambda(A+\tilde{A})]$ is deduced, the Wigner function of $S_{n}\left\|\mathbf{0}\right\rangle$ can be calculated. ## 5 Some examples of calculating the Wigner function For $n=2,$ form Eq.(7) we have $S_{2}^{\prime}=\exp\left[\mathtt{i}2\lambda\left(Q_{1}P_{2}+Q_{2}P_{1}\right)\right]$ which exhibits clearly the stronger squeezing than the usual two-mode squeezing operator $S_{2}^{\prime}.$ For $n=3,$ the three-mode operator [11] $S_{3}$, from Eq.(9) we see that the matrix $A$ is $\left(\begin{array}[]{ccc}0&1&1\\\ 1&0&1\\\ 1&1&0\end{array}\right),$ thus we have $\Lambda\tilde{\Lambda}=\allowbreak\left(\begin{array}[]{ccc}u&v&\allowbreak v\\\ \allowbreak v&u&\allowbreak v\\\ v&v&u\end{array}\right),\text{ }u=\frac{2}{3}e^{2\lambda}+\frac{1}{3e^{4\lambda}},\text{ }v=\frac{1}{3e^{4\lambda}}-\frac{1}{3}e^{2\lambda},$ (33) and$\ \left(\Lambda\tilde{\Lambda}\right)^{-1}$ is obtained by replacing $\lambda$ with $-\lambda$ in $\Lambda\tilde{\Lambda}.$ Thus the squeezing state $S_{3}\left\|000\right\rangle$ is $S_{3}\left\|000\right\rangle=A_{3}\exp\left[\frac{1}{6}A_{1}\sum_{i=1}^{3}a_{i}^{\dagger 2}-\frac{2}{3}A_{2}\sum_{i<j}^{3}a_{i}^{\dagger}a_{j}^{\dagger}\right]\left\|000\right\rangle,$ (34) where $A_{1}=\left(1-\text{sech}2\lambda\right)\tanh\lambda,\text{ }A_{2}=\frac{\sinh 3\lambda}{2\cosh\lambda\cosh 2\lambda},A_{3}=\text{sech}\lambda\cosh^{-1/2}2\lambda.$ (35) In particular, for the case of the infinite squeezing $\lambda\rightarrow\infty$, Eq.(36) reduces to $S_{3}\left\|000\right\rangle\sim\exp\left\\{\frac{1}{6}\left[\sum_{i=1}^{3}a_{i}^{\dagger 2}-4\sum_{i<j}^{3}a_{i}^{\dagger}a_{j}^{\dagger}\right]\right\\}\left\|000\right\rangle\equiv\left\|\ \right\rangle_{s_{3}},$ (36) which is just the common eigenvector of the three compatible Jacobian operators in three-body case with zero eigenvalues [14], i.e., $\displaystyle\left(P_{1}+P_{2}+P_{3}\right)\left\|\ \right\rangle_{s_{3}}$ $\displaystyle=0,\text{ }\left(Q_{3}-Q_{2}\right)\left\|\ \right\rangle_{s_{3}}=0,$ $\displaystyle\text{ }\left(\frac{\mu_{3}Q_{3}+\mu_{2}Q_{2}}{\mu_{3}+\mu_{2}}-Q_{1}\right)\left\|\ \right\rangle_{s_{3}}$ $\displaystyle=0,\text{ }\left(\mu_{i}=\frac{m_{i}}{m_{1}+m_{2}+m_{3}}\right),$ (37) as common eigenvector $\left[P_{1}+P_{2}+P_{3},Q_{3}-Q_{2}\right]=0,\left[\frac{\mu_{3}Q_{3}+\mu_{2}Q_{2}}{\mu_{3}+\mu_{2}}-Q_{1},P_{1}+P_{2}+P_{3}\right]=0.$ (38) Since the common eigenvector of three compatible Jacobian operators is an entangled state, the state $\left\|\ \right\rangle_{s_{3}}$ is also an entangled state. By using Eq.(32) the Wigner function is $\displaystyle\left\langle\mathbf{0}\right\|S_{3}^{-1}\Delta_{3}\left(\vec{q},\vec{p}\right)S_{3}\left\|\mathbf{0}\right\rangle$ (39) $\displaystyle=$ $\displaystyle\frac{1}{\pi^{3}}\exp\left[-\frac{2}{3}\left(\cosh 4\lambda+2\cosh 2\lambda\right)\sum_{i=1}^{3}\left\|\alpha_{i}\right\|^{2}\right]$ $\displaystyle\times\exp\left\\{-\frac{1}{3}\allowbreak\left(\sinh 4\lambda-2\sinh 2\lambda\right)\sum_{i=1}^{3}\alpha_{i}^{2}\right.$ $\displaystyle-\left.\frac{2}{3}\sum_{j>i=1}^{3}\left[\left(\cosh 4\lambda-\cosh 2\lambda\right)\alpha_{i}\alpha_{j}^{\ast}+\left(\allowbreak\sinh 2\lambda+\sinh 4\lambda\right)\alpha_{i}\alpha_{j}\right]+c.c.\right\\}.$ For $n=4$ case, the four-mode operator $S_{4}$ is $S_{4}=\exp\\{\mathtt{i}\lambda\left[\left(Q_{1}+Q_{3}\right)\left(P_{4}+P_{2}\right)+\left(Q_{2}+Q_{4}\right)\left(P_{1}+P_{3}\right)\right]\\}$ (40) the matrix $A=\left(\begin{array}[]{cccc}0&1&0&1\\\ 1&0&1&0\\\ 0&1&0&1\\\ 1&0&1&0\end{array}\right)$ , thus we have $\Lambda\tilde{\Lambda}=\allowbreak\left(\begin{array}[]{cccc}r&t&s&t\\\ t&r&t&s\\\ s&t&r&t\\\ t&s&t&r\end{array}\right),$ (41) where $r=\cosh^{2}2\lambda,s=\sinh^{2}2\lambda,t=-\sinh 2\lambda\cosh 2\lambda.$ Then substituting Eq.(41) into Eq.(32) we obtain $\left\langle\mathbf{0}\right\|S_{4}^{-1}\Delta_{4}\left(\vec{q},\vec{p}\right)S_{4}\left\|\mathbf{0}\right\rangle=\frac{1}{\pi^{4}}\exp\left\\{-2\cosh^{2}2\lambda\left[\sum_{i=1}^{4}\left\|\alpha_{i}\right\|^{2}+\left(M+M^{\ast}\right)\tanh^{2}2\lambda+\left(R^{\ast}+R\right)\allowbreak\tanh 2\lambda\right]\right\\},$ (42) where $M=\alpha_{1}\alpha_{3}^{\ast}+\alpha_{2}\alpha_{4}^{\ast},$ $R=\alpha_{1}\alpha_{2}+\alpha_{1}\alpha_{4}+\alpha_{2}\alpha_{3}+\alpha_{3}\alpha_{4}.$ This form differs evidently from the Wigner function of the direct-product of usual two two-mode squeezed states’ Wigner functions. In addition, using Eq. (41) we can check Eqs.(25) and (26). Further, using Eq.(41) we have $N^{-1}=\frac{1}{2}\allowbreak\left(\begin{array}[]{cccc}2&\tanh 2\lambda&0&\tanh 2\lambda\\\ \tanh 2\lambda&2&\tanh 2\lambda&0\\\ 0&\tanh 2\lambda&2&\tanh 2\lambda\\\ \tanh 2\lambda&0&\tanh 2\lambda&2\end{array}\right),\text{ }\det N=\cosh^{2}2\lambda.$ (43) Then substituting Eqs.(43) into Eq.(17) yields the four-mode squeezed state [11, 15], $S_{4}\left\|0000\right\rangle=\text{sech}2\lambda\exp\left[-\frac{1}{2}\left(a_{1}^{{\dagger}}+a_{3}^{{\dagger}}\right)\left(a_{2}^{{\dagger}}+a_{4}^{{\dagger}}\right)\tanh 2\lambda\right]\left\|0000\right\rangle,$ (44) from which one can see that the four-mode squeezed state is not the same as the direct product of two two-mode squeezed states in Eq.(1). In summary, by virtue of Dirac’s coordinate representation and the IWOP technique: we have shown that an n-mode squeezing operator $S_{n}\equiv\exp[i\lambda\sum_{i=1}^{n}(Q_{i}P_{i+1}+Q_{i+1}P_{i}))],$ $(Q_{n+1}=Q_{1},\ P_{n+1}=P_{1}),$ is an n-mode squeezing operator which enhances the stronger squeezing for the n-mode quadratures [16]. $S_{n}$’s normally ordered expansion and new n-mode squeezed vacuum states are obtained. ACKNOWLEDGEMENT Work supported by the National Natural Science Foundation of China under grants 10775097 and 10874174. ## References * [1] D. Stoler, _Phys. Rev. D_ , 1 (1970) 3217. * [2] Bouwmeester D. et al., _The Physics of Quantum Information_ , (Springer, Berlin) 2000; Nielsen M. A. and Chuang I. L., _Quantum Computation and Quantum Information_ (Cambridge University Press) 2000. * [3] Buzek V., _J. Mod. Opt._ , 37 (1990) 303. * [4] Loudon R., Knight P. L., _J. Mod. Opt._ , 34 (1987) 709\. * [5] Dodonov V. V., _J. Opt. B: Quantum Semiclass. Opt._ , 4 (2002) R1. * [6] Fan H.-y and Klauder J. R., _Phys. Rev. A_ 49 (1994) 704; Fan H.-y and Fan Y., _Phys. Rev. A_ , 54 (1996) 958; Fan H.-y, _Europhys. Lett._ , 23 (1993) 1; Fan H.-y, _Europhys. Lett._ , 17 (1992) 285; Fan H.-y, _Europhys. Lett._ , 19 (1992) 443. * [7] Einstein A., Poldolsky B. and Rosen N., _Phys. Rev._ , 47 (1935) 777. * [8] Fan H.-y, _Ann. Phys._ , 323 (2008) 500; _Ann. Phys._ , 323 (2008) 1502. * [9] Fan H.-y, _J Opt B: Quantum Semiclass. Opt._ , 5 (2003) R147. * [10] Dirac P. A. M., The Principle of Quantum Mechanics, Fourth ed., Oxford University Press, 1958. * [11] Hu L.-y. and Fan H.-y., _Mod. Phys. Lett. B._ , 22 (2008) 2055. * [12] Wigner E. P., _Phys. Rev._ , 40 (1932) 749; O’Connell R. F. and Wigner E. P., _Phys. Lett. A_ , 83 (1981) 145;Schleich W., _Quantum Optics_ (New York: Wiley) 2001. * [13] Fan H.-y, _J. Phys. A_ , 25 (1992) 3443; Fan H.-y, Fan Y., _Int. J. Mod. Phys. A_ , 17 (2002) 701;Fan H.-y, _Mod. Phys. Lett. A_ , 15 (2000) 2297; * [14] Fan H.-y and Zhang Y., _Phys. Rev. A_ 57 (1998) 3225. * [15] Hu L.-y. and Fan H.-y., _J_. _Mod. Opt._ , 55 (2008) 1065. * [16] Hu L.-y. and Fan H.-y., _Europhys. Lett_., (2009) in press.	arxiv-papers	2009-03-05T14:10:09	2024-09-04T02:49:01.013555	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xue-xiang Xu, Li-yun Hu and Hong-yi Fan", "submitter": "Liyun Hu", "url": "https://arxiv.org/abs/0903.0997" }
0903.1422	# Multiple teleportation via the partially entangled states Meiyu Wang, Fengli Yan flyan@mail.hebtu.edu.cn College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang 050016, China Hebei Advanced Thin Films Laboratory, Shijiazhuang 050016, China ###### Abstract We investigate the multiple teleportation with some nonmaximally entangled channels. The efficiencies of two multiple teleportation protocols, the separate multiple teleportation protocol (SMTP) and the global multiple teleportation protocol (GMTP), are calculated. We show that GMTP is more efficient than SMTP. ###### pacs: 03.67.Hk Quantum teleportation s1 is one of the most significant components in quantum information processing, which allows indirect transmission of quantum information between distant parties by using previously shared entanglement and classical communication between them. Indeed, it is considered as a basic building block of quantum communication nowadays. Not only is it one of the most intriguing phenomena in the quantum world, but also a very useful tool to perform various tasks in quantum information processing and quantum computing s14 ; s15 . For example, controlled quantum gates are implemented by means of quantum teleportation, which is very important in linear optical quantum computation s2 ; s3 . Recently, the original scheme for teleporting a qubit has been widely generalized in many different ways s5 ; s6 ; s7 ; s9 ; s10 ; s11 ; s12 ; s13 ; s16 . In the previous teleportation protocols and in many other applications of teleportation, we want to construct an unknown input state with unity fidelity at another location while destroying the original copy, which is always achieved if two parties share a maximally entangled state. However, it might happen that our parties do not share a maximally entangled state. This limitation can be overcome by distilling out of an ensemble of partially entangled states a maximally entangled one s4 . But this approach requires a large amount of copies of partially entangled states to succeed. Another way to achieve unity fidelity teleportation with limited resources is based on the probabilistic quantum teleportation protocols of Refs. s5 ; s6 ; s7 . Recently, in an interesting work, Modławska and Grudka s8 showed that if the qubit is teleported several times via some nonmaximally entangled states, then the “errors” introduced in the previous teleportations can be corrected by the “errors” introduced in the following teleportations. Their strategy was developed in the framework of the scheme proposed in Ref.s3 for linear optical teleportation. In this paper, we show that this feature of the multiple teleportation of Ref.s8 is not restricted to the teleportation scheme stated in Ref.s3 . Based on the general teleportation language of the original proposal shown in Ref.s1 , we compare the efficiencies of two multiple teleportation protocols, the separate multiple teleportation and the global multiple teleportation. In the former protocol, a complete teleportation including error correction is strictly executed by neighboring parties. On the other hand, in the latter protocol, all errors introduced in the teleportation are corrected by the final receiver. We find the global multiple teleportation is more efficient than the separate multiple teleportation. To illustrate two protocols clearly, let us first begin with the multiple teleportation in the case of three parties. Alice wants to teleport an unknown quantum state $\|\psi\rangle=a\|0\rangle+b\|1\rangle$ (1) to Bob, where $a,b\in C$ and $\|a\|^{2}+\|b\|^{2}=1$. There is no direct entanglment resource between Alice and Bob, fortunately, Alice and the third party Charlie have a partially entangled state $\|\Psi\rangle=\alpha\|00\rangle+\beta\|11\rangle,$ (2) while Charlie and Bob share the same entanglment resource, where $\alpha$ and $\beta$ are real numbers and satisfy $\alpha^{2}+\beta^{2}=1$. Without loss of generality, we suppose $\|\alpha\|\leq\|\beta\|$. The simplest and directest strategy is to perform two separate teleportations, i.e., Alice teleports the quantum state $\|\psi\rangle$ to Charlie via the first teleportation. Then Charlie teleports it to Bob via the second teleportation. Because this procotol consists of two separate teleportations, we call it the separate multiple teleportation procotol (SMTP). According to the standard probabilistic teleportation protocol, in the first separate teleportation, Alice performs the Bell-basis measurement (BM) on the teleported qubit and the entangled qubit in her side. Charlie can apply the corresponding Pauli transformation conditioned on the result of BM, i.e., $I$ if the BM yields $\|\Phi^{+}\rangle$, $\sigma_{z}$ for $\|\Phi^{-}\rangle$, $\sigma_{x}$ for $\|\Psi^{+}\rangle$, and ${\rm i}\sigma_{y}$ for $\|\Psi^{-}\rangle$ , where $I$ is the identity, $\sigma_{x},\sigma_{y},\sigma_{z}$ are standard Pauli matrices and $\|\Phi^{\pm}\rangle=\frac{1}{\sqrt{2}}\left(\|00\rangle\pm\|11\rangle\right),$ (3) $\|\Psi^{\pm}\rangle=\frac{1}{\sqrt{2}}\left(\|01\rangle\pm\|10\rangle\right).$ (4) Finally, the state Charlie received becomes $\|\psi_{\rm 1}\rangle=\frac{1}{\sqrt{p_{1}}}(\alpha a\|0\rangle+\beta b\|1\rangle)$ (5) with the probability $p_{1}=\|a\alpha\|^{2}+\|b\beta\|^{2}$ or $\|\psi_{\rm 2}\rangle=\frac{1}{\sqrt{p_{2}}}(\beta a\|0\rangle+\alpha b\|1\rangle)$ (6) with the probability $p_{2}=\|a\beta\|^{2}+\|b\alpha\|^{2}$. These states are in accordance with the original state $\|\psi\rangle$ only if the quantum channel is a maximally entangled state, i.e. $\alpha=\beta$. For the case of non- maximally entangled channel, there exists the “error” in $\|\psi_{1}\rangle$ and $\|\psi_{2}\rangle$. These states can be returned to the original state with certain probability by performing the generalized measurerment given by Kraus operators: $\displaystyle E_{S1}$ $\displaystyle=\|0\rangle\langle 0\|+\frac{\alpha}{\beta}\|1\rangle\langle 1\|,$ (7a) $\displaystyle E_{F1}$ $\displaystyle=\sqrt{1-\frac{\alpha^{2}}{\beta^{2}}}\|1\rangle\langle 1\|$ (7b) for $\|\psi_{1}\rangle$ and $\displaystyle E_{S2}$ $\displaystyle=\frac{\alpha}{\beta}\|0\rangle\langle 0\|+\|1\rangle\langle 1\|,$ (8a) $\displaystyle E_{F2}$ $\displaystyle=\sqrt{1-\frac{\alpha^{2}}{\beta^{2}}}\|0\rangle\langle 0\|$ (8b) for $\|\psi_{2}\rangle$. When $E_{S}$ is obtained, the qubit ends in its original state $\|\psi\rangle=a\|0\rangle+b\|1\rangle$. The success probability in the first teleportation is $p=\sum_{i=1}^{2}p_{i}\langle\psi_{i}\|E^{\dagger}_{Si}E_{Si}\|\psi_{i}\rangle=2\alpha^{2}.$ (9) Next, Charlie teleports the recovered quantum state to Bob by the similar process. Combining these two teleportations, the total probability that Bob receives the quantum state $\|\psi\rangle$ is $P_{S}=p^{2}=4\alpha^{4}.$ (10) However, the above teleportation protocol is not the optimal strategy. In fact, the third party Charlie does not need to recover the quantum state to be teleported, but teleports the “error state” to Bob directly. Lastly, Bob corrects all “errors” of the quantum state in the teleportation process. Formally, either Alice and Charlie or Charlie and Bob do not complete an intact separate teleportation, so we call it the global multiple teleportation protocol (GMTP). Let us, thus, assume that Charlie does not correct the “error” introduced in the first teleportation, he only makes a Pauli transformation according to Alice’s measurement outcome, then he also performs BM on his two qubits and broadcasts the measurement outcome to Bob. After making the corresponding Pauli transformation conditioned on Charlie’s measurement outcome, Bob’s qubit will collapse into one of the following states $\displaystyle\|\phi_{1}\rangle=\frac{1}{\sqrt{p^{\prime}_{1}}}(\alpha^{2}a\|0\rangle+\beta^{2}b\|1\rangle),$ (11a) $\displaystyle\|\phi_{2}\rangle=\frac{1}{\sqrt{p^{\prime}_{2}}}(\beta^{2}a\|0\rangle+\alpha^{2}b\|1\rangle),$ (11b) $\displaystyle\|\phi_{3}\rangle=a\|0\rangle+b\|1\rangle)$ (11c) with the probabilities $p^{\prime}_{1}=\alpha^{4}\|a\|^{2}+\beta^{4}\|b\|^{2},$ $p^{\prime}_{2}=\beta^{4}\|a\|^{2}+\alpha^{4}\|b\|^{2},$ $p^{\prime}_{3}=2\alpha^{2}\beta^{2}$ respectively. When the state is in $\|\phi_{3}\rangle$, we do not have to perform the error correction. It is very joyful to see that the second teleportation corrects the “error” introduced by the first teleportation. This effect is called error self-correction. For $\|\phi_{1}\rangle$ and $\|\phi_{2}\rangle$, one can recover the original state by performing generalized measurement given by Kraus operators: $\displaystyle E^{\prime}_{S1}$ $\displaystyle=\|0\rangle\langle 0\|+\frac{\alpha^{2}}{\beta^{2}}\|1\rangle\langle 1\|,$ (12a) $\displaystyle E^{\prime}_{F1}$ $\displaystyle=\sqrt{1-\frac{\alpha^{4}}{\beta^{4}}}\|1\rangle\langle 1\|$ (12b) and $\displaystyle E^{\prime}_{S2}$ $\displaystyle=\frac{\alpha^{2}}{\beta^{2}}\|0\rangle\langle 0\|+\|1\rangle\langle 1\|,$ (13a) $\displaystyle E^{\prime}_{F2}$ $\displaystyle=\sqrt{1-\frac{\alpha^{4}}{\beta^{4}}}\|0\rangle\langle 0\|$ (13b) respectively. The total probability of successfully recovering the original state is $P_{G}(3)=2\alpha^{2}\beta^{2}+\sum_{i=1}^{2}p^{\prime}_{i}\langle\phi_{i}\|E^{\prime{\dagger}}_{Si}E^{\prime}_{Si}\|\phi_{i}\rangle=2\alpha^{2}.$ (14) The ratio of efficiency of GMTP to that of SMTP $P_{G}(3)/P_{S}=\frac{1}{2\alpha^{2}}.$ (15) We can easily see $P_{G}/P_{S}\geq 1$ because of $\alpha\leq\frac{1}{\sqrt{2}}$ . It is obvious that for the maximally entangled channel, the two protocols are equivalent, but for the partially entangled channel, GMTP is more efficient than SMTP. Moreover, the less $\alpha$ is, the more efficient the GMTP is. It is straightforward to generalize the above two protocols to arbitrary parties. Let us first discuss the GMTP. Since error self-correction only appears in the even times Bell-basis measurements, so here we discuss the $(2N+1)$-party teleportation. Suppose that Alice 1 wants to teleport a quantum state $\|\psi\rangle=a\|0\rangle+b\|1\rangle$ to Alice $2N+1$. There is no direct entanglement resource between them, but they can link through $2N-1$ intermediaries called Alice 2, Alice 3, $\cdots$, Alice $2N$, respectively. Two neighboring parties share the partially entangled state described by Eq.(2). They can complete the task through cooperative teleportation. After $2N$ Bell-basis measurements and corresponding Pauli transformations conditioned on previous parties, the final receiver’s qubit will be in one of the states $\|\phi_{i}\rangle=\frac{1}{\sqrt{p^{G}_{i}}}(\alpha^{2N-i}\beta^{i}a\|0\rangle+\alpha^{i}\beta^{2N-i}b\|1\rangle),\\\ $ (16) with the probability $C_{2N}^{i}p_{i}^{G}\equiv C_{2N}^{i}(\alpha^{2(2N-i)}\beta^{2i}\|a\|^{2}+\alpha^{2i}\beta^{2(2N-i)}\|b\|^{2})$, $i=0,1,2,\cdots,2N$. By correcting the error, the total success probability is $P_{G}(2N+1)=C_{2N}^{N}\alpha^{2N}\beta^{2N}+2\sum_{i=0}^{N-1}C_{2N}^{i}\alpha^{2(2N-i)}\beta^{2i}.$ (17) On the other hand, in the case of SMTP, we must perform $2N$ separate teleportations, then the total success probability equals $P_{S}(2N+1)=p^{2N}=2^{2N}\alpha^{4N}.$ (18) It is easy to verify that $P_{G}(2N+1)\geq P_{S}(2N+1)$. In order to show how the total success probabilities of two protocols depend on the entanglement of channels for different $N$, we will choose concurrence $C$ defined by Wootters as a convenient measure of entanglement Wootters . The concurrence varies from $C=0$ of a separable state to $C=1$ of a maximally entangled state. For a pure partially entangled state described by Eq. (2), the concurrence may be expressed explicitly by $C=2\|\alpha\beta\|$. In Fig.1, we plot $P_{S}$ and $P_{G}$ as the function of concurrence $C$ for different $N$. We can see that both the total success probabilities of two protocols declines with the decrease of the entanglement of channels. Moreover, the greater $N$ is, the more sharper the success probabilities declines. It shows that the quantum channel with small entanglement will become unpractical with the increase of $N$. Fig.1 also indicates explicitly that the GMTP is more efficient than SMTP. For example, for the case of $N=10$, the total success probability of GMTP $P_{G}\approx 21\%$ while the total success probability of SMTP $P_{S}$ only attains $0.14\%$ when the concurrence of channels is $C=0.96$. Figure 1: The total success probability $P_{G}$ and $P_{S}$ versus concurrence $C$ for different $N$ (Solid line: $P_{S}$, dashed line: $P_{G}$). From top to bottom, $N$ corresponding takes $1,5,10$. The ratio of $P_{G}$ to $P_{S}$ as a function of $C$ for different $N$ is illustrated in Fig.2. Here we only take the concurrence from $0.9$ to $1$ because the small entanglement channels are unpractical for large $N$. From Fig.2, we can see that the greater $N$ is, the larger $P_{G}/P_{S}$ is. In other words, the efficiency of GMTP is far higher than that of SMTP when the steps of teleportation increase. Figure 2: The ratio $P_{G}$ to $P_{S}$versus concurrence $C$ for different $N$ . From bottom to top, $N$ corresponding takes $1,5,10$. When the entanglement of the quantum channel is different between neighboring parties, the circumstance becomes complicated. Here we only consider the case of three parties . Alice wants to teleport an unknown quantum state $\|\psi\rangle=a\|0\rangle+b\|1\rangle$ to Bob. There is no direct entanglement resource between Alice and Bob, fortunately, Alice and the third party Charlie share a partially entangled state $\|\Psi_{1}\rangle=\alpha_{1}\|00\rangle+\beta_{1}\|11\rangle,$ (19) while Charlie and Bob share another entanglement resource $\|\Psi_{2}\rangle=\alpha_{2}\|00\rangle+\beta_{2}\|11\rangle,$ (20) where $\alpha_{i}$ and $\beta_{i}$ are real numbers and satisfy $\|\alpha_{i}\|\leq\|\beta_{i}\|$ and $\alpha_{i}^{2}+\beta_{i}^{2}=1$. After two Bell-basis measurements and Pauli operations, the qubit of Bob will be in one of the following states $\|\psi_{ij}\rangle=\frac{1}{\sqrt{p_{ij}}}(\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}a\|0\rangle+\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}b\|1\rangle)$ (21) with the probabilities $p_{ij}=\|\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}a\|^{2}+\|\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}b\|^{2}(i,j=0,1)$ respectively. The qubit can be returned to its original state by performing the generalized measurement given by Kraus operators: $\displaystyle E_{S}$ $\displaystyle=\|0\rangle\langle 0\|+\frac{\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}}{\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}}\|1\rangle\langle 1\|,$ (22a) $\displaystyle E_{F}$ $\displaystyle=\sqrt{1-\frac{\|\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}\|^{2}}{\|\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}\|^{2}}}\|1\rangle\langle 1\|$ (22b) for $\|\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}\|\leq\|\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}\|$. A similar measurement exists if $\|\alpha_{1}^{1-i}\alpha_{2}^{1-j}\beta_{1}^{i}\beta_{2}^{j}\|\geq\|\alpha_{1}^{i}\alpha_{2}^{j}\beta_{1}^{1-i}\beta_{2}^{1-j}\|$. By tedious but standard calculation we can obtain the success probability of teleportation $P=\min\\{2\alpha_{1}^{2},2\alpha_{2}^{2}\\}.$ (23) It is an interesting result, the success probability of teleportation is completely determined by the channel of less entanglement. For another channel of more entanglement, its entanglement does not affect the success probability at all. In other words, the channel of more entanglement is equivalent to the maximally entangled channel in the total teleportation process. In summary, we have presented two multiple teleportation protocols via some partially entangled state, the separate multiple teleportation and the globe multiple teleportation. In the former protocol, a complete teleportation including error correction is strictly executed by neighboring parties. However, in the latter protocol, all errors introduced in the teleportation are corrected by the final receiver. It has been shown that the property of self error-correction is a general feature of multiple teleportations, not being restricted to the scheme proposed in Ref.s3 . We also have compared the efficiencies of the two multiple teleportation protocols and found the globe multiple teleportation is more efficient than the separate multiple teleportation due to the property of self error-correction. ## References * (1) C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). * (2) X.B. Wang, T. Hiroshima, A. Tomita, and M. Hayashi, Phys. Rep. 448, 1 (2007). * (3) G.L. Long, F.G. Deng, C. Wang, X.H. Li, K. Wen, and W.Y. Wang, Front. Phys. China 2, 251 (2007). * (4) D. Gottesman and I.L. Chuang, Nature (London) 402, 390 (1999). * (5) E. Knill, R. Laflamme, and G.J. Milburn, Nature (London) 409, 46 (2001). * (6) P. Agrawal and A.K. Pati, Phys. Lett. A 305, 12 (2002). * (7) G. Gordon and G. Rigolin, Phys. Rev. A 73, 042309 (2006). * (8) W.L. Li, C.F. Li, and G.C. Guo, Phys. Rev. A 61, 034301 (2000). * (9) F.G. Deng, C.Y. Li, Y.S. Li, H.Y. Zhou, and Y. Wang, Phys. Rev. A 72, 022338 (2005). * (10) F.L. Yan and D. Wang, Phys. Lett. A 316, 297 (2003). * (11) T. Gao, F.L. Yan, and Y.C. Li, Europhys. Lett. 84, 50001 (2008). * (12) M.Y. Wang and F.L. Yan, Phys. Lett. A 355, 94 (2006). * (13) M.Y. Wang, F.L. Yan, T. Gao, and Y.C. Li, International Journal of Quantum Information 6, 201 (2008). * (14) T. Gao, F.L. Yan, and Z.X. Wang, Quantum Information and Computation 4, 186 (2004). * (15) C.H. Bennett1, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, and W.K. Wootters, Phys. Rev. Lett. 76, 722 (1996). * (16) J. Modławska and A. Grudka, Phys. Rev. Lett. 100, 110503 (2008). * (17) W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)	arxiv-papers	2009-03-08T13:25:39	2024-09-04T02:49:01.024076	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Meiyu Wang, Fengli Yan", "submitter": "Ting Gao", "url": "https://arxiv.org/abs/0903.1422" }
0903.1429	# Remote Preparation of the Two-Particle State YAN Feng-Li, ZHANG Guo-Hua College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang 050016, China ###### Abstract We present a scheme of remote preparation of the two-particle state by using two Einstein-Podolsky-Rosen pairs or two partial entangled two-particle states as the quantum channel. The probability of the successful remote state preparation is obtained. ###### pacs: 03.65.Ta, 03.67.Hk, 03.67.Lx Quantum entanglement is a valuable resource for the implementation of quantum computation and quantum communication protocols, like quantum teleportation [1,2], quantum key distribution [3,4], quantum secure direct communication [5], dense coding [6-8], quantum computation [9], remote state preparation [10-15] and so on. Quantum teleportation is regarded as one of the most profound results of quantum information theory. In the original quantum teleportation protocol of Bennett et al. [1], it was showed that an unknown state of a qubit can be perfectly transported from a sender Alice to a remote receiver Bob with the aid of long-range Einstein-Podolsky-Rosen correlation and transmission of two bits of classical information without transmission of the carrier of the quantum state. They have also generalized the protocol for an unknown qubit by using a maximally entangled state in $d\times d$ dimensional Hilbert space and sending $2{\rm log}_{2}d$ bits of classical information. Another important application of quantum entanglement is remote state preparation, where two spatially distant people Alice and Bob can prepare an quantum state known to Alice but unknown to Bob, with the aid of previously shared quantum entanglement and the classical communication. Recently, Pati [10] presented a protocol of remotely preparing a special ensemble of states. Lo [11] showed that remote state preparation requires less classical communication than teleportation for the special ensembles of states, but for general states the classical communication cost of teleportation would be equal to that of remote state preparation. Bennett et al. [12] studied the trade off between entanglement cost and classical communication cost in remote state preparation. Since then, some people have investigated various theoretical protocols about generalization of remote state preparation. Zhan [13] gave a scheme for preparing remotely a three-particle GHZ state. Huang et al. [14] put forward a protocol for preparing remotely the multipartite pure state. A scheme for preparing remotely a two-particle entangled state via two pairs of entangled particles was presented by Liu et al. [15]. In this paper, we propose a scheme of preparing remotely the two-particle state. Two Einstein-Podolsky-Rosen pairs and two partial entangled two- particle states as the quantum channel are considered, respectively. The probability of the successful remote state preparation is calculated. Let us first begin our remote state preparation with two Einstein-Podolsky- Rosen pairs as the quantum channel. We suppose that a sender Alice wants to help a remote receiver Bob to prepare a two-particle state in the following formation $\|\phi\rangle=\alpha\|00\rangle+\beta\|01\rangle+\gamma\|10\rangle+\delta\|11\rangle,$ (1) where $\alpha$ and $\gamma$ are real numbers, $\beta$ and $\delta$ are complex numbers and $\beta^{}\delta$ is real, $\|\alpha\|^{2}+\|\beta\|^{2}+\|\gamma\|^{2}+\|\delta\|^{2}=1$. We suppose that Alice knows $\alpha$, $\beta$, $\gamma$ and $\delta$ completely, but Bob does not know them at all. We also assume that the quantum channel shared by Alice and Bob is composed of two Einstein-Podolsky-Rosen pairs $\|\psi\rangle_{12}=\frac{1}{\sqrt{2}}(\|00\rangle+\|11\rangle)_{12},$ (2) $\|\psi\rangle_{34}=\frac{1}{\sqrt{2}}(\|00\rangle+\|11\rangle)_{34},$ (3) where particles 1 and 3 belong to Alice while Bob has particles 2 and 4. In order to help Bob to remotely prepare a two-particle state stated in Eq.(1) on the particles 2 and 4, Alice must make a measurement on her two particles 1 and 3. The measurement basis chosen by Alice is a set of mutually orthogonal basis vectors $\\{\|\varphi\rangle_{13},\|\varphi_{\perp}\rangle_{13},\|\psi\rangle_{13},\|\psi_{\perp}\rangle_{13}\\}$, where $\displaystyle\|\varphi\rangle_{13}=(\alpha\|00\rangle+\beta\|01\rangle+\gamma\|10\rangle+\delta\|11\rangle)_{13},$ $\displaystyle\|\varphi_{\perp}\rangle_{13}=(-\delta^{}\|00\rangle+\gamma\|01\rangle-\beta^{}\|10\rangle+\alpha\|11\rangle)_{13},$ $\displaystyle\|\psi\rangle_{13}=(\gamma\|00\rangle+\delta\|01\rangle-\alpha\|10\rangle-\beta\|11\rangle)_{13},$ $\displaystyle\|\psi_{\perp}\rangle_{13}=(\beta^{}\|00\rangle-\alpha\|01\rangle-\delta^{}\|10\rangle+\gamma\|11\rangle)_{13}.$ Here $\\{\|00\rangle_{13},\|01\rangle_{13},\|10\rangle_{13},\|11\rangle_{13}\\}$ is the computation basis. Then we have $\displaystyle\|\psi\rangle_{12}\otimes\|\psi\rangle_{34}$ $\displaystyle=$ $\displaystyle\frac{1}{2}[\|\varphi\rangle_{13}(\alpha\|00\rangle+\beta^{}\|01\rangle+\gamma\|10\rangle+\delta^{}\|11\rangle)_{24}$ $\displaystyle+\|\varphi_{\perp}\rangle_{13}(-\delta\|00\rangle+\gamma\|01\rangle-\beta\|10\rangle+\alpha\|11\rangle)_{24}$ $\displaystyle+\|\psi\rangle_{13}(\gamma\|00\rangle+\delta^{}\|01\rangle-\alpha\|10\rangle-\beta^{}\|11\rangle)_{24}$ $\displaystyle+\|\psi_{\perp}\rangle_{13}(\beta\|00\rangle-\alpha\|01\rangle-\delta\|10\rangle+\gamma\|11\rangle)_{24}].$ Thus if Alice performs a measurement in the basis $\\{\|\varphi\rangle_{13},\|\varphi_{\perp}\rangle_{13},\|\psi\rangle_{13},\|\psi_{\perp}\rangle_{13}\\}$ on her two particles 1 and 3, then each outcome will occur with the equal probability $\frac{1}{4}$. If Alice’s measurement result is $\|\varphi_{\perp}\rangle$, then particles 2 and 4 will collapse into the state $(-\delta\|00\rangle+\gamma\|01\rangle-\beta\|10\rangle+\alpha\|11\rangle)_{24}.$ (6) After that if Alice communicates to Bob of her actual measurement outcome via a classical channel, then Bob will be able to apply the following unitary transformation $U=(\|0\rangle\langle 1\|+\|1\rangle\langle 0\|)_{2}\otimes(\|0\rangle\langle 1\|-\|1\rangle\langle 0\|)_{4}$ (7) on his particles 2 and 4. The resulting state of Bob’s particles will be the original state $\|\phi\rangle$. Likewise if the actual result of Alice’s measurement is $\|\psi_{\perp}\rangle_{13}$, then Bob gets the state $(\beta\|00\rangle-\alpha\|01\rangle-\delta\|10\rangle+\gamma\|11\rangle)_{24}.$ (8) When Bob received the classical information of the actual measurement result sent by Alice, he can perform an appropriate operation $U^{\prime}=(\|0\rangle\langle 0\|-\|1\rangle\langle 1\|)_{2}\otimes(\|0\rangle\langle 1\|-\|1\rangle\langle 0\|)_{4}$ (9) on his particles 2 and 4 to obtain the state $\|\phi\rangle$. So when these two measurement outcomes happen, Alice can help Bob to remotely prepare the two- particle state $\|\phi\rangle$. However, when the measurement outcome is $\|\varphi\rangle_{13}$ ($\|\psi\rangle_{13}$), the remote state preparation can not be successful, as the state of the particles 2 and 4 will be $(\alpha\|00\rangle+\beta^{}\|01\rangle+\gamma\|10\rangle+\delta^{}\|11\rangle)_{24}$ ( $(\gamma\|00\rangle+\delta^{}\|01\rangle-\alpha\|10\rangle-\beta^{}\|11\rangle)_{24}$). Because Bob does not know the coefficients $\alpha$, $\beta$, $\gamma$ and $\delta$ at all, he cannot transform either the state $(\alpha\|00\rangle+\beta^{}\|01\rangle+\gamma\|10\rangle+\delta^{}\|11\rangle)_{24}$ or $(\gamma\|00\rangle+\delta^{}\|01\rangle-\alpha\|10\rangle-\beta^{}\|11\rangle)_{24}$ into the state $\|\phi\rangle$. But, if $\alpha$, $\beta$, $\gamma$ and $\delta$ are real numbers, the situation would be changed. When Alice’s measurement result $\|\varphi\rangle_{13}$ or $\|\psi\rangle_{13}$ occurs, it is not difficult for Bob to prepare the two-particle state $\|\phi\rangle$ by performing the suitable unitary operation determined by the outcome of Alice’s measurement. Here we omit the concrete steps. Now, we present a scheme for preparing remotely a two-particle state via two non-maximally entangled states. Suppose that Alice still wishes to help Bob to prepare remotely the state $\|\phi\rangle$ in Eq.(1), but the two entangled states shared by Alice and Bob are two non-maximally entangled states $\|\psi\rangle_{12}=(a\|00\rangle+b\|11\rangle)_{12},$ (10) $\|\psi\rangle_{34}=(c\|00\rangle+d\|11\rangle)_{34},$ (11) where the parameters $a$, $b$, $c$ and $d$ are real numbers, $\|a\|^{2}+\|b\|^{2}=1$, $\|c\|^{2}+\|d\|^{2}=1$, and $\|a\|\leq\|b\|,$ $\|c\|\leq\|d\|$. We also assume that the particles 1 and 3 belong to Alice while Bob has particles 2 and 4. Since Alice knows the parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ exactly, she can perform a measurement on particles 1 and 3 in the basis $\\{\|\varphi\rangle_{13},\|\varphi_{\perp}\rangle_{13},\|\psi\rangle_{13},\|\psi_{\perp}\rangle_{13}\\}$. A simple algebraic rearrangement of the expression $\|\psi\rangle_{12}\otimes\|\psi\rangle_{34}$ in terms of the states $\|\varphi\rangle_{13},$ $\|\varphi_{\perp}\rangle_{13},$ $\|\psi\rangle_{13},$ $\|\psi_{\perp}\rangle_{13}$ yields $\displaystyle\|\psi\rangle_{12}\otimes\|\psi\rangle_{34}$ $\displaystyle=$ $\displaystyle\|\varphi\rangle_{13}(ac\alpha\|00\rangle+ad\beta^{}\|01\rangle+bc\gamma\|10\rangle+bd\delta^{}\|11\rangle)_{24}$ $\displaystyle+\|\varphi_{\perp}\rangle_{13}(-ac\delta\|00\rangle+ad\gamma\|01\rangle- bc\beta\|10\rangle+bd\alpha\|11\rangle)_{24}$ $\displaystyle+\|\psi\rangle_{13}(ac\gamma\|00\rangle+ad\delta^{}\|01\rangle- bc\alpha\|10\rangle-bd\beta^{}\|11\rangle)_{24}$ $\displaystyle+\|\psi_{\perp}\rangle_{13}(ac\beta\|00\rangle-ad\alpha\|01\rangle- bc\delta\|10\rangle+bd\gamma\|11\rangle)_{24}.$ Therefore, if the actual result of Alice’s measurement on the two particles 1 and 3 is $\|\varphi_{\perp}\rangle_{13}$ with the probability $\|bd\alpha\|^{2}+\|bc\beta\|^{2}+\|ad\gamma\|^{2}+\|ac\delta\|^{2}$ then the state of particles 2 and 4 will be $\|\varphi\rangle_{24}=\frac{(-ac\delta\|00\rangle+ad\gamma\|01\rangle- bc\beta\|10\rangle+bd\alpha\|11\rangle)_{24}}{\sqrt{\|bd\alpha\|^{2}+\|bc\beta\|^{2}+\|ad\gamma\|^{2}+\|ac\delta\|^{2}}}.$ (13) When Bob is informed the actual measurement outcome $\|\varphi_{\perp}\rangle_{13}$ by Alice via a classical channel, he can get the original state described in Eq.(1) with certain probability. Firstly, Bob operates a unitary operation $U_{1}=(\|0\rangle\langle 1\|+\|1\rangle\langle 0\|)_{2}\otimes(\|0\rangle\langle 1\|-\|1\rangle\langle 0\|)_{4}$ (14) on particles 2 and 4. Obviously $U_{1}$ will transform the state $\|\varphi\rangle_{24}$ into $\|\varphi^{\prime}\rangle_{24}=\frac{(bd\alpha\|00\rangle+bc\beta\|01\rangle+ad\gamma\|10\rangle+ac\delta\|11\rangle)_{24}}{\sqrt{\|bd\alpha\|^{2}+\|bc\beta\|^{2}+\|ad\gamma\|^{2}+\|ac\delta\|^{2}}}.$ (15) Secondly, Bob introduces an auxiliary two-level particle $a$ with the initial state $\|0\rangle_{a}$ and performs a collective unitary transformation $\displaystyle U_{2}=\left(\begin{array}[]{cccccccc}\frac{ac}{bd}&A&0&0&0&0&0&0\\\ A&-\frac{ac}{bd}&0&0&0&0&0&0\\\ 0&0&\frac{a}{b}&B&0&0&0&0\\\ 0&0&B&-\frac{a}{b}&0&0&0&0\\\ 0&0&0&0&\frac{c}{d}&C&0&0\\\ 0&0&0&0&C&-\frac{c}{d}&0&0\\\ 0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&-1\\\ \end{array}\right)$ (24) on particles 2, 4 and $a$ under the basis $\\{\|000\rangle_{24a},\|001\rangle_{24a},\|010\rangle_{24a},\|011\rangle_{24a},\|100\rangle_{24a},\|101\rangle_{24a},$ $\|110\rangle_{24a},\|111\rangle_{24a}\\},$ where $\begin{array}[]{ccc}A=\sqrt{1-(\frac{ac}{bd})^{2}},&B=\sqrt{1-(\frac{a}{b})^{2}},&C=\sqrt{1-(\frac{c}{d})^{2}}.\end{array}$ Since it has been assumed that $\|a\|\leq\|b\|$ and $\|c\|\leq\|d\|$, so one has $\|ac\|^{2}\leq\|bd\|^{2}$. The unitary transformation $U_{2}$ will transform $\|\varphi^{\prime}\rangle_{24}\|0\rangle_{a}$ into $\displaystyle\|\varphi^{\prime\prime}\rangle_{24a}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{\|bd\alpha\|^{2}+\|bc\beta\|^{2}+\|ad\gamma\|^{2}+\|ac\delta\|^{2}}}$ $\displaystyle{[ac(\alpha\|00\rangle+\beta\|01\rangle+\gamma\|10\rangle+\delta\|11\rangle)_{24}}\|0\rangle_{a}$ $\displaystyle+(\alpha\sqrt{(bd)^{2}-(ac)^{2}}\|00\rangle+c\beta\sqrt{b^{2}-a^{2}}\|01\rangle$ $\displaystyle+a\gamma\sqrt{d^{2}-c^{2}}\|10\rangle)_{24}\|1\rangle_{a}].$ Finally, Bob performs a measurement on auxiliary particle $a$ in the basis $\\{\|0\rangle_{a},\|1\rangle_{a}\\}$. If the result of his measurement is $\|1\rangle_{a}$, then the remote preparation of the original state fails. If the measurement outcome $\|0\rangle_{a}$ occurs with probability $\frac{\|ac\|^{2}}{\|bd\alpha\|^{2}+\|bc\beta\|^{2}+\|ad\gamma\|^{2}+\|ac\delta\|^{2}}$, then the remote preparation of a two-particle state $\|\phi\rangle$ is successfully realized. Evidently, when actual measurement outcome $\|\varphi_{\perp}\rangle_{13}$ is obtained, the probability of successfully remote state preparation is $(\|bd\alpha\|^{2}+\|bc\beta\|^{2}+\|ad\gamma\|^{2}+\|ac\delta\|^{2})\frac{\|ac\|^{2}}{\|bd\alpha\|^{2}+\|bc\beta\|^{2}+\|ad\gamma\|^{2}+\|ac\delta\|^{2}}=\|ac\|^{2}$. Similarly, by Eq.(12), if Alice’s measurement result on particles 1 and 3 is $\|\psi_{\perp}\rangle_{13}$ with the probability $\|ad\alpha\|^{2}+\|ac\beta\|^{2}+\|bd\gamma\|^{2}+\|bc\delta\|^{2}$, the state of particles 2 and 4 will become $\|\psi\rangle_{24}=\frac{(ac\beta\|00\rangle-ad\alpha\|01\rangle- bc\delta\|10\rangle+bd\gamma\|11\rangle)_{24}}{\sqrt{\|ad\alpha\|^{2}+\|ac\beta\|^{2}+\|bd\gamma\|^{2}+\|bc\delta\|^{2}}}.$ (26) Now Bob operates the following unitary transformation $U^{\prime}_{1}=(\|0\rangle\langle 0\|-\|1\rangle\langle 1\|)_{2}\otimes(-\|0\rangle\langle 1\|+\|1\rangle\langle 0\|)_{4}$ (27) on particles 2 and 4. Hence the state shown in Eq.(18) was transformed into $\|\psi^{\prime}\rangle_{24}=\frac{(ad\alpha\|00\rangle+ac\beta\|01\rangle+bd\gamma\|10\rangle+bc\delta\|11\rangle)_{24}}{\sqrt{\|ad\alpha\|^{2}+\|ac\beta\|^{2}+\|bd\gamma\|^{2}+\|bc\delta\|^{2}}}.$ (28) Next, Bob introduces an auxiliary particle $a$ with the initial state $\|0\rangle_{a}$ and performs a unitary transformation $\displaystyle U^{\prime}_{2}=\left(\begin{array}[]{cccccccc}\frac{c}{d}&C&0&0&&0&0&0\\\ C&-\frac{c}{d}&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&-1&0&0&0&0\\\ 0&0&0&0&\frac{ac}{bd}&A&0&0\\\ 0&0&0&0&A&-\frac{ac}{bd}&0&0\\\ 0&0&0&0&0&0&\frac{a}{b}&B\\\ 0&0&0&0&0&0&B&-\frac{a}{b}\\\ \end{array}\right)$ (37) on particles 2, 4 and $a$ under the basis $\\{\|000\rangle_{24a},$ $\|001\rangle_{24a},$ $\|010\rangle_{24a},\|011\rangle_{24a},$ $\|100\rangle_{24a},$ $\|101\rangle_{24a},$ $\|110\rangle_{24a},$ $\|111\rangle_{24a}\\}$. It is straightforward to verify that the resulting state of of particles 2, 4 and $a$ is $\displaystyle\|\psi^{\prime\prime}\rangle_{24a}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{\|ad\alpha\|^{2}+\|ac\beta\|^{2}+\|bd\gamma\|^{2}+\|bc\delta\|^{2}}}$ $\displaystyle[ac(\alpha\|00\rangle+\beta\|01\rangle+\gamma\|10\rangle+\delta\|11\rangle)_{24}\|0\rangle_{a}$ $\displaystyle+(a\alpha\sqrt{d^{2}-c^{2}}\|00\rangle+\gamma\sqrt{(bd)^{2}-(ac)^{2}}\|10\rangle$ $\displaystyle+c\delta\sqrt{b^{2}-a^{2}}\|11\rangle)_{24}\|1\rangle_{a}].$ The above equation shows that Bob can construct a two-particle state, which Alice wishes to prepare remotely, with certain probability by performing a measurement on auxiliary particle $a$ in the basis $\\{\|0\rangle_{a},\|1\rangle_{a}\\}$. If Bob’s actual measurement result is $\|0\rangle_{a}$, then remote state preparation is successful; otherwise remote state preparation fails. It is easy to prove that the successful probability of remote state preparation in this case is $(\|ad\alpha\|^{2}+\|ac\beta\|^{2}+\|bd\gamma\|^{2}+\|bc\delta\|^{2})\frac{\|ac\|^{2}}{\|ad\alpha\|^{2}+\|ac\beta\|^{2}+\|bd\gamma\|^{2}+\|bc\delta\|^{2}}=\|ac\|^{2}$. However, if the Alice’s actual measurement outcome on particles 1 and 3 is $\|\varphi\rangle_{13}$ ($\|\psi\rangle_{13}$), Bob will obtain the state $(ac\alpha\|00\rangle+ad\beta^{}\|01\rangle+bc\gamma\|10\rangle+bd\delta^{}\|11\rangle)_{24}$ ($(ac\gamma\|00\rangle+ad\delta^{}\|01\rangle-bc\alpha\|10\rangle- bd\beta^{}\|11\rangle)_{24}$). Since Bob has no knowledge of these states, he can not unitary convert each of them into the original state, so remote state preparation fails. But, when $\alpha$, $\beta$, $\gamma$ and $\delta$ are real numbers, the situation is not the same. In this case Bob can prepare the state in every Alice’s measurement outcome with certain probability. For saving space we omit the concrete steps for preparing. Synthesizing two cases (the Alice’s the actual measurement outcome is either $\|\varphi_{\perp}\rangle_{13}$ or $\|\psi_{\perp}\rangle_{13}$) the probability of the successful remote state preparation is $2\|ac\|^{2}$. If $\|a\|=\|b\|=\|c\|=\|d\|=\frac{1}{\sqrt{2}},$ namely, the quantum channel consists of two Einstein-Podolsky-Rosen pairs, the probability equals to 50%. In summary, we have presented a scheme of the remote preparation of a two- particle state via two Einstein-Podolsky-Rosen pairs and two partial entangled two-particle state, respectively. The two-particle state can be prepared probabilistically if the sender performs a measurement and the receiver introduces an appropriate unitary transformation with the aid of long-range Einstein-Podolsky-Rosen correlation and transmission of two bits of classical information. Furthermore we obtained the probability of the successful remote state preparation of the two-particle state. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant No: 10671054 and Hebei Natural Science Foundation of China under Grant Nos: A2005000140;07M006, and the Key Project of Science and Technology Research of Education Ministry of China under Grant No:207011. ## References (1) Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A, and Wootters W K 1993 Phys. Rev. Lett. 70 1895 * (2) Gao T, Wang Z X and Yan F L 2003 Chin. Phys. Lett. 20 2094 * (3) Ekert A K 1991 Phys. Rev. Lett. 67 661 * (4) Wang X B, Hiroshima T, Tomita A and Hayashi M 2007 Phys. Rep. 448 1 * (5) Long G L, Deng F G, Wang C, Li X H, Wen K and Wang W Y 2007 Front. Phys. China, 2 251 * (6) Bennett C H and Wiesner S J 1992 Phys. Rev. Lett. 69 2881 * (7) Wang M Y, Yang L G and Yan F L 2005 Chin. Phys. Lett. 22 1053 * (8) Yan F L, Wang M Y 2004 Chin. Phys. Lett. 21 1195 * (9) Barenco A 1996 Contemp. Phys. 37 375 * (10) Pati A K 2000 Phys. Rev. A 63 014302 * (11) Lo H K 2000 Phys. Rev. A 62 012313 * (12) Bennett C H, DiVincenzo D P, Shor P W, Smolin J A, Trehal B M and Wootters W K 2001 Phys. Rev. Lett. 87 077902 * (13) Zhan Y B 2005 Commun. Theor. Phys. 43 637 * (14) Huang Y X and Zhan M S 2004 Phys. Lett. A 327 404 * (15) Liu J M and Wang Y Z 2003 Phys. Lett. A 316 159	arxiv-papers	2009-03-08T14:06:52	2024-09-04T02:49:01.027986	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yan Feng-Li, Zhang Guo-Hua", "submitter": "Ting Gao", "url": "https://arxiv.org/abs/0903.1429" }
0903.1449	# Alloy Stabilized Wurtzite Ground State Structures of Zinc-Blende Semiconducting Compounds H. J. Xiang National Renewable Energy Laboratory, Golden, Colorado 80401, USA Su-Huai Wei National Renewable Energy Laboratory, Golden, Colorado 80401, USA Shiyou Chen Surface Science Laboratory and Department of Physics, Fudan University, Shanghai 200433, China X. G. Gong Surface Science Laboratory and Department of Physics, Fudan University, Shanghai 200433, China ###### Abstract The ground state structures of the AxB1-xC wurtzite (WZ) alloys with $x=$0.25, 0.5, and 0.75 are revealed by a ground state search using the valence-force field model and density-functional theory total energy calculations. It is shown that the ground state WZ alloy always has a lower strain energy and formation enthalpy than the corresponding zinc-blende (ZB) alloy. Therefore, we propose that the WZ phase can be stabilized through alloying. This novel idea is supported by the fact that the WZ AlP0.5Sb0.5, AlP0.75Sb0.25, ZnS0.5Te0.5, and ZnS0.75Te0.25 alloys in the lowest energy structures are more stable than the corresponding ZB alloys. To our best knowledge, this is the first example where the alloy adopts a structure distinct from both parent phases. ###### pacs: 61.50.Ah,61.66.Dk,64.70.kg,71.15.Nc III-V and II-VI semiconductors usually crystallize into one of two forms: hexagonal wurtzite (WZ) and cubic zinc blende (ZB) structures. The ZB and WZ structures have the same local tetrahedral environment and start to differ only in their third-nearest-neighbor atomic arrangement. Despite the structural similarity, there are some significant differences in the electronic and optical properties Yeh1994 ; Schlfgaarde1997 . Compared to the hexagonal structure, the cubic phase has a more isotropic property, higher carrier mobility, lower phonon scattering, and often better doping efficiency. In contrast, the WZ phase has a larger band gap (usually direct), a spontaneous electric polarization, and a lower propagating speed of dislocations and thus an improved lifetime of the laser diodes Sugiura1997 . For certain device applications, one phase is preferred over the other. To have a controllable way to synthesize the desired phase, it is important to understand the mechanism for stabilizing a certain structure. In general, the WZ structure is preferred over the ZB structure when the ionicity of a compound is high Garcia1993 . This is because the ideal WZ structure has a larger Coulomb interaction energy with a larger Madelung constant, whereas the ZB structure leads to a better covalent bond formation Phillips1973 ; John1974 ; Chelikowsky1978 ; Yeh1992 . To change the stability, one often grows materials into different forms. For example, many ZB compounds can adopt the hexagonal WZ structure when forming nanowires (NWs) Koguchi1992 ; Shan2006 ; Patriarche2008 . Empirical calculations suggested that the stability of the WZ NW is due to the fact that the WZ NW has less surface atoms than the ZB NW with a similar diameter Akiyama2006 ; Dubrovskii2008 . Theoretical calculations also showed that stability of WZ compounds such as GaN can be changed when carriers are introduced through doping Dalpian2004 ; Dalpian2006 . Moreover, metastable phases can be synthesized by employing non- equilibrium growth techniques. For example, metastable ZB GaN can be grown on cubic substrates Lazarov2005 . In this paper, we show for the first time that the ground state (GS) WZ alloy (WZA) always has a lower strain energy than the corresponding ZB alloy (ZBA). Therefore, if strain energy is dominant in alloy formation, stable GS ternary WZAs can form even though the binary constituents are more stable in the ZB phase. This provides an opportunity to form desired WZAs through alloying. Our first principles calculations confirm this idea, showing that WZ AlP0.5Sb0.5, AlP0.75Sb0.25, ZnS0.5Te0.5, and ZnS0.75Te0.25 have lower total energies than the ZB counterparts. The GS structures of ZBAs have been extensively studied Wei1990 ; Ferreira1989 ; Lu1994 ; Liu2007 ; Chen2008 . For instance, it was shown that the GS ZB A0.5B0.5C alloy (Without loss of generality, B ion is assumed to have a larger radius than A ion, and C could be anion or cation) adopts the tetragonal chalcopyrite structure (space group I$\bar{4}$2d, No. 122) Wei1990 . However, the knowledge of the GS structures of WZAs remains incomplete. Our previous work Xiang2008 showed that the GS structure of the A0.5B0.5C WZA is of the $\beta-$NaFeO2 type with the space group $Pna2_{1}$ (No. 33) as shown in Fig. 1(a). Here, in this work, we identify that the GS structures of A0.25B0.75C and A0.75B0.25C WZAs have the structures shown in Fig. 1(c) or (d) with the space group $P2_{1}$ (No. 4). The formation enthalpy of isovalent semiconductor alloys AxB1-xC is defined as $\Delta H_{f}=E(x)-[xE_{AC}+(1-x)E_{BC}],$ (1) where $E_{AC}$, $E_{BC}$, and $E(x)$ are the total energies of bulk AC and BC, and the AxB1-xC alloy with the same crystal structure (WZ or ZB). It is well known that for lattice-mismatched isovalent semiconductor alloys, the major contribution to the formation enthalpy is the strain energy. The strain energy ($E_{s}$) could be described well by the VFF model Keating1966 ; Martin1970 ; Martins1984 , which considers the deviation of the nearest-neighbor bond lengths and bond angles from the ideal bulk values. Here, we consider all possible supercells with up to 32 atoms per unit cell. For each supercell, we consider all possible configurations of alloys with x=0.25 and 0.75. The VFF model is used to relax the structure and predict the energy of the configuration. We considered GaxIn1-xN, AlPxSb1-x, ZnSxTe1-x, and GaPxAs1-x. They have various degrees of lattice mismatch: 10.1%, 11.5%, 11.8%, and 3.7%. Our calculations reveal the Lazarevicite structure (space group $Pmn2_{1}$, No. 31) shown in Fig. 1(b) has the lowest strain energy for A0.25B0.75C for all four different sets of VFF parameters Martins1984 ; Kim1996 . The $Pmn2_{1}$ A0.25B0.75C structure has the same supercell as the $Pna2_{1}$ A0.5B0.5C structure. One can get the $Pmn2_{1}$ A0.25B0.75C structure by replacing one half of the A atoms in the $Pna2_{1}$ A0.5B0.5C structure with B atoms so that each C atom has one neighbor A atom and three neighbor B atoms. The $Pmn2_{1}$ WZ A0.25B0.75C structure is similar to the famatinite ZB A0.25B0.75C structure Liu2007 in that they both have similar local environment for C atoms. For the WZ A0.75B0.25C alloy, we identify two low strain energy structures with the $P2_{1}$ space group (No. 4) [$P2_{1}$-I: Fig. 1(c), and $P2_{1}$-II: Fig. 1(d)]. In contrast to the $Pmn2_{1}$ A0.75B0.25C structure, there are some C atoms which have four A neighbor atoms in both structures. In this sense, the $P2_{1}$-I and $P2_{1}$-II WZ A0.75B0.25C structures are similar to the Q8 and Q16 ZB A0.75B0.25C structures Wei1990 ; Lu1994 . As shown in Table 1, the $P2_{1}$-I structure has the lowest strain energy. However, the strain energy difference between the $P2_{1}$-II and $P2_{1}$-I structures is very small, less than 0.3 meV/atom. To see if the GS structures predicted by VFF strain energy calculations are consistent to the density functional theory (DFT) total energy calculations, we performed DFT calculations DFT ; PAW ; VASP ; LDA on the WZ A0.25B0.75C and A0.75B0.25C alloys with the $Pmn2_{1}$, $P2_{1}$-I, and $P2_{1}$-II structures. Our results are shown in Table 1. We can see that the $Pmn2_{1}$ structure is not the GS of the WZ A0.25B0.75C alloy because the $P2_{1}$ structures have a slightly lower total energy, even though the $Pmn2_{1}$ structure has a lower strain energy. This can be explained in terms of the Coulomb interaction. For the AxB1-xC alloy, the charge of A ions is different from that of B ions due to the different electronegativity. In this case, the Coulomb interaction is found to stablize the $P2_{1}$ structures over the $Pmn2_{1}$ structure because the $P2_{1}$ structures has larger charge fluctuation Magri1990 . Similar situation also occurs in ZBAs Chen2008 . After knowing the GS structures, we now compare the strain energy of the ZBA and WZA using the VFF model. Our results are shown in Table 2. We can see that for all considered systems (GaxIn1-xN, AlPxSb1-x, ZnSxTe1-x, and GaPxAs1-x with $x=0.25$, $0.5$, and $0.75$), the GS WZAs always have a lower strain energy than the GS ZBAs. The difference in the formation enthalpy mainly depends on the size of the lattice mismatch of alloy: For the first three AxB1-xC alloys with large lattice mismatch ($\Delta a>10\%$), the strain energy difference $dE_{s}$ at $x=0.5$ is around 5 meV/atom, whereas, the difference $dE_{s}$ for GaP0.5As0.5 ($\Delta a<4\%$), is only 0.7 meV/atom. Our above VFF calculations show that the WZ structure has a better ability to accomodate the strain in a lattice mismatched alloy than the ZB structure. This is due to the fact that the WZ structure has a larger degree of freedom to release the strain. First, for the binary compound, the four-atoms unit- cell WZ structure has three free parameters ($a$, $c$, $u$). In contrast, the two-atoms unit-cell ZB structure only has one free parameter ($a$). Second, the WZA is also more flexible than the ZBA. As an example, we compare the 16-atoms WZ Pna21 and 8-atoms ZB chalcopyrite structures. In both structures, each C atom bonds with two A and two B atoms. In the ZB A0.5B0.5C chalcopyrite structure, there are three free parameters. However, there are fifteen free parameters in the WZ Pna21 structure. The larger number of degree of freedom in the WZ Pna21 structure leads to an enhanced flexibility in strain relaxation. For a better understanding of the strain relaxation in WZAs, we can also decompose the total strain energy into the contributions from each atom decompose . In this way, we can tell which kind of atoms are mainly responsible for the different behavior between the WZ and ZB alloys. This analysis shows that the main difference comes from the B ions with a large size. For example, the total contributions to the strain energy in the chalcopyrite (Pna21) AlP0.5Sb0.5 alloy (here A$=$P, B$=$Sb, and C$=$Al) from Al, P, and Sb are 23.7 (22.6) meV/atom, 2.2 (1.4) meV/atom, and 7.4 (1.9) meV/atom, respectively. We can see that the strain energy difference from Sb ions contributes 74% to the total strain energy difference. In addition, we find that the difference mainly comes from the deviation of the Al-Sb-Al bond angles from the ideal value (109.47∘). In chalcopyrite AlP0.5Sb0.5 alloy, the maximum deviation of the Al-Sb-Al bond angles is 5.4∘, much larger than that (2.7∘) in WZ AlP0.5Sb0.5 alloy. The calculated DFT formation enthalpy difference $d\Delta H_{f}=\Delta H_{f}(WZA)-\Delta H_{f}(ZBA)$, where $\Delta H_{f}(WZA)$ [$\Delta H_{f}(ZBA)$] is the formation enthalpy of the WZA (ZBA) defined in Eq. 1, are shown in Table 2. We see that it follows the same trend as the strain energy difference, i.e., the GS WZA always has lower formation enthalpy than the corresponding ZBA. However, the lower formation enthalpy in the WZA does not necessarily mean that the WZA has lower total energy than the ZBA because the formation enthalpy are defined with respect to the pure bulk compounds with the same lattice structure, whereas the total energy difference between the WZ and ZB AxB1-xC alloys should also include the bond energy difference ($dE_{b}$) between the WZ and ZB phases of the parent binary compounds. We define $E_{WZ-ZB}(AC)$ [$E_{WZ-ZB}(BC)$] as the energy difference between the WZ and ZB phases of the AC (BC) compound. The bond energy difference $dE_{b}(x)$ between the WZA and ZBA as a function of $x$ are then defined as: $dE_{b}(x)=xE_{WZ-ZB}(AC)+(1-x)E_{WZ-ZB}(BC).$ (2) The total energy difference between the WZA and ZBA can then be calculated as $dE_{tot}=d\Delta H_{f}+dE_{b}$ (3) It is clear from Eq. (3) that only when the formation enthalpy difference ($d\Delta H_{f}$) is more negative than $-dE_{b}$, the WZA can be more stable than the ZBA. The DFT total energy calculations are performed to determine which alloy structure is the GS phase of GaxIn1-xN, AlPxSb1-x, ZnSxTe1-x, and GaPxAs1-x with $x=0.25$, $0.5$, and $0.75$. For the parent compounds, we find that the energy differences $E_{WZ-ZB}$ between the WZ and ZB phases are $-5.6$, $-10.8$, $3.5$, $6.5$, $3.2$, $6.0$, $8.8$, and $11.4$ meV/atom for GaN, InN, AlP, AlSb, ZnS, ZnTe, GaP, and GaAs, respectively. In agreement with previous first principles calculations Yeh1992 and experimental observations, we find that GaN and InN have the WZ GS structure, whereas the other compounds take the ZB phase as the most stable structure. The DFT results from the alloy calculations are summarized in Table 2. For alloys with WZ binary constituents (InN and GaN) or small lattice-mismatched ZB binary constituents (GaP and GaAs), the GS alloy structure (GaxIn1-xN and GaPxAs1-x) is the same as the parent compounds. However, for AlP0.5Sb0.5, AlP0.75Sb0.25, ZnS0.5Te0.5, and ZnS0.75Te0.25, the WZA structure is the GS phase despite that the alloys are formed from ZB parent compounds. It is interesting to note that compounds such as MnTe (CdO), which has the stable NiAs (Rocksalt) structure can be stabilized in the ZB phase by alloying it with ZB compounds Wei1986 ; Zhu2008 . Here we show that the alloy can be stabilized in a structure that is different from both parent structures. This remarkable alloy stabilized wurtzite structures originate from the fact that the gain in the strain energy relaxation when forming the WZA is larger than the average of the bond energy difference between the ZB and WZ phases. For example, $d\Delta H_{f}=-6.50$ meV/atom and $dE_{b}=5.01$ meV/atom for AlP0.5Sb0.5. It is also interesting to see that, the alloy stabilization energy $d\Delta H_{f}$ for A0.75B0.25C is larger than A0.25B0.75C, i.e., the WZA is more favored when a large atom is mixed into a smaller host than a smaller atom is mixed into a large host. In order to determine the concentration $x$ at which $dE_{tot}<0$, the dependence of the difference in the formation enthalpy [$d\Delta H_{f}(x)$] between the WZA and ZBA on the concentration $x$ is essential. By definition, $d\Delta H_{f}(0)=0$ and $d\Delta H_{f}(1)=0$. The $x$ dependence of $d\Delta H_{f}(x)$ can be obtained by fitting the data in Table. 2 to a fourth order polynomial. The fitted result for the AlPxSb1-x alloy is shown in Fig. 2. We can see that the curve is asymmetric with respect to $x=0.5$; the minimum of $d\Delta H_{f}(x)$ occurs at $x=0.61$. Following Eq. 3, we obtain the dependence of $dE_{tot}$ on $x$ (Fig. 2). It is seen that the minimum of $dE_{tot}$ occurs at $x=0.66$. And when $0.34<x<0.89$, the WZ AlPxSb1-x alloy is more stable than the ZBA. For ZnSxTe1-x, the result is similar, and the lowest concentration and highest concentration for a stable WZ ZnSxTe1-x alloy are 0.39 and 0.87, respectively. In summary, we have identified the GS structures of the AxB1-xC WZAs with $x=0.25$, 0.5, and 0.75. Using VFF and DFT calculations, we show that the GS WZA always has a lower strain energy and formation enthalpy than the corresponding ZBA, and thus the strain relaxation favors the formation of the WZA. We confirm this idea by showing that GS WZ AlPxSb1-x (ZnSxTe1-x) with $0.34<x<0.89$ ($0.39<x<0.87$) is more stable than the corresponding ZBA although their parent structures crystallize in the ZB phase. Work at NREL was supported by the U.S. Department of Energy, under Contract No. DE-AC36-08GO28308. The work in Fudan (FU) is partially supported by the National Sciences Foundation of China, the Basic Research Program of MOE and Shanghai, the Special Funds for Major State Basic Research, and Postgraduate Innovation Fund of FU. ## References * (1) C.-Y. Yeh, S.-H. Wei, and A. Zunger, Phys. Rev. B 50, 2715 (1994). * (2) M. van Schlfgaarde, A. Sher, and A.-B. Chen, J. Cryst. Growth 178, 8 (1997). * (3) L. Sugiura, J. Appl. Phys. 81, 1633 (1997). * (4) A. Garcia and M. L. Cohen, Phys. Rev. B 47, 4215 (1993). * (5) Chin-Yu Yeh, Z. W. Lu, S. Froyen, and A. Zunger, Phys. Rev. B 45, 12130 (1992); ibid 46, 10086 (1992). * (6) J. C. Phillips, Bonds and Bands in Semiconductors (Academic, New York, 1973). * (7) J. St. John and A. N. Bloch, Phys. Rev. Lett. 33, 1095 (1974) * (8) J. R. Chelikowsky and J. C. Phillips, Phys. Rev. B 17, 2453 (1978). * (9) G. Patriarche, F. Glas, M. Tchernycheva, C. Sartel, L. Largeau, and J.-C. Harmand, Nano Lett. 8, 1638 (2008). * (10) M. Koguchi, H. Kakibayashi, M. Yazawa, K. Hiruma, and T. Katsuyama, Jpn. J. Appl. Phys. 31, 2061 (1992). * (11) C. X. Shan, Z. Liu, X. T. Zhang, C. C. Wong, and S. K. Hark, Nanotechnology 17, 5561 (2006). * (12) T. Akiyama, K. Sano, K. Nakamura, and T. Ito, Jpn. J. Appl. Phys. 45, L275 (2006). * (13) V. G. Dubrovskii and N. V. Sibirev, Phys. Rev. B 77, 035414 (2008). * (14) G. M. Dalpian and S.-H. Wei, Phys. Rev. Lett. 93, 216401 (2004). * (15) G. M. Dalpian, Y. Yan, and S.-H. Wei, Appl. Phys. Lett. 89, 011907 (2006). * (16) V. K. Lazarov, J. Zimmerman, S. H. Cheung, L. Li, M. Weinert, and M. Gajdardziska-Josifovska, Phys. Rev. Lett. 94, 216101 (2005). * (17) S.-H. Wei, L. G. Ferreira, and A. Zunger, Phys. Rev. B 41, 8240 (1990). * (18) L. G. Ferreira, S.-H. Wei, and A. Zunger, Phys. Rev. B 40, 3197 (1989). * (19) Z.W. Lu, D. B. Laks, S.-H.Wei, and A. Zunger, Phys. Rev. B 50, 6642 (1994). * (20) J. Z. Liu, G. Trimarchi, and A. Zunger, Phys. Rev. Lett. 99, 145501 (2007). * (21) S. Chen, X. G. Gong, and S.-H. Wei, Phys. Rev. B 77, 073305 (2008). * (22) H. J. Xiang, S.-H. Wei, J. L. F. Da Silva, and J. Li, Phys. Rev. B 78, 193301 (2008). * (23) P. Keating, Phys. Rev. 145, 637 (1966). * (24) R. Martin, Phys. Rev. B 1, 4005 (1970). * (25) J. L. Martins and A. Zunger, Phys. Rev. B 30, 6217 (1984). * (26) K. Kim, W. R. L. Lambrecht, and B. Segall, Phys. Rev. B 53, 16310 (1996). * (27) Our first-principles density functional theory (DFT) calculations were performed on the basis of the projector augmented wave method PAW encoded in the Vienna ab initio simulation package VASP using the local density approximation LDA . For relaxed structures, the atomic forces are less than 0.01 eV/Å. * (28) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994); G. Kresse and D. Joubert, ibid 59, 1758 (1999). * (29) G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996); Phys. Rev. B 54, 11169 (1996). * (30) J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). * (31) R. Magri, S.-H. Wei, and A. Zunger, Phys. Rev. B 42, 11388 (1990). * (32) For each bond involving the atom $i$, half of bond stretching energy is contributed to the atom $i$. The bond bending energy contributions from atom $i$ include all bond angles which are centered at atom $i$. * (33) S.-H. Wei and A. Zunger, Phys. Rev. Lett. 56, 2391 (1986). * (34) Y. Z. Zhu, G. D. Chen, H. Ye, A. Walsh, C. Y. Moon, and S.-H. Wei, Phys. Rev. B 77, 245209 (2008). Figure 1: (a) The GS $Pna2_{1}$ structure of the WZ A0.5B0.5C alloy. (b) The $Pmn2_{1}$ structure, which is the lowest strain energy structure of the WZ A0.25B0.75C alloy. (c) The lowest strain energy structure ($P2_{1}$-I) of the WZ A0.75B0.25C alloy. (d) The low strain energy structure ($P2_{1}$-II) of the WZ A0.75B0.25C alloy. Figure 2: (Color online) Differences in the formation enthalpy ($d\Delta H_{f}$), total energy ($dE_{tot}$), and bond energy ($dE_{b}$) between the WZ and ZB AlPxSb1-x alloys. Table 1: VFF-calculated strain energy (in meV/atom) of WZ GaxIn1-xN, AlPxSb1-x, ZnSxTe1-x, and GaPxAs1-x alloys for the $Pmn2_{1}$, $P2_{1}$-I, and $P2_{1}$-II structures at $x=0.25$ and $0.75$. The numbers in parenthesis are the DFT calculated formation enthalpies. $*$ and ${\ddagger}$ indicate the GS structures obtained from the VFF and DFT calculations, respectively. Structures \| $Pmn2_{1}$ \| $P2_{1}$-I \| $P2_{1}$-II ---\|---\|---\|--- Ga0.25In0.75N \| 12.54∗ (13.13) \| 14.13 (12.38‡) \| 14.17 (12.50) Ga0.75In0.25N \| 20.16 (16.91) \| 19.01∗ (13.25‡) \| 19.28 (13.47) AlP0.25Sb0.75 \| 19.43∗ (18.88) \| 21.02 (18.19‡) \| 21.20 (18.37) AlP0.75Sb0.25 \| 24.41 (27.09) \| 23.46∗ (23.35) \| 23.69 (23.32‡) ZnS0.25Te0.75 \| 13.19∗ (21.36) \| 14.19 (20.76‡) \| 14.35 (21.05) ZnS0.75Te0.25 \| 15.14 (29.36) \| 14.73∗ (26.96) \| 14.81 (26.86‡) GaP0.25As0.75 \| 2.32∗ (2.26) \| 2.42 (1.99) \| 2.44 (1.96‡) GaP0.75As0.25 \| 2.57 (2.48) \| 2.55∗ (2.09‡) \| 2.58 (2.12) Table 2: Differences in the VFF strain energy ($dE_{s}$), DFT formation enthalpy ($d\Delta H_{f}$), DFT bond energy ($dE_{b}$), and DFT total energy ($dE_{tot}$) between the GS WZAs and ZBAs. Energy is in meV/atom. \| $dE_{s}$ \| $d\Delta H_{f}$ \| $dE_{b}$ \| $dE_{tot}$ ---\|---\|---\|---\|--- Ga0.25In0.75N \| $-4.03$ \| $-5.41$ \| $-9.53$ \| $-14.94$ Ga0.5In0.5N \| $-4.61$ \| $-5.79$ \| $-8.21$ \| $-14.00$ Ga0.75In0.25N \| $-4.77$ \| $-7.25$ \| $-6.91$ \| $-14.16$ AlP0.25Sb0.75 \| $-6.18$ \| $-4.68$ \| $5.76$ \| 1.08 AlP0.5Sb0.5 \| $-7.35$ \| $-6.50$ \| $5.01$ \| $-1.49$ AlP0.75Sb0.25 \| $-5.52$ \| $-6.14$ \| $4.27$ \| $-1.87$ ZnS0.25Te0.75 \| $-4.42$ \| $-3.89$ \| 5.33 \| 1.44 ZnS0.5Te0.5 \| $-5.18$ \| $-5.43$ \| 4.62 \| $-0.81$ ZnS0.75Te0.25 \| $-3.67$ \| $-5.12$ \| 3.90 \| $-1.22$ GaP0.25As0.75 \| $-0.62$ \| $-0.78$ \| 10.75 \| 9.97 GaP0.5As0.5 \| $-0.74$ \| $-1.05$ \| 10.12 \| 9.07 GaP0.75As0.25 \| $-0.39$ \| $-0.85$ \| 9.48 \| 8.63	arxiv-papers	2009-03-08T19:30:03	2024-09-04T02:49:01.032630	{ "license": "Public Domain", "authors": "H. J. Xiang, Su-Huai Wei, Shiyou Chen, and X. G. Gong", "submitter": "H. J. Xiang", "url": "https://arxiv.org/abs/0903.1449" }
0903.1504	# Two Fixed-Point Theorems For Special Mappings 1112000 Mathematics Subject Classification: Primary 46J10, 46J15, 47H10. A. Beiranvand, S. Moradi222First author, M. Omid and H. Pazandeh Faculty of Science, Department of Mathematics Arak University, Arak, Iran ###### Abstract In this paper, we study the existence of fixed points for mappings defined on complete (compact) metric space ($X,d$) satisfying a general contractive (contraction) inequality depended on another function. These conditions are analogous to Banach conditions. Keywords: Fixed point, contraction mapping, contractive mapping, sequentially convergent, subsequentially convergent. ## 1 Introduction The first important result on fixed points for contractive-type mapping was the well-known Banach’s Contraction Principle appeared in explicit form in Banach’s thesis in 1922, where it was used to establish the existence of a solution for an integral equation. This paper published for the first time in 1922 in [1]. In the general setting of complete metric spaces, this theorem runs as follows (see [3, Theorem 2.1] or [8, Theorem 1.2.2]). ###### Theorem 1.1. $($Banach’s Contraction Principle$)$ Let $(X,d)$ be a complete metric space and $S:X\longrightarrow X$ be a contraction $($there exists $k\in]0,1[$ such that for each $x,y\in X$; $d(Sx,Sy)\leq kd(x,y)$$)$. Then $S$ has a unique fixed point in $X$, and for each $x_{0}\in X$ the sequence of iterates $\\{S^{n}x_{0}\\}$ converges to this fixed point. After this classical result Kannan in [2] analyzed a substantially new type of contractive condition. Since then there have been many theorems dealing with mappings satisfying various types of contractive inequalities. Such conditions involve linear and nonlinear expressions (rational, irrational, and of general type). The intrested reader who wants to know more about this matter is recommended to go deep into the survey articles by Rhoades [5,6,7] and Meszaros [4], and into the references therein. Another result on fixed points for contractive-type mapping is generally attributed to Edelstein (1962) who actually obtained slightly more general versions. In the general setting of compact metric spaces this result runs as followes (see [3, Theorem 2.2]). ###### Theorem 1.2. Let $(X,d)$ be a compact metric space and $S:X\longrightarrow X$ be a contractive $($for every $x,y\in X$ such that $x\neq y$; $d(Sx,Sy)<d(x,y)$$)$. Then $S$ has a unique fixed point in $X$, and for any $x_{0}\in X$ the sequence of iterates $\\{S^{n}x_{0}\\}$ converges to this fixed point. The aim of this paper is to analyze the existence of fixed points for mapping $S$ defined on a complete (compact) metric space $(X,d)$ such that is $T-contraction$ ($T-contractive$). See Theorem 2.6 and Theorem 2.9 below. First we introduce the $T-contraction$ and $T-contractive$ functions and then we extend the Banach-Contraction Principle and Theorem 1.2. At the end of paper some properties and examples concerning this kind of contractions and contractives are given. In the sequel, $\mathbb{N}$ will represent the set of natural numbers. ## 2 Definitions and Main Results The following theorems (Theorem 2.6 and Theorem 2.9) are the main results of this paper. In the first, we define some new definitions. ###### Definition 2.1. Let $(X,d)$ be a metric space and $T,S:X\longrightarrow X$ be two functions. A mapping $S$ is said to be a $T-contraction$ if there exists $k\in]0,1[$ such that for $d(TSx,TSy)\leq kd(Tx,Ty)\quad\quad\forall x,y\in X.$ Note 1. By taking $Tx=x$ (T is identity function) $T-contraction$ and contraction are equivalent. The following example shows that $T-contraction$ functions maybe not contraction. ###### Example 2.2. Let $X=[1,+\infty)$ with metric induced by $\mathbb{R}$: $d(x,y)=\|x-y\|$. We consider two mappings $T,S:X\longrightarrow X$ by $Tx=\frac{1}{x}+1$ and $Sx=2x$. Obviously $S$ is not contraction but $S$ is $T-contraction$, because: $\big{\|}TSx- TSy\big{\|}=\big{\|}\frac{1}{2x}+1-\frac{1}{2y}-1\big{\|}=\big{\|}\frac{1}{2x}-\frac{1}{2y}\big{\|}\leq\frac{1}{2}\big{\|}\frac{1}{x}-\frac{1}{y}\big{\|}=\frac{1}{2}\big{\|}\frac{1}{x}+1-\frac{1}{y}-1\big{\|}=\frac{1}{2}\big{\|}Tx- Ty\big{\|}.$ ###### Definition 2.3. Let $(X,d)$ be a metric space. A mapping $T:X\longrightarrow X$ is said sequentially convergent if we have, for every sequence $\\{y_{n}\\}$, if $\\{Ty_{n}\\}$ is convergence then $\\{y_{n}\\}$ also is convergence. ###### Definition 2.4. Let $(X,d)$ be a metric space. A mapping $T:X\longrightarrow X$ is said subsequentially convergent if we have, for every sequence $\\{y_{n}\\}$, if $\\{Ty_{n}\\}$ is convergence then $\\{y_{n}\\}$ has a convergent subsequence. ###### Proposition 2.5. If $(X,d)$ be a compact metric space, then every function $T:X\longrightarrow X$ is subsequentially convergent and every continuous function $T:X\longrightarrow X$ is sequentially convergent. ###### Theorem 2.6. Let $(X,d)$ be a complete metric space and $T:X\longrightarrow X$ be a one-to- one, continuous and subsequentially convergent mapping. Then for every $T-contraction$ continuous function $S:X\longrightarrow X$, $S$ has a unique fixed point. Also if $T$ is a sequentially convergent, then for each $x_{0}\in X$, the sequence of iterates $\\{S^{n}x_{0}\\}$ converges to this fixed point. ###### Proof. For every $x_{1}$ and $x_{2}$ in $X$, $\displaystyle d(Tx_{1},Tx_{2})$ $\displaystyle\leq d(Tx_{1},TSx_{1})+d(TSx_{1},TSx_{2})+d(TSx_{2},Tx_{2})$ $\displaystyle\leq d(Tx_{1},TSx_{1})+kd(Tx_{1},Tx_{2})+d(TSx_{2},Tx_{2}),$ so $d(Tx_{1},Tx_{2})\leq\frac{1}{1-k}[d(Tx_{1},TSx_{1})+d(TSx_{2},Tx_{2})]$ (2.0.1) Now select $x_{0}\in X$ and define the iterative sequence $\\{x_{n}\\}$ by $x_{n+1}=Sx_{n}$ (equivalently, $x_{n}=S^{n}x_{0}$), $n=1,2,3,...$. By (2.0.1) for any indices $m,n\in\mathbb{N}$, $\displaystyle d(Tx_{n},Tx_{m})=d(TS^{n}x_{0},TS^{m}x_{0})$ $\displaystyle\leq\frac{1}{1-k}[d(TS^{n}x_{0},TS^{n+1}x_{0})+d(TS^{m+1}x_{0},TS^{m}x_{0})]$ $\displaystyle\leq\frac{1}{1-k}[k^{n}d(Tx_{0},TSx_{0})+k^{m}d(TSx_{0},Tx_{0})]$ hence $d(TS^{n}x_{0},TS^{m}x_{0})\leq\frac{k^{n}+k^{m}}{1-k}d(Tx_{0},TSx_{0}).$ (2.0.2) Relation (2.0.2) and condition $0<k<1$ show that $\\{TS^{n}x_{0}\\}$ is a Cauchy sequence, and since $X$ is complete there exists $a\in X$ such that $\underset{n\rightarrow\infty}{\lim}TS^{n}x_{0}=a.$ (2.0.3) Since $T$ is subsequentially convergent $\\{S^{n}x_{0}\\}$ has a convergent subsequence. So, there exist $b\in X$ and $\\{n_{k}\\}_{k=1}^{\infty}$ such that $\underset{k\rightarrow\infty}{\lim}S^{n_{k}}x_{0}=b$. Hence, $\underset{k\rightarrow\infty}{\lim}TS^{n_{k}}x_{0}=Tb$, and by (2.0.3), we conclude that $Tb=a.$ (2.0.4) Since $S$ is continuous and $\underset{k\rightarrow\infty}{\lim}S^{n_{k}}x_{0}=b$, then $\underset{k\rightarrow\infty}{\lim}S^{n_{k}+1}x_{0}=Sb$ and so $\underset{k\rightarrow\infty}{\lim}TS^{n_{k}+1}x_{0}=TSb.$ Again by (2.0.3), $\underset{k\rightarrow\infty}{\lim}TS^{n_{k}+1}x_{0}=a$ and therefore $TSb=a$. Since $T$ is one-to-one and by (2.0.4), Sb=b. So, $S$ has a fixed point. Since $T$ is one-to-one and $S$ is $T-contraction$, $S$ has a unique fixed point. ∎ ###### Remark 2.7. By above theorem and taking $Tx=x$ (T is identity function), we can conclude Theorem 1.1. ###### Definition 2.8. Let $(X,d)$ be a metric space and $T,S:X\longrightarrow X$ be two functions. A mapping $S$ is said to be a $T-contractive$ if for every $x,y\in X$ such that $Tx\neq Ty$ then $d(TSx,TSy)<d(Tx,Ty)$. Obviously, every $T-contraction$ function is $T-contractive$ but the converse is not true. For example if $X=[1,+\infty)$, $d(x,y)=\|x-y\|$, $Sx=\sqrt{x}$ and $Tx=x$ then $S$ is $T-contractive$ but $S$ is not $T-contraction$. ###### Theorem 2.9. Let $(X,d)$ be a compact metric space and $T:X\longrightarrow X$ be a one-to- one and continuous mapping. Then for every $T-contractive$ function $S:X\longrightarrow X$, $S$ has a unique fixed point. Also for any $x_{0}\in X$ the sequence of iterates $\\{S^{n}x_{0}\\}$ converges to this fixed point. ###### Proof. Step 1. In the first we show that $S$ is continuous. Let $\underset{n\rightarrow\infty}{\lim}x_{n}=x$. We prove that $\underset{n\rightarrow\infty}{\lim}Sx_{n}=Sx$. Since $S$ is $T-contractive$ $d(TSx_{n},TSx)\leq d(Tx_{n},Tx)$ and this shows that $\underset{n\rightarrow\infty}{\lim}TSx_{n}=TSx$ (because $T$ is continuous). Let $\\{Sx_{n_{k}}\\}$ be an arbitary convergence subsequence of $\\{Sx_{n}\\}$. There exists a $y\in X$ such that $\underset{k\rightarrow\infty}{\lim}Sx_{n_{k}}=y$. Since $T$ is continuous so, $\underset{k\rightarrow\infty}{\lim}TSx_{n_{k}}=Ty$. By $\underset{n\rightarrow\infty}{\lim}TSx_{n}=TSx$, we conclude that $TSx=Ty$. Since $T$ is one-to-one so, $Sx=y$. Hence, every convergence subsequence of $\\{Sx_{n}\\}$ converge to $Sx$. Since $X$ is a compact metric space $S$ is continuous. Step 2. Since $T$ and $S$ are continuous, the function $\varphi:X\longrightarrow[0,+\infty)$ defined by $\varphi(y)=d(TSy,Ty)$ is continuous on $X$ and hence by compactness attains its minimum, say at $x\in X$. If $Sx\neq x$ then $\varphi(Sx)=d(TS^{2}x,TSx)<d(TSx,Tx)$ is a contradiction. So $Sx=x$. Now let $x_{0}\in X$ and set $a_{n}=d(TS^{n}x_{0},Tx)$. Since $a_{n+1}=d(TS^{n+1}x_{0},Tx)=d(TS^{n+1}x_{0},TSx)\leq d(TS^{n}x_{0},Tx)=a_{n},$ then $\\{a_{n}\\}$ is a nonincreasing sequence of nonnegative real numbers and so has a limit, say $a$. By compactness, $\\{TS^{n}x_{0}\\}$ has a convergent subsequence $\\{TS^{n_{k}}x_{0}\\}$; say $\lim TS^{n_{k}}x_{0}=z.$ (2.0.5) Since $T$ is sequentially convergence (by Note 2) for a $w\in X$ we have $\lim S^{n_{k}}x_{0}=w.$ (2.0.6) By (2.0.5) and (2.0.6), $Tw=z$. So $d(Tw,Tx)=a$. Now we show that $Sw=x$. If $Sw\neq x$, then $\displaystyle a=\lim d(TS^{n}x_{0},Tx)$ $\displaystyle=\lim d(TS^{n_{k}}x_{0},Tx)=d(TSw,Tx)$ $\displaystyle=d(TSw,TSx)<d(Tw,Tx)=a$ that is contradiction. So $Sw=x$ and hence, $a=\lim d(TS^{n_{k}+1}x_{0},Tx)=d(TSw,Tx)=0.$ Therefore, $\lim TS^{n}x_{0}=Tx_{0}$. Since $T$ is sequentially convergence (by Proposition 2), then $\lim S^{n}x_{0}=x$. ∎ Similar to Remark 2.7, we can conclude Theorem 1.2. ###### Remark 2.10. In Theorem 2.6 (Theorem 2.9) if $S^{n}$ is $T-contraction$$(T-contractive)$, then $S^{n}$ has a unique fixed point and we conclude that $S$ has a unique fixed point. So, we can replace $S$ by $S^{n}$ in Theorem 2.6 (Theorem 2.9). We know that for some function $S$, $S$ is not $T-contraction$$(T-contractive)$, but for some $n\in\mathbb{N}$ $S^{n}$ is $T-contraction$ $(T-contractive)$ (see the following example). ## 3 Examples and Applications In this section we have some example about Theorem 2.6 and Theorem 2.9 and the conditions of these theorems, and show that we can not omit the conditions of these theorems. ###### Example 3.1. Let $X=[0,1]$ with metric induced by $\mathbb{R}$: $d(x,y)=\|x-y\|$. Obviously $(X,d)$ be a complete metric space and the function $S:X\longrightarrow X$ by $Sx=\frac{x^{2}}{\sqrt{2}}$ is not contractive. If $T:X\longrightarrow X$ define by $Tx=x^{2}$ then $S$ is $T-contractive$, because: $\big{\|}TSx- TSy\big{\|}=\big{\|}\frac{x^{4}}{2}-\frac{y^{4}}{2}\big{\|}=\frac{1}{2}\big{\|}x^{2}+y^{2}\big{\|}\big{\|}Tx- Ty\big{\|}<\big{\|}Tx-Ty\big{\|}.$ So by Theorem 2.9 $S$ has a unique fixed point. ###### Example 3.2. Let $X=[0,1]$ with metric induced by $\mathbb{R}$: $d(x,y)=\|x-y\|$. Obviously $(X,d)$ is a compact metric space. Let $T,S:X\longrightarrow X$ define by $Tx=x^{2}$ and $Sx=\frac{1}{2}\sqrt{1-x^{2}}$. Clearly $S$ is not contraction, but $S$ is $T-contraction$ and hence is $T-contractive$. Also $T$ is one-to- one. So by Theorem 2.8 $S$ has a unique fixed point. ###### Example 3.3. Let $X=[1,+\infty)$ with metric induced by $\mathbb{R}$: $d(x,y)=\|x-y\|$, thus, since $X$ is a closed subset of $\mathbb{R}$, it is a complete metric space. We define $T,S:X\longrightarrow X$ by $Tx=\ln x+1$ and $Sx=2\sqrt{x}$. Obviously, for every $n\in\mathbb{N}$, $S^{n}$ is not contraction. But we have, $\big{\|}TSx-TSy\big{\|}=\frac{1}{2}\big{\|}\ln x-\ln y\big{\|}=\frac{1}{2}\big{\|}Tx-Ty\big{\|}\leq\frac{1}{2}\big{\|}Tx-Ty\big{\|}.$ Hence, $S$ is $T-contraction$. Also $T$ is one-to-one and subsequentially convergent. Therefore, by Theorem 2.5 $S$ has a unique fixed point. The following examples show that we can not omit the conditions of Theorem 2.6 and Theorem 2.9. In the following note we have two examples such that show that we can not omit the one-to-one of $T$ in Theorem 2.6 and Theorem 2.9. In first example $S$ has more than one fixed point and in the second example $S$ has not a fixed point. Note 2. Let $X=\\{0,\frac{1}{2},1\\}$ with metric $d(x,y)=\|x-y\|$. For functions $T_{1},S_{1}:X\longrightarrow X$ defined by $T_{1}x=\left\\{\begin{array}[]{c l}0&\text{ $x=0,1$}\\\ \frac{1}{2}&\text{$x=\frac{1}{2}$}\end{array}\right.$ and $S_{1}x=\left\\{\begin{array}[]{c l}0&\text{ $x=0,\frac{1}{2}$}\\\ 1&\text{$x=1$}\end{array}\right.$ we have $T_{1}$ is subsequentially convergent and since $\big{\|}T_{1}S_{1}x-T_{1}S_{1}y\big{\|}\leq\frac{1}{2}\big{\|}T_{1}x-T_{1}y\big{\|}\>\>(\forall x,y\in X),$ $S_{1}$ is $T_{1}-contraction$. But $T_{1}$ is not one-to-one and $S_{1}$ has two fixed points. If we define the functions $T_{2},S_{2}:X\longrightarrow X$ by $T_{2}x=\left\\{\begin{array}[]{c l}0&\text{ $x=0,1$}\\\ \frac{1}{2}&\text{$x=\frac{1}{2}$}\end{array}\right.$ and $S_{2}x=\left\\{\begin{array}[]{c l}1&\text{ $x=0,\frac{1}{2}$}\\\ 0&\text{$x=1$}\end{array}\right.$ then we have $T_{2}$ is subsequentially convergent and since $\big{\|}T_{2}S_{2}x-T_{2}S_{2}y\big{\|}\leq\frac{1}{2}\big{\|}T_{2}x-T_{2}y\big{\|}\>\>(\forall x,y\in X),$ $S_{2}$ is $T_{2}-contraction$. But $T_{2}$ is not one-to-one and $S_{2}$ has not a fixed point. The following example shows that we can not omit the subsequentially convergent of $T$ in Theorem 2.6. ###### Example 3.4. Let $X=[0,+\infty)$ with metric induced by $\mathbb{R}$: $d(x,y)=\|x-y\|$. Obviously $(X,d)$ be a complete metric space. For functions $T,S:X\longrightarrow X$ defined by $Sx=2x+1$ and $Tx=\exp(-x)$ we have, $T$ is one-to-one and $S$ is $T-contraction$ because: $\displaystyle\big{\|}TSx-TSy\big{\|}$ $\displaystyle=\big{\|}\exp(-2x-1)-\exp(-2y-1)\big{\|}=\frac{1}{e}\big{\|}\exp(-x)+\exp(-y)\big{\|}$ $\displaystyle\big{\|}\exp(-x)-\exp(-y)\big{\|}\leq\frac{2}{e}\big{\|}\exp(-x)-\exp(-y)\big{\|}=\frac{2}{e}\big{\|}Tx- Ty\big{\|}.$ But $T$ is not subsequentially convergent $(Tn\underset{n\longrightarrow\infty}{\longrightarrow}0$ but $\\{n\\}_{1}^{\infty}$ has not any convergence subsequence$)$ and $S$ has not a fixed point. ## References * [1] S. Banach, Sur Les Operations Dans Les Ensembles Abstraits et Leur Application Aux E’quations Inte’grales, Fund. Math. 3(1922), 133-181(French). * [2] R. Kannan, Some Results on Fixed Points, Bull. Calcutta Math. Soc. 60(1968), 71-76. * [3] Kazimierz Goebel and W.A.Kirk, Topiqs in Metric Fixed Point Theory, Combridge University Press, New York, 1990. * [4] J. Meszaros, A Comparison of Various Definitions of Contractive Type Mappings, Bull. Calcutta Math. Soc. 84(1992), no. 2, 167-194. * [5] B. E. Rhoades, A Comparison of Various Definitions of Contractive Mappings, Trans. Amer. Math. Soc. 226(1977), 257-290. * [6] B. E. Rhoades, Contractive definitions revisited, Topological Methods in Nonlinear Functional Analysis (Toronto, Ont.,1982), Contemp. Math., Vol. 21, American Mathematical Society, Rhode Island, 1983, pp. 189-203. * [7] B. E. Rhoades, Contractive Definitions, Nonlinear Analysis, World Science Publishing, Singapore, 1987, pp. 513-526. * [8] O. R. Smart, Fixed Point Theorems, Cambridge University Press, London, 1974. Email: A-Beiranvand@Arshad.araku.ac.ir S-Moradi@araku.ac.ir M-Omid@Arshad.araku.ac.ir H-Pazandeh@Arshad.araku.ac.ir	arxiv-papers	2009-03-09T13:28:41	2024-09-04T02:49:01.037270	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "A. Beiranvand, S. Moradi, M. Omid and H. Pazandeh", "submitter": "Sirous Moradi", "url": "https://arxiv.org/abs/0903.1504" }
0903.1569	# A Fixed-Point Theorem For Mapping Satisfying a General Contractive Condition Of Integral Type Depended an Another Function 1112000 Mathematics Subject Classification: Primary 46J10, 46J15, 47H10. S. Moradi222First author and A. Beiranvand Faculty of Science, Department of Mathematics Arak University, Arak, Iran ###### Abstract In this paper, we study the existence of fixed points for mappings defined on complete metric space ($X,d$) satisfying a general contractive inequality of integral type depended on another function. This conditions is analogous of Banach conditions and Branciari Theorem. Keywords: Fixed point, contraction mapping, contractive mapping, sequently convergent, subsequently convergent, integral type. ## 1 Introduction The first important result on fixed points for contractive-type mapping was the well-known Banach’s Contraction Principle appeared in explicit form in Banach’s thesis in 1922, where it was used to establish the existence of a solution for an integral equation [1]. In the general setting of complete metric space this theorem runs as follows(see[5,Theorem 2.1] or[10,Theorem1.2.2]). ###### Theorem 1.1. $($Banach’s Contraction Principle$)$ Let $(X,d)$ be a complete metric space and $f:X\longrightarrow X$ be a contraction $($there exists $k\in(0,1)$ such that for each $x,y\in X$; $d(fx,fy)\leq kd(x,y)$$)$. Then $f$ has a unique fixed point in $X$, and for each $x_{0}\in X$ the sequence of iterates $\\{f^{n}x_{0}\\}$ converges to this fixed point. After this classical result Kannan in [4] analyzed a substantially new type of contractive condition. Since then there have been many theorems dealing with mappings satisfying various types of contractive inequalities. Such conditions involve linear and nonlinear expressions (rational, irrational, and of general type). The intrested reader who wants to know more about this matter is recommended to go deep into the survey articles by Rhoades [7,8,9] and Meszaros [6], and into the references therein. Another result on fixed points for contractive-type mapping is generally attributed to Edelstein (1962) who actually obtained slightly more general versions. In the general setting of compact metric spaces this result runs as followes (see [5, Theorem 2.2]). ###### Theorem 1.2. Let $(X,d)$ be a compact metric space and $f:X\longrightarrow X$ be a contractive $($for every $x,y\in X$ such that $x\neq y$; $d(fx,fy)<d(x,y)$$)$. Then $f$ has a unique fixed point in $X$, and for any $x_{0}\in X$ the sequence of iterates $\\{f^{n}x_{0}\\}$ converges to this fixed point. Also in 2002 in [3] A. Branciari analyzed the existence of fixed point for mapping $f$ defined on a complete metric space $(X,d)$ satisfying a contractive condition of integral type.(see the following theorem). ###### Theorem 1.3. Let $(X,d)$ be a complete metric space, $\alpha\in(0,1)$ and $f:X\longrightarrow X$ be a mapping such that for each $x,y\in X$, $\int_{0}^{d(fx,fy)}\phi(t)dt\leq\alpha{\int_{0}^{d(x,y)}\phi(t)dt}$, where $\phi:[0,+\infty)\longrightarrow[0,+\infty)$ is a Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of $[0,+\infty)$, nonnegative, and such that for each $\epsilon>0,\int_{0}^{\epsilon}\phi(t)dt>0$; then $f$ has a unique fixed point $a\in X$ such that for each $x\in X$, $\underset{n\rightarrow\infty}{\lim}f^{n}x=a$. The aim of this paper is to study the existence of fixed point for mapping $f$ defined on a compact metric space$(X,d)$ such that is $T_{\int\phi}-contraction$. In particular, we extend the main theorem due to A. Branciari [3] (Theorem 1.3) and the main theorem in [2] (2008). First we introduce the $T_{\int\phi}-contraction$ function and then extended the A.Branciari Theorem and the main theorem in [2] and Banach-contraction principle, by the same metod for proof of the A. Branciari Theorem. At the end of paper some examples and applications concerning this kind of contractions. In [3] A. Branciari gave an example (Example 3.6) such that we can conclude this example by theorem 1.2. (because $X=\\{1/n:n\in\mathbb{N}\\}\bigcup\\{0\\}$, with metric induced by $\mathbb{R}$, $d(x,y)=\|x-y\|$, is a compact metric space and $f$ is a contractive mapping). In the end of this paper we give an example (Example 3.5) such that we can not conclude this example by Theorem 1.1, Theorem 1.2. Branciari Theorem and the main theorem in [2], but we can conclude this example by the main theorem (Theorem 2.5 ) in this paper. In the sequel, $\mathbb{N}$ will represent the set of natural numbers, $\mathbb{R}$ the set of real number and $\mathbb{R}^{+}$ the set of nonnegative real number. ## 2 Definitions and Main Result The following theorem (Theorem 2.5) is the main result of this paper. In the first, we define some new definitions. ###### Definition 2.1. Let $(X,d)$ be a metric space and $f,T:X\longrightarrow X$ be two functions and $\phi:[0,+\infty)\longrightarrow[0,+\infty)$ be a Lebesgue-integrable mapping. A mapping $f$ is said to be a $T_{\int\phi}-contraction$ if there exists $\alpha\in(0,1)$ such that for all $x,y\in X$ $\int_{0}^{d(Tfx,Tfy)}\phi(t)dt\leq\alpha{\int_{0}^{d(Tx,Ty)}\phi(t)dt}$ ###### Remark 2.2. By taking $Tx=x$ and $\phi=1$, $T_{\int\phi}-contraction$ and contraction are equivalent. Also by taking $Tx=x$ we can define $\int\phi-contraction$. ###### Example 2.3. Let $X=[1,+\infty)$ with metric induced by $\mathbb{R}$: $d(x,y)=\|x-y\|$. We consider two mappings $T,f:X\longrightarrow X$ by $Tx=\frac{1}{x}+1$ and $fx=2x$. Obviously $f$ is not contraction but $f$ is $T_{\int{1}}-contraction$. ###### Definition 2.4. $[2]$ Let $(X,d)$ be a metric space. A mapping $T:X\longrightarrow X$ is said sequentially convergent if we have, for every sequence $\\{y_{n}\\}$, if $\\{Ty_{n}\\}$ is convergence then $\\{y_{n}\\}$ also is convergence. $T$ is said subsequentially convergent if we have, for every sequence $\\{y_{n}\\}$, if $\\{Ty_{n}\\}$ is convergence then $\\{y_{n}\\}$ has a convergent subsequence. ###### Theorem 2.5. $[$Main theorem $]$ Let $(X,d)$ be a complete metric space, $\alpha\in(0,1)$, $T,f:X\longrightarrow X$ be mapping such that $T$ is continuous, one-to-one and subsequentially convergent and $f$ is $T_{\int\phi}-contraction$ where $\phi:[0,+\infty)\longrightarrow[0,+\infty)$ is a Lebesgue-integrable mapping which is summable on each compact subset of $[0,+\infty)$, nonnegative and such that for each $\epsilon>0,\int_{0}^{\epsilon}\phi(t)dt>0$; then $f$ has a unique fixed point $a\in X$. Also if $T$ is sequentially convergent, then for each $x_{0}\in X$, the sequence of iterates $\\{f^{n}x_{0}\\}$ converges to this fixed point. ###### Proof. STEP 1. Let $\alpha\in(0,1)$ such that for all $x,y\in X$ $\int_{0}^{d(Tfx,Tfy)}\phi(t)dt\leq\alpha{\int_{0}^{d(Tx,Ty)}\phi(t)dt}.\hskip 56.9055pt(2.1)$ So if for $a,b>0$, $\int_{0}^{a}\phi(t)dt\leq\alpha{\int_{0}^{b}\phi(t)dt}$ then $a<b.$ STEP 2. We show that $f$ is a continuouse mapping. If $\underset{n\rightarrow\infty}{\lim}x_{n}=x$ then by $\int_{0}^{d(Tfx_{n},Tfx)}\phi(t)dt\leq\alpha{\int_{0}^{d(Tx_{n},Tx)}\phi(t)dt}$ and $\underset{n\rightarrow\infty}{\lim}d(Tx_{n},Tx)=0$, we conclude that: $\lim_{n\rightarrow\infty}d(Tfx_{n},Tfx)=0.$ Since $T$ is subsequentially convergent, $\\{fx_{n}\\}$ has a subsequence such $\\{{fx_{n}}_{k}\\}_{k=1}^{\infty}$ converge to a $y\in X$. So $d(Ty,Tfx)=0$. Since $T$ is one-to-one, $y=fx$. Hence, $\\{fx_{n}\\}$ has a subsequence converge to $fx$. Therefore for every sequence $\\{x_{n}\\}$ converge to $x$, the sequence $\\{fx_{n}\\}$ has a subsequence converge to $fx$. This shows that $f$ is continuouse at $x$. STEP 3. Since (2.1) is holds, for all $n\in\mathbb{N}:$ $\int_{0}^{d(Tf^{n+1}x,Tf^{n}x)}\phi(t)dt\leq\alpha^{n}{\int_{0}^{d(Tfx,Tx)}\phi(t)dt}\qquad\forall x\in X.$ As a consequence, since $\alpha\in(0,1)$, we further have $\int_{0}^{d(Tf^{n+1}x,Tf^{n}x)}\phi(t)dt\rightarrow 0^{+}\hskip 14.22636ptas\hskip 14.22636ptn\rightarrow\infty\hskip 56.9055pt(2.2)$ Since $\hskip 14.22636pt\int_{0}^{\epsilon}\phi(t)dt>0,\hskip 14.22636pt\forall\epsilon>0\hskip 56.9055pt(2.3)$ is holds we conclude that $\lim_{n\rightarrow\infty}d(Tf^{n+1}x,Tf^{n}x)=0\hskip 85.35826pt(2.4)$ Step 4. $\\{Tf^{n}x\\}$ is a bounded sequence. If $\\{Tf^{n}x\\}_{n=1}^{\infty}$is not a bounded sequence then, we choose the sequence $\\{n_{k}\\}_{k=1}^{\infty}$ such that $n_{1}=1$ and for each $k\in\mathbb{N}$, $n_{k+1}$ is ”minimal” in the sense that $d(Tf^{n_{k+1}}x,Tf^{n_{k}}x)>1.$ So, $\displaystyle 1$ $\displaystyle<$ $\displaystyle d(Tf^{n_{k+1}}x,Tf^{n_{k}}x)$ $\displaystyle\leq$ $\displaystyle d(Tf^{n_{k+1}}x,Tf^{n_{k+1}-1}x)+d(Tf^{n_{k+1}-1}x,Tf^{n_{k}}x)$ $\displaystyle\leq$ $\displaystyle d(Tf^{n_{k+1}}x,Tf^{n_{k+1}-1}x)+1.\hskip 71.13188pt(2.5)$ Hence, by (2.4) and (2.5) we conclude that $d(Tf^{n_{k+1}}x,Tf^{n_{k}}x)\rightarrow 1\hskip 14.22636ptas\hskip 14.22636ptk\rightarrow\infty\hskip 56.9055pt(2.6)$ Also by step 1, $d(Tf^{n_{k+1}}x,Tf^{n_{k}+1}x)\leq d(Tf^{n_{k+1}-1}x,Tf^{n_{k}}x).$ Therefore, $\displaystyle 1-d(Tf^{n_{k}+1}x,Tf^{n_{k}}x)$ $\displaystyle<$ $\displaystyle d(Tf^{n_{k+1}}x,Tf^{n_{k}}x)-d(Tf^{n_{k}+1}x,Tf^{n_{k}}x)$ $\displaystyle\leq$ $\displaystyle d(Tf^{n_{k+1}}x,Tf^{n_{k}+1}x)$ $\displaystyle\leq$ $\displaystyle d(Tf^{n_{k+1}-1}x,Tf^{n_{k}}x)$ $\displaystyle\leq$ $\displaystyle 1.$ Hence, by (2.4), $d(Tf^{n_{k+1}}x,Tf^{n_{k}+1}x)\rightarrow 1\hskip 14.22636ptas\hskip 14.22636ptk\rightarrow\infty.\hskip 56.9055pt(2.7)$ Therefore, $\displaystyle\int_{0}^{d(Tf^{n_{k+1}}x,Tf^{n_{k}+1}x)}\phi(t)dt$ $\displaystyle\leq$ $\displaystyle\alpha{\int_{0}^{d(Tf^{n_{k+1}-1}x,Tf^{n_{k}}x)}\phi(t)dt}$ $\displaystyle\leq$ $\displaystyle\alpha{\int_{0}^{1}\phi(t)dt}.\hskip 56.9055pt(2.8)$ By (2.7) and (2.8) we conclude that $\displaystyle\int_{0}^{1}\phi(t)dt$ $\displaystyle=$ $\displaystyle\lim_{k\rightarrow\infty}\int_{0}^{d(Tf^{n_{k+1}}x,Tf^{n_{k}+1}x)}\phi(t)dt$ $\displaystyle\leq$ $\displaystyle\alpha{\int_{0}^{1}\phi(t)dt}.$ So $\int_{0}^{1}\phi(t)dt=0$ and this is contradiction. STEP 5. By (2.1) for every $m,n\in\mathbb{N}(m>n)$, $\int_{0}^{d(Tf^{m}x,Tf^{n}x)}\phi(t)dt\leq\alpha^{n}{\int_{0}^{d(Tf^{m-n}x,Tx)}\phi(t)dt}.\hskip 56.9055pt(2.9)$ By step 4, (2.9) and $\alpha\in(0,1),$ $\underset{m,n\rightarrow\infty}{\lim}\int_{0}^{d(Tf^{m}x,Tf^{n}x)}=0\hskip 56.9055pt(2.10)$ Since (2.3) is hold $\underset{m,n\rightarrow\infty}{\lim}d(Tf^{m}x,Tf^{n}x)=0$, and this shows that $\\{Tf^{n}x\\}_{n=1}^{\infty}$ is a Cauchy sequence. Hence there exists $a\in X$ such that $\underset{n\rightarrow\infty}{\lim}Tf^{n}x=a\hskip 56.9055pt(2.11)$ STEP 6. Since $T$ is a subsequentially convergent, $\\{f^{n}x\\}$ has a convergent subsequence. So there exists $b\in X$ and $\\{n_{k}\\}_{k=1}^{\infty}$ such that $\underset{k\rightarrow\infty}{\lim}f^{n_{k}}x=b$. Since $T$ is continuouse $\underset{k\rightarrow\infty}{\lim}Tf^{n_{k}}x=Tb$, and by (2.11) we conclude that $Tb=a.\hskip 56.9055pt(2.12)$ Since $f$ is continuouse (step 2) and $\underset{k\rightarrow\infty}{\lim}f^{n_{k}}x=b,\underset{k\rightarrow\infty}{\lim}f^{n_{k}+1}x=fb$ and so $\underset{k\rightarrow\infty}{\lim}Tf^{n_{k}+1}x=Tfb.$ Again by (2.11) we have $\underset{k\rightarrow\infty}{\lim}Tf^{n_{k}+1}x=a$ and therefore, $Tfb=a.$ So by (2.12), $Tfb=Tb.$ Since $T$ is one-to one, $fb=b.$ Therefore $f$ has a fixed point. STEP 7. Since $T$ is one-to-one and $f$ is $T_{\int\phi}-contraction$, $f$ has a unique fixed point. ∎ ## 3 Examples and Applications In this section, we give some applications and some examples concerning these contractive mapping of integral type, which clarify the connection between our result and the classical ones. ###### Remark 3.1. Theorem 2.5 is a generalization of the Banach’s contraction principle (Theorem 1.1), letting $\phi(t)=1$ for each $t\geq 0$ and $Tx=x$ for each $x\in X$ in Theorem 2.5, we have $\displaystyle\int_{0}^{d(Tfx,Tfy)}\phi(t)dt$ $\displaystyle=$ $\displaystyle d(fx,fy)$ $\displaystyle\leq$ $\displaystyle\alpha d(x,y)$ $\displaystyle=$ $\displaystyle\alpha{\int_{0}^{d(Tx,Ty)}\phi(t)dt}$ ###### Remark 3.2. Theorem 2.5 is a generalization of the A. Branciari theorem (Theorem 1.3), letting $Tx=x$ for each $x\in X$ in Theorem 2.5, so $\displaystyle\int_{0}^{d(Tfx,Tfy)}\phi(t)dt$ $\displaystyle=$ $\displaystyle\int_{0}^{d(fx,fy)}\phi(t)dt$ $\displaystyle\leq$ $\displaystyle\alpha{\int_{0}^{d(x,y)}\phi(t)dt}$ $\displaystyle=$ $\displaystyle\alpha{\int_{0}^{d(Tx,Ty)}\phi(t)dt}.$ We can conclude the following theorem ( the main Theorem in [2]) by Theorem 2.5. ###### Theorem 3.3. Let $(X,d)$ be a complete metric space and $T:X\longrightarrow X$ be a one-to- one, continuouse and subsequentially convergent mapping. Then for every $T-contraction$ function $f:X\longrightarrow X$, $f$ has a unique fixed point. Also if $T$ is sequentially convergent, then for each $x_{0}\in X$, the sequence of iterates $\\{f^{n}x\\}$ converges to this fixed point. ($f:X\longrightarrow X$ is $T-contraction$ if there exist $\alpha\in(0,1)$ such that for all $x,y\in X$ $d(Tfx,Tfy)\leq\alpha{d(Tx,Ty)}.)$ ###### Proof. By taking $\phi(t)=1$ for each $t\in[0,+\infty)$ in Theorem 2.5 we can conclude this theorem. ∎ ###### Example 3.4. Let $X=[1,+\infty)$ with metric induced by $\mathbb{R}:d(x,y)=\|x-y\|,$ thus, since $X$ is a closed subset of $\mathbb{R},$ it is a complete metric space. we define $T,f:X\longrightarrow X$ by $Tx=\ln{x}+1$ and $fx=k\sqrt{x}$ such that $k\geq 1$ be a fixed element of $\mathbb{R}.$ Obviousely $f$ is not contraction, but $f$ is $T_{\int 1}-contraction$ and $T$ is one-to-one, continuouse and sequentially convergent. So $f$ has a unique fixed point by Theorem 2.5. The following example is the main example of this paper. In the following we show that, we can not conclude this example by Theorem 1.1, Theorem 1.2, Theorem 1.3 (Branciari Theorem) and Theorem 3.3. ###### Example 3.5. Let $X:={\\{\frac{1}{n}\ \ \|\ \ n\in\mathbb{N}\\}}\bigcup{\\{0}\\}$ with metric induced by $\mathbb{R}:d(x,y):=\|x-y\|$, thus, since $X$ is a closed subset of $\mathbb{R}$, it is a complete metric space. We consider a mapping $f:X\longrightarrow X$ defined by $fx=\left\\{\begin{array}[]{ll}\frac{1}{n+3}&;x=\frac{1}{n},\>n\>is\>odd\\\ 0&;x=0\\\ \frac{1}{n-1}&;x=\frac{1}{n},\>n\>is\>even\end{array}\right.$ and defined $\phi:[0,+\infty)\longrightarrow[0,+\infty)$ by $\phi(t)=\left\\{\begin{array}[]{ll}t^{{\frac{1}{t}}-2}[1-\log{t}]&;t>0\\\ 0&;t=0\end{array}\right.$ we have $\int_{0}^{\tau}\phi(t)dt=\tau^{\frac{1}{\tau}}.$ By taking $n=2$ and $m=4$, $\|f(1/m)-f(1/n)\|>\|1/m-1/n\|$, so $f$ is not contraction and contractive. Hence, we can not conclude that, $f$ has a fixed point by Theorem 1.1 and Theorem 1.2. Now we show that we can not use Branciari Theorem for this example. For $x=1/m$, $y=1/n$ where $m$ and $n$ are even if $\int_{0}^{\|fx-fy\|}\phi(t)dt\leq\alpha{\int_{0}^{\|x-y\|}\phi(t)dt}$ then $\|\frac{1}{m-1}-\frac{1}{n-1}\|^{\frac{1}{\|\frac{1}{m-1}-\frac{1}{n-1}\|}}\leq\alpha{\|\frac{1}{m}-\frac{1}{n}\|^{\frac{1}{\|\frac{1}{m}-\frac{1}{n}\|}}}$ $\Rightarrow\hskip 28.45274pt\|\frac{m-n}{(m-1)(n-1)}\|^{\|\frac{(m-1)(n-1)}{m-n}\|}\leq\alpha{\|\frac{m-n}{mn}\|^{\|\frac{mn}{m-n}\|}}$ For $m=4$ and $n=2$ we conclude that $1<\alpha.$ So we can not use Branciari Theorem. Now we defined $T:X\longrightarrow X$ by $Tx=\left\\{\begin{array}[]{ll}\frac{1}{n-1}&;x=\frac{1}{n},\>n\>is\>even\\\ 0&;x=0\\\ \frac{1}{n+1}&;x=\frac{1}{n},\>n\>is\>odd\end{array}\right.$ Obviously $T$ is one-to-one and sequentially convergent and continuouse. we have $Tfx=\left\\{\begin{array}[]{ll}\frac{1}{n+2}&;x=\frac{1}{n},\>n\>is\>odd\\\ 0&;x=0\\\ \frac{1}{n}&;x=\frac{1}{n},\>n\>is\>even\end{array}\right.$ Since $\sup{\frac{\|Tfx-Tfy\|}{\|Tx-Ty}\|}=1$, $f$ is not $T-contraction$, and so we can not use Theorem 3.3 for this example. Now we show that the condition of Theorem 2.5 are holds. We show that $f$ is $T_{\int\phi}-contraction$ and $\int_{0}^{\|Tfx-Tfy\|}\phi(t)dt\leq{\frac{1}{2}\int_{0}^{\|Tx- Ty\|}\phi(t)dt}\hskip 14.22636ptfor\ \ all\ \ x,y\in X.\hskip 56.9055pt(2.13)$ Case 1. Let $x=\frac{1}{m},y=\frac{1}{n}$ and $m$ and $n$ are even. Then $\displaystyle\qquad\qquad\int_{0}^{\|Tfx- Tfy\|}\phi(t)dt\leq{\frac{1}{2}\int_{0}^{\|Tx-Ty\|}\phi(t)dt}$ $\displaystyle\Leftrightarrow\hskip 28.45274pt\|\frac{1}{m}-\frac{1}{n}\|^{\frac{1}{\|\frac{1}{m}-\frac{1}{n}\|}}\leq\frac{1}{2}{\|\frac{1}{m-1}-\frac{1}{n-1}\|^{\frac{1}{\|\frac{1}{m-1}-\frac{1}{n-1}\|}}}$ $\displaystyle\Leftrightarrow\hskip 28.45274pt\|\frac{m-n}{mn}\|^{\|\frac{mn}{m-n}\|}.\|\frac{(m-1)(n-1)}{m-n}\|^{\|\frac{(m-1)(n-1)}{m-n}\|}\leq\frac{1}{2}$ $\displaystyle\Leftrightarrow\hskip 28.45274pt\|\frac{(m-1)(n-1)}{mn}\|^{\|\frac{(m-1)(n-1)}{m-n}\|}.\|\frac{(m-n)}{mn}\|^{\|\frac{(m+n-1)}{m-n}\|}\leq\frac{1}{2}$ Obviously the last inequality is holds, because $\|\frac{(m-1)(n-1)}{mn}\|\leq 1\>\>and\>\>\|\frac{(m-1)(n-1)}{m-n}\|\geq 1$ and so $\|\frac{(m-1)(n-1)}{mn}\|^{\|\frac{(m-1)(n-1)}{m-n}\|}\leq 1,$ and $\|\frac{m-n}{mn}\|^{\|\frac{m+n-1}{m-n}\|}\leq\frac{1}{2}.$ Therefore for this case (2.13) is holds. Case 2. Let $x=\frac{1}{m},y=\frac{1}{n}$ and $m$ and $n$ are odd. Case 3. Let $x=\frac{1}{m},y=\frac{1}{n}$, $m$ is even and $n$ is odd. By the same argument in case 1 we conclude that (2.13) for case 2 and case 3 is holds. Case 4. Let $x=0,y=\frac{1}{n}$ such that $n$ is even. Then $\displaystyle\int_{0}^{\|Tfx-Tfy\|}\phi(t)dt\leq\frac{1}{2}{\int_{0}^{\|Tx- Ty\|}\phi(t)dt}$ $\displaystyle\Leftrightarrow(\frac{1}{n})^{n}\leq\frac{1}{2}{(\frac{1}{n-1})^{n-1}}$ $\displaystyle\Leftrightarrow(\frac{1}{n})^{n}(n-1)^{n-1}\leq\frac{1}{2}$ $\displaystyle\Leftrightarrow(\frac{n-1}{n})^{n-1}.\frac{1}{n}\leq\frac{1}{2}.$ The last inequality is holds, because, $(\frac{n-1}{n})^{n-1}\leq 1\qquad and\qquad\frac{1}{n}\leq\frac{1}{2}.$ Therefore (2.13) is true for this case. Case 5. Let $x=0\ \ \ y=\frac{1}{n}$ such that $n$ is odd. By the same argument in case 4 we conclude that, (2.13) is holds for this case. Hence, (2.13) is holds for all $x,y\in X.$ Therefore the condition of Theorem 2.5 are hold and so $f$ has a unique fixed point. ## References * [1] S. Banach, Sur Les Operations Dans Les Ensembles Abstraits et Leur Application Aux E’quations Inte’grales, Fund. Math. 3(1922), 133-181(French). * [2] A. Beiranvand, S. Moradi, M. Omid and H. Pazandeh , Two Fixed-Point Theorem For Special Mapping, to appear. * [3] A. Branciari, A fixed point theorem for mapping satisfying a general contractive condition of integral type Int. J. M. and M. since, 29:9 (2002), 531-536. * [4] R. Kannan, Some results on fixed points, Bull.Calcutta Math. Soc. 60(1968),71-76. * [5] Kazimierz Goebel and W. A. Kirk, Topiqs in Metric Fixed Point Theory, Combridge University Press, New York, 1990. * [6] J. Meszaros, A Comparison of Various Definitions of Contractive Type Mappings, Bull. Calcutta Math. Soc. 84(1992), no. 2, 167-194. * [7] B. E. Rhoades, A Comparison of Various Definitions of Contractive Mappings, Trans. Amer. Math. Soc. 226(1977), 257-290. * [8] B. E. Rhoades, Contractive definitions revisited, Topological Methods in Nonlinear Functional Analysis (Toronto, Ont.,1982), Contemp. Math., Vol. 21, American Mathematical Society, Rhode Island, 1983, pp. 189-203. * [9] B. E. Rhoades, Contractive Definitions, Nonlinear Analysis, World Science Publishing, Singapore, 1987, pp. 513-526. * [10] O. R. Smart, Fixed Point Theorems, Cambridge University Press, London, 1974. Email: S-Moradi@araku.ac.ir A-Beiranvand@Arshad.araku.ac.ir	arxiv-papers	2009-03-09T15:08:39	2024-09-04T02:49:01.041964	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "S. Moradi and A. Beiranvand", "submitter": "Sirous Moradi", "url": "https://arxiv.org/abs/0903.1569" }
0903.1574	# Fixed-Point Theorem For Mappings Satisfying a General Contractive Condition Of Integral Type Depended an Another Function 1112000 Mathematics Subject Classification: Primary 46J10, 46J15, 47H10. S. Moradi Faculty of Science, Department of Mathematics Arak University, Arak, Iran ###### Abstract We established a fixed-point theorem for mapping satisfying a general contractive inequality of integral type depended an another function. This theorem substantially extend the theorem due to Branciari (2003) and Rhoades (2003). Keywords: Fixed point, contractive mapping, sequently convergent, subsequently convergent, integral type. ## 1 Introduction In 2002 [2], Branciari established the Banach Contractive Principle in the following theorem. ###### Theorem 1.1. Let $(X,d)$ be a complete metric space, $k\in[0,1)$ and $S:X\longrightarrow X$ be a mapping such that, for each $x,y\in X$, $\int_{0}^{d(Sx,Sy)}\phi(t)dt\leq k{\int_{0}^{d(x,y)}\phi(t)dt},\qquad\qquad\qquad(1)$ where $\phi:[0,+\infty)\longrightarrow[0,+\infty)$ is a Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of $[0,+\infty)$, nonnegative, and such that for each $\epsilon>0,\int_{0}^{\epsilon}\phi(t)dt>0$; then $S$ has a unique fixed point $b\in X$ such that for each $x\in X$, $\underset{n\rightarrow\infty}{\lim}S^{n}x=b$. After this result in (2003), Rhoades established the Branciari Theorem in the following. ###### Theorem 1.2. Let $(X,d)$ be a complete metric space, $k\in[0,1)$ and $S:X\longrightarrow X$ a mapping such that, for each $x,y\in X$, $\int_{0}^{d(Sx,Sy)}\phi(t)dt\leq k{\int_{0}^{m(x,y)}\phi(t)dt},\qquad\qquad\qquad(2)$ where $m(x,y)=\max\\{d(x,y),d(x,Sx),d(y,Sy),\frac{d(x,Sy)+d(y,Sx)}{2}\\}\qquad\qquad(3)$ and $\phi:[0,+\infty)\longrightarrow[0,+\infty)$ is a Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of $[0,+\infty)$, nonnegative, and such that for each $\epsilon>0,\int_{0}^{\epsilon}\phi(t)dt>0$. Then $S$ has a unique fixed point $b\in X$ such that for each $x\in X$, $\underset{n\rightarrow\infty}{\lim}S^{n}x=b$. In 2009 [1] A. Beiranvand, S. Moradi, M. Omid and H. Pazandeh introduced a new class of contractive mapping and extend the Banach Contractive Principle. Also in 2009 [4] A. Beiranvand and S. Moradi established the Branciari Theorem for these classes of mappings. It is the purpose of this paper to make an extension the Rhoades Theorem (Theorem 1.2). For the main theorem (Theorem 2.1) we need the following definition. ###### Definition 1.3. $[1]$ Let $(X,d)$ be a metric space. A mapping $T:X\longrightarrow X$ is said sequentially convergent if we have, for every sequence $\\{y_{n}\\}$, if $\\{Ty_{n}\\}$ is convergence then $\\{y_{n}\\}$ also is convergence. $T$ is said subsequentially convergent if we have, for every sequence $\\{y_{n}\\}$, if $\\{Ty_{n}\\}$ is convergence then $\\{y_{n}\\}$ has a convergent subsequence. ## 2 Main Result The following theorem (Theorem 2.1) is the main result of this paper. ###### Theorem 2.1. Let $(X,d)$ be a complete metric space, $k\in[0,1)$ and $S:X\longrightarrow X$ a mapping such that, for each $x,y\in X$, $\int_{0}^{d(TSx,TSy)}\phi(t)dt\leq k{\int_{0}^{m^{\prime}(Tx,Ty)}\phi(t)dt},\qquad\qquad\qquad(4)$ where $m^{\prime}(Tx,Ty)=\max\\{d(Tx,Ty),d(Tx,TSx),d(Ty,TSy),\frac{d(Tx,TSy)+d(Ty,TSx)}{2}\\}\qquad(5)$ and $\phi:[0,+\infty)\longrightarrow[0,+\infty)$ is a Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of $[0,+\infty)$, nonnegative, and such that $for\>each\>\epsilon>0\qquad\int_{0}^{\epsilon}\phi(t)dt>0\qquad\qquad\qquad(6)$ and $T:X\longrightarrow X$ is a continuous, one-to-one and subsequentially convergent. Then $S$ has a unique fixed point $b\in X$ and, if $T$ is sequentially convergent then for each $x\in X$, $\underset{n\rightarrow\infty}{\lim}S^{n}x=b$. ###### Proof. From (4) $S$ is continuous and if $x\neq y$ then, $d(TSx,TSy)<m^{\prime}(x,y).\qquad\qquad\qquad\qquad(7)$ Let $x\in X$. Define $x_{n}=TS^{n}x$. From (5) we conclude that: $\displaystyle m^{\prime}(x_{m},x_{n})=m^{\prime}(TS^{m}x,TS^{n}x)=$ $\displaystyle\max\\{d(x_{m},x_{n}),d(x_{m},x_{m+1}),d(x_{n},x_{n+1}),\frac{d(x_{n},x_{m+1})+d(x_{m},x_{n+1})}{2}\\}.\qquad\qquad(8)$ We break the argument into four steps. STEP 1. $\underset{n\rightarrow\infty}{\lim}d(x_{n},x_{n+1})=0$. proof. For each integer $n\geq 1$, from (4), $\int_{0}^{d(x_{n},d_{n+1})\phi(t)dt}\leq k{\int_{0}^{m^{\prime}(x_{n-1},x_{n})}\phi(t)dt},\hskip 142.26378pt(9)$ and by (8), $\displaystyle m^{\prime}(x_{n-1},x_{n})$ $\displaystyle=$ $\displaystyle\max\\{d(x_{n-1},x_{n}),d(x_{n-1},x_{n}),d(x_{n},x_{n+1}),\frac{d(x_{n-1},x_{n+1})+d(x_{n},x_{n})}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{d(x_{n-1},x_{n}),d(x_{n},x_{n+1}),\frac{d(x_{n-1},x_{n+1})}{2}\\}$ $\displaystyle\leq$ $\displaystyle\max\\{d(x_{n-1},x_{n}),d(x_{n},x_{n+1}),\frac{d(x_{n-1},x_{n})+d(x_{n},x_{n+1})}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{d(x_{n-1},x_{n}),d(x_{n},x_{n+1})\\}\>\>(from\>(6)\>and\>(7))$ $\displaystyle=$ $\displaystyle d(x_{n-1},x_{n}).\hskip 199.16928pt(10)$ Hence, by (9) and (10) we have, $\int_{0}^{d(x_{n},d_{n+1})\phi(t)dt}\leq k^{n}{\int_{0}^{d(x,x_{1})}\phi(t)dt}.\hskip 85.35826pt(11)$ Taking the limit of (11), as $n\rightarrow\infty$, gives $\underset{n\rightarrow\infty}{\lim}\int_{0}^{d(x_{n},d_{n+1})\phi(t)dt}=0$. Since (6) is holds, $\underset{n\rightarrow\infty}{\lim}d(x_{n},x_{n+1})=0.\qquad\qquad\qquad\qquad\qquad\qquad\qquad(12)$ STEP 2. $\\{x_{n}\\}$ is a bounded sequence. proof. If $\\{x_{n}\\}$ is not a bounded sequence then, we choose a sequence $\\{n(k)\\}_{k=1}^{\infty}$ such that $n(1)=1$ and for each $k\in\mathbb{N}$; $n(k+1)$ is ”minimal” in the sense such that $d(x_{n(k+1)},x_{n(k)})>1$. Obviously $n(k)\geq k$ for all $k\in\mathbb{N}$. By step 1, there exists $k_{0}\in\mathbb{N}$ such that for every $k\geq k_{0}$; $d(x_{k+1},x_{k})<\frac{1}{2}$. So for each $k\geq k_{0}$; $\displaystyle 1<d(x_{n(k+1)},x_{n(k)})$ $\displaystyle\leq$ $\displaystyle d(x_{n(k+1)},x_{n(k+1)-1})+d(x_{n(k+1)-1},x_{n(k)})$ $\displaystyle\leq$ $\displaystyle d(x_{n(k+1)},x_{n(k+1)-1})+1.\qquad\qquad\qquad\qquad(13)$ By (12) and (13) we conclude that, $\underset{n\rightarrow\infty}{\lim}d(x_{n(k+1)},x_{n(k)})=1.\qquad\qquad\qquad\qquad\qquad\qquad(14)$ Also, $\displaystyle d(x_{n(k+1)},x_{n(k)})-d(x_{n(k+1)+1},x_{n(k+1)})-d(x_{n(k)+1},x_{n(k)})$ $\displaystyle\leq d(x_{n(k+1)+1},x_{n(k)+1})\leq d(x_{n(k+1)+1},x_{n(k+1)})$ $\displaystyle+d(x_{n(k+1)},x_{n(k)})+d(x_{n(k)},x_{n(k)+1}).\hskip 71.13188pt(15)$ Since (12), (14) and (15) are hold, $\underset{n\rightarrow\infty}{\lim}d(x_{n(k+1)+1},x_{n(k)+1})=1.\qquad\qquad\qquad\qquad\qquad\qquad(16)$ Therefore by (8), $\displaystyle m^{\prime}(x_{n(k+1)},x_{n(k)})=\max\\{d(x_{n(k+1)},x_{n(k)}),d(x_{n(k+1)},x_{n(k+1)+1}),$ $\displaystyle d(x_{n(k)},x_{n(k)+1}),\frac{d(x_{n(k)},x_{n(k+1)+1})+d(x_{n(k+1)},x_{n(k)+1})}{2}\\},\qquad\qquad(17)$ from (12) and (14), for large enough $k$, $\displaystyle m^{\prime}(x_{n(k+1)},x_{n(k)})=\max\\{d(x_{n(k+1)},x_{n(k)}),\frac{d(x_{n(k)},x_{n(k+1)+1})+d(x_{n(k+1)},x_{n(k)+1})}{2}\\}$ $\displaystyle=\max\\{d(x_{n(k+1)},x_{n(k)}),\frac{[d(x_{n(k)},x_{n(k)+1})+d(x_{n(k)+1},x_{n(k+1)+1})]}{2}+$ $\displaystyle\frac{[d(x_{n(k+1)},x_{n(k+1)+1})+d(x_{n(k+1)+1},x_{n(k)+1})]}{2}\\}\underset{k\rightarrow\infty}{\longrightarrow}1.\qquad\qquad\qquad\qquad\qquad(18)$ So by (16) and (18) and $\int_{0}^{d(x_{n(k+1)+1},x_{n(k)+1})}\phi(t)dt\leq k\int_{0}^{m^{\prime}(x_{n(k+1)},x_{n(k)}))}\phi(t)dt,\qquad\qquad\qquad(19)$ we conclude that, $\int_{0}^{1}\phi(t)dt\leq k\int_{0}^{1}\phi(t)dt.\qquad\qquad\qquad\qquad\qquad\qquad(20)$ Since $k\in[0,1)$, $\int_{0}^{1}\phi(t)dt=0$ and this is contradiction with (6). STEP 3. $\\{x_{n}\\}$ is a Cauchy sequence. proof. For every $m,n\in\mathbb{N}(m>n)$ by (4) $\displaystyle\int_{0}^{d(x_{m},x_{n})}\phi(t)dt\leq\int_{0}^{m^{\prime}(x_{m-1},x_{n-1})}\phi(t)dt$ $\displaystyle=k\int_{0}^{\max\\{d(x_{m-1},x_{n-1}),d(x_{m-1},x_{m}),d(x_{n-1},x_{n}),\frac{d(x_{m-1},x_{n})+d(x_{n-1},x_{m})}{2}\\}}\phi(t)dt$ $\displaystyle\leq k\int_{0}^{\max\\{d(x_{m-1},x_{n-1}),d(x_{m-1},x_{m}),d(x_{n-1},x_{n}),d(x_{m-1},x_{n}),d(x_{n-1},x_{m})\\}}\phi(t)dt$ $\displaystyle=k\int_{0}^{d(x_{r(1)},x_{s(1)})}\phi(t)dt,\qquad\qquad\qquad\qquad\qquad\qquad\qquad(21)$ where $s(1)\geq n-1$ and $r(1)>s(1)$. By the same argument, there exist $r(2),s(2)\in\mathbb{N}$ such that $r(2)>s(2)$ and $s(2)\geq s(1)-1\geq n-2$ such that $\int_{0}^{d(x_{r(1)},x_{s(1)})}\phi(t)dt\leq k\int_{0}^{d(x_{r(2)},x_{s(2)})}\phi(t)dt.\qquad\qquad\qquad\qquad\qquad(22)$ So, by (21) and (22), $\int_{0}^{d(x_{m},x_{n})}\phi(t)dt\leq k^{2}\int_{0}^{d(x_{r(2)},x_{s(2)})}\phi(t)dt.\qquad\qquad\qquad\qquad\qquad(23)$ By the same argument, there exist $r(n),s(n)\in\mathbb{N}$ such that $r(n)>s(n)$ and $s(n)\geq s(n)-n\geq n-n=0$ and $\int_{0}^{d(x_{m},x_{n})}\phi(t)dt\leq k^{n}\int_{0}^{d(x_{r(n)},x_{s(n)})}\phi(t)dt.\qquad\qquad\qquad\qquad\qquad(24)$ Since $\\{x_{n}\\}$ is a bounded sequence and (24) is holds, $\underset{m,n\rightarrow\infty}{\lim}\int_{0}^{d(x_{m},x_{n})}\phi(t)dt=0.\qquad\qquad\qquad\qquad\qquad(25)$ Hence, from (6), $\underset{m,n\rightarrow\infty}{\lim}d(x_{m},x_{n})=0.\qquad\qquad\qquad\qquad\qquad\qquad(26)$ Therefore $\\{x_{n}\\}$ is a Cauchy sequence. Step 4. $S$ has a fixed point. proof. Since $(X,d)$ is a complete metric space and $\\{x_{n}\\}$ is a Cauchy sequence there exists $a\in X$ such that $\underset{n\rightarrow\infty}{\lim}TS^{n}(x)=a.\hskip 142.26378pt(27)$ Since $T$ is subsequentially convergent, $\\{S^{n}(x)\\}$ has a convergent subsequence alike $\\{S^{n(k)}(x)\\}_{k=1}^{\infty}$. Suppose that $\underset{k\rightarrow\infty}{\lim}S^{n(k)}(x)=b.\hskip 142.26378pt(28)$ Since $T$ is continuous, $\underset{k\rightarrow\infty}{\lim}TS^{n(k)}(x)=Tb.\hskip 128.0374pt(29)$ From (27) and (29) we conclude that $Tb=a.\hskip 184.9429pt(30)$ Since $S$ is continuous and (28) is holds, $\underset{k\rightarrow\infty}{\lim}S^{n(k)+1}(x)=Sb.\hskip 128.0374pt(31)$ So, $\underset{k\rightarrow\infty}{\lim}TS^{n(k)+1}(x)=TSb.\hskip 113.81102pt(32)$ Again from (27) and (30) $TSb=a=Tb.\hskip 156.49014pt(33)$ Since $T$ is one-to-one, $Sb=b$. Therefore $S$ has a fixed point. Obviously, by (4) and (6) we conclude that $S$ has a unique fixed point. ∎ ###### Remark 2.2. Theorem 2.1 is a generalization of the Rhoades theorem (Theorem 1.2), letting $Tx=x$ for each $x\in X$ in Theorem 2.5, so $\displaystyle\int_{0}^{d(Sx,Sy)}\phi(t)dt$ $\displaystyle=$ $\displaystyle\int_{0}^{d(TSx,TSy)}\phi(t)dt$ $\displaystyle\leq$ $\displaystyle k{\int_{0}^{m^{\prime}(x,y)}\phi(t)dt}=k{\int_{0}^{m(x,y)}\phi(t)dt}.\qquad(34)$ The following example shows that (4) is indeed a proper extension of (2). ###### Example 2.3. Let $X=[1,+\infty)$ endowed with the Euclidean metric. Define $S:X\longrightarrow X$ by $Sx=4\sqrt{x}$. Obviously $S$ has a unique fixed point $b=16$. If (2) holds for some $k\in[0,1)$, then for every $x,y\in X$ such that $x\neq y$, we have $\|Sx-Sy\|<m(x,y).\hskip 156.49014pt(35)$ But by taking $x=1$ and $y=4$ we have, $\|Sx-Sy\|=m(x,y)=4$ and this is contradiction. Therefore we can not use the Rhoades theorem (Theorem 1.2) for this example. Now we define $T:X\longrightarrow X$ by $Tx=\ln(e.x)$. Obviously $T$ is one- to-one, continuous and sequentially convergent and $\|TSx-TSy\|=\frac{1}{2}\|\ln(\frac{e.x}{e.y})\|=\frac{1}{2}\|Tx- Ty\|\leq\frac{1}{2}m^{\prime}(Tx,Ty).\qquad\qquad(36)$ By taking $\phi\equiv 1$, all conditions of Theorem 2.1 are hold and therefore $S$ has a unique fixed point. ## References * [1] A. Beiranvand, S. Moradi, M. Omid and H. Pazandeh , Two Fixed-Point Theorem For Special Mapping, to appear. * [2] A. Branciari, A fixed point theorem for mapping satisfying a general contractive condition of integral type Int. J. M. and M. since, 29 (2002), 531-536. * [3] B. E. Rhoades, Two fixed-point theorems for mappings satisfying a general contractive condition of integral type Int. J. M. and M. since, 63 (2003), 4007-4013. * [4] S. Moradi and A. Beiranvand, A fixed-point theorem for mapping satisfying a general contractive condition of integral type depended an another function, to appear. Email: S-Moradi@araku.ac.ir	arxiv-papers	2009-03-09T15:19:35	2024-09-04T02:49:01.045218	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "S. Moradi", "submitter": "Sirous Moradi", "url": "https://arxiv.org/abs/0903.1574" }
0903.1577	# Kannan Fixed-Point Theorem On Complete Metric Spaces And On Generalized Metric Spaces Depended an Another Function 1112000 Mathematics Subject Classification: Primary 46J10, 46J15, 47H10. S. Moradi Faculty of Science, Department of Mathematics Arak University, Arak, Iran ###### Abstract We obtain sufficient conditions for existence of unique fixed point of Kannan type mappings on complete metric spaces and on generalized complete metric spaces depended an another function. Keywords: Fixed point, contractive mapping, sequently convergent, subsequently convergent. ## 1 Introduction The fixed point theorem most be frequently cited in Banach condition mapping principle (see [4] or [6]), which asserts that if $(X,d)$ is a complete metric space and $S:X\longrightarrow X$ is a contractive mapping ($S$ is contractive if there exists $k\in[0,1)$ such that for all $x,y\in X$, $d(Sx,Sy)\leq kd(x,y)$) then $S$ has a unique fixed point. In 1968 [5] Kannan established a fixed point theorem for mapping satisfying: $d(Sx,Sy)\leq\lambda\big{[}d(x,Sx)+d(y,Sy)\big{]},\hskip 199.16928pt(1)$ for all $x,y\in X$, where $\lambda\in[0,\frac{1}{2})$. Kannan’s paper [5] was followed by a spate of papers containing a variety of contractive definitions in metric spaces. Rhoades [7] in 1977 considered 250 type of contractive definitions and analyzed the relationship among them. In 2000 Branciari [3] introduced a class of generalized metric spaces by replacing triangular inequality by similar ones which involve four or more points instead of three and improved Banach contraction mapping principle. Recently Azam and Arshad [1] in 2008 extended the Kannan’s theorem for this kind of generalized metric spaces. In 2009 [2] A. Beiranvand, S. Moradi, M. Omid and H. Pazandeh introduced new classes of contractive functions and established the Banach contractive principle. In the present paper at first we extend the Kannan’s theorem [5] and then extend the theorem due to Azam and Arshad [1] for these new classes of functions. From the main results we need some new definitions. ###### Definition 1.1. $[2]$ Let $(X,d)$ be a metric space. A mapping $T:X\longrightarrow X$ is said sequentially convergent if we have, for every sequence $\\{y_{n}\\}$, if $\\{Ty_{n}\\}$ is convergence then $\\{y_{n}\\}$ also is convergence. $T$ is said subsequentially convergent if we have, for every sequence $\\{y_{n}\\}$, if $\\{Ty_{n}\\}$ is convergence then $\\{y_{n}\\}$ has a convergent subsequence. ###### Definition 1.2. $([1]$ or $[3])$ Let $X$ be a nonempty set. Suppose that the mapping $d:X\longrightarrow X$, satisfies: (i) $d(x,y)\geq 0$, for all $x,y\in X$ and $d(x,y)=0$ if and only if $x=y$; (ii)$d(x,y)=d(y,x)$ for all $x,y\in X$; (iii)$d(x,y)\leq d(x,w)+d(w,z)+d(z,y)$ for all $x,y\in X$ and for all distinct points $w,z\in X\backslash\\{x,y\\}$[rectangular property]. Then $d$ is called a generalized metric and $(X,d)$ is a generalized metric space. For more information can see [1] and [3]. ## 2 Main Results In this section at first we extend the Kannan’s theorem [5] and then extend the Azam and Arshad theorem [1]. ###### Theorem 2.1. $[$Extended Kannan’s Theorem$]$ Let $(X,d)$ be a complete metric space and $T,S:X\longrightarrow X$ be mappings such that $T$ is continuous, one-to-one and subsequentially convergent. If $\lambda\in[0,\frac{1}{2})$ and $d(TSx,TSy)\leq\lambda\big{[}d(Tx,TSx)+d(Ty,TSy)\big{]},\hskip 142.26378pt(2)$ for all $x,y\in X$, then $S$ has a unique fixed point. Also if $T$ is sequentially convergent then for every $x_{0}\in X$ the sequence of iterates $\\{S^{n}x_{0}\\}$ converges to this fixed point. ###### Proof. Let $x_{0}$ be and arbitrary point in $X$. We define the iterative sequence $\\{x_{n}\\}$ by $x_{n+1}=Sx_{n}$ (equivalently, $x_{n}=S^{n}x_{0}$), $n=1,2,...$. Using the inequality (2), we have $\displaystyle d(Tx_{n},Tx_{n+1})$ $\displaystyle=$ $\displaystyle d(TSx_{n-1},TSx_{n})$ $\displaystyle\leq$ $\displaystyle\lambda\big{[}d(Tx_{n-1},TSx_{n-1})+d(Tx_{n},TSx_{n})\big{]},\hskip 85.35826pt(3)$ so, $d(Tx_{n},Tx_{n+1})\leq\frac{\lambda}{1-\lambda}d(Tx_{n-1},Tx_{n}).\hskip 190.63338pt(4)$ By the same argument, $\displaystyle d(Tx_{n},Tx_{n+1})$ $\displaystyle\leq$ $\displaystyle\frac{\lambda}{1-\lambda}d(Tx_{n-1},Tx_{n})\leq(\frac{\lambda}{1-\lambda})^{2}d(Tx_{n-2},Tx_{n-1})$ $\displaystyle\leq$ $\displaystyle...\leq(\frac{\lambda}{1-\lambda})^{n}d(Tx_{0},Tx_{1}).\hskip 147.95424pt(5)$ By (5), for every $m,n\in\mathbb{N}$ such that $m>n$ we have, $\displaystyle d(Tx_{m},Tx_{n})$ $\displaystyle\leq$ $\displaystyle d(Tx_{m},Tx_{m-1})+d(Tx_{m-1},Tx_{m-2})+...+d(Tx_{n+1},Tx_{n})$ $\displaystyle\leq$ $\displaystyle\big{[}(\frac{\lambda}{1-\lambda})^{m-1}+(\frac{\lambda}{1-\lambda})^{m-2}+...+(\frac{\lambda}{1-\lambda})^{n}\big{]}d(Tx_{0},Tx_{1})$ $\displaystyle\leq$ $\displaystyle\big{[}(\frac{\lambda}{1-\lambda})^{n}+(\frac{\lambda}{1-\lambda})^{n+1}+...\big{]}d(Tx_{0},Tx_{1})$ $\displaystyle=$ $\displaystyle(\frac{\lambda}{1-\lambda})^{n}\frac{1}{1-(\frac{\lambda}{1-\lambda})}d(Tx_{0},Tx_{1}).\hskip 128.0374pt(6)$ Letting $m,n\longrightarrow\infty$ in (6), we have $\\{Tx_{n}\\}$ is a Cauchy sequence, and since $X$ is a complete metric space, there exists $v\in X$ such that $\underset{n\rightarrow\infty}{\lim}Tx_{n}=v.\hskip 312.9803pt(7)$ Since $T$ is a subsequentially convergent, $\\{x_{n}\\}$ has a convergent subsequence. So there exists $u\in X$ and $\\{x_{n(k)}\\}_{k=1}^{\infty}$ such that $\underset{k\rightarrow\infty}{\lim}x_{n(k)}=u$. Since $T$ is continuous and $\underset{k\rightarrow\infty}{\lim}x_{n(k)}=u$, $\underset{k\rightarrow\infty}{\lim}Tx_{n(k)}=Tu$. By (7) we conclude that $Tu=v$. So $\displaystyle d(TSu,Tu)$ $\displaystyle\leq$ $\displaystyle d(TSu,TS^{n(k)}x_{0})+d(TS^{n(k)}x_{0},TS^{n(k)+1}x_{0})+d(TS^{n(k)+1}x_{0},Tu)$ $\displaystyle\leq$ $\displaystyle\lambda\big{[}d(Tu,TSu)+d(TS^{n(k)-1}x_{0},TS^{n(k)}x_{0})\big{]}$ $\displaystyle+$ $\displaystyle(\frac{\lambda}{1-\lambda})^{n(k)}d(TSx_{0},Tx_{0})+d(Tx_{n(k)+1},Tu)$ $\displaystyle=$ $\displaystyle\lambda d(Tu,TSu)+\lambda d(Tx_{n(k)-1},Tx_{n(k)})$ $\displaystyle+$ $\displaystyle(\frac{\lambda}{1-\lambda})^{n(k)}d(Tx_{1},Tx_{0})+d(Tx_{n(k)+1},Tu),\hskip 85.35826pt(8)$ hence, $\displaystyle d(TSu,Tu)$ $\displaystyle\leq$ $\displaystyle\frac{\lambda}{1-\lambda}d(Tx_{n(k)-1},Tx_{n(k)})+\frac{1}{1-\lambda}(\frac{\lambda}{1-\lambda})^{n(k)}d(Tx_{1},Tx_{0})$ $\displaystyle+$ $\displaystyle\frac{1}{1-\lambda}d(Tx_{n(k)+1},Tu)\underset{k\rightarrow\infty}{\longrightarrow}0.\hskip 142.26378pt(9)$ Therefore $d(TSu,Tu)=0$. Since $T$ is one-to-one $Su=u$. So $S$ has a fixed point. Since (2) holds and $T$ is one-to-one, $S$ has a unique fixed point. Now if $T$ is sequentially convergent, by replacing $\\{n\\}$ with $\\{n(k)\\}$ we conclude that $\underset{n\rightarrow\infty}{\lim}x_{n}=u$ and this shows that $\\{x_{n}\\}$ converges to the fixed point of $S$. ∎ ###### Remark 2.2. By taking $Tx\equiv x$ in Theorem 2.1, we can conclude the Kannan’s theorem[5]. The following example shows that Theorem 2.1 is indeed a proper extension on Kannan’s theorem. ###### Example 2.3. Let $X=\\{0\\}\cup\\{\frac{1}{4},\frac{1}{5},\frac{1}{6},...\\}$ endowed with the Euclidean metric. Define $S:X\longrightarrow X$ by $S(0)=0$ and $S(\frac{1}{n})=\frac{1}{n+1}$ for all $n\geq 4$. Obviously the condition (1) is not true for every $\lambda>0$. So we can not use the Kannan’s theorem [5]. By define $T:X\longrightarrow X$ by $T(0)=0$ and $T(\frac{1}{n})=\frac{1}{n^{n}}$ for all $n\geq 4$ we have, for $m,n\in\mathbb{N}$ ($m>n$), $\displaystyle\|TS(\frac{1}{m})-TS(\frac{1}{n})\|$ $\displaystyle=$ $\displaystyle\frac{1}{(n+1)^{n+1}}-\frac{1}{(m+1)^{m+1}}$ $\displaystyle<$ $\displaystyle\frac{1}{(n+1)^{n+1}}\leq\frac{1}{3}\big{[}\frac{1}{n^{n}}-\frac{1}{(n+1)^{n+1}}\big{]}$ $\displaystyle\leq$ $\displaystyle\frac{1}{3}\big{[}\frac{1}{n^{n}}-\frac{1}{(n+1)^{n+1}}+\frac{1}{m^{m}}-\frac{1}{(m+1)^{m+1}}\big{]}$ $\displaystyle=$ $\displaystyle\frac{1}{3}\big{[}\|T(\frac{1}{n})-TS\frac{1}{n}\|+\|T(\frac{1}{m})-TS\frac{1}{m}\|\big{]}.\hskip 71.13188pt(10)$ The inequality (10) shows that (2) is true for $\lambda=\frac{1}{3}$. Therefore by Theorem 2.1 $S$ has a unique fixed point. In the following theorem we extend the Azam and Arshad theorem [1]. ###### Theorem 2.4. Let $(X,d)$ be a complete generalizes metric space and $T,S:X\longrightarrow X$ be mappings such that $T$ is continuous, one-to-one and subsequentially convergent. If $\lambda\in[0,\frac{1}{2})$ and $d(TSx,TSy)\leq\lambda\big{[}d(Tx,TSx)+d(Ty,TSy)\big{]},\hskip 142.26378pt(11)$ for all $x,y\in X$, then $S$ has a unique fixed point. Also if $T$ is sequentially convergent then for every $x_{0}\in X$ the sequence of iterates $\\{S^{n}x_{0}\\}$ converges to this fixed point. ###### Proof. ∎ ###### Remark 2.5. By taking $Tx\equiv x$ in Theorem 2.4, we can conclude the Azam and Arshad theorem [1]. The following example shows that Theorem 2.4 is indeed a proper extension on Azam and Arshad theorem. ###### Example 2.6. $[1]$ Let $X=\\{1,2,3,4\\}$. Define $d:X\times X\longrightarrow\mathbb{R}$ as follows: $d(1,2)=d(2,1)=3,$ $d(2,3)=d(3,2)=d(1,3)=d(3,1)=1,$ $d(1,4)=d(4,1)=d(2,4)=d(4,2)=d(3,4)=d(4,3)=4$. Obviously $(X,d)$ is a generalized metric space and is not a metric space. Now define a mapping $S:X\longrightarrow X$ as follows: $Sx=\left\\{\begin{array}[]{ll}2&;x\neq 1\\\ 4&;x=1\\\ \end{array}\right.$ Obviously the inequality (1) is not holds for $S$ for every $\lambda\in[0,\frac{1}{2})$. So we can not use the Azam and Arshad theorem for $S$. By define $T:X\longrightarrow X$ by: $Tx=\left\\{\begin{array}[]{ll}2&;x=4\\\ 3&;x=2\\\ 4&;x=1\\\ 1&;x=3\\\ \end{array}\right.$ we have $TSx=\left\\{\begin{array}[]{ll}3&;x\neq 1\\\ 2&;x=1\\\ \end{array}\right.$ We can show that $d(TSx,TSy)\leq\frac{1}{3}\big{[}d(Tx,TSx)+d(Ty,TSy)\big{]}.\hskip 142.26378pt(12)$ Therefore by Theorem 2.4, $S$ has a unique fixed point. ## References * [1] A. Azam and M. Arshad, Kannan fixed point theorem on generalized metric spaces, The J. Nonlinear Sci., 1(2008), no. 1, 45-48. * [2] A. Beiranvand, S. Moradi, M. Omid and H. Pazandeh , Two Fixed-Point Theorem For Special Mapping, to appear. * [3] A. Branciari, A fixed point theorem of Banach-Caccippoli type on a class of generalized metric spaces Publ. Math. Debrecen, 57 1-2(2000), 31-37. 1, 1, 1.2 * [4] K. Goebel and W. A. Kirk, Topiqs in Metric Fixed Point Theory, Combridge University Press, New York, 1990. * [5] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60(1968),71-76. 1, 1. * [6] M. A. Khamsi and W. A. Kirk, An introduction to metric spaces and fixed point theory, John Wiley and Sons, Inc., 2001. * [7] B. E. Rhoades, A Comparison of Various Definitions of Contractive Mappings, Trans. Amer. Math. Soc. 226(1977), 257-290. Email: S-Moradi@araku.ac.ir	arxiv-papers	2009-03-09T15:24:20	2024-09-04T02:49:01.048210	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "S. Moradi", "submitter": "Sirous Moradi", "url": "https://arxiv.org/abs/0903.1577" }
0903.1580	# Birkhoff’s invariant and Thorne’s Hoop Conjecture G. W. Gibbons D.A.M.T.P., Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K ###### Abstract I propose a sharp form of Thorne’s hoop conjecture which relates Birkhoff’s invariant $\beta$ for an outermost apparent horizon to its $ADM$ mass, $\beta\leq 4\pi M_{ADM}$. I prove the conjecture in the case of collapsing null shells and provide further evidence from exact rotating black hole solutions. Since $\beta$ is bounded below by the length $l$ of the shortest non-trivial geodesic lying in the apparent horizon, the conjecture implies $l\leq 4\pi M_{ADM}$. The Penrose conjecture, $\sqrt{\pi A}\leq 4\pi M_{ADM}$, and Pu’s theorem imply this latter consequence for horizons admitting an antipodal isometry. Quite generally, Penrose’s inequality and Berger’s isembolic inequality, $\sqrt{\pi A}\geq{2\over\sqrt{\pi}}i$, where $i$ is the injectivity radius, imply $4c\leq 2i\leq 4\pi M_{ADM}$, where $c$ is the convexity radius. ## 1 The Hoop Conjecture Thorne’s original Hoop Conjecture [1] was that > Horizons form when and only when a mass $m$ gets compacted into a region > whose circumference in EVERY direction is $C\leq 4\pi M$. The capitalization “EVERY ”was intended to emphasis the fact that while the collapse of oblate shaped bodies the circumferences are all roughly equal, in the prolate case, a the collapse of a long almost cylindrically shaped body whose girth was nevertheless small would not necessarily produce a horizon. However, as proposed, the statement is so imprecise as to render either proof or disproof impossible. Presumably for the mass we could take the ADM mass, $M_{ADM}$, but what about the circumference of the hoop? Since Thorne’s article, there have been many attempts to tighten up the formulation of the conjecture, the most recent of which is [2] to which the reader is referred for an extensive list of previous contributions to this question. The present note was inspired by [3] and in many respects takes forward a suggestion of Tod [4]. It is closely related to recent work of Yau and others on the idea of quasi-local mass. ## 2 Birkhoff’s Invariant and Birkhoff’s Theorem We can assume that the apparent horizon is topologically spherical [6, 7, 8, 9]. In what follows we follow [3] fairly closely. Suppose that $S=\\{S^{2},g\\}$ is a sphere with arbitrary metric $g$ and $f:S\rightarrow{{R}}$ a function on $S$ with just two critical points, a maximum and a minimum. Each level set $f^{-1}(c),\,c\in{{R}}$ has a length $l(c)$ and for any given function $f$ we define $\beta(f)={\rm max}_{c}\,l(c)\,.$ (1) We now define the Birkhoff invariant $\beta(S,g)$ by minimising $\beta(f)$ over all such functions $\beta=\inf_{f}\beta(f)\,.$ (2) The intuitive meaning of $\beta$ is the least length of a length of a closed (elastic) string or rubber band which may be slipped over over the surface $S$ [5]. To understand why, note that each function $f$ gives a foliation of $S$ by a one parameter family of curves $f=c$ which we may think of as the string or rubber band at each “moment of time ” $c$. $\beta(f)$ is the longest length of the band during that process. If we change the foliation we can hope to reduce this longest length and the infinum is the best that we can do. The phrase “moment of time ”is in quotation marks because we are not regarding $f$ as a physical time function, merely a convenient way of thinking about the geometry of $S$. Birkhoff’s Theorem [5] then assures us that there exist a closed geodesic $\gamma$ on $S$ with length $l(\gamma)=\beta(g)$. Clearly, if $l(g)$ is the length of the smallest non-trivial closed (i.e periodic) geodesic then $l(g)\leq\beta(g)\,.$ (3) It seems therefore that the Birkhoff invariant $\beta(g)$ should be taken as a precise formulation of Thorne’s rather vague notion of circumference. We shall proceed on this basis. Thus we make the following > Conjecture: For an outermost marginally trapped surface $S$ lying in a > Cauchy hypersurface surface $\Sigma$ with ADM mass $M_{\rm ADM}$ on which > the Dominant Energy condition holds, then > > $\beta(g)\leq 4\pi M_{\rm ADM}\,.$ (4) In other words, (4) is conjectured to be a necessary condition for a marginally outermost trapped surface. Bearing in mind Thorne’s comments about very prolate shaped surfaces for which $\beta(g)$ can be extremely small, it is not claimed that (4) is a sufficient condition for a closed surface $S$ to be trapped or marginally. Clearly, from (3), this form of the hoop conjecture implies $l(g)\leq 4\pi M_{\rm ADM}\,,$ (5) and therefore a counter example to (5) would be a counter example to (4). ### 2.1 The Kerr-Newman Horizon We first test the conjecture on the general charged rotating black hole. In standard notation, the metric on the horizon is [10] $g=ds^{2}=(r_{+}^{2}+a^{2})\Bigl{(}(1-x^{2}\sin^{2}\theta)d\theta^{2}+{\sin^{2}\theta d\phi^{2}\over 1-x^{2}\sin^{2}\theta}\Bigr{)}\,,$ (6) with $x^{2}={a^{2}\over r_{+}^{2}+a^{2}}\,.$ (7) This is clearly foliated by the orbits of the group of rotations generated by by ${\partial\over\partial\phi}$ and we take $f=\cos\theta$. That is, we are thinking of the coordinate $\theta$ as a function on $S$. In this case the greatest length of the small circles, i.e. of the orbits, is $l_{e}$, the length of the of the equatorial geodesic at $\theta={\pi\over 2}$ and we have $\beta(\cos\theta)=l_{e}=2\pi(r_{+}+{a^{2}\over r_{+}})=2\pi(2M-{Q^{2}\over r_{+}})\leq 4\pi M\,.$ (8) The right hand side of (8) is certainly an upper bound for the Birkhoff invariant and so the conjecture certainly holds in this case. However the horizon is prolate in character, in the sense that the polar circumference $l_{p}$ which is the length of a meridional geodesic $l_{p}$ (i.e. one with $\phi={\rm constant}$ and $\phi={\rm constant}+\pi$ ), is $l_{p}=\sqrt{r_{+}^{2}+a^{2}}\int^{\pi}_{0}\sqrt{1-x^{2}\sin^{2}\theta}\,d\theta\,.$ (9) In fact taking $f=\sin\theta\cos\phi$, we have $\beta(\sin\theta\cos\phi)=l_{p}\,,$ (10) and since $\sqrt{r_{+}^{2}+a^{2}}\leq r_{+}+{a^{2}\over r_{+}}\,,$ (11) we have $\beta(g)\leq l_{p}\leq l_{e}\leq 4\pi M\,.$ (12) Despite being prolate, the Gaussian curvature $K$ of the surface is given by $K={({r_{+}}^{2}+a^{2})(r_{+}^{2}-3a^{2}\cos^{2}\theta)\over({r_{+}}^{2}+a^{2}\cos^{2}\theta)^{3}}\,,$ (13) and can become negative at the poles $\theta=0,\pi$. The Kerr-Newman metrics have been generalised to include up to four different charges associated with four different abelian vector fields [12]. In the subclass for which only two charges are non-vanishing we can use the results of [13] to examine the conjecture. The energy momentum tensor of the system satisfies the Dominant Energy Condition and the horizon geometry may be extracted from eqn(45) of [13] $ds^{2}=Wd\theta^{2}+{(r_{+1}r_{+2}+a^{2})^{2}\over W}\sin^{2}\theta\,d\phi^{2}\,,$ (14) with $W=r_{+1}r_{+2}+a^{2}\cos^{2}\theta\,.$ (15) and $r_{+1}=r_{+}+2m\sinh^{2}\delta_{1}\,,\qquad r_{+2}=r_{+}+2m\sinh^{2}\delta_{2}$ (16) with $r_{+}$ the larger root of $r^{2}-2mr+a^{2}=0$ and $\delta_{1}$ and $\delta_{2}$ two parameters specifying the two charges. If $\delta_{1}=\delta_{2}$ we obtain the Kerr-Newman case. Just as the horizon geometry of the Kerr-Newman solution is isometric to that of the neutral Kerr, so in this more general case, we find an isometric horizon geometry. Of course the interpretation of the parameters occurring in the metric is different, but the geometry is the same. Thus $\beta(g)\leq l_{p}\leq l_{e}=2\pi\bigl{(}\sqrt{r_{+1}r_{+2}}+{a^{2}\over\sqrt{r_{+1}r_{+2}}}\bigr{)}\,.$ (17) Now for positive $x,y,z$, $xy\leq{1\over 4}(x+y)^{2}\,,\quad\Longrightarrow\quad\sqrt{(z+x)(z+y)}\leq z+{\textstyle{1\over 2}}(x+y)\,.$ (18) Thus, $\sqrt{r_{+1}r_{+2}}\leq r_{+}+m\bigl{(}\sinh^{2}\delta_{1}+\sinh^{2}\delta_{2}\bigr{)}$ (19) and ${a^{2}\over\sqrt{r_{+1}r_{+2}}}\leq{a^{2}\over r_{+}}\,,$ (20) Thus $\bigl{(}\sqrt{r_{+1}r_{+2}}+{a^{2}\over\sqrt{r_{+1}r_{+2}}}\bigr{)}\leq r_{+}+m\bigl{(}\sinh^{2}\delta_{1}+\sinh^{2}\delta_{2}\bigr{)}+{a^{2}\over r_{+}}.$ (21) But $2m=r_{+}+{a^{2}\over r_{+}}\,,$ (22) and the ADM mass is given by $M_{ADM}=2m+2m\bigl{(}\sinh^{2}\delta_{1}+\sinh^{2}\delta_{2}\bigr{)}$ (23) Thus $\beta(g)\leq 4\pi M_{ADM}\,,$ (24) and the conjecture holds in this case. It would be interesting to check it in the four charge case, but the algebra appears to be rather more complicated. ## 3 Collapsing Shells and Convex Bodies There is a class of examples [11] in which a shell of null matter collapses at the speed of light in which the apparent horizon $S$ may be thought of as a convex body isometrically embedded in Euclidean space ${{E}}^{3}$. In this case one has $8\pi M_{\rm ADM}\geq\int_{S}HdA\,,$ (25) where $H={\textstyle{1\over 2}}({1\over R_{1}}+{1\over R_{2}})$ is the mean curvature and $R_{1}$ and $R_{2}$ the principal radii of curvature of $S$ and $dA$ is the area element on $S$. The right hand side is called the total mean curvature and it was shown by Álvarez Paiva [3] in this case that $\beta(g)\leq{\textstyle{1\over 2}}\int_{S}HdA\,.$ (26) Combining Álvarez Paiva’s (26) with (25) establishes the conjecture (4) in this case. In fact the proof is close to the ideas in [4] and so we briefly review it. If ${\bf n}$ is a unit vector we define the height function on $S\subset{{E}}^{3}$ by $h={\bf n}.{\bf x}\,,\qquad{\bf x\in S}\,.$ (27) Let $S_{\bf n}$ be the orthogonal projection of the body $S$ onto a plane with unit normal ${\bf n}$ and let $C({\bf n})=l(\partial S_{\bf n})$ be the perimeter of $S_{\bf n}$. Then $\beta(g)\leq\beta(h)\leq C({\bf n})\,.$ (28) Now [4] $\int_{S}HdA={1\over 2\pi}\int_{S^{2}}C({\bf n})d\omega\,,$ (29) where $d\omega$ is the standard volume element on the round two-sphere $S^{2}$ of unit radius. Thus averaging (28) over $S^{2}$ and using (29) gives (26). ## 4 Shadows and widths The total mean curvature of a convex surface in Euclidean space has a number of interpretations. The width $w({\bf n})=w(-{\bf n})$ is the distance between two parallel tangent planes with normals $\pm{\bf n}$. One has ${1\over 2\pi}\int_{S}HdA=\langle w\rangle={1\over 4\pi}\int_{S}^{2}w({\bf n})d\omega\,.$ (30) Thus if $M={1\over 8\pi}\int_{S}HdA\,,$ (31) then $W\geq 4M\geq w$ (32) where $W$ is the greatest and $w$ the smallest width Similarly the Tod points out [4] that ${1\over 16\pi}C_{m}\leq M\leq{1\over 4\pi}C_{m}\,,$ (33) where $C_{m}$ is the largest perimeter of any orthogonal projection of the body. ## 5 Quasi-local masses Recent suggestions for a quasi-local mass expression [14, 15, 16, 17, 18] have involved isometrically embedding the horizon into Euclidean space ${{E}}^{3}$. This will, by results of Weyl and Pogorelov, certainly be possible if the Gauss curvature of $S$ is positive. In that case, since the embedding is isometric, the Birkhoff invariant can be calculated as if the surface is in flat Euclidean space and the inequality (26) holds. The total mean curvature $H$ associated with the embedding into ${{E}}^{3}$ also enters into the suggested expression for the quasi-local mass of a trapped surface, $4\pi M_{KLY}=\int_{S}\bigl{(}H-\sqrt{2\rho\mu}\bigr{)}dA$ (34) where $-2\rho$ and $2\mu$ are the expansions of outward and ingoing null normals, suitably normalised. Thus it is possible that some progress could be made there. However, as we have seen above the Gaussian curvature of the Kerr-Newman horizon can become negative near the poles, and as a consequence it cannot be isometrically embedded into ${{E}}^{3}$. Ignoring this difficulty for the time being, we observe that in the time symmetric case for a marginally outer trapped surface $4\pi M_{KLY}=\int_{S}HdA\leq\beta(g)\,.$ (35) ## 6 Areas It is now well established that the area $A(g)$ of the outermost marginally trapped surface should satisfy Penrose’s isoperimetric type conjecture that $\sqrt{\pi A(g)}\leq 4\pi M_{ADM}\,.$ (36) Evidently, if we could bound $\beta(g)$ above by by $\sqrt{\pi A(g)}$ we would have a proof of my version (4) of the hoop conjecture. On the other hand, if we can bound $\sqrt{\pi A(g)}$ above by $\beta(g)$, then the hoop conjecture would imply the Penrose conjecture. This raises the question of what is known about bounds for $A(g)$, $\beta(g)$, $l(g)$ and other invariants, either for a surface in general, or one with some additional restrictions. We begin by noting that the Riemannian metric $g$ on $S$ allows us to define a distance $d(x,y)=d(y,x),x,y\in S$ which is the infinum of the length of all curves from $x$ to $y$. Then $b(x)=\max_{y}\,d(x,y)$ (37) is the furthest we can get from $x$. We then define $\displaystyle e(g)$ $\displaystyle=$ $\displaystyle\min_{x}b(x)=\min_{x}\max_{y}d(x,y)$ (38) $\displaystyle E(g)$ $\displaystyle=$ $\displaystyle\max_{x}b(x)=\max_{x}\max_{y}\,d(x,y)$ (39) Hebda [21] provides a lower bound for $A$: $\sqrt{A(g)}\geq{1\over\sqrt{2}}\bigl{(}2e(s)-E(g)\bigr{)}\,.$ (40) Using (36) we get $4\pi M_{ADM}\geq{\sqrt{\pi\over 2}}\bigl{(}2e(s)-E(g)\bigr{)}\,,$ (41) For the sphere the right hand side of (40) is $\sqrt{2\pi^{3}}M_{ADM}$ which is satisfied but not sharp. There seems therefore no reason to choose $C(g)={\sqrt{\pi\over 2}}\bigl{(}2e(s)-E(g)\bigr{)}$, in order to sharpen Thorne’s conjecture. Another lower bound for the area has been given by Croke [22]. If, as above, $l(g)$ is the length of the shortest non-trivial geodesic on $S$, then Croke proves that $\sqrt{A(g)}\geq{1\over 31}l(g)\,.$ (42) This is again, far from the best possible result, which Croke conjectures to be $\sqrt{A(g)}\geq{1\over 3^{1\over 4}2^{{\textstyle{1\over 2}}}}l(g)\,,$ (43) which is attained for two flat equilateral triangles glued back to back. If we use (36 ) and (43 we obtain $\Bigl{(}{\pi^{2}\over 12}\Bigr{)}^{1\over 4}l(g)\leq 4\pi M_{ADM}\,.$ (44) If one takes $C(g)=l(g)$, then (44) is weaker than Thorne’s suggestion and taking $C(g)=\Bigl{(}{\pi^{2}\over 12}\Bigr{)}^{1\over 4}l(g)$ looks rather perverse, and in any case there is a problem about when it is attained. Moreover, since $\beta(g)\geq l(g)$, we cannot easily relate (44) to my form of the conjecture (4). Curiously however, for a special class of surfaces, we can improve considerably on (40) or (44). ### 6.1 Horizons admitting an anti-podal map Many results for general surfaces rely on on the existence of non-null homotopic closed curves. For a surface with spherical topology no such curves exist. However it is possible to restrict attention to the special class of surfaces for which ${{Z}}_{2}$ acts freely and isometrically such that $x\rightarrow Ix$. The quotient $S^{2}/I\equiv{{R}P}^{2}$ and Pu provides a lower bound for$A(S/I)$ in terms of the the systole ${\rm sys}(S/I)$, i.e. the length of the shortest non-null homotopic curve: $\sqrt{A(S/I)}\geq\sqrt{2\over\pi}{\rm sys}(S/I)\,.$ (45) Now the shortest non-null homotopic curve on $S/I$ is a closed geodesic which lifts to a closed geodesic of twice the length on $S$, thus ${\rm sys}(S/I)=\min_{x}\,d(x,Ix)\leq b(x)\leq e(g)\,,$ (46) where $b(x)$ and $e(g)$ are taken on the spherical double cover. If, as before, $l(g)$ is the length of the shortest non-trivial geodesic on $S$, then for this class of metrics $\sqrt{A(g)}\geq{2\over\sqrt{\pi}}\,{\rm sys}(S/I)\geq{l(g)\over\sqrt{\pi}}$ (47) and hence, using (36) we obtain for this class of metrics, $l(g)\leq 4\pi M_{ADM}\,,$ (48) i.e. the inequality (5) which is a consequence of my version of the hoop conjecture (4). Thus no counter example to to my conjecture can be constructed within the class of horizons admitting an antipodal isometry. Of course (5) is of the form of Thorne’s suggestion, if we take the circumference $C=l(g)$. However $l(g)$ does not carry with it the idea of the least circumference in all directions. I have argued above that it is $\beta(g)$ which better captures that notion, and so I prefer to think of (5) of the more basic inequality (4) and the fact that (5) holds in this special case as a confirmation of the general plausibility of this line of argument. It will perhaps be felt instructive to recall some of the details of Pu’s proof. He makes use of the fact that any metric $g$ on ${{R}}{{P}}^{2}$ may be written as $ds^{2}=\Omega^{2}(\theta,\phi)\bigl{(}d\theta^{2}+\sin^{2}\theta d\phi^{2}\bigr{)},$ (49) where $\Omega(\theta,\phi)=\Omega(\pi-\theta,\phi+\pi)$. Thus $A(S/I))=\int\Omega^{2}\sin\theta d\theta d\phi,$ (50) and ${\rm sys}(g)=2\inf_{x}\inf_{\gamma}\int_{\gamma}\Omega d\sigma$ where $\gamma$ is a curve running from $x\equiv(\theta,\phi)$ to $-x\equiv(\pi-\theta,\phi+\pi)$ and $d\sigma$ is the element of length calculated using the round metric. Now Pu considers the effect of averaging the conformal factor $\Omega$ with respect to the action of $SO(3)$, using the Haar or bi-invariant measure on $SO(3)$. If the averaged metric, which is of course the round metric, is $\overline{g}$, one has $A({\overline{g}})\leq A(g)$ (51) but $s({\overline{g}})\geq s(g).$ (52) Thus the ratio $A/s^{2}$ is never smaller than for the round metric ${\overline{g}}$, and this case it is ${1\over\pi}$ and so his inequality follows. ### 6.2 Injectivity and Convexity Radii The mathematical literature on area, the lengths of geodesics etc is often couched in terms of the injectivity radius $i(g)$ and the convexity radius $c(g)$. In the sequel we mainly follow the papers of Berger [24, 25, 26]. The definitions are valid for any dimension. The injectivity radius $i(x)$ of a point $x\in S$ is the supremum of the distances out to which which the exponential map is a diffeomorphism onto its image. The injectivity radius $i(g)$ of the manifold is the infinum over all points in $S$ of $i(x)$. In the case of an axisymmetric body for which the metric may be written as $ds^{2}=R^{2}\Bigl{\\{}d\theta^{2}+a^{2}(\theta)d\phi^{2}\Bigr{\\}}$ (53) with $R$ an overall constant setting the scale, $0\leq\theta\leq\pi$, the injectivity radius of the north ($\theta=0$) or south ($\theta=\pi$pole is ${1\over 2}l_{p}=\pi R$ and $i(g)\leq{1\over 2}l_{p}\,.$ (54) Now local extrema of $a(\theta)$ correspond to azimuthal geodesics. If the Gaussian curvature is positive, there will only be one, and define $l_{e}$ as its length. Otherwise $l_{e}$ as the smallest such length. $l(g)\leq l_{p}\,,\qquad l(g)\leq l_{e}$ (55) The convexity radius $c(x)$ of a point $x$ is the largest radius for which the geodesics ball $B_{c}(x)$ centred on $x$ is geodesically convex, that is every point in $B_{c}(x)$ is connected by a unique geodesic interval lying entirely within $B(x)$. The convexity radius $c(g)$ of the manifold is the infinum over all points in $S$ of $c(x)$. On the round unit-sphere ($R=1$) we have $i=\pi$ and $c={\pi\over 2}$. Now Berger proves that $l(g)\geq 2c(g)\,.$ (56) and hence $\beta(g)\geq 2c(g)$ (57) Thus in the case of horizons admitting an antipodal map, we can combine Pu’s result and (36) to obtain $2c(g)\leq 4\pi M_{ADM}\,,$ (58) in the case that my form of the hoop conjecture (4) holds we obtain from (57) the same result. In fact Klingenberg has shown that either $l(g)=2i(g)$ (59) or there is a geodesic segment of length $l(g)$ whose end points are conjugate. Finally for metrics on $S^{2}$ we have [27, 28] the so-called isembolic inequality $\sqrt{\pi A}\geq 2i(g)\geq 4c(g)$ (60) and hence by (36) $4\pi M_{ADM}\geq 2i(g)\geq 4c(g)\,.$ (61) ## 7 Higher dimensions Birkhoff’s invariant has a natural generalisation to higher dimensions (see e.g. [19] ) In the simplest case of an $n$-dimensional horizon, topologically equivalent to $S^{n}$, one considers foliations whose leaves are topologically $S^{n-1}$’s. There is then an obvious generalisation of (4) relating the ADM mass to the infinum over all foliations of the $(n-1)$-volume of the leaf of greatest $(n-1)$-volume. Pu [23] points out the obvious generalisation to metrics on ${{R}}{{P}}^{n}$ which are conformal to the round metric. However except in the case $n=2$, not every metric on ${{R}}{{P}}^{n}$ is conformal to the flat round metric and so for $n>2$ this is a very special case. However Berger’s isembolic inequality does generalise in an obvious way to all dimensions. Of course it is by now notorious [20] that in five dimensional spacetimes, horizons need not be topologically spherical and in particular one has black rings with horizon topology $S^{1}\times S^{2}$. The methods and ideas of [19] should also be relevant in that case. Further discussion of the higher dimensional situation will be deferred for a future publication. ## 8 Acknowledgements I am grateful to Gabor Domokos for bringing [3] to my attention and hence re- igniting my interest in these questions. I thank Harvey Reall, Gabriel Paternain, Chris Pope, Paul Tod and Shing-Tung Yau and Claude Warnick for helpful comments and suggestions. ## References * [1] K S Thorne, Nonspherical Gravitational Collapse: A Short Review in Magic without Magic ed. J Klauder (San Francisco: Freeman) (1972) * [2] J. M. M. Senovilla, A Reformulation of the Hoop Conjecture Europhys. Lett. 81, 20004 (2008) [arXiv:0709.0695 [gr-qc]]. * [3] J .C. Álvarez Paiva, Total mean curvature and closed geodesics. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 373–377. * [4] K. P. Tod, The hoop conjecture and the Gibbons-Penrose construction of trapped surfaces Class Quantum Grav 9 (1992) 1581-1591 * [5] G. D. Birkhoff, Dynamical systems with two degrees of freedom Trans. Amer. Math. Soc. 18 (1918) * [6] S. W. Hawking, Black holes in general relativity. Comm. Math. Phys. 25 (1972), 152-166. * [7] G. W. Gibbons, The time symmetric initial value problem for black holes, Commun. Math. Phys. 27 (1972) 87. * [8] G. W. Gibbons, Some Aspects of Gravitational Radiation and Gravitational Collapse Ph.D. Thesis, University of Cambridge (1972) * [9] S. W. Hawking, The event horizon , in Black holes (Les astres occlus) C. M. and B. De Witt(1973) 1-55 * [10] L. Smarr, Surface Geometry of Charged Rotating Black Holes, Phys Rev D 7 (1973) 289 * [11] G. W. Gibbons, Collapsing Shells and the Isoperimetric Inequality for Black Holes, Class. Quant. Grav. 14 (1997) 2905 [arXiv:hep-th/9701049]. * [12] M. Cvetic and D. Youm, Entropy of Non-Extreme Charged Rotating Black Holes in String Theory, Phys. Rev. D 54, (1996) 2612 [arXiv:hep-th/9603147]. * [13] Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope, Charged rotating black holes in four-dimensional gauged and ungauged supergravities, Nucl. Phys. B 717 (2005) 246 [arXiv:hep-th/0411045]. * [14] C. C. Liu and S. T. Yau, New definition of quasilocal mass and its positivity,” Phys. Rev. Lett. 90 (2003) 231102 [arXiv:gr-qc/0303019]. * [15] C. C. Liu and S. T. Yau, Positivity of quasi-local mass II, J. Amer. Math. Soc. 19 (2006) 181-204 [arXiv:math/0412292]. * [16] M. T. Wang and S. T. Yau, A generalization of Liu-Yau’s quasi-local mass, [ arXiv:math/0602321]. * [17] M. T. Wang and S. T. Yau Quasi-local mass in general relativity Phys.Rev.Lett102(2009) 021101 [arXiv:0804.1174[gr-qc]]. * [18] M. T. Wang and S. T. Yau Isometric embeddings into the Minkoswksi spactime and new quasi-local mass Commun Math Phys in press [arXiv:0805.1370[math.DG]]. * [19] T Colding and W.P. Minicozzi II, Width and Finite Extinction Time of Ricci Flow [arXiv:0707.0108 [math.DG]]. * [20] R. Emparan and H. S. Reall, A rotating black ring in five dimensions, Phys. Rev. Lett. 88 (2002) 101101 [arXiv:hep-th/0110260]. * [21] J Hebda, Some Lower Bounds for the Area of Surfaces, Inventiones Mathematicae 65 (1982) 485-490 * [22] C. B. Croke, Area and the length of the shortest geodesic J Differential Geometry 27 (1988) 1-21 * [23] P.M. Pu, Some inequalities in certain non-orientable Riemannian manifolds, Pacific J Math 2 (1952) 55-71 * [24] M Berger, Some relations between volume, injectivity radius and convexity radius in Riemannian manifolds, in Cahen and Flato eds. Differential Geometry and Relativity , Dordrecht ( 1976) * [25] M Berger, Filling Riemannian manifolds or isosytolic inequalities, in T J Wilmore and N J Hitchin (eds) Global Riemannian Geometry, Ellis-Horwood(1984) * [26] M Berger, Riemannian Geometry during the second half of the twentieth century, Jahresber. Deutsh. Math-Verein 100 (1998) 45-208 * [27] M. Berger and B. Gostiaux, Differential Geometry Manifolds, Curves and Surfaces Graduate Texts in Mathematics 115 Springer-Verlag (1988) section 11.4.4 p.413 * [28] C Croke, Curvature free volume estimates Inventiones Mathematicae 75(1984) 515-512	arxiv-papers	2009-03-09T15:40:05	2024-09-04T02:49:01.051765	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "G.W. Gibbons", "submitter": "Gary Gibbons", "url": "https://arxiv.org/abs/0903.1580" }
0903.1663	# Atmospheric Sulfur Photochemistry on Hot Jupiters K. Zahnle NASA Ames Research Center, Moffett Field, CA 94035 Kevin.J.Zahnle@NASA.gov M. S. Marley NASA Ames Research Center, Moffett Field, CA 94035 Mark.S.Marley@NASA.gov R. S. Freedman NASA Ames Research Center, Moffett Field, CA 94035 freedman@darkstar.arc.nasa.gov K. Lodders Washington University - St. Louis J. J. Fortney Department of Astronomy and Astrophysics, University of California - Santa Cruz ###### Abstract We develop a new 1D photochemical kinetics code to address stratospheric chemistry and stratospheric heating in hot Jupiters. Here we address optically active S-containing species and CO2 at $1200\leq T\leq 2000$ K. HS (mercapto) and S2 are highly reactive species that are generated photochemically and thermochemically from H2S with peak abundances between 1-10 mbar. S2 absorbs UV between 240 and 340 nm and is optically thick for metallicities $[{\rm S}/{\rm H}]>0$ at $T\geq 1200$ K. HS is probably more important than S2, as it is generally more abundant than S2 under hot Jupiter conditions and it absorbs at somewhat redder wavelengths. We use molecular theory to compute an HS absorption spectrum from sparse available data and find that HS should absorb strongly between 300 and 460 nm, with absorption at the longer wavelengths being temperature sensitive. When the two absorbers are combined, radiative heating (per kg of gas) peaks at 100 $\mu$bars, with a total stratospheric heating of $\sim\\!8\times 10^{4}$ W/m2 for a jovian planet orbiting a solar- twin at 0.032 AU. Total heating is insensitive to metallicity. The CO2 mixing ratio is a well-behaved quadratic function of metallicity, ranging from $1.6\times 10^{-8}$ to $1.6\times 10^{-4}$ for $-0.3<[{\rm M}/{\rm H}]<1.7$. CO2 is insensitive to insolation, vertical mixing, temperature ($1200<T<2000$), and gravity. The photochemical calculations confirm that CO2 should prove a useful probe of planetary metallicity. planetary systems — stars: individual(HD 209458, HD 149026) ††slugcomment: submitted to Ap. J. Lett. ## 1 Introduction Stratospheric temperature inversions are ubiquitous in the Solar System, and it is beginning to look as if they are commonplace on hot Jupiters as well. Stratospheric temperature inversions form when substantial amounts of light are absorbed at low pressures (high altitudes) where radiative cooling is inefficient. Hubeny et al. (2003) pointed out that efficient absorption of visible light by gaseous TiO and VO would greatly heat the upper atmospheres of those planets already hot enough for these molecules to be present as vapor. Thermal inversions on transiting hot Jupiters were first seen by Richardson et al. (2007) for HD 209458b and Harrington et al. (2007) for HD 149026b. The observed flux ratio at 8 $\mu$m for HD 149026b agreed only with models that included a thermal inversion (Fortney et al. 2006). Temperature inversions have since been confirmed by _Spitzer_ observations of HD 209458b (Knutson et al. 2008a), XO-1b (Machalek et al. 2008), and TrES-4 (Knutson et al. 2009), all of which show distinctive flux ratios in IRAC bands that suggest inversions (Fortney et al. 2006; Burrows et al. 2007). More circumstantial evidence exists for HD 179949b (Barnes et al. 2008). On the other hand TrES-1, the least irradiated planet with published _Spitzer_ observations, does not appear to have a pronounced inversion (Burrows et al. 2008). Nor, seemingly, does HD 189733b, which is also modestly irradated (Charbonneau et al. 2008; Barman et al. 2008). One suggestion is that temperature inversions are triggered by irradiation reaching a critical level that is hot enough to evaporate TiO and VO from grains, as discussed by Burrows et al. (2007), Fortney et al. (2008), and Burrows et al. (2008). However, irradiation of XO-1b and HD 189733b is within uncertainties the same (Torres et al. 2008), which poses a challenge to the irradiation trigger. In the Solar System, stratospheric temperature inversions are often caused by absorption of UV light by gases or aerosols produced by photochemistry. Here we ask if atmospheric chemistry might play a similar role in hot Jupiters. Speculation has tended to focus on sulfur-containing species (Tinetti 2008), as the reservoir species H2S is expected to be abundant (Visscher et al 2006) in these atmospheres and many of its breakdown products (S2, in particular) absorb violet and ultraviolet light. ## 2 The Photochemical Model Previous photochemical modeling of hot Jupiters addressed the abundance of photochemical H (Liang et al 2003) and the absence of photochemical smogs (Liang et al 2004). Liang et al (2003) focused on the high H/H2 ratio that arises from H2O photolysis. In their second paper, Liang et al (2004) argued that simple hydrocarbons would not condense to form photochemical smogs in hot solar composition atmospheres. Neither study considered sulfur. We have developed a new general purpose 1D photochemical kinetics code applicable to hot extrasolar planets. The code is based on the sulfur photochemistry model for early Earth originally described by Kasting et al (1989) and Kasting (1990), and subsequently adapted by Zahnle et al (2006) and Claire et al (2006) to address sulfur photochemistry of Earth’s atmosphere during the Archean, and by Zahnle et al. (2008) to address martian atmospheric chemistry. Steady state solutions are found by integrating the system through time using a fully implicit backward-difference method. Our chemical network has been upgraded from that used by Zahnle et al (1995) to address the chemistry generated when the fragments of Comet Shoemaker Levy 9 struck Jupiter. We have assembled a reasonably complete list of the reactions that can take place between the small molecules and free radicals that can be made from H, C, O, N, and S. The code solves 507 chemical reactions for 49 chemical species: H, H2O, OH, O, O2, CO, CO2, HCO, H2CO, C, CH, CH2, CH3, CH4, CH3O, C2, C2H, C2H2, C2H3, C2H4, C 2H5, C2H6, C4H, C4H2, CN, HCN, N, N2, NO, NH, NH2, NH3, NS, H2S, HS, S, S2, S3, S4, S8, SO, HSO, SO2, OCS, CS, HCS, H2CS, CS2, and H2. Reaction rates, when known, are selected from the publicly available NIST database (http://kinetics.nist.gov/kinetics). In order of decreasing priority, we choose between reported reaction rates according to relevant temperature range, newest review, newest experiment, and newest theory. Reverse reaction rates $k_{r}=K_{\rm eq}k_{f}$ of two-body reactions are determined from the forward reaction rate $k_{f}$ and the equilibrium $K_{\rm eq}=\exp{\left\\{\left(-\Delta H+T\Delta S\right)/RT\right\\}}$ by using $H^{\circ}(T)$ and $S^{\circ}(T)$ as available ($R$ is the gas constant). Rates are not available for all reactions, especially for reactions involving elemental sulfur. We will present a full listing of the chemical reactions important to sulfur in a more general followup study. Here we use simple descriptions of atmospheric properties. The background atmosphere is 84% H2 and 16% He. We include Rayleigh scattering by H2 (Dalgarno and Williams 1962). For our base case we assume an isothermal atmosphere with T=1400 K; constant vertical eddy diffusivity $K_{zz}=1\times 10^{7}$ cm2/s; a surface gravity of 20 m/s2; and insolation levels $I$ by a solar twin that are 1000$\times$ greater than at Earth. Metallicity proved to be the most interesting parameter and was varied $-0.3\leq[{\rm M}/{\rm H}]\leq 1.7$. In these units, solar metallicity is $[{\rm M}/{\rm H}]=0$, Jupiter’s is $[{\rm M}/{\rm H}]=0.5$, and Saturn’s is $[{\rm M}/{\rm H}]=0.8$. Short chemical lifetimes of S-containing species make our results insensitive to $K_{zz}$. Model parameters are listed in Table 1. At the upper boundary we set a zero flux lid at 1 $\mu$bar, with neither escape nor exogenous supply. For the lower boundary we use fixed equilibrium mixing ratios of the most abundant species at 1 bar of H2 and temperature $T$ (Lodders and Fegley 2002, Visscher et al 2006). For other species we force the mixing ratio at 1 bar to approach zero. We scale the lower boundary conditions such that the total mixing ratios of C, O, N, and S all scale linearly with metallicity. Absorption by S2 between 240 nm and $\sim$360 nm from the ground state is analogous to the Schumann-Runge system in O2 (Okabe 1978). Strong, distinctive S2 emission near 300 nm was observed on Jupiter after the impact with Shoemaker-Levy 9 with Jupiter in 1994 (Noll et al 1995). Subsequent thermochemical modeling showed that S2 readily forms as a major product in a shock-heated ($T>1000$ K) gas of either cometary or jovian composition (Zahnle et al 1995, Zahnle 1996). S2 has also been seen in gases vented by volcanoes on Io (Spencer et al, 2000; Moses et al 2002). For S2, we use absorption cross sections at 1500 K computed by van der Heijden and van der Mullen (2001). The HS (mercapto) radical absorbs from its ground state at 324 nm (Okabe 1978). Visscher et al (2006) predicted that HS would be very abundant in equilibrium at hot Jupiter conditions. We find the same. We therefore calculated absorption cross sections of HS at four temperatures at 30 mbar pressure using literature values of the molecular properties. The ground $X^{2}\Pi$ state has been well studied (Ram et al 1995) but the upper level $A^{2}\Sigma^{+}$ is subject to strong pre-dissociation (Resende and Ornellas 2001, Wheeler et al 1997, Schneider et al 1990, Henneker and Popkie 1971), and only the value of the rotational constant $B$ and the spacing of the lowest vibrational energy levels have been well measured. Using these constants and a value for the electronic band oscillator strength of the 0-0 transition derived from a study of HS in the solar spectra by Berdyugina and Livingston (2002), a line list was computed using the RLS code developed by R.N. Zare and D. Albritton (Zare et al 1973). This RLS code uses the molecular constants and band strengths to predict line positions and strengths by fitting to an RKR potential (Zare et al 1973). Other needed data — Franck-Condon factors, partition functions, etc. — were derived either from the cited literature, the program itself, or from Sauval and Tatum (1984) or Larsson (1983). The calculations were carried out for values of $v^{\prime\prime}$(0-4) and $v^{\prime}(012)$. Because the excited vibrational levels of the $A^{2}\Sigma^{+}$ state are unstable with respect to predissociation, the corresponding optical transitions are likely to be broad and shallow, or even continuous. These uncertainties principally affect the absorption spectrum at wavelengths shorter than 324 nm, which is in the range that is absorbed strongly by S2. Results are shown in Figure 1. In the photochemical model we used only the 1500 K absorption coefficients. Other sulfur allotropes are better absorbers than S2 but less abundant. S3 absorbs strongly between 350 and 500 nm, and S4 absorbs between 450 nm and 600 nm, but more weakly (Billmers and Smith 1991). Unfortunately, the chemistries of S3 and S4 are very uncertain, and we have had to estimate the important reaction rates. In an earlier version of this study, we focused on the heats of formation, and we tentatively concluded that S3 heating would be important for metallicities $[{\rm S}/{\rm H}]>0.7$. We have since learned that reactions of the form H + Sn $\rightarrow$ HS + Sn-1, where $n\geq 2$, are strongly favored by entropy. The revised model predicts less S2 and much less S3, which reduces the importance of S3 heating considerably. Sulfanes (H2Sn, hydropolysulfides) will be present in cooler hot Jupiters. At low temperatures sulfanes absorb VUV between 260 nm and 330 nm (Steudel and Eckert 2003). Absorption may extend beyond 400 nm at higher temperatures as the ground state becomes vibrationally excited, as in HS, but to first approximation these wavelengths are covered by the more abundant S2 and HS. We have not included sulfanes in this study. ## 3 Results Figure 2 shows how CO2 and the abundant S-containing species vary as a function of altitude. This particular case shows a hot Jupiter at 1400 K with a “planetary” metallicity of $[{\rm M}/{\rm H}]=0.7$. Figure 2 is broadly representative of all our models with $1200\leq T\leq 2000$ K and $-0.3<[{\rm M}/{\rm H}]<1.7$. In particular, S2 and HS show well-defined peaks at $\sim\\!2$ mbars that coincide with the altitude where H2S photolysis becomes important. At lower altitudes H2S is the main S-containing species, and at higher altitudes S is. It is also notable that the atmosphere becomes more oxidizing at higher altitudes where H2O photolysis is important. Table 1 lists some key results pertinent to sulfur for several variations of basic model parameters. The models assume that $K_{zz}=10^{7}$ cm2/s and $g=2000$ cm/s2 unless otherwise noted. In this temperature range the models are insensitive to $K_{zz}$ (results not shown). Model G shows that, as expected, column densities vary inversely with $g$. Column densities of S2 and HS are sensitive to metallicity. To first approximation, species with one metal atom, such as H2O and H2S, increase linearly with metallicity, and species with two metal atoms, such as SO and S2, increase as the square of metallicity (VIsscher et al 2006). A slight complication is that CO and N2 increase linearly with metallicity because these are the major reservoir species for C and N, respectively; hence CO2 increases as the square of metallicity (as CO $\times$ O), rather than as the cube. The models are not sensitive to temperature and insolation over the parameter ranges ($1200\leq T\leq 2000$ K and $1\leq I\leq 1000$) presented here. Insensitivity of the chemistry to $T$ and $I$ surprised us, and suggests that thermochemical equilibrium is more important for sulfur than photochemistry or kinetics. Minor differences are that HS is favored by higher temperatures and SO and S2 are favored by high $I$. Not shown here is that the chemistry changes markedly for $T<1100$ K: hydrocarbons, CS, and CS2 become abundant, and the results become sensitive to $K_{zz}$. Cooler atmospheres introduce a variety of new topics best left for another study. Carbon dioxide, a robust molecule and a potential observable, has been reported in HD 189733b by Swain et al (2009). CO2 is generated from CO by reaction with OH radicals. The chief source of OH is the reaction of H2O with atomic hydrogen; at high altitudes UV photolysis of H2O is also important. We find that CO2 mixing ratios range from $1.6\times 10^{-8}$ to $1.6\times 10^{-4}$ for $-0.3\leq[M/H]\leq 1.7$, scaling as the square of metallicity. Table 1 lists computed CO2 mixing ratios in the models discussed here. These results are insensitive to insolation, vertical mixing, temperature between 1200 K and 2000 K, and gravity. The CO2/CO ratio is nearly independent of pressure, as seen in Figure 2. Pressure independence is expected because the controlling reactions, CO2+H $\leftrightarrow$ CO + OH and H+H2O $\leftrightarrow$ H2 \+ OH, and the controlling equilibrium, CO2+H2 $\leftrightarrow$ CO + H2O, all leave the total pressure unchanged. (At very high altitudes photochemistry alters the CO2/CO ratio.) The computed CO2 abundances are in good agreement with the reported observation of CO2 at the ppmv level in HD 189733b (Swain et al. 2009). The sensitivity of CO2 to metallicity and insensitivity to other atmospheric parameters makes CO2 a good probe of planetary metallicity, as pointed out by Lodders and Fegley (2002). ### 3.1 Optical depth and stratospheric heating Figure 3 shows the pressure levels where the solar and planetary metallicity atmospheres of Models A, M, and MM become optically thick. Opacity is dominated by HS, with some contribution by S2 at wavelengths shorter than 300 nm. The twin peaks between 300 nm and 320 nm may be fictitious, but the peak at 324 nm could prove diagnostic of HS. A solar and a K0V stellar spectrum are shown for comparison. Figure 4 shows the magnitude of stratospheric heating and the pressure level where the heating occurs for a solar-twin primary at 0.032 AU ($I=1000$) for 3 metallicities (Models A, M, and MM). Radiative heating is dominated by HS, and is nearly saturated through the stratosphere for all these models (see also Table 1). By contrast, peak heating at $\sim\\!100\,\mu$bars takes place where SO and SO2 are significant. The sensitivity of SO and SO2 to metallicity is reflected in greater heating rates at $\sim\\!100\,\mu$bars. Cumulative stratospheric heating rates for these models are listed in Table 1. For a solar-twin at 0.032 AU, cumulative heating above 1 mbar is typically $4\times 10^{4}$ W/m2 and above 0.1 bars is typically $8\times 10^{4}$ W/m2, i.e., about half the energy is absorbed in the lower stratosphere. Burrows et al. (2008) modeled hot stratospheres by adding an unknown gray absorber. They found that gray cross-sections of $0.05-0.6$ cm2/g, averaged over 430 to 1000 nm for altitudes above 0.03 bars, could produce the observed heating. Heating profiles using gray opacities in this range are plotted for comparison on Fig 4 for the same planet and star. The gray opacities produce more heating in total (indeed, the stratospheres in both these models are optically thick), and more heating at low altitudes, but at higher altitudes sulfur generates heating at levels quite similar to what Burrows et al find useful. ## 4 Conclusions We develop a new 1D photochemical model for stratospheric modeling of hydrogen-rich atmospheres of warm or hot exoplanets. This model is applicable to any H-rich planet subject to high insolation, including hot Neptunes, superearths, and waterworlds. Here we apply the model to sulfur chemistry, stratospheric heating, and CO2 abundance. We find that hot stratospheres of hot Jupiters could be explained by absorption of UV and violet visible light by HS and S2, two highly reactive species that are generated chemically from H2S. For a hot Jupiter orbiting a solar-twin at 0.032 AU, for a wide range of possible planetary compositions, HS and S2 together absorb $4\times 10^{4}$ W/m2 at altitudes above 1 mbar and another $4\times 10^{4}$ W/m2 at altitudes between 1 mbar and 0.1 bar. This level of heating approaches what Fortney et al (2006) and Burrows et al (2008) use in their most successful LTE spectral models. Non-LTE mechanisms may improve the agreement, because LTE models systematically overestimate radiative cooling and thus underestimate the temperature. Chemiluminescence by H2O, formed by the exothermic reaction of OH+H2, might also be expected. Although our computed HS and S2 column densities increase with metallicity, optically thick columns are predicted for all plausible atmospheric compositions, which means that millibar-level temperature inversions are expected to be commonplace. The distinctive interaction of S2 and HS with near ultraviolet light could make these species detectable in transit by the refurbished HST; there is evidence for a blue absorber in legacy HST data of HD 209458b (Sing et al 2008). On the other hand, sulfur does not give an easy answer to why some hot Jupiters have superheated stratospheres, and others not. In an earlier draft of this study, we speculated that S3—which is very sensitive to metallicity—might be part of the explanation. This no longer appears likely. We have since developed a better understanding of HS’s opacity, which turns out to be considerable. We no longer see a strong connection between metallicity and radiative heating, save at very low pressures ($<\\!100\mu$bars) where SO and SO2 become important. It now seems that sulfur chemistry by itself is unlikely to explain differences between planets, although planetary metallicity may still be key. Heating by sulfur compounds does not preclude heating by TiO and VO on hotter planets. Sulfur species provide considerable heating from below 1000 K to above 2000 K, but they do not provide the spectral coverage at visible wavelengths that TiO and VO provide. For TiO and VO to be abundant enough to explain stratospheric heating, the temperature needs to be very high, in excess of 2000 K, and not just in the stratosphere but also at deeper levels in the planet where these two refractory oxides would otherwise be cold- trapped in silicate clouds. OGLE-TR-56b (Sing and López-Morales 2009) seems to meet the TiO-VO threshold. CO2 is generated by the reaction of CO with OH and destroyed by the reverse (endothermic) reaction with H, CO + OH $\leftrightarrow$ CO2 \+ H. At low altitudes OH is generated by the reaction of H2O with atomic H, supplemented at high altitudes by UV photolysis of H2O. As both the major source and major sink of CO2 are proportional to atomic hydrogen densities, the kinetic inhibition against hydrogen recombination does not disturb CO2’s thermochemical equilibrium. We find that CO2 mixing ratios vary quadratically with metallicity from $1.6\times 10^{-8}$ to $1.6\times 10^{-4}$ for $0<[{\rm M}/{\rm H}]<0.7$. This result is insensitive to insolation, vertical mixing, temperature (for $1200\leq T\leq 2000$ K), and gravity. Because the reactions that form and destroy CO2 leave the total number of molecules unchanged, the CO2/CO ratio is also pressure independent. The computed CO2 abundances are in good agreement with the observation of CO2 at the ppmv level in HD 189733b (Swain et al 2009). Therefore we confirm Lodders and Fegley’s (2002) suggestion that CO2 is a promising probe of planetary metallicity. ## 5 Acknowledgements We thank R. V. Yelle for discussions regarding the potential importance of S3, and G. Tinetti for an insightful review. We thank NASA’s Exobiology and Planetary Atmospheres Programs for support. KL was also supported by NSF Grant AST-0707377. ## References * (1) * (2) Allard, F., Hauschildt, P. H., Alexander, D. R., Tamanai, A., & Schweitzer, A. 2001, ApJ, 556, 357 * (3) * (4) Barman, T. S. 2008, ApJ, 676, L61 * (5) * (6) Barnes, J. R., Barman, T. S., Jones, H. R. A., Leigh, C. J., Cameron, A. C., Barber, R. J., & Pinfield, D. J. 2008, MNRAS, 390, 1258 * (7) * (8) Berdyugina, S.V. and Livingston, W.C. (2002) Astron. Astrophys. 387, L6-L9. * (9) * (10) Billmers RI and Smith AL (1991). J. Phys. Chem. 95, 4242-4245. * Burrows et al. (2007) Burrows, A., Hubeny, I., Budaj, J., Knutson, H. A., & Charbonneau, D. 2007, ApJ, 668, L171 * Burrows et al. (2008) Burrows, A., Budaj, J., & Hubeny, I. 2008, ApJ, 678, 1436 * Burrows et al. (2008) Burrows, A., Budaj, J., & Hubeny, I. 2008, ApJ, 678, 1436 * Charbonneau et al. (2008) Charbonneau, D., Knutson, H. A., Barman, T., Allen, L. E., Mayor, M., Megeath, S. T., Queloz, D., & Udry, S. 2008, ApJ, 686, 1341 * (15) Claire MW, Catling DC, Zahnle KJ (2006). Geobiology 4, 239-269. * (16) Dalgarno A and Williams DA (1962). . Astrophy. J. 136, 690-692. * (17) Désert, J.-M., Vidal-Madjar, A., Lecavelier Des Etangs, A., Sing, D., Ehrenreich, D., Hébrard, G., & Ferlet, R. 2008, A&A, 492, 585 * Fegley & Lodders (1994) Fegley, B. J. & Lodders, K. 1994, Icarus, 110, 117 * Fortney et al. (2006) Fortney, J. J., Saumon, D., Marley, M. S., Lodders, K., & Freedman, R. S. 2006, ApJ, 642, 495 * Fortney et al. (2008) Fortney, J. J., Lodders, K., Marley, M. S., & Freedman, R. S. 2008, ApJ, 678, 1419 * Harrington et al. (2007) Harrington, J., Luszcz, S., Seager, S., Deming, D., & Richardson, L. J. 2007, Nature, 447, 691 * (22) Henneker, W. H. and Popkie, H.E. (1971) J. Chem. Phys. 54, 1763-1778. * Hubeny et al. (2003) Hubeny, I., Burrows, A., & Sudarsky, D. 2003, ApJ, 594, 1011 * (24) Kasting JF, Zahnle KJ, Pinto JP, Young AT (1989) Origins of Life, 19, 95-108. * (25) Kasting JF (1990). Origins of Life 20, 199-231. * Knutson et al. (2008) Knutson, H. A., Charbonneau, D., Allen, L. E., Burrows, A., & Megeath, S. T. 2008, ApJ, 673, 526 * Knutson et al. (2009) Knutson, H. A., Charbonneau, D., Burrows, A., O’Donovan, F. T., & Mandushev, G. 2009, ApJ, 691, 866 * (28) Larsson, M. (1983) Astron. Astrophys. 128, 291-298. * (29) Liang M-C, Parkinson CD, Lee AYT, Yung YL, and Seager S, (2003) “Source of atomic hydrogen in the atmosphere of HD 209458b” Astrophys. J. 596, L247 L250. * (30) Liang M-C, Seager S, Parkinson CD, Lee AYT, and Yung YL (2004) “On the insignificance of photochemical hydrocarbon aerosols in the atmospheres of close-in extrasolar giant planets” Astrophys. J. 605, L61 L64. * (31) Lodders, K. and Fegley B. (1999) The Planetary Scientist’s Companion. Oxford. * Lodders (1999) Lodders, K. 1999, ApJ, 519, 793 * Lodders (2002) —. 2002, ApJ, 577, 974 * (34) Lodders, K. and Fegley, B.J. (2002). Icarus 155, 393-424. * (35) Machalek P, McCullough PR, Burke CJ, Valenti JA, Burrows A, and Hora JL (2008) Astrophys. J. 684, 1427-1432. * Marley et al. (2007) Marley, M. S., Fortney, J., Seager, S., & Barman, T. 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, 733–747 * (37) Moses JI, Zolotov MY, and Fegley B (2002). Icarus 156, 76 106. * (38) Nicholas, J.E., Amodio, C.A., Baker, M.J. (1979) J. Chem. Soc. Faraday Trans. 1:75, 1868-1880. * (39) Noll KS, McGrath MA, Trafton LM, Atreya SK, Caldwell JJ, Weaver HA, Yelle RV, Barnet C, and Edgington S (1995) Science, 267, 1307\. * (40) Okabe H (1978) The Photochemistry of Small Molecules. Wiley-Interscience, New York, 431 pp. * (41) Ram, R S, Bernath P. F., Engleman R. and Brault J. W. (1995) J. Molec. Spect. 172, 34-42. * (42) Resende, S.M., and Ornellas, F.R. (2001) J. Chem. Phys. 115, 2178-2187. * Richardson et al. (2007) Richardson, L. J., Deming, D., Horning, K., Seager, S., & Harrington, J. 2007, Nature, 445, 892 * (44) Sauval, A.J. and Tatum, J.B. (1984) Astrophs. J. Supp. 56, 193-209. * (45) Schneider, L., Meier W, and Weige KH (1990) J. Chem. Phys. 92, 7027-7037. * Showman et al. (2008) Showman, A. P., Fortney, J. J., Lian, Y., Marley, M. S., Freedman, R. S., Knutson, H. A., & Charbonneau, D. 2008, ArXiv e-prints * (47) Sing, D.K., Vidal-Madjar A., Désert, J.-M., Lecavelier des Etangs, A., and Ballester, G. (2008). Astrophys. J. 686, 658-666. * (48) Sing, D.K., and López-Morales, M. (2009). Astron. Astrophys. 493, L31-L34. * (49) Spencer JR., Jessup, KL, McGrath MA, Ballester GE, amd Yelle RV (2000). Science 288, 1208-1210. * (50) Steudel R and Eckert B (2003). Elemental sulfur and sulfur-rich compounds. Springer, 202 pp. * (51) Swain, M. R., Vasisht, G., Tinetti, G., Bouwman, J., Chen, P., Yung, Y., Deming, D., & Deroo, P. 2009, Astrophys. J. Lett. 690, L114-L117. * (52) Tinetti, G. (2008). Bull. Am. Astron. Soc. 40, 463\. * Torres et al. (2008) Torres, G., Winn, J. N., & Holman, M. J. 2008, ApJ, 677, 1324 * (54) van der Heijden and van der Mullen (2001). J. Phys. B. Atom. Mol. Opt. Phys. 34, 4183-4201. * (55) Visscher C, Lodders K, and Fegley B. (2006). Astrophys. J. 648, 1181-1195. * (56) Wheeler, M.D., Orr-Ewing, AJ, and Ashfold, MNR (1997) J. Chem. Phys. 107, 7591-7600. * (57) Zahnle KJ, Mac Low M-M, Lodders K, and Fegly B. (1995). Geophys. Res. Lett. 22, 1593-1596. * (58) Zahnle KJ, Claire MW, Catling DC (2006). Geobiology 4, 271-282. * (59) Zahnle KJ, Haberle RM, Catling DC, Kasting JF (2008). J. Geophys. Res. 113, E11004, doi:10.1029/2008JE003160. * (60) Zare, R.N., Schmeltekopf AL, Harrop WJ, and Albritton DL (1973) J. Molec. Spect. 46, 37-66. Model \| $[{\rm M}/{\rm H}]^{a}$ \| $I^{b}$ \| $T$ \| S2 [cm-2]c \| HS [cm-2]c \| HSd \| SOd \| CO${}_{2}^{d}$ \| Heatinge \| Heatingf ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- A \| $0$ \| 1000 \| 1400 \| $4.2\times 10^{18}$ \| $1.2\times 10^{20}$ \| 6 \| $0.07$ \| $0.065$ \| $6.4\times 10^{4}$ \| $2.7\times 10^{4}$ M \| $0.7$ \| 1000 \| 1400 \| $2.2\times 10^{20}$ \| $1.0\times 10^{21}$ \| 26 \| $1.7$ \| $1.6$ \| $8.4\times 10^{4}$ \| $4.1\times 10^{4}$ MM \| $1.4$ \| 1000 \| 1400 \| $6.0\times 10^{21}$ \| $5.4\times 10^{21}$ \| 100 \| $27$ \| $41$ \| $1.1\times 10^{5}$ \| $5.1\times 10^{4}$ H \| $0.7$ \| 1000 \| 1600 \| $2.2\times 10^{20}$ \| $2.3\times 10^{21}$ \| 43 \| $1.2$ \| $1.4$ \| $9.0\times 10^{4}$ \| $4.3\times 10^{4}$ HH \| $0.7$ \| 1000 \| 1800 \| $1.9\times 10^{20}$ \| $4.0\times 10^{21}$ \| 52 \| $1.3$ \| $1.5$ \| $9.5\times 10^{4}$ \| $4.4\times 10^{4}$ HHH \| $0.7$ \| 1000 \| 2000 \| $1.3\times 10^{20}$ \| $5.0\times 10^{21}$ \| 41 \| $1.4$ \| $1.3$ \| $9.6\times 10^{4}$ \| $4.1\times 10^{4}$ C \| $0.7$ \| 1000 \| 1200 \| $1.6\times 10^{20}$ \| $2.5\times 10^{20}$ \| 11 \| $1.9$ \| $1.9$ \| $7.5\times 10^{4}$ \| $3.7\times 10^{4}$ G \| $0.7$ \| 1000 \| 1400 \| $4.3\times 10^{20}$ \| $2.0\times 10^{21}$ \| 32 \| $1.3$ \| $1.6$ \| $9.3\times 10^{4}$ \| $4.7\times 10^{4}$ I \| $0.7$ \| 200 \| 1400 \| $2.1\times 10^{20}$ \| $1.0\times 10^{21}$ \| 37 \| $0.9$ \| $1.6$ \| $1.7\times 10^{4}$ \| $8.8\times 10^{3}$ SSC \| $0.7$ \| 1 \| 1200 \| $1.1\times 10^{20}$ \| $2.4\times 10^{20}$ \| 16 \| $0.06$ \| $1.9$ \| $72$ \| $44$ $a$ – Metallicity. This notation means that the planet is $10^{[{\rm M}/{\rm H}]}$ richer in C, S, N, and O than the Sun. $b$ – Insolation. $I=1000$ corresponds to a solar twin primary at 0.032 AU. $c$ – Column densities above 1 bar. $d$ – Mixing ratio in ppmv at 1 mbar. $e$ – Total atmospheric heating [W/m2] above 0.1 bar for a solar twin source. $f$ – Total atmospheric heating [W/m2] above 1 mbar for a solar twin source. Figure 1: Theoretical absorption cross sections of HS radicals at near UV, violet and indigo wavelengths at four temperatures at 30 mbar pressure. Cross sections were computed from the lowest five vibrational levels of the ground electronic state $X^{2}\Pi$ to the lowest three vibrational levels of the upper level $A^{2}\Sigma^{+}$. The excited vibrational levels of $A^{2}\Sigma^{+}$ are strongly predissociating, which suggests that absorption at wavelengths shorter than 324 nm is probably continuous rather than allocated into the well-defined bands shown here. Figure 2: Important sulfur species, CO, and CO2 in the atmosphere of a hot Jupiter with a “planetary” metallicity of $[{\rm M}/{\rm H}]=0.7$. The atmosphere is assumed isothermal at 1400 K and insolated 1000$\times$ more strongly than Earth. Other model M parameters are listed in Table 1. The prominent transition at $\sim$2 mbar — where the S2 mixing ratio peaks — is associated with photolysis of H2S. The bump in CO2 at 6 $\mu$bars is attributable to photochemistry. Abundance profiles in the 1400 K atmosphere are generally representative of atmospheres with $1200\leq T\leq 2000$ K. Figure 3: Pressure levels of the $\tau=1$ surface as a function of wavelength for three metallicities, $[{\rm S}/{\rm H}]=0$, 0.7, and 1.4. These metallicities correspond to models A, M, and MM of Table Atmospheric Sulfur Photochemistry on Hot Jupiters. Absorption between 250 and 300 nm is mostly by S2 and absorption between 300 and 460 nm is by HS. Structure blueward of 324 nm is associated with transitions to predissociating states and is probably fictitious. The $\tau=1$ surface of a pure H2 Rayleigh scattering atmosphere and two incident stellar spectra, one for the Sun and another for a generic K0V dwarf, are shown for comparison. Figure 4: Radiative heating at different altitudes for three metallicities, $[{\rm S}/{\rm H}]=0$, 0.7, and 1.4. These correspond to models A, M, and MM of Table Atmospheric Sulfur Photochemistry on Hot Jupiters. Heating rates are given in W/kg, which emphasizes the potential impact on temperature. Heating peaks at 100 $\mu$bars but extends through the stratosphere. Heating with constant gray opacities of $0.05$ and $0.6$ cm2/g for $430<\lambda<1000$ nm is shown for comparison.	arxiv-papers	2009-03-09T23:41:23	2024-09-04T02:49:01.058271	{ "license": "Public Domain", "authors": "K. Zahnle, M.S. Marley, R.S. Freedman, K. Lodders, J.J. Fortney", "submitter": "Mark S. Marley", "url": "https://arxiv.org/abs/0903.1663" }
0903.1769	# Wigner operator’s new transformation in phase space quantum mechanics and its applications ††thanks: Work supported by the National Natural Science Foundation of China under grant: 10775097, 10874174, and Specialized research fund for the doctoral program of higher education of China 1,2Hong-yi Fan 1Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, China 2Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui 230026, China ###### Abstract Using operators’ Weyl ordering expansion formula (Hong-yi Fan,__ J. Phys. A 25 (1992) 3443) we find new two-fold integration transformation about the Wigner operator $\Delta\left(q^{\prime},p^{\prime}\right)$ ($q$-number transform) in phase space quantum mechanics, $\iint_{-\infty}^{\infty}\frac{\mathtt{d}p^{\prime}\mathtt{d}q^{\prime}}{\pi}\Delta\left(q^{\prime},p^{\prime}\right)e^{-2i\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}=\delta\left(p-P\right)\delta\left(q-Q\right),$ and its inverse $\iint_{-\infty}^{\infty}\mathtt{d}q\mathtt{d}p\delta\left(p-P\right)\delta\left(q-Q\right)e^{2i\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}=\Delta\left(q^{\prime},p^{\prime}\right),$ where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to studying mutual converting formulas among $Q-P$ ordering, $P-Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. PACS: 03.65.-w, 02.90.+p Keywords: Wigner operator; Weyl ordering; two-fold integration transformation ## 1 Introduction Phase space quantum mechanics (PSQM) pioneered by Wigner [1] and Weyl [2] has been paid more and more attention since the foundation of quantum mechanics, because it has wide applications in quantum statistics, quantum optics, and quantum chemistry. In PSQM observables and states are replaced by functions on classical phase space so that expected values are calculated, as in classical statistical physics, by averaging over the phase space. The phase-space approaches provides valuable physical insight and allows us to describe alike classical and quantum processes using the similar language. Development of phase space quantum mechanics [3-5] always accompanies with solving operator ordering problems. Weyl proposed a scheme for quantizing classical coordinate and momentum quantity $q^{m}p^{n}$ ($c$-number) as the quantum operators ($q$-number) in the following way $q^{m}p^{n}\rightarrow\left(\frac{1}{2}\right)^{m}\sum_{l=0}^{m}\binom{m}{l}Q^{m-l}P^{n}Q^{l},$ (1) where $Q,$ $P$ are the coordinate and momentum operators, respectively, $[Q,P]=\mathtt{i}\hbar.$ (Later in this work we set $\hbar=1).$ The right-hand side of (1) is in Weyl ordering, so we introduced the symbol $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$ to characterize it [6-7], and $\displaystyle q^{m}p^{n}$ $\displaystyle\rightarrow\left(\frac{1}{2}\right)^{m}\sum_{l=0}^{m}\binom{m}{l}Q^{m-l}P^{n}Q^{l}$ $\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\left(\frac{1}{2}\right)^{m}\sum_{l=0}^{m}\binom{m}{l}Q^{m-l}P^{n}Q^{l}\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}Q^{m}P^{n}\genfrac{}{}{0.0pt}{}{:}{:},$ (2) where in the second step we have used the property that Bose operators are commutative within $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}.$ This is like the fact that Bose operators are commutative within the normal ordering symbol $:$ $:$. The Weyl quantization rule between an operator $H\left(P,Q\right)$ and its classical correspondence is $H\left(P,Q\right)=\iint_{-\infty}^{\infty}\mathtt{d}q\mathtt{d}ph\left(p,q\right)\Delta\left(q,p\right),$ (3) where $\Delta\left(q,p\right)$ is the Wigner operator [2-5] [8]. Using $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}$ we have invented the integration technique within Weyl ordered product of operators with which we constructed an operators’ Weyl ordering expansion formula (see Eq. (21) below), which is the same as Eq. (53) in Ref. [6]). In this work we shall use this formula to find new two-fold $q$-number integration transformation about the Wigner operator $\Delta\left(q^{\prime},p^{\prime}\right)$ in phase space quantum mechanics (see Eqs. (33) and (34) below), which helps to convert P-Q ordering and Q-P ordering to Weyl ordering, and vice versa. The work is arranged as follows: In Sec. 2 we briefly review the Weyl ordered form of Wigner operator. In Sec. 3 we derive the Weyl ordering forms of $\delta\left(p-P\right)\delta\left(q-Q\right)$ and $\delta\left(q-Q\right)\delta\left(p-P\right),$ their transformation to the Wigner operator is shown in Sec. 4. Based on Sec. 4 we in Sec. 5 propose a new $c$-number integration transformation in $p-q$ phase space, see Eq. (35) below, and its inverse transformation, which possesses Parsval-like theorem. Secs. 6-8 are devoted to deriving mutual converting formulas among $Q-P$ ordering, $P-Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. ## 2 The Weyl ordered form of Wigner operator According to Eq. (3) we can rewrite Eq. (2) as $\genfrac{}{}{0.0pt}{}{:}{:}Q^{m}P^{n}\genfrac{}{}{0.0pt}{}{:}{:}=\iint\mathtt{d}q\mathtt{d}pq^{m}p^{n}\Delta\left(q,p\right),$ (4) which implies that the integration kernel (the Wigner operator) is [6-7] $\Delta\left(q,p\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q-Q\right)\delta\left(p-P\right)\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(p-P\right)\delta\left(q-Q\right)\genfrac{}{}{0.0pt}{}{:}{:}.$ (5) Substituting (5) into (3) yields $H\left(P,Q\right)=\genfrac{}{}{0.0pt}{}{:}{:}h\left(P,Q\right)\genfrac{}{}{0.0pt}{}{:}{:},$ where $\genfrac{}{}{0.0pt}{}{:}{:}h\left(P,Q\right)\genfrac{}{}{0.0pt}{}{:}{:}$ is just the result of replacing $p\rightarrow P,q\rightarrow Q$ in $h\left(p,q\right)$ and then putting it within $\genfrac{}{}{0.0pt}{}{:}{:}\genfrac{}{}{0.0pt}{}{:}{:}.$ Further, using $Q=\frac{a+a^{\dagger}}{\sqrt{2}},\text{ \ }P=\frac{a-a^{\dagger}}{\sqrt{2}\mathtt{i}},\text{ }\alpha=\frac{q+\mathtt{i}p}{\sqrt{2}},\text{ }\left[a,a^{\dagger}\right]=1,$ (6) we can express $\Delta\left(q,p\right)\rightarrow\Delta\left(\alpha,\alpha^{\ast}\right)=\frac{1}{2}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\alpha-a\right)\delta\left(\alpha^{\ast}-a^{\dagger}\right)\genfrac{}{}{0.0pt}{}{:}{:}.$ (7) It then follows $\displaystyle\genfrac{}{}{0.0pt}{}{:}{:}K\left(a^{\dagger},a\right)\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\int\mathtt{d}^{2}\alpha K\left(\alpha^{\ast},\alpha\right)\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(\alpha-a\right)\delta\left(\alpha^{\ast}-a^{\dagger}\right)\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=2\int\mathtt{d}^{2}\alpha K\left(\alpha^{\ast},\alpha\right)\Delta\left(\alpha,\alpha^{\ast}\right),$ (8) Thus the neat expression of $\Delta\left(q,p\right)$ in Dirac’s delta function form is very useful, one of its uses is that the marginal distributions of Wigner operator can be clearly shown, due to the coordinate and momentum projectors are respectively $\left\|q\right\rangle\left\langle q\right\|=\delta\left(q-Q\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q-Q\right)\genfrac{}{}{0.0pt}{}{:}{:},$ (9) $\left\|p\right\rangle\left\langle p\right\|=\delta\left(p-P\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(p-P\right)\genfrac{}{}{0.0pt}{}{:}{:},$ (10) we immediately know that the following marginal integration $\int_{-\infty}^{\infty}\mathtt{d}q\Delta\left(q,p\right)=\int_{-\infty}^{\infty}\mathtt{d}q\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q-Q\right)\delta\left(p-P\right)\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(p-P\right)\genfrac{}{}{0.0pt}{}{:}{:}=\left\|p\right\rangle\left\langle p\right\|,$ (11) similarly, $\int_{-\infty}^{\infty}\mathtt{d}p\Delta\left(q,p\right)=\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q-Q\right)\genfrac{}{}{0.0pt}{}{:}{:}=\left\|q\right\rangle\left\langle q\right\|.$ (12) It then follows the completeness of $\Delta\left(q,p\right),$ $\iint\limits_{-\infty}^{\infty}\mathtt{d}q\mathtt{d}p\Delta\left(q,p\right)=1,$ (13) so the Weyl rule for $H\left(P,Q\right)$ in (3) can also be viewed as $H$’s expansion in terms of $\Delta\left(q,p\right).$ When $H\left(P,Q\right)$ is in Weyl ordered, which means $H\left(P,Q\right)=\genfrac{}{}{0.0pt}{}{:}{:}H\left(P,Q\right)\genfrac{}{}{0.0pt}{}{:}{:},$ then using the completeness (13) we see $\genfrac{}{}{0.0pt}{}{:}{:}H\left(P,Q\right)\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}H\left(P,Q\right)\genfrac{}{}{0.0pt}{}{:}{:}\iint\limits_{-\infty}^{\infty}\mathtt{d}q\mathtt{d}p\Delta\left(q,p\right)=\iint\limits_{-\infty}^{\infty}\mathtt{d}q\mathtt{d}pH\left(q,p\right)\Delta\left(q,p\right),$ (14) as if $\Delta\left(q,p\right)$ was the ”eigenvector” of $\genfrac{}{}{0.0pt}{}{:}{:}H\left(P,Q\right)\genfrac{}{}{0.0pt}{}{:}{:}.$ On the other hand, due to the normally ordered forms of $\left\|q\right\rangle\left\langle q\right\|$ and $\left\|p\right\rangle\left\langle p\right\|$ [8] $\left\|q\right\rangle\left\langle q\right\|=\frac{1}{\sqrt{\pi}}\colon e^{-\left(q-Q\right)^{2}}\colon,$ (15) $\left\|p\right\rangle\left\langle p\right\|=\frac{1}{\sqrt{\pi}}\colon e^{-\left(p-P\right)^{2}}\colon,$ (16) we know the normally ordered form of $\Delta\left(q,p\right)$ [9] $\Delta\left(q,p\right)=\frac{1}{\pi}\colon e^{-\left(q-Q\right)^{2}-\left(p-P\right)^{2}}\colon=\frac{1}{\pi}\colon e^{-2\left(\alpha^{\ast}-a^{\dagger}\right)\left(\alpha-a\right)}\colon=\Delta\left(\alpha,\alpha^{\ast}\right).$ (17) Using the completeness relation of the coherent state $\left\|\beta\right\rangle,$ $\int\frac{d^{2}\beta}{\pi}\left\|\beta\right\rangle\left\langle\beta\right\|=1,\text{\ }\left\|\beta\right\rangle=\exp[-\frac{\|\beta\|^{2}}{2}+\beta a^{\dagger}]\left\|0\right\rangle,\text{ \ }a\left\|\beta\right\rangle=\beta\left\|\beta\right\rangle,$ (18) where $\left[a,a^{\dagger}\right]=1,$ $\left\|\beta\right\rangle$ is the coherent state [10-11], we have $\displaystyle 2\pi\mathtt{Tr}\Delta\left(\alpha,\alpha^{\ast}\right)$ $\displaystyle=2\mathtt{Tr}\left[\colon e^{-2\left(\alpha^{\ast}-a^{\dagger}\right)\left(\alpha-a\right)}\colon\int\frac{\mathtt{d}^{2}\beta}{\pi}\left\|\beta\right\rangle\left\langle\beta\right\|\right]$ $\displaystyle=2\int\frac{\mathtt{d}^{2}\beta}{\pi}e^{-2\left(\alpha^{\ast}-\beta^{\ast}\right)\left(\alpha-\beta\right)}=1,$ (19) this is equivalent to (13). Using (17) we also easily obtain $\displaystyle\mathtt{Tr}\left[\Delta\left(\alpha,\alpha^{\ast}\right)\Delta\left(\alpha^{\prime},\alpha^{\prime\ast}\right)\right]$ $\displaystyle=\frac{1}{\pi^{2}}\mathtt{Tr}\left[\colon e^{-2\left(\alpha^{\ast}-a^{\dagger}\right)\left(\alpha-a\right)}\colon\int\frac{\mathtt{d}^{2}\beta}{\pi}\left\|\beta\right\rangle\left\langle\beta\right\|\colon e^{-2\left(\alpha^{\prime\ast}-a^{\dagger}\right)\left(\alpha^{\prime}-a\right)}\colon\right]$ $\displaystyle=\mathtt{Tr}\left[\int\frac{\mathtt{d}^{2}\beta}{\pi^{3}}e^{-2\left(\alpha^{\ast}-a^{\dagger}\right)\left(\alpha-\beta\right)}\left\|\beta\right\rangle\left\langle\beta\right\|e^{-2\left(\alpha^{\prime\ast}-\beta^{\ast}\right)\left(\alpha^{\prime}-a\right)}\right]$ $\displaystyle=\int\frac{\mathtt{d}^{2}\beta}{\pi}\left\langle\beta\right\|e^{-2\left(\alpha^{\prime\ast}-\beta^{\ast}\right)\left(\alpha^{\prime}-a\right)}e^{-2\left(\alpha^{\ast}-a^{\dagger}\right)\left(\alpha-\beta\right)}\left\|\beta\right\rangle$ $\displaystyle=\int\frac{\mathtt{d}^{2}\beta}{\pi}e^{-2\left(\alpha^{\ast}-\beta^{\ast}\right)\left(\alpha-\beta\right)-2\left(\alpha^{\prime\ast}-\beta^{\ast}\right)\left(\alpha^{\prime}-\beta\right)}e^{4\left(\alpha-\beta\right)\left(\alpha^{\prime\ast}-\beta^{\ast}\right)}$ $\displaystyle=\int\frac{\mathtt{d}^{2}\beta}{\pi^{3}}e^{2\beta^{\ast}\left(\alpha^{\prime}-\alpha\right)-2\beta\left(\alpha^{\prime\ast}-\alpha^{\ast}\right)-2\|\alpha\|^{2}-2\|\alpha^{\prime}\|^{2}+4\alpha\alpha^{\prime\ast}}$ $\displaystyle=\frac{1}{4\pi}\delta\left(\alpha-\alpha^{\prime}\right)\delta\left(\alpha^{\ast}-\alpha^{\prime\ast}\right).$ (20) ## 3 Weyl ordering of $\delta\left(p-P\right)\delta\left(q-Q\right)$ and $\delta\left(q-Q\right)\delta\left(p-P\right)$ In Refs. [6-7] we have presented operators’ Weyl ordering expansion formula $\rho=2\int\frac{\mathtt{d}^{2}\beta}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\left\langle-\beta\right\|\rho\left\|\beta\right\rangle\exp\left[2\left(\beta^{\ast}a-a^{\dagger}\beta+a^{\dagger}a\right)\right]\genfrac{}{}{0.0pt}{}{:}{:}.$ (21) For the pure coherent state density operator $\left\|\alpha\right\rangle\left\langle\alpha\right\|,$ using (21) and the overlap $\left\langle\alpha\right\|\left.\beta\right\rangle=\exp[-\frac{1}{2}\left(\|\alpha\|^{2}+\|\beta\|^{2}\right)+\alpha^{\ast}\beta]$ we derive $\displaystyle\left\|\alpha\right\rangle\left\langle\alpha\right\|$ $\displaystyle=2\genfrac{}{}{0.0pt}{}{:}{:}\int\frac{\mathtt{d}^{2}\beta}{\pi}\left\langle-\beta\right\|\left.\alpha\right\rangle\left\langle\alpha\right\|\left.\beta\right\rangle\exp[2\left(\beta^{\ast}a-a^{\dagger}\beta+a^{\dagger}a\right)]\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=2\genfrac{}{}{0.0pt}{}{:}{:}\exp\left[-2\left(\alpha-a\right)\left(\alpha^{\ast}-a^{\dagger}\right)\right]\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=2\genfrac{}{}{0.0pt}{}{:}{:}\exp\left[-\left(p-P\right)^{2}-\left(q-Q\right)^{2}\right]\genfrac{}{}{0.0pt}{}{:}{:},$ (22) thus the Weyl ordered form of pure coherent state $\left\|\alpha\right\rangle\left\langle\alpha\right\|$ is a Gaussian in $p-q$ space. Combining Eqs. (21), (8) and (20) yields $\displaystyle 2\pi\mathtt{Tr}\left[\rho\Delta\left(\alpha,\alpha^{\ast}\right)\right]$ $\displaystyle=4\int\mathtt{d}^{2}\beta\left\langle-\beta\right\|\rho\left\|\beta\right\rangle\mathtt{Tr}\left\\{\genfrac{}{}{0.0pt}{}{:}{:}\exp\left[2\left(\beta^{\ast}a-a^{\dagger}\beta+a^{\dagger}a\right)\right]\genfrac{}{}{0.0pt}{}{:}{:}\Delta\left(\alpha,\alpha^{\ast}\right)\right\\}$ $\displaystyle=4\int\mathtt{d}^{2}\beta\left\langle-\beta\right\|\rho\left\|\beta\right\rangle\mathtt{Tr}\left[2\int\mathtt{d}^{2}\alpha^{\prime}\exp\left[2\left(\beta^{\ast}\alpha^{\prime}-\alpha^{\prime\ast}\beta+\alpha^{\prime\ast}\alpha^{\prime}\right)\right]\Delta\left(\alpha^{\prime},\alpha^{\prime\ast}\right)\Delta\left(\alpha,\alpha^{\ast}\right)\right]$ $\displaystyle=2\int\frac{\mathtt{d}^{2}\beta}{\pi}\left\langle-\beta\right\|\rho\left\|\beta\right\rangle\int\mathtt{d}^{2}\alpha^{\prime}\exp\left[2\left(\beta^{\ast}\alpha^{\prime}-\alpha^{\prime\ast}\beta+\alpha^{\prime\ast}\alpha\right)\right]\delta\left(\alpha-\alpha^{\prime}\right)\delta\left(\alpha^{\ast}-\alpha^{\prime\ast}\right)$ $\displaystyle=2\int\frac{\mathtt{d}^{2}\beta}{\pi}\left\langle-\beta\right\|\rho\left\|\beta\right\rangle\exp\left[2\left(\beta^{\ast}\alpha-\alpha^{\ast}\beta+\alpha^{\ast}\alpha\right)\right],$ (23) which is just an alternate expression of the Wigner function of $\rho,$ comparing (21) with (23) we see that the latter is just the result of replacing $a\rightarrow\alpha,$ $a^{\dagger}\rightarrow\alpha^{\ast},$ in the former, this is because that the right hand side of (21) is in Weyl ordering. Now we examine what is the Weyl ordering of $\delta\left(p-P\right)\delta\left(q-Q\right).$ Using the completeness relation of $\left\|q\right\rangle,$ the coordinate eigenstate, and the completeness relation of the momentum eigenstate $\left\|p\right\rangle,$ $\left\langle q\right.\left\|p\right\rangle=\frac{1}{\sqrt{2\pi}}e^{\mathtt{i}pq},$ we have $\displaystyle\delta\left(p-P\right)\delta\left(q-Q\right)$ $\displaystyle=\int\mathtt{d}p^{\prime}\left\|p^{\prime}\right\rangle\left\langle p^{\prime}\right\|\delta\left(p-P\right)\delta\left(q-Q\right)\int\mathtt{d}q^{\prime}\left\|q^{\prime}\right\rangle\left\langle q^{\prime}\right\|$ $\displaystyle=\frac{1}{\sqrt{2\pi}}\int\mathtt{d}p^{\prime}\left\|p^{\prime}\right\rangle\int\mathtt{d}q^{\prime}\left\langle q^{\prime}\right\|\delta\left(p-p^{\prime}\right)\delta\left(q-q^{\prime}\right)e^{-\mathtt{i}p^{\prime}q^{\prime}}$ $\displaystyle=\frac{1}{\sqrt{2\pi}}\left\|p\right\rangle\left\langle q\right\|e^{-\mathtt{i}pq}.$ (24) The overlap between $\left\langle q\right\|$ and the coherent state is $\left\langle q\right\|\left.\beta\right\rangle=\pi^{-1/4}\exp\left\\{-\frac{q^{2}}{2}+\sqrt{2}q\beta-\frac{1}{2}\beta^{2}-\frac{1}{2}\|\beta\|^{2}\right\\},$ (25) and $\left\langle-\beta\right.\left\|p\right\rangle=\pi^{-1/4}\exp\left\\{-\frac{p^{2}}{2}-\sqrt{2}ip\beta^{\ast}+\frac{1}{2}\beta^{\ast 2}-\frac{1}{2}\|\beta\|^{2}\right\\}.$ (26) Substituting (24) into (21) and using (25)-(26) lead to $\displaystyle\delta\left(p-P\right)\delta\left(q-Q\right)$ $\displaystyle=\frac{\sqrt{2}}{\pi}\int\frac{d^{2}\beta}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\left\langle-\beta\right\|\left.p\right\rangle\left\langle q\right\|e^{-\mathtt{i}pq}\left\|\beta\right\rangle\exp\left[2\left(\beta^{\ast}a-a^{\dagger}\beta+a^{\dagger}a\right)\right]\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\frac{\sqrt{2}}{\pi}e^{-\frac{q^{2}+p^{2}}{2}-\mathtt{i}pq}\int\frac{\mathtt{d}^{2}\beta}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\exp\left\\{-\|\beta\|^{2}+\sqrt{2}q\beta-\sqrt{2}\mathtt{i}p\beta^{\ast}\right\\}$ $\displaystyle\times\exp\left[2\left(\beta^{\ast}a-a^{\dagger}\beta+a^{\dagger}a\right)-\frac{\beta^{2}}{2}+\frac{\beta^{\ast 2}}{2}\right]\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\frac{1}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\exp\\{\sqrt{2}q\left(a-a^{\dagger}\right)+\sqrt{2}\mathtt{i}p\left(a+a^{\dagger}\right)-2\mathtt{i}pq+a^{\dagger 2}-a^{2}-a^{\dagger}a\\}\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\frac{1}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\exp[-2\mathtt{i}\left(q-Q\right)\left(p-P\right)]\genfrac{}{}{0.0pt}{}{:}{:}.$ (27) Similarly, we can derive $\displaystyle\delta\left(q-Q\right)\delta\left(p-P\right)$ $\displaystyle=2\int\frac{\mathtt{d}^{2}\beta}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\left\langle-\beta\right\|\left.q\right\rangle\left\langle p\right\|e^{\mathtt{i}pq}\left\|\beta\right\rangle\exp\left[2\left(\beta^{\ast}a-a^{\dagger}\beta+a^{\dagger}a\right)\right]\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\frac{1}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\exp[2\mathtt{i}\left(q-Q\right)\left(p-P\right)]\genfrac{}{}{0.0pt}{}{:}{:}.$ (28) Eqs. (27)-(28) are the Weyl ordered forms of $\delta\left(p-P\right)\delta\left(q-Q\right)$ and $\delta\left(q-Q\right)\delta\left(p-P\right),$ respectively. ## 4 The new transformation of Wigner operator Taking $\frac{1}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\exp[-2\mathtt{i}\left(q-Q\right)\left(p-P\right)]\genfrac{}{}{0.0pt}{}{:}{:}$as an integration kernel of the following integration transformation with the result $\genfrac{}{}{0.0pt}{}{:}{:}K\left(P,Q\right)\genfrac{}{}{0.0pt}{}{:}{:},$ $\iint_{-\infty}^{\infty}\frac{\mathtt{d}p\mathtt{d}q}{\pi}f\left(p,q\right)\genfrac{}{}{0.0pt}{}{:}{:}\exp[-2\mathtt{i}\left(q-Q\right)\left(p-P\right)]\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}K\left(P,Q\right)\genfrac{}{}{0.0pt}{}{:}{:},$ (29) then from (27) we have $\genfrac{}{}{0.0pt}{}{:}{:}K\left(P,Q\right)\genfrac{}{}{0.0pt}{}{:}{:}=\iint_{-\infty}^{\infty}\mathtt{d}p\mathtt{d}qf\left(p,q\right)\delta\left(p-P\right)\delta\left(q-Q\right)=f\left(p,q\right)\|_{p\rightarrow P,\text{ }q\rightarrow Q,\text{ }P\text{ before }Q},$ (30) this is the integration formula for quantizing classical function $f(p,q)$ as $P-Q$ ordering of operators. On the other hand, from (28) we have $\displaystyle\iint_{-\infty}^{\infty}\frac{\mathtt{d}p\mathtt{d}q}{\pi}f(p,q)\genfrac{}{}{0.0pt}{}{:}{:}\exp[2\mathtt{i}\left(q-Q\right)\left(p-P\right)]\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\iint_{-\infty}^{\infty}\mathtt{d}p\mathtt{d}qf(p,q)\delta\left(q-Q\right)\delta\left(p-P\right)=f\left(p,q\right)\|_{q\rightarrow Q,\text{ }p\rightarrow P,\text{ }Q\text{ before }P},$ (31) this is the scheme of quantizing classical function $f(p,q)$ as $Q-P$ ordering of operators. By noticing (5) we see $\displaystyle\frac{1}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\exp[-2\mathtt{i}\left(q-Q\right)\left(p-P\right)]\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\frac{1}{\pi}\iint\mathtt{d}p^{\prime}\mathtt{d}q^{\prime}e^{-2\mathtt{i}\left(q-q^{\prime}\right)\left(p-p^{\prime}\right)}\genfrac{}{}{0.0pt}{}{:}{:}\delta\left(q^{\prime}-Q\right)\delta\left(p^{\prime}-P\right)\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\frac{1}{\pi}\iint\mathtt{d}p^{\prime}\mathtt{d}q^{\prime}\Delta\left(q^{\prime},p^{\prime}\right)e^{-2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}.$ (32) It then follows from (32) and (27) that $\frac{1}{\pi}\iint\mathtt{d}p^{\prime}\mathtt{d}q^{\prime}\Delta\left(q^{\prime},p^{\prime}\right)e^{-2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}=\delta\left(p-P\right)\delta\left(q-Q\right).$ (33) Similarly we can derive $\frac{1}{\pi}\iint\mathtt{d}p^{\prime}\mathtt{d}q^{\prime}\Delta\left(q^{\prime},p^{\prime}\right)e^{2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}=\delta\left(q-Q\right)\delta\left(p-P\right),$ (34) so $e^{\pm 2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}/\pi$ can be considered the classical Weyl correspondence of $\delta\left(q-Q\right)\delta\left(p-P\right)$ and $\delta\left(p-P\right)\delta\left(q-Q\right),$ respectively. Moreover, the inverse transform of (32) is $\displaystyle\iint\frac{\mathtt{d}q\mathtt{d}p}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\exp[-2\mathtt{i}\left(q-Q\right)\left(p-P\right)]\genfrac{}{}{0.0pt}{}{:}{:}e^{2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}$ $\displaystyle=\iint\frac{\mathtt{d}q\mathtt{d}p}{\pi}\iint dp^{\prime\prime}dq^{\prime\prime}\Delta\left(q^{\prime\prime},p^{\prime\prime}\right)e^{-2\mathtt{i}\left(p-p^{\prime\prime}\right)\left(q-q^{\prime\prime}\right)+2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}$ $\displaystyle=\iint dp^{\prime\prime}dq^{\prime\prime}\Delta\left(q^{\prime\prime},p^{\prime\prime}\right)e^{-2i\left(p^{\prime\prime}q^{\prime\prime}-p^{\prime}q^{\prime}\right)}\delta\left(q^{\prime}-q^{\prime\prime}\right)\delta\left(p^{\prime}-p^{\prime\prime}\right)=\Delta\left(q^{\prime},p^{\prime}\right).$ (35) which means $\iint\mathtt{d}q\mathtt{d}p\delta\left(p-P\right)\delta\left(q-Q\right)e^{2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}=\Delta\left(q^{\prime},p^{\prime}\right),$ (36) or $\iint\mathtt{d}q\mathtt{d}p\delta\left(q-Q\right)\delta\left(p-P\right)e^{-2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}=\Delta\left(q^{\prime},p^{\prime}\right).$ (37) Eqs. (33)-(37) are new transformations of the Wigner operator in $q-p$ phase space. ## 5 The new transformation in phase space Further, multiplying both sides of (35) from the left by $\iint\mathtt{d}q^{\prime}\mathtt{d}p^{\prime}h\left(p^{\prime},q^{\prime}\right)$ we obtain $\displaystyle\iint\mathtt{d}q^{\prime}\mathtt{d}p^{\prime}h\left(p^{\prime},q^{\prime}\right)\Delta\left(q^{\prime},p^{\prime}\right)$ $\displaystyle=\iint\mathtt{d}q^{\prime}\mathtt{d}p^{\prime}h\left(p^{\prime},q^{\prime}\right)\iint\frac{\mathtt{d}q\mathtt{d}p}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\exp[-2i\left(q-Q\right)\left(p-P\right)]\genfrac{}{}{0.0pt}{}{:}{:}e^{2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}$ $\displaystyle=\iint\frac{\mathtt{d}q\mathtt{d}p}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\exp[-2\mathtt{i}\left(q-Q\right)\left(p-P\right)]\genfrac{}{}{0.0pt}{}{:}{:}G\left(p,q\right),$ (38) where we have introduced $G\left(p,q\right)\equiv\frac{1}{\pi}\iint\mathtt{d}q^{\prime}\mathtt{d}p^{\prime}h\left(p^{\prime},q^{\prime}\right)e^{2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)},$ (39) this is a new interesting transformation, because when $h\left(p^{\prime},q^{\prime}\right)=1,$ $\frac{1}{\pi}\iint\mathtt{d}q^{\prime}\mathtt{d}p^{\prime}e^{2\mathtt{i}\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}=\int_{-\infty}^{\infty}\mathtt{d}q^{\prime}\delta\left(q-q^{\prime}\right)e^{2\mathtt{i}p\left(q-q^{\prime}\right)}=1.$ (40) The inverse of (39) is $\iint\frac{dqdp}{\pi}e^{-2i\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)}G\left(p,q\right)=h\left(p^{\prime},q^{\prime}\right).$ (41) In fact, substituting (39) into the the left-hand side of (41) yields $\displaystyle\iint_{-\infty}^{\infty}\frac{\mathtt{d}q\mathtt{d}p}{\pi}\iint\frac{\mathtt{d}q^{\prime\prime}\mathtt{d}p^{\prime\prime}}{\pi}h(p^{\prime\prime},q^{\prime\prime})e^{2\mathtt{i}\left[\left(p-p^{\prime\prime}\right)\left(q-q^{\prime\prime}\right)-\left(p-p^{\prime}\right)\left(q-q^{\prime}\right)\right]}$ $\displaystyle=\iint_{-\infty}^{\infty}\mathtt{d}q^{\prime\prime}\mathtt{d}p^{\prime\prime}h(p^{\prime\prime},q^{\prime\prime})e^{2\mathtt{i}\left(p^{\prime\prime}q^{\prime\prime}-p^{\prime}q^{\prime}\right)}\delta\left(p^{\prime\prime}-p^{\prime}\right)\delta\left(q^{\prime\prime}-q^{\prime}\right)=h(p^{\prime},q^{\prime}).$ (42) This transformation’s Parsval-like theorem is $\displaystyle\iint_{-\infty}^{\infty}\frac{\mathtt{d}q\mathtt{d}p}{\pi}\|h(p,q)\|^{2}$ $\displaystyle=\iint\frac{\mathtt{d}q^{\prime}\mathtt{d}p^{\prime}}{\pi}\|G\left(p^{\prime},q^{\prime}\right)\|^{2}\iint\frac{\mathtt{d}p^{\prime\prime}\mathtt{d}q^{\prime\prime}}{\pi}e^{2i\left(p^{\prime\prime}q^{\prime\prime}-p^{\prime}q^{\prime}\right)}\iint_{-\infty}^{\infty}\frac{\mathtt{d}q\mathtt{d}p}{\pi}e^{2i\left[\left(-p^{\prime\prime}p-q^{\prime\prime}q\right)+\left(pp^{\prime}+q^{\prime}q\right)\right]}$ $\displaystyle=\iint\frac{\mathtt{d}q^{\prime}\mathtt{d}p^{\prime}}{\pi}\|G\left(p^{\prime},q^{\prime}\right)\|^{2}\iint\mathtt{d}p^{\prime\prime}\mathtt{d}q^{\prime\prime}e^{2i\left(p^{\prime\prime}q^{\prime\prime}-p^{\prime}q^{\prime}\right)}\delta\left(q^{\prime}-q^{\prime\prime}\right)\delta\left(p^{\prime}-p^{\prime\prime}\right)=\iint\frac{\mathtt{d}q^{\prime}\mathtt{d}p^{\prime}}{\pi}\|G\left(p^{\prime},q^{\prime}\right)\|^{2}.$ (43) ## 6 P-Q ordering and Q-P ordering to Weyl ordering We now use the above transformation to discuss some operator ordering problems. For instance, from the integration formula $\iint\limits_{-\infty}^{\infty}\frac{\mathtt{d}x\mathtt{d}y}{\pi}x^{m}y^{r}\exp[2\mathtt{i}\left(y-s\right)\left(x-t\right)]=\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(-\mathtt{i}\right)^{r}H_{m,r}\left(\sqrt{2}t,\mathtt{i}\sqrt{2}s\right),$ (44) where $H_{m,r\text{ }}$is the two-variable Hermite polynomials [12-13], $H_{m,r}(t,s)=\sum_{l=0}^{\min(m,r)}\frac{m!r!(-1)^{l}}{l!(m-l)!(r-l)!}t^{m-l}s^{r-l}.$ (45) Eq. (44) can be proved as follows: L.H.S. of (44) $\displaystyle=e^{2\mathtt{i}st}\left(\frac{\partial}{\partial t}\right)^{r}\left(\frac{\partial}{\partial s}\right)^{m}\iint\limits_{-\infty}^{\infty}\frac{\mathtt{d}x\mathtt{d}y}{\pi}e^{2\mathtt{i}xy}\exp[-2\mathtt{i}yt-2\mathtt{i}sx]$ $\displaystyle=e^{2\mathtt{i}st}\left(\frac{\partial}{\partial t}\right)^{r}\left(\frac{\partial}{\partial s}\right)^{m}\int_{-\infty}^{\infty}\mathtt{d}xe^{-2\mathtt{i}sx}\delta\left(x-t\right)$ $\displaystyle=e^{2\mathtt{i}st}\left(\frac{\partial}{\partial t}\right)^{r}\left(\frac{\partial}{\partial s}\right)^{m}e^{-2\mathtt{i}st}=\text{R.H.S. of (44).}$ (46) Using (28) and (44) we know $\displaystyle Q^{m}P^{r}$ $\displaystyle=\iint_{-\infty}^{\infty}\mathtt{d}p\mathtt{d}qq^{m}p^{r}\delta\left(q-Q\right)\delta\left(p-P\right)$ $\displaystyle=\iint_{-\infty}^{\infty}\frac{\mathtt{d}p\mathtt{d}q}{\pi}q^{m}p^{r}\genfrac{}{}{0.0pt}{}{:}{:}\exp[2\mathtt{i}\left(p-P\right)\left(q-Q\right)]\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(-\mathtt{i}\right)^{r}\genfrac{}{}{0.0pt}{}{:}{:}H_{m,r}\left(\sqrt{2}Q,\mathtt{i}\sqrt{2}P\right)\genfrac{}{}{0.0pt}{}{:}{:},$ (47) this is a simpler way to put $Q^{m}P^{r}$ into its Weyl ordering. Similarly, using (27) and the complex conjugate of (44) we see that the Weyl ordered form of $P^{r}Q^{m}$ is $\displaystyle P^{r}Q^{m}$ $\displaystyle=\iint_{-\infty}^{\infty}\mathtt{d}p\mathtt{d}qp^{r}q^{m}\delta\left(p-P\right)\delta\left(q-Q\right)$ $\displaystyle=\iint_{-\infty}^{\infty}\frac{\mathtt{d}p\mathtt{d}q}{\pi}\genfrac{}{}{0.0pt}{}{:}{:}\exp[-2\mathtt{i}\left(q-Q\right)\left(p-P\right)]\genfrac{}{}{0.0pt}{}{:}{:}q^{m}p^{r}$ $\displaystyle=\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(\mathtt{i}\right)^{r}\genfrac{}{}{0.0pt}{}{:}{:}H_{m,r}\left(\sqrt{2}Q,-\mathtt{i}\sqrt{2}P\right)\genfrac{}{}{0.0pt}{}{:}{:}.$ (48) ## 7 Weyl ordering to P-Q ordering and Q-P ordering According to (39) and (41) we know that the inverse transform of (44) is $\iint\frac{\mathtt{d}s\mathtt{d}t}{\pi}\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(-\mathtt{i}\right)^{r}H_{m,r}\left(\sqrt{2}t,\mathtt{i}\sqrt{2}s\right)e^{-2\mathtt{i}\left(y-s\right)\left(x-t\right)}=x^{m}y^{r},$ (49) which is a new integration formula. Then from (27) and (49) we have $\displaystyle\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(-\mathtt{i}\right)^{r}H_{m,r}\left(\sqrt{2}Q,\mathtt{i}\sqrt{2}P\right)\|_{P\text{ before }Q}$ $\displaystyle=\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(-\mathtt{i}\right)^{r}\iint\mathtt{d}p\mathtt{d}q\delta\left(p-P\right)\delta\left(q-Q\right)H_{m,r}\left(\sqrt{2}q,\mathtt{i}\sqrt{2}p\right)$ $\displaystyle=\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(-\mathtt{i}\right)^{r}\iint\frac{\mathtt{d}p\mathtt{d}q}{\pi}H_{m,r}\left(\sqrt{2}q,\mathtt{i}\sqrt{2}p\right)\genfrac{}{}{0.0pt}{}{:}{:}e^{-2\mathtt{i}\left(q-Q\right)\left(p-P\right)}\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}Q^{m}P^{r}\genfrac{}{}{0.0pt}{}{:}{:}.$ (50) Due to (45) we see $\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(-\mathtt{i}\right)^{r}H_{m,r}\left(\sqrt{2}Q,\mathtt{i}\sqrt{2}P\right)\|_{P\text{ before }Q}=\sum_{l=0}\left(\frac{\mathtt{i}}{2}\right)^{l}l!\binom{r}{l}\binom{m}{l}P^{r-l}Q^{m-l},$ (51) so (50)-(51) leads to $\genfrac{}{}{0.0pt}{}{:}{:}Q^{m}P^{r}\genfrac{}{}{0.0pt}{}{:}{:}=\sum_{l=0}\left(\frac{\mathtt{i}}{2}\right)^{l}l!\binom{r}{l}\binom{m}{l}P^{r-l}Q^{m-l},$ (52) Eq. (50) or Eq. (52) is the fundamental formula of converting Weyl ordered operator to its $P-Q$ ordering. Similarly, from (28) and the hermite conjugate of (49) we have $\displaystyle\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(\mathtt{i}\right)^{r}H_{m,r}\left(\sqrt{2}Q,-\mathtt{i}\sqrt{2}P\right)\|_{Q\text{ before }P\text{ }}$ $\displaystyle=\iint\mathtt{d}p\mathtt{d}q\delta\left(q-Q\right)\delta\left(p-P\right)\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(\mathtt{i}\right)^{r}H_{m,r}\left(\sqrt{2}q,-\mathtt{i}\sqrt{2}p\right)$ $\displaystyle=\iint\frac{\mathtt{d}p\mathtt{d}q}{\pi}\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(\mathtt{i}\right)^{r}H_{m,r}\left(\sqrt{2}q,-\mathtt{i}\sqrt{2}p\right)\genfrac{}{}{0.0pt}{}{:}{:}e^{2\mathtt{i}\left(q-Q\right)\left(p-P\right)}\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}Q^{m}P^{r}\genfrac{}{}{0.0pt}{}{:}{:}=\genfrac{}{}{0.0pt}{}{:}{:}P^{r}Q^{m}\genfrac{}{}{0.0pt}{}{:}{:},$ (53) so $\genfrac{}{}{0.0pt}{}{:}{:}Q^{m}P^{r}\genfrac{}{}{0.0pt}{}{:}{:}=\sum_{l=0}\left(\frac{-\mathtt{i}}{2}\right)^{l}l!\binom{r}{l}\binom{m}{l}Q^{m-l}P^{r-l},$ (54) this is the fundamental formula of converting Weyl ordered operator to its $Q-P$ ordering, which is in contrast to (52). ## 8 Q-P ordering to P-Q ordering and vice versa Combining (47) and (52) together we derive $\displaystyle Q^{m}P^{r}$ $\displaystyle=\sum_{l=0}\frac{m!r!}{l!(m-l)!(r-l)!}(\frac{\mathtt{i}}{2})^{l}\genfrac{}{}{0.0pt}{}{:}{:}Q^{m-l}P^{r-l}\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\sum_{l=0}\frac{m!r!}{l!(m-l)!(r-l)!}(\frac{\mathtt{i}}{2})^{l}\sum_{k=0}\left(\frac{\mathtt{i}}{2}\right)^{k}k!\binom{r-l}{k}\binom{m-l}{k}P^{r-l-k}Q^{m-l-k}$ $\displaystyle=\sum_{l=0}\sum_{k=0}\frac{m!r!}{l!(m-l-k)!(r-l-k)!k!}(\frac{\mathtt{i}}{2})^{l+k}P^{r-l-k}Q^{m-l-k}$ $\displaystyle=\sum_{k=0}\frac{m!r!}{(m-k)!(r-k)!k!}(\mathtt{i})^{k}P^{r-k}Q^{m-k},$ (55) which puts $Q^{m}P^{r}$ to its $P-Q$ ordering. It then follows the commutator $\left[Q^{m},P^{r}\right]=\sum_{k=1}\frac{m!r!}{(m-k)!(r-k)!k!}(\mathtt{i})^{k}P^{r-k}Q^{m-k}.$ (56) On the other hand, from (48), (45) and (54) we have $\displaystyle P^{r}Q^{m}$ $\displaystyle=\left(\frac{1}{\sqrt{2}}\right)^{m+r}\left(\mathtt{i}\right)^{r}\genfrac{}{}{0.0pt}{}{:}{:}H_{m,r}\left(\sqrt{2}Q,-\mathtt{i}\sqrt{2}P\right)\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\sum_{l=0}\frac{m!r!}{l!(m-l)!(r-l)!}(\frac{-\mathtt{i}}{2})^{l}\genfrac{}{}{0.0pt}{}{:}{:}Q^{m-l}P^{r-l}\genfrac{}{}{0.0pt}{}{:}{:}$ $\displaystyle=\sum_{l=0}\frac{m!r!}{l!(m-l)!(r-l)!}(\frac{-\mathtt{i}}{2})^{l}\sum_{k=0}\left(\frac{-\mathtt{i}}{2}\right)^{k}k!\binom{r-l}{k}\binom{m-l}{k}Q^{m-l-k}P^{r-l-k}$ $\displaystyle=\sum_{k=0}\frac{m!r!}{(m-k)!(r-k)!k!}(-\mathtt{i})^{k}Q^{m-k}P^{r-k},$ (57) which puts $P^{r}Q^{m}$ to its $Q-P$ ordering. Thus (56) is also equal to $\left[Q^{m},P^{r}\right]=\sum_{k=1}\frac{m!r!}{(m-k)!(r-k)!k!}(-\mathtt{i})^{k}Q^{m-k}P^{r-k}.$ (58) ## 9 $P-Q$ ordering or $Q-P$ ordering expansion of $\left(P+Q\right)^{n}$ Due to $\displaystyle\left(P+Q\right)^{n}$ $\displaystyle=\frac{\mathtt{d}^{n}}{\mathtt{d}\lambda^{n}}\left.e^{\lambda\left(P+Q\right)}\right\|_{\lambda=0}=\frac{\mathtt{d}^{n}}{\mathtt{d}\lambda^{n}}\left.\genfrac{}{}{0.0pt}{}{:}{:}e^{\lambda\left(P+Q\right)}\genfrac{}{}{0.0pt}{}{:}{:}\right\|_{\lambda=0}$ $\displaystyle=\genfrac{}{}{0.0pt}{}{:}{:}\left(P+Q\right)^{n}\genfrac{}{}{0.0pt}{}{:}{:}=\sum_{l=0}^{n}\binom{n}{l}\genfrac{}{}{0.0pt}{}{:}{:}Q^{l}P^{n-l}\genfrac{}{}{0.0pt}{}{:}{:},$ (59) substituting (52) into (59) we derive $\left(P+Q\right)^{n}=\sum_{l=0}^{n}\binom{n}{l}\sum_{k=0}\left(\frac{\mathtt{i}}{2}\right)^{k}k!\binom{l}{k}\binom{n-l}{k}P^{l-k}Q^{n-l-k},$ (60) or using (54) we have $\left(P+Q\right)^{n}=\sum_{l=0}^{n}\binom{n}{l}\sum_{k=0}\left(\frac{-\mathtt{i}}{2}\right)^{k}k!\binom{l}{k}\binom{n-l}{k}Q^{l-k}P^{n-l-k}.$ (61) In sum, by virtue of the formula of operators’ Weyl ordering expansion and the technique of integration within Weyl ordered product of operators we have found new two-fold integration transformation about the Wigner operator $\Delta\left(q^{\prime},p^{\prime}\right)$ in phase space quantum mechanics, which provides us with a new approach for deriving mutual converting formulas among $Q-P$ ordering, $P-Q$ ordering and Weyl ordering of operators. A new $c$-number two-fold integration transformation in $p-q$ phase space (Eq. (39)-(41)) is also proposed, we expect that it may have other uses in theoretical physics. In this way, the contents of phase space quantum mechanics [14] can be enriched. ## References * [1] H. Z. Weyl, Physics, 46 (1927) 1 * [2] E. Wigner, Phys. Rev. 40 (1932) 749; G. S. Agarwal and E. Wolf, Phys. Rev. D 2 (1970) 2161; M. Hillery, R. Connel, M. Scully and E. Wigner, Phys. Rep. 106 (1984) 121;V. Bužek, C. H. Keitel and P. L. Knight, Phys. Rev. A 51 (1995) 2575; H. Moyal, Proc. Camb. Phil. Soc. 45 (1949) 99 * [3] H. Lee, Phys. Rep. 259 (1995) 150 * [4] N. L. Balazs and B. K. Jennings, Phys. Rep. 104 (1984) 347; C. Zachos, Inter. J. Mod. Phys. A 17, (2002) 297 * [5] W. Schleich, Quantum Optics in Phase Space, Wiley-VCH, Berlin 2001 and many references therein * [6] Hong-yi Fan,__ J. Phys. A 25 (1992) 3443 * [7] Hong-yi Fan, Ann. Phys. 323 (2008) 500 * [8] A. Wünsche, J. Opt. B: Quantum Semiclass. Opt. 1 (1999) R11; Hong-yi Fan, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) R147 * [9] Hong-yi Fan and H. R. Zaidi, Phys. Lett. A 123 (1987) 303; Hong-yi Fan and Tu-nan Ruan, Commun. Theor. Phys. 2 (1983) 1563; 3 (1984) 345 * [10] J. R. Klauder and B. -S. Skagerstam, Coherent States, World Scientific, Singapore, 1985 * [11] R. J. Glauber, Phys. Rev. 130 (1963) 2529; 131 (1963) 2766 * [12] A. Erdèlyi, Higher Transcendental Functions, The Bateman Manuscript Project, McGraw Hill, 1953 * [13] Hong-yi Fan and Jun-hua Chen, Phys. Lett. A 303 (2002) 311 * [14] K. Vogel and H. Risken, Phys. Rev. A 40, (1989) 2847; P. Kasperkovitz and M. Peev, Ann. Phys. 230 (1994) 21	arxiv-papers	2009-03-10T13:39:35	2024-09-04T02:49:01.064102	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hong-yi Fan", "submitter": "Hong-yi Fan", "url": "https://arxiv.org/abs/0903.1769" }
0903.1965	# Experimental investigation of nodal domains in the chaotic microwave rough billiard Nazar Savytskyy, Oleh Hul and Leszek Sirko Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland (August 26, 2004) ###### Abstract We present the results of experimental study of nodal domains of wave functions (electric field distributions) lying in the regime of Shnirelman ergodicity in the chaotic half-circular microwave rough billiard. Nodal domains are regions where a wave function has a definite sign. The wave functions $\Psi_{N}$ of the rough billiard were measured up to the level number $N=435$. In this way the dependence of the number of nodal domains $\aleph_{N}$ on the level number $N$ was found. We show that in the limit $N\rightarrow\infty$ a least squares fit of the experimental data reveals the asymptotic number of nodal domains $\aleph_{N}/N\simeq 0.058\pm 0.006$ that is close to the theoretical prediction $\aleph_{N}/N\simeq 0.062$. We also found that the distributions of the areas $s$ of nodal domains and their perimeters $l$ have power behaviors $n_{s}\propto s^{-\tau}$ and $n_{l}\propto l^{-\tau^{\prime}}$, where scaling exponents are equal to $\tau=1.99\pm 0.14$ and $\tau^{\prime}=2.13\pm 0.23$, respectively. These results are in a good agreement with the predictions of percolation theory. Finally, we demonstrate that for higher level numbers $N\simeq 220-435$ the signed area distribution oscillates around the theoretical limit $\Sigma_{A}\simeq 0.0386N^{-1}$. ###### pacs: 05.45.Mt,05.45.Df In recent papers Blum et al. Blum2002 and Bogomolny and Schmit Bogomolny2002 have considered the distribution of the nodal domains of real wave functions $\Psi(x,y)$ in 2D quantum systems (billiards). The condition $\Psi(x,y)=0$ determines a set of nodal lines which separate regions (nodal domains) where a wave function $\Psi(x,y)$ has opposite signs. Blum et al. Blum2002 have shown that the distributions of the number of nodal domains can be used to distinguish between systems with integrable and chaotic underlying classical dynamics. In this way they provided a new criterion of quantum chaos, which is not directly related to spectral statistics. Bogomolny and Schmit Bogomolny2002 have shown that the distribution of nodal domains for quantum wave functions of chaotic systems is universal. In order to prove it they have proposed a very fruitful, percolationlike, model for description of properties of the nodal domains of generic chaotic system. In particular, the model predicts that the distribution of the areas $s$ of nodal domains should have power behavior $n_{s}\propto s^{-\tau}$, where $\tau=187/91$ Ziff1986 . In this paper we present the first experimental investigation of nodal domains of wave functions of the chaotic microwave rough billiard. We tested experimentally some of important findings of papers by Blum et al. Blum2002 and Bogomolny and Schmit Bogomolny2002 such as the signed area distribution $\Sigma_{A}$ or the dependence of the number of nodal domains $\aleph_{N}$ on the level number $N$. Additionally, we checked the power dependence of nodal domain perimeters $l$, $n_{l}\propto l^{-\tau^{\prime}}$, where according to percolation theory the scaling exponent $\tau^{\prime}=15/7$ Ziff1986 , which was not considered in the above papers. In the experiment we used the thin (height $h=8$ mm) aluminium cavity in the shape of a rough half-circle (Fig. 1). The microwave cavity simulates the rough quantum billiard due to the equivalence between the Schrödinger equation and the Helmholtz equation Hans ; Hans2 . This equivalence remains valid for frequencies less than the cut-off frequency $\nu_{c}=c/2h\simeq 18.7$ GHz, where c is the speed of light. The cavity sidewalls are made of 2 segments. The rough segment 1 is described by the radius function $R(\theta)=R_{0}+\sum_{m=2}^{M}{a_{m}\sin(m\theta+\phi_{m})}$, where the mean radius $R_{0}$=20.0 cm, $M=20$, $a_{m}$ and $\phi_{m}$ are uniformly distributed on [0.269,0.297] cm and [0,2$\pi$], respectively, and $0\leq\theta<{\pi}$. It is worth noting that following our earlier experience Hlushchuk01b ; Hlushchuk01 we decided to use a rough half-circular cavity instead of a rough circular cavity because in this way we avoided nearly degenerate low-level eigenvalues, which could not be possible distinguished in the measurements. As we will see below, a half-circular geometry of the cavity was also very suitable in the accurate measurements of the electric field distributions inside the billiard. Figure 1: Sketch of the chaotic half-circular microwave rough billiard in the $xy$ plane. Dimensions are given in cm. The cavity sidewalls are marked by 1 and 2 (see text). Squared wave functions $\|\Psi_{N}(R_{c},\theta)\|^{2}$ were evaluated on a half-circle of fixed radius $R_{c}=17.5$ cm. Billiard’s rough boundary $\Gamma$ is marked with the bold line. The surface roughness of a billiard is characterized by the function $k(\theta)=(dR/d\theta)/R_{0}$. Thus for our billiard we have the angle average $\tilde{k}=(\left<k^{2}(\theta)\right>_{\theta})^{1/2}\simeq 0.488$. In such a billiard the dynamics is diffusive in orbital momentum due to collisions with the rough boundary because $\tilde{k}$ is much above the chaos border $k_{c}=M^{-5/2}=0.00056$ Frahm97 . The roughness parameter $\tilde{k}$ determines also other properties of the billiard Frahm . The eigenstates are localized for the level number $N<N_{e}=1/128\tilde{k}^{4}$. Because of a large value of the roughness parameter $\tilde{k}$ the localization border lies very low, $N_{e}\simeq 1$. The border of Breit-Wigner regime is $N_{W}=M^{2}/48\tilde{k}^{2}\simeq 35$. It means that between $N_{e}<N<N_{W}$ Wigner ergodicity Frahm ought to be observed and for $N>N_{W}$ Shnirelman ergodicity should emerge. In 1974 Shnirelman Shnirelman proved that quantum states in chaotic billiards become ergodic for sufficiently high level numbers. This means that for high level numbers wave functions have to be uniformly spread out in the billiards. Frahm and Shepelyansky Frahm showed that in the rough billiards the transition from the exponentially localized states to the ergodic ones is more complicated and can pass through an intermediate regime of Wigner ergodicity. In this regime the wave functions are nonergodic and compose of rare strong peaks distributed over the whole energy surface. In the regime of Shnirelman ergodicity the wave functions should be distributed homogeneously on the energy surface. In this paper we focus our attention on Shnirelman ergodicity regime. One should mention that rough billiards and related systems are of considerable interest elsewhere, e.g. in the context of dynamic localization Sirko00 , localization in discontinuous quantum systems Borgonovi , microdisc lasers Yamamoto ; Stone and ballistic electron transport in microstructures Blanter . In order to investigate properties of nodal domains knowledge of wave functions (electric field distributions inside the microwave billiard) is indispensable. To measure the wave functions we used a new, very effective method described in Savytskyy2003 . It is based on the perturbation technique and preparation of the “trial functions”. Below we will describe shortly this method. The wave functions $\Psi_{N}(r,\theta)$ (electric field distribution $E_{N}(r,\theta)$ inside the cavity) can be determined from the form of electric field $E_{N}(R_{c},\theta)$ evaluated on a half-circle of fixed radius $R_{c}$ (see Fig. 1). The first step in evaluation of $E_{N}(R_{c},\theta)$ is measurement of $\|E_{N}(R_{c},\theta)\|^{2}$. The perturbation technique developed in Slater52 and used successfully in Slater52 ; Sridhar91 ; Richter00 ; Anlage98 was implemented for this purpose. In this method a small perturber is introduced inside the cavity to alter its resonant frequency according to $\nu-\nu_{N}=\nu_{N}(aB_{N}^{2}-bE_{N}^{2}),$ $None$ where $\nu_{N}$ is the $N$th resonant frequency of the unperturbed cavity, $a$ and $b$ are geometrical factors. Equation (1) shows that the formula can not be used to evaluate $E_{N}^{2}$ until the term containing magnetic field $B_{N}$ vanishes. To minimize the influence of $B_{N}$ on the frequency shift $\nu-\nu_{N}$ a small piece of a metallic pin (3.0 mm in length and 0.25 mm in diameter) was used as a perturber. The perturber was moved by the stepper motor via the Kevlar line hidden in the groove (0.4 mm wide, 1.0 mm deep) made in the cavity’s bottom wall along the half-circle $R_{c}$. Using such a perturber we had no positive frequency shifts that would exceed the uncertainty of frequency shift measurements (15 kHz). We checked that the presence of the narrow groove in the bottom wall of the cavity caused only very small changes $\delta\nu_{N}$ of the eigenfrequencies $\nu_{N}$ of the cavity $\|\delta\nu_{N}\|/\nu_{N}\leq 10^{-4}$. Therefore, its influence into the structure of the cavity’s wave functions was also negligible. A big advantage of using hidden in the groove line was connected with the fact that the attached to the line perturber was always vertically positioned what is crucial in the measurements of the square of electric field $E_{N}$. To eliminate the variation of resonant frequency connected with the thermal expansion of the aluminium cavity the temperature of the cavity was stabilized with the accuracy of 0.05 $\deg$. Figure 2: Panel (a): Squared wave function $\|\Psi_{435}(R_{c},\theta)\|^{2}$ (in arbitrary units) measured on a half-circle with radius $R_{c}=17.5$ cm ($\nu_{435}\simeq 14.44$ GHz). Panel (b): The “trial wave function” $\Psi_{435}(R_{c},\theta)$ (in arbitrary units) with the correctly assigned signs, which was used in the reconstruction of the wave function $\Psi_{435}(r,\theta)$ of the billiard (see Fig. 3). The regime of Shnirelman ergodicity for the experimental rough billiard is defined for $N>35$. Using a field perturbation technique we measured squared wave functions $\|\Psi_{N}(R_{c},\theta)\|^{2}$ for 156 modes within the region $80\leq N\leq 435$. The range of corresponding eigenfrequencies was from $\nu_{80}\simeq 6.44$ GHz to $\nu_{435}\simeq 14.44$ GHz. The measurements were performed at 0.36 mm steps along a half-circle with fixed radius $R_{c}=17.5$ cm. This step was small enough to reveal in details the space structure of high-lying levels. In Fig. 2 (a) we show the example of the squared wave function $\|\Psi_{N}(R_{c},\theta)\|^{2}$ evaluated for the level number $N=435$. The perturbation method used in our measurements allows us to extract information about the wave function amplitude $\|\Psi_{N}(R_{c},\theta)\|$ at any given point of the cavity but it doesn’t allow to determine the sign of $\Psi_{N}(R_{c},\theta)$ Stein95 . Our results presented in Savytskyy2003 suggest the following sign-assignment strategy: We begin with the identification of all close to zero minima of $\|\Psi_{N}(R_{c},\theta)\|$. Then the sign “minus” maybe arbitrarily assigned to the region between the first and the second minimum, “plus” to the region between the second minimum and the third one, the next “minus” to the next region between consecutive minima and so on. In this way we construct our “trial wave function” $\Psi_{N}(R_{c},\theta)$. If the assignment of the signs is correct we should reconstruct the wave function $\Psi_{N}(r,\theta)$ inside the billiard with the boundary condition $\Psi_{N}(r_{\Gamma},\theta_{\Gamma})=0$. The wave functions of a rough half-circular billiard may be expanded in terms of circular waves (here only odd states in expansion are considered) $\Psi_{N}(r,\theta)=\sum_{s=1}^{L}a_{s}C_{s}J_{s}(k_{N}r)\sin(s\theta),$ $None$ where $C_{s}=(\frac{\pi}{2}\int_{0}^{r_{max}}\|J_{s}(k_{N}r)\|^{2}rdr)^{-1/2}$ and $k_{N}=2\pi\nu_{N}/c$. Figure 3: The reconstructed wave function $\Psi_{435}(r,\theta)$ of the chaotic half-circular microwave rough billiard. The amplitudes have been converted into a grey scale with white corresponding to large positive and black corresponding to large negative values, respectively. Dimensions of the billiard are given in cm. In Eq. (2) the number of basis functions is limited to $L=k_{N}r_{max}=l_{N}^{max}$, where $r_{max}=21.4$ cm is the maximum radius of the cavity. $l_{N}^{max}=k_{N}r_{max}$ is a semiclassical estimate for the maximum possible angular momentum for a given $k_{N}$. Circular waves with angular momentum $s>L$ correspond to evanescent waves and can be neglected. Coefficients $a_{s}$ may be extracted from the “trial wave function” $\Psi_{N}(R_{c},\theta)$ via $a_{s}=[\frac{\pi}{2}C_{s}J_{s}(k_{N}R_{c})]^{-1}\int_{0}^{\pi}\Psi_{N}(R_{c},\theta)\sin(s\theta)d\theta.$ $None$ Since our “trial wave function” $\Psi_{N}(R_{c},\theta)$ is only defined on a half-circle of fixed radius $R_{c}$ and is not normalized we imposed normalization of the coefficients $a_{s}$: $\sum_{s=1}^{L}\|a_{s}\|^{2}=1$. Now, the coefficients $a_{s}$ and Eq. (2) can be used to reconstruct the wave function $\Psi_{N}(r,\theta)$ of the billiard. Due to experimental uncertainties and the finite step size in the measurements of $\|\Psi_{N}(R_{c},\theta)\|^{2}$ the wave functions $\Psi_{N}(r,\theta)$ are not exactly zero at the boundary $\Gamma$. As the quantitative measure of the sign assignment quality we chose the integral $\gamma\int_{\Gamma}\|\Psi_{N}(r,\theta)\|^{2}dl$ calculated along the billiard’s rough boundary $\Gamma$, where $\gamma$ is length of $\Gamma$. In Fig. 2 (b) we show the “trial wave function” $\Psi_{435}(R_{c},\theta)$ with the correctly assigned signs, which was used in the reconstruction of the wave function $\Psi_{435}(r,\theta)$ of the billiard (see Fig. 3). Using the method of the “trial wave function” we were able to reconstruct 138 experimental wave functions of the rough half-circular billiard with the level number $N$ between 80 and 248 and 18 wave functions with $N$ between 250 and 435. The wave functions were reconstructed on points of a square grid of side $4.3\cdot 10^{-4}$ m. The remaining wave functions from the range $N=80-435$ were not reconstructed because of the accidental near-degeneration of the neighboring states or due to the problems with the measurements of $\|\Psi_{N}(R_{c},\theta)\|^{2}$ along a half-circle coinciding for its significant part with one of the nodal lines of $\Psi_{N}(r,\theta)$. These problems are getting much more severe for $N>250$. Furthermore, the computation time $t_{r}$ required for reconstruction of the ”trial wave function” scales like $t_{r}\propto 2^{n_{z}-2}$, where $n_{z}$ is the number of identified zeros in the measured function $\|\Psi_{N}(R_{c},\theta)\|$. For higher $N$, the computation time $t_{r}$ on a standard personal computer with the processor AMD Athlon XP 1800+ often exceeds several hours, what significantly slows down the reconstruction procedure. Figure 4: Structure of the energy surface in the regime of Shnirelman ergodicity. Here we show the moduli of amplitudes $\|C^{(N)}_{nl}\|$ for the wave functions: (a) $N=86$, (b) $N=435$. The wave functions are delocalized in the $n,l$ basis. Full lines show the semiclassical estimation of the energy surface (see text). Ergodicity of the billiard’s wave functions can be checked by finding the structure of the energy surface Frahm97 . For this reason we extracted wave function amplitudes $C^{(N)}_{nl}=\left<n,l\|N\right>$ in the basis $n,l$ of a half-circular billiard with radius $r_{max}$, where $n=1,2,3\ldots$ enumerates the zeros of the Bessel functions and $l=1,2,3\ldots$ is the angular quantum number. The moduli of amplitudes $\|C^{(N)}_{nl}\|$ and their projections into the energy surface for the representative experimental wave functions $N=86$ and $N=435$ are shown in Fig. 4. As expected, in the regime of Shnirelman ergodicity the wave functions are extended homogeneously over the whole energy surface Hlushchuk01 . The full lines on the projection planes in Fig. 4(a) and Fig. 4(b) mark the energy surface of a half-circular billiard $H(n,l)=E_{N}=k^{2}_{N}$ estimated from the semiclassical formula Hlushchuk01b : $\sqrt{(l^{max}_{N})^{2}-l^{2}}-l\arctan(l^{-1}\sqrt{(l^{max}_{N})^{2}-l^{2}})+\pi/4=\pi n$. The peaks $\|C^{(N)}_{nl}\|$ are spread almost perfectly along the lines marking the energy surface. Figure 5: Amplitude distribution $P(\Psi A^{1/2})$ for the eigenstates: (a) $N=86$ and (b) $N=435$ constructed as histograms with bin equal to 0.2. The width of the distribution $P(\Psi)$ was rescaled to unity by multiplying normalized to unity wave function by the factor $A^{1/2}$, where $A$ denotes billiard’s area. Full line shows standard normalized Gaussian prediction $P_{0}(\Psi A^{1/2})=(1/\sqrt{2\pi})e^{-\Psi^{2}A/2}$. An additional confirmation of ergodic behavior of the measured wave functions can be also sought in the form of the amplitude distribution $P(\Psi)$ Berry77 ; Kaufman88 . For irregular, chaotic states the probability of finding the value $\Psi$ at any point inside the billiard, without knowledge of the surrounding values, should be distributed as a Gaussian, $P(\Psi)\sim e^{-\beta\Psi^{2}}$. It is worth noting that in the above case the spatial intensity should be distributed according to Porter-Thomas statistics Hans2 . The amplitude distributions $P(\Psi A^{1/2})$ for the wave functions $N=86$ and $N=435$ are shown in Fig. 5. They were constructed as normalized to unity histograms with the bin equal to 0.2. The width of the amplitude distributions $P(\Psi)$ was rescaled to unity by multiplying normalized to unity wave functions by the factor $A^{1/2}$, where $A$ denotes billiard’s area (see formula (23) in Kaufman88 ). For all measured wave functions in the regime of Shnirelman ergodicity there is a good agreement with the standard normalized Gaussian prediction $P_{0}(\Psi A^{1/2})=(1/\sqrt{2\pi})e^{-\Psi^{2}A/2}$. Figure 6: The number of nodal domains $\aleph_{N}$ (full circles) for the chaotic half-circular microwave rough billiard. Full line shows a least squares fit $\aleph_{N}=a_{1}N+b_{1}\sqrt{N}$ to the experimental data (see text), where $a_{1}=0.058\pm 0.006$, $b_{1}=1.075\pm 0.088$. The prediction of the theory of Bogomolny and Schmit Bogomolny2002 $a_{1}=0.062$. The number of nodal domains $\aleph_{N}$ vs. the level number $N$ in the chaotic microwave rough billiard is plotted in Fig. 6. The full line in Fig. 6 shows a least squares fit $\aleph_{N}=a_{1}N+b_{1}\sqrt{N}$ of the experimental data, where $a_{1}=0.058\pm 0.006$, $b_{1}=1.075\pm 0.088$. The coefficient $a_{1}=0.058\pm 0.006$ coincides with the prediction of the percolation model of Bogomolny and Schmit Bogomolny2002 $\aleph_{N}/N\simeq 0.062$ within the error limits. The second term in a least squares fit corresponds to a contribution of boundary domains, i.e. domains, which include the billiard boundary. Numerical calculations of Blum et al. Blum2002 performed for the Sinai and stadium billiards showed that the number of boundary domains scales as the number of the boundary intersections, that is as $\sqrt{N}$. Our results clearly suggest that in the rough billiard, at low level number $N$, the boundary domains also significantly influence the scaling of the number of nodal domains $\aleph_{N}$, leading to the departure from the predicted scaling $\aleph_{N}\sim N$. Figure 7: Distribution of nodal domain areas. Full line shows the prediction of percolation theory $\log_{10}(\langle n_{s}/n\rangle)=-\frac{187}{91}\log_{10}(\langle s/s_{min}\rangle)$. A least squares fit $\log_{10}(\langle n_{s}/n\rangle)=a_{2}-\tau\log_{10}(\langle s/s_{min}\rangle)$ of the experimental results lying within the vertical lines yields the scaling exponent $\tau=1.99\pm 0.14$ and $a_{2}=-0.05\pm 0.04$. The result of the fit is shown by the dashed line. The bond percolation model Bogomolny2002 at the critical point $p_{c}=1/2$ allows us to apply other results of percolation theory to the description of nodal domains of chaotic billiards. In particular, percolation theory predicts that the distributions of the areas $s$ and the perimeters $l$ of nodal clusters should obey the scaling behaviors: $n_{s}\propto s^{-\tau}$ and $n_{l}\propto l^{-\tau^{\prime}}$, respectively. The scaling exponents Ziff1986 are found to be $\tau=187/91$ and $\tau^{\prime}=15/7$. In Fig. 7 we present in logarithmic scales nodal domain areas distribution $\langle n_{s}/n\rangle$ vs. $\langle s/s_{min}\rangle$ obtained for the microwave rough billiard. The distribution $\langle n_{s}/n\rangle$ was constructed as normalized to unity histogram with the bin equal to 1. The areas $s$ of nodal domains were calculated by summing up the areas of the nearest neighboring grid sites having the same sign of the wave function. In Fig. 7 the vertical axis $\langle n_{s}/n\rangle=\frac{1}{N_{T}}\sum_{i=1}^{N_{T}}n_{s}^{(N)}/n^{(N)}$ represents the number of nodal domains $n_{s}^{(N)}$ of size $s$ divided by the total number of domains $n^{(N)}$ averaged over $N_{T}=18$ wave functions measured in the range $250\leq N\leq 435$. In these calculations we used only the highest measured wave functions in order to minimize the influence of boundary domains on nodal domain areas distribution. Following Bogomolny and Schmit Bogomolny2002 , the horizontal axis is expressed in the units of the smallest possible area $s_{min}^{(N)}$, $\langle s/s_{min}\rangle=\frac{1}{N_{T}}\sum_{i=1}^{N_{T}}s/s_{min}^{(N)}$, where $s_{min}^{(N)}=\pi(j_{01}/k_{N})^{2}$ and $j_{01}\simeq 2.4048$ is the first zero of the Bessel function $J_{0}(j_{01})=0$. The full line in Fig. 7 shows the prediction of percolation theory $\log_{10}(\langle n_{s}/n\rangle)=-\frac{187}{91}\log_{10}(\langle s/s_{min}\rangle)$. In a broad range of $\log_{10}(\langle s/s_{min}\rangle)$, approximately from 0.2 to 1.3, which is marked by the two vertical lines the experimental results follow closely the theoretical prediction. Indeed, a least squares fit $\log_{10}(\langle n_{s}/n\rangle)=a_{2}-\tau\log_{10}(\langle s/s_{min}\rangle)$ of the experimental results lying within the vertical lines yields the scaling exponent $\tau=1.99\pm 0.14$ and $a_{2}=-0.05\pm 0.04$, which is in a good agreement with the predicted $\tau=187/91\simeq 2.05$. The dashed line in Fig. 7 shows the results of the fit. In the vicinity of $\log_{10}(\langle s/s_{min}\rangle)\simeq 1$ and $1.2$ small excesses of large areas are visible. A similar situation, but for larger $\log_{10}(s/s_{min})>4$, can be also observed in the nodal domain areas distribution presented in Fig. 5 in Ref. Bogomolny2002 for the random wave model. The exact cause of this behavior is not known but we can possible link it with the limited number of wave functions used for the preparation of the distribution. Figure 8: Distribution of nodal domain perimeters. Full line shows the prediction of percolation theory $\log_{10}(\langle n_{l}/n\rangle)=-\frac{15}{7}\log_{10}(\langle l/l_{min}\rangle)$. A least squares fit $\log_{10}(\langle n_{l}/n\rangle)=a_{3}-\tau^{\prime}\log_{10}(\langle l/l_{min}\rangle)$ of the experimental results lying within the range marked by the vertical lines yields $\tau^{\prime}=2.13\pm 0.23$ and $a_{3}=0.04\pm 0.21$. The result of the fit is shown by the dashed line. Nodal domain perimeters distribution $\langle n_{l}/n\rangle$ vs. $\langle l/l_{min}\rangle$ is shown in logarithmic scales in Fig. 8. The distribution $\langle n_{l}/n\rangle$ was constructed as normalized to unity histogram with the bin equal to 1 . The perimeters of nodal domains $l$ were calculated by identifying the continues paths of grid sites at the domains boundaries. The averaged values $\langle n_{l}/n\rangle$ and $\langle l/l_{min}\rangle$ are defined similarly as previously defined $\langle n_{s}/n\rangle$ and $\langle s/s_{min}\rangle$, e.g. $\langle l/l_{min}\rangle=\frac{1}{N_{T}}\sum_{i=1}^{N_{T}}l/l_{min}^{(N)}$, where $l_{min}^{(N)}=2\pi\sqrt{s_{min}^{(N)}/\pi}=2\pi(j_{01}/k_{N})$ is the perimeter of the circle with the smallest possible area $s_{min}^{(N)}$. The full line in Fig. 8 shows the prediction of percolation theory $\log_{10}(\langle n_{l}/n\rangle)=-\frac{15}{7}\log_{10}(\langle l/l_{min}\rangle)$. Also in this case the agreement between the experimental results and the theory is good what is well seen in the range $0.2<\log_{10}(\langle l/l_{min}\rangle)<1.2$, which is marked by the two vertical lines. A least squares fit $\log_{10}(\langle n_{l}/n\rangle)=a_{3}-\tau^{\prime}\log_{10}(\langle l/l_{min}\rangle)$ of the experimental results lying within the marked range yields $\tau^{\prime}=2.13\pm 0.23$ and $a_{3}=0.04\pm 0.21$. The result of the fit is shown in Fig. 8 by the dashed line. As we see the scaling exponent $\tau^{\prime}=2.13\pm 0.23$ is close to the exponent predicted by percolation theory $\tau^{\prime}=15/7\simeq 2.14$. The above results clearly demonstrate that percolation theory is very useful in description of the properties of wave functions of chaotic billiards. Figure 9: The normalized signed area distribution $N\Sigma_{A}$ for the chaotic half-circular microwave rough billiard. Full line shows predicted by the theory asymptotic limit $N\Sigma_{A}\simeq 0.0386$, Blum et al. Blum2002 . Another important characteristic of the chaotic billiard is the signed area distribution $\Sigma_{A}$ introduced by Blum et al. Blum2002 . The signed area distribution is defined as a variance: $\Sigma_{A}=\langle(A_{+}-A_{-})^{2}\rangle/A^{2}$, where $A_{\pm}$ is the total area where the wave function is positive (negative) and $A$ is the billiard area. It is predicted Blum2002 that the signed area distribution should converge in the asymptotic limit to $\Sigma_{A}\simeq 0.0386N^{-1}$. In Fig. 9 the normalized signed area distribution $N\Sigma_{A}$ is shown for the microwave rough billiard. For lower states $80\leq N\leq 250$ the points in Fig. 9 were obtained by averaging over 20 consecutive eigenstates while for higher states $N>250$ the averaging over 5 consecutive eigenstates was applied. For low level numbers $N<220$ the normalized distribution $N\Sigma_{A}$ is much above the predicted asymptotic limit, however, for $220<N\leq 435$ it more closely approaches the asymptotic limit. This provides the evidence that the signed area distribution $\Sigma_{A}$ can be used as a useful criterion of quantum chaos. A slow convergence of $N\Sigma_{A}$ at low level numbers $N$ was also observed for the Sinai and stadium billiards Blum2002 . In the case of the Sinai billiard this phenomenon was attributed to the presence of corners with sharp angles. According to Blum et al. Blum2002 the effect of corners on the wave functions is mainly accentuated at low energies. The half-circular microwave rough billiard also possesses two sharp corners and they can be responsible for a similar behavior. In summary, we measured the wave functions of the chaotic rough microwave billiard up to the level number $N=435$. Following the results of percolationlike model proposed by Bogomolny2002 we confirmed that the distributions of the areas $s$ and the perimeters $l$ of nodal domains have power behaviors $n_{s}\propto s^{-\tau}$ and $n_{l}\propto l^{-\tau^{\prime}}$, where scaling exponents are equal to $\tau=1.99\pm 0.14$ and $\tau^{\prime}=2.13\pm 0.23$, respectively. These results are in a good agreement with the predictions of percolation theory Ziff1986 , which predicts $\tau=187/91\simeq 2.05$ and $\tau^{\prime}=15/7\simeq 2.14$, respectively. We also showed that in the limit $N\rightarrow\infty$ a least squares fit of the experimental data yields the asymptotic number of nodal domains $\aleph_{N}/N\simeq 0.058\pm 0.006$ that is close to the theoretical prediction $\aleph_{N}/N\simeq 0.062$ Bogomolny2002 . Finally, we found out that the signed area distribution $\Sigma_{A}$ approaches for high level number $N$ theoretically predicted asymptotic limit $0.0386N^{-1}$ Blum2002 . Acknowledgments. This work was partially supported by KBN grant No. 2 P03B 047 24. We would like to thank Szymon Bauch for valuable discussions. ## References * (1) G. Blum, S. Gnutzmann, and U. Smilansky, Phys. Rev. Lett. 88, 114101-1 (2002). * (2) E. Bogomolny and C. Schmit, Phys. Rev. Lett. 88, 114102-1 (2002). * (3) R. M. Ziff, Phys. Rev. Lett. 56, 545 (1986). * (4) H.-J. Stöckmann, J. Stein, Phys. Rev. Lett. 64, 2215 (1990). * (5) H.-J. Stöckmann, Quantum Chaos, an Introduction, (Cambridge University Press, 1999). * (6) Y. Hlushchuk, A. Błȩdowski, N. Savytskyy, and L. Sirko, Physica Scripta 64, 192 (2001). * (7) Y. Hlushchuk, L. Sirko, U. Kuhl, M. Barth, H.-J. Stöckmann, Phys. Rev. E 63, 046208-1 (2001). * (8) K.M. Frahm and D.L. Shepelyansky, Phys. Rev. Lett. 78, 1440 (1997). * (9) K.M. Frahm and D.L. Shepelyansky, Phys. Rev. Lett. 79, 1833 (1997). * (10) A. Shnirelman, Usp. Mat. Nauk. 29, N6, 18 (1974). * (11) L. Sirko, Sz. Bauch, Y. Hlushchuk, P.M. Koch, R. Blümel, M. Barth, U. Kuhl, and H.-J. Stöckmann, Phys. Lett. A 266, 331 (2000). * (12) F. Borgonovi, Phys. Rev. Lett. 80, 4653 (1998). * (13) Y. Yamamoto and R.E. Sluster, Phys. Today 46, 66 (1993). * (14) J.U. Nöckel and A.D. Stone, Nature 385, 45 (1997). * (15) Ya. M. Blanter, A.D. Mirlin, and B.A. Muzykantskii, Phys. Rev. Lett. 80, 4161 (1998). * (16) N. Savytskyy and L. Sirko, Phys. Rev. E 65, 066202-1 (2002). * (17) L.C. Maier and J.C. Slater, J. Appl. Phys. 23, 68 (1952). * (18) S. Sridhar, Phys. Rev. Lett. 67, 785 (1991). * (19) C. Dembowski, H.-D. Gräf, A. Heine, R. Hofferbert, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 84, 867 (2000). * (20) D.H. Wu, J.S.A. Bridgewater, A. Gokirmak, and S.M. Anlage, Phys. Rev. Lett. 81, 2890 (1998). * (21) J. Stein, H.-J. Stöckmann, and U. Stoffregen, Phys. Rev. Lett. 75, 53 (1995). * (22) S.W. McDonald and A.N. Kaufman, Phys. Rev A 37, 3067 (1988). * (23) M.V. Berry, J. Phys. A 10, 2083 (1977).	arxiv-papers	2009-03-11T12:58:30	2024-09-04T02:49:01.070935	{ "license": "Public Domain", "authors": "Nazar Savytskyy, Oleh Hul and Leszek Sirko", "submitter": "Oleh Hul", "url": "https://arxiv.org/abs/0903.1965" }
0903.1991	# Glueballs at Finite Temperature in $SU(3)$ Yang-Mills Theory Xiang-Fei Mengab, Gang Licd, Yuan-Jiang Zhangcd, Ying Chencd, Chuan Liue, Yu- Bin Liua, Jian-Ping Maf, and Jian-Bo Zhangg (CLQCD Collaboration) aSchool of Physics, Nankai University, Tianjin 300071, People s Republic of China bNational Supercomputing Center, Tianjin 300457, People s Republic of China cInstitute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People s Republic of China dTheoretical Center for Science Facilities, Chinese Academy of Sciences, Beijing 100049, People s Republic of China eSchool of Physics, Peking University, Beijing 100871, People s Republic of China fInstitute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, People s Republic of China gDepartment of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, People s Republic of China ###### Abstract Thermal properties of glueballs in $SU(3)$ Yang-Mills theory are investigated in a large temperature range from $0.3T_{c}$ to $1.9T_{c}$ on anisotropic lattices. The glueball operators are optimized for the projection of the ground states by the variational method with a smearing scheme. Their thermal correlators are calculated in all 20 symmetry channels. It is found in all channels that the pole masses $M_{G}$ of glueballs remain almost constant when the temperature is approaching the critical temperature $T_{c}$ from below, and start to reduce gradually with the temperature going above $T_{c}$. The correlators in the $0^{++}$, $0^{-+}$, and $2^{++}$ channels are also analyzed based on the Breit-Wigner Ansatz by assuming a thermal width $\Gamma$ to the pole mass $\omega_{0}$ of each thermal glueball ground state. While the values of $\omega_{0}$ are insensitive to $T$ in the whole temperature range, the thermal widths $\Gamma$ exhibit distinct behaviors at temperatures below and above $T_{c}$. The widths are very small (approximately few percent of $\omega_{0}$ or even smaller) when $T<T_{c}$, but grow abruptly when $T>T_{c}$ and reach values of roughly $\Gamma\sim\omega_{0}/2$ at $T\approx 1.9T_{c}$. ###### pacs: 12.38.Gc, 11.15.Ha, 14.40.Rt, 25.75.Nq ## I Introduction The past two or three decades witnessed intensive and extensive studies on the phase transition of quantum chromodynamics(QCD) cpod2007 , which is believed to be the fundamental theory of strong interaction. Based on the two characteristics of QCD, namely the conjectured color confinement at low energies and the asymptotic freedom of gluons and quarks at high energies, QCD at finite temperature is usually described by two extreme pictures. One is with the weakly interacting meson gas in the low temperature regime and another is with perturbative quark gluon plasma (QGP) in the high temperature regime. The two regimes are bridged by a deconfinement phase transition (or crossover). The study of the equation of state shows that the perturbative picture of QGP can only be achieved at very high temperatures $T\geq 2T_{c}$. In other words, the dynamical degrees of freedom up to the temperature of a few times of $T_{c}$ are not just the quasifree gluons and quarks EOS . Some other theoretical studies also support this scenario and conjecture that in the intermediate temperature range above $T_{c}$ there may exist different types of excitations corresponding to different distance scales soft_modes ; quark_number rendering the thermal states much more complicated. Apart from the quasifree quarks and gluons at the small distance scale, the large scale excitations can be effective low-energy modes in the mesonic channels as a result of the strongly interacting partons gupta . The properties of the interaction among quarks and gluons at low and high temperatures can be studied with thermal correlators. There have been many works on the correlators of charmonia at finite temperature. Phenomenological studies predicted the binding between quarks is reduced to dissolve $J/\psi$ at temperatures close to $T_{c}$ and proposed the suppression of charmonia as a signal of QGP plb178 ; prl57 . For example, potential model studies show that excited states like $\psi^{{}^{\prime}}$ and $\chi_{c}$ are dissociated at $T_{c}$, while the ground state charmonia $J/\Psi$ and $\eta_{c}$ survive up to $T=1.1T_{c}$ zpc37 ; plb514 ; prd64 ; prd70 ; prc72 ; epjc43 . However, it is unclear whether the potential model works well at finite temperatures epjc43b . In contrast, many recent numerical studies indicate that $J/\Psi$ and $\eta_{c}$ might still survive above $1.5T_{c}$ prl92 ; prd69 ; prd74 ; jhw2005 ; npa783 . Of course, it is possible that the $\bar{c}c$ states observed in lattice QCD are just scattering states. A further lattice study on spatial boundary-condition dependence of the energy of low-lying $\bar{c}c$ system concludes that they are spatially localized (quasi)bound states in the temperature region of $1.11\sim 2.07T_{c}$ ptps174 . Obviously, the results of numerical lattice QCD studies are coincident to the picture of the QCD transition in the intermediate temperature regime. Until now most of the lattice studies on hadronic correlators are in the quenched approximation. Because of the lack of dynamical quarks in quenched QCD the binding of quark-antiquark systems must be totally attributed to the nonperturbative properties of gluons, which are the unique dynamical degree of freedom in the theory. Since glueballs are the bound states of gluons, a natural question is how glueballs respond to the varying temperatures. At low temperature $T\sim 0$, the existence of quenched glueballs have been verified by extensive lattice numerical studies, and their spectrum are also established quite well prd56 ; prd60 ; prd73 ; npb221 ; plb309 ; npb314 ; prl75 ; outp . An investigation of the evolution of glueballs versus the increasing temperature is important to understand the QCD transition ptn58 ; npa637 and the hadronization of quark-gluon plasma prd75 . From the point of view of QCD sum rules, glueball masses are closely related to the gluon condensate. Lattice studies lap583 and model calculations pan70 indicate that the gluon condensate keeps almost constant below $T_{c}$ and reduces gradually with the increasing temperature above $T_{c}$. Based on this picture, it is expected intuitively that glueball masses should show a similar behavior also until they melt into gluons qcd20 . In fact, there has already been a lattice study on the scalar and tensor glueball properties at finite temperature prd66 . In contrast to the expectation and the finite $T$ behavior of charmonium spectrum, it is interestingly observed that the pole-mass reduction starts even below $T_{c}$ ($m_{G}$(T $\sim$ $T_{c}$) $\simeq$ 0.8$m_{G}$(T $\sim$ 0)). It is known that the spatial symmetry group on the lattice is the 24-element cubic point group $O$, whose irreducible representations are $R=$ $A_{1}$, $A_{2}$, $E$, $T_{1}$, and $T_{2}$. Along with the parity $P$ and charge conjugate transformation $C$, all the possible quantum numbers that glueballs can catch are $R^{PC}$ with $PC=++,-+,+-,$, and $++$, which add up to 20 symmetry channels. Motivated by the different temperature behaviors of $\bar{c}c$ systems with different quantum numbers, we would like to investigate the temperature dependence of glueballs in this paper. Our numerical study in this work is carried out on anisotropic lattices with much finer lattice in the temporal direction than in spatial ones. In order to explore the temperature evolution of glueball spectrum, the temperature range studied here extends from $0.3T_{c}$ to $1.9T_{c}$, which is realized by varying the temporal extension of the lattice. Using anisotropic lattices, the lattice parameters are carefully determined so that there are enough time slices for a reliable data analysis even at the highest temperature. In the present study, we are only interested in the ground state in each symmetry channel $R^{PC}$. For the study optimized glueball operators that couple mostly to the ground states are desired. Practically, these optimized operators are built up by the combination of smearing schemes and the variational method prd56 ; prd60 ; prd73 . In the data processing, the correlators of these optimized operators are analyzed through two approaches. First, the thermal masses $M_{G}$ of glueballs are extracted in all the channels and all over the temperature range by fitting the correlators with a single-cosh function form, as is done in the standard hadron mass measurements. Thus the $T_{-}evolution$ of the thermal glueball spectrums is obtained. Secondly, with respect that the finite temperature effects may result in mass shifts and thermal widths of glueballs, we also analyze the correlators in $A_{1}^{++}$, $A_{1}^{-+}$, $E^{++}$, and $T_{2}^{++}$ channels with the Breit-Wigner Ansatz which assumes these glueball thermal widths, say, change $M_{G}$ into $\omega_{0}-i\Gamma$ in the spectral function (see below). As a result, the temperature dependence of $\omega_{0}$ and $\Gamma$ can shed some light on the scenario of the QCD transition. This paper is organized as follows. In Sec. II, a description of the determination of working parameters, such as the critical temperature $T_{c}$, temperature range, and lattice spacing $a_{s}$, as well as a brief introduction to the variational method is given. In Sec. III, after a discussion of its feasibility, the results of the single-cosh fit to the thermal correlators are described in details. The procedure of the Breit- Wigner fit is also given in this section. Section IV gives the conclusion and some further discussions. ## II Numerical Details For heavy particles such as charmonia and glueballs, the implementation of anisotropic lattices is found to be very efficient in the previous numerical lattice QCD studies both at low and finite temperatures. On the other hand, the Symanzik improvement and tadpole improvement schemes of the gauge action are verified to have better continuum extrapolation behaviors for many physical quantities. In other words, the finite lattice spacing artifacts are substantially reduced by these improvements. With these facts, we adopt the following improved gauge action which has been extensively used in the study of glueballs prd56 ; prd60 ; prd73 , $\displaystyle S_{IA}={\beta}{\\{\frac{5}{3}\frac{\Omega_{sp}}{{\xi}u_{s}^{4}}+\frac{4}{3}\frac{\xi\Omega_{tp}}{u_{t}^{2}u_{s}^{2}}-\frac{1}{12}\frac{\Omega_{sr}}{{\xi}u_{s}^{6}}-\frac{1}{12}\frac{{\xi}\Omega_{str}}{u_{s}^{4}u_{t}^{2}}\\}}$ (1) where $\beta$ is related to the bare QCD coupling constant, $\xi=a_{s}/a_{t}$ is the aspect ratio for anisotropy (we take $\xi=5$ in this work), $u_{s}$ and $u_{t}$ are the tadpole improvement parameters of spatial and temporal gauge links, respectively. $\Omega_{C}=\sum_{C}\frac{1}{3}ReTr(1-W_{C})$, with $W_{C}$ denoting the path-ordered product of link variables along a closed contour $C$ on the lattice. $\Omega_{sp}$ includes the sum over all spatial plaquettes on the lattice, $\Omega_{tp}$ includes the temporal plaquettes , $\Omega_{sr}$ denotes the product of link variables about planar $2{\times}1$ spatial rectangular loops, and $\Omega_{str}$ refers to the short temporal rectangles(one temporal link, two spatial). Practically, $u_{t}$ is set to 1, and $u_{s}$ is defined by the expectation value of the spatial plaquette, $u_{s}=<\frac{1}{3}TrP_{ss^{{}^{\prime}}}>^{1/4}$. ### II.1 Determination of critical temperature Since the temperature $T$ on the lattice is defined by $T=\frac{1}{N_{t}a_{t}},$ (2) where $N_{t}$ is the temporal lattice size, $T$ can be changed by varying either $N_{t}$ or the coupling constant $\beta$ which is related directly to the lattice spacing. In order for the critical temperature to be determined with enough precision, for a given $N_{t}=24$, we first determine the critical coupling $\beta_{c}$, because $\beta$ can be changed continuously. The order parameter is chosen as the susceptibility $\chi_{P}$ of Polyakov line, which is defined as $\chi_{P}=\langle\Theta^{2}\rangle-\langle\Theta\rangle^{2}$ (3) where $\Theta$ is the $Z(3)$ rotated Polyakov line, $\displaystyle\Theta$ $\displaystyle=$ $\displaystyle\left\\{\ \begin{array}[]{ll}{\rm Re}P\exp[-2\pi i/3];&\arg P\in[\pi/3,\pi)\\\ {\rm Re}P;&\arg P\in[-\pi/3,\pi/3)\\\ {\rm Re}P\exp[2\pi i/3];&\arg P\in[-\pi,-\pi/3)\end{array}\right.,$ (7) and $P$ represents the trace of the spatially averaged Polyakov line on each gauge configuration. After a $\beta$-scanning on $L^{4}=24^{4}$ anisotropic lattices with $\xi=5$, the critical point is trapped in a very narrow window $\beta_{c}\in[2.800,2.820]$. In order to determine $T_{c}$ more precisely, a more refined study is carried out in the $\beta$ window mentioned above with much larger statistics through the spectral density method. Practically, the spectral density method prl61 ; npb17 is applied to extrapolate the simulated $\chi_{P}$’s at $\beta=2.805,2.810$, and 2.815. In table 1 are the numbers of heat-bath sweeps for each $\beta$. The extrapolation results are illustrated in Fig. 1 where the open triangles denote the simulated values of $\chi_{P}$, while the filled squares are the extrapolated values. Finally, the peak position gives the critical coupling constant ${\beta}_{c}=2.808$, which corresponds to the critical temperature $T_{c}\approx 0.724r_{0}^{-1}=296~{}{\rm MeV}$ with the lattice spacing $r_{0}/a_{s}=3.476$ mpla21 and $r_{0}^{-1}=410(20)\,{\rm MeV}$. Table 1: The simulation parameters for the determination of the critical point. The configurations are selected every ten sweeps. $\beta$ \| Total configurations \| Thermalization \| Bin size ---\|---\|---\|--- 2.80 \| 20000 \| 5000 \| 1000 2.805 \| 30000 \| 10000 \| 1000 2.81 \| 30000 \| 10000 \| 1000 2.815 \| 20000 \| 5000 \| 1000 2.82 \| 8000 \| 3000 \| 500 Figure 1: The $\chi_{P}$ extrapolation based on the spectral density method. The open triangles denote the simulated values of $\chi_{P}$, while the filled squares are the extrapolated values. The peak position gives the critical $\beta_{c}=2.808$. With $T_{c}$ fairly determined, the working coupling constant $\beta$ is set based on two requirements. First, the spatial volume of the lattice should be large enough in order for the glueballs to be free of any sizable finite volume effects. Secondly, we require that temporal lattice has a good resolution even at the temperature $T\sim 2T_{c}$. Practically the working coupling constant is finally set to be $\beta=3.2$. The lattice spacing at this $\beta$ is set by calculating the static potential $V(r)$ on an anisotropic lattice $24^{3}\times 128$. With the conventional parametrization of V(r), $V(r)=V_{0}+\sigma r+\frac{e_{c}}{r},$ (8) the lattice spacing $a_{s}$ is determined in the units of $r_{0}$ to be $\frac{a_{s}}{r_{0}}=\sqrt{\frac{\sigma a_{s}^{2}}{1.65+e_{c}}}=0.1825(7)$ (9) where $r_{0}$ is the hadronic scale parameter. If we take $r_{0}^{-1}=410(20)\,{MeV}$, we have $a_{s}=0.0878(4)\,{\rm fm}$. The spatial volume at $L=24$ is therefore estimated to be $(2.1\,{\rm fm})^{3}$. On the other hand, using $T_{c}=296$MeV obtained at $\beta=2.808$ as a rough estimate of $T_{c}$ and ignoring the systematic error due to finite lattice spacings, $T_{c}$ and $2T_{c}$ at $\beta=3.2$ are expected to be achieved around $N_{t}\sim 40$ and $N_{t}\sim 20$, respectively. Obviously, the above two requirements are all satisfied. Table 2: Listed are the parameters used to check the critical behavior for $\beta$=3.2. The configurations are selected every ten sweeps. $N_{t}$ \| Total configurations \| Thermalization \| $<P>$ \| $\chi_{P}$ ---\|---\|---\|---\|--- 60 \| 2000 \| 500 \| -8.73$\times 10^{-5}$ \| 6.65$\times 10^{-5}$ 48 \| 2000 \| 500 \| 6.01$\times 10^{-5}$ \| 1.81$\times 10^{-4}$ 44 \| 8000 \| 2000 \| 2.25$\times 10^{-3}$ \| 3.12$\times 10^{-3}$ 40 \| 8000 \| 2000 \| 1.72$\times 10^{-2}$ \| 9.14$\times 10^{-3}$ 36 \| 8000 \| 2000 \| 5.21$\times 10^{-2}$ \| 3.10$\times 10^{-3}$ 32 \| 3000 \| 1000 \| 8.51$\times 10^{-2}$ \| 2.23$\times 10^{-3}$ 28 \| 2000 \| 500 \| 0.1253 \| 2.00$\times 10^{-3}$ 24 \| 2000 \| 500 \| 0.1817 \| 2.09$\times 10^{-3}$ 20 \| 2000 \| 500 \| 0.2571 \| 1.82$\times 10^{-3}$ Figure 2: $\chi_{P}$ is plotted versus $N_{t}$ at $\beta=3.2$. There is a peak of $\chi_{P}$ near $N_{t}=40$. Based on the discussions above, with a fixed $\beta=3.2$, the calculations of the thermal correlators of glueballs are carried out on a series of lattice $24^{3}\times N_{t}$ with $N_{t}=$ 20, 24, 28, 32, 36, 40, 44, 48, 60, 80, and 128, which cover the temperature range $0.3T_{c}<T<2T_{c}$. As a cross-check, $\chi_{P}$ at different $N_{t}$ are calculated first and the results are shown in Fig. 2 and Table 2. It is clear that the expectation value of the Polyakov line drops to zero near $N_{t}=40$ and the peak position of $\chi_{P}$, which gives the critical temperature, is trapped between $N_{t}=36$ and $N_{t}=40$. In practice, we do not carry out a precise determination of $T_{c}$ at $\beta=3.2$, but take the temperature at $N_{t}=38$, $T\approx(38a_{t})^{-1}=(38a_{s}/\xi)^{-1}=296$MeV, as an approximation of $T_{c}~{}(\beta=3.2)$, to scale the temperatures involved in this work. It should be noted that, owing to the lattice artifact, the critical temperature $T_{c}$ determined at different lattice spacing (or $\beta$) may differ from each other. The closeness of $T_{c}(\beta=2.808)$ and $T_{c}(\beta=3.2)$ may signal that the lattice spacing dependence of $T_{c}$ is mild in this work due to the application of the improved gauge action. ### II.2 Variational method It is known that many states contribute to a hadronic two-point function. Ideally one can extract the information of the lowest-lying states from the two-point function in the large time region if it lasts long enough in the time direction. This is the case for some light hadron states, such as $\pi$ meson, $K$ meson, etc. However, for heavy particles, especially for glueballs whose correlation function are much more noisy than that of conventional hadrons made up of quarks, their two-point functions damp so fast with time that they are always undermined by noise rapidly before the ground states dominate. Practically, in the study of the glueball sector, in order to enhance the overlap of the glueball operators to the ground state, the commonly used techniques are the smearing schemes and the variational techniques. In this work, we adopt the sophisticated strategy implemented by the studies of the zero-temperature glueball spectrum prd56 ; prd60 ; prd73 , which is outlined below. Figure 3: Prototype Wilson loops used in making the smeared glueball operatorsprd60 . First, for each gauge configuration, we perform six smearing/fuzzing schemes to the spatial links, which are various combinations of the single-link procedure (smearing) and the double-link procedure (fuzzing) $\displaystyle U_{j}^{s}(x)$ $\displaystyle=$ $\displaystyle P_{SU(3)}\\{U_{j}(x)+\lambda_{s}\sum\limits_{\pm(k\neq j)}U_{k}(x)U_{j}(x+\hat{k})U_{k}^{\dagger}(x+\hat{j})\\},$ $\displaystyle U_{j}^{f}(x)$ $\displaystyle=$ $\displaystyle P_{SU(3)}\\{U_{j}(x)U_{j}(x+\hat{j})+\lambda_{f}\sum\limits_{\pm(k\neq j)}U_{k}(x)U_{j}(x+\hat{k})U_{j}(x+\hat{j}+\hat{k})U_{k}(x+2\hat{j})\\},$ (10) where $P_{SU(3)}$ denotes the projection into $SU(3)$ and is realized by the Jacobi method liu2 . The six schemes are given explicitly as $s_{\lambda_{s}}^{10}$, $s_{\lambda_{s}}^{18}$, $s_{\lambda_{s}}^{26}$, $f_{\lambda_{f}}\bigotimes s_{\lambda_{s}}^{10}$, $f_{\lambda_{f}}\bigotimes s_{\lambda_{s}}^{18}$, $f_{\lambda_{f}}\bigotimes s_{\lambda_{s}}^{26}$, where $s/f$ denotes the smearing/fuzzing procedure defined in Eq. (II.2), and $\lambda_{s}/\lambda_{f}$ the tunable parameter which we take $\lambda_{s}=0.1$ and $\lambda_{f}=0.5$ in this work. Secondly, we choose the same prototype Wilson loops as that in Ref. prd60 (as shown in Fig. 3), such that for each smearing/fuzzing scheme, all the different spatially oriented copies of these prototypes are calculated from the smeared gauge configurations. Thus for a given irreducible representation $R$ of the spatial symmetry group $O$, say, $R=A_{1},A_{2},E,T_{1}$, or $T_{2}$, a realization of $R$ can be a specific combination of differently oriented Wilson loops generated from the same prototype loop (one can refer to Ref. prd73 for the concrete combinational coefficients). The glueball operators $\phi$ with the quantum number $R^{PC}$ are thereby constructed along with the spatial reflection and the time inversion operations. In practice, we establish four realizations of each $R^{PC}$ which are based on four different prototypes, respectively. Therefore, along with the six smearing/fuzzing schemes, an operator set of the same specific quantum number $R^{PC}$ is composed of 24 different operators, $\\{\phi_{\alpha},\alpha=1,2,\ldots,24\\}$. The last step is the implementation of the variational method (VM). The main goal of VM is to find an optimal combination of the set of operators, $\Phi=\sum v_{\alpha}\phi_{\alpha}$, which overlaps most to a specific state (in this work, we only focus on the ground states). The combinational coefficients ${\bf v}=\\{v_{\alpha},\alpha=1,2,\ldots,n\\}$ can be obtained by minimizing the effective mass, $\tilde{m}(t_{D})=-\frac{1}{t_{D}}\ln\frac{\sum\limits_{\alpha\beta}v_{\alpha}v_{\beta}\tilde{C}_{\alpha\beta}(t_{D})}{\sum\limits_{\alpha\beta}v_{\alpha}v_{\beta}\tilde{C}_{\alpha\beta}(0)},$ (11) at $t_{D}=1$, where $\tilde{C}_{\alpha\beta}(t)$ is the correlation matrix of the operator set, $\tilde{C}_{\alpha\beta}(t)=\sum\limits_{\tau}\langle 0\|{\phi}_{\alpha}(t+\tau){\phi}_{\beta}(\tau)\|0\rangle.$ (12) This is equivalent to solving the generalized eigenvalue equation $\tilde{C}(t_{D}){\bf v}^{(R)}=e^{-t_{D}\tilde{m}(t_{D})}\tilde{C}(0){\bf v}^{(R)},$ (13) and the eigenvector ${\bf v}$ gives the desired combinational coefficients. Thus, the optimal operator that couples most to a specific states (the ground state in this work) can be built up as $\Phi=\sum\limits_{\alpha}v_{\alpha}\phi_{\alpha},$ (14) whose correlator $C(t)$ is expected to be dominated by the contribution of this state. ## III DATA ANALYSIS OF THE THERMAL CORRELATORS OF GLUEBALLS All 20 $R^{PC}$ channels, with $R=A_{1},A_{2},E,T_{1},T_{2}$ and $PC=++,+-,-+,--$, are considered in the calculation of the thermal correlators of glueballs on anisotropic lattices mentioned in Sec. II. At each temperature, after 10000 pseudo-heat-bath sweeps of thermalization, the measurements are carried out every three compound sweeps, with each compound sweep composed of one pseudo-heat-bath and five micro-canonical over- relaxation(OR) sweeps. In order to reduce the possible autocorrelations, the measured data are divided into bins of the size $n_{\rm mb}=400$, and each bin is regarded as an independent measurement in the data analysis procedure. The numbers of bins $N_{\rm bin}$ and $n_{mb}$ at various temperature are listed in Table 3. Table 3: Simulation parameters to calculate glueball spectrum. $\beta=3.2$, $a_{s}=0.0878\,{\rm fm}$, $L_{s}=2.11\,{\rm fm}$. $N_{t}$ \| $T/T_{c}$ \| $n_{\rm mb}$ \| $N_{\rm bin}$ ---\|---\|---\|--- 128 \| 0.30 \| 400 \| 24 80 \| 0.47 \| 400 \| 30 60 \| 0.63 \| 400 \| 44 48 \| 0.79 \| 400 \| 40 44 \| 0.86 \| 400 \| 44 40 \| 0.95 \| 400 \| 40 36 \| 1.05 \| 400 \| 40 32 \| 1.19 \| 400 \| 56 28 \| 1.36 \| 400 \| 40 24 \| 1.58 \| 400 \| 40 20 \| 1.90 \| 400 \| 40 Theoretically, under the periodic boundary condition in the temporal direction, the temporal correlators $C(t,T)$ at the temperature $T$ can be written in the spectral representation as $\displaystyle C(t,T)$ $\displaystyle\equiv$ $\displaystyle\frac{1}{Z(T)}{\rm Tr}\left(e^{-H/T}\Phi(t)\Phi(0)\right)$ (15) $\displaystyle=$ $\displaystyle\sum\limits_{m,n}\frac{\|\langle n\|\Phi\|m\rangle\|^{2}}{2Z(T)}\exp\left(-\frac{E_{m}+E_{n}}{2T}\right)$ $\displaystyle\times\cosh\left[\left(t-\frac{1}{2T}\right)(E_{n}-E_{m})\right]$ $\displaystyle=$ $\displaystyle\int\limits_{-\infty}^{\infty}d\omega\rho(\omega)K(\omega,T),$ with a $T$-dependent kernel $K(\omega,T)=\frac{\cosh(\omega/(2T)-\omega t)}{\sinh(\omega/(2T))}$ (16) and the spectral function, $\displaystyle\rho(\omega)$ $\displaystyle=$ $\displaystyle\sum\limits_{m,n}\frac{\|\langle n\|\Phi\|m\rangle\|^{2}}{2Z(T)}e^{-E_{m}/T}$ (17) $\displaystyle\times$ $\displaystyle(\delta(\omega-(E_{n}-E_{m})-\delta(\omega-(E_{m}-E_{n})),$ where $Z(T)$ is the partition function at $T$, and $E_{n}$ the energy of the thermal state $\|n\rangle$ ($\|0\rangle$ represents the vacuum state). In the zero-temperature limit($T\rightarrow 0$), due to the factor $\exp(-E_{m}/T)$, the spectral function $\rho(\omega)$ degenerates to $\rho(\omega)=\sum\limits_{n}\frac{\|\langle 0\|\Phi\|n\rangle\|^{2}}{2Z(0)}\left(\delta(\omega- E_{n})-\delta(\omega+E_{n})\right),$ (18) thus we have the function form of the correlation function, $C(t,T=0)=\sum\limits_{n}W_{n}e^{-E_{n}\tau}$ (19) with $W_{n}=\|\langle 0\|\Phi\|n\rangle\|^{2}/Z(0)$. However, for any finite temperature (this is always the case for finite lattices), all the thermal states with the nonzero matrix elements $\langle m\|\Phi\|n\rangle$ may contribute to the spectral function $\rho(\omega)$. Intuitively in the confinement phase, the fundamental degrees of freedom are hadronlike modes, thus the thermal states should be multihadron states. If they interact weakly with each other, we can treat them as free particles at the lowest order approximation and consider $E_{m}$ as the sum of the energies of hadrons including in the thermal state $\|m\rangle$. Since the contribution of a thermal state $\|m\rangle$ to the spectral function is weighted by the factor $\exp(-E_{m}/T)$, apart from the vacuum state, the maximal value of this factor is $\exp(-M_{\rm min}/T)$ with $M_{\rm min}$ the mass of the lightest hadron mode in the system. As far as the quenched glueball system is concerned, the lightest glueball is the scalar, whose mass at the low temperature is roughly $M_{0^{++}}\sim 1.6$ GeV, which gives a very tiny weight factor $\exp(-M_{0^{++}}/T_{c})\sim 0.003$ at $T_{c}$ in comparison with unity factor of the vacuum state. That is to say, for the quenched glueballs, up to the critical temperature $T_{c}$, the contribution of higher spectral components beyond the vacuum to the spectral function are much smaller than the statistical errors (the relative statistical errors of the thermal glueball correlators are always a few percent) and can be neglected. As a result, the function form of $\rho(\omega)$ in Eq. 18 can be a good approximation for the spectral function of glueballs at least up to $T_{c}$. Accordingly, considering the finite extension of the lattice in the temporal direction, the function form of the thermal correlators can be approximated as $C(t,T)=\sum\limits_{n}W_{n}\frac{\cosh(M_{n}(1/(2T)-t))}{\sinh(M_{n}/(2T))},$ (20) which is surely the commonly used function form for the study of hadron masses at low temperatures on the lattice. As is always done, the glueball masses $M_{n}$ derived by this function are called the pole masses in this work. ### III.1 Results of the single-cosh fit Even though the above discussion are based on the weak-interaction approximation for the hadronlike modes below $T_{c}$, we would like to apply Eq. 20 to analyzing the thermal correlators all over the temperature in concern. The interest of doing so is twofold. First, the thermal scattering of the glueball-like modes would result in a mass shift, say the deviation of the pole mass from the glueball mass at zero-temperature, which reflects the strength of the interaction at different temperature. Secondly, the breakdown of this function form would signal the dominance of new degrees of freedom instead of the hadronlike modes in the thermal states. In practice, after the thermal correlators $C(t,T)$ of the optimal operators are obtained according to the steps described in Sec. II(B), the pole masses of the ground state (or the lowest spectral component) can be extracted straightforwardly. First, for each $R^{PC}$ channel and at each temperature $T$, the effective mass $M_{\rm eff}(t)$ as a function of $t$ is derived by solving the equation $\frac{C(t+1,T)}{C(t,T)}=\frac{\cosh((t+1-N_{t}/2)a_{t}M_{\rm eff}(t))}{\cosh((t-N_{t}/2)a_{t}M_{\rm eff}(t))},$ (21) Secondly, the effective masses are plotted versus $t$ and the plateaus give the fit windows $[t_{1},t_{2}]$. Finally, the pole masses of the ground states are obtained by fitting $C(t,T)$ through a single-cosh function form. As a convention in this work, we use $M_{G}$ to represent the mass of a glueball state in the physical units and $M$ to represent the dimensionless mass parameter in the data processing with the relation $M=M_{G}a_{t}$. \| ---\|--- \| \| Figure 4: Effective masses at different temperatures in $A_{1}^{++}$ channel. Data points are the effective masses with jackknife error bars. The vertical lines indicate the time window $[t_{1},t_{2}]$ over which the single-cosh fittings are carried out, while the horizontal lines illustrate the best-fit result of pole masses (in each panel the double horizontal lines represent the error band estimated by jackknife analysis) \| ---\|--- \| \| Figure 5: Similar to Fig. 4, but in $A_{1}^{-+}$ channel. \| ---\|--- \| \| Figure 6: Similar to Fig. 4, but in $E^{++}$ channel. \| ---\|--- \| \| Figure 7: Similar to Fig. 4, but in $T_{2}^{++}$ channel. In Fig. 4, Fig. 5, Fig. 6, and Fig. 7 are shown the effective masses with jackknife errors at various temperatures in $A_{1}^{++}$, $A_{1}^{-+}$, $E^{++}$, and $T_{2}^{++}$ channels, respectively. The vertical lines indicate the time window $[t_{1},t_{2}]$ over which the single-cosh fittings are carried out, while the horizontal lines illustrate the best-fit result of pole masses (in each figure panel the double horizontal lines give the error band estimated by jackknife analysis). These figures exhibit some common features: At the temperatures below $T_{c}$ ($N_{t}=$ 128, 80, 40), the effective mass plateaus show up almost from right the beginning of $t$, as it should be for the optimal glueball operators, while at $T>T_{c}$ ($N_{t}=$ 36, 24, and 20), the plateaus appear later and later in time, and even do not exist at $N_{t}=20$ ($T=1.90T_{c}$). This observation can be interpreted as follows. Since the effective masses are calculated based on Eq. 20, the very early appearance of the plateaus below $T_{c}$ implies that the thermal correlators $C(t,T)$ of the optimal operators are surely dominated by the ground state and can be well described by the function form of Eq. 20. In other words, the picture of weakly interacting glueball-like modes makes sense for the state of matter below $T_{c}$. While at $T>T_{c}$, the later appearance and the narrower size plateaus signal that the picture of the state of matter is distinct from that at $T<T_{c}$. However, because of the existence of effective mass plateaus up to $T\sim 1.58T_{c}$($N_{t}=24$), the possibility that glueball-like modes survive at this high temperature cannot be excluded. Table 4: The pole masses (in units of $a_{t}^{-1}$) in all the 20 $R^{PC}$ channels are extracted at all the temperatures. $R^{PC}$ \| 128 \| 80 \| 60 \| 48 \| 44 \| 40 \| 36 \| 32 \| 28 \| 24 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $A_{1}^{++}$ \| 0.140( 2) \| 0.144( 3) \| 0.144( 2) \| 0.143( 3) \| 0.140(2) \| 0.140( 3) \| 0.132( 4) \| 0.126( 2) \| 0.122( 4) \| 0.116( 3) $A_{1}^{+-}$ \| 0.441( 3) \| 0.435( 3) \| 0.434( 5) \| 0.437( 4) \| 0.432( 4) \| 0.435( 5) \| 0.399( 6) \| 0.322( 9) \| 0.267(16) \| 0.241(13) $A_{1}^{-+}$ \| 0.221( 3) \| 0.225( 2) \| 0.222( 2) \| 0.225( 2) \| 0.218( 3) \| 0.222( 2) \| 0.183( 5) \| 0.174( 3) \| 0.155( 4) \| 0.146( 4) $A_{1}^{--}$ \| 0.475( 6) \| 0.453( 8) \| 0.447( 9) \| 0.464( 7) \| 0.473( 6) \| 0.468( 6) \| 0.426(12) \| 0.417(10) \| 0.287(19) \| 0.253(18) $A_{2}^{++}$ \| 0.323( 4) \| 0.327( 4) \| 0.326( 4) \| 0.330( 2) \| 0.326( 4) \| 0.332( 3) \| 0.282( 7) \| 0.249( 8) \| 0.224( 9) \| 0.208( 9) $A_{2}^{+-}$ \| 0.302( 5) \| 0.308( 3) \| 0.308( 5) \| 0.312( 3) \| 0.312( 5) \| 0.308( 6) \| 0.268( 6) \| 0.241( 7) \| 0.220( 8) \| 0.201( 6) $A_{2}^{-+}$ \| 0.450( 5) \| 0.449( 7) \| 0.446( 5) \| 0.440( 6) \| 0.452( 4) \| 0.448( 5) \| 0.396(10) \| 0.340(11) \| 0.330(12) \| 0.250(14) $A_{2}^{--}$ \| 0.387( 3) \| 0.388( 3) \| 0.385( 4) \| 0.390( 5) \| 0.376( 4) \| 0.375( 4) \| 0.354( 7) \| 0.293( 7) \| 0.268(10) \| 0.214( 9) $E^{++}$ \| 0.210( 1) \| 0.205( 1) \| 0.207( 1) \| 0.209(2) \| 0.206(1) \| 0.189( 4) \| 0.167( 4) \| 0.153( 3) \| 0.143( 3) \| 0.139( 2) $E^{+-}$ \| 0.401( 2) \| 0.403( 2) \| 0.401( 2) \| 0.394( 4) \| 0.400( 2) \| 0.395( 3) \| 0.375( 4) \| 0.311( 6) \| 0.261( 7) \| 0.230( 7) $E^{-+}$ \| 0.273( 1) \| 0.266( 1) \| 0.264( 2) \| 0.273( 2) \| 0.275( 1) \| 0.262( 2) \| 0.218( 4) \| 0.196( 4) \| 0.183( 4) \| 0.181( 4) $E^{--}$ \| 0.374( 1) \| 0.368( 2) \| 0.360( 2) \| 0.361( 3) \| 0.363( 3) \| 0.352( 4) \| 0.308( 8) \| 0.262( 6) \| 0.231( 6) \| 0.213( 6) $T_{1}^{++}$ \| 0.327( 2) \| 0.326( 4) \| 0.327( 2) \| 0.334( 2) \| 0.331( 2) \| 0.312( 5) \| 0.287( 7) \| 0.266( 3) \| 0.227( 6) \| 0.215( 4) $T_{1}^{+-}$ \| 0.278( 1) \| 0.274( 2) \| 0.265( 3) \| 0.278( 2) \| 0.281( 1) \| 0.261( 3) \| 0.207( 6) \| 0.199( 2) \| 0.181( 4) \| 0.175( 2) $T_{1}^{-+}$ \| 0.372( 2) \| 0.377( 4) \| 0.371( 3) \| 0.380( 2) \| 0.374( 2) \| 0.370( 3) \| 0.331( 5) \| 0.289( 7) \| 0.248( 7) \| 0.230( 5) $T_{1}^{--}$ \| 0.350( 4) \| 0.349( 2) \| 0.344( 3) \| 0.351( 2) \| 0.350( 2) \| 0.343( 3) \| 0.272( 8) \| 0.252( 5) \| 0.212( 6) \| 0.201( 5) $T_{2}^{++}$ \| 0.205( 1) \| 0.209( 1) \| 0.206( 1) \| 0.205( 1) \| 0.207( 2) \| 0.191( 3) \| 0.160( 3) \| 0.152( 2) \| 0.148( 2) \| 0.143( 2) $T_{1}^{+-}$ \| 0.322( 2) \| 0.317( 2) \| 0.310( 4) \| 0.317( 3) \| 0.320( 2) \| 0.303( 5) \| 0.276( 5) \| 0.250( 3) \| 0.201( 4) \| 0.190( 4) $T_{1}^{-+}$ \| 0.265( 2) \| 0.260( 3) \| 0.264( 2) \| 0.273( 3) \| 0.272( 2) \| 0.264( 2) \| 0.240( 3) \| 0.213( 3) \| 0.187( 4) \| 0.183( 4) $T_{1}^{--}$ \| 0.368( 2) \| 0.358( 3) \| 0.364( 3) \| 0.358( 4) \| 0.367( 2) \| 0.353( 5) \| 0.282(13) \| 0.254( 6) \| 0.235( 6) \| 0.220( 4) The pole masses in all 20 $R^{PC}$ channels are extracted in units $a_{t}^{-1}$ at all temperatures and are shown in Table 4. Specifically, with the lattice spacing determined in Sec. II, the pole masses of $A_{1}^{++}$, $A_{1}^{-+}$, $E^{++}$ and $T_{2}^{++}$ at $T\simeq 0$ in physical units are $M_{A_{1}^{++}}$=1.576(22)GeV, $M_{A_{1}^{-+}}$=2.488(34)GeV, $M_{E^{++}}\simeq M_{T_{2}^{++}}$=2.364(11)GeV, respectively, which are in agreement with that of previous studies prd56 ; prd60 ; prd73 ; npb221 ; plb309 ; npb314 ; prl75 ; outp . From the table, one can see that the behaviors of the pole masses with respect to the temperature in all 20 channels are uniform: the pole masses keep almost constant with the temperature increasing from $0.30T_{c}$ to right below $T_{c}$ ($0.95T_{c}$), and start to reduce gradually when $T>T_{c}$. When $T$ increases up to $1.90T_{c}$, the pole masses cannot be extract reliably through the single- cosh fit for the lack of clear effective mass plateaus. Figure 8 illustrates these behavior of pole masses in $A_{1}^{++}$, $A_{1}^{-+}$, $E^{++}$ and $T_{2}^{++}$ channels. Figure 8: The $T$-dependence of pole masses $A_{1}^{++}$, $A_{1}^{-+}$ $E^{++}$, and $T_{2}^{++}$ glueballs. These results imply that glueballs can be very stable below $T_{c}$ and survive up to $1.6T_{c}$. This coincides with the thermal properties of heavy quarkonia observed by model calculation and lattice numerical studies soft_modes ; quark_number ; gupta ; prl92 ; prd69 ; prd74 ; jhw2005 ; npa783 ; prd63 ; ijmpa16 , but different from the observation of a previous lattice study on glueballs where the observed pole-mass reduction start even at $T\simeq 0.8T_{c}$ prd66 . ### III.2 Breit-Wigner analysis In the single-cosh analysis, it is seen that, when the temperature increases up to $T_{c}$, the thermal correlators can be well described by Eq. 20 and the pole masses of glueballs are insensitive to $T$. This is in agreement with the picture that the state of matter below $T_{c}$ are made up of weakly interacting glueball-like modes. When $T>T_{c}$, the thermal correlators deviate from Eq. 20 more and more. This observation implies that the degrees of freedom are very different from that when $T<T_{c}$. Theoretically in the deconfined phase, gluons can be liberated from hadrons. However, the study of the equation of state shows that the state of the matter right above $T_{c}$ is far from a perturbative gluon gas. In other words, the gluons in the intermediate temperature above $T_{c}$ may interact strongly with each other and glueball-like resonances can possibly be formed. Thus different from bound states at low temperature, thermal glueballs can acquire thermal width due to the thermal scattering between strongly interacting gluons and the magnitudes of the thermal widths can signal the strength of these types of interactions at different temperatures. In order to take the thermal width into consideration, we also adopt the Breit-Wigner Ansatz, which is suggested by the pioneering work Ref. prd66 , to analyze the thermal correlators once more. First, we treat thermal glueballs as resonance objects which correspond to the poles (denoted by $\omega=\omega_{0}-i\Gamma$) of the retarded and advanced Green functions in the complex $\omega\\_$plane (note that conventionally in particle physics, a resonance pole is always denoted as $M-i\Gamma/2$ where $M$ is the mass of the resonance and $\Gamma$ is its width.) $\omega_{0}$ is called the mass of the resonance glueball and $\Gamma$ its thermal width in this work. Secondly, we assume that the spectral function $\rho(\omega)$ is dominated by these resonance glueballs. Thus the spectral function is parametrized as $\rho(\omega)=A(\delta_{\Gamma}(\omega-\omega_{0})-\delta_{\Gamma}(\omega+\omega_{0})+\ldots,$ (22) where $\delta_{\epsilon}$ is the Lorentzian function $\delta_{\epsilon}(x)=\frac{1}{\pi}{\rm Im}\left(\frac{1}{x-i\epsilon}\right)=\frac{1}{\pi}\frac{\epsilon}{x^{2}+\epsilon^{2}},$ (23) and ”$\ldots$” represents the terms of excited states. With this spectral function, the thermal glueball correlator $G(t,T)$ can be expressed as $\displaystyle C(t,T)$ $\displaystyle=$ $\displaystyle\int\limits_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{\cosh(\omega(\frac{1}{2T}-t))}{2\sinh(\frac{\omega}{2T})}$ (24) $\displaystyle\times$ $\displaystyle 2\pi A\left(\delta_{\Gamma}(\omega-\omega_{0})-\delta_{\Gamma}(\omega+\omega_{0})\right)+\ldots.$ (a) $N_{t}=128$($T/T_{c}=0.32$) \| (b) $N_{t}=36$($T/T_{c}=1.09$) \| (c) $N_{t}=20$($T/T_{c}=1.97$) ---\|---\|--- \| \| \| \| Figure 9: Determinations of the fit range $[t_{1},t_{2}]$ in $T_{2}^{++}$ channel at $N_{t}=$ 128, 36, and 20. In each row, $\omega_{0}^{\rm(eff)}(t)$ and $\Gamma^{\rm(eff)}(t)$ obtained by solving Eq. III.2 are plotted by data points with jackknife error bars. $[t_{1},t_{2}]$ are chosen to include the time slices between the two vertical lines, where $\omega_{0}^{\rm(eff)}(t)$ and $\Gamma^{\rm(eff)}(t)$ show up plateaus simultaneously. The best-fit results of $\omega_{0}$ and $\Gamma$ through the function $g_{\Gamma}(t)$ are illustrated by the horizontal lines. The integral on the right hand side of above equation, denoted by $g_{\Gamma}(t)$, can be calculated explicitly as $g_{\Gamma}(t)=A\left[{\rm Re}\left(\frac{\cosh((\omega_{0}+i\Gamma)(\frac{1}{2T}-t))}{\sinh(\frac{(\omega_{0}+i\Gamma)}{2T})}\right)+2\omega_{0}T\sum\limits_{n=1}^{\infty}\cos\left(2\pi nTt\right)\left\\{\frac{1}{(2\pi nT+\Gamma)^{2}+\omega_{0}^{2}}-(n\rightarrow-n)\right\\}\right],$ (25) which can be used as the fit function to extract $\omega_{0}$ and $\Gamma$ from the thermal correlators obtained from the numerical calculation. Practically, the infinite series in the above equation is truncated by setting the upper limit of the summation to be 50, which is tested to be enough for all the cases considered in this work. In the present study, we carry out the Breit-Wigner analysis in $A_{1}^{++}$, $A_{1}^{-+}$, $E^{++}$, and $T_{2}^{++}$ channels, whose continuum correspondences are $0^{++}$, $0^{-+}$, and $2^{++}$. Although the variational method is exploited to enhance the contribution of the ground state to the thermal correlators, the contributions from higher spectral components cannot be eliminated completely. Therefore, the fit range must be chosen properly where the contribution of the ground state dominates. We take the strategy advocated in Ref.prd66 as follows. For a given correlator $C(t,T)$, the effective peak position $\omega_{0}^{\rm(eff)}(t)$ and the effective width $\Gamma^{\rm(eff)}(t)$ are obtained by solving the equations $\displaystyle\frac{g_{\Gamma}(t)}{g_{\Gamma}(t+1)}$ $\displaystyle=$ $\displaystyle\frac{C(t,T)}{C(t+1,T)},$ $\displaystyle\frac{g_{\Gamma}(t+1)}{g_{\Gamma}(t+2)}$ $\displaystyle=$ $\displaystyle\frac{C(t+1,T)}{C(t+2,T)}.$ (26) The statistical errors of $\omega_{0}^{\rm(eff)}(t)$ and $\Gamma^{\rm(eff)}(t)$ can be estimated through the jackknife analysis. Thus the fit range, denoted by $[t_{1},t_{2}]$, is chosen to be the time range where $\omega_{0}^{\rm(eff)}(t)$ and $\Gamma^{\rm(eff)}(t)$ show up plateaus simultaneously. For example, the procedure in $T_{2}^{++}$ channel is illustrated in Fig. 9 for $N_{t}=128,36,20$ (corresponding to the temperature $T/T_{c}=0.30,1.05,1.90$), where the fit ranges $[t_{1},t_{2}]$ are determined to include the time slices between the two vertical lines in each figure. Figure 10: $\omega_{0}$’s are plotted versus $T/T_{c}$ for $A_{1}^{++}$, $A_{1}^{-+}$ $E^{++}$, and $T_{2}^{++}$ channels. The vertical lines indicate the critical temperature. Figure 11: $\Gamma$’s are plotted versus $T/T_{c}$ for $A_{1}^{++}$, $A_{1}^{-+}$ $E^{++}$, and $T_{2}^{++}$ channels. The vertical lines indicate the critical temperature. (a) $A_{1}^{-+}$ \| (b) $E^{++}$ \| (c) $T_{2}^{++}$ ---\|---\|--- \| \| Figure 12: Plotted are the spectral function $\rho(\omega)$ at $T/T_{c}=0.30$, 0.95,1.05 and 1.58 with the best-fit parameters. Panel (a)$\\_$(c) are for $A_{1}^{-+}$, $E^{++}$, and $T_{2}^{++}$ channels, respectively. Table 5: The best-fit $\omega_{0}$ and $\Gamma$ of $A_{1}^{++}$ channel at different $T$ through the Breit-Wigner fit. Also listed are the fit window $[t_{1},t_{2}]$ and the chi-square per degree of freedom, $\chi^{2}/d.o.f$. $N_{t}$ \| $T/T_{c}$ \| $\omega_{0}$ \| $\Gamma$ \| $[t_{1},t_{2}]$ \| $\chi^{2}/d.o.f$ ---\|---\|---\|---\|---\|--- 128 \| 0.30 \| 0.142(2) \| 0.008(10) \| (3, 6) \| 1.266 80 \| 0.47 \| 0.146(2) \| 0.013(9) \| (2, 4) \| 0.921 60 \| 0.63 \| 0.144(2) \| 0.008(3) \| (2, 7) \| 0.135 48 \| 0.79 \| 0.143(2) \| 0.014(6) \| (2, 4) \| 0.639 44 \| 0.86 \| 0.142(2) \| 0.004(3) \| (1, 7) \| 0.758 40 \| 0.95 \| 0.143(2) \| 0.003(4) \| (4, 7) \| 0.850 36 \| 1.05 \| 0.146(2) \| 0.028(4) \| (3, 6) \| 0.960 32 \| 1.19 \| 0.141(2) \| 0.053(2) \| (1, 4) \| 0.393 28 \| 1.36 \| 0.142(2) \| 0.056(4) \| (2, 5) \| 1.253 24 \| 1.58 \| 0.143(1) \| 0.059(3) \| (1, 4) \| 0.302 20 \| 1.90 \| 0.146(2) \| 0.077(5) \| (2, 4) \| 0.918 Table 6: The best-fit $\omega_{0}$ and $\Gamma$ of $A_{1}^{-+}$ channel at different $T$ through the Breit-Wigner fit. Also listed are the fit window $[t_{1},t_{2}]$ and the chi-square per degree of freedom, $\chi^{2}/d.o.f$. $N_{t}$ \| $T/T_{c}$ \| $\omega_{0}$ \| $\Gamma$ \| $[t_{1},t_{2}]$ \| $\chi^{2}/d.o.f$ ---\|---\|---\|---\|---\|--- 128 \| 0.30 \| 0.226(2) \| 0.006(5) \| (3, 9) \| 0.509 80 \| 0.47 \| 0.228(2) \| 0.008(4) \| (2, 6) \| 0.640 60 \| 0.63 \| 0.227(1) \| 0.013(5) \| (2, 7) \| 0.216 48 \| 0.79 \| 0.228(2) \| 0.012(5) \| (2, 6) \| 0.177 44 \| 0.86 \| 0.229(2) \| 0.013(4) \| (2, 8) \| 0.184 40 \| 0.95 \| 0.224(2) \| 0.004(6) \| (3, 6) \| 0.549 36 \| 1.05 \| 0.221(2) \| 0.037(3) \| (1, 8) \| 0.935 32 \| 1.19 \| 0.219(2) \| 0.047(3) \| (1, 6) \| 0.250 28 \| 1.36 \| 0.211(3) \| 0.068(6) \| (2, 5) \| 0.091 24 \| 1.58 \| 0.208(3) \| 0.089(7) \| (2, 4) \| 0.003 20 \| 1.90 \| 0.211(3) \| 0.099(9) \| (2, 6) \| 0.083 Table 7: The best-fit $\omega_{0}$ and $\Gamma$ of $E^{++}$ channel at different $T$ through the Breit-Wigner fit. Also listed are the fit window $[t_{1},t_{2}]$ and the chi-square per degree of freedom, $\chi^{2}/d.o.f$. $N_{t}$ \| $T/T_{c}$ \| $\omega_{0}$ \| $\Gamma$ \| ${t_{1},t_{2}}$ \| $\chi^{2}/DOF$ ---\|---\|---\|---\|---\|--- 128 \| 0.30 \| 0.212(1) \| 0.012(4) \| (2, 5) \| 0.274 80 \| 0.47 \| 0.211(1) \| 0.006(3) \| (2, 8) \| 0.616 60 \| 0.63 \| 0.212(1) \| 0.010(3) \| (2, 9) \| 0.844 48 \| 0.79 \| 0.213(1) \| 0.011(3) \| (2, 5) \| 0.206 44 \| 0.86 \| 0.211(1) \| 0.011(3) \| (2, 8) \| 1.268 40 \| 0.95 \| 0.207(1) \| 0.022(3) \| (2, 6) \| 0.250 36 \| 1.05 \| 0.205(2) \| 0.034(2) \| (1, 8) \| 0.183 32 \| 1.19 \| 0.200(1) \| 0.049(2) \| (1, 6) \| 0.478 28 \| 1.36 \| 0.191(2) \| 0.067(4) \| (2, 6) \| 0.297 24 \| 1.58 \| 0.189(2) \| 0.083(5) \| (2, 4) \| 0.253 20 \| 1.90 \| 0.196(2) \| 0.091(5) \| (2, 4) \| 0.046 Table 8: The best-fit $\omega_{0}$ and $\Gamma$ of $T_{2}^{++}$ channel at different $T$ through the Breit-Wigner fit. Also listed are the fit window $[t_{1},t_{2}]$ and $\chi^{2}/d.o.f$. $N_{t}$ \| $T/T_{c}$ \| $\omega_{0}$ \| $\Gamma$ \| $[t_{1},t_{2}]$ \| $\chi^{2}/d.o.f$ ---\|---\|---\|---\|---\|--- 128 \| 0.30 \| 0.210(1) \| 0.008(2) \| (3,10) \| 0.442 80 \| 0.47 \| 0.213(1) \| 0.009(3) \| (3, 9) \| 0.696 60 \| 0.63 \| 0.213(1) \| 0.012(3) \| (3, 7) \| 0.326 48 \| 0.79 \| 0.210(1) \| 0.007(4) \| (3, 6) \| 0.288 44 \| 0.86 \| 0.214(1) \| 0.012(2) \| (2, 6) \| 0.437 40 \| 0.95 \| 0.208(1) \| 0.023(1) \| (1, 7) \| 0.743 36 \| 1.05 \| 0.204(1) \| 0.039(2) \| (1, 7) \| 0.606 32 \| 1.19 \| 0.199(1) \| 0.047(1) \| (1, 6) \| 0.527 28 \| 1.36 \| 0.196(2) \| 0.064(3) \| (2, 5) \| 0.022 24 \| 1.58 \| 0.194(1) \| 0.077(4) \| (2, 7) \| 0.119 20 \| 1.90 \| 0.196(2) \| 0.093(4) \| (2, 5) \| 0.027 After the fit ranges for all the thermal correlators are chosen, the jackknife analysis can be carried out straightforward and the detailed procedures are omitted here. Table 5, 6, 7, and 8 show the fit windows $[t_{1},t_{2}]$, the chi-square per degree of freedom $\chi^{2}/d.o.f$, and the best-fit results of $\omega_{0}$ and $\Gamma$ at various temperature in $A_{1}^{++}$, $A_{1}^{-+}$, $E^{++}$, and $T_{2}^{++}$ channels. In almost all the cases, the fit ranges start from $t_{1}=1$, 2, or 3, and last for quite a few time slices. This reflects that, as is expected, the optimal glueball operators couple almost exclusively to the lowest spectral components after the implementation of the variational method. All the $\chi^{2}/d.o.f$’s are $\sim O(1)$ or even smaller, which reflect the reliability of the fits. The main features of the best fit $\omega_{0}$ and $\Gamma$ based on Breit- Wigner Ansatz are described as follows: * • The peak positions $\omega_{0}$ of the spectral functions $\rho(\omega)$ are insensitive to the temperature in all the considered channels. In particular, the $\omega_{0}$ in $A_{1}^{++}$ channel keeps almost constant all over the temperature range from $0.30T_{c}$ to $1.90T_{c}$. In the other three channels, the $\omega_{0}$’s do not change within errors below $T_{c}$, but reduce mildly with the increasing temperature above $T_{c}$. The reduction of $\omega_{0}$ at the highest temperature $T=1.90T_{c}$ is less than 5% in these three channels. * • In all four channels, the thermal widths $\Gamma$ are small and do not vary much below $T_{c}$, but grow rapidly with the increasing temperature when $T>T_{c}$. Below $T_{c}$, the thermal widths are of order $\Gamma\sim 5\%$ or even smaller (especially for the $A_{1}^{++}$ $\Gamma$ is consistent with zero). The thermal widths increase abruptly when the temperature passes $T_{c}$ and reach values $\sim\omega_{0}/2$ at $T=1.90T_{c}$. These features can be seen easily in Fig. 10 and 11, where the behaviors of $\omega_{0}$ and $\Gamma$ with respect to the temperature $T$ are plotted for all four channels. The line$\\_$shapes of the spectral functions with the best-fit parameters at different $T$ are shown in Fig. 12 for $A_{1}^{-+}$, $E^{++}$, and $T_{2}^{++}$ channels (we do not plot the spectral function of $A_{1}^{++}$ channel due to the small thermal widths). ## IV Summary and Discussions Table 9: The ”pole masses” $M_{G}$ obtained by single-cosh analysis and ($\omega_{0},\Gamma$) obtained based on the Breit-Wigner ansatz are combined together for comparison. Listed in the table are the results in $A_{1}^{++}$ and $A_{1}^{-+}$ channels (all the data are converted into the physical units). \| \| \| $A_{1}^{++}$ \| \| \| \| $A_{1}^{-+}$ \| ---\|---\|---\|---\|---\|---\|---\|---\|--- $N_{t}$ \| $T/T_{c}$ \| $m_{G}$[GeV] \| $\omega_{0}$[GeV] \| $\Gamma$[GeV] \| \| $m_{G}$[GeV] \| $\omega_{0}$[GeV] \| $\Gamma$[GeV] 128 \| 0.30 \| 1.576(22) \| 1.602(14) \| 0.091(113) \| \| 2.488(31) \| 2.549(17) \| 0.069(52) 80 \| 0.47 \| 1.621(29) \| 1.644(22) \| 0.145(100) \| \| 2.533(24) \| 2.560(17) \| 0.086(44) 60 \| 0.63 \| 1.627(18) \| 1.621(16) \| 0.098(38) \| \| 2.499(27) \| 2.559(16) \| 0.147(54) 48 \| 0.79 \| 1.616(21) \| 1.612(26) \| 0.156(67) \| \| 2.533(23) \| 2.564(20) \| 0.135(55) 44 \| 0.86 \| 1.577(25) \| 1.598(17) \| 0.045(34) \| \| 2.454(34) \| 2.577(18) \| 0.144(48) 40 \| 0.95 \| 1.576(39) \| 1.621(20) \| 0.034(46) \| \| 2.499(25) \| 2.525(26) \| 0.042(71) 36 \| 1.05 \| 1.486(43) \| 1.638(28) \| 0.315(48) \| \| 2.060(48) \| 2.490(23) \| 0.413(32) 32 \| 1.19 \| 1.418(21) \| 1.588(25) \| 0.586(28) \| \| 1.959(37) \| 2.464(20) \| 0.529(28) 28 \| 1.36 \| 1.373(48) \| 1.599(26) \| 0.619(43) \| \| 1.745(43) \| 2.375(28) \| 0.768(71) 24 \| 1.58 \| 1.306(41) \| 1.613(28) \| 0.664(37) \| \| 1.644(45) \| 2.308(32) \| 0.999(83) 20 \| 1.90 \| - \| 1.642(32) \| 0.873(59) \| \| - \| 2.380(35) \| 1.114(99) Table 10: The pole masses $M_{G}$ obtained by single-cosh analysis and ($\omega_{0},\Gamma$) obtained based on the Breit-Wigner Ansatz are combined together for comparison. Listed in the table are the results in $E^{++}$ and $T_{2}^{++}$ channels (all the data are converted into physical units). \| \| \| $E^{++}$ \| \| \| \| $T_{2}^{++}$ \| ---\|---\|---\|---\|---\|---\|---\|---\|--- $N_{t}$ \| $T/T_{c}$ \| $m_{G}$[GeV] \| $\omega_{0}$[GeV] \| $\Gamma$[GeV] \| \| $m_{G}$[GeV] \| $\omega_{0}$[GeV] \| $\Gamma$[GeV] 128 \| 0.30 \| 2.364(11) \| 2.385(12) \| 0.140(42) \| \| 2.308(14) \| 2.363(10) \| 0.091(25) 80 \| 0.47 \| 2.308(13) \| 2.368(12) \| 0.069(29) \| \| 2.353(15) \| 2.387(10) \| 0.105(32) 60 \| 0.63 \| 2.330(11) \| 2.383(12) \| 0.116(32) \| \| 2.319(13) \| 2.396(10) \| 0.140(32) 48 \| 0.79 \| 2.353(19) \| 2.393(15) \| 0.129(34) \| \| 2.308(15) \| 2.362(14) \| 0.083(43) 44 \| 0.86 \| 2.319(15) \| 2.379(12) \| 0.119(32) \| \| 2.330(21) \| 2.405(10) \| 0.136(26) 40 \| 0.95 \| 2.263(14) \| 2.327(15) \| 0.247(38) \| \| 2.218(16) \| 2.344(11) \| 0.259(16) 36 \| 1.05 \| 1.880(41) \| 2.305(17) \| 0.382(23) \| \| 1.925(33) \| 2.298(14) \| 0.437(17) 32 \| 1.19 \| 1.722(35) \| 2.247(16) \| 0.549(20) \| \| 1.801(25) \| 2.244(11) \| 0.532(15) 28 \| 1.36 \| 1.610(31) \| 2.155(19) \| 0.754(43) \| \| 1.666(23) \| 2.205(17) \| 0.717(36) 24 \| 1.58 \| 1.565(23) \| 2.132(21) \| 0.937(55) \| \| 1.610(27) \| 2.184(16) \| 0.870(41) 20 \| 1.90 \| - \| 2.201(23) \| 1.023(60) \| \| - \| 2.209(20) \| 0.935(48) On $24^{3}\times N_{t}$ anisotropic lattices with the anisotropy $\xi=5$ at the gauge coupling $\beta=3.2$, the thermal glueball correlators are calculated in a large temperature range from $0.30T_{c}$ to $1.90T_{c}$, which are realized by varying $N_{t}$ to represent different temperatures. Based on the lattice spacing $a_{s}=0.0878(4)\,{\rm fm}$ determined by $r_{0}^{-1}=(410(20)\,{\rm MeV})$, the spatial extension of the lattices are estimated to be $(2.1\,{\rm fm})^{3}$, which is large enough to be free of the finite volume effects. On the other hand, because of the large anisotropy, there are enough data point in the temporal direction for the thermal correlators to be analyzed comfortably even at the highest temperature $T\sim 2T_{c}$ concerned in this work. With the implementation of the smearing scheme and the variational method, we can construct the optimal glueball operators in all the symmetry channel, which couple mostly to the lowest-lying states (or more precisely, the lowest-lying spectral components). As a result, the thermal correlators of these operators can be considered to be contributed dominantly from these lowest-lying states. The thermal correlators are analyzed based on two ansatz, say, the single-cosh function form and the Breit-Wigner Ansatz. In Table 9 and Table 10, the ”pole masses” $M_{G}$ obtained by single-cosh analysis and ($\omega_{0},\Gamma$) obtained based on the Breit-Wigner Ansatz are combined together for comparison (all the data are converted into physical units). The most striking observation from the single-cosh analysis is that, in all 20 $R^{PC}$ channels, the best-fit pole-masses $M_{G}$ are almost constant within errors from the low temperature up to right below the critical temperature $T_{c}$. This is what should be from the point of view of deconfinement phase transition of QCD: Since below $T_{c}$ the system is in the confinement phase, the fundamental degrees of freedom must be hadrons. Above $T_{c}$, the reduction of the pole masses does signal the QCD transition, after which the state of the matter is very different from that below $T_{c}$. However, the existence of effective mass plateaus, from which the pole masses are extracted, also implies that color singlet objects, the glueball-like modes, can also survive at the intermediate temperature above $T_{c}$. The results of the Breit-Wigner fit are consistent with this picture. In the Breit-Wigner Ansatz, thermal widths $\Gamma$ are introduced to glueball states to account for the effects of finite temperature, such as the thermal scattering and the thermal fluctuations. As shown in Table 9 and 10, below $T_{c}$ (or in the confinement phase), the best-fit $\omega_{0}$’s are very close to the pole masses, and the thermal widths $\Gamma$ are very tiny and are always of a few percent of $\omega_{0}$. This means the glueball states are surely stable in the confinement phase and the thermal interaction among them are weak. With the temperature increasing above $T_{c}$, while the temperature dependence of $\omega_{0}$’s is very mild, the thermal widths $\Gamma$ grow rapidly and reach values of roughly half of $\omega$’s at $T\sim 1.9T_{c}$. This clearly reflects that glueballs act as resonances are unstable more and more, and the reduction of pole masses above $T_{c}$ can be taken as the effect of these growing thermal widths. To summarize, in pure gauge theory, the state of matter is dominated by weakly interacting hadronlike states below $T_{c}$; when $T>T_{c}$, glueball states survive as resonancelike modes up to a temperature $T\sim 1.9T_{c}$ with their thermal widths growing with increasing $T$, which implies that in this intermediate temperature range, glueballs are unstable and may decay into gluons, and reversely gluons also interact strongly enough to form glueball- like resonances. The two procedure may reach the thermal equilibrium at a given temperature, such that the gluon degree of freedom become more and more important with $T$ increasing. At very high temperature, the glueball-like resonances may disappear finally and the state of matter can thereby be described by a perturbative gluon plasma. This picture is coincident with the observations both in the study of equation of state of QCD and the thermal properties of heavy quarkonia. On the other hand, the surprising results of RHIC experiments may also support this picture to some extent. First, the data of RHIC experiments are well described by the hydrodynamical modelprl86 . Secondly, the investigation of elliptic flow data using a Boltzmann-type equation for gluon scattering is not consistent with the perturbative QCD apparentlynpa697 . So the quark-gluon plasma at the RHIC temperature is most likely a strongly interacting system. ## Acknowledgments This work is supported in part by NSFC (Grant No. 10347110, 10421003, 10575107, 10675005, 10675101, 10721063, and 10835002) and CAS (Grant No. KJCX3-SYW-N2 and KJCX2-YW-N29). The numerical calculations were performed on DeepComp 6800 supercomputer of the Supercomputing Center of Chinese Academy of Sciences, Dawning 4000A supercomputer of Shanghai Supercomputing Center, and NKstar2 Supercomputer of Nankai University. ## References * (1) P. Petreczky, arXiv:hep-lat/0506012, and references therein; J. Schaffner-Bielich, PoS (CPOD2007) 062 (2007), arXiv:astro-ph/0709.1043, and references therein. * (2) F. Karsch, E. Laermann, and A. Peikert, Phys. Lett. B 478, 447 (2000). * (3) S. Gottlieb, W. Liu, D. Toussaint, R.L. Renken, R.L. Sugar, Phys. Rev. Lett. 59, 2247 (1987); S. Gottlieb, W. Liu, R.L. Renken, R.L. Sugar, D. Toussaint, Phys. Rev. D 38, 2888 (1988); S. Gottlieb, U.M. Heller, A.D. Kennedy, S. Kim, J.B. Kogut, C. Liu, R.L. Renken, D.K. Sinclair, R.L. Sugar, D. Toussaint, K.C. Wang, Phys. Rev. D 55, 6852 (1997). * (4) C. DeTar, Phys. Rev. D 32, 276 (1985); Phys. Rev. D 37, 2328 (1988). * (5) G. Boyd, S. Gupta, F. Karsch, and E. Laermann, Z. Phys. C 64, 331 (1994). * (6) T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986). * (7) T. Hashimoto, O. Miyamura, K. Hirose, and T. Kanki, Phys. Rev. Lett. 57, 2123 (1986). * (8) F. Karsch, M.T. Mehr, and H. Satz, Z. Phys. C 37, 617 (1988). * (9) S. Digal, P. Petreczky, and H. Satz, Phys. Lett. B 514, 57 (2001). * (10) S. Digal, P. Petreczky, and H. Satz, Phys. Rev. D 64, 094015 (2001). * (11) E.V. Shuryak and I. Zahed, Phys. Rev. D 70, 054507 (2004). * (12) C.-Y. Wong, Phys. Rev. C 72, 034906 (2005), arXiv:hep-ph/0408020. * (13) Á. Mócsy, P. Petreczky, Eur. Phys. J. C 43, 77 (2005), arXiv:hep-ph/0512156. * (14) P. Petreczky, Eur. Phys. J. C 43, 51 (2005). * (15) M. Asakawa and T. Hatsuda, Phys. Rev. Lett. 92, 012001 (2004), arXiv:hep-lat/0308034. * (16) S. Datta, F. Karsch, P. Petreczky, and I. Wetzorke , Phys. Rev. D 69, 094507 (2004), arXiv:hep-lat/0312037; S. Datta, F. Karsch, P. Petreczky, and I. Wetzorke , J.Phys. G 30, S1347 (2004). * (17) H. Iida, T. Doi, N. Ishii, H. Suganuma, and K. Tsumura, Phys. Rev. D 74, 074502 (2006), arXiv:hep-lat/0602008 . * (18) A. Jakovac, Proc. Sci., JHW2005(2005)023; A. Jakovac, P. Petreczky, K. Petrov, and A. Velytsky, PoS(JHW 2005), arXiv:hep-lat/0603005. * (19) B. Berg and A. Billoire, Nucl. Phys. A783, 477 (2007). * (20) H. Iida , T. Doi, N. Ishii, H. Suganuma, and K. Tsumura, Prog. Theor. Phys. Suppl. 174, 238 (2008), arXiv:hep-lat/0806.0126 . * (21) C.J. Morningstar and M. Peardon, Phys. Rev. D 56, 4043 (1997). * (22) C.J. Morningstar and M. Peardon, Phys. Rev. D 60, 034509 (1999). * (23) Y. Chen et al., Phys. Rev. D 73, 014516 (2006). * (24) B. Berg and A. Billoire, Nucl. Phys. B221, 109 (1983). * (25) G. Bali, et al. (UKQCD Collaboration), Phys. Lett. B 309, 378 (1993). * (26) C. Michael and M. Teper, Nucl. Phys. B314, 347 (1989). * (27) J. Sexton, A.Vaccarino and D. Weingarten, Phys. Rev. Lett. 75, 4563 (1995). * (28) M. J. Teper, arXiv:hep-th/9812187. * (29) Yu.A. Simonov, Phys. At. Nucl. 58, 309 (1995),hep-ph/9311216. * (30) N.O. Agasian, D. Ebert, E.-M. Ilgenfritz, Nucl. Phys. A637, 135 (1998); A. Drago, M. Gibilisco, C. Ratti, Nucl. Phys. A742, 165 (2004). * (31) V. Vento, Phys. Rev. D 75, 055012 (2007), arXiv:hep-ph/0609219. * (32) P. Petreczky, Nucl. Phys. B, Proc. Suppl. 140, 78 (2005); F. Karsch, Lect. Notes Phys. 583, 209 (2002), arXiv:hep-lat/0106019; D.E. Miller, Acta Physica Polonica B 28, 2937 (1997), arXiv:hep-ph/9807304. * (33) J. Sollfrank and U. Heinz, Z. Phys. C 65, 111 (1995); A. Drago, M. Gibilisco, and C. Ratti, Nucl. Phys. A742, 165 (2004); B.J. Schaefer, O. Bohr and J. Wambach, Phys. Rev. D 65, 105008 (2002); A. Di Giacomo, E. Meggiolaro, Yu.A. Simonov, and A.I. Veselov, Phys. Atom. Nucl. 70, 908 (2007). * (34) H. Leutwyler, Deconfinemente and Chiral symmetry in QCD 20 Years later, P.M. Zerwas and H.A. Castrup (Eds.) (World Scientific, Singapore 1993). * (35) N. Ishii, H. Suganuma, and H. Matsufuru, Phys. Rev. D 66, 094506 (2002), arXiv:hep-lat/0206020 ; N. Ishii, H. Suganuma, and H. Matsufuru, Phys. Rev. D 66, 014507 (2002), arXiv:hep-lat/0309102. * (36) A.M Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988). * (37) S. Huang et al., Nucl.Phys. B (Proc. Suppl.) 17, 281 (1990). * (38) W. Liu et al., Mod. Phys. Lett. A 21, 2313 (2006). * (39) Y. Liang, K.F. Liu, B.A. Li, S.J. Dong, and K. Ishikawa, Phys. Lett. B307, 375 (1993). * (40) QCD-TARO Collaboration, Ph. de Forcrand, M. García Pérez, T. Hashimoto, S. Hioki, H. Matsufuru, O. Miyamura, A. Nakamura, I. O. Stamatescu, T. Takaishi, T. Umeda, Phys. Rev. D 63, 054501 (2001). * (41) T. Umeda, R. Katayama,O. Miyamura and H. Matsufuru, Int. J. Mod. Phys. A 16, 2215 (2001). * (42) STAR Collaboration: K.H. Ackermann, et al., Phys. Rev. Lett. 86, 402 (2001), arXiv:nucl-ex/0009011 . * (43) D. Molnar and M. Gyulassy, Nucl. Phys. A697 , 495 (2002) ;A703, 893 (2002); D. Molnar, arXiv:hep-ph/0408044.	arxiv-papers	2009-03-11T15:00:45	2024-09-04T02:49:01.076108	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xiang-Fei Meng, Gang Li, Ying Chen, Chuan Liu, Yu-Bin Liu, Jian-Ping\n Ma, and Jian-Bo Zhang", "submitter": "Xiangfei Meng", "url": "https://arxiv.org/abs/0903.1991" }
0903.2093	# extension functors of local cohomology modules M. Aghapournahr1 1 Arak University, Beheshti St, P.O. Box: 879, Arak, Iran. m-aghapour@araku.ac.ir , A. J. Taherizadeh2 2,3 Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran, Iran. taheri@saba.tmu.ac.ir and Alireza Vahidi3 3 Payame Noor University (PNU), Iran. vahidi.ar@gmail.com ###### Abstract. Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of $R$, and $X$ an $R$–module. In this paper, for fixed integers $s,t$ and a finite $\mathfrak{a}$–torsion $R$–module $N$, we first study the membership of $\mbox{Ext}\,^{s+t}_{R}(N,X)$ and $\mbox{Ext}\,^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))$ in Serre subcategories of the category of $R$–modules. Then we present some conditions which ensure the existence of an isomorphism between them. Finally, we introduce the concept of Serre cofiniteness as a generalization of cofiniteness and study this property for certain local cohomology modules. ###### Key words and phrases: Local cohomology modules, Serre subcategories, Cofinite modules. ###### 2000 Mathematics Subject Classification: 13D45, 13D07. ## 1\. Introduction Throughout $R$ will denote a commutative Noetherian ring with non-zero identity and $\mathfrak{a}$ an ideal of $R$. Also $N$ will be a finite $\mathfrak{a}$–torsion module and $X$ an $R$–module. For unexplained terminology from homological and commutative algebra we refer the reader to [10] and [11]. The following conjecture was made by Grothendieck in [19]. ###### Conjecture 1.1. For any ideal $\mathfrak{a}$ and finite $R$–module $X$, the module $\emph{\mbox{Hom}\,}_{R}(R/\mathfrak{a},H^{n}_{\mathfrak{a}}(X))$ is finite for all $n\geq 0$. This conjecture is false in general as shown by Hartshorne in [21]. However, he defined an $R$–module $X$ to be $\mathfrak{a}$–cofinite if $\mbox{Supp}\,_{R}(X)\subseteq V(\mathfrak{a})$ and $\mbox{Ext}\,^{i}_{R}(R/\mathfrak{a},X)$ is finite for each $i$, and he asked the following question. ###### Question 1.2. If $\mathfrak{a}$ is an ideal of $R$ and $X$ is a finite $R$–module when is $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},H^{j}_{\mathfrak{a}}(X))$ finite for every $i$ and $j$? There are some attempts to show that under some conditions, for fixed integers $s$ and $t$, the $R$–module $\mbox{Ext}\,^{s}_{R}(R/\mathfrak{a},H^{t}_{\mathfrak{a}}(X))$ is finite, for example see [3, Theorem 3.3], [16, Theorems A and B], [17, Theorem 6.3.9] and [24, Theorem 3.3]. Recently, the first author and Melkersson in [1] and [2], and Asgharzadeh and Tousi in [5] approached the study of local cohomology modules by means of Serre subcategories and it is noteworthy that their approach enables us to deal with several important problems on local cohomology modules comprehensively. For more information, we refer the reader to [23] to see a survey of some important problems on finiteness, vanishing, Artinianness, and finiteness of associated primes of local cohomology modules. In this paper, we study some properties of extension functors of local cohomology modules by using Serre classes. Recall that a class of $R$–modules is a Serre subcategory of the category of $R$–modules when it is closed under taking submodules, quotients and extensions. Always, $\mathcal{S}$ stands for a Serre subcategory of the category of $R$–modules. The crucial points of Section 2 are Theorems 2.1 and 2.3 which show that when $R$–modules $\mbox{Ext}\,^{s+t}_{R}(N,X)$ and $\mbox{Ext}\,^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))$ belong to $\mathcal{S}$. These two theorems, which are frequently used through the paper, enable us to demonstrate some new facts and improve some older facts about the extension functors of local cohomology modules. We find the weakest possible conditions for finiteness of associated primes of local cohomology modules and, improve and give a new proof for [24, Theorem 3.3] in Corollaries 2.5 and 2.7. The relation between $R$–modules $\mbox{Ext}\,^{s+t}_{R}(N,X)$ and $\mbox{Ext}\,^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))$ to be in a Serre subcategory of the category of $R$–modules is shown in Corollary 2.8. In Section 3, we first introduce the class of Melkersson subcategory as a special case of Serre classes and next investigate the extension functors of local cohomology modules in these subcategories. In Propositions 3.2, 3.3 and 3.4, we give new proofs for [1, Theorems 2.9 and 2.13] and study the membership of the local cohomology modules of an $R$–module $X$ with respect to different ideals in Melkersson subcategories. Our main result in this section is Theorem 3.5 which provides an isomorphism between the $R$–modules $\mbox{Ext}\,^{s+t}_{R}(N,X)$ and $\mbox{Ext}\,^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))$. Corollaries 3.6 through 3.9 are some applications of this theorem. In Section 4, we present a generalization of the concept of cofiniteness with respect to an ideal to Serre subcategories of the category of $R$–modules. Theorems 4.2, 4.4 and 4.6 generalize [26, Proposition 2.5], [27, Proposition 3.11], [14, Theorem 3.1], [16, Theorems A and B] and [13, Corollary 2.7]. The Change of ring principle for Serre cofiniteness is presented in Theorem 4.8. We also give a proposition about $\mathfrak{a}$–cofinite minimax local cohomology modules in Proposition 4.10. Corollaries 4.11 and 4.12 are immediate results of this proposition where Corollary 4.11 improves [6, Theorem 2.3]. ## 2\. Local cohomology modules and Serre subcategories Let $\mathfrak{a}$ be an ideal of $R$, $N$ a finite $\mathfrak{a}$–torsion module and $s,t$ non-negative integers. In this section, we present sufficient conditions which convince us the $R$–modules $\mbox{Ext}\,^{t}_{R}(N,X)$ and $\mbox{Ext}\,^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))$ are in a Serre subcategory of the category of $R$–modules. Even though we can provide elementary proofs by using induction for our main theorems, for shortening the proofs we use spectral sequences argument. ###### Theorem 2.1. Let $X$ be an $R$–module and $t$ be a non-negative integer such that $\emph{\mbox{Ext}\,}^{t-r}_{R}(N,H^{r}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ for all $r$, $0\leq r\leq t.$ Then $\emph{\mbox{Ext}\,}^{t}_{R}(N,X)$ is in $\mathcal{S}.$ ###### Proof. By [29, Theorem 11.38], there is a Grothendieck spectral sequence $E^{p,q}_{2}:=\mbox{Ext}\,^{p}_{R}(N,H^{q}_{\mathfrak{a}}(X))_{\stackrel{{\scriptstyle\Longrightarrow}}{{p}}}\mbox{Ext}\,^{p+q}_{R}(N,X).$ For all $r$, $0\leq r\leq t$, we have $E_{\infty}^{t-r,r}=E_{t+2}^{t-r,r}$ since $E_{i}^{t-r-i,r+i-1}=0=E_{i}^{t-r+i,r+1-i}$ for all $i\geq t+2$; so that $E_{\infty}^{t-r,r}$ is in $\mathcal{S}$ from the fact that $E_{t+2}^{t-r,r}$ is a subquotient of $E_{2}^{t-r,r}$ which is in $\mathcal{S}$ by assumption. There exists a finite filtration $0=\phi^{t+1}H^{t}\subseteq\phi^{t}H^{t}\subseteq\cdots\subseteq\phi^{1}H^{t}\subseteq\phi^{0}H^{t}=\mbox{Ext}\,^{t}_{R}(N,X)$ such that $E_{\infty}^{t-r,r}=\phi^{t-r}H^{t}/\phi^{t-r+1}H^{t}$ for all $r$, $0\leq r\leq t$. Now the exact sequences $0\longrightarrow\phi^{t-r+1}H^{t}\longrightarrow\phi^{t-r}H^{t}\longrightarrow E_{\infty}^{t-r,r}\longrightarrow 0,$ for all $r$, $0\leq r\leq t,$ yield the assertion. ∎ Recall that, an $R$–module $X$ is said to be weakly Laskerian if the set of associated primes of any quotient module of $X$ is finite (see [13, Definition 2.1]). Also, we say that $X$ is $\mathfrak{a}$–weakly cofinite if $\mbox{Supp}\,_{R}(X)\subseteq V(\mathfrak{a})$ and $\mbox{Ext}\,^{i}_{R}(R/\mathfrak{a},X)$ is weakly Laskerian for all $i\geq 0$ (see [14, Definition 2.4]). We denote the category of $R$–modules (resp. the category of finite $R$–modules, the category of weakly Laskerian $R$–modules) by $\mathcal{C}(R)$ (resp. $\mathcal{C}_{f.g}(R)$, $\mathcal{C}_{w.l}(R)$). ###### Corollary 2.2. (cf. [17, Theorem 6.3.9(i)]) Let $X$ be an $R$–module and $n$ be a non- negative integer such that for all $r$, $0\leq r\leq n,$ $\emph{\mbox{Ext}\,}^{n-r}_{R}(N,H^{r}_{\mathfrak{a}}(X))$ is weakly Laskerian (resp. finite). Then $\emph{\mbox{Ext}\,}^{n}_{R}(N,X)$ is weakly Laskerian (resp. finite) and so $\emph{\mbox{Ass}\,}_{R}(Ext^{n}_{R}(N,X))$ is finite. The next theorem is related to the $R$–module $\mbox{Ext}\,^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))$ to be in a Serre subcategory of the category of $R$–modules. ###### Theorem 2.3. Let $X$ be an $R$–module and $s,t$ be non-negative integers such that * (i) $\emph{\mbox{Ext}\,}^{s+t}_{R}(N,X)$ is in $\mathcal{S}$, * (ii) $\emph{\mbox{Ext}\,}^{s+t+1-i}_{R}(N,H^{i}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ for all $i$, $0\leq i<t,$ and * (iii) $\emph{\mbox{Ext}\,}^{s+t-1-i}_{R}(N,H^{i}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ for all $i$, $t+1\leq i<s+t.$ Then $\emph{\mbox{Ext}\,}^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$. ###### Proof. Consider the Grothendieck spectral sequence $E^{p,q}_{2}:=\mbox{Ext}\,^{p}_{R}(N,H^{q}_{\mathfrak{a}}(X))_{\stackrel{{\scriptstyle\Longrightarrow}}{{p}}}\mbox{Ext}\,^{p+q}_{R}(N,X).$ For all $r\geq 2$, let $Z_{r}^{s,t}=\ker(E_{r}^{s,t}\longrightarrow E_{r}^{s+r,t+1-r})$ and $B_{r}^{s,t}=\mbox{Im}\,(E_{r}^{s-r,t+r-1}\longrightarrow E_{r}^{s,t})$. We have the exact sequences: $0\longrightarrow Z_{r}^{s,t}\longrightarrow E_{r}^{s,t}\longrightarrow E_{r}^{s,t}/Z_{r}^{s,t}\longrightarrow 0$ and $0\longrightarrow B_{r}^{s,t}\longrightarrow Z_{r}^{s,t}\longrightarrow E_{r+1}^{s,t}\longrightarrow 0.$ Since, by assumptions (ii) and (iii), $E_{2}^{s+r,t+1-r}$ and $E_{2}^{s-r,t+r-1}$ are in $\mathcal{S}$, $E_{r}^{s+r,t+1-r}$ and $E_{r}^{s-r,t+r-1}$ are also in $\mathcal{S}$, and so $E_{r}^{s,t}/Z_{r}^{s,t}$ and $B_{r}^{s,t}$ are in $\mathcal{S}$. It shows that $E_{r}^{s,t}$ is in $\mathcal{S}$ whenever $E_{r+1}^{s,t}$ is in $\mathcal{S}$. We have $E_{r}^{s-r,t+r-1}=0=E_{r}^{s+r,t+1-r}$ for all $r$, $r\geq t+s+2$. Therefore we obtain $E_{t+s+2}^{s,t}=E_{\infty}^{s,t}$. To complete the proof, it is enough to show that $E_{\infty}^{s,t}$ is in $\mathcal{S}$. There exists a finite filtration $0=\phi^{s+t+1}H^{s+t}\subseteq\phi^{s+t}H^{s+t}\subseteq\cdots\subseteq\phi^{1}H^{s+t}\subseteq\phi^{0}H^{s+t}=\mbox{Ext}\,^{s+t}_{R}(N,X)$ such that $E_{\infty}^{s+t-j,j}=\phi^{s+t-j}H^{s+t}/\phi^{s+t-j+1}H^{s+t}$ for all $j$, $0\leq j\leq s+t$. Since $\mbox{Ext}\,^{s+t}_{R}(N,X)$ is in $\mathcal{S}$, $\phi^{s}H^{s+t}$ is in $\mathcal{S}$ and so $E_{\infty}^{s,t}=\phi^{s}H^{s+t}/\phi^{s+1}H^{s+t}$ is in $\mathcal{S}$ as we desired. ∎ ###### Corollary 2.4. (cf. [5, Theorem 2.2]) Suppose that $X$ is an $R$–module and $n$ is a non- negative integer such that * (i) $\emph{\mbox{Ext}\,}^{n}_{R}(N,X)$ is in $\mathcal{S}$, and * (ii) $\emph{\mbox{Ext}\,}^{n+1-i}_{R}(N,H^{i}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ for all $i$, $0\leq i<n.$ Then $\emph{\mbox{Hom}\,}_{R}(N,H^{n}_{\mathfrak{a}}(X))$ is in $\mathcal{S}.$ ###### Proof. Apply Theorem 2.3 with $s=0$ and $t=n$. ∎ We can deduce from the above corollary the main results of [25, Theorem B], [9, Theorem 2.2], [28, Theorem 5.6], [13, Corollary 2.7], [17, Theorem 6.3.9(ii)], [7, Theorem 2.3], [15, Corollary 3.2], [8, Corollary 2.3] and [6, Lemma 2.2] concerning the finiteness of associated primes of local cohomology modules. We just state the weakest possible conditions which yield the finiteness of associated primes of local cohomology modules in the next corollary. ###### Corollary 2.5. Suppose that $X$ is an $R$–module and $n$ is a non-negative integer such that * (i) $\emph{\mbox{Ext}\,}^{n}_{R}(R/\mathfrak{a},X)$ is weakly Laskerian, and * (ii) $\emph{\mbox{Ext}\,}^{n+1-i}_{R}(R/\mathfrak{a},H^{i}_{\mathfrak{a}}(X))$ is weakly Laskerian for all $i$, $0\leq i<n.$ Then $\emph{\mbox{Hom}\,}_{R}(R/\mathfrak{a},H^{n}_{\mathfrak{a}}(X))$ is weakly Laskerian, and so $\emph{\mbox{Ass}\,}_{R}(H^{n}_{\mathfrak{a}}(X))$ is finite. ###### Proof. Apply Corollary 2.4 with $N=R/\mathfrak{a}$ and $\mathcal{S}=\mathcal{C}_{w.l}(R)$, and note that we have the equality $\mbox{Ass}\,_{R}(\mbox{Hom}\,_{R}(R/\mathfrak{a},H^{n}_{\mathfrak{a}}(X)))=V(\mathfrak{a})\cap\mbox{Ass}\,_{R}(H^{n}_{\mathfrak{a}}(X))=\mbox{Ass}\,_{R}(H^{n}_{\mathfrak{a}}(X))$. ∎ It is easy to see that if $R$ is a local ring and $\mathcal{S}$ is a non-zero Serre subcategory of the category of $R$–modules, then every $R$–module with finite length belongs to $\mathcal{S}$. ###### Corollary 2.6. (cf. [5, Theorem 2.12]) Let $R$ be a local ring with maximal ideal $\mathfrak{m}$ and $X$ be an $R$–module. Assume also that $\mathcal{S}$ is a non-zero Serre subcategory of $\mathcal{C}(R)$ and $n$ is a non-negative integer such that * (i) $\emph{\mbox{Ext}\,}^{n}_{R}(R/\mathfrak{m},X)$ is finite, and * (ii) $\emph{\mbox{Ext}\,}^{n+1-i}_{R}(R/\mathfrak{m},H^{i}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ for all $i$, $0\leq i<n.$ Then $\emph{\mbox{Hom}\,}_{R}(R/\mathfrak{m},H^{n}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$. ###### Proof. Since $\mathcal{S}\neq 0$, $\mbox{Ext}\,^{n}_{R}(R/\mathfrak{m},X)$ is in $\mathcal{S}$. Now, the assertion follows from Corollary 2.4. ∎ Khashayarmanesh, in [24, Theorem 3.3], by using the concept of $\mathfrak{a}$–filter regular sequence, proved the following corollary with stronger assumptions. His assumptions were $X$ is a finite $R$–module with finite Krull dimension and $N=R/\mathfrak{b},$ where $\mathfrak{b}$ is an ideal of $R$ contains $\mathfrak{a}$, while it is a simple conclusion of Theorem 2.3 for an arbitrary $R$–module $X$ and a finite $\mathfrak{a}$–torsion module $N.$ ###### Corollary 2.7. (cf. [24, Theorem 3.3]) Suppose that $X$ is an $R$–module and $s,t$ are non- negative integers such that * (i) $\emph{\mbox{Ext}\,}^{s+t}_{R}(N,X)$ is finite, * (ii) $\emph{\mbox{Ext}\,}^{s+t+1-i}_{R}(N,H^{i}_{\mathfrak{a}}(X))$ is finite for all $i$, $0\leq i<t,$ and * (iii) $\emph{\mbox{Ext}\,}^{s+t-1-i}_{R}(N,H^{i}_{\mathfrak{a}}(X))$ is finite for all $i$, $t+1\leq i<s+t.$ Then $\emph{\mbox{Ext}\,}^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))$ is finite. ###### Proof. Apply Theorem 2.3 for $\mathcal{S}=\mathcal{C}_{f.g}(R)$. ∎ Theorem 2.1 in conjunction with Theorem 2.3 arise the following corollary. ###### Corollary 2.8. Let $X$ be an $R$–module and $n,m$ be non-negative integers such that $n\leq m$. Assume also that $H^{i}_{\mathfrak{a}}(X)$ is in $\mathcal{S}$ for all $i$, $i\neq n$ (resp. $0\leq i\leq n-1$ or $n+1\leq i\leq m$). Then, for all $i$, $i\geq 0$ (resp. $0\leq i\leq m-n$), $\emph{\mbox{Ext}\,}^{i}_{R}(N,H^{n}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ if and only if $\emph{\mbox{Ext}\,}^{i+n}_{R}(N,X)$ is in $\mathcal{S}$. In the course of the remaining parts of the paper by $\mbox{cd}\,_{\mathcal{S}}(\mathfrak{a},X)$ ($\mathcal{S}$–cohomological dimension of $X$ with respect to $\mathfrak{a}$) we mean the largest integer $i$ in which $H^{i}_{\mathfrak{a}}(X)$ is not in $\mathcal{S}$ (see [5, Definition 3.4] or [1, Definition 3.5]). Note that when $\mathcal{S}=0$, then $\mbox{cd}\,_{\mathcal{S}}(\mathfrak{a},X)=\mbox{cd}\,(\mathfrak{a},X)$ as in [20]. ###### Corollary 2.9. Let $X$ be an $R$–module and $n$ be a non-negative integer. Then the following statements hold true. 1. _(i)_ If $\emph{\mbox{cd}\,}_{\mathcal{S}}(\mathfrak{a},X)=0$, then $\emph{\mbox{Ext}\,}_{R}^{n}(N,\Gamma_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ if and only if $\emph{\mbox{Ext}\,}_{R}^{n}(N,X)$ is in $\mathcal{S}$. 2. _(ii)_ If $\emph{\mbox{cd}\,}_{\mathcal{S}}(\mathfrak{a},X)=1$, then $\emph{\mbox{Ext}\,}_{R}^{n}(N,H^{1}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ if and only if $\emph{\mbox{Ext}\,}_{R}^{n+1}(N,X/\Gamma_{\mathfrak{a}}(X))$ is in $\mathcal{S}$. 3. _(iii)_ If $\emph{\mbox{cd}\,}_{\mathcal{S}}(\mathfrak{a},X)=2$, then $\emph{\mbox{Ext}\,}_{R}^{n}(N,H^{2}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ if and only if $\emph{\mbox{Ext}\,}_{R}^{n+2}(N,D_{\mathfrak{a}}(X))$ is in $\mathcal{S}$. ###### Proof. (i) This is clear from Corollary 2.8. (ii) For all $i\neq 1$, $H^{i}_{\mathfrak{a}}(X/\Gamma_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ by assumption. Now, the assertion follows from Corollary 2.8. (iii) By [10, Corollary 2.2.8], $H^{i}_{\mathfrak{a}}(D_{\mathfrak{a}}(X))$ is in $\mathcal{S}$ for all $i\neq 2$. Again, use Corollary 2.8. ∎ ## 3\. Special Serre subcategories In this section, we study the extension functors of local cohomology modules in some special Serre subcategories of the category of $R$–modules. We begin with a definition. ###### Definition 3.1. (see [1, Definition 2.1]) Let $\mathcal{M}$ be a Serre subcategory of the category of $R$–modules. We say that $\mathcal{M}$ is a Melkersson subcategory with respect to the ideal $\mathfrak{a}$ if for any $\mathfrak{a}$–torsion $R$–module $X$, $0:_{X}\mathfrak{a}$ is in $\mathcal{M}$ implies that $X$ is in $\mathcal{M}$. $\mathcal{M}$ is called Melkersson subcategory when it is a Melkersson subcategory with respect to all ideals of $R$. In honor of Melkersson who proved this property for Artinian category (see [10, Theorem 7.1.2]) and Artinian $\mathfrak{a}$–cofinite category (see [27, Proposition 4.1]), we named the above subcategory as Melkersson subcategory. To see some examples of Melkersson subcategories, we refer the reader to [1, Examples 2.4 and 2.5]. The next two propositions show that how properties of Melkersson subcategories behave similarly at the initial points of Ext and local cohomology modules. These propositions give new proofs for [1, Theorems 2.9 and 2.13] based on Theorems 2.1 and 2.3. ###### Proposition 3.2. (see [1, Theorem 2.13]) Let $X$ be an $R$–module, $\mathcal{M}$ be a Melkersson subcategory with respect to the ideal $\mathfrak{a}$, and $n$ be a non-negative integer such that $\emph{\mbox{Ext}\,}^{j-i}_{R}(R/\mathfrak{a},H^{i}_{\mathfrak{a}}(X))$ is in $\mathcal{M}$ for all $i,j$ with $0\leq i\leq n-1$ and $j=n,n+1.$ Then the following statements are equivalent. * (i) $\emph{\mbox{Ext}\,}^{n}_{R}(R/\mathfrak{a},X)$ is in $\mathcal{M}$. * (ii) $H^{n}_{\mathfrak{a}}(X)$ is in $\mathcal{M}$. ###### Proof. (i) $\Rightarrow$ (ii). Apply Theorem 2.3 with $s=0$ and $t=n$. It shows that $\mbox{Hom}\,_{R}(R/\mathfrak{a},H^{n}_{\mathfrak{a}}(X))$ is in $\mathcal{M}$. Thus $H^{n}_{\mathfrak{a}}(X)$ is in $\mathcal{M}$. (ii) $\Rightarrow$ (i). Apply Theorem 2.1 with $t=n.$ ∎ ###### Proposition 3.3. (see [1, Theorem 2.9]) Let $X$ be an $R$–module, $\mathcal{M}$ be a Melkersson subcategory with respect to the ideal $\mathfrak{a}$, and $n$ be a non- negative integer. Then the following statements are equivalent. * (i) $H^{i}_{\mathfrak{a}}(X)$ is in $\mathcal{M}$ for all $i$, $0\leq i\leq n$. * (ii) $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},X)$ is in $\mathcal{M}$ for all $i$, $0\leq i\leq n$. ###### Proof. (i) $\Rightarrow$ (ii). Let $0\leq t\leq n.$ Since $H^{r}_{\mathfrak{a}}(X)$ is in $\mathcal{M}$ for all $r$, $0\leq r\leq t,$ $\mbox{Ext}\,^{t-r}_{R}(R/\mathfrak{a},H^{r}_{\mathfrak{a}}(X))$ is in $\mathcal{M}$ for all $r$, $0\leq r\leq t.$ Hence $\mbox{Ext}\,^{t}_{R}(R/\mathfrak{a},X)$ is in $\mathcal{M}$ by Theorem 2.1. (ii) $\Rightarrow$ (i). We prove by using induction on $n$. Let $n=0$ and consider the isomorphism $\mbox{Hom}\,_{R}(R/\mathfrak{a},X)\cong\mbox{Hom}\,_{R}(R/\mathfrak{a},\Gamma_{\mathfrak{a}}(X)).$ Since $\mbox{Hom}\,_{R}(R/\mathfrak{a},X)$ is in $\mathcal{M}$, $\mbox{Hom}\,_{R}(R/\mathfrak{a},\Gamma_{\mathfrak{a}}(X))$ is in $\mathcal{M}$. Thus $\Gamma_{\mathfrak{a}}(X)$ is in $\mathcal{M}.$ Now, suppose that $n>0$ and that $n-1$ is settled. Since $\mbox{Ext}\,^{i}_{R}(R/\mathfrak{a},X)$ is in $\mathcal{M}$ for all $i$, $0\leq i\leq n-1$, $H^{i}_{\mathfrak{a}}(X)$ is in $\mathcal{M}$ for all $i$, $0\leq i\leq n-1$ by the induction hypothesis. Now, by the above proposition, $H^{n}_{\mathfrak{a}}(X)$ is in $\mathcal{M}$. ∎ In the next proposition, we study the membership of the local cohomology modules of an $R$–module $X$ with respect to different ideals in Melkersson subcategories which, among other things, shows that $\mbox{cd}\,_{\mathcal{M}}(\mathfrak{b},X)\leq\mbox{cd}\,_{\mathcal{M}}(\mathfrak{a},X)+\mbox{ara}(\mathfrak{b}/\mathfrak{a})$ where $\mathcal{M}$ is a Melkersson subcategory of $\mathcal{C}(R)$ and $\mathfrak{b}$ is an ideal of $R$ contains $\mathfrak{a}$. ###### Proposition 3.4. Let $X$ be an $R$–module and $\mathfrak{b}$ be an ideal of $R$ such that $\mathfrak{a}\subseteq\mathfrak{b}$. Assume also that $\mathcal{M}$ is a Melkersson subcategory of $\mathcal{C}(R)$ and $n$ is a non-negative integer such that $H^{i}_{\mathfrak{a}}(X)$ is in $\mathcal{M}$ for all $i$, $0\leq i\leq n$ _(_ resp. $i\geq n$_)_. Then $H^{i}_{\mathfrak{b}}(X)$ is in $\mathcal{M}$ for all $i$, $0\leq i\leq n$ _(_ resp. $i\geq n+\emph{\mbox{ara}}(\mathfrak{b}/\mathfrak{a})$_)_. ###### Proof. Let $r=\mbox{ara}(\mathfrak{b}/\mathfrak{a})$. There exist $x_{1},...,x_{r}\in R$ such that $\sqrt{\mathfrak{b}}=\sqrt{\mathfrak{a}+(x_{1},...,x_{r})}$. We can, and do, assume that $\mathfrak{b}=\mathfrak{a}+\mathfrak{c}$ where $\mathfrak{c}=(x_{1},...,x_{r})$. By [29, Theorem 11.38], there is a Grothendieck spectral sequence $E^{p,q}_{2}:=H^{p}_{\mathfrak{c}}(H^{q}_{\mathfrak{a}}(X))_{\stackrel{{\scriptstyle\Longrightarrow}}{{p}}}H^{p+q}_{\mathfrak{b}}(X).$ Assume that $t$ is a non-negative integer such that $0\leq t\leq n$ (resp. $t\geq n+r$). For all $i$, $0\leq i\leq t$, $E_{\infty}^{t-i,i}=E_{t+2}^{t-i,i}$ since $E_{j}^{t-i-j,i+j-1}=0=E_{j}^{t-i+j,i-j+1}$ for all $j\geq t+2$. Therefore $E_{\infty}^{t-i,i}$ is in $\mathcal{M}$ from the fact that $E_{t+2}^{t-i,i}$ is a subquotient of $E_{2}^{t-i,i}=H^{t-i}_{\mathfrak{c}}(H^{i}_{\mathfrak{a}}(X))$ which belongs to $\mathcal{M}$ by assumption and Proposition 3.3. There exists a finite filtration $0=\phi^{t+1}H^{t}\subseteq\phi^{t}H^{t}\subseteq\cdots\subseteq\phi^{1}H^{t}\subseteq\phi^{0}H^{t}=H^{t}_{\mathfrak{b}}(X)$ such that $E_{\infty}^{t-i,i}=\phi^{t-i}H^{t}/\phi^{t-i+1}H^{t}$ for all $i$, $0\leq i\leq t.$ Now the exact sequences $0\longrightarrow\phi^{t-i+1}H^{t}\longrightarrow\phi^{t-i}H^{t}\longrightarrow E_{\infty}^{t-i,i}\longrightarrow 0,$ for all $i$, $0\leq i\leq t$, show that $H^{t}_{\mathfrak{b}}(X)$ is in $\mathcal{M}$. ∎ Let $\mathfrak{a}$ be an ideal of $R$, $N$ a finite $\mathfrak{a}$–torsion module and $s,t$ non-negative integers. In the following theorem, we find some sufficient conditions for validity of the isomorphism $\mbox{Ext}\,^{s+t}_{R}(N,X)\cong\mbox{Ext}\,^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))$ which concerns to the case $\mathcal{S}=0$. ###### Theorem 3.5. Let $X$ be an $R$–module and $s,t$ be non-negative integers such that * (i) $\emph{\mbox{Ext}\,}^{s+t-i}_{R}(N,H^{i}_{\mathfrak{a}}(X))=0$ for all $i$, $0\leq i<t$ or $t<i\leq s+t$, * (ii) $\emph{\mbox{Ext}\,}^{s+t+1-i}_{R}(N,H^{i}_{\mathfrak{a}}(X))=0$ for all $i$, $0\leq i<t,$ and * (iii) $\emph{\mbox{Ext}\,}^{s+t-1-i}_{R}(N,H^{i}_{\mathfrak{a}}(X))=0$ for all $i$, $t+1\leq i<s+t.$ Then we have $\emph{\mbox{Ext}\,}^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))\cong\emph{\mbox{Ext}\,}^{s+t}_{R}(N,X)$. ###### Proof. Consider the Grothendieck spectral sequence $E^{p,q}_{2}:=\mbox{Ext}\,^{p}_{R}(N,H^{q}_{\mathfrak{a}}(X))_{\stackrel{{\scriptstyle\Longrightarrow}}{{p}}}\mbox{Ext}\,^{p+q}_{R}(N,X)$ and, for all $r\geq 2$, the exact sequences $0\rightarrow B_{r}^{s,t}\rightarrow Z_{r}^{s,t}\rightarrow E_{r+1}^{s,t}\rightarrow 0\textmd{ \ and \ }0\rightarrow Z_{r}^{s,t}\rightarrow E_{r}^{s,t}\rightarrow E_{r}^{s,t}/Z_{r}^{s,t}\rightarrow 0$ as we used in Theorem 2.3. Since $E_{2}^{s+r,t+1-r}=0=E_{2}^{s-r,t+r-1}$, $E_{r}^{s+r,t+1-r}=0=E_{r}^{s-r,t+r-1}$. Therefore $E_{r}^{s,t}/Z_{r}^{s,t}=0=B_{r}^{s,t}$ which shows that $E^{s,t}_{r}=E^{s,t}_{r+1}$. Hence we have $\mbox{Ext}\,^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))=E^{s,t}_{2}=E^{s,t}_{3}=\cdots=E^{s,t}_{s+t+1}=E^{s,t}_{s+t+2}=E^{s,t}_{\infty}.$ There is a finite filtration $0=\phi^{s+t+1}H^{s+t}\subseteq\phi^{s+t}H^{s+t}\subseteq\cdots\subseteq\phi^{1}H^{s+t}\subseteq\phi^{0}H^{s+t}=\mbox{Ext}\,^{s+t}_{R}(N,X)$ such that $E^{s+t-j,j}_{\infty}=\phi^{s+t-j}H^{s+t}/\phi^{s+t-j+1}H^{s+t}$ for all $j$, $0\leq j\leq s+t.$ Note that for each $j$, $0\leq j\leq t-1$ or $t+1\leq j\leq s+t$, by assumption (i), we have $E^{s+t-j,j}_{\infty}=0$. Therefore we get $0=\phi^{s+t+1}H^{s+t}=\phi^{s+t}H^{s+t}=\cdots=\phi^{s+2}H^{s+t}=\phi^{s+1}H^{s+t}$ and $\phi^{s}H^{s+t}=\phi^{s-1}H^{s+t}=\cdots=\phi^{1}H^{s+t}=\phi^{0}H^{s+t}=\mbox{Ext}\,^{s+t}_{R}(N,X).$ Thus $\mbox{Ext}\,^{s}_{R}(N,H^{t}_{\mathfrak{a}}(X))=E^{s,t}_{\infty}=\phi^{s}H^{s+t}/\phi^{s+1}H^{s+t}=\mbox{Ext}\,^{s+t}_{R}(N,X)$ as desired. ∎ The following corollaries are immediate applications of the above theorem which give us some useful isomorphisms and equalities about the extension functors and Bass numbers of local cohomology modules, respectively. ###### Corollary 3.6. (cf. [2, Corollary 4.2.(c)]) Let $X$ be an $R$–module and $n$ be a non- negative integer. Then the isomorphism $\emph{\mbox{Hom}\,}_{R}(N,H^{n}_{\mathfrak{a}}(X))\cong\emph{\mbox{Ext}\,}^{n}_{R}(N,X)$ holds in either of the following cases: * (i) $\emph{\mbox{Ext}\,}^{j-i}_{R}(N,H^{i}_{\mathfrak{a}}(X))=0$ for all $i,j$ with $0\leq i\leq n-1$ and $j=n,n+1;$ * (ii) $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},X)=0$ for all $i$, $0\leq i\leq n-1.$ ###### Proof. (i) Apply Theorem 3.5 with $s=0$ and $t=n$. (ii) By Proposition 3.3, $H^{i}_{\mathfrak{a}}(X)=0$ for all $i$, $0\leq i\leq n-1$. Now, use case (i). ∎ ###### Corollary 3.7. Let $X$ be an $R$–module and $n,m$ be non-negative integers such that $n\leq m$. Assume also that $H^{i}_{\mathfrak{a}}(X)=0$ for all $i$, $i\neq n$ (resp. $0\leq i\leq n-1$ or $n+1\leq i\leq m$). Then we have $\emph{\mbox{Ext}\,}^{i}_{R}(N,H^{n}_{\mathfrak{a}}(X))\cong\emph{\mbox{Ext}\,}^{i+n}_{R}(N,X)$ for all $i$, $i\geq 0$ (resp. $0\leq i\leq m-n$). ###### Proof. For all $i$, $i\geq 0$ (resp. $0\leq i\leq m-n$), apply Theorem 3.5 with $s=i$ and $t=n$. ∎ ###### Corollary 3.8. (cf. [18, Proposition 3.1]) Let $X$ be an $R$–module and $n$ be a non-negative integer such that $H^{i}_{\mathfrak{a}}(X)=0$ for all $i$, $i\neq n.$ Then we have $\mu^{i}(\mathfrak{p},H^{n}_{\mathfrak{a}}(X))=\mu^{i+n}(\mathfrak{p},X)$ for all $i\geq 0$ and all $\mathfrak{p}\in V(\mathfrak{a}).$ ###### Proof. Let $\mathfrak{p}\in V(\mathfrak{a}).$ By assumption, $H^{i}_{\mathfrak{a}R_{\mathfrak{p}}}(X_{\mathfrak{p}})=0$ for all $i$, $i\neq n$; so that $\mbox{Ext}\,^{i}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}},H^{n}_{\mathfrak{a}R_{\mathfrak{p}}}(X_{\mathfrak{p}}))\cong\mbox{Ext}\,^{i+n}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}},X_{\mathfrak{p}})$ for all $i\geq 0$ by Corollary 3.7. Hence $\mu^{i}(\mathfrak{p},H^{n}_{\mathfrak{a}}(X))=\mu^{i+n}(\mathfrak{p},X)$ for all $i\geq 0$. ∎ ###### Corollary 3.9. For an arbitrary $R$–module $X$, the following statements hold true. 1. _(i)_ If $\emph{\mbox{cd}\,}(\mathfrak{a},X)=0$, then $\emph{\mbox{Ext}\,}^{i}_{R}(N,\Gamma_{\mathfrak{a}}(X))\cong\emph{\mbox{Ext}\,}^{i}_{R}(N,X)$ for all $i\geq 0$. 2. _(ii)_ If $\emph{\mbox{cd}\,}(\mathfrak{a},X)=1$, then $\emph{\mbox{Ext}\,}^{i}_{R}(N,H^{1}_{\mathfrak{a}}(X))\cong\emph{\mbox{Ext}\,}^{i+1}_{R}(N,X/\Gamma_{\mathfrak{a}}(X))$ for all $i\geq 0$. 3. _(iii)_ If $\emph{\mbox{cd}\,}(\mathfrak{a},X)=2$, then $\emph{\mbox{Ext}\,}^{i}_{R}(N,H^{2}_{\mathfrak{a}}(X))\cong\emph{\mbox{Ext}\,}^{i+2}_{R}(N,D_{\mathfrak{a}}(X))$ for all $i\geq 0$. ###### Proof. By considering Corollary 3.7, this is similar to that of Corollary 2.9. ∎ ## 4\. Cofinite modules We first introduce the class of cofinite modules with respect to an ideal and a Serre subcategory of the category of $R$-modules. ###### Definition 4.1. Let $\mathfrak{a}$ be an ideal of $R$, $X$ be an $R$–module and $\mathcal{S}$ be a Serre subcategory of $\mathcal{C}(R)$. We say that $X$ is $\mathcal{S}$–cofinite with respect to the ideal $\mathfrak{a}$ if $\emph{\mbox{Supp}\,}_{R}(X)\subseteq V(\mathfrak{a})$ and $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},X)$ is in $\mathcal{S}$ for all $i\geq 0.$ We will denote this concept by $(\mathcal{S},\mathfrak{a})$–cofinite. Note that when $\mathcal{S}$ is $\mathcal{C}_{f.g}(R)$ (resp. $\mathcal{C}_{w.l}(R)$), $X$ is $(\mathcal{S},\mathfrak{a})$–cofinite exactly when $X$ is $\mathfrak{a}$–cofinite (resp. $\mathfrak{a}$–weakly cofinite). ###### Theorem 4.2. Let $X$ be an $R$–module and $n$ be a non-negative integer such that $H^{i}_{\mathfrak{a}}(X)$ is $(\mathcal{S},\mathfrak{a})$–cofinite for all $i$, $i\neq n$. Then the following statements are equivalent. * (i) $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},X)$ is in $\mathcal{S}$ for all $i\geq 0$. * (ii) $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},X)$ is in $\mathcal{S}$ for all $i\geq n$. * (iii) $H^{n}_{\mathfrak{a}}(X)$ is $(\mathcal{S},\mathfrak{a})$–cofinite. ###### Proof. (i) $\Rightarrow$ (ii). This is clear. (ii) $\Rightarrow$ (iii). For all $i\geq 0$, apply Theorem 2.3 with $N=R/\mathfrak{a}$, $s=i$ and $t=n$. (iii) $\Rightarrow$ (i). Apply Theorem 2.1 with $N=R/\mathfrak{a}$. ∎ As an immediate result, the following corollary recovers and improves [26, Proposition 2.5], [27, Proposition 3.11] and [14, Theorem 3.1]. ###### Corollary 4.3. (cf. [26, Proposition 2.5], [27, Proposition 3.11] and [14, Theorem 3.1]) Let $X$ be an $R$–module and $n$ be a non-negative integer such that $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$–cofinite (resp. $\mathfrak{a}$–weakly cofinite) for all $i$, $i\neq n$. Then the following statements are equivalent. * (i) $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},X)$ is finite (resp. weakly Laskerian) for all $i\geq 0$. * (ii) $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},X)$ is finite (resp. weakly Laskerian) for all $i\geq n$. * (iii) $H^{n}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$–cofinite (resp. $\mathfrak{a}$–weakly cofinite). ###### Theorem 4.4. Suppose that $X$ is an $R$–module and $n$ is a non-negative integer such that * (i) $H^{i}_{\mathfrak{a}}(X)$ is $(\mathcal{S},\mathfrak{a})$–cofinite for all $i$, $0\leq i\leq n-1$, and * (ii) $\emph{\mbox{Ext}\,}^{1+n}_{R}(N,X)$ is in $\mathcal{S}$. Then $\emph{\mbox{Ext}\,}^{1}_{R}(N,H^{n}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$. ###### Proof. Consider [22, Proposition 3.4] and apply Theorem 2.3 with $s=1$ and $t=n$. ∎ The following result is an application of the above theorem. ###### Corollary 4.5. (cf. [16, Theorem A] and [13, Corollary 2.7]) Let $X$ be an $R$–module and $n$ be a non-negative integer. Assume also that * (i) $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$–cofinite (resp. $\mathfrak{a}$–weakly cofinite) for all $i$, $0\leq i\leq n-1$, and * (ii) $\emph{\mbox{Ext}\,}^{1+n}_{R}(N,X)$ is finite (resp. weakly Laskerian). Then $\emph{\mbox{Ext}\,}^{1}_{R}(N,H^{n}_{\mathfrak{a}}(X))$ is finite (resp. weakly Laskerian). ###### Theorem 4.6. Let $X$ be an $R$–module and $n$ be a non-negative integer such that $\emph{\mbox{Ext}\,}^{n+1}_{R}(N,X)$ and $\emph{\mbox{Ext}\,}^{n+2}_{R}(N,X)$ are in $\mathcal{S}$, and $H^{i}_{\mathfrak{a}}(X)$ is $(\mathcal{S},\mathfrak{a})$–cofinite for all $i$, $0\leq i<n.$ Then the following statements are equivalent. * (i) $\emph{\mbox{Hom}\,}_{R}(N,H^{n+1}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$. * (ii) $\emph{\mbox{Ext}\,}^{2}_{R}(N,H^{n}_{\mathfrak{a}}(X))$ is in $\mathcal{S}$. ###### Proof. (i) $\Rightarrow$ (ii). Consider [22, Proposition 3.4] and apply Theorem 2.3 with $s=2$ and $t=n$. (ii) $\Rightarrow$ (i). Again consider [22, Proposition 3.4] and apply Theorem 2.3 with $s=0$ and $t=n+1$. ∎ Asadollahi and Schenzel proved that over local ring $(R,\mathfrak{m})$, if $X$ is a Cohen-Macaulay $R$-module and $t=\mbox{grade}\,(\mathfrak{a},X)$ then $\mbox{Hom}\,_{R}(R/\mathfrak{a},H^{t+1}_{\mathfrak{a}}(X))$ is finite if and only if $\mbox{Ext}\,^{2}_{R}(R/\mathfrak{a},H^{t}_{\mathfrak{a}}(X))$ is finite (see [4, Theorem 1.2]). Dibaei and Yassemi, in [16], generalized this result with weaker assumptions on $R$ and $X$. As an immediate consequence of Theorem 4.6, the following is a generalization of [16, Theorem B]. ###### Corollary 4.7. (cf. [16, Theorem B]) Let $X$ be an $R$–module and $n$ be a non-negative integer. Assume also that $\emph{\mbox{Ext}\,}^{n+1}_{R}(N,X)$ and $\emph{\mbox{Ext}\,}^{n+2}_{R}(N,X)$ are finite (resp. weakly Laskerian), and $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$–cofinite (resp. $\mathfrak{a}$–weakly cofinite) for all $i$, $0\leq i<n.$ Then the following statements are equivalent. * (i) $\emph{\mbox{Hom}\,}_{R}(N,H^{n+1}_{\mathfrak{a}}(X))$ is finite (resp. weakly Laskerian). * (ii) $\emph{\mbox{Ext}\,}^{2}_{R}(N,H^{n}_{\mathfrak{a}}(X))$ is finite (resp. weakly Laskerian). In [12, Proposition 2], Delfino and Marley proved the Change of ring principle for cofiniteness. In the following theorem, we prove it for Serre cofiniteness. The proof is an adaption of the proof of [12, Proposition 2]. ###### Theorem 4.8. Let $\phi:A\longrightarrow B$ be a homomorphism between Noetherian rings such that $B$ is a finite $A$–module, $\mathfrak{a}$ be an ideal of $A$ and $X$ be a $B$–module. Let $\mathcal{S}$ and $\mathcal{T}$ be Serre subcategories of $\mathcal{C}(A)$ and $\mathcal{C}(B)$, respectively. Assume also that for any $B$–module $Y$, $Y$ is in $\mathcal{T}$ exactly when $Y$ is in $\mathcal{S}$ _(_ as an $A$–module _)_. Then $X$ is $(\mathcal{T},\mathfrak{a}B)$–cofinite if and only if $X$ is $(\mathcal{S},\mathfrak{a})$–cofinite _(_ as an $A$–module _)_. ###### Proof. By [29, Theorem 11.65], there is a Grothendieck spectral sequence $E^{p,q}_{2}:=\mbox{Ext}\,^{p}_{B}(\mbox{Tor}\,^{A}_{q}(B,A/\mathfrak{a}),X)_{\stackrel{{\scriptstyle\Longrightarrow}}{{p}}}\mbox{Ext}\,^{p+q}_{A}(A/\mathfrak{a},X).$ $(\Rightarrow)$. For all $p$ and $q$, by [22, Proposition 3.4], $E^{p,q}_{2}$ is in $\mathcal{S}$. Therefore $E^{p,q}_{\infty}$ belongs to $\mathcal{S}$ since $E^{p,q}_{\infty}=E^{p,q}_{p+q+2}$ and $E^{p,q}_{p+q+2}$ is a subquotient of $E^{p,q}_{2}$. Let $n$ be a non-negative integer. There exists a finite filtration $0=\phi^{n+1}H^{n}\subseteq\phi^{n}H^{n}\subseteq\cdots\subseteq\phi^{1}H^{n}\subseteq\phi^{0}H^{n}=\mbox{Ext}\,^{n}_{A}(A/\mathfrak{a},X)$ such that $E_{\infty}^{n-i,i}=\phi^{n-i}H^{n}/\phi^{n-i+1}H^{n}$ for all $i$, $0\leq i\leq n$. Now, by the exact sequences $0\longrightarrow\phi^{n-i+1}H^{n}\longrightarrow\phi^{n-i}H^{n}\longrightarrow E_{\infty}^{n-i,i}\longrightarrow 0,$ for all $i$, $0\leq i\leq n$, $\mbox{Ext}\,^{n}_{A}(A/\mathfrak{a},X)$ is in $\mathcal{S}$. $(\Leftarrow)$. By using induction on $n$, we show that $E^{n,0}_{2}=\mbox{Ext}\,^{n}_{B}(B/\mathfrak{a}B,X)$ is in $\mathcal{T}$ for all $n\geq 0$. The case $n=0$ is clear from the isomorphism $\mbox{Hom}\,_{B}(B/\mathfrak{a}B,X)\cong\mbox{Hom}\,_{A}(A/\mathfrak{a},X)$. Assume that $n>0$ and that $E^{p,0}_{2}$ is in $\mathcal{T}$ for all $p$, $0\leq p\leq n-1$. For all $r\geq 2$, we have $E^{n,0}_{r+1}\cong E^{n,0}_{r}/\mbox{Im}\,(E^{n-r,r-1}_{r}\longrightarrow E^{n,0}_{r})$. Thus $E^{n,0}_{r}$ is in $\mathcal{T}$ whenever $E^{n,0}_{r+1}$ is in $\mathcal{T}$ because $E^{n-r,r-1}_{r}$ is in $\mathcal{T}$ by the induction hypotheses and [22, Proposition 3.4]. Since $E^{n,0}_{\infty}=E^{n,0}_{n+2}$, to complete the proof it is enough to show that $E^{n,0}_{\infty}$ is in $\mathcal{T}$. By assumption, $\mbox{Ext}\,^{n}_{A}(A/\mathfrak{a},X)$ is in $\mathcal{T}$ and hence $\phi^{n}H^{n}$ is in $\mathcal{T}$. That is $E^{n,0}_{\infty}$ belongs to $\mathcal{T}$ as desired. ∎ ###### Definition 4.9. (see [30]) The $R$–module $X$ is a minimax module if it has a finite submodule $X^{\prime}$ such that $X/X^{\prime}$ is Artinian. The class of minimax modules thus includes all finite and all Artinian modules. Note that the category of minimax modules and the category of $\mathfrak{a}$–cofinite minimax modules are two Serre subcategories of the category of $R$–modules (see [27, Corollary 4.4]). ###### Proposition 4.10. Let $X$ be an $R$–module and $n,m$ be non-negative integers such that $n\leq m$. Assume also that * (i) $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$–cofinite for all $i$, $0\leq i\leq n-1$, * (ii) $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},X)$ is finite for all $i$, $n\leq i\leq m$, and * (iii) $H^{i}_{\mathfrak{a}}(X)$ is minimax for all $i$, $n\leq i\leq m$. Then $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$–cofinite for all $i$, $0\leq i\leq m$. ###### Proof. Apply Theorem 2.3 with $s=0$ and $t=n$ for $N=R/\mathfrak{a}$ and $\mathcal{S}=\mathcal{C}_{f.g}(R).$ It shows that $\mbox{Hom}\,_{R}(R/\mathfrak{a},H^{n}_{\mathfrak{a}}(X))$ is finite. Thus $H^{n}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$–cofinite from [27, Proposition 4.3]. ∎ ###### Corollary 4.11. (cf. [6, Theorem 2.3]) Let $X$ be an $R$–module and $n$ be a non-negative integer such that * (i) $H^{i}_{\mathfrak{a}}(X)$ is minimax for all $i$, $0\leq i\leq n-1$, and * (ii) $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},X)$ is finite for all $i$, $0\leq i\leq n.$ Then $\emph{\mbox{Hom}\,}_{R}(R/\mathfrak{a},H^{n}_{\mathfrak{a}}(X))$ is finite. ###### Proof. By [27, Proposition 4.3], $\Gamma_{\mathfrak{a}}(X)$ is $\mathfrak{a}$–cofinite. Hence $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$–cofinite for all $i$, $0\leq i\leq n-1$, from Proposition 4.10. Thus, by Theorem 2.3, $\mbox{Hom}\,_{R}(R/\mathfrak{a},H^{n}_{\mathfrak{a}}(X))$ is finite. ∎ ###### Corollary 4.12. Suppose that $X$ is an $R$–module and that $n$ is a non-negative integer. Then the following statements are equivalent. * (i) $H^{i}_{\mathfrak{a}}(X)$ is Artinian $\mathfrak{a}$–cofinite for all $i$, $0\leq i\leq n$. * (ii) $\emph{\mbox{Ext}\,}^{i}_{R}(R/\mathfrak{a},X)$ has finite length for all $i$, $0\leq i\leq n.$ ###### Proof. (i) $\Rightarrow$ (ii). Let $0\leq t\leq n.$ Since $\mbox{Ext}\,^{t-i}_{R}(R/\mathfrak{a},H^{i}_{\mathfrak{a}}(X))$ has finite length for all $i$, $0\leq i\leq t,$ $\mbox{Ext}\,^{t}_{R}(R/\mathfrak{a},X)$ has also finite length by Theorem 2.1. (ii) $\Rightarrow$ (i). By Proposition 3.3, $H^{i}_{\mathfrak{a}}(X)$ is Artinian for all $i$, $0\leq i\leq n.$ Let $0\leq t\leq n$ and consider Corollary 4.11. It shows that $\mbox{Hom}\,_{R}(R/\mathfrak{a},H^{t}_{\mathfrak{a}}(X))$ is finite and so has finite length. Now, the assertion follows from [27, Proposition 4.3]. ∎ ## References * [1] M. Aghapournahr, L. Melkersson, Local cohomology and Serre subcategories, J. Algebra, 320 (2008), 1275–1287. * [2] M. Aghapournahr, L. Melkersson, A natural map in local cohomology, To appear in Ark. Mat. * [3] J. Asadollahi, K. Khashyarmanesh, Sh. Salarian, A generalization of the cofiniteness problem in local cohomology, J. Aust. Math. Soc., 75 (2003), 313–324. * [4] J. Asadollahi and P. Schenzel, Some results on associated primes of local cohomology modules, Japan. J. Math., 29 (2003), 285–296. * [5] M. Asgharzadeh, M. Tousi, A unified approach to local cohomology modules using serre classes, arXiv: 0712.3875v2 [math.AC]. * [6] K. Bahmanpour, R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc., 136 (2008), 2359–2363. * [7] K. Borna Lorestani, P. Sahandi, T. Sharif, A note on the associated primes of local cohomology modules, Comm. Algebra, 34 (2006), 3409–3412. * [8] K. Borna Lorestani, P. Sahandi, S. Yassemi, Artinian local cohomology, Can. Math. Bull., 50 (2007), 598–602. * [9] M. P. Brodmann, A. Lashgari, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128(10) (2000), 2851–2853. * [10] M .P. Brodmann, R. Y. Sharp, Local cohomology : an algebraic introduction with geometric applications, Cambridge University Press, 1998. * [11] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge University Press, revised ed., 1998. * [12] D. Delfino, T. Marley, Cofinite modules of local cohomology, J. Pure Appl. Algebra, 121(1) (1997), 45–52. * [13] K. Divaani-Aazar, A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc., 133 (2005), 655–660. * [14] K. Divaani-Aazar, A. Mafi, Associated primes of local cohomology modules of weakly Laskerian modules, Comm. Algebra, 34 (2006), 681–690. * [15] M. T. Dibaei, A. Nazari, Graded local cohomology : attached and associated primes, asymptotic behaviors, Comm. Algebra, 35 (2007), 1567–1576. * [16] M. T. Dibaei, S. Yassemi, Finiteness of extention functors of local cohomology modules. Comm. Algebra, 34 (2006), 3097–3101. * [17] M. T. Dibaei, S. Yassemi, Associated primes of the local cohomology modules, Abelian groups, rings, modules, and homological algebra, 49–56, Chapman and Hall/CRC, 2006. * [18] M. T. Dibaei, S. Yassemi, Bass numbers of local cohomology modules with respect to an ideal, Algebr Represent Theor., 11 (2008), 299–306. * [19] A. Grothendieck, Cohomologie locale des faisceaux coh$\acute{e}$rents et th$\acute{e}$or$\grave{e}$mes de Lefschetz locaux et globaux (SGA 2), North-Holland, Amsterdam, 1968. * [20] R. Hartshorne, Cohomological dimension of algeraic varieties, Ann. of Math., 88 (1968), 403–450. * [21] R. Hartshorne, Affine duality and cofiniteness, Invent. Math., 9 (1970), 145–164. * [22] S. H. Hasanzadeh, A. Vahidi, On vanishing and cofinitness of generalized local cohomology modules, to appear in Comm. Algebra. * [23] C. Huneke, Problems on local cohomology : Free resolutions in commutative algebra and algebraic geometry, (Sundance, UT, 1990), 93–108, Jones and Bartlett, 1992. * [24] K. Khashyarmanesh, On the finiteness properties of extention and torsion functors of local cohomology modules, Proc. Amer. Math. Soc., 135 (2007), 1319–1327. * [25] K. Khashyarmanesh, Sh. Salarian, On the associated primes of local cohomology modules, Comm. Algebra, 27 (1999), 6191–6198. * [26] T. Marley and J. C. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra, 256(1) (2002), 180–193. * [27] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra, 285 (2005), 649–668. * [28] L. T. Nhan, On generalized regular sequences and the finiteness for associated primes of local cohomology modules, Comm. Algebra, 33 (2005), 793–806. * [29] J. Rotman, An introduction to homological algebra, Academic Press, 1979. * [30] H. Zöschinger, Minimax Moduln, J. Algebra, 102 (1986), 1–32.	arxiv-papers	2009-03-12T18:23:40	2024-09-04T02:49:01.083831	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. Aghapournahr, A. J. Taherizadeh, A. Vahidi", "submitter": "Moharram Aghapournahr", "url": "https://arxiv.org/abs/0903.2093" }
0903.2097	# Anomalous Expansion of the Copper-Apical Oxygen Distance in Superconducting La2CuO4 $-$ La1.55Sr0.45CuO4 Bilayers Hua Zhou1 Yizhak Yacoby2 Ron Pindak1 pindak@bnl.gov Vladimir Butko1 Gennady Logvenov1 Ivan Bozovic1 1Brookhaven National Laboratory, Upton, NY 11973 USA 2Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel ###### Abstract We have introduced an improved X-ray phase-retrieval method with unprecedented speed of convergence and precision, and used it to determine with sub-Ångstrom resolution the complete atomic structure of an ultrathin superconducting bilayer film, composed of La1.55Sr0.45CuO4 and La2CuO4 neither of which is superconducting by itself. The results show that phase-retrieval diffraction techniques enable accurate measurement of structural modifications in near- surface layers, which may be critically important for elucidation of surface- sensitive experiments. Specifically we find that close to the sample surface the unit cell size remains constant while the copper-apical oxygen distance shows a dramatic increase, by as much as 0.45 Å. The apical oxygen displacement is known to have a profound effect on the superconducting transition temperature. ###### pacs: 68.35.Ct, 61.05.Cm, 74.78.Fk, 74.72.Dn ††preprint: prepared for Arxiv Preprint The exciting discovery of interface superconductivity in complex oxides Reyren et al. (2007); Bozovic et al. (2002); Gozar et al. (2008); Ueno et al. (2008); Yuli et al. (2008) has triggered intense debate about its origin and the possibility to enhance Tc even further Kivelson (2002); Yunoki et al. (2007); Okamoto and Maier (2008); Berg et al. (2008). The interfacial enhancement Gozar et al. (2008) of the superconducting critical temperature (Tc) is influenced by the crystal structure. The Z-axis lattice constant (c0) varies significantly among La2CuO4 (LCO), La1.55Sr0.45CuO4 (LSCO), and bilayer LSCO/LCO films, depending even on the deposition sequence, and it affects superconductivity: Tc scales with c0 almost perfectly linearly Butko et al. (2008). The reason for this is not understood at present, but notice that in (La,Sr)2CuO4 the change in c0 goes together with the change in cA, the distance between copper and the nearest apical oxygen, which some believe to play a key role in the high temperature superconductivity (HTS) phenomenon Fernandes et al. (1991); Ohta et al. (1991); Feiner et al. (1992); Lubrittoy et al. (1996); Pavarini et al. (2001); Slezaka et al. (2008). In any case, it is certain that (i) from one cuprate to another, cA varies more than any other bond length, and (ii) at least in simple cuprates with a single CuO2 layer in the unit cell it correlates with the maximal Tc \- the longer cA, the higher Tc. At least, the first fact can be understood: cA is ’soft’ because apical oxygen has no hard contact with the nearest copper ion; rather, it ”levitates” on the electrostatic potential - a structural feature peculiar to certain layered oxides with alternating ionic planes of opposite charge Radovic et al. (2008). This makes apical oxygen prone to very large displacements - e.g., in HgBa2CuO6 one finds cA $\approx$ 2.8 Å, longer by 0.9 Å than the in-plane Cu-O bond; coincidentally, this compound has the highest Tc = 97 K among all single-CuO2-layer cuprates Wagner et al. (1993). It is thus important to find out what happens to the apical oxygen in LSCO-LCO bilayers; however, standard X-ray diffraction (XRD) is not well suited for this $-$ one needs to ”get inside the unit cell” and look for individual atomic displacements. For this, the most suitable technique is the Coherent Bragg Rod Analysis (COBRA) method Yacoby et al. (2002); Sowwan et al. (2002); Willmott et al. (2007); Yacoby et al. (2008); Cionca et al. (2008); Fong et al. (2005). However, COBRA is most effective for films that are just a few unit cells (UCs) thick, and in the case of HTS compounds fabrication of ultrathin films with bulk properties has been proven to be extremely challenging. Fortunately, we had a technical solution at hand $-$ a unique atomic layer-by-layer molecular beam epitaxy (ALL-MBE) system with proven capability of fabricating ultrathin HTS layers Bozovic (2001); Bozovic et al. (2003). For this study we have synthesized by ALL-MBE a number of (n $\times$ LSCO + m $\times$ LCO) bilayers, where (n,m) determine the thickness of the respective layers expressed as the number of UCs. In this paper, we show the COBRA results for two of these, (2.5,2.5) and (2,3). The films were deposited at T = 650∘C and p = 9 $\times$ 10-6 Torr of ozone and subsequently cooled down under high vacuum to drive out all the interstitial oxygen. We used 10 $\times$ 10 $\times$ 1 mm 3 single-crystal LaSrAlO4 (LSAO) substrates polished with the large surface perpendicular to the (001) direction. The substrate lattice constants are a0 = b0 = 3.755 Å, c0 = 12.56 Å; the films are pseudomorphic with LSAO and under compressive strain. The crystal structure of LCO is illustrated in Fig. 1a. Atomic force microscopy scans over a large (10 $\times$ 10 $\mu$m2) area showed root-mean-square surface roughness of 0.25 nm in the (2,3) and 0.11 nm in the (2.5,2.5) bilayer sample; this is significantly less than the 1 UC step height which in LSCO is 1.32 nm. Magnetic susceptibility was measured via two-coil mutual inductance technique and revealed sharp superconducting transitions at Tc = 34 K in the (2,3), and Tc = 36 K in the (2.5,2.5) bilayer, significantly higher than the values reported for (n,m) bilayers in Ref. [3], which is remarkable given that these films are only 5 UC thick. This was also confirmed by measuring the electric resistance (see Fig. 1b) after the X-ray scattering experiments were completed and gold pads were evaporated to enable four-point-contact measurements. Figure 1: (Color online) A simplified structure model (one-half the crystallographic unit cell) of La2CuO4 and the transport property for a bilayer film. (a) At room temperature, the structure is tetragonal and the space group is I4/mmm. Noted that the La(Sr)-O layers are strongly corrugated, exaggerated in this sketch for clarity. (b) The electric resistance of the (2.5,2.5) bilayer, measured by the four-point-contact technique, as a function of temperature. Inset: a schematic of the bilayer on a LSAO substrate. The atomic structure of the LSCO/LCO bilayer film was investigated at beamline ID-33 of the Advanced Photon Source by measuring the diffraction intensities along the substrate-defined Bragg rods. The sample and a PILATUS 100k photon- counting pixel detector Schlepütz et al. (2005) were mounted on a six-circle goniometer in Kappa geometry. The experimental set-up and procedures were described in detail in previous works Sowwan et al. (2002); Yacoby et al. (2008). Ten symmetry inequivalent Bragg rods were recorded with a maximum value for the vertical reciprocal space coordinate of Lmax = 10.5 r.l.u. (reciprocal lattice units) and a sampling density of 50 points per r.l.u.. The X-ray flux after the Si (1,1,1) monochromator crystal was 3$\times 10^{12}$ photons/sec at a wavelength of $\lambda$ = 0.8266 Å. For all Bragg rod measurements, except for the (0,0,L) rod, the angle of incidence had a fixed value of 3.5∘. The X-ray beam was focused to 0.1 mm (V) $\times$ 0.2 mm (H), resulting in a 2 mm long X-ray footprint. The background and diffuse X-ray scattering contribution were removed efficiently and accurately using the PILATUS detector images. The final results were then normalized by taking into account the beam polarization and Lorentz factors. The results were subsequently analyzed using the COBRA method Yacoby et al. (2002); Willmott et al. (2007); Yacoby et al. (2008); Cionca et al. (2008); Fong et al. (2005). In general, COBRA uses the measured diffraction intensities and the fact that the complex structure factors (CSFs) vary continuously along the substrate-defined Bragg rods to determine the diffraction phases and the CSFs. The CSFs are then Fourier transformed into real space to obtain the 3-dimensional electron density of the film and the substrate with sub-Ångstrom resolution. Figure 2: (Color online) Representative Bragg rods of LSCO/LCO system (open diamond) and calculated diffraction intensity obtained from COBRA-determined electron density (solid line). The experimental data of three representative Bragg rods are shown in Fig. 2(a). Notice that the diffraction intensity along the Bragg rods (excluding the Bragg peaks) vary over more than 4 orders of magnitude with excellent signal-to-noise ratio. The reference structures chosen as the starting point for the COBRA analysis were the bulk LSAO structure and the tetragonal LSCO/LCO bilayer with the nominal layer atomic positions. In our numerical simulations, the topmost 4 UCs of the substrate were allowed to deform, however the resulting deformations turned out to be very small. The COBRA method uses the approximation that at two adjacent points along the Bragg rod the change in CSFs contributed by the unknown part of electron density is negligible compared to the change in CSFs contributed from the reference structure Sowwan et al. (2002). The use of this approximation allows COBRA to converge very quickly to approximately the right solution but not to the exact one. To overcome this limitation we further refined the CSFs using the Difference-Map algorithm introduced by Elser Elser (2003) and recently applied to thin films Björck et al. (2008). Using the COBRA solution as the starting point for the Difference-Map algorithm and using a proper filter program that takes advantage of the fact that the CSFs vary continuously along the Bragg rods, the Difference-Map algorithm converges after about 20 iterations; the convergence accelerates by about two orders of magnitude. As seen in Fig. 2, the final calculated and measured intensities are in very good agreement. Similar agreement was found for all other Bragg rods and the overall X-ray reliability factor $R=\frac{\Sigma\|\|F_{0}\|-\|F_{c}\|\|}{\Sigma\|F_{0}\|}=0.02$; here, F0 and Fc are the observed and the calculated diffraction amplitudes, respectively. To the best of our knowledge, so far there has been only one attempt to determine the structure of a thin film using the Difference Map method Björck et al. (2008). In that study, the atomicity constraint was imposed and over 2,000 iterations were needed to achieve convergence. Our analysis shows that the combined COBRA/Difference Map method combines the best features of both methods and ensures rapid convergence to the correct solution without the need to use the atomicity constraint. Figure 3: (Color online) The electron density variation, determined by COBRA, along the [0, 0, Z] column of atoms as indicated by the dashed line in Figure 1a. Note that the topmost four unit cells of the substrate are included in the structure refinement. The left and right dashed lines represent the nominal LSAO/LSCO and LSCO/LCO interfaces, respectively. The CSFs obtained have been Fourier transformed into real space yielding the 3D electron density (ED). As an example, we show in Fig. 3 the ED of a (2,3) bilayer sample along the [0,0,Z] line that goes through the La(Sr), O, and Cu(Al) atoms. Two points should be stressed. First, as seen the ED has almost no negative parts. Together with the excellent agreement between the calculated and measured diffraction intensities, this suggests that the ED is very close to the correct one. Second, all the atoms including the oxygens can be clearly identified and their positions determined with sub-Ångstrom resolution. The small ED intensity fluctuation below -53 Å provides a measure of the inaccuracy in the ED and as seen it is small even compared to the oxygen ED. The atomic positions in the Z direction were accurately determined by fitting a Gaussian to each peak. We determined the size of the unit cell in the Z direction by measuring the distance between consecutive pairs of La(Sr) and Cu atoms. The results are shown in Fig. 4 inset. Each point corresponds to an average of 4 La-La distances and 2 Cu-Cu distances. The measured lattice constant of the bilayer film is 13.304 $\pm$ 0.016 Å and is larger by 0.148 Å than that of the bulk LCO (c0 = 13.156 Å). Figure 4: (Color online) Evolution of the inter-atomic distances. (a) The measured Cu(Al)$-$ apical O distance, cA, varies as a function of the nominal position of Cu(Al) atoms inside the refined structure. The data from two representative bilayer samples and the average over the two are presented. The lower and upper arrows represent the bulk values of cA for LSAO and for LCO, respectively. (b) The comparison of cA, c1, and c2, averaged for each unit cell, as a function of Z position. Inset: the lattice constant c0 as a function of Z. The dotted line represents the bulk LSAO value. The horizontal dashed line is the average value of c0 in bilayers extracted from the electron density, as described in the text. In both (a), (b) and the inset, the vertical dashed lines represent the nominal LSAO/LSCO interfaces, respectively. While the changes observed in the unit cell sizes are as expected from the strain and the elastic parameters Ledbetter et al. (1992), the variations in Cu-apical O and La-apical O distances are quite unexpected. The distances cA,c1 and c2 are defined in Fig. 1a; the distances labeled c${}_{A}^{{}^{\prime}}$,c${}_{1}^{{}^{\prime}}$ and c${}_{2}^{{}^{\prime}}$ would be their symmetry equivalents in bulk samples, but in thin films they could differ in principle. For our samples, the measured values for the primed and unprimed distances were in fact equal within the experimental error, except at the LSAO/LSCO interface. The measured values averaged over cA and c${}_{A}^{{}^{\prime}}$ are shown in Fig. 4a. The diamond and circular dots represent the distances measured in (2,3) and (2.5,2.5) bilayers, respectively. The triangular dots are averages over the two samples. Every pair of triangular dots corresponds to one UC. The dashed vertical lines represent the nominal LSAO/LSCO interfaces. The arrows on the right indicate cA as measured in the bulk samples. The results show that, within the experimental error, the values of cA in the substrate are equal to those in the bulk but they are very different in the film. In both LSAO and LCO bulk crystals Radovic et al. (2008), cA is equal to 2.41 Å. In the metal layer closest to the substrate cA = 2.3 Å, and it then rises steadily all the way to cA = 2.75 Å $-$ a change of 0.45 Å. In Fig. 4b we display cA as well as the La-apical O distance, c2, and the La- CuO2 plane distance, c1. Each point represents an average over the two bulk- symmetry-equivalent distances and over the two measured samples. Notice that c1 changes by less than 0.1 Å. On the other hand, cA increases by about 0.45 Å, while c2 decreases by about 0.25 Å. Close to the film surface, the apical oxygen atoms are displaced away from the nearest Cu atoms. The La atoms are displaced towards the closest CuO2 plane, but by a smaller amount, while the separation between two adjacent CuO2 planes remains constant. According to Ref. 15, cA = 2.7 Å should correspond to a Tc of 80 K at the optimum doping. However, from Ref. 31 we know that the hole density drops sharply on the I side of the interface and the screening length is equal to 6 $\pm$ 2 Å. This implies that on the I side and next to the M-I interface only one or two CuO2 layers are doped via carrier accumulation while the others remain insulating. Thus, unfortunately, we have a mismatch: in the optimally- doped LCO layer, cA is close to its standard (bulk) value, while it is greatly elongated only in insulating layers. It is tempting to speculate that one could create LSCO-based samples with Tc much higher than 36 K, perhaps as high as 80-90 K, if only one could achieve cA elongation and optimal doping in the same LCO layer. An obvious avenue for further research is to try making I layers even thinner, thus bringing the interface superconductivity closer to the film surface. Another is to try engineering more sophisticated hetero- structures and superlattices combining LCO with other metallic oxides (nickelates, zincates, etc.). In summary, we have used ALL-MBE to synthesize precise ultrathin bilayers using metallic but non-superconducting LSCO and insulating LCO blocks, and observed interface superconductivity with Tc = 34-36 K, significantly higher than before. We have used synchrotron X-ray diffraction and the combined COBRA/Difference-Map phase-retrieval method to determine accurately the atomic structure and found the unit cell size to be constant despite dramatic atomic displacements within the cell. In particular, near the surface the Cu$-$apical O increases greatly, by as much as 0.45 Å, while it is known that variations in the apical oxygen position strongly affect Tc. We conclude that in cuprates the crystal structure can be modified in near-surface layers, and in such a way that superconductivity properties can be dramatically altered. This result amplifies the importance of high quality surface structure determination in conjunction with surface sensitive probes of electronic states such as scanning tunneling microscopy or angle-resolved photoemission spectroscopy. This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Project MA-509-MACA, Contract No. DE- AC02-98CH10886, and use of the Advanced Photon Source under Contract No. DE- AC02-06CH11357. One of us (Y. Yacoby) would like to acknowledge with thanks fruitful discussions with M. Björck. ## References * Reyren et al. (2007) N. Reyren et al., Science 317, 1196 (2007). * Bozovic et al. (2002) I. Bozovic et al., Phys. Rev. Lett. 89, 107001 (2002). * Gozar et al. (2008) A. Gozar et al., Nature 455, 782 (2008). * Ueno et al. (2008) K. Ueno et al., Nat. Mater. 7, 855 (2008). * Yuli et al. (2008) O. Yuli et al., Phys. Rev. Lett. 101, 057005 (2008). * Kivelson (2002) S. A. Kivelson, Physica B 318, 61 (2002). * Yunoki et al. (2007) S. Yunoki et al., Phys. Rev. B 76, 064532 (2007). * Okamoto and Maier (2008) S. Okamoto and T. A. Maier, Phys. Rev. Lett. 101, 156401 (2008). * Berg et al. (2008) E. Berg, D. Orgad, and S. A. Kivelson, Phys. Rev. B 78, 094509 (2008). * Butko et al. (2008) V. Butko et al., preprint (2008). * Fernandes et al. (1991) A. A. R. Fernandes et al., Phys. Rev. B 44, 7601 (1991). * Ohta et al. (1991) Y. Ohta, T. Tohyama, and S. Maekawa, Phys. Rev. B 43, 2968 (1991). * Feiner et al. (1992) L. F. Feiner, M. Grilli, and C. D. Castro, Phys. Rev. B 45, 10647 (1992). * Lubrittoy et al. (1996) C. Lubrittoy, K. Rosciszewski, and A. M. Olesz, J. Phys.: Condens. Matter 8, 11053 (1996). * Pavarini et al. (2001) E. Pavarini et al., Phys. Rev. Lett. 87, 047003 (2001). * Slezaka et al. (2008) J. A. Slezaka et al., Proc. Nat. Acad. Sci. 105, 3203 (2008). * Radovic et al. (2008) Z. Radovic, N. Bozovic, and I. Bozovic, Phys. Rev. B 77, 092508 (2008). * Wagner et al. (1993) J. L. Wagner et al., Physica C 210, 447 (1993). * Yacoby et al. (2002) Y. Yacoby et al., Nat. Mater. 1, 99 (2002). * Sowwan et al. (2002) M. Sowwan et al., Phys. Rev. B 66, 205311 (2002). * Willmott et al. (2007) P. R. Willmott et al., Phys. Rev. Lett. 99, 155502 (2007). * Yacoby et al. (2008) Y. Yacoby et al., Phys. Rev. B 77, 195426 (2008). * Cionca et al. (2008) C. N. Cionca et al., Appl. Phys. Lett. 92, 151914 (2008). * Fong et al. (2005) D. D. Fong et al., Phys. Rev. B 71, 144112 (2005). * Bozovic (2001) I. Bozovic, IEEE Trans. Appl. Supercond. 11, 2686 (2001). * Bozovic et al. (2003) I. Bozovic et al., Nature 422 422, 873 (2003). * Schlepütz et al. (2005) C. M. Schlepütz et al., Acta Crystallogr. Sect. A 61, 418 (2005). * Elser (2003) V. Elser, Acta Crystallogr. Sect. A 59, 201 (2003). * Björck et al. (2008) M. Björck et al., J. Phys.: Condens. Matter 20, 445006 (2008). * Ledbetter et al. (1992) H. Ledbetter, S. Kim, and A. Roshko, Z. Phys. B - Condensed Matter 89, 275 (1992). * Smadici et al. (2009) S. Smadici et al., Phys. Rev. Lett. 102, 107004 (2009).	arxiv-papers	2009-03-12T05:21:44	2024-09-04T02:49:01.088713	{ "license": "Public Domain", "authors": "Hua Zhou, Yizhak Yacoby, Ron Pindak, Vladimir Butko, Gennady Logvenov,\n and Ivan Bozovic", "submitter": "Hua Zhou", "url": "https://arxiv.org/abs/0903.2097" }
0903.2129	# Investigation of nodal domains in the chaotic microwave ray-splitting rough billiard Oleh Hul, Nazar Savytskyy, Oleg Tymoshchuk, Szymon Bauch and Leszek Sirko Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland (October 20, 2005) ###### Abstract We study experimentally nodal domains of wave functions (electric field distributions) lying in the regime of Shnirelman ergodicity in the chaotic microwave half-circular ray-splitting rough billiard. For this aim the wave functions $\Psi_{N}$ of the billiard were measured up to the level number $N=415$. We show that in the regime of Shnirelman ergodicity ($N>208$) wave functions of the chaotic half-circular microwave ray-splitting rough billiard are extended over the whole energy surface and the amplitude distributions are Gaussian. For such ergodic wave functions the dependence of the number of nodal domains $\aleph_{N}$ on the level number $N$ was found. We show that in the limit $N\rightarrow\infty$ the least squares fit of the experimental data yields $\aleph_{N}/N\simeq 0.063\pm 0.023$ that is close to the theoretical prediction $\aleph_{N}/N\simeq 0.062$. We demonstrate that for higher level numbers $N\simeq 215-415$ the variance of the mean number of nodal domains $\sigma^{2}_{N}/N$ is scattered around the theoretical limit $\sigma^{2}_{N}/N\simeq 0.05$. We also found that the distribution of the areas $s$ of nodal domains has power behavior $n_{s}\propto s^{-\tau}$, where the scaling exponent is equal to $\tau=2.14\pm 0.12$. This result is in a good agreement with the prediction of percolation theory. ###### pacs: 05.45.Mt,05.45.Df In recent theoretical papers by Bogomolny and Schmit Bogomolny2002 and Blum et al. Blum2002 the distributions of the nodal domains of real wave functions $\Psi(x,y)$ in 2D quantum systems (billiards) have been considered. Nodal domains are regions where a wave function $\Psi(x,y)$ has a definite sign. The condition $\Psi(x,y)=0$ determines a set of nodal lines which separate nodal domains. Bogomolny and Schmit Bogomolny2002 have proposed a very fruitful, percolationlike, model for description of properties of the nodal domains of generic chaotic system. Using this model they have shown that the distribution of nodal domains of quantum wave functions of chaotic systems is universal. Blum et al. Blum2002 have shown that the systems with integrable and chaotic underlying classical dynamics can be distinguished by different distributions of the number of nodal domains. In this way they provided a new criterion of quantum chaos, which is not directly related to spectral statistics. Theoretical findings of Bogomolny and Schmit Bogomolny2002 and Blum et al. Blum2002 have been recently tested in the experiment with the microwave half- circular rough billiard by Savytskyy et al. Savytskyy2004 . In this paper we present the first experimental investigation of nodal domains of wave functions of the chaotic microwave ray-splitting rough billiard. Ray- splitting systems are a new class of chaotic systems in which the underlying classical mechanics is non-Newtonian and non-deterministic BLUM96 ; SIR97 ; BLUMEL2001 . In ray-splitting systems a wave which encounters a discontinuity in the propagation medium splits into two or more rays travelling usually away from the discontinuity. Ray splitting occurs in many fields of physics, whenever the wave length is large in comparison with the range over which the potential changes. Ideal model systems for the investigation of ray-splitting phenomena are ray-splitting billiards COUC92 ; BLUMEL2001 and microwave cavities with dielectric inserts SIR97 ; BAUCH98 ; HLUSH2000 ; Savytskyy2001 . Measurements of wave functions of ray-splitting systems are very demanding because in principle they require the direct access to the all parts of the system Stoeckmann2001 including those filled with ray-splitting media, such as dielectric in the case of ray-splitting microwave billiards. This is one of the main reasons for which only low wave functions ($N\leq 100$) of ray- splitting billiards have been measured so far Stoeckmann2001 . In this paper we use a new method of the reconstruction of wave functions introduced by Savytskyy and Sirko Savytskyy2002 which in the case of the half-circular microwave ray-splitting rough billiard allowed for the reconstruction of wave functions with the level numbers $N\leq 415$. Figure 1: Sketch of the chaotic half-circular microwave ray-splitting rough billiard which consists a half-circular Teflon insert of radius $R_{d}=8.465$ cm. Dimensions are given in cm. The cavity sidewalls are marked by 1 and 2 (see text). Squared wave functions $\|\Psi_{N}(R_{c},\theta)\|^{2}$ were evaluated on a half-circle of fixed radius $R_{c}=19.25$ cm. Billiard’s rough boundary is marked by $\Gamma$. In the experiment we used the thin (height $h=8$ mm) aluminium cavity in the shape of a rough half-circle (Fig. 1) which consisted a half-circular Teflon insert of radius $R_{d}=8.465$ cm. The insert had the same height as the rough cavity. The microwave cavity simulates the rough ray-splitting quantum billiard due to the equivalence between the Schrödinger equation and the Helmholtz equation BLUMEL2001 . This equivalence remains valid for frequencies less than the cut-off frequency $\nu_{c}=c/2\eta h\simeq 13.1$ GHz, where c is the speed of light and $\eta=1.425$ is the index of refraction of the Teflon insert. The cavity sidewalls were made of two segments. The rough segment 1 is described by the radius function $R(\theta)=R_{0}+\sum_{m=2}^{M}{a_{m}\sin(m\theta+\phi_{m})}$, where the mean radius $R_{0}$=20.0 cm, $M=20$, $a_{m}$ and $\phi_{m}$ are uniformly distributed on [0.084,0.091] cm and [0,2$\pi$], respectively, and $0\leq\theta<{\pi}$. It is important to note that we used a rough half- circular cavity instead of a rough circular cavity because in this way we avoided nearly degenerate low-level eigenvalues Hlushchuk01b ; Hlushchuk01 . Additionally, a half-circular geometry of the cavity was necessary for the accurate measurements of the electric field distributions inside the billiard. According to Frahm97 the roughness of a billiard may be characterized by the function $k(\theta)=(dR/d\theta)/R_{0}$. The roughness parameter $\tilde{k}$ defined as the angle average of the function $k(\theta)$ was for our billiard $\tilde{k}=(\left<k^{2}(\theta)\right>_{\theta})^{1/2}\simeq 0.200$. In such a billiard the dynamics is diffusive in orbital momentum due to collisions with the rough boundary because the roughness parameter $\tilde{k}$ is much larger the chaos border parameter $k_{c}=M^{-5/2}=0.00056$ Frahm97 . The roughness parameter $\tilde{k}$ determines also other properties of the billiard Frahm . The eigenstates are localized for the level number $N<N_{e}=1/128\tilde{k}^{4}=5$. The border of Breit-Wigner regime is given by $N_{W}=M^{2}/48\tilde{k}^{2}\simeq 208$. It means that between $N_{e}<N<N_{W}$ Wigner ergodicity Frahm ought to be observed and for $N>N_{W}$ Shnirelman ergodicity should emerge. In the regime of Shnirelman ergodicity wave functions have to be uniformly spread out in the billiard Shnirelman . In this paper we focus our attention on Shnirelman ergodicity regime. It is worth noting that rough billiards and related systems are of considerable interest elsewhere, e.g. in the context of microdisc lasers Yamamoto ; Stone , light scattering in optical fibers Doya2002 , ballistic electron transport in microstructures Blanter , dynamic localization Sirko00 and localization in discontinuous quantum systems Borgonovi . In order to measure the wave functions (electric field distributions inside the microwave billiard), which are indispensable in investigation of nodal domains, we used a new, very effective method described in Savytskyy2002 . It is based on the perturbation technique and construction of the “trial functions”. Following Savytskyy2002 we will show that the wave functions $\Psi_{N}(r,\theta)$ (electric field distribution $E_{N}(r,\theta)$ inside the cavity) of the billiard can be determined from the form of electric field $E_{N}(R_{c},\theta)$ evaluated on a half-circle of fixed radius $R_{c}$ (see Fig. 1). The first step in evaluation of $E_{N}(R_{c},\theta)$ is measurement of $\|E_{N}(R_{c},\theta)\|^{2}$. For this purpose the perturbation technique developed in Slater52 and used successfully in Slater52 ; Sridhar91 ; Richter00 ; Anlage98 was applied. In this method a small perturber is introduced inside the cavity to alter its resonant frequency according to $\nu-\nu_{N}=\nu_{N}(aB_{N}^{2}-bE_{N}^{2}),$ $None$ where $\nu_{N}$ is the $N$th resonant frequency of the unperturbed cavity, $a$ and $b$ are geometrical factors. Equation (1) can be used to evaluate $E_{N}^{2}$ only when the term containing magnetic field $B_{N}$ is sufficiently small. In order to minimize the influence of $B_{N}$ on the frequency shift $\nu-\nu_{N}$ a small piece of a metallic pin (3.0 mm in length and 0.25 mm in diameter) was used as a perturber. The perturber was attached to the micro filament line hidden in the groove (0.4 mm wide, 1.0 mm deep) made in the cavity’s bottom wall along the half-circle $R_{c}$ and moved by the stepper motor. Application of such a small pin perturber reduced the largest positive frequency shifts to the uncertainty of frequency shift measurements (15 kHz). It was verified that the presence of the narrow groove in the bottom wall of the cavity caused only very small changes $\delta\nu_{N}$ of the eigenfrequencies $\nu_{N}$ of the cavity $\|\delta\nu_{N}\|/\nu_{N}\leq 10^{-4}$. Therefore, its influence into the structure of the cavity’s wave functions was also negligible. A big advantage of using the perturber that was attached to the line, was connected with the fact that the perturber was always vertically positioned, which is crucial in the measurements of the square of electric field $E_{N}$. The influence of the thermal expansion of the Teflon insert and the aluminium cavity into its resonant frequencies was eliminated by stabilization of the temperature of the cavity with the accuracy of $0.05^{\circ}$. Figure 2: Panel (a): Squared wave function $\|\Psi_{415}(R_{c},\theta)\|^{2}$ (in arbitrary units) measured on a half-circle with radius $R_{c}=19.25$ cm ($\nu_{415}\simeq 12.98$ GHz). Panel (b): The “trial wave function” $\Psi_{415}(R_{c},\theta)$ (in arbitrary units) with the correctly assigned signs, which was used in the reconstruction of the wave function $\Psi_{415}(r,\theta)$ of the billiard (see Fig. 3). The regime of Shnirelman ergodicity for the experimental rough billiard is defined for $N>208$. Using a field perturbation technique we measured squared wave functions $\|\Psi_{N}(R_{c},\theta)\|^{2}$ for 30 modes within the region $215\leq N\leq 415$. The range of corresponding eigenfrequencies was from $\nu_{215}\simeq 9.42$ GHz to $\nu_{415}\simeq 12.98$ GHz. The measurements were performed at 0.36 mm steps along a half-circle with fixed radius $R_{c}=19.25$ cm. This step was small enough to reveal in details the space structure of high-lying levels. In Fig. 2 (a) we show the example of the squared wave function $\|\Psi_{N}(R_{c},\theta)\|^{2}$ evaluated for the level number $N=415$. The perturbation method used in our measurements allows us to extract information about the wave function amplitude $\|\Psi_{N}(R_{c},\theta)\|$ at any given point of the cavity but it doesn’t allow to determine the sign of $\Psi_{N}(R_{c},\theta)$ Stein95 . However, the determination of the sign of the wave function $\Psi_{N}(R_{c},\theta)$ is crucial in the procedure of the reconstruction of the full wave function $\Psi_{N}(r,\theta)$ of the billiard. The papers Savytskyy2002 ; Savytskyy2004 suggest the following sign-assignment strategy. First one should identify of all close to zero minima of $\|\Psi_{N}(R_{c},\theta)\|$. Then the sign “minus” is arbitrarily assigned to the region between the first and the second minimum, “plus” to the region between the second minimum and the third one and so on. In this way the “trial wave function” $\Psi_{N}(R_{c},\theta)$ is constructed. If the assignment of the signs is correct the wave function $\Psi_{N}(r,\theta)$ should be reconstructed inside the billiard with the boundary condition $\Psi_{N}(r_{\Gamma},\theta_{\Gamma})=0$. The wave function of a rough ray-splitting half-circular billiard outside of the half-circular Teflon insert ($r\geq R_{d}$) may be expanded in terms of Hankel functions $\Psi^{out}_{N}(r,\theta)=\sum_{s=1}^{L}a_{s}\Omega_{s}(k_{N}r)\sin(s\theta),$ $None$ where $\Omega_{s}(x)=Re\\{H^{(2)}_{s}(x)+S_{ss}(k_{N}R_{d})H^{(1)}_{s}(x)\\}$ and $k_{N}=2\pi\nu_{N}/c$. $H^{(1)}_{s}(x)$ and $H^{(2)}_{s}(x)$ are Hankel functions of the first and the second kind, respectively. The matrix $S_{ss^{\prime}}(k_{N}R_{d})$ is defined as follows Hentschel2002 $S_{ss^{\prime}}(k_{N}R_{d})=-\frac{H^{(2)^{\prime}}_{s}(k_{N}R_{d})-\eta[J^{\prime}_{s}(\eta k_{N}R_{d})/J_{s}(\eta k_{N}R_{d})]H^{(2)}_{s}(k_{N}R_{d})}{H^{(1)^{\prime}}_{s}(k_{N}R_{d})-\eta[J^{\prime}_{s}(\eta k_{N}R_{d})/J_{s}(\eta k_{N}R_{d})]H^{(1)}_{s}(k_{N}R_{d})}\delta_{ss^{\prime}},$ $None$ where the derivatives of Hankel and Bessel functions are marked by primes. In Eq. (2) the number of basis functions is limited to $L=k_{N}r_{max}+3$, where $r_{max}=20.7$ cm is the maximum radius of the cavity. $l_{N}^{max}=k_{N}r_{max}$ is a semiclassical estimate for the maximum possible angular momentum for a given $k_{N}$. The functions with angular momentum $s>l_{N}^{max}$ describe evanescent waves. We checked that the basis of $L$ wave functions was large enough to properly reconstruct billiard’s wave functions. The coefficients $a_{s}$ may be determined from the “trial wave functions” $\Psi_{N}(R_{c},\theta)$ via $a_{s}=[\frac{\pi}{2}\Omega_{s}(k_{N}R_{c})]^{-1}\int_{0}^{\pi}\Psi_{N}(R_{c},\theta)sin(s\theta)d\theta.$ $None$ The wave functions of the billiard inside the Teflon insert ($r\leq R_{d}$) may be expanded in terms of circular waves $\Psi^{in}_{N}(r,\theta)=\sum_{s=1}^{L^{\prime}}a^{{}^{\prime}}_{s}J_{s}(\eta k_{N}r)\sin(s\theta).$ $None$ In Eq. (5) the number of basis functions was limited to $L^{\prime}=\eta k_{N}R_{d}$. The coefficients $a_{s}$ given by Eq. (4) and the continuity condition fulfilled at the border of the dielectric insert $\Psi^{out}_{N}(R_{d},\theta)=\Psi^{in}_{N}(R_{d},\theta)$ may be used to evaluate the coefficients $a^{{}^{\prime}}_{s}$ in Eq. (5) allowing in this way to reconstruct the full wave function $\Psi_{N}(r,\theta)$ of the billiard. In the evaluation of the coefficients $a^{{}^{\prime}}_{s}$ in Eq. (5) an important role plays the value of the refraction index $n$ of the Teflon insert. We measured the refraction index $\eta=1.425\pm 0.002$ of Teflon by measuring the set of resonant frequencies of a microwave circular cavity of radius $R_{T}=3.25$ cm entirely filled by it. Figure 3: The reconstructed wave function $\Psi_{415}(r,\theta)$ of the chaotic half-circular microwave rough billiard. The amplitudes have been converted into a grey scale with white corresponding to large positive and black corresponding to large negative values, respectively. Dimensions of the billiard are given in cm. The position of the half-circular Teflon insert of radius $R_{d}=8.465$ cm is marked with a solid line. Using the method of the “trial wave function” we were able to reconstruct 30 experimental wave functions of the rough half-circular billiard with the level number $N$ between 215 and 415\. The wave functions were reconstructed on points of a square grid of side $4.2\cdot 10^{-4}$ m. As the quantitative measure of the sign assignment quality we chose the integral $\gamma\int_{\Gamma}\|\Psi_{N}(r,\theta)\|^{2}dl$ calculated along the billiard’s rough boundary $\Gamma$, where $\gamma$ is length of $\Gamma$. In Fig. 2 (b) we show the “trial wave function” $\Psi_{415}(R_{c},\theta)$ with the correctly assigned signs, which was used in the reconstruction of the wave function $\Psi_{415}(r,\theta)$ of the billiard (see Fig. 3). It is worth noting that inside of the Teflon insert the size of nodal domains are much smaller than outside of it. The remaining wave functions from the range $N=215-415$ were not reconstructed because of the accidental near-degeneration of the neighboring states or due to the problems with the measurements of $\|\Psi_{N}(R_{c},\theta)\|^{2}$ along a half-circle coinciding for its significant part with one or several of the nodal lines of $\Psi_{N}(r,\theta)$. The problem of the near-degenerated states is important because in the presence of the perturber the resonances are shifted, which may cause the initially non-overlapping states to become near-degenerated at certain positions of the perturber. Such a situation prevents us from the reconstruction of the wave functions. The problems mentioned are getting much more severe for $N>200$. Furthermore, the computation time $t_{r}$ required for reconstruction of the ”trial wave function” scales like $t_{r}\propto 2^{n_{z}-2}$, where $n_{z}$ is the number of identified zeros in the measured function $\|\Psi_{N}(R_{c},\theta)\|$. Figure 4: Structure of the energy surface in the regime of Shnirelman ergodicity. Here we show the moduli of amplitudes $\|C^{(N)}_{nl}\|$ for the wave functions: (a) $N=215$, (b) $N=415$. The wave functions are delocalized in the $n,l$ basis. Full lines show the energy surface (see text). The structure of the energy surface Frahm97 of the billiard’s wave functions plays an important role in the identification of their ergodicity. To check it we extracted wave function amplitudes $C^{(N)}_{nl}=\left<n,l\|N\right>$ in the basis $n,l$ of a half-circular ray-splitting billiard (desymmetrized annular ray-splitting billiard) Kohler1998 with radius $r_{max}$ and a half-circular Teflon insert of radius $R_{d}$ . The normalized eigenfunctions of the half- circular ray-splitting billiard are given by $\Phi_{nl}(r,\theta)=\left\\{\begin{array}[]{cc}A_{ln}J_{l}(\eta\kappa_{ln}r)\sin(l\theta),&0\leq r\leq R_{d},\\\ A_{ln}\left[C_{ln}J_{l}(\kappa_{ln}r)+D_{ln}Y_{l}(\kappa_{ln}r)\right]\sin(l\theta),&R_{d}\leq r\leq r_{max},\end{array}\right.$ $None$ where $A_{ln}=\left\\{\frac{\pi}{2}\left(\int_{0}^{R_{d}}rJ_{l}(\eta\kappa_{ln}r)^{2}dr+\int_{R_{d}}^{r_{max}}r\left[C_{ln}J_{l}(\kappa_{ln}r)+D_{ln}Y_{l}(\kappa_{ln}r)\right]^{2}dr\right)\right\\}^{-\frac{1}{2}}$. $J_{l}(\kappa_{ln}r)$ and $Y_{l}(\kappa_{ln}r)$ are Bessel and Neumann functions, respectively. The main quantum number $n=1,2,3\ldots$ enumerates the zeros $y_{ln}=\kappa_{ln}r_{max}$ of the radial function $C_{ln}J_{l}(y_{ln})+D_{ln}Y_{l}(y_{ln})=0,$ $None$ and $l=1,2,3\ldots$ is the angular momentum quantum number. The coefficients $C_{ln}$ and $D_{ln}$ can be determined from the continuity conditions of the wave function $\Phi_{nl}(r,\theta)$ and it’s derivative $\Phi_{nl}^{{}^{\prime}}(r,\theta)$ on Teflon’s boundary $R_{d}$ $\left\\{\begin{array}[]{cc}J_{l}(\eta\kappa_{ln}R_{d})=C_{ln}J_{l}(\kappa_{ln}R_{d})+D_{ln}Y_{l}(\kappa_{ln}R_{d}),\\\ \eta J_{l}^{{}^{\prime}}(\eta\kappa_{ln}R_{d})=C_{ln}J_{l}^{{}^{\prime}}(\kappa_{ln}R_{d})+D_{ln}Y_{l}^{{}^{\prime}}(\kappa_{ln}R_{d}).\end{array}\right.$ $None$ The moduli of amplitudes $\|C^{(N)}_{nl}\|$ and their projections into the energy surface for the representative experimental wave functions $N=215$ and $N=415$ are shown in Fig. 4. As expected, in the regime of Shnirelman ergodicity the wave functions are extended over the whole energy surface Hlushchuk01 . The full lines on the projection planes in Fig. 4(a) and Fig. 4(b) mark the energy surface of a half-circular annular ray-splitting billiard $H(n,l)\simeq E_{N}=k^{2}_{N}$ estimated from the formula $\|H(n,l)-E_{N}\|/E_{N}\leq 0.12$. The peaks $\|C^{(N)}_{nl}\|$ are spread almost perfectly along the lines marking the energy surface. Figure 5: Panel (a): The amplitude distribution $P(\Psi_{N}A^{1/2})$ for the wave function $N=215$. Panel (b): The distribution $P(\Psi_{N}A^{1/2})$ for the wave function $N=415$. The amplitude distributions were constructed as histograms with bin equal to 0.2. The width of the distribution $P(\Psi)$ was rescaled to unity by multiplying normalized to unity wave function by the factor $A^{1/2}$, where $A$ denotes billiard’s area. Full lines show standard normalized Gaussian prediction $P_{0}(\Psi A^{1/2})=(1/\sqrt{2\pi})e^{-\Psi^{2}A/2}$. Ergodic behavior of the measured wave functions can be also tested by evaluation of the amplitude distribution $P(\Psi_{N})$ Berry77 ; Kaufman88 . For irregular, chaotic states the probability of finding the value $\Psi_{N}$ at any point inside the billiard should be distributed as a Gaussian, $P(\Psi_{N})\sim e^{-\beta\Psi_{N}^{2}}$. In Fig. 5(a) we show the amplitude distribution $P(\Psi_{N}A^{1/2})$ for the wave function $N=215$ while in Fig. 5(b) the distribution $P(\Psi_{N}A^{1/2})$ for the wave function $N=415$ is presented. The distributions were constructed as normalized to unity histograms with the bin equal to 0.2. The width of the amplitude distributions $P(\Psi_{N})$ was rescaled to unity by multiplying normalized to unity wave functions by the factor $A^{1/2}$, where $A$ denotes billiard’s area (see formula (23) in Kaufman88 ). For all measured wave functions lying in the regime of Shnirelman ergodicity the distributions of $P(\Psi_{N}A^{1/2})$ were in good agreement with the standard normalized Gaussian prediction $P_{0}(\Psi A^{1/2})=(1/\sqrt{2\pi})e^{-\Psi^{2}A/2}$. Figure 6: The number of nodal domains $\aleph_{N}$ (full circles) for the chaotic half-circular microwave ray-splitting rough billiard. Full line shows the least squares fit $\aleph_{N}=a_{1}N+b_{1}\sqrt{N}$ to the experimental data (see text), where $a_{1}=0.063\pm 0.023$, $b_{1}=0.77\pm 0.40$. The prediction of the theory of Bogomolny and Schmit Bogomolny2002 $a_{1}=0.062$. The number of nodal domains $\aleph_{N}$ vs. the level number $N$ in the chaotic microwave ray-splitting rough billiard is plotted in Fig. 6. The full line in Fig. 6 shows the least squares fit $\aleph_{N}=a_{1}N+b_{1}\sqrt{N}$ of the experimental data, where $a_{1}=0.063\pm 0.023$, $b_{1}=0.77\pm 0.40$. The coefficient $a_{1}=0.063\pm 0.023$ coincides with the prediction of the percolation model of Bogomolny and Schmit Bogomolny2002 $\aleph_{N}/N\simeq 0.062$ within the error limits. The errors of the coefficients $a_{1}$ and $b_{1}$ are relatively high because the number of nodal domains fluctuates significantly in the function of the level number $N$, what was also demonstrated in Blum2002 (see Fig .(5)). It is worth mention that in the paper Savytskyy2004 the coefficient $a_{1}$ was estimated in the experiment with the microwave rough billiard without the ray-splitting Teflon insert. Its value $a_{1}=0.058\pm 0.006$ was also close to the theoretical prediction. The second term in the least squares fit corresponds to a contribution of boundary domains, i.e. domains that include the billiard boundary. Numerical calculations of Blum et al. Blum2002 performed for the Sinai and stadium billiards showed that the number of boundary domains scales as the number of the boundary intersections, that is as $\sqrt{N}$. Present results together with the results of Savytskyy2004 clearly suggest that in the rough billiards (with and without ray-splitting), at low level number $N$, the boundary domains also significantly influence the scaling of the number of nodal domains $\aleph_{N}$, leading to the departure from the predicted scaling $\aleph_{N}\sim N$. Figure 7: The variance of the mean number of nodal domains divided by the level number $\sigma^{2}_{N}/N$ for the chaotic half-circular microwave ray- splitting rough billiard. Full line shows predicted by the theory limit $\sigma^{2}_{N}/N\simeq 0.05$, Bogomolny and Schmit Bogomolny2002 . Measured wave functions of the ray-splitting billiard may be also used for the calculations of the variance $\sigma^{2}_{N}$ of the mean number of nodal domains. It was predicted in Bogomolny2002 that for chaotic wave functions the variance of the mean number of nodal domains should converge to the theoretical limit $\sigma^{2}_{N}\simeq 0.05N$. In Fig. 7 the variance of the mean number of nodal domains divided by the level number $\sigma^{2}_{N}/N$ is shown for the microwave ray-splitting rough billiard. The variance $\sigma^{2}_{N}=\frac{1}{N_{w}-1}\sum_{i=1}^{N_{w}}(\aleph_{N_{i}}-\langle\aleph_{N}\rangle)^{2}$ was calculated in the window of $N_{w}=5$ consecutive eigenstates measured between $215\leq N\leq 415$, where the mean number of nodal domains was defined as $\langle\aleph_{N}\rangle=\frac{1}{N_{w}}\sum_{i=1}^{N_{w}}\aleph_{N_{i}}$. For level numbers $N<300$ the variance $\sigma^{2}_{N}/N$ is above the predicted theoretical limit, however, for $300<N\leq 415$ it is slightly below it. A similar erratic behavior of $\sigma^{2}_{N}/N$ was also observed in Bogomolny2002 . Figure 8: Distribution of nodal domain areas. Full line shows the prediction of percolation theory $\log_{10}(\langle n_{s}/n\rangle)=-\frac{187}{91}\log_{10}(\langle s/s_{min}\rangle)$. The least squares fit $\log_{10}(\langle n_{s}/n\rangle)=a_{2}-\tau\log_{10}(\langle s/s_{min}\rangle)$ of the experimental results lying within the vertical lines yields the scaling exponent $\tau=2.14\pm 0.12$ and $a_{2}=-0.06\pm 0.12$. The result of the fit is shown by the dashed line. The percolation model Bogomolny2002 allows for applying the results of percolation theory to the description of nodal domains of chaotic billiards. The percolation theory predicts that the distribution of the areas $s$ of nodal clusters should obey the scaling behavior: $n_{s}\propto s^{-\tau}$. The scaling exponent Ziff1986 is found to be $\tau=187/91$. In Fig. 8 we present in logarithmic scales nodal domain areas distribution $\langle n_{s}/n\rangle$ vs. $\langle s/s_{min}\rangle$ obtained for the microwave ray-splitting rough billiard. The distribution $\langle n_{s}/n\rangle$ was constructed as normalized to unity histogram with the bin equal to 1. The areas $s$ of nodal domains were calculated by summing up the areas of the nearest neighboring grid sites having the same sign of the wave function. In Fig. 8 the vertical axis $\langle n_{s}/n\rangle=\frac{1}{N_{T}}\sum_{i=1}^{N_{T}}n_{s}^{(N)}/n^{(N)}$ represents the number of nodal domains $n_{s}^{(N)}$ of size $s$ divided by the total number of domains $n^{(N)}$ averaged over $N_{T}=30$ wave functions measured in the range $215\leq N\leq 415$. In these calculations we took into account only the nodal domains which entirely lied outside or inside of the Teflon insert for which percolation theory Ziff1986 should be applicable. The horizontal axis in Fig. 8 is expressed in the units of the smallest possible area $s_{min}^{(N)}$ Bogomolny2002 , $\langle s/s_{min}\rangle=\frac{1}{N_{T}}\sum_{i=1}^{N_{T}}s/s_{min}^{(N)}$, where $s_{min}^{(N)}=\pi(j_{01}/\eta k_{N})^{2}$ and $j_{01}\simeq 2.4048$ is the first zero of the Bessel function $J_{0}(j_{01})=0$. For nodal domains lying inside the Teflon insert the refraction index was according to our measurements $\eta=1.425$ while outside of the insert we assumed $\eta=1$. The full line in Fig. 8 shows the prediction of percolation theory $\log_{10}(\langle n_{s}/n\rangle)=-\frac{187}{91}\log_{10}(\langle s/s_{min}\rangle)$. In a broad range of $\log_{10}(\langle s/s_{min}\rangle)$, approximately from 0.2 to 1.4, which is marked by the two vertical lines the experimental results follow closely the theoretical prediction. The least squares fit $\log_{10}(\langle n_{s}/n\rangle)=a_{2}-\tau\log_{10}(\langle s/s_{min}\rangle)$ of the experimental results lying within the vertical lines gives the scaling exponent $\tau=2.14\pm 0.12$ and $a_{2}=-0.06\pm 0.12$, which is in a good agreement with the predicted $\tau=187/91\simeq 2.05$. The result of the fit is shown in Fig. 8 by the dashed line. In summary, for the first time we measured high-lying wave functions of the chaotic microwave ray-splitting rough billiard. We showed that in the limit $N\rightarrow\infty$ the least squares fit of the experimental data reveals the asymptotic number of nodal domains $\aleph_{N}/N\simeq 0.063\pm 0.023$ that is close to the theoretical prediction $\aleph_{N}/N\simeq 0.062$ Bogomolny2002 . We demonstrate that for higher level numbers $N\simeq 215-415$ the variance of the mean number of nodal domains $\sigma^{2}/N$ is scattered around the theoretical limit $\sigma^{2}/N\simeq 0.05$. Following the results of percolationlike model proposed by Bogomolny2002 we confirmed that the distribution of the areas $s$ of nodal domains has power behavior $n_{s}\propto s^{-\tau}$, where scaling exponent is equal to $\tau=2.14\pm 0.12$. This result is in a good agreement with the prediction of percolation theory Ziff1986 , which predicts $\tau=187/91\simeq 2.05$. The experimental results presented in this paper strongly suggest that many properties of nodal domains in chaotic ray-splitting billiards are the same, like in conventional chaotic billiards without ray-splitting. Acknowledgments. This work was supported by Ministry of Science and Information Society Technologies grant No. 2 P03B 047 24. ## References * (1) E. Bogomolny and C. Schmit, Phys. Rev. Lett. 88, 114102-1 (2002). * (2) G. Blum, S. Gnutzmann, and U. Smilansky, Phys. Rev. Lett. 88, 114101-1 (2002). * (3) N. Savytskyy, O. Hul, and L. Sirko Phys. Rev. E 70, 056209 (2004). * (4) R. Blümel, T. M. Antonsen, B. Georgeot, E. Ott, and R. E. Prange, Phys. Rev. Lett. 76, 2476 (1996); Phys. Rev. E 53, 3284 (1996). * (5) L. Sirko, P. M. Koch and R. Blümel, Phys. Rev. Lett. 78, 2940 (1997). * (6) R. Blümel, P.M. Koch, and L. Sirko, Found. Phys. 31, 269 (2001). * (7) L. Couchman, E. Ott, and T. M. Antonsen, Jr., Phys. Rev. A 46, 6193 (1992). * (8) N. Savytskyy, A. Kohler, Sz. Bauch, R. Bl mel, and L. Sirko Phys. Rev. E 64, 036211 (2001). * (9) Sz. Bauch, A. Błȩdowski, L. Sirko, P. M. Koch, and R. Blümel, Phys. Rev. E 57, 304 (1998). * (10) Y. Hlushchuk, A. Kohler, Sz. Bauch, L. Sirko, R. Blümel, M. Barth, and H.-J. Stöckmann, Phys. Rev. E 61, 366 (2000). * (11) R. Schäfer, U. Kuhl, M. Barth, and H.-J. Stöckmann, Found. Phys. 31, 475 (2001). * (12) N. Savytskyy and L. Sirko, Phys. Rev. E 65, 066202-1 (2002). * (13) Y. Hlushchuk, A. Błȩdowski, N. Savytskyy, and L. Sirko, Physica Scripta 64, 192 (2001). * (14) Y. Hlushchuk, L. Sirko, U. Kuhl, M. Barth, H.-J. Stöckmann, Phys. Rev. E 63, 046208-1 (2001). * (15) K.M. Frahm and D.L. Shepelyansky, Phys. Rev. Lett. 78, 1440 (1997). * (16) K.M. Frahm and D.L. Shepelyansky, Phys. Rev. Lett. 79, 1833 (1997). * (17) A. Shnirelman, Usp. Mat. Nauk. 29, N6, 18 (1974). * (18) Y. Yamamoto and R.E. Sluster, Phys. Today 46(6), 66 (1993). * (19) J.U. Nöckel and A.D. Stone, Nature 385, 45 (1997). * (20) V. Doya, O. Legrand, and F. Mortessagne, Phys. Rev. Lett. 88, 014102 (2002). * (21) Ya. M. Blanter, A.D. Mirlin, and B.A. Muzykantskii, Phys. Rev. Lett. 80, 4161 (1998). * (22) L. Sirko, Sz. Bauch, Y. Hlushchuk, P.M. Koch, R. Blümel, M. Barth, U. Kuhl, and H.-J. Stöckmann, Phys. Lett. A 266, 331 (2000). * (23) F. Borgonovi, Phys. Rev. Lett. 80, 4653 (1998). * (24) L.C. Maier and J.C. Slater, J. Appl. Phys. 23, 68 (1952). * (25) S. Sridhar, Phys. Rev. Lett. 67, 785 (1991). * (26) C. Dembowski, H.-D. Gräf, A. Heine, R. Hofferbert, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 84, 867 (2000). * (27) D.H. Wu, J.S.A. Bridgewater, A. Gokirmak, and S.M. Anlage, Phys. Rev. Lett. 81, 2890 (1998). * (28) A. Kohler and R. Blümel, Phys. Lett. A 238, 271 (1998). * (29) J. Stein, H.-J. Stöckmann, and U. Stoffregen, Phys. Rev. Lett. 75, 53 (1995). * (30) M. Hentschel and K. Richter, Phys. Rev. E 66, 056207 (2002). * (31) S.W. McDonald and A.N. Kaufman, Phys. Rev A 37, 3067 (1988). * (32) M.V. Berry, J. Phys. A 10, 2083 (1977). * (33) R. M. Ziff, Phys. Rev. Lett. 56, 545 (1986).	arxiv-papers	2009-03-12T09:42:49	2024-09-04T02:49:01.093286	{ "license": "Public Domain", "authors": "Oleh Hul, Nazar Savytskyy, Oleg Tymoshchuk, Szymon Bauch and Leszek\n Sirko", "submitter": "Oleh Hul", "url": "https://arxiv.org/abs/0903.2129" }
0903.2130	# Surface magnetoinductive breathers in two-dimensional magnetic metamaterials Maria Eleftheriou1,2, Nikos Lazarides3,4, George P. Tsironis3 and Yuri S. Kivshar5 1Department of Materials Science and Technology, University of Crete, P.O. Box 2208, Heraklion 71003, Crete, Greece 2Department of Music Technology and Acoustics, Technological Educational Institute of Crete, Rethymno 74100, Crete, Greece 3Department of Physics, University of Crete and Institute of Electronic Structure and Laser Foundation for Research and Technology-Hellas, P.O. Box 2208, Heraklion 71003, Greece 4Department of Electrical Engineering, Technological Educational Institute of Crete, P.O. Box 140, Heraklion 71500, Crete, Greece 5Nonlinear Physics Center, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia ###### Abstract We study discrete surface breathers in two-dimensional lattices of inductively-coupled split-ring resonators with capacitive nonlinearity. We consider both Hamiltonian and dissipative systems and analyze the properties of the modes localized in space and periodic in time (discrete breathers) located in the corners and at the edges of the lattice. We find that surface breathers in the Hamiltonian systems have lower energy than their bulk counterparts, and they are generally more stable. ###### pacs: 63.20.Pw, 75.30.Kz, 78.20.Ci Theoretical results on the existence of novel types of discrete surface solitons localized in the corners or at the edges of two-dimensional photonic lattices makris_2D ; pla_our ; pre_2D have been recently confirmed by the experimental observation of two-dimensional surface solitons in optically- induced photonic lattices prl_1 and two-dimensional waveguide arrays laser- written in fused silica prl_2 ; ol_szameit . These two-dimensional nonlinear surface modes demonstrate novel features in comparison with their counterparts in truncated one-dimensional waveguide arrays OL_george ; PRL_george ; OL_molina . In particular, in a sharp contrast to one-dimensional surface solitons, the mode threshold is lower at the surface than in a bulk making the mode excitation easier pla_our . Recently, it was shown LTK that, similar to discrete solitons analyzed extensively for optical systems, surface discrete breathers can be excited near the edge of a one-dimensional metamaterial created by a truncated array of nonlinear split-ring resonators. Networks of split-ring resonators (SRRs) that have nonlinear capacitive elements can support nonlinear localized modes or discrete breathers (DB’s) under rather general conditions that depend primarily on the inductive coupling between SRRs and their resonant frequency LET ; ELT . The corresponding one-dimensional surface modes have somewhat lower energy (in the Hamiltonian case) and can easily be generated in one- dimensional SRR lattices LTK . In this Brief Communication, we develop further those ideas and analyze two- dimensional lattices of split-ring resonators. Similar to the optical systems, we find that two-dimensional lattices of inductively-coupled split-ring resonators with capacitive nonlinearity can support the existence of long- lived two-dimensional discrete breathers localized in the corners or at the edge of the lattice. We consider a two-dimensional lattice of SRRs in both planar and planar-axial configuration [see Figs. 1(a,b)]. In the planar configuration, all SRR loops are in the same plane with their centers forming an orthogonal lattice, while in the planar-axial configuration the loops have a planar arrangement in one direction and an axial configuration in the other direction. Each SRR is equivalent to a nonlinear RLC circuit, with an ohmic resistance $R$, self- inductance $L$, and capacitance $C$. We assume that the capacitor $C$ contains a nonlinear Kerr-type dielectric, so that the permittivity $\epsilon$ can be presented in the form, $\displaystyle\epsilon(\|{\bf E}\|^{2})=\epsilon_{0}\left(\epsilon_{\ell}+\alpha\frac{\|{\bf E}\|^{2}}{E_{c}^{2}}\right),$ (1) where ${\bf E}$ is the electric field with the characteristic value $E_{c}$, $\epsilon_{\ell}$ is linear permittivity, $\epsilon_{0}$ is the permittivity of the vacuum, and $\alpha=+1~{}~{}(-1)$ corresponding to self-focusing (self- defocusing) nonlinearity, respectively. As a result, each SRR acquires the field-dependent capacitance $C(\|{\bf E}\|^{2})=\epsilon(\|{\bf E}_{g}\|^{2})\,A/d_{g}$, where $A$ is the area of the cross-section of the SRR wire, ${\bf E}_{g}$ is the electric field induced along the SRR slit, and $d_{g}$ is the size of the slit. The field ${\bf E}_{g}$ is induced by the magnetic and/or electric component of the applied electromagnetic field, depending on the relative orientation of the field with respect to the SRR plane and the slits Shardivov . Below we assume that the magnetic component of the incident (applied) electromagnetic field is always perpendicular to the SRR plane, so that the electric field component is transverse to the slit. With this assumption, only the magnetic component of the field excites an electromotive force in SRRs, resulting in an oscillating current in each SRR loop. This results in the development of an oscillating voltage difference $U$ across the slits or, equivalently, of an oscillating electric field ${\bf E}_{g}$ in the slits. Figure 1: Schematic of a two-dimensional lattice of split-ring resonators for (a) planar and (b) planar-axial geometries. In both the geometries the magnetic component of the applied field is directed along the SRR axes, while the electric field component is transverse to the slits. If $Q$ is a charge stored in teach SRR capacitor, from a general relation of a voltage-dependent capacitance $C(U)=dQ/dU$ and Eq. (1), we obtain $\displaystyle Q=C_{\ell}\left(1+\alpha\frac{U^{2}}{3\epsilon_{\ell}\,U_{c}^{2}}\right)U,$ (2) where $U=d_{g}E_{g}$, $C_{\ell}=\epsilon_{0}\epsilon_{\ell}(A/d_{g})$ is the linear capacitance, and $U_{c}=d_{g}E_{c}$. We assume that the arrays are placed in a time-varying and spatially uniform magnetic field of the form $\displaystyle H=H_{0}\,\cos(\omega t),$ (3) where $H_{0}$ is the field amplitude, $\omega$ is the field frequency, and $t$ is the time variable. The excited electromotive force ${\cal E}$ , which is the same in all SRRs, is given by the expression $\displaystyle{\cal E}={\cal E}_{0}\,\sin(\omega t),\qquad{\cal E}_{0}\equiv\mu_{0}\,\omega\,S\,H_{0},$ (4) where $S$ is the area of each SRR loop, and $\mu_{0}$ the permittivity of the vacuum. Each SRR exposed to the external field given by Eq. (3) is a nonlinear oscillator exhibiting a resonant magnetic response at a particular frequency which is very close to its linear resonance frequency $\omega_{\ell}=1/\sqrt{L\,C_{\ell}}$ (for $R\simeq 0$). All SRRs in an array are coupled together due to magnetic dipole-dipole interaction through their mutual inductances. However, we assume below only the nearest-neighbor interactions, so that neighboring SRRs are coupled through their mutual inductances $M_{x}$ and $M_{y}$. This is a good approximation in the planar configurations [see Fig.1(a)], even if SRRs are located very close. Validity of the nearest-neighbor approximation for the planar-axial configuration [see Fig.1(b)] has been verified by taking into account the interaction of SRRs with their four nearest neighbors. Assumimg that the mutual inductance $M_{x,y}^{(s)}$ between an SRR and its $s-$th neighbor decays with distance as $M_{x,y}^{(s)}\simeq M_{x,y}/s^{3}$ ELT , we find practically the same results. Therefore, the electrical equivalent of an SRR array in an alternating magnetic field is an array of nonlinear RLC oscillators coupled with their nearest neighbors through their mutual inductances; the latter are being driven by identical alternating voltage sources. Equations describing the dynamics of the charge $Q_{n,m}$ and the current $I_{n,m}$ circulating in the $n,m-$th SRR may be derived from Kirchhoff’s voltage law for each SRR LET ; Shardivov $\displaystyle\frac{dQ_{n,m}}{dt}$ $\displaystyle=$ $\displaystyle I_{n,m}$ (5) $\displaystyle L\frac{dI_{n,m}}{dt}$ $\displaystyle+$ $\displaystyle RI_{n,m}+f(Q_{n,m})=$ (6) $\displaystyle-$ $\displaystyle M_{x}\left(\frac{dI_{n-1,m}}{dt}+\frac{dI_{n+1,m}}{dt}\right)$ $\displaystyle-$ $\displaystyle M_{y}\left(\frac{dI_{n,m-1}}{dt}+\frac{dI_{n,m+1}}{dt}\right)+{\cal E},$ where $f(Q_{n,m})=U_{n,m}$ is given implicitly from Eq. (2). Using the relations $\displaystyle\omega_{\ell}^{-2}$ $\displaystyle=$ $\displaystyle LC_{\ell},~{}~{}\tau=t\omega_{\ell},~{}~{}I_{c}=U_{c}\omega_{\ell}C_{\ell},~{}~{}Q_{c}=C_{\ell}U_{c}$ (7) $\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle U_{c}\varepsilon,~{}~{}I_{n,m}=I_{c}i_{n,m},~{}~{}Q_{n,m}=Q_{c}q_{n,m},$ (8) and Eq. (4), we normalize Eqs. (5) and (6) to the form $\displaystyle\frac{dq_{n,m}}{d\tau}$ $\displaystyle=$ $\displaystyle{i_{n,m}}$ (9) $\displaystyle\frac{di_{n,m}}{d\tau}$ $\displaystyle+$ $\displaystyle\gamma\,i_{n,m}+f(q_{n,m})+\lambda_{x}\left(\frac{di_{n-1,m}}{d\tau}+\frac{di_{n+1,m}}{d\tau}\right)$ (10) $\displaystyle+$ $\displaystyle\lambda_{y}\left(\frac{di_{n,m-1}}{d\tau}+\frac{di_{n,m+1}}{d\tau}\right)=\varepsilon_{0}\,\sin(\Omega\tau),$ where $\gamma=RC_{\ell}\omega_{\ell}$ is the loss coefficient, $\lambda_{x,y}=M_{x,y}/L$ are the the coupling parameters in the $x-$ and $y-$direction, respectively, and $\varepsilon_{0}={\cal E}_{0}/U_{c}$. Note that the loss coefficient $\gamma$, which is usually small ($\gamma\ll 1$), may account both for Ohmic and radiative losses Kourakis . Neglecting losses and without applied field, Eqs. (9) and (10) can be derived from the Hamiltonian $\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle\sum_{n,m}\left\\{\frac{1}{2}\dot{q}_{n,m}^{2}+V_{n,m}\right\\}$ (11) $\displaystyle-$ $\displaystyle\sum_{n,m}\left\\{\lambda_{x}\,\dot{q}_{n,m}\,\dot{q}_{n+1,m}+\lambda_{y}\,\dot{q}_{n,m}\,\dot{q}_{n,m+1}\right\\},$ where the nonlinear on-site potential $V_{n,m}$ is given by $\displaystyle V_{n,m}\equiv V(q_{n,m})=\int_{0}^{q_{n,m}}f(q_{n,m}^{\prime})\,dq_{n,m}^{\prime},$ (12) and $\dot{q}_{n,m}\equiv d{q}_{n,m}/d\tau$. Analytical solution of Eq. (2) for $u_{n,m}=f(q_{n,m})$ with the conditions of $u_{n,m}$ being real and $u_{n,m}(q_{n,m}=0)=0$, gives the approximate expression $\displaystyle f(q_{n,m})\simeq q_{n,m}-\frac{\alpha}{3\epsilon_{\ell}}q_{n,m}^{3}+3\left(\frac{\alpha}{3\epsilon_{\ell}}\right)^{2}q_{n,m}^{5},$ (13) which is valid for relatively low $q_{n}$ ($q_{n}<1,~{}~{}n=1,2,...,N$). Thus, the on-site potential is soft for $\alpha=+1$ and hard for $\alpha=-1$. In the 2D case the mutual inductances $M_{x}$ and $M_{y}$ may differ both in their sign, depending on the configuration, and their magnitude. Actually, even in the planar 2D configuration with $d_{x}=d_{y}$ a small anisotropy should be expected because we consider SRRs having only one slit. This anisotropy can be effectively taken into account by considering slightly different coupling parameters $\lambda_{x}$ and $\lambda_{y}$. The coupling parameters $\lambda_{x,y}$ as well as the loss coefficient $\gamma$ can be calculated numerically for this specific model with high accuracy. However, for our purposes, it is sufficient to estimate these parameters for realistic (experimental) array parameters, ignoring the nonlinearity of the SRRs and the effects due to the weak coupling as in Refs. LET ; ELT with the following typical values $\lambda\approx 0.02$ and $\gamma\approx 0.01$. We construct discrete breathers located in the corner of a two-dimensional lattice of $15\times 15$ sites using the anti-continuous limit method as in Ref. LET for the set of Eqs. (9)-(10), setting $\gamma=0$ and $\varepsilon_{0}=0$ (Hamiltonian discrete breathers). For the case of $\alpha=+1$ corresponding to self-focusing nonlinearity and period $T_{b}=6.69$, we may construct linearly stable breathers for parameters up to $\lambda_{x}=\lambda_{y}=0.029$. Breather stability has been checked through the Floquet monodromy matrix throughout the paper. For the case where an anisotropy is introduced, $\lambda_{x}<\lambda_{y}$, linearly stable discrete breathers can be constructed up to $\lambda_{x}=0.028$ and simultaneously $\lambda_{y}=0.031$, or for the case of planar-axial configuration up to $\lambda_{x}=0.031$ and at the same time $\lambda_{y}=-0.028$. If we look for discrete breathers constructed in the middle of the upper edge of the lattice for example, we find that the values of the coupling where an instability occurs are slightly decreased (e.g. the upper stability limit of coupling for planar geometry is $\lambda_{x}=\lambda_{y}=0.028$). Several cases of linearly stable discrete breathers are shown in Fig. 2 for $\alpha=+1$. The same analysis holds for $\alpha=-1$ (defocusing nonlinearity) where the upper stability limit for the values of couplings are of the same order of magnitude as for $\alpha=+1$, both for the corner and edge breathers (see Fig. 3). The breather period in the latter case is $T_{b}=5.8$. Figure 2: Density amplitudes $q_{n,m}$ for discrete Hamiltonian breathers constructed in (a-c) upper left corner or (d-f) upper edge of the lattice of $15\times 15$ sites, $\alpha=+1$ and $T_{b}=6.69$. (a,c) $\lambda_{x}=\lambda_{y}=0.028$, (b,e) $\lambda_{x}=0.026$ and $\lambda_{y}=0.029$, (c,f) $\lambda_{x}=0.029$ and $\lambda_{y}=-0.026$. All plots depict a $5\times 5$ sublattice that includes the breather zones. Figure 3: Density amplitude $q_{n,m}$ for discrete Hamiltonian breathers constructed in (a-c) upper left corner or in (d-f) upper edge of a lattice of $15\times 15$ sites, $\alpha=-1$ and $T_{b}=5.8$. (a,d) $\lambda_{x}=\lambda_{y}=0.030$, (b,e) $\lambda_{x}=0.028$ and $\lambda_{y}=0.031$, (c,f) $\lambda_{x}=0.028$ and $\lambda_{y}=-0.025$. All plots depict the $5\times 5$ sublattice around the linearly stable breathers. Localized modes in the damped-driven case are constructed for $\gamma=0.01$, $\varepsilon_{0}=0.04$ and $\alpha=+1$ with the method described in Ref. LET . The resulting localized modes are called dissipative breathers, and their examples are shown in Fig. 4 for $T_{b}=5.8$ and (a) $\lambda_{x}=\lambda_{y}=0.0007$, (b) $\lambda_{x}=0.0022$ and $\lambda_{y}=0.0052$, and (c) $\lambda_{x}=0.0052$ and $\lambda_{y}=-0.0022$. The dissipative modes have been evolved in time, and we found that at long times some dissipative breathers constructed for relatively large couplings loose their initial shape and finally decay. Figure 4: Density amplitude $q_{n,m}$ for discrete dissipative breathers for $\gamma=0.01$, $\varepsilon_{0}=0.04$, $\alpha=+1$ and $T_{b}=6.82$, constructed in the upper left corner for (a) $\lambda_{x}=\lambda_{y}=0.0007$, (b) $\lambda_{x}=0.0022$ and $\lambda_{y}=0.0052$, and (c) $\lambda_{x}=0.0052$ and $\lambda_{y}=-0.0022$. All plots depict the $5\times 5$ sublattice around the breather. Dissipative breathers are very narrow and essentially confined on one lattice site. Additionally, we calculate the total energy of discrete breathers in a lattice with planar and planar-axial configuration for $\alpha=+1$ and $T_{b}=6.69$ (Hamiltonian case). Figure 5 shows the energy histograms of the relevant corner of the lattice normalized to the energy of the corner $(1,1)$ breather. In order to construct the histograms centered in each of the lattice sites, we normalized it to the edge breather energy. In the case (a) the discrete breather is constructed in a lattice of coupling $\lambda_{x}=\lambda_{y}=0.028$, in (b) the case with anisotropy in couplings $\lambda_{x}=0.026$ and $\lambda_{y}=0.029$, while in the case (c), couplings are $\lambda_{x}=0.029$ and $\lambda_{y}=-0.026$. The energy of the discrete breathers as a function of the lattice site increases, i.e, as the discrete breather is constructed in the interior of the lattice energy is larger compared to the discrete breather that is located in the corner of the lattice. Figure 5: Histogram of the breather Hamiltonian. Breather difference energies $\Delta E$ for $\alpha=+1$, $Tb=6.69$ constructed in the upper left $3\times 3$ corner of the lattice. Case (a) $\lambda_{x}=\lambda_{y}=0.028$, case (b) $\lambda_{x}=0.026$ and $\lambda_{y}=0.029$, and case (c) $\lambda_{x}=0.029$ and $\lambda_{y}=-0.026$. To evaluate $\Delta E$ we calculate the energy of the breathers centered at different sites and subtract the energy of the corner breather. Figure 6: Amplitudes $q_{n,m}$ of the breather for $\alpha=+1$, $T_{b}=6.69$ and $\lambda_{x}=\lambda_{y}=0.028$, constructed on the site (1,1) for (a) t=0 and (b) $t=1450T_{b}$, and the breather constructed on site (3,3), for (c) $t=0$ and (d) $t=1450T_{b}$. We note that in the one-dimensional case the bulk breathers have lower energy compared to the surface ones LTK while in two-dimensional lattice the behavior is the contrary. We thus find that two-dimensional surface and especially edge breathers form easier. Finally, we study the time evolution of the discrete breather that is constructed in the corner site (1,1) and compare this case with a discrete breather centered at the (3,3) site for the coupling $\lambda_{x}=\lambda_{y}=0.028$. The breather of the latter case after $t=95T_{b}$ starts to loose its shape, in contrast to the breather of (1,1) site which survives for much longer times, viz. $t=1450T_{b}$ [see Fig. (6)]. For different coupling values such as $\lambda_{x}=\lambda_{y}=0.01$ we find that both the corner (1,1) and inner (3,3) breathers remain stable for at least $t=1450T_{b}$. This feature, while compatible with the fact that the corner breathers are more stable than inner ones, shows additionally that in finite lattices small changes in parameters may affect the stability properties of the breathers Morgante . In conclusion, we have studied surface discrete breathers located in the corner and at the edge of the two-dimensional lattices of the split-ring resonators. Using standard numerical methods, we have found nonlinear localized modes both in the Hamiltonian and dissipative systems. Two- dimensional breathers in conservative lattices have been found to be linearly stable for up to certain (large) values of the coupling coefficient, in both planar and planar-axial configurations of the split-ring-resonator lattices. Dissipative discrete surface breather can retain their shapes for several periods of time, and they depending critically on the lattice coupling. Finally, we have found that the discrete breathers located deep inside the lattice have higher energy compared to the breathers located in the corners and at the edges. This distinct two-dimensional feature of nonlinear localized modes contrasts with the one-dimensional behavior being attributed to the larger number of neighbors of the two-dimensional lattice. Furthermore, the two-dimensional breathers located inside the lattice loose rapidly their initial shape as they evolve in time while the surface breathers are seen to be stable at least for $t\approx 1500T_{b}$. ## References * (1) K.G. Makris, J. Hudock, D.N. Christodoulides, G. Stegeman, O. Manela, and M. Segev, Opt. Lett. 31, 2774 (2006). * (2) R.A. Vicencio, S. Flach, M.I. Molina, and Yu.S. Kivshar, Phys. Lett. A 364, 274 (2007). * (3) H. Susanto, P.G. Kevrekidis, B.A. Malomed, R. Carretero-González, and D.J. Franzeskakis, Phys. Rev. E 75, 056605 (2007). * (4) X. Wang, A. Bezryadina, Z. Chen, K.G. Makris, D.N. Christodoulides, and G.I. Stegeman, Phys. Rev. Lett. 98, 123903 (2007). * (5) A. Szameit, Y.V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and L. Torner, Phys. Rev. Lett. 98, 173903 (2007). * (6) A. Szameit, Y. V. Kartashov, V.A. Vysloukh, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, and L. Torner, Opt. Lett. 33, 1542 (2008). * (7) K.G. Makris, S. Suntsov, D.N. Christodoulides, G.I. Stegeman, and A. Haché, Opt. Lett. 30, 2466 (2005). * (8) S. Suntsov, K.G. Makris, D.N. Christodoulides, G.I. Stegeman, A. Haché, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901 (2006). * (9) M. Molina, R. Vicencio, and Yu. S. Kivshar, Opt. Lett. 31, 1693 (2006). * (10) N. Lazarides, G.P. Tsironis and Yu. S. Kivshar, Phys. Rev. E 77, 065601 (2008). * (11) N. Lazarides, M. Eleftheriou, and G.P. Tsironis, Phys. Rev. Lett. 97, 157406 (2006). * (12) M. Eleftheriou, N. Lazarides and G.P. Tsironis, Phys. Rev. E. 77, 036608 (2008). * (13) A. A. Zharov, I. V. Shardivov and Y. S. Kivshar, Phys. Rev. Lett. 91, 037401 (2003); I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Y. S. Kivshar, Photonics Nanostruct. Fundam. Appl. 4, 69 (2006). * (14) I. Kourakis, N. Lazarides, and G.P. Tsironis, Phys. Rev. E 75, 067601 (2007). * (15) A. M. Morgante, M. Johansson, S. Aubry, and G. Kopidakis, J. Phys. A: Math. Gen. 35, 4999 (2002).	arxiv-papers	2009-03-12T09:44:20	2024-09-04T02:49:01.097619	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Maria Eleftheriou, Nikos Lazarides, George P. Tsironis and Yuri S.\n Kivshar", "submitter": "Maria Eleftheriou", "url": "https://arxiv.org/abs/0903.2130" }
0903.2133	# Experimental investigation of Wigner’s reaction matrix for irregular graphs with absorption Oleh Hul1, Oleg Tymoshchuk1, Szymon Bauch1, Peter M. Koch2, and Leszek Sirko1 1Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland 2Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800 USA (September 22, 2005) ###### Abstract We use tetrahedral microwave networks consisting of coaxial cables and attenuators connected by $T$-joints to make an experimental study of Wigner’s reaction $K$ matrix for irregular graphs in the presence of absorption. From measurements of the scattering matrix $S$ for each realization of the microwave network we obtain distributions of the imaginary and real parts of $K$. Our experimental results are in good agreement with theoretical predictions. ###### pacs: 05.45.Mt,03.65.Nk Pauling introduced quantum graphs of connected one-dimensional wires almost seven decades ago Pauling . Kuhn used the same idea a decade later Kuhn to describe organic molecules by free electron models. Quantum graphs can be considered as idealizations of physical networks in the limit where the lengths of the wires greatly exceed their widths; this corresponds to assuming that propagating waves remain in a single transverse mode. Among the systems modeled by quantum graphs are, e.g., electromagnetic optical waveguides Flesia ; Mitra , mesoscopic systems Imry ; Kowal , quantum wires Ivchenko ; Sanchez and excitation of fractons in fractal structures Avishai ; Nakayama . Recent work has shown that quantum graphs provide an excellent system for studies of quantum chaos Kottossmilansky ; Kottos ; Prlkottos ; Zyczkowski ; Kus ; Tanner ; Kottosphyse ; Kottosphysa ; Gaspard ; Blumel ; Hul2004 . Quantum graphs with external leads (antennas) have been analyzed in detail in Kottosphyse ; Kottosphysa . Quantum graphs with absorption, a more realistic but more complicated system, have been studied numerically in Hul2004 , but until now there have been no experimental studies of the effect of absorption. This paper presents results of our experimental study of distributions of Wigner’s reaction matrix Akguc2001 (often called in the literature just the $K$ matrix Fyodorov2004 ) for microwave networks that correspond to graphs with time reversal symmetry ($\beta=1$ symmetry class of random matrix theory Mehta ) in the presence of absorption. For the case of an experiment having a single-channel antenna, the $K$ matrix and scattering matrix $S$ are related by $S=\frac{1-iK}{1+iK}.$ (1) The function $Z=iK$ has direct physical meaning as the electrical impedance, which has been recently measured in a microwave cavity experiment Anlage2005 . For the one-channel case the $S$ matrix can be parameterized as $S=\sqrt{R}e^{i\theta},$ (2) where $R$ is the reflection coefficient and $\theta$ is the phase. After seminal work of López, Mello and Seligman Lopez1981 came theoretical studies of the properties of statistical distributions of the $S$ matrix with direct processes and imperfect coupling Doron1992 ; Brouwer1995 ; Savin2001 . A recent experiment investigated the distribution of the $S$ matrix for chaotic microwave cavities with absorption Kuhl2005 . The distribution $P(R)$ of the reflection coefficient $R$ in Eq. (2), at the beginning investigated in the strong absorption limit Kogan , has been recently known for any dimensionless absorption strength $\gamma=2\pi\Gamma/\Delta$, where $\Gamma$ is the absorption width and $\Delta$ is the mean level spacing. For systems with time reversal symmetry ($\beta=1$) Méndez-Sánchez et al. Sanchez2003 studied $P(R)$ experimentally, and Savin et al. Savin2005 found an exact formula for $P(R)$. For systems violating time reversal symmetry ($\beta=2$), Beenakker and Brouwer Beenakker2001 calculated $P(R)$ for the case of a perfectly coupled, single-channel lead. In our experiment we simulate quantum graphs with microwave networks. The analogy between them is based on the Schrödinger equation for the former being equivalent to the telegraph equation for the latter Hul2004 . We call them microwave graphs. Measurements of the scattering matrix for them were stimulated by Blumel88 and the pioneering measurements in Doron90 . A simple microwave graph, the tetrahedral case, consists of six coaxial cables (bonds) that meet three-at-a-time at $N=4$ different $T$-joints (vertices). Each coaxial cable consists of an inner conductor with radius $r_{1}$ separated from a concentric outer conductor with inner radius $r_{2}$ by a homogeneous, non-magnetic material with dielectric constant $\varepsilon$. The fundamental $TEM$ mode that propagates (the so-called Lecher wave) down to zero frequency exists because the cross section of the cable is doubly connected (Jones, , p. 253). For frequencies $\nu$ below the onset of the TE11 mode in a coaxial cable, which propagates above $\nu_{c}\simeq\frac{c}{\pi(r_{1}+r_{2})\sqrt{\varepsilon}}$ Jones , the cable is single mode: only the TEM mode propagates. For SMA-RG-402 coaxial cable, which has $r_{1}=0.05$ cm, $r_{2}=0.15$ cm, and $\varepsilon\simeq 2.08$ (teflon dielectric), single-mode propagation occurs below 32.9 GHz. An (ideal) microwave graph with no aborption and no leads to the outside world is a closed (bound) system. The presence of absorption and/or leads to the outside world creates an open system. Because the coaxial cables are lossy, we may vary absorption in the microwave graphs by changing the length of cable(s), Hul2004 , by adding one or more (coaxial) microwave attenuators, or by changing the coupling to the outside world. Figure 1: A diagram of experimental setup used to measure the scattering matrix $S$ of tetrahedral microwave graphs with absorption. Absorption in the graphs was varied by changing the attenuator. The vector network analyzer used for all measurements was an HP model 8722D. Figure 1 shows our experimental setup for measurements of the single-channel scattering matrix $S$ for tetrahedral microwave graphs. We used a Hewlett- Packard model 8722D microwave vector network analyzer to measure the scattering matrix $S$ of such graphs in two different frequency windows, viz., 3.5–7.5 GHz and 12-16 GHz. As the figure shows, at one of the four vertices we used a 4-joint connector to couple the microwave graph to the vector network analyzer via a single-channel lead realized with an HP model 85133-60017, low- loss, flexible microwave cable; the other three vertices consisted of $T$-joints. The plane of calibration in the measurements was at the entrance to the 4-joint connector. Note that the microwave graph in Fig. 1 has a microwave attenuator in one of its bonds. To investigate the distributions of imaginary and real parts of the $K$ matrix we measured the scattering matrix $S$ for $184$ different realizations of tetrahedral microwave graphs having a microwave (SMA) attenuator in one of the bonds. For each graph realization, which was obtained either by the replacement of the bonds or putting an attenuator to a different bond, the scattering matrix $S$ was measured in 1601 equally spaced steps. The total optical lengths of the microwave graphs, including joints and the single attenuator, was 196.2 cm when a 3 dB, 6 dB, or 20 dB attenuator was used, whereas it was 197.4 cm when the 10 dB attenuator was used. To avoid degeneracy of eigenvalues in the graphs, we chose optical lengths for the bonds that were not simply commensurable. Figure 2: Panels (a) and (b) show, respectively, the modulus $\|S\|$ and the phase $\theta$ of the scattering matrix $S$ measured for the graph (see Fig. 1) with absorption parameter $\gamma=3.6$ (see the text) over the frequency range 12 - 13.4 GHz and with use of a 3 dB attenuator. Panels (c) and (d) show corresponding measurements for a graph with $\gamma=6.8$ over the same frequency range and with use of a 20 dB attenuator. The total optical length, 196.2 cm, of both microwave graphs was the same, including joints and the attenuator. Figure 2 shows the modulus $\|S\|$ and the phase $\theta$ of the scattering matrix $S$ of a tetrahedral graph with $\gamma=3.6$ (in panels (a) and (b), obtained with use of a 3 dB attenuator) and one with $\gamma=6.8$ (in panels (c) and (d), obtained with use of a 20 dB attenuator); both cases cover the frequency range 12–13.4 GHz. The lengths of corresponding bonds in the two graphs were the same. Direct processes are present in the scattering because the microwave vector analyzer was connected to the graphs by the 4-joint connector. For each, individual realization of the graph we may estimate them from the average value of the scattering matrix $\langle S\rangle$. Our measurements averaged over all realizations of microwave graphs gave $\langle S\rangle_{av}\simeq 0.47\pm 0.03+i(-0.01\pm 0.04)$, where $i=\sqrt{-1}$. The experimental value for $\|\langle S\rangle_{av}\|\simeq 0.47$ is close to a theoretical estimate Kottosphyse ; Kottosphysa for the modulus of the vertex reflection amplitude $\|\rho\|=0.5$ for a 4-joint connector with Neumann boundary conditions, $\rho=\frac{2}{n_{v}}-1,$ (3) where $n_{v}=4$ is the number of bonds meeting at the vertex in question. Equation (1) holds for systems with absorption but without direct processes. The case of imperfect coupling $\|\langle S\rangle\|>0$ and direct processes present can be mapped onto that of perfect one Fyodorov2004 by making the following parametrization, $S_{0}=\frac{S-\|\langle S\rangle\|}{1-\|\langle S\rangle\|S},$ (4) where $S_{0}$ is the scattering matrix of a graph for the perfect-coupling case (no direct processes present). For systems with time reversal symmetry ($\beta=1$ in equations below), the distributions $P(v)$ of the imaginary and $P(u)$ of the real parts of the $K$ matrix Fyodorov2004 are given by the following interpolation formulas: $P(v)=\frac{N_{\beta}e^{-a}}{\pi\sqrt{2\gamma}v^{3/2}}(A[K_{0}(a)+K_{1}(a)]a+\sqrt{\pi}Be^{-a}),$ (5) and $P(u)=\frac{N_{\beta}e^{-\gamma/4}}{2\pi\bar{u}}[\frac{A}{2}\sqrt{\frac{\gamma}{4}}D(\frac{\bar{u}}{2})+BK_{1}(\frac{\gamma\bar{u}}{4})],$ (6) where $-v=\textrm{Im}\,K<0$ and $u=\textrm{Re}\,K$ are, respectively, the imaginary and real parts of the $K$ matrix. The normalization constant is $N_{\beta}=\alpha\left(A\Gamma(\beta/2+1,\alpha)+Be^{-\alpha}\right)^{-1}$, where $\alpha=\gamma\beta/2$, $\Gamma(x,\alpha)=\int_{\alpha}^{\infty}dtt^{x-1}e^{-t}$ is the upper incomplete Gamma function, $A=e^{\alpha}-1$ and $B=1+\alpha-e^{\alpha}$. In Eq. (5) the variable $a=\frac{\gamma}{16}(\sqrt{v}+1/\sqrt{v})^{2}$ and $K_{0}$, $K_{1}$ are MacDonald functions. In Eq. (6) $D(z)=\int_{0}^{\infty}dq\sqrt{1+z(q+q^{-1})}e^{-\gamma z(q+q^{-1})/4}$ and $\bar{u}=\sqrt{u^{2}+1}$. Figure 3: Experimental distribution $P(v)$ of the imaginary part of the $K$ matrix at different values of the mean absorption parameter: $\bar{\gamma}=3.8$ (open circles), $\bar{\gamma}=5.2$ (full circles) and $\bar{\gamma}=6.7$ (open triangles), respectively. Each corresponding theoretical distribution $P(v)$ evaluated from Eq. (5) is also shown: $\gamma=3.8$ (dotted line), $\gamma=5.2$ (solid line), and $\gamma=6.7$ (dashed line), respectively. Figure 3 shows experimental distributions $P(v)$ for three mean values of the parameter $\bar{\gamma}$, viz., 3.8, 5.2, and 6.7. The distribution for $\bar{\gamma}=3.8$ is obtained by averaging over 69 realizations of microwave graphs having $\gamma$ within the window $[3.5,\,4.1]$. The distribution for $\bar{\gamma}=5.2$ is obtained by averaging over 60 realizations of microwave graphs having $\gamma$ within the window $[4.7,\,5.6]$. The distribution for $\bar{\gamma}=6.7$ is obtained by averaging over 55 realizations of microwave graphs having $\gamma$ within the window $[6.3,\,7.1]$. We estimated the experimental values of the $\gamma$ parameter by adjusting the theoretical mean reflection coefficient $\langle R\rangle_{th}$ to the experimental one $\langle R_{0}\rangle=\langle S_{0}S_{0}^{{\dagger}}\rangle$, where $\langle R\rangle_{th}=\int_{0}^{1}dRRP(R).$ (7) We also applied the following interpolation formula Kuhl2005 for the distribution $P(R)$: $P(R)=N_{\beta}\frac{e^{-\frac{\alpha}{1-R}}}{(1-R)^{2+\beta/2}}[A\alpha^{\beta/2}+B(1-R)^{\beta/2}].$ (8) We offer the following comment on the validity of the Eq. (8). We used it instead of exact formulas (12-14) recently presented in Savin2005 , which may be used to find the distribution $P(R)$, because Eq. (8) is sufficiently accurate (see Fig. 1 in Savin2005 ) while allowing for much faster numerical calculations. Figure 3 also presents for comparison with each experimental distribution $P(v)$ (symbols) the corresponding numerical distribution (lines) evaluated from Eq. (5). We see that the experimental distribution $P(v)$ at $\bar{\gamma}=3.8$ and at 5.2 agree well with their theoretical counterparts. However, the comparison for $\bar{\gamma}=6.7$ shows some discrepancies, particularly in the range $0.3<v<0.8$. Figure 4: Experimental distribution $P(u)$ of the real part of the $K$ matrix at different values of the mean absorption parameter: $\bar{\gamma}=3.8$ (open circles), $\bar{\gamma}=5.2$ (full circles) and $\bar{\gamma}=6.7$ (open triangles), respectively. Each corresponding theoretical distribution $P(u)$ evaluated from Eq. (6) is also shown: $\gamma=3.8$ (dotted line), $\gamma=5.2$ (solid line), and $\gamma=6.7$ (dashed line), respectively. We may use measurements of the distribution $P(u)$ of the real part of Wigner’s reaction matrix for an imporant and natural consistency check on our determination of $\gamma$. Figure 4 compares experimental and theoretical $P(u)$ distributions at the aforementioned values of $\bar{\gamma}$, viz., 3.8, 5.2, and 6.7. Though each case shows good overall agreement between experimental and theoretical results, for all three cases the middle ($-0.25<u<0.25$) of the theoretical distribution is slightly higher than its experimental counterpart. According to the definition of the $K$ matrix (see Eq. (1)), such behavior of the experimental distribution $P(u)$ suggests a deficit of small values of $\|\textrm{Im}\,S_{0}\|$. We do not yet know the origin of this deficit. Though there are the small discrepancies we have mentioned, the good overall agreement between experimental and theoretical results justifies a posteriori the procedure we have used to determine the experimental values of $\gamma$. The distributions $P(v)$ and $P(u)$ of imaginary and real parts of Wigner’s reaction matrix may be also found using the alternative approach described in Anlage2005 ; Anlage2005b . In these papers the radiation impedance approach was developed and used to obtaining the distributions of real and imaginary parts of the normalized impedance $Z=\frac{\textrm{Re }Z_{c}+i(\textrm{Im }Z_{c}-\textrm{Im }Z_{r})}{\textrm{Re }Z_{r}}$ (9) of a chaotic microwave cavity, where $Z_{c(r)}=Z_{0}(1+S_{c(r)})/(1-S_{c(r)})$ is the cavity (radiation) impedance expressed by the cavity (radiation) scattering matrix $S_{c(r)}$ and $Z_{0}$ is the characteristic impedance of the transmission line. The radiation impedance $Z_{r}$ is the impedance seen at the input of the coupling structure for the same coupling geometry, but with the sidewalls removed to infinity. This interesting approach is especially useful in the studies of microwave systems, in which, in general, both the system and radiation impedances are measurable. However, it is not obvious how to use in practice this approach in the case of quantum systems. We used this alternative approach to find distributions $P(v)$ and $P(u)$ of imaginary and real parts of Wigner’s reaction matrix for irregular tetrahedral microwave graphs. Wigner’s reaction matrix can be simply expressed by the normalized impedance $K=-iZ$. The radiation impedance $Z_{r}$ was found experimentally by measuring in two different frequency windows, viz., 3.5–7.5 GHz and 12-16 GHz of the scattering matrix $S_{r}$ of the 4-joint connector with three joints terminated by 50 $\Omega$ terminators. Figure 5: Experimental distribution $P(v)$ of the imaginary part of the $K$ matrix at different values of the mean absorption parameter: $\bar{\gamma}=3.8$ (open circles), $\bar{\gamma}=5.2$ (full circles) and $\bar{\gamma}=6.7$ (open triangles), respectively, calculated using the radiation impedance approach Anlage2005 ; Anlage2005b . Each corresponding theoretical distribution $P(v)$ evaluated from Eq. (5) is also shown: $\gamma=3.8$ (dotted line), $\gamma=5.2$ (solid line), and $\gamma=6.7$ (dashed line), respectively. Figure 6: Experimental distribution $P(u)$ of the real part of the $K$ matrix at different values of the mean absorption parameter: $\bar{\gamma}=3.8$ (open circles), $\bar{\gamma}=5.2$ (full circles) and $\bar{\gamma}=6.7$ (open triangles), respectively, calculated using the radiation impedance approach Anlage2005 ; Anlage2005b . Each corresponding theoretical distribution $P(u)$ evaluated from Eq. (6) is also shown: $\gamma=3.8$ (dotted line), $\gamma=5.2$ (solid line), and $\gamma=6.7$ (dashed line), respectively. In Fig. 5 and Fig. 6 we show the distributions $P(v)$ and $P(u)$ calculated using the radiation impedance approach Anlage2005 ; Anlage2005b . As in the case of the scattering matrix approach, the experimental distributions are obtained at three values of the parameter $\bar{\gamma}=3.8$, 5.2 and 6.7. Figure 5 shows that the distribution $P(v)$ of the imaginary part of Wigner’s reaction matrix for $\bar{\gamma}=5.2$ is in good agreement with the theoretical prediction Fyodorov2004 . However, for $\bar{\gamma}=3.8$ and 6.7 the theoretical results are slightly higher than the experimental ones, what is especially noticeable at the peaks of the distributions. The experimental distribution $P(u)$ of the real part of Wigner’s reaction matrix presented in Figure 6 at three values of the parameter $\bar{\gamma}=3.8$, 5.2 and 6.7 displays a very good agreement with the theoretical result. The comparison of Figures 3 and 5 and Figures 4 and 6 show that the distributions $P(v)$ and $P(u)$ evaluated by means of the radiation impedance approach are at the peaks slightly higher than the ones obtained by the scattering matrix approach, what may suggest that the influence of the phase of $S_{r}$ on the distributions is not negligible Anlage2005b . In summary, using the scattering matrix approach and the radiation impedance approach we have measured distributions $P(v)$ and $P(u)$ of imaginary and real parts of Wigner’s reaction matrix for irregular tetrahedral microwave graphs consisting of SMA cables, connectors, and attenuators. Use of different attenuators allowed us to vary absorption in the graphs in a controlled, quantitative way. For the case of time reversal symmetry ($\beta=1$), the experimental results for $P(v)$ and $P(u)$ calculated for both approaches at the same three values of the mean parameter $\bar{\gamma}$ are in good overall agreement with theoretical predictions. Acknowledgments: This work was supported by KBN grant No. 2 P03B 047 24 and an equipment grant from ONR(DURIP). ## References * (1) L. Pauling, J. Chem. Phys. 4, 673 (1936). * (2) H. Kuhn, Helv. Chim. Acta, 31, 1441 (1948). * (3) C. Flesia, R. Johnston, and H. Kunz, Europhys. Lett. 3 , 497 (1987). * (4) R. Mitra and S. W. Lee, Analytical techniques in the Theory of Guided Waves (Macmillan, New York, 1971). * (5) Y. Imry, Introduction to Mesoscopic Systems (Oxford, New York, 1996). * (6) D. Kowal, U. Sivan, O. Entin-Wohlman, Y. Imry, Phys. Rev. B 42, 9009 (1990). * (7) E. L. Ivchenko, A. A. Kiselev, JETP Lett. 67, 43 (1998). * (8) J.A. Sanchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, Phys. Rev. Lett. 80 , 948 (1998). * (9) Y. Avishai and J.M. Luck, Phys. Rev. B 45, 1074 (1992). * (10) T. Nakayama, K. Yakubo, and R. L. Orbach, Rev. Mod. Phys. 66, 381 (1994). * (11) T. Kottos and U. Smilansky, Phys. Rev. Lett. 79, 4794 (1997). * (12) T. Kottos and U. Smilansky, Annals of Physics 274, 76 (1999). * (13) T. Kottos and U. Smilansky, Phys. Rev. Lett. 85, 968 (2000). * (14) T. Kottos and H. Schanz, Physica E 9, 523 (2003). * (15) T. Kottos and U. Smilansky, J. Phys. A 36, 3501 (2003). * (16) F. Barra and P. Gaspard, Journal of Statistical Physics 101, 283 (2000). * (17) G. Tanner, J. Phys. A 33, 3567 (2000). * (18) P. Pakoński, K. Życzkowski and M. Kuś, J. Phys. A 34, 9303 (2001). * (19) P. Pakoński, G. Tanner and K. Życzkowski, J. Stat. Phys. 111, 1331 (2003). * (20) R. Blümel, Yu Dabaghian, and R.V. Jensen, Phys. Rev. Lett. 88, 044101 (2002). * (21) O. Hul, S. Bauch, P. Pakoński, N. Savytskyy, K. Życzkowski, and L. Sirko, Phys. Rev. E 69, 056205 (2004). * (22) G. Akguc and L. E. Reichl, Phys. Rev. E 64, 056221 (2001). * (23) Y.V. Fyodorov and D.V. Savin, JETP Letters 80, 725 (2004). * (24) M.L. Mehta, Random Matrices, (Academic Press, New York, 1991). * (25) S. Hemmady, X. Zheng, E. Ott, T.M. Antonsen, and S.M. Anlage, Phys. Rev. Lett. 94, 014102 (2005). * (26) S. Hemmady, X. Zheng, T.M. Antonsen, E. Ott, and S.M. Anlage, Phys. Rev. E 71, 056215 (2005). * (27) G. López, P.A. Mello, and T.H. Seligman, Z. Phys. A 302, 351 (1981). * (28) E. Doron and U. Smilansky, Nucl. Phys. A 545, 455 (1992). * (29) P.W. Brouwer, Phys. Rev. B 51, 16878 (1995). * (30) D.V. Savin, Y.V. Fyodorov, and H.-J. Sommers, Phys. Rev. E 63, 035202 (2001). * (31) U. Kuhl, M. Martinez-Mares, R.A. Méndez-Sánchez, and H.-J. Stöckmann, Phys. Rev. Lett. 94, 144101 (2005). * (32) E. Kogan, P.A. Mello, H. Liqun, Phys. Rev. E 61, R17 (2000). * (33) R.A. Méndez-Sánchez, U. Kuhl, M. Barth, C.V. Lewenkopf, and H.-J. Stöckmann, Phys. Rev. Lett. 91, 174102-1 (2003). * (34) D.V. Savin, H.-J. Sommers, and Y.V. Fyodorov, arXiv:cond-mat/0502359 v1, 15 Feb 2005. * (35) C.W.J. Beenakker and P.W. Brouwer, Physica E 9, 463 (2001). * (36) D. S. Jones, Theory of Electromagnetism (Pergamon Press, Oxford, 1964), p. 254. * (37) E. Doron, U. Smilansky, and A. Frenkel, Phys. Rev. Lett. 65, 3072 (1990). * (38) R. Blümel and U. Smilansky, Phys. Rev. Lett. 60, 477 (1988).	arxiv-papers	2009-03-12T10:01:38	2024-09-04T02:49:01.101541	{ "license": "Public Domain", "authors": "Oleh Hul, Oleg Tymoshchuk, Szymon Bauch, Peter M. Koch and Leszek\n Sirko", "submitter": "Oleh Hul", "url": "https://arxiv.org/abs/0903.2133" }
0903.2235	# On the vanishing, artinianness and finiteness of local cohomology modules Moharram Aghapournahr Moharram Aghapournahr Arak University Beheshti St, P.O. Box:879, Arak, Iran m-aghapour@araku.ac.ir and Leif Melkersson Leif Melkersson Department of Mathematics Linköping University S-581 83 Linköping Sweden lemel@mai.liu.se ###### Abstract. Let $R$ be a noetherian ring, $\mathfrak{a}$ an ideal of $R$, and $M$ an $R$–module. We prove that for a finite module $M$, if $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is minimax for all $i\geq r\geq 1$, then $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is artinian for $i\geq r$. A Local-global Principle for minimax local cohomology modules is shown. If $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is coatomic for $i\leq r$ ($M$ finite) then $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is finite for $i\leq r$. We give conditions for a module, which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules. ###### Key words and phrases: Local cohomology, minimax module, coatomic module. ###### 2000 Mathematics Subject Classification: 13D45, 13D07 ## 1\. Introduction Throughout $R$ is a commutative noetherian ring. For unexplained items from homological and commutative algebra we refer to [2] and [13]. Huneke gave in [10] a survey of some important problems on finiteness, vanishing and artinianness of local cohomology modules. We give some further contributions to the study of certain finiteness, vanishing and artinianness results for the local cohomology modules $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ for an $R$–module $M$ with respect to an ideal $\mathfrak{a}$. A thorough treatment of local cohomology is given by Brodmann and Sharp in [1]. A module $M$ is a _minimax_ module if there is a finite (i.e. finitely generated) submodule $N$ of $M$ such that the quotient module $M/N$ is artinian. Thus the class of minimax modules includes all finite and all artinian modules. Moreover, it is closed under taking submodules, quotients and extensions, i.e., it is a Serre subcategory of the category of $R$–modules. Minimax modules have been studied by Zink in [17] and Zöschinger in [19, 20]. See also [15]. Many equivalent conditions for a module to be minimax are given by them. We summarize some of those as follows: ###### Theorem 1.1. For a module $M$ over the commutative noetherian ring $R$, the following conditions are equivalent: 1. (i) $M/N$ has finite Goldie dimension for each submodule $N$ of $M$. 2. (ii) $M/N$ has finite socle for each submodule $N$ of $M$. 3. (iii) $M/N$ is an artinian module whenever $N$ is a submodule of $M$, such that $\operatorname{Supp}_{R}(M/N)\subset\operatorname{Max}{R}$. 4. (iv) $M/N$ is artinian for some finite submodule $N$ of $M$. 5. (v) For each increasing sequence $N_{1}\subset N_{2}\subset\dots$ of submodules of $M$ there is $l$ such that $N_{n+1}/N_{n}$ is artinian for all $n\geq l$. 6. (vi) For each decreasing sequence $N_{1}\supset N_{2}\supset\dots$ of submodules of $M$ there is $l$ such that $N_{n+1}/N_{n}$ is finite for all $n\geq l$. 7. (vii) (When $(R,\mathfrak{m})$ is a complete local ring) $M$ is Matlis reflexive. An $R$–module $M$ has _finite Goldie dimension_ if $M$ contains no infinite direct sum of submodules. For a commutative noetherian ring this can be expressed in two other ways, namely that the injective hull $\operatorname{E}(M)$ of $M$ decomposes as a finite direct sum of indecomposable injective modules or that $M$ is an essential extension of a finite submodule. In 2.2 we will give another equivalent condition for a module to be minimax. We prove in 2.3, that when $M$ is a finite $R$–module such that the local cohomology modules $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ are minimax modules for all $i\geq r$, where $r\geq 1$ then they must be artinian. An $R$–module $M$ is called $\mathfrak{a}$– _cofinite_ if $\operatorname{Supp}_{R}(M)\subset\operatorname{V}{(\mathfrak{a})}$ and $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is finite for each $i$. Hartshorne introduced this notion in [9], where he gave a negative answer to a question by Grothendieck in [8], by giving an example of a local cohomology module which is not $\mathfrak{a}$–cofinite. If an $R$–module $M$ with support in $\operatorname{V}{(\mathfrak{a})}$ is known to be a minimax module, then it suffices to know that $0:_{M}{\mathfrak{a}}$ is finite in order to conclude that $M$ is $\mathfrak{a}$–cofinite, [14, Proposition 4.3]. If we know that $0:_{M}{\mathfrak{a}}$ is finite, then of course in general $M$ is neither minimax nor $\mathfrak{a}$–cofinite, but if $M$ is assumed to be locally minimax, then $M$ is $\mathfrak{a}$–cofinite and minimax as we show in 2.6. This is applied to prove a Local-global Principle for minimax modules in 2.8. A prime ideal $\mathfrak{p}$ is said to be _coassociated_ to $M$ if $\mathfrak{p}=\operatorname{Ann}_{R}({M/N})$ for some $N\subset M$ such that $M/N$ is artinian and is said to be _attached_ to $M$ if $\mathfrak{p}=\operatorname{Ann}_{R}({M/N})$ for some arbitrary submodule $N$ of $M$, (equivalently $\mathfrak{p}=\operatorname{Ann}_{R}({M/{\mathfrak{p}}M})$). The set of these prime ideals are denoted by $\operatorname{Coass}_{R}(M)$ and $\operatorname{Att}_{R}(M)$ respectively. Thus $\operatorname{Coass}_{R}(M)\subset\operatorname{Att}_{R}(M)$ and the two sets are equal when $M$ is an artinian module. An alternative description for coassociated primes is given by $\operatorname{Coass}_{R}(M)=\underset{\mathfrak{m}\in\operatorname{Max}{R}}{\bigcup}\operatorname{Ass}_{R}(\operatorname{Hom}_{R}(M,\operatorname{E}{(R/\mathfrak{m})})).$ Thus when $(R,\mathfrak{m})$ is a local ring the coassociated primes of an $R$–module are just the associated primes of its Matlis dual. $M$ is called _coatomic_ when each proper submodule $N$ of $M$ is contained in a maximal submodule $N^{\prime}$ of $M$ (i.e. such that $M/N^{\prime}\cong R/\mathfrak{m}$ for some $\mathfrak{m}\in\operatorname{Max}{R}$). This property can also be expressed by $\operatorname{Coass}_{R}(M)\subset\operatorname{Max}{R}$ or equivalently that any artinian homomorphic image of $M$ must have finite length. In particular all finite modules are coatomic. Coatomic modules have been studied by Zöschinger [18]. A module $M$ which is minimax or coatomic has the property that the localization $M_{\mathfrak{p}}$ is a finitely generated $R_{\mathfrak{p}}$–module for each non-maximal prime ideal $\mathfrak{p}$. When $M$ is a minimax module this follows from condition (iv) of1.1. We show in 3.8 that if $M$ is finite and all local cohomology modules $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ are coatomic for all $i<n$, then they are actually finite in this range. In fact this is another condition equivalent to Falting’s Local-global Principle for the finiteness of local cohomology modules, [1, Theorem 9.6.1 and Proposition 9.1.2]. A vanishing theorem of Yoshida [16] is generalized in 3.9 and 3.10. For an $R$–module $M$ and an ideal $\mathfrak{a}$ of $R$, we let $\operatorname{cd}{(\mathfrak{a},M)}=\min\\{n\geq 0\mid\operatorname{H}^{i}_{\mathfrak{a}}(M)=0\text{ for all }i>n\,\\}$ and $\operatorname{q}(\mathfrak{a},M)=\min\\{n\geq 0\mid\operatorname{H}^{i}_{\mathfrak{a}}(M)\text{ is artinian for all }i>n\,\\}.$ We show that if $M$ is a coatomic $R$–module, then for any $R$–module $N$ such that $\operatorname{Supp}_{R}(N)\subset\operatorname{Supp}_{R}(M)$, we have $\operatorname{cd}{(\mathfrak{a},N)}\leq\operatorname{cd}{(\mathfrak{a},M)}$. This generalizes a result by Dibaei and Yassemi in [5, Theorem 1.4] who proved it when $M$ is finite. ## 2\. Artinianness of local cohomology modules ###### Lemma 2.1. Let $M$ be an $R$–module. The following statements are equivalent: 1. (i) $M$ is an artinian $R$–module. 2. (ii) $M_{\mathfrak{m}}$ is an artinian $R_{\mathfrak{m}}$–module for all $\mathfrak{m}\in{\operatorname{Max}{R}}$ and $\operatorname{Ass}_{R}(M)$ is a finite set. A module $M$ is _weakly laskerian_ when each quotient $M/N$ has just finitely many associated primes. For a study of such modules, see [6]. Every minimax module is trivially weakly laskerian. The converse holds under the additional condition that the module is locally minimax. ###### Proposition 2.2. Let $M$ be an $R$–module. The following statements are equivalent: 1. (i) $M$ is a minimax $R$–module. 2. (ii) $M_{\mathfrak{m}}$ is a minimax $R_{\mathfrak{m}}$–module for all $\mathfrak{m}{\in}\operatorname{Max}{R}$ and $M$ is a weakly laskerian $R$–module. ###### Proof. The only nontrivial part is (ii)$\Rightarrow$ (i). We show that if $\operatorname{Supp}_{R}(M/N)\subset\operatorname{Max}{R}$ then $M/N$ is artinian. By hypothesis $\operatorname{Ass}_{R}(M/N)$ is a finite set and consists of maximal ideals. For each maximal ideal $\mathfrak{m}$, the $R_{\mathfrak{m}}$–module $(M/N)_{\mathfrak{m}}$ is a minimax module with support at the maximal ideal of $R_{\mathfrak{m}}$. Therefore by part (iii) of 1.1 $(M/N)_{\mathfrak{m}}$ is an artinian $R_{\mathfrak{m}}$–module for all $\mathfrak{m}\in{\operatorname{Max}{R}}$. By 2.1, $M/N$ is an artinian $R$–module. ∎ The following theorem is the main result of this section. ###### Theorem 2.3. Let $R$ be a noetherian ring, $\mathfrak{a}$ an ideal of $R$ and $M$ a finite $R$–module. If $r\geq 1$ is an integer such that $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is a minimax module for all $i\geq r$, then $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is an artinian module for all $i\geq r$. ###### Proof. Suppose $\mathfrak{p}$ is a nonmaximal prime ideal of $R$. Then $\operatorname{H}^{i}_{\mathfrak{a}}(M)_{\mathfrak{p}}\cong\operatorname{H}^{i}_{{\mathfrak{a}}R_{\mathfrak{p}}}(M_{\mathfrak{p}})$ is a finite $R_{\mathfrak{p}}$–module for all $i\geq r$, since as we remarked in the introduction, when we localize at nonmaximal prime ideals, we obtain finitely generated modules. Therefore from [16, Proposition 3.1] we get that $\operatorname{H}^{i}_{\mathfrak{a}}(M)_{\mathfrak{p}}=0$ for all $i\geq r$. Hence $\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M))\subset{\operatorname{Max}{R}}$ for all $i\geq r$. By the condition (iii) of 1.1, the modules $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ are artinian for all $i\geq r$. ∎ ###### Corollary 2.4. Let $\mathfrak{a}$ an ideal of $R$ and $M$ a finite $R$–module. If $q=\operatorname{q}(\mathfrak{a},M)>0$, then the module $\operatorname{H}^{q}_{\mathfrak{a}}(M)$ is not minimax, in particular it is not finite. ###### Proposition 2.5. Let $M$ be a minimax module and $\mathfrak{a}$ an ideal of $R$. If $M$ is $\mathfrak{a}$–cofinite and socle-free, then there is $l$ such that $M=0:_{M}{\mathfrak{a}^{l}}$ and $M$ is finite. ###### Proof. Given $n$, let $\mathfrak{a}^{n}=(c_{1},\dots,c_{r})$. We define $h:M\to M^{r}$ by $h(m)=({c_{i}}m)_{i=1}^{r}$. Clearly $\operatorname{Ker}{h}=0:_{M}{\mathfrak{a}^{n}}$, so the module $M_{n}=M/(0:_{M}{\mathfrak{a}^{n}})$ is isomorphic to a submodule of $M^{r}$. In particular $M_{n}$ is socle-free. Consider the increasing sequence $0:_{M}{\mathfrak{a}}\subset 0:_{M}{\mathfrak{a}^{2}}\subset\dots$ of submodules of $M$, whose union is equal to $M$. Since $M$ is minimax, 1.1 (v) implies that there is $l$ such that $0:_{M}{\mathfrak{a}^{n+1}}/(0:_{M}{\mathfrak{a}^{n}})$ is artinian for all $n\geq l$. But $M/(0:_{M}{\mathfrak{a}^{n}})$ is socle-free. Hence $0:_{M}{\mathfrak{a}^{n+1}}=0:_{M}{\mathfrak{a}^{n}}$ for all $n\geq l$, and therefore $M=0:_{M}{\mathfrak{a}^{l}}$. ∎ The following theorem generalizes [14, Proposition 4.3]. ###### Theorem 2.6. Let $M$ be an $R$–module such that $\operatorname{Supp}_{R}(M)\subset\operatorname{V}{(\mathfrak{a})}$ and $M$ is locally minimax. If $0:_{M}{\mathfrak{a}}$ is finite, then $M$ is an $\mathfrak{a}$–cofinite minimax module. In particular this is the case, if there exists an element $x\in\mathfrak{a}$ such that $0:_{M}{x}$ is $\mathfrak{a}$–cofinite. ###### Proof. Let $L$ be the sum of the artinian submodules of $M$. Then $0:_{L}{\mathfrak{a}}$ is finite and therefore has finite length. Hence by [14, Proposition 4.1] $L$ is artinian and $\mathfrak{a}$–cofinite. The module $\overline{M}=M/L$ is locally minimax and furthermore it is socle- free. From the exactness of $0\to 0:_{L}{\mathfrak{a}}\to 0:_{M}{\mathfrak{a}}\to 0:_{\overline{M}}{\mathfrak{a}}\to\operatorname{Ext}^{1}_{R}(R/\mathfrak{a},L),$ we get that $0:_{\overline{M}}{\mathfrak{a}}$ is finite. We may therefore replace $M$ by $\overline{M}$, and assume that $M$ is socle-free. Let $\mathfrak{m}$ be any maximal ideal. Then $M_{\mathfrak{m}}$ is a socle- free minimax module over $R_{\mathfrak{m}}$, in fact it is ${\mathfrak{a}}R_{\mathfrak{m}}$–cofinite by [14, Proposition 4.3]. We are therefore able to apply proposition 2.5, so there is $n$ such that $(M_{n})_{\mathfrak{m}}=0$ where $M_{n}=M/(0:_{M}{\mathfrak{a}}^{n})$. Since as noted in the proof of 2.5 for each $n$, there is $r$ such that $M_{n}$ is isomorphic to a submodule of $M^{r}$, $\operatorname{Ass}_{R}(M_{n})\subset\operatorname{Ass}_{R}(M)=\operatorname{Ass}_{R}(0:_{M}{\mathfrak{a}})$ and $\operatorname{Ass}_{R}(M_{n})$ is therefore finite. Consequently $\operatorname{Supp}_{R}(M_{n})$ must be a closed subset of $X=\operatorname{Spec}R$. Therefore $U_{n}=X\setminus{\operatorname{Supp}_{R}(M_{n})}$ is an increasing sequence of open subsets of $X$. Since for each maximal ideal $\mathfrak{m}$, there is $n$ such that $(M_{n})_{\mathfrak{m}}=0$ i.e. $\mathfrak{m}\in{U_{n}}$, $X=\cup_{n=0}^{\infty}{U_{n}}$. By the quasi-compactness of $X$, we get that $X=U_{n}$ for some $n$. Hence $M=0:_{M}{\mathfrak{a}}^{n}$ which is finite. ∎ The following corollary describes a relation between the properties of cofiniteness and minimaxness for local cohomology. ###### Corollary 2.7. Let $n$ be a non-negative integer and $M$ a finite $R$–module 1. (a) If $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$–cofinite for all $i<n$ and $\operatorname{H}^{t}_{\mathfrak{a}}(M)$ is a locally minimax module, then it is also $\mathfrak{a}$–cofinite minimax. 2. (b) If $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$–cofinite for all $i<n$ and a locally minimax module for all $i$ $\geq$ $n$, then $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is $\mathfrak{a}$–cofinite for all $i$. ###### Proof. It is enough to prove that $\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{n}_{\mathfrak{a}}(M))$ is finite by 2.6 and this is immediate by use of [4, Theorem 2.1] (b) Use part (a). ∎ The following theorem, which is one of our main results shows that the Local- global Principle is valid for minimax local cohomology modules. ###### Theorem 2.8. Let $\mathfrak{a}$ be an ideal of $R$, $M$ a finite $R$–module and $t$ a non- negative integer. The following statements are equivalent: 1. (i) $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is a minimax $R$–module for all $i\leq t$. 2. (ii) $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is an $\mathfrak{a}$–cofinite minimax $R$–module for all $i\leq t$. 3. (iii) $\operatorname{H}^{i}_{\mathfrak{a}}(M)_{\mathfrak{m}}$ is a minimax $R_{\mathfrak{m}}$–module for all $\mathfrak{m}\in\operatorname{Max}{R}$ and for all $i\leq t$. ###### Proof. The only non-trivial part is the implication $(iii)\Rightarrow(ii)$. We prove this by induction on $t$. When $t=0$ there is nothing to prove. Suppose $t>0$ and the case $t-1$ is settled. So we may assume that $\operatorname{\Gamma}_{\mathfrak{a}}(M)=0$. Thus there exists $x\in{\mathfrak{a}}$ such that $0\to M\overset{x}{\to}M\to M/{x}M\to 0$ is exact. We get the exact sequence $\operatorname{H}^{i}_{\mathfrak{a}}(M)_{\mathfrak{m}}\to\operatorname{H}^{i}_{\mathfrak{a}}(M/{x}M)_{\mathfrak{m}}\to\operatorname{H}^{i+1}_{\mathfrak{a}}(M)_{\mathfrak{m}}$ It follows that $\operatorname{H}^{i}_{\mathfrak{a}}(M/{xM})_{\mathfrak{m}}$ is a minimax $R_{\mathfrak{m}}$–module for $i\leq{t-1}$. By the induction hypothesis $\operatorname{H}^{i}_{\mathfrak{a}}(M/{xM})$ is an $\mathfrak{a}$–cofinite $R$–module for $i\leq t$. It follows that $0\underset{\operatorname{H}^{t}_{\mathfrak{a}}(M)}{:}x$ is $\mathfrak{a}$–cofinite and from 2.6 we conclude that $\operatorname{H}^{t}_{\mathfrak{m}}(M)$ is $\mathfrak{a}$–cofinite minimax. ∎ ###### Example 2.9. Suppose the set $\Omega$ of maximal ideals of $R$ is infinite. Then the module $\oplus_{\mathfrak{m}\in\Omega}R/\mathfrak{m}$ is locally a minimax module, but it is not a minimax module. ## 3\. Finiteness, vanishing and non vanishing ###### Lemma 3.1. Let $R$ be a noetherian ring, $\mathfrak{a}$ an ideal of $R$, $M$ an $R$–module. Then $\mathfrak{a}M$ is finite if and only if $M/(0:_{M}{\mathfrak{a}})$ is finite. ###### Proof. $(\Rightarrow)$ Suppose $\mathfrak{a}=(a_{1},\dots,a_{n})$ and define $f:M\to(\mathfrak{a}M)^{n}$ by $f(m)=(a_{i}m)_{i=1}^{n}$. Since $\ker{f}=0:_{M}{\mathfrak{a}}$, the module $M/(0:_{M}{\mathfrak{a}})$ is isomorphic to a submodule of $(\mathfrak{a}M)^{n}$. $(\Leftarrow)$ Define a homomorphism $g:M^{n}\to\mathfrak{a}M$ by $g((m_{i})_{i=1}^{n})=\sum\limits_{i=1}^{n}{a_{i}m_{i}}$. Then $g$ is surjective and $(0:_{M}{\mathfrak{a}})^{n}\subset\operatorname{Ker}{g}$, so $\mathfrak{a}M$ is a homomorphic image of $(M/(0:_{M}{\mathfrak{a}}))^{n}$. By the way, when $\mathfrak{a}={x}R$ is a principal ideal, the modules ${x}M$ and $M/(0:_{M}{x})$ are in fact isomorphic. ∎ ###### Theorem 3.2 (Nonvanishing for coatomic modules). Let $(R,\mathfrak{m})$ be a noetherian local ring. If $M$ is a nonzero coatomic $R$–module of dimension $n$, then $\operatorname{H}^{n}_{\mathfrak{m}}(M)\neq 0$. ###### Proof. If $n=0$, there is nothing to prove. Suppose $n\geq 1$ then from [18, Satz 2.4 $(i)\Rightarrow(iii)$] there is an integer $t\geq 1$ such that $\mathfrak{m}^{t}M$ is finite and by 3.1 equivalently $M/(0:_{M}{\mathfrak{m}^{t}})$ is finite. On the other hand $\dim_{R}M/(0:_{M}{\mathfrak{m}^{t}})=\dim_{R}M=n$, and $\operatorname{H}^{i}_{\mathfrak{m}}(0:_{M}{\mathfrak{m}^{t}})=0$ for all $i\geq 1$. Making use of the exact sequence $0\to 0:_{M}{\mathfrak{m}^{t}}\to M\to M/(0:_{M}{\mathfrak{m}^{t}})\to 0$ we get $\operatorname{H}^{n}_{\mathfrak{m}}(M)\cong\operatorname{H}^{n}_{\mathfrak{m}}(M/(0:_{M}{\mathfrak{m}^{t}}))$, which is $\neq 0$, by [1, Theorem 6.1.4]. ∎ ###### Lemma 3.3. [See also [12, Corollary 2.5].] If $R$ and $\mathfrak{a}$ are as before and $M$ is a finite $R$–module of dimension $n$, then 1. (a) $\dim_{R}\operatorname{H}^{n-i}_{\mathfrak{a}}(M)\leq i$. 2. (b) If $(R,\mathfrak{m})$ is a local ring, then $\operatorname{Supp}_{R}(\operatorname{H}^{n-1}_{\mathfrak{a}}(M))$ is a finite set consisting of prime ideals $\mathfrak{p}$ such that $\dim{R/\mathfrak{p}}\leq 1$. ###### Proof. (a) For $\mathfrak{p}\in\operatorname{Supp}_{R}(\operatorname{H}^{n-i}_{\mathfrak{a}}(M))$, we get $\operatorname{H}^{n-i}_{\mathfrak{a}}(M)_{\mathfrak{p}}\cong\operatorname{H}^{n-i}_{{\mathfrak{a}}R_{\mathfrak{p}}}(M_{\mathfrak{p}})\neq 0$.Hence [1, Theorem 6.1.2] implies that $\dim{M_{\mathfrak{p}}}\geq n-i$ and therefore we have $\dim{R/\mathfrak{p}}\leq n-\dim{M_{\mathfrak{p}}}\leq i.$ (b) Let $\mathfrak{m}=(x_{1},\dots,x_{r})$. Then $\dim{M_{x_{i}}}\leq n-1$ for $1\leq i\leq r$. Hence by [1, Exercise 7.1.7] $\operatorname{H}^{n-1}_{\mathfrak{a}{R}_{x_{i}}}(M_{x_{i}})$ is an artinian $R_{x_{i}}$–module and $\operatorname{Supp}_{R_{x_{i}}}(\operatorname{H}^{n-1}_{\mathfrak{a}}(M)_{x_{i}})$ is finite. If $\mathfrak{p}\in\operatorname{Supp}_{R}(\operatorname{H}^{n-1}_{\mathfrak{a}}(M))$ and $\mathfrak{p}\neq\mathfrak{m}$ then there is $i$ such that ${x_{i}}\notin\mathfrak{p}$, i.e. ${\mathfrak{p}}R_{x_{i}}\in\operatorname{Supp}_{R_{x_{i}}}(\operatorname{H}^{n-1}_{\mathfrak{a}}(M)_{x_{i}})$. Hence $\operatorname{Supp}_{R}(\operatorname{H}^{n-1}_{\mathfrak{a}}(M))$ must be finite. ∎ ###### Proposition 3.4. Let $M$ be a coatomic module of dimension $n\geq 1$ over the local ring $(R,\mathfrak{m})$ and let $\mathfrak{a}$ be an ideal of $R$. Then we have that 1. (a) $\operatorname{H}^{n}_{\mathfrak{a}}(M)$ is artinian and $\mathfrak{a}$–cofinite. 2. (b) $\operatorname{Att}_{R}(\operatorname{H}^{n}_{\mathfrak{a}}(M))=\\{\mathfrak{p}\in{\operatorname{Supp}_{R}(M)}\|\operatorname{cd}{(\mathfrak{a},R/\mathfrak{p})}=n\\}$. 3. (c) $\operatorname{Supp}_{R}(\operatorname{H}^{n-1}_{\mathfrak{a}}(M))$ is a finite set consisting of prime ideals $\mathfrak{p}$ such that $\dim{R/\mathfrak{p}}\leq 1$. ###### Proof. (a): As in the proof of 3.2 we have (1) $\operatorname{H}^{n}_{\mathfrak{a}}(M)\cong\operatorname{H}^{n}_{\mathfrak{a}}(M/(0:_{M}{\mathfrak{m}^{t}}))$ for some $t\geq 1$ such that $M/(0:_{M}{\mathfrak{m}^{t}})$ is a finite $R$–module. Consequently by [14, Proposition 5.1] $\operatorname{H}^{n}_{\mathfrak{a}}(M)$ is artinian and $\mathfrak{a}$–cofinite. (b): Put $L=M/(0:_{M}{\mathfrak{m}^{t}})$ and note that $\operatorname{Supp}_{R}(L)=\operatorname{Supp}_{R}(M)$. But by [3, Theorem A] $\operatorname{Att}_{R}(\operatorname{H}^{n}_{\mathfrak{a}}(L))=\\{\mathfrak{p}\in{\operatorname{Supp}_{R}(L)}\|\operatorname{cd}{(\mathfrak{a},R/\mathfrak{p})}=n\\},$ so the assertion holds. (c). Use the isomorphism (1) and part (b) of 3.3. ∎ However when $n=0$, $\operatorname{H}^{n}_{\mathfrak{a}}(M)$ may not be artinian. ###### Example 3.5. $M={(R/\mathfrak{m})}^{(\mathbb{N})}$ is an $\mathfrak{m}$–torsion coatomic module of dimension zero but is not artinian. ###### Lemma 3.6. Let $M$ be a finite or more generally a coatomic $R$–module. Then $\operatorname{cd}{(\mathfrak{a},M)}=0$ if and only if $\operatorname{Supp}_{R}(M)\subset\operatorname{V}{(\mathfrak{a})}$. ###### Proof. $(\Leftarrow)$. Trivial. $(\Rightarrow)$. We may assume that $(R,\mathfrak{m})$ is local. First assume that $M$ is finite. If $\operatorname{Supp}_{R}(M)\not\subset\operatorname{V}{(\mathfrak{a})}$, then the module $\overline{M}=M/\operatorname{\Gamma}_{\mathfrak{a}}(M)$ is nonzero, and $\operatorname{\Gamma}_{\mathfrak{a}}(\overline{M})=0$. Hence we have $r=\operatorname{depth}_{\mathfrak{a}}\overline{M}>0$, but by [1, Theorem 6.2.7] $\operatorname{H}^{r}_{\mathfrak{a}}(\overline{M})\neq 0$. On the other hand $\operatorname{H}^{r}_{\mathfrak{a}}(M)\cong\operatorname{H}^{r}_{\mathfrak{a}}(\overline{M})$ and this is a contradiction. Now suppose $M$ is coatomic. As before for any $r>0$ we have $\operatorname{H}^{r}_{\mathfrak{a}}(M)\cong\operatorname{H}^{r}_{\mathfrak{a}}(M/(0:_{M}{\mathfrak{m}^{t}}))$ for some $t\geq 1$ such that $M/(0:_{M}{\mathfrak{m}^{t}})$ is finite. Note that $\operatorname{Supp}_{R}(M/(0:_{M}{\mathfrak{m}^{t}}))=\operatorname{Supp}_{R}(M)$ and use the result just shown for finite modules. ∎ We next generalize [5, Theorem 1.4]. See also [7, Theorem 2.2]. ###### Proposition 3.7. Let $\mathfrak{a}$ be an ideal of $R$ and $M$ a coatomic $R$–module. Let $N$ be an arbitrary module such that $\operatorname{Supp}_{R}(N)\subset\operatorname{Supp}_{R}(M)$, then $\operatorname{cd}{(\mathfrak{a},N)}\leq\operatorname{cd}{(\mathfrak{a},M)}$ ###### Proof. We may assume that $(R,\mathfrak{m})$ is local. Suppose $\operatorname{cd}{(\mathfrak{a},M)}=0$, then by 3.6 $\operatorname{Supp}_{R}(M)\subset\operatorname{V}{(\mathfrak{a})}$. Hence $\operatorname{Supp}_{R}(N)\subset\operatorname{V}{(\mathfrak{a})}$ and therefore $\operatorname{H}^{i}_{\mathfrak{a}}(N)=0$ for all $i>0$, i.e. $\operatorname{cd}{(\mathfrak{a},N)}=0$. Let $\operatorname{cd}{(\mathfrak{a},M)}\geq 1$, Then as before $\operatorname{H}^{r}_{\mathfrak{a}}(M)\cong\operatorname{H}^{r}_{\mathfrak{a}}(M/(0:_{M}{\mathfrak{m}^{t}}))$ for some $t\geq 1$ such that $M/(0:_{M}{\mathfrak{m}^{t}})$ is finite. Since $\operatorname{Supp}_{R}(N)\subset\operatorname{Supp}_{R}(M)=\operatorname{Supp}_{R}(M/(0:_{M}{\mathfrak{m}^{t}})),$ we get from [5, Theorem 1.4] $\operatorname{cd}{(\mathfrak{a},N)}\leq\operatorname{cd}{(\mathfrak{a},M/(0:_{M}{\mathfrak{m}^{t}}))}=\operatorname{cd}{(\mathfrak{a},M)}.$ ∎ Next we prove some vanishing and finiteness results for local cohomology. ###### Theorem 3.8. Let $R$ be a noetherian ring, $\mathfrak{a}$ an ideal of $R$ and $M$ a finite $R$–module. The following statements are equivalent: 1. (i) $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is coatomic for all $i<n$. 2. (ii) $\operatorname{Coass}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M))\subset\operatorname{V}{(\mathfrak{a})}$ for all $i<n$. 3. (iii) $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is finite for all $i<n$. ###### Proof. By [1, Theorem 9.6.1] and[18, 1.1, Folgerung] we may assume that $(R,\mathfrak{m})$ is a local ring. $\Rightarrow$ (ii) It is trivial by the definition of coatomic modules. $\Rightarrow$ (iii) By [21, Satz 1.2] there is $t\geq 1$ such that $\mathfrak{a}^{t}\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is finite for all $i<n$. Therefore there is $s\geq t$ such that $\mathfrak{a}^{s}\operatorname{H}^{i}_{\mathfrak{a}}(M)=0$ for all $i<n$. Then apply [1, Proposition 9.1.2]. $\Rightarrow$ (i) Any finite $R$–module is coatomic. ∎ The following results are generalizations of [16, Proposition 3.1] ###### Theorem 3.9. Let $\mathfrak{a}$ be an ideal of $R$ and $M$ a finite $R$–module and let $r\geq 1$. The following statements are equivalent: 1. (i) $\operatorname{H}^{i}_{\mathfrak{a}}(M)=0$ for all $i\geq r$. 2. (ii) $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is finite for all $i\geq r$. 3. (iii) $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is coatomic for all $i\geq r$. ###### Proof. $(i)\Rightarrow(ii)\Rightarrow(iii)$ Trivial. $(iii)\Rightarrow(i)$ By use of [16, Proposition 3.1] and [18, 1.1, Folgerung] we may assume that $(R,\mathfrak{m})$ is a local ring. Note that coatomic modules satisfies Nakayama’s lemma. So the proof is the same as in [16, Proposition 3.1]. ∎ ###### Corollary 3.10. Let $M$ be a coatomic $R$–module. If $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is coatomic for all $i\geq r$, where $r\geq 1$, then $\operatorname{H}^{i}_{\mathfrak{a}}(M)=0$ for all $i\geq r$. ###### Proof. We may assume that $(R,\mathfrak{m})$ is a local ring. So as before there is an isomorphism $\operatorname{H}^{r}_{\mathfrak{a}}(M)\cong\operatorname{H}^{r}_{\mathfrak{a}}(M/(0:_{M}{\mathfrak{m}^{t}}))$ for some $t\geq 1$ such that $M/(0:_{M}{\mathfrak{m}^{t}})$ is finite, and then use $(iii)\Rightarrow(i)$ of 3.9. ∎ ###### Corollary 3.11. Let $\mathfrak{a}$ an ideal of $R$ and $M$ a finite $R$–module. If $c=\operatorname{cd}(\mathfrak{a},M)>0$, then $\operatorname{H}^{c}_{\mathfrak{a}}(M)$ is not coatomic in particular it is not finite. ###### Corollary 3.12. If $M$ is coatomic and $r\geq 1$, the following are equivalent: 1. (i) $\operatorname{H}^{i}_{\mathfrak{a}}(M)=0$ for all $i\geq r$. 2. (ii) $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is finite for all $i\geq r$. 3. (iii) $\operatorname{H}^{i}_{\mathfrak{a}}(M)$ is coatomic for all $i\geq r$. ## References * [1] M.P. Brodmann, R.Y. Sharp, _Local cohomology: an algebraic introduction with geometric applications_ , Cambridge University Press, 1998. * [2] W. Bruns, J. Herzog, _Cohen-Macaulay rings_ , Cambridge University Press, revised ed., 1998. * [3] M.T. Dibaei, S. Yassemi, _Attached primes of the top local cohomology modules with respect to an ideal_ , Arch. Math. 84 (2005), 292–297. * [4] M.T . Dibaei, S. Yassemi, _Associated primes and cofiniteness of local cohomology modules_ , manuscripta math, 117(2005), 199–205. * [5] M.T. Dibaei, S. Yassemi, _Cohomological dimension of complexes_ , Comm. Algebra 32 (2004), 4375–4386. * [6] K. Divaani-Aazar, A. Mafi, _Associated primes of local cohomology modules_ , Proc. Amer. Math. Soc. 133 (2005), 655–660. * [7] K. Divaani-Aazar, R. Naghipour, M. Tousi, _Cohomological dimension of certain algebraic varieties_ , Proc. Amer. Math. Soc. 130(2002), 3537–3544. * [8] A. Grothendieck, _Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2)_ , North-Holland, Amsterdam, 1968. * [9] R. Hartshorne, _Affine duality and cofiniteness_ , Invent. Math. 9 (1970), 145–164. * [10] C. Huneke, _Problems on local cohomology_ :Free resolutions in commutative algebra and algebraic geometry, (Sundance, UT, 1990), 93–108, Jones and Bartlett, 1992. * [11] C. Huneke, J. Koh, _Cofiniteness and vanishing of local cohomology modules_ , Math. Proc. Cambridge Philos. Soc. 110(1991), 421–429. * [12] T. Marley, _The associated primes of local cohomology modules over rings of small dimension_ , manuscripta math. 104(2001), 519–525 * [13] H. Matsumura, _Commutative ring theory_ , Cambridge University Press, 1986. * [14] L. Melkersson, _Modules cofinite with respect to an ideal_ , J. Algebra. 285(2005), 649–668. * [15] P. Rudlof, _On minimax and related modules_ , Can. J. Math. 44 (1992), 154–166. * [16] K. I. Yoshida, _Cofiniteness of local cohomology modules for ideals of dimension one_ , Nagoya Math. J. 147(1997), 179–191. * [17] T. Zink, _Endlichkeitsbedingungen für Moduln über einem Noetherschen Ring_ , Math. Nachr. 164 (1974), 239–252. * [18] H. Zöschinger, _Koatomare Moduln_ , Math. Z. 170(1980) 221–232. * [19] H. Zöschinger, _Minimax Moduln_ , J. Algebra. 102(1986), 1–32. * [20] H. Zöschinger, _Über die Maximalbedingung für radikalvolle Untermoduln_ , Hokkaido Math. J. 17 (1988), 101–116. * [21] H. Zöschinger, _Über koassoziierte Primideale_ , Math Scand. 63(1988), 196–211.	arxiv-papers	2009-03-12T18:10:53	2024-09-04T02:49:01.106889	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Moharram Aghapournahr, Leif Melkersson", "submitter": "Moharram Aghapournahr", "url": "https://arxiv.org/abs/0903.2235" }
0903.2304	# Distinction of Tripartite Greenberger-Horne-Zeilinger and W States Entangled in Time (or Energy) and Space Jianming Wen111Electronic address: jianm1@umbc.edu Current address: Physics Department, University of Virginia, Charlottesville, Virginia 22904, USA and Morton H. Rubin Physics Department, University of Maryland, Baltimore County, Baltimore, Maryland 21250, USA ###### Abstract In tripartite discrete systems, two classes of genuine tripartite entanglement have been discovered, namely, the Greenberger-Horne-Zeilinger (GHZ) class and the W class. To date, much research effort has been concentrated on the polarization entangled three-photon GHZ and W states. Most studies of continuous variable multiparticle entanglement have been focused on Gaussian states. In this Brief Report, we examine two classes of three-photon entangled states in space and time. One class is a three-mode three-photon entangled state and the other is a two-mode triphoton state. These states show behavior similar to the GHZ and W states when one of the photons is not detected. The three-mode entangled state resembles a W state, while a two-mode three-photon state resembles a GHZ state when one of the photons is traced away. We characterize the distinction between these two states by comparing the second- order correlation functions $G^{(2)}$ with the third-order correlation function $G^{(3)}$. ###### pacs: 42.50.Dv, 03.65.Ud, 03.67.Mn, 01.55.+b ## I Introduction Generating entangled states is a primary task for the application of quantum information processing. The experimental preparation, manipulation, and detection of multiphoton entangled states is of great interest for the implementation of quantum communication schemes quantum cryptographic protocols, and for fundamental tests of quantum theory. Generation of entangled photon pairs has been demonstrated from the processes of spontaneous parametric down conversion (SPDC) SPDC1 ; SPDC2 ; SPDC3 and four-wave mixing wen . These paired photons have proved to be key elements in many research fields such as quantum computing, quantum imaging, and quantum lithography. Although entanglement of bipartite systems is well understood, the characterization of entanglement for multipartite systems is still under intense study. In entangled three-qubit states it has been shown that there are two inequivalent classes of states, under stochastic local operations and classical communications, namely, the Greenberger-Horne-Zeilinger (GHZ) class GHZ and the W class wstate . The GHZ class is a three qubit state of the form $\|\mathrm{GHZ}\rangle=\frac{1}{\sqrt{2}}(\|000\rangle+\|111\rangle)$, which leads to a conflict between local realism and nonstatistical predictions of quantum theory. Another three-qubit state, the W state, takes the form $\|\mathrm{W}\rangle=\frac{1}{\sqrt{3}}(\|100\rangle+\|010\rangle+\|001\rangle)$. It has been shown that this state is inequivalent to the GHZ state under stochastic local measurements and classical exchange of messages duer . The entanglement in the W state is robust against the loss of one qubit, while the GHZ state is reduced to a product of two qubits. That is, tracing over one of the three qubits in the GHZ state leaves $\frac{1}{2}(\|00\rangle\langle 00\|+\|11\rangle\langle 11\|)$, which is an unentangled mixture state. However, tracing out one qubit in the W state and the density matrix of the remaining qubits becomes $\frac{2}{3}\|\Psi^{+}\rangle\langle\Psi^{+}\|+\frac{1}{3}\|00\rangle\langle 00\|$, with $\|\Psi^{+}\rangle=\frac{1}{\sqrt{2}}(\|01\rangle+\|10\rangle)$ being a maximally entangled state of two qubits. It has been further shown that the W state allows for a generalized GHZ-like argument against the Einstein- Podolsky-Rosen type of elements of reality cabello . To date, much effort has been concentrated on the polarization entangled three-photon GHZ and W states. Experimental realizations of polarization entangled GHZ states and more recently W states have been performed in optical and trapped ion experiments experimentGHZ1 ; experimentGHZ2 ; experimentGHZ3 ; experimentGHZ4 ; experimentW1 ; experimentW2 ; experimentW3 . Recently, the study of continuous-variable (CV) multipartite entanglement was initiated in gaussian2 , where a scheme was suggested to create pure CV $N$-party entanglement using squeezed light and $N-1$ beam splitters. In gaussian a complete classification of trimode Gaussian states was with a necessary and sufficient condition for the separability to determine to which class a given state belongs. The CV analysis requires quadrature-type measurement; in this Brief Report we shall be interested in studying three-photon states using direct photon counting detection. We here consider three-photon GHZ-type and W-like states entangled in time and space, which differ from the CV characterization of gaussian2 . We will show that three-mode states, which we denote by $\|1,1,1\rangle$, are similar to W states, while two-mode states, denoted by $\|1,2\rangle$, resemble GHZ-type states. The distinction between these two states has been demonstrated by looking at the second-order coherence function $G^{(2)}$. For related work with emphasis on the entanglement properties of CV three particle Gaussian GHZ and W states, see gaussian3 ; njp . This research is of importance, not only for testing foundations of quantum theory, but also for many promising applications based on quantum entanglement CV ; imaging . ## II Triphoton W State To illustrate the distinction between $\|1,1,1\rangle$ and $\|1,2\rangle$ states, we start with the case in which the source produces three-photon entangled states in different modes. For simplicity, a monochromatic plane- wave pump beam is assumed to travel along the $\hat{z}$ direction in the medium producing a state at the output face of the medium given by $\displaystyle\|\Psi_{1}\rangle=\int{d}\omega_{1}d\omega_{2}d\omega_{3}\int{d}\vec{\alpha}_{1}d\vec{\alpha}_{2}d\vec{\alpha}_{3}\Phi(L\Delta)\delta(\omega_{1}+\omega_{2}+\omega_{3}-\Omega)H(\vec{\alpha}_{1}+\vec{\alpha}_{2}+\vec{\alpha}_{3})\|1_{\vec{k}_{1}},1_{\vec{k}_{2}},1_{\vec{k}_{3}}\rangle,$ (1) where $\Omega$ is the pump frequency, and $\omega_{j}$ with $\vec{\alpha}_{j}$ are the frequencies and transverse wave vectors of photons in mode $\vec{k}_{j}$, respectively. $\delta(\omega_{1}+\omega_{2}+\omega_{3}-\Omega)$ is the steady-state or the frequency phase-matching condition. The integral over the finite length $L$ of the system gives the longitudinal detuning function, $\Phi(L\Delta)$, which determines the natural spectral width of the triphoton state. The longitudinal detuning function, in the non-depleted pump approximation usually takes the form of $\displaystyle\Phi(x)=\frac{1-e^{-ix}}{ix}=\mathrm{sinc}\Big{(}\frac{x}{2}\Big{)}e^{-i(x/2)},$ (2) with $x=L\Delta$ and $\Delta=(\vec{k}_{p}-\vec{k}_{1}-\vec{k}_{2}-\vec{k}_{3})\cdot\hat{z}$, and $\vec{k}_{p}$ is the wave vector of the input pump field. Let $\omega_{j}=\Omega_{j}+\nu_{j}$ with fixed frequency $\Omega_{j}$. Choosing the central frequencies so that $\Omega=\Omega_{1}+\Omega_{2}+\Omega_{3}$, frequency phase matching now becomes $\nu_{1}+\nu_{2}+\nu_{3}=0$. Assuming $\|\nu_{j}\|<<\Omega_{j}$, and that the crystal is cut for collinear phase matching, $k_{p}=K_{1}+K_{2}+K_{3}$, we can expand $k_{j}$ in powers of $\nu_{j}$, $k_{j}=K_{j}+\nu_{j}/u_{j}+\cdots$ where $1/u_{j}$ is the group velocity of the photon $j$ evaluated at $\Omega_{j}$. Then to leading order we may write $x$ as $x=-\sum_{j=1}^{3}L\nu_{j}/u_{j}=-\nu_{1}L/D_{12}-\nu_{3}L/D_{32},$ (3) where we have used frequency phase matching to eliminate $\nu_{2}$, and $1/D_{ij}$ is the time difference between the $i$th photon and the $j$th one passing through a unit length material. With a slight abuse of notation, we shall write $\Phi(L\Delta)=\Phi(\nu_{1},\nu_{3})$. The integration over the transverse coordinates ($\vec{\rho}$) on the output surface(s) of the source gives the transverse detuning function as $\displaystyle H(\vec{\alpha}_{1}+\vec{\alpha}_{2}+\vec{\alpha}_{3})=\frac{1}{A}\int{d}\vec{\rho}e^{i\vec{\rho}\cdot(\vec{\alpha}_{1}+\vec{\alpha}_{2}+\vec{\alpha}_{3})}.$ (4) In the ideal case, $H$ becomes a $\delta$-function, $\delta(\vec{\alpha}_{1}+\vec{\alpha}_{2}+\vec{\alpha}_{3})$. In Eq. (1) we use the paraxial approximation, which is a good approximation for quantum imaging and lithography rubin ; wen12 . With the quasi-monochromatic assumption $\|\nu_{j}\|<<\Omega_{j}$ this leads to the factoring of the state into longitudinal and transverse degrees of freedom in the quasi-monochromatic approximation. We are interested in examining the temporal and spatial correlations between two subsystems by tracing the third in the free- propagation geometry. The second-order [$G^{(2)}$] and third-order [$G^{(3)}$] correlation functions are defined, respectively, as $\displaystyle G^{(2)}$ $\displaystyle=$ $\displaystyle\sum_{\vec{k}_{3}}\|\langle 0\|a_{\vec{k}_{3}}E^{(+)}_{2}E^{(+)}_{1}\|\Psi_{1}\rangle\|^{2},$ (5) $\displaystyle G^{(3)}$ $\displaystyle=$ $\displaystyle\|\langle 0\|E^{(+)}_{3}E^{(+)}_{2}E^{(+)}_{1}\|\Psi_{1}\rangle\|^{2},$ (6) with freely propagating electric fields given by $\displaystyle E^{(+)}_{j}(\vec{\rho}_{j},z_{j},t_{j})=\int{d}\omega_{j}\int{d}\vec{\alpha}_{j}E_{j}f_{j}(\omega_{j})e^{-i\omega_{j}t_{j}}e^{i(k_{j}z_{j}+\vec{\alpha}_{j}\cdot\vec{\rho}_{j})}a_{\vec{k}_{j}},$ (7) where $E_{j}=\sqrt{\hbar\omega_{j}/2\epsilon_{0}}$, $k_{j}=\omega_{j}/c$ is the wave number, $z_{j}$ and $\vec{\rho}_{j}$ are spatial coordinates of the $j$th detector, and $a_{\vec{k}_{j}}$ is a photon annihilation operator at the output surface of the source and obeys $[a_{\vec{k}},a^{\dagger}_{\vec{k}^{\prime}}]=\delta(\vec{\alpha}-\vec{\alpha}^{\prime})\delta(\omega-\omega^{\prime})$, respectively. The function $f_{j}(\omega)$ is a narrow bandwidth filter function which is assumed to be peaked at $\Omega_{j}$. In Eq. (7) we have decomposed $\vec{k}_{j}$ into $k_{j}\hat{z}+\vec{\alpha}_{j}$. Substituting Eqs. (1) and (7) into (5) gives $\displaystyle G^{(2)}=C_{0}G^{(2)}_{l}(\tau_{1}-\tau_{2})\times{G}^{(2)}_{t}(\vec{\rho}_{1}-\vec{\rho}_{2}),$ (8) where $C_{0}$ is a slowly varying constant, and the temporal and spatial correlations, respectively, are $\displaystyle G^{(2)}_{l}(\tau_{1}-\tau_{2})$ $\displaystyle=$ $\displaystyle\int{d}\nu_{3}\bigg{\|}\int{d}\nu_{1}f_{1}(\nu_{1})f_{2}(\nu_{1}+\nu_{3})\Phi(\nu_{1},\nu_{3})e^{-i\nu_{1}(\tau_{1}-\tau_{2})}\bigg{\|}^{2},$ (9) $\displaystyle G^{(2)}_{t}(\vec{\rho}_{1}-\vec{\rho}_{2})$ $\displaystyle=$ $\displaystyle\int{d}\vec{\alpha}_{3}\bigg{\|}\int{d}\vec{\alpha}_{1}e^{i\vec{\alpha}_{1}\cdot(\vec{\rho}_{1}-\vec{\rho}_{2})}\bigg{\|}^{2},$ (10) where $\tau_{j}=t_{j}-z_{j}/c$ and $\omega_{j}=\Omega_{j}+\nu_{j}$. Similarly, plugging Eqs. (1) and (7) into (6) yields $\displaystyle G^{(3)}=C_{1}G^{(3)}_{l}(\tau_{1}-\tau_{2},\tau_{3}-\tau_{2})\times{G}^{(3)}_{t}(\vec{\rho}_{1}-\vec{\rho}_{2},\vec{\rho}_{3}-\vec{\rho}_{2}),$ (11) where the third-order temporal and spatial correlations are $\displaystyle G^{(3)}_{l}(\tau_{1}-\tau_{2},\tau_{3}-\tau_{2})$ $\displaystyle=$ $\displaystyle\bigg{\|}\int{d}\nu_{1}d\nu_{3}f_{1}(\nu_{1})f_{2}(\nu_{1}+\nu_{3})f_{3}(\nu_{3})\Phi(\nu_{1},\nu_{3})e^{-i\nu_{1}(\tau_{1}-\tau_{2})}e^{-i\nu_{3}(\tau_{3}-\tau_{2})}\bigg{\|}^{2},$ (12) $\displaystyle G^{(3)}_{t}(\vec{\rho}_{1}-\vec{\rho}_{2},\vec{\rho}_{3}-\vec{\rho}_{2})$ $\displaystyle=$ $\displaystyle\bigg{\|}\int{d}\vec{\alpha}_{1}d\vec{\alpha}_{3}e^{i\vec{\alpha}_{1}\cdot(\vec{\rho}_{1}-\vec{\rho}_{2})}e^{i\vec{\alpha}_{3}\cdot(\vec{\rho}_{3}-\vec{\rho}_{2})}\bigg{\|}^{2},$ (13) and $C_{1}$ is constant. By comparing Eq. (9) with (12), it is clear that although one photon is not detected (traced away) in the two-photon detection, there remains a correlation between the remaining two photons. The width of the two-photon temporal correlation depends on the three photon bandwidth. The comparison between Eqs. (10) and (13) indicates that the spatial correlation between two photons is limited by the bandwidth of the transverse modes. Ideally, point-to-point correlation is achieved by assuming infinite transverse bandwidth. Combining the temporal and spatial properties together show that the $\|1,1,1\rangle$ state (1) is a W state entangled in time and space, which is robust against one-photon loss. Figure 1: (color online) Temporal correlations of $G^{(3)}_{l}$ and $G^{(2)}_{l}$ for the $\|1,1,1\rangle$ state normalized to unity at their origin. The units of $\tau_{ij}=\tau_{i}-\tau_{j}$ are 10 ps. (a) Third-order temporal correlation $G^{(3)}_{l}(\tau_{12},\tau_{32})$. (b) Conditional third-order correlation $G^{(3)}_{l}(\tau_{12})$ obtained by setting $\tau_{32}=-\tau_{12}+\|L/D_{12}\|$. (c) Second-order temporal correlation $G^{(2)}_{l}(\tau_{12})$. The corresponding parameters are chosen as $L/2D_{ij}=10$ ps and all the filters are Gaussian with the same bandwidth of 0.4 THZ. There are several schemes which might produce such a state. One scheme is three-photon cascade emission whose spectral properties have been analyzed in chekhova . Another configuration utilizes two parametric down conversions and one up-conversion to create a triphoton state, as proposed by Keller et al keller . The transverse properties of triphotons generated from such a case have been studied in wen1 by considering quantum imaging experiments. It was shown that by implementing two-photon imaging, the quality of the images is limited by the bandwidth of the transverse modes of the non-detected third photon. In Fig. 1 we have compared the temporal correlations between the third-order correlation function $G^{(3)}_{l}(\tau_{12},\tau_{32})$ and the second-order $G^{(2)}_{l}(\tau_{12})$ with Gaussian filters in Eqs. (9) and (12). The filters were taken to the same bandwidth which is large compared to the width of the $\Phi(L\Delta)$ function. The plots have been normalized with respect to their maximum value. In generating the figure $D_{12}$ has been taken equal to $D_{32}$ and they have both been taken to be negative. Because of this the plot of $G^{(3)}$ is symmetric around the line $\tau_{12}=\tau_{32}$, and only positive values of the $\tau_{ij}$ are physically allowed. The length of $G^{(3)}$ is determined by the phase matching function $\Phi$ as illustrated in Fig. 1(a); Fig. 1(b) shows the conditional measurement of $G^{(3)}_{l}(\tau_{12})$ obtained by setting $\tau_{32}=-\tau_{12}+\|L/D_{12}\|$. The width of $G^{(3)}_{l}$ is determined by the filters. In Fig. 1(c) the second-order temporal correlation $G^{(2)}_{l}(\tau_{12})$ is plotted. The width of $G^{(2)}_{l}(\tau_{12})$ is larger than that of the conditional $G^{(3)}_{l}(\tau_{12})$ reflecting the lack of cutoff of the bandwidth for the non-detected third photon. ## III Triphoton GHZ State After analyzing the properties of the $\|1,1,1\rangle$ state, we now consider the case in which the source produces three-photon entangled states with a pair of degenerate photons of the form wen2 $\displaystyle\|\Psi_{2}\rangle=\int{d}\omega_{1}d\omega_{2}\int{d}\vec{\alpha}_{1}d\vec{\alpha}_{2}\Phi(x)\delta(2\omega_{1}+\omega_{2}-\Omega)\delta(2\vec{\alpha}_{1}+\vec{\alpha}_{2})\|2_{\vec{k}_{1}},1_{\vec{k}_{2}}\rangle,$ (14) where $\Phi$ characterizes the natural bandwidth of triphotons and has the same form as Eq. (2), with $x=-2L\nu_{1}/D_{12}$, $\omega_{j}=\Omega_{j}+\nu_{j}$, and $\vec{\alpha}_{j}$ are the frequencies and transverse wave vectors of the degenerate $(j=1)$ and nondegenerate $(j=2)$ photons. In wen2 we show that by sending two degenerate photons to the target while keeping the non-degenerate one traversing the imaging lens, a factor-of-2 spatial resolution improvement can be obtained, beyond the Rayleigh diffraction limit. Before proceeding with the discussion, we note that the major difference between the $\|1,2\rangle$ state [Eq. (14)] and $\|1,1,1\rangle$ [Eq. (1)] is that the $\|1,1,1\rangle$ state has more degrees of freedom than the $\|1,2\rangle$ state. This is the source of the difference between two states when performing two-photon detection, as we shall see. Physically, because two of the photons are degenerate, the measurement of one of them separately uniquely determines the state of the other one and the two photon state becomes a product state. This is true even if the photon is not measured but can be measured separately in principle. The effect of this is that the state generated is a mixed state. Note that for the completely degenerate case, a similar argument implies that tracing away one of the photons gives a mixed two-photon state. For the two-photon measurement here, we first assume that one of the degenerate photons is not detected. The second-order $G^{(2)}$ and third-order $G^{(3)}$ correlation functions now become $\displaystyle G^{(2)}$ $\displaystyle=$ $\displaystyle\sum_{\vec{k}_{1}}\|\langle 0\|a_{\vec{k}_{1}}E^{(+)}_{2}E^{(+)}_{1}\|\Psi_{2}\rangle\|^{2},$ (15) $\displaystyle G^{(3)}$ $\displaystyle=$ $\displaystyle\|\langle 0\|E^{(+)}_{2}[E^{(+)}_{1}]^{2}\|\Psi_{2}\rangle\|^{2},$ (16) where $E^{(+)}_{j}$ is the free-space electric field given in Eq. (7). Note that because of the degeneracy, a two-photon detector is necessary for three- photon joint detection wen2 . Following the same procedure for the $\|1,1,1\rangle$ calculation, it is easy to show that the second-order and third-order correlation functions are $\displaystyle G^{(2)}_{l}$ $\displaystyle=$ $\displaystyle\int{d}\nu_{1}\bigg{\|}f_{1}(\nu_{1})f_{2}(\nu_{1})\Phi(-2\nu_{1}/D_{12})\bigg{\|}^{2},$ (17) $\displaystyle G^{(3)}_{l}(\tau_{12})$ $\displaystyle=$ $\displaystyle\bigg{\|}\int{d}\nu_{1}f^{2}_{1}(\nu_{1})f_{2}(\nu_{1})\Phi(-2\nu_{1}/D_{12})e^{-2i\nu_{1}\tau_{12}}\bigg{\|}^{2},$ (18) in the temporal domain, and $\displaystyle G^{(2)}_{t}$ $\displaystyle=$ $\displaystyle\int{d}\vec{\alpha}_{1},$ (19) $\displaystyle G^{(3)}_{t}(\vec{\rho}_{1}-\vec{\rho}_{2})$ $\displaystyle=$ $\displaystyle\bigg{\|}\int{d}\vec{\alpha}_{1}e^{2i\vec{\alpha}_{1}\cdot(\vec{\rho}_{1}-\vec{\rho}_{2})}\bigg{\|}^{2},$ (20) in the spatial space. Comparing Eqs. (17) and (19) with (18) and (20) shows that if one of the degenerate photons is traced away, there will be no correlation between the remaining photons, which is the property of tripartite GHZ state. Indeed, one can easily show that the $\|1,2\rangle$ state (14) always reduces to a product state, if one photon is not measured. The reason for this is that if one photon is traced away, then the remaining photons is put into a definite mode because of our assumption of perfect phase matching and the resulting state is a mixed state of the form $\rho=\sum_{\vec{k}}\|F(\vec{k})\|^{2}\|\vec{k}_{p}-\vec{k},\vec{k}\rangle\langle\vec{k}_{p}-\vec{k},\vec{k}\|.$ (21) Recently, we have found that to some extent, the $\|1,1,1\rangle$ state can mimic some properties of the $\|1,2\rangle$ state, e.g., by sending two nearly degenerate photons in the $\|1,1,1\rangle$ state to the object while propagating the third one through the imaging lens in the quantum imaging configuration, a factor-of-2 spatial resolution enhancement is achievable in the coincidence counting measurement. However, the Gaussian lens equation is not the same as that with the $\|1,2\rangle$ state and more importantly, the physics behind these two imaging processes is quite different. ## IV Conclusion In summary, we have shown that the triphoton $\|1,1,1\rangle$ state is analogous to a W state, while the $\|1,2\rangle$ state is analogus to a GHZ state by comparing the third-order and second-order correlation functions in both temporal and spatial domains. Our analysis on these state properties may be important to not only the understanding of multipartite systems but also the technologies based on quantum entanglement. For example, in Refs. wen1 and wen2 we have discussed quantum imaging using these two classes of states and have found different spatial resolutions in application. The essential difference between these two states is that $\|1,1,1\rangle$ has a larger Hilbert spaces than $\|1,2\rangle$. Specifically, measurement of one of the degenerate photons in the GHZ-type state allows for the possibility of a separate measurement of the degenerate photon state. This reduces the two- photon state to a mixed state. For the W-like state, only partial information can in principle be obtained and so some entanglement remains. The authors wish to thank their colleague Kevin McCann for help with the numerical computations for the figure. We thank one of the referee’s for pointing out reference njp to us. We acknowledge the financial support in part by U.S. ARO MURI Grant W911NF-05-1-0197. ## References * (1) M. H. Rubin, D. N. Klyshko, Y.-H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122 (1994). * (2) Y.-H. Shih, Rep. Prog. Phys. 66, 1009 (2003). * (3) D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach Science, New York, 1988). * (4) S. Du, J.-M. Wen, and M. H. Rubin, J. Opt. Soc. Am. B 25, C98 (2008); J.-M. Wen and M. H. Rubin, Phys. Rev. A 74, 023808 (2006); 74, 023809 (2006); J.-M. Wen, S. Du, and M. H. Rubin, ibid. 75, 033809 (2007); 76, 013825 (2007); J.-M. Wen, S. Du, Y. P. Zhang, M. Xiao, and M. H. Rubin, ibid. 77, 033816 (2008). * (5) D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos (Dordrecht, Kluwer, 1989). * (6) A. Zeilinger, M. A. Horne, and D. M. Greenberger, NASA conf. Publ. No. 3135 (National Aeronautics and Space Administration, code NTT, Washington D.C., 1997). * (7) W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000). * (8) A. Cabello, Phys. Rev. A 65, 032108 (2002). * (9) D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345 (1999). * (10) J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, Nature (London) 403, 515 (2000). * (11) K. J. Resch, P. Walther, and A. Zeilinger, Phys. Rev. Lett. 94, 070402 (2005). * (12) Y.-A. Chen, T. Yang, A.-N. Zhang, Z. Zhao, A. Cabello, and J.-W. Pan, Phys. Rev. Lett. 97, 170408 (2006). * (13) M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter, Phys. Rev. Lett. 92, 077901 (2004). * (14) C. F. Roos, M. Riebe, H. Häffner, W. Hänsel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, Sience 304, 1478 (2004). * (15) H. Mikami, Y. Li, K. Fukuoka, and T. Kobayashi, Phys. Rev. Lett. 95, 150404 (2005). * (16) P. van Loock and S. L. Braunstein, Phys. Rev. Lett. 84, 3482 (2000); Phys. Rev. A 63, 022106 (2001). * (17) G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, Phys. Rev. A 64, 052303 (2001). * (18) P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 (2003). * (19) G. Adesso and F. Illluminati, New J. Phys. 8, 15 (2006). * (20) S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005). * (21) Y.-H. Shih, IEEE J. Sel. Top. Quantum Electron. 13, 1016 (2007). * (22) M. H. Rubin, Phys. Rev. A 54, 5349 (1996). * (23) J.-M. Wen, M. H. Rubin, and Y.-H. Shih, Phys. Rev. A 76, 045802 (2007). * (24) M. V. Chekhova, O. A. Ivanova, V. Berardi, and A. Garuccio, Phys. Rev. A 72, 023818 (2005). * (25) T. E. Keller, M. H. Rubin, Y.-H. Shih, and L.-A. Wu, Phys. Rev. A 57, 2076 (1998). * (26) J.-M. Wen, P. Xu, M. H. Rubin, and Y.-H. Shih, Phys. Rev. A 76, 023828 (2007). * (27) J.-M. Wen, M. H. Rubin, and Y.-H. Shih, submitted to Phys. Rev. A (2008), quant-ph/arXiv:0812.2032	arxiv-papers	2009-03-13T01:29:57	2024-09-04T02:49:01.112079	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jianming Wen and Morton H. Rubin", "submitter": "Jianming Wen", "url": "https://arxiv.org/abs/0903.2304" }
0903.2334	1–5 # Habitable Zones for Earth-mass Planets in Multiple Planetary Systems Jianghui JI1 Lin LIU2 Hiroshi KINOSHITA3 Guangyu LI1 1Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China email: jijh@pmo.ac.cn 2Department of Astronomy, Nanjing University, Nanjing 210093, China email: xhliao@nju.edu.cn 3National Astronomical Observatory, Mitaka, Tokyo 181-8588, Japan (2007) ###### Abstract We perform numerical simulations to study the Habitable zones (HZs) and dynamical structure for Earth-mass planets in multiple planetary systems. For example, in the HD 69830 system, we extensively explore the planetary configuration of three Neptune-mass companions with one massive terrestrial planet residing in 0.07 AU $\leq a\leq$ 1.20 AU, to examine the asteroid structure in this system. We underline that there are stable zones of at least $10^{5}$ yr for low-mass terrestrial planets locating between 0.3 and 0.5 AU, and 0.8 and 1.2 AU with final eccentricities of $e<0.20$. Moreover, we also find that the accumulation or depletion of the asteroid belt are also shaped by orbital resonances of the outer planets, for example, the asteroidal gaps at 2:1 and 3:2 mean motion resonances (MMRs) with Planet C, and 5:2 and 1:2 MMRs with Planet D. In a dynamical sense, the proper candidate regions for the existence of the potential terrestrial planets or HZs are 0.35 AU $<a<$ 0.50 AU, and 0.80 AU $<a<$ 1.00 AU for relatively low eccentricities, which makes sense to have the possible asteroidal structure in this system. ###### keywords: methods:$n$-body simulations-planetary systems-stars:individual (HD69830, 47 UMa) ††volume: 249††journal: Exoplanets: Detection, Formation and Dynamics††editors: Sylvio Ferraz-Mello, Yi-Sui Sun & Ji-lin Zhou, eds. ## 1 Introduction To date, over 260 extrasolar planets have been discovered around the nearby stars within 200 pc (Butler et al. 2006; The Extrasolar Planets Encyclopaedia111As of Nov. 8, 2007, see http://exoplanet.eu/catalog.php and http://exoplanets.org/) mostly by the measurements of Doppler surveys and transiting techniques. The increasing numbers of known extrasolar planets are largely attributed to increasing precision in measurement techniques. Observational improvements will likely lead to more substantial discoveries, including: (1)diverse multi-planetary systems, of which more than 20 multiple systems with orbital resonance or secular interactions are already known;(2)low-mass companions around main-sequence stars (so-called super- Earths), e.g., 55 Cancri (McArthur et al. 2004), GJ 876 (Rivera et al. 2005), HD 160691 (Santos et al. 2004; Gozdziewski et al. 2007); (3)a true Solar System analog, with several terrestrial planets, asteroidal structure and a dynamical environment consistent with terrestrial planets in the Habitable Zone (HZ) (Kasting et al. 1993) that could permit the development of life, e.g., Gl 581 (von Bloh et al. 2007); (4)a comprehensive census of a diversity of planetary systems, which will provide abundant clues for theorists to more accurately model planetary formation processes (Ida & Lin 2004; Boss 2006). Lovis et al. (2006) (hereafter Paper I) reported the discovery of an interesting system of three Neptune-mass planets orbiting about HD 69830 through high precision measurements with the HARPS spectrograph at La Silla, Chile. The nearby star HD 69830 is of spectral type K0V with an estimated mass of $0.86\pm 0.03M_{\odot}$ and a total luminosity of $0.60\pm 0.03L_{\odot}$ (Paper I), about 12.6 pc away from the Sun. In addition, Beichman et al. (2005) announced the detection of a large infrared excess owing to hot grains of crystalline silicates orbiting the star HD 69830 and inferred that there could be a massive asteroid within 1 AU. Subsequently, Alibert et al. (2006) and Paper I performed lots of calculations to simulate the system and revealed that the innermost planet may possess a rocky core surrounded by a tiny gaseous envelope. This planet probably formed inside the ice line in the beginning, whereas the two outer companions formed outside the ice line from a rocky embryo and then accreted the water and gas onto the envelope in the subsequent formation process. Hence, it is important for one to understand the dynamical structure in the final assemblage of the planetary system (Asghari et al. 2004; Ji et al. 2005), and to investigate suitable HZs for life-bearing terrestrial planets (Jones et al. 2005; Raymond et al. 2006; von Bloh et al. 2007; Gaidos et al. 2007) advancing the space missions (such as CoRot, Kepler and TPF) aiming at detecting them, thus one of our goals is to focus on the issues of the potential Earth-mass planets in the system. ## 2 Dynamical Structure and Habitable Zones in HD 69830 system Modern observations by Spitzer and HST indicate that circumstellar debris disks (e.g., AU Mic and $\beta$ Pic) are quite common in the early planetary formation. Beichman et al. (2006) used Spitzer to show that $13\pm 3\%$ of mature main sequence stars exhibit Kuiper Belt analogs. They further point out that the existence of debris disks is extremely important for the resulting detection of individual planets, and related to the formation and evolution of planetary systems. As mentioned previously, Beichman et al. (2005) also provide clear evidence of the presence of the disk in HD 69830. Subsequently Paper I’s best-fit orbital solutions were for three Neptune-mass planets with well-separated nearly-circular orbits, which may imply that the HD69830 system is similar to our Solar System in that it is dynamically consistent with the possible presence of terrestrial planets and asteroidal and Kuiper belt structures. Hence, it deserves to make a detailed investigation from a numerical perspective. Figure 1: Left panel: Contour of the surviving time for Earth-like planets for the integration of $10^{5}$ yr. Right panel: Status of their final eccentricities. Horizontal and vertical axes are the initial a and e. Stable zones for the low-mass planets in the region between 0.3 and 0.5 AU, and 0.8 and 1.2 AU with final low eccentricities. To investigate the dynamical structure and potential HZs in this system, we performed additional simulations with HD69830’s three Neptune-mass companions in coplanar orbits, and one massive Earth-like planet. In the runs, the mass of the assumed terrestrial planet ranges from 0.01 $M_{\oplus}$ to 1 $M_{\oplus}$. The initial orbital parameters are as follows: the numerical investigations were carried out in $[a,e]$ parameter space by direct integrations, and for a uniform grid of 0.01 AU in semi-major axis (0.07 AU $\leq a\leq$ 1.20 AU) and 0.01 in eccentricity ($0.0\leq e\leq 0.20$), the inclinations are $0^{0}<I<5^{0}$, and the angles of the nodal longitude, the argument of periastron, and the mean anomaly are randomly distributed between $0^{0}$ and $360^{0}$ for each orbit, then each terrestrial mass body was numerically integrated with three Neptune-mass planets in the HD 69830 system. In total, about 2400 simulations were exhaustively run for typical integration time spans from $10^{5}$ to $10^{6}$ yr (about $10^{6}-10^{7}$ times the orbital period of the innermost planet) (see also Ji et al. 2007 for details). Figure 1 shows the contours of the survival time for Earth-like planets (left panel) and the status of their final eccentricities (right panel) for the integration over $10^{5}$ yr, where horizontal and vertical axes are the initial $a$ and $e$. The left panel displays that there are stable zones for a terrestrial planet in the regime between 0.3 and 0.5 AU, and 0.8 and 1.2 AU with final eccentricities of $e<0.20$. Obviously, unstable zones exist near the orbits of the three Neptune-mass planets where the planetary embryos have short dynamical survival time, and their eccentricities can quickly be pumped up to a high value $\sim$ 0.9 (right panel). In these regions the evolution is insensitive to the initial masses. The terrestrial bodies are related to many of the mean motion resonances of the Neptunian planets and the overlapping resonance mechanism (Murray & Dermott 1999) can reveal their chaotic behaviors of being ejected from the system in short dynamical lifetime. Furthermore, most of terrestrial orbits are within $3R_{hill}$ sphere of the Neptune-mass planets, and others are involved in the secular resonance with two inner companions. Analogous to our Solar system, if we consider the middle planet (HD 69830 c) as the counterpart as Jupiter, we will have the regions of mean motion resonances: 2:1 (0.117 AU), 3:2 (0.142 AU), 3:1 (0.089 AU) and 5:2 (0.101 AU), 2:3 (0.244 AU). In Fig. 1, we notice there indeed exist the apparent asteroidal gaps about or within the above MMRs (e.g., 3:1 and 5:2 MMRs), while in the region between 0.10 AU and 0.14 AU for $e<0.10$, there are stable islands where the planetary embryos can last at least $10^{5}$ yr. In addition, for Planet D, most of the terrestrial planets in 0.50 AU $<a<$ 0.80 AU are chaotic and their eccentricities are excited to moderate and even high values, the characterized MMRs with respect to the accumulation or depletion of the asteroid belt are 3:2 (0.481 AU), 2:1 (0.397 AU), 5:2 (0.342 AU), 4:3 (0.520 AU), 1:1 (0.630 AU), 2:3 (0.826 AU), 1:2 (1.000 AU), and our results enrich those of Paper I for massless bodies over two consecutive 1000-year intervals, showing a broader stable region beyond 0.80 AU. Note that there exist stable Trojan terrestrial bodies in a narrow stripe about 0.630 AU, involved in 1:1 MMR with Planet D, and they can survive at least $10^{6}$ yr with resulting small eccentricities in the extended integrations. The stable Trojan configurations may possibly appear in the extrasolar planetary systems (see also Dvorak 2007; Psychoyos & Hadjidemetriou 2007 in this issue), e.g., Ji et al. (2005) explored such Trojan planets orbiting about 47 Uma, and Gozdziewski & Konacki (2006) also argued that there may exist Trojan pair configurations in the HD 128311 and HD 82943 systems. Ford & Gaudi (2006) developed a novel method of detecting Trojan companions to transiting close-in extrasolar planets and argue that the terrestrial-mass Trojans may be detectable with present ground-based observatories. Terrestrial Trojan planets with low eccentricity orbits close to 1 AU could potentially be habitable, and are worthy of further investigation in the future. ## 3 Summary and Discussion In this work, we investigated the planetary configuration of three Neptune- mass companions similar to those surrounding HD 69830 and added one massive terrestrial planet in the region of 0.07 AU $\leq a\leq$ 1.20 AU to examine the dynamical stability of terrestrial mass planets and to explore the asteroid structure in this system. We show that there are stable zones of at least $10^{5}$ yr for the low-mass terrestrial planets located between 0.3 and 0.5 AU, and 0.8 and 1.2 AU with final eccentricities of $e<0.20$. Moreover, we also find that the accumulation or depletion of the asteroid belt is also shaped by orbital resonances of the outer planets, for example, the asteroidal gaps of 2:1 and 3:2 MMRs with Planet C, and 5:2 and 1:2 resonances with Planet D. On the other hand, the stellar luminosity of HD 69830 is lower than that of the Sun, thus the HZ should shift inwards compared to our Solar System. In a dynamical consideration, the proper candidate regions for the existence of the potential terrestrial planets or HZs are 0.35 AU $<a<0.50$ AU, and 0.80 AU $<a<1.00$ AU for relatively low eccentricities. Finally, we may summarize that the HD 69830 system can possess an asteroidal architecture resembling the Solar System and both the mean motion resonance (MMR) and secular resonances will work together to influence the distribution of the small bodies in the planetary system. In other simulations, we also show the potential Habitable zones for Earth-mass planets in the 47 UMa planetary system (see Ji et al. 2005), and the results imply that future space-based observations, e.g., CoRot, Kelper and TPF will hopefully produce a handful of samples belonging to the category of the terrestrial bodies. ###### Acknowledgements. We thank the anonymous referee for informative comments and suggestions that helped to improve the contents. This work is financially supported by the National Natural Science Foundations of China (Grants 10573040, 10673006, 10203005, 10233020) and the Foundation of Minor Planets of Purple Mountain Observatory. ## References * [Alibert et al.(2006)] Alibert, Y., et al. 2006, A&A, 455, L25 * [Asghari (2004)] Asghari, N., et al. 2004, A&A, 426, 353 * [Beichman et al.(2005)] Beichman, C. A., et al. 2005, ApJ, 626, 1061 * [Beichman et al.(2006)] Beichman, C. A., et al. 2006, ApJ, 652, 1674 * [Boss(2006)] Boss, A. P. 2006, ApJ, 644, L79 * [Butler et al. (2006)] Butler, R. P., et al. 2006, ApJ, 646, 505 * [Dvorak et al. (2007)] Dvorak, R. 2007, IAU S249, this issue * [Ford et al. (2006)] Ford, E. B., &, Gaudi, B. S. 2006, ApJ, 652, L137 * [Gaidos et al.(2007)] Gaidos, E., et al. 2007, Science, 318, 210 * [Gozdziewski(2006)] Gozdziewski, K., & Konacki, M. 2006, ApJ, 647, 573 * [Gozdziewski (2007)] Gozdziewski, K, et al. 2007, ApJ, 657, 546 * [Ida & Lin(2004)] Ida, S., & Lin, D. N. C. 2004, ApJ, 604, 388 * [Ji et al.(2005)] Ji, J., Liu, L., Kinoshita, H., & Li, G.Y. 2005, ApJ, 631, 1191 * [Ji et al.(2007)] Ji, J., Kinoshita, H., Liu, L., & Li, G.Y. 2007, ApJ, 657, 1092 * [Jones et al.(2005)] Jones, B. W., Underwood, D. R., & Sleep, P. N. 2005, ApJ, 622, 1091 * [Kasting et al.(1993)] Kasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993, Icarus, 101, 108 * [Lovis et al. (2006)] Lovis, C., et al. 2006, Nature, 441, 305 (Paper I) * [McArthur et al. (2004)] McArthur, B.E., et al. 2004, ApJ, 614, L81 * [Murray (1999)] Murray, C. D., & Dermott, S. F. 1999, Solar System Dynamics (New York: Cambridge Univ. Press) * [Psychoyos et al. (2007)] Psychoyos, D., &, Hadjidemetriou, J. D. 2007, IAU S249, this issue * [Raymond et al.(2006)] Raymond, S. N., Mandell, A. M., & Sigurdsson, S. 2006, Science, 313, 1413 * [Rivera et al.(2005)] Rivera, E. J., et al. 2005, ApJ, 634, 625 * [Santos et al. (2004)] Santos, N.C., et al. 2004, A&A, 426, L19 * [von Bloh et al.(2007)] von Bloh, W., Bounama, C., Cuntz, M., & Franck, S. 2007, arXiv:0705.3758	arxiv-papers	2009-03-13T09:05:12	2024-09-04T02:49:01.116302	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ji Jianghui (1,2), Liu Lin (3), H. Kinoshita (4), Li Guangyu (1,2)\n ((1)Purple Mountain Observatory, CAS (2)NAOC, (3)Nanjing Univ., (4)NAOJ)", "submitter": "Jianghui Ji", "url": "https://arxiv.org/abs/0903.2334" }
0903.2456	# Atomistic origins of the phase transition mechanism in Ge2Sb2Te5 Juarez L. F. Da Silva,1 Aron Walsh,1 Su-Huai Wei,1 and Hosun Lee2 1National Renewable Energy Laboratory, 1617 Cole Blvd., Golden, CO 80401, USA 2Dept. of Applied Physics, Kyung Hee University, Suwon 446-701, South Korea ###### Abstract Combined static and molecular dynamics first-principles calculations are used to identify a direct structural link between the metastable crystalline and amorphous phases of Ge2Sb2Te5. We find that the phase transition is driven by the displacement of Ge atoms along the rocksalt [111] direction from the stable-octahedron to high-energy-unstable tetrahedron sites close to the intrinsic vacancy regions, which give rise to the formation of local 4-fold coordinated motifs. Our analyses suggest that the high figures of merit of Ge2Sb2Te5 are achieved from the optimal combination of intrinsic vacancies provided by Sb2Te3 and the instability of the tetrahedron sites provided by GeTe. Crystalline-amorphous phase transition, mechanism, density functional theory, Ge2Sb2Te5 ###### pacs: 61.43.-j,61.50.Ks,71.15.Nc ††preprint: GST compounds Ternary (GeTe)m(Sb2Te3)n materials, in particular the Ge2Sb2Te5 (GST) composition, have been considered as the most natural candidates for non- volatile memory applications through exploiting the fast and reversible resistance change between a metastable (m-GST) crystalline phase (low resistivity) and an amorphous (a-GST) phase (high resistivity). Ovshinsky (1968); Yamada et al. (1991); Wuttig and Yamada (2007); Chong et al. (2008) However, the mechanism of the phase transition is still under intense debate. The existing models,Kolobov et al. (2004, 2006); Welnic et al. (2006); Hegedüs and Elliott (2008) have provided a preliminary understanding of the transition mechanism, but fail to provide a clear and direct structural link between the m-GST and a-GST phases, which play a key role in the understanding of the reservible transition at an atomistic level. The m-GSTYamada and Matsunaga (2000); Njoroge et al. (2002); Matsunaga and Yamada (2004); Park et al. (2005); Sun et al. (2006); Wuttig et al. (2007); Shportko et al. (2008) phase crystallizes in a rocksalt-type (RS) structure, in which the Te atoms occupy the anion sites and Ge, Sb, and the naturally occurring intrinsic vacancies from Sb2Te3 (20% in GST) occupy the cation sites. It has been suggested that a-GST is characterized by the presence of 4-fold coordinated Ge atoms,Kolobov et al. (2004); Baker et al. (2006); Akola and Jones (2007); Caravati et al. (2007); Hegedüs and Elliott (2008); Jóv’ari et al. (2008) in which the sum of the occurrences GeTe4, Ge(SbTe3), and Ge(GeTe3) is about 66%.Jóv’ari et al. (2008) Ge$-$Ge and Ge$-$Sb bonds are found in those motifs, which is assumed to be due to disorder effects, since they are not present in the crystalline phases. Furthermore, a-GST shows a volume expansion of $6-7$% compared with the m-GST phase, Weidenhof et al. (1999); Njoroge et al. (2002) and it has a higher energy ($28-40$ meV/atom) with respect to m-GST.Kalb et al. (2003) Theoretically, first-principles molecular dynamics (MD) starting from a liquid phase with slow cooling rates have been used to generate a metastable phase (RS-type structure), however, no direct transition path was identified to link the proposed m-GST and a-GST phases. Thus, a new approach that connects the two phases at the atomistic level becomes highly desirable. Figure 1: (Color online) Structure models of the GST phases. (a) Metastable crystalline GST (m-GST). (b) m-GST (shift) structure, in which the Ge atoms occupy the 4-fold tetrahedron sites with lowest energy. (c) Amorphous GST obtained at zero temperature (am-GST) using modified m-GST structures (m-GST with Ge shift), in which the tetrahedron Ge sites were initially occupied. (d) Amorphous GST obtained by high temperature molecular dynamics DFT calculations (a-GST). The Ge, Sb, and Te atoms are indicated in green, blue, and red, respectively. In this work, using first-principles methods, we will address the following open questions: Is there a dominant structure link between both phases? What are the roles of GeTe and Sb2Te3 in GST? We will show in this Letter that combined static (zero temperature) and MD (high-temperature) first-principles calculations can explain the phase transition mechanism between the m-GST and a-GST phases. Moreover, our study shows that generating the amorphous phase from a known crystalline phase provides a better understanding of the structural relationship between both phases. Thus, it provides a new avenue for further study of amorphous materials phase change transitions. Our static total energy and MD calculations are based on the all-electron projected augmented wave (PAW) methodBlöchl (1994); Kresse and Joubert (1999) and density functional theory (DFT) within the generalized gradient approximation (GGA-PBE)Perdew et al. (1996) as implemented in VASP. Kresse and Hafner (1993); Kresse and Furthmüller (1996) To represent the metastable phase (RS-type structure), we employ a hexagonal $(2{\times}2{\times}1)$ unit cell, in which the Te atoms are stacked along of the [0001] direction. Da Silva et al. (2008) The MD calculations were performed employing cubic and hexagonal cells with 108 to 126 atoms. The total energies and equilibrium volumes for all structures in both crystalline and amorphous phases were obtained by full relaxation of the volume, shape, and atomic positions of the unit cell to minimize the quantum mechanical stresses and forces. To understand the phase transition, we first established the crystal structure of the m-GST phase,Da Silva et al. (2008) as shown in Fig. 1. The obtained structure is consistent with experimental results and provides new insights into m-GST.Yamada and Matsunaga (2000); Matsunaga and Yamada (2004); Park et al. (2005); Kolobov et al. (2004); Matsunaga et al. (2007) In this layered- structure the ordered intrinsic vacancies separate the building block units (GST), in which the Ge and Sb atoms are intermixed in planes. All Ge atoms are 6-fold coordinated in m-GST. However, it has been reported that up to one fifth of the Ge atoms are 4-fold coordinated with Te atoms in a-GST (GeTe4), while the remaining Ge atoms form 4-fold motifs with combined Ge, Sb, and Te atoms.Jóv’ari et al. (2008) We notice that tetrahedral Ge atoms can be obtained by shifting the octahedral Ge atoms in m-GST along the hexagonal $c$ direction, i.e., there are two tetrahedron sites for each Ge atom, Fig. 1. In order to identify the lowest energy tetrahedron sites, we calculated the energetics for the occupation of each site by Ge atoms. The lowest energy sites are located in the intrinsic vacancy regions, while the highest energy sites are located in the center of the GST building blocks, i.e., there is a strong preference for the four-fold Ge atoms to be located in or near intrinsic vacancy regions. Assuming that all the Ge atoms are shifted from their octahedron sites and occupy the lowest energy tetrahedron sites, we find that 50% of Ge will shift from the octahedra to tetrahedra along the [0001] direction, while the remaining 50% shift along of the opposite direction. The m-GST (shift) structure in which all Ge atoms occupy the tetrahedral sites according to the distribution of intrinsic vacancies and energy barriers is shown in Fig. 1b, which leads to the formation of Ge$-$Ge bonds. This configuration is highly unstable, and the system will relax without energy barrier to a lower energy phase (see am-GST structure in Fig. 1c). Figure 2: Pair-correlation functions of various GST phases. Amorphous GST obtained by molecular dynamic calculations (a-GST, black lines). Amorphous GST obtained from occupation of tetrahedron sites in m-GST and complete relaxation (am-GST, red lines). Meta-stable GST phase (m-GST, blue lines, scaled by 0.50). Table 1: Bond lengths (in Å) of Ge2Sb2Te5 (GST) in the amorphous and crystalline phases. \| Amorphous GST \| Meta-stable GST ---\|---\|--- \| a-GST \| am-GST \| Exp. \| m-GST \| Exp. Ge$-$Te \| 2.74 \| 2.79 \| $2.60-2.63$111Exp. Reference Jóv’ari et al., 2008; Kolobov et al., 2004; Baker et al., 2006. \| $2.87-3.24$ \| $2.83-3.15$222Exp. Reference Kolobov et al., 2004. Sb$-$Te \| 2.91 \| 2.96 \| $2.82-2.85$111Exp. Reference Jóv’ari et al., 2008; Kolobov et al., 2004; Baker et al., 2006. \| $2.96-3.30$ \| 2.91222Exp. Reference Kolobov et al., 2004. Te$-$Te \| 4.16 \| 4.28 \| \| $4.14-4.38$ \| 4.26222Exp. Reference Kolobov et al., 2004. Ge$-$Ge \| 2.63 \| 2.64 \| $2.47-2.48$111Exp. Reference Jóv’ari et al., 2008; Kolobov et al., 2004; Baker et al., 2006. \| $4.27-4.62$ \| Ge$-$Sb \| 2.79 \| 2.78 \| 2.69111Exp. Reference Jóv’ari et al., 2008; Kolobov et al., 2004; Baker et al., 2006. \| $4.23-4.53$ \| Sb$-$Sb \| 2.93 \| 2.92 \| \| $4.27-4.62$ \| To provide a more direct structural link between the m-GST and a-GST phases, we first generated a-GST structures using first-principles MD simulations at high temperatures, $T$, using the same approach adopted in previous a-GST studies.Akola and Jones (2007); Sun et al. (2007); Caravati et al. (2007); Hegedüs and Elliott (2008) Secondly, we generated several amorphous structures from modified m-GST structures, in which a percentage of the Ge atoms (100%, 75%, 50%, 25%) are shifted to the tetrahedron sites from the lower energy octahedron sites (m-GST with shift Ge). The goal is to show that amorphous structures obtained in this way (am-GST in Fig. 1) can preserve most of the structural features present in the a-GST generated by conventional MD calculations (a-GST in Fig. 1), and therefore provide a direct structural link and solid evidence to support the mechanisms that determines the phase transition from m-GST to a-GST. The amorphous structures obtained by both approaches are shown in Fig. 1. To quantify our analysis, we calculated the pair correlation (PC) functions, which are shown in Fig. 2. For the a-GST structures, the PC functions were averaged over five structures, while the PC functions of the am-GST structures were calculated for ten structures with different initial occupation of the Ge tetrahedron sites; the structure that provided the best agreement with the a-GST PC functions is shown in Fig. 2. Figure 3: Potential energy path for atomic displacements of Ge atoms along of the rocksalt (RS) [111] direction of GeTe. (a) Distorted RS structure. (b) Perfect RS structure. (c) Long Ge$-$Te bonds zincblende (ZB) structure. (d) Graphite like structure. (e) Perfect ZB structure. (f) RS-layer structure. Our PC function analysis shows that for all the am-GST structures, the one in which 50% of the Ge atoms are shifted from the octahedron to the tetrahedron sites along the hexagonal [0001] direction and the rest 50% moves along the $[000\bar{1}]$ direction reproduces almost all features present in the PC functions of a-GST, although some minor differences still exist. Furthermore, even minor features are well-described by both structures, with the formation of Ge$-$Ge bonds and cavity regions, both of which have been identified as key characteristics of a-GST.Baker et al. (2006); Akola and Jones (2007); Caravati et al. (2007); Hegedüs and Elliott (2008) We observe that am-GST structures in which less than half of the Ge atoms are moved to the tetrahedron sites do not yield PC functions similar to the a-GST structures, instead they show strong similarity to the PC function calculated for m-GST (see Fig. 2). Furthermore, we observed that only the am-GST structures in which the Ge atoms initially occupy four-fold sites in or near the intrinsic vacancies lead to structure properties in good agreement with the calculated MD a-GST structures. Thus, it suggests that the location of the intrinsic vacancies plays an important role in the phase transition, which can be explained by the lower energy barriers for Ge displacements close to intrinsic vacancy regions. For the lowest energy m-GST structures, in which the intrinsic vacancies are ordered in a plane perpendicular to $c$. However, at high temperature or under non-equilibrium growth conditions the intrinsic vacancies may distribute more randomly among the cation sites, which is expected to play an important role in the pattern of shifted Ge atoms from their stable octahedra. Our predicted results are in good agreement with available experimental data. For example, using the calculated equilibrium volumes for both phases, we obtained a density of 5.89 g/cm3 (m-GST) and 5.35 g/cm3 (a-GST and am-GST), i.e. the amorphization gives rise to a volume expansion, which decreases the density by about 9.20%. The experimentally observed expansion is on the order of 6.4%.Njoroge et al. (2002) The volume expansion upon amorphization is a consequence of the Ge atoms moving to the lower coordination sites in the a-GST structures. Therefore, the smaller volume deformation observed in the experimental sample may indicate that the amorphization process in not complete, Yamada et al. (1991); Jóv’ari et al. (2008) i.e. not all the Ge atoms are moved away from their stable octahedron sites. Comparison of the total energies reveals that the a-GST structure is about $140-182$ meV/atom higher in energy than the lowest energy m-GST structure, which corresponds to the energy limit between the fully amorphized (100% shift of the Ge atoms) and the ordered m-GST structure. Differential scanning calorimetry measurements obtained $28-42$ meV/atom.Kalb et al. (2003) We found that the calculated energy differences decrease by about 30 meV/atom if the intrinsic vacancies become disordered in m-GST. Furthermore, the energy difference could be much smaller (e.g. about 50 meV/atom) if only a fraction of the Ge atoms undergo site transitions. This again suggests that full scale amorphorization of GST or a complete ordering of Ge, Sb, and intrinsic vacancies in m-GST may not be typical in the GST phases. The averaged bond lengths calculated for both phases are summarized in Table 1 along with available experimental results. Jóv’ari et al. (2008); Kolobov et al. (2004); Baker et al. (2006) The calculated bond lengths deviate by about $3-6$% from the experimental results; however, most of the error is due to the use of GGA in our calculations, which systematically overestimates the lattice constants by about 3%. Furthermore, it is important to notice that the nearest-neighbor distances are spread over a large range of values, e.g. Ge$-$Te is from 2.67 to 2.94 Å and Sb$-$Te is from 2.86 to 3.23 Å. We found a contraction in the averaged Ge$-$Te bond lengths in a-GST of up to 10% compared with m-GST, e.g. Ge$-$Te decreases from $2.87-3.24$ Å (m-GST) to $2.67-2.94$ Å (a-GST), while experimental measurements obtained a decrease of about 12%. Similar trends exist for Sb$-$Te. To understand the relaxation effects introduced by the shift of Ge atoms from octahedron to tetrahedron sites, we calculated the potential energy path along the RS [111] direction for GeTe as a function of Ge shift from octahedron (perfect RS) to the tetrahedron (zincblende) sites. The results are shown in Fig. 3. As expected, the distorted RS structure has the lowest energy (54 meV lower than the perfect RS structure), in which the distortion is driven by Peierls-type level repulsion near the band edge. Unexpectedly, the zincblende (ZB) structure in which the Ge atoms occupy the tetrahedron sites with bond angles (Ge$-$Te$-$Ge) of $109.47^{\circ}$, is not a local minimum as would be expected based on the general trends for binary semiconductors. In fact, we found that the ZB structure relaxes without energy barrier to the ‘graphite- like’ or to the ‘long-Ge$-$Te’ structures, which have lower energies than the high-symmetry ZB phase. Thus, Ge at ideal tetrahedral sites are intrinsically unstable in GeTe, which drives the Ge atoms at tetrahedral sites in GST to move away and adopt a variety of lower symmetry coordination environments. The variety of coordination environments found in the GeTe energy surface is remarkable. From Ge site occupation of (0.25,0.25,0.25) to (0.40,0.40,0.40), three structures have similar energies, i.e. ‘long-Ge$-$Te’, layered-ZB, and layered-RS. In ‘long-Ge$-$Te’, the Ge atoms form three short bonds (2.77 Å) and one long bond (4.68 Å) with the Te atoms. However, Ge is only three-fold coordinated in the layered structures with bond lengths of 2.76 Å, which is 2.90% (14.51%) smaller than the short (longer) Ge$-$Te bond lengths in the distorted RS structure. As the layered GeTe structures are lower in energy than the graphite-like phase and only about 100 meV/f.u. higher than the distorted RS structure, it indicates a strong tendency of Ge atoms to form four-fold motifs with three short Ge$-$Te bonds (about 2.76 Å) and bond angles of about $90^{\circ}$. Similar results, e.g. short bond lengths and average bond angles of about $90^{\circ}$, are observed by our calculations for a-GST, which is also consistent with previous MD results for a-GST,Akola and Jones (2007); Caravati et al. (2007); Hegedüs and Elliott (2008) as well as by experimental observations.Kolobov et al. (2004, 2006); Jóv’ari et al. (2008) Therefore, the inherent instability of Ge at the tetrahedral sites, low displacement energy, and unique coordination preferences of GeTe plays an important role in the formation of a-GST. In summary, using first-principles calculations, we obtained a direct structural link between the meta-stable and amorphous GST phases, as well as the role of the parent compounds. The Sb2Te3 provides intrinsic lattice vacancies, while GeTe contributes its RS-type structure in which Ge displacements along the RS [111] direction can be realized at low energy cost. The instability at the tetrahedral sites leads to the generation of disordered GST structures in which the Ge atoms are mostly four-fold coordinated with three short Ge$-$Te bond lengths. As the displacement has the lowest energy near intrinsic vacancy sites, our analysis suggests that a high degree of amorphization can be achieved most easily when the system has a composition of (GeTe)2(Sb2Te3), i.e., is consistent with the observation that GST has the highest figure of merit of all Ge$-$Sb$-$Te compounds. Moreover, we show that generating amorphous materials directly from its crystalline counterpart provides a better approach to understand these type of phase transitions present in phase change materials. ## References * Ovshinsky (1968) S. R. Ovshinsky, Phys. Rev. Lett. 21, 1450 (1968). * Yamada et al. (1991) N. Yamada, E. Ohno, K. Nishiuchi, N. Akahira, and T. Takao, J. Appl. Phys. 69, 2849 (1991). * Wuttig and Yamada (2007) M. Wuttig and N. Yamada, Nature Mater. 6, 824 (2007). * Chong et al. (2008) T. C. Chong, L. P. Shi, X. Q. Wei, R. Zhao, H. K. Lee, P. Yang, and A. Y. Du, Phys. Rev. Lett. 100, 136101 (2008). * Kolobov et al. (2004) A. V. Kolobov, P. Fons, A. I. Frenkel, A. L. Ankudinov, J. Tominaga, and T. Uruga, Nature Mater. 3, 703 (2004). * Kolobov et al. (2006) A. V. Kolobov, J. Haines, A. Pradel, M. Ribes, P. Fons, J. Tominaga, Y. Katayama, T. Hammouda, and T. Uruga, Phys. Rev. Lett. 97, 035701 (2006). * Welnic et al. (2006) W. Welnic, A. Pamungkas, R. Detemple, C. Steimer, S. Blügel, and M. Wuttig, Nature Mater. 5, 56 (2006). * Hegedüs and Elliott (2008) J. Hegedüs and S. R. Elliott, Nature Mater. 7, 399 (2008). * Yamada and Matsunaga (2000) N. Yamada and T. Matsunaga, J. Appl. Phys. 88, 7020 (2000). * Njoroge et al. (2002) W. K. Njoroge, H.-W. Wöltgens, and M. Wuttig, J. Vac. Sci. Technol. A 20, 230 (2002). * Matsunaga and Yamada (2004) T. Matsunaga and N. Yamada, Phys. Rev. B 69, 104111 (2004). * Park et al. (2005) Y. J. Park, J. Y. Lee, M. S. Youm, Y. T. Kim, and H. S. Lee, J. Appl. Phys. 97, 093506 (2005). * Sun et al. (2006) Z. Sun, J. Zhou, and R. Ahuja, Phys. Rev. Lett. 96, 055507 (2006). * Wuttig et al. (2007) M. Wuttig, D. Lüsebrink, D. Wamwangi, W. Welnic, M. Gillessen, and R. Dronskowski, Nature Mater. 6, 122 (2007). * Shportko et al. (2008) K. Shportko, S. Kremers, M. Woda, D. Lencer, J. Robertson, and M. Wuttig, Nature Mat. 7, 653 (2008). * Baker et al. (2006) D. A. Baker, M. A. Paesler, G. Lucovsky, S. C. Agarwal, and P. C. Taylor, Phys. Rev. Lett. 96, 255501 (2006). * Akola and Jones (2007) J. Akola and R. O. Jones, Phys. Rev. B 76, 235201 (2007). * Caravati et al. (2007) S. Caravati, M. Bernasconi, T. D. Kühne, M. Krack, and M. Parrinello, Appl. Phys. Lett. 91, 171906 (2007). * Jóv’ari et al. (2008) P. Jóv’ari, I. Kaban, J. Steiner, B. Beuneu, A. Schöps, and M. A. Webb, Phys. Rev. B 77, 035202 (2008). * Weidenhof et al. (1999) V. Weidenhof, I. Friedrich, S. Ziegler, and M. Wuttig, J. Appl. Phys. 86, 5879 (1999). * Kalb et al. (2003) J. Kalb, F. Spaepen, and M. Wuttig, J. Appl. Phys. 93, 2389 (2003). * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). * Perdew et al. (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). * Kresse and Hafner (1993) G. Kresse and J. Hafner, Phys. Rev. B 48, 13115 (1993). * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). * Da Silva et al. (2008) J. L. F. Da Silva, A. Walsh, and H. Lee, Phys. Rev. B 78, 224111 (2008). * Matsunaga et al. (2007) T. Matsunaga, R. Kojima, N. Yamada, K. Kifune, Y. Kubota, and M. Takata, Appl. Phys. Lett. 90, 161919 (2007). * Sun et al. (2007) Z. Sun, J. Zhou, and R. Ahuja, Phys. Rev. Lett. 98, 055505 (2007).	arxiv-papers	2009-03-13T18:58:24	2024-09-04T02:49:01.122547	{ "license": "Public Domain", "authors": "Juarez L. F. Da Silva, Aron Walsh, Su-Huai Wei, and Hosun Lee", "submitter": "Juarez L. F. Da Silva", "url": "https://arxiv.org/abs/0903.2456" }
0903.2462	0.4pt0.2pt A Systematic Study of Gröbner Basis Methods Vom Fachbereich Informatik der Technischen Universität Kaiserslautern genehmigte Habilitationsschrift von Dr. Birgit Reinert Datum der Einreichung: 6. Januar 2003 Datum des wissenschaftlichen Vortrags: 9. Februar 2004 Dekan: Prof. Dr. Hans Hagen Habilitationskommission: Vorsitzender: Prof. Dr. Otto Mayer Berichterstatter: Prof. Dr. Klaus E. Madlener Prof. Dr. Teo Mora Prof. Dr. Volker Weispfenning ## Vorwort Die vorliegende Arbeit ist die Quintessenz meiner Ideen und Erfahrungen, die ich in den letzten Jahren bei meiner Forschung auf dem Gebiet der Gröbnerbasen gemacht habe. Meine geistige Heimat war dabei die Arbeitsgruppe von Professor Klaus Madlener an der Technischen Universität Kaiserslautern. Hier habe ich bereits im Studium Bekanntschaft mit der Theorie der Gröbnerbasen gemacht und mich während meiner Promotion mit dem Spezialfall dieser Theorie für Monoid- und Gruppenringe beschäftigt. Nach der Promotion konnte ich im Rahmen eines DFG-Forschungsstipendiums zusätzlich Problemstellungen und Denkweisen anderer Arbeitsgruppen kennenlernen - die Arbeitsgruppe von Professor Joachim Neubüser in Aachen und die Arbeitsgruppe von Professor Theo Mora in Genua. Meine Aufenthalte in diesen Arbeitsgruppen haben meinen Blickwinkel für weitergehende Fragestellungen erweitert. An dieser Stelle möchte ich mich bei allen jenen bedanken, die mich in dieser Zeit begleitet haben und so zum Entstehen und Gelingen dieser Arbeit beigetragen haben. Mein besonderer Dank gilt meinem akademischen Lehrer Professor Klaus Madlener, der meine akademische Ausbildung schon seit dem Grundstudium begleitet und meine Denk- und Arbeitsweise wesentlich geprägt hat. Durch ihn habe ich gelernt, mich intensiv mit diesem Thema zu beschäftigen und mich dabei nie auf nur einen Blickwinkel zu beschränken. Insbesondere sein weitreichenden Literaturkenntnisse und die dadurch immer neu ausgelösten Fragen aus verschiedenen Themengebieten bewahrten meine Untersuchungen vor einer gewissen Einseitigkeit. Er hat mich gelehrt, selbständig zu arbeiten, Ideen und Papiere zu hinterfragen, mir meine eigene Meinung zu bilden, diese zu verifizieren und auch zu vertreten. Professor Teo Mora und Professor Volker Weispfenning danke ich für die Übernahme der weiteren Begutachtungen dieser Arbeit. Professor Teo Mora danke ich insbesondere auch für die fruchtbare Zeit in seiner Arbeitsgruppe in Genua. Seine Arbeiten und seine Fragen haben meine Untersuchungen zum Zusammenhang zwischen Gröbnerbasen in Gruppenringen und dem Todd-Coxeter Ansatz für Gruppen und die Fragestellungen dieser Arbeit wesentlich geprägt. Meinen Kollegen aus unserer Arbeitsgruppe danke ich für ihre Diskussionsbereitschaft und ihre geduldige Anteilnahme an meinen Gedanken. Insbesondere Andrea Sattler-Klein, Thomas Deiß, Claus-Peter Wirth und Bernd Löchner haben immer an mich geglaubt und mich ermutigt, meinen Weg weiter zu gehen. Mit ihnen durfte ich nicht nur Fachliches sondern das Leben teilen. Um eine solche Arbeit fertigzustellen braucht man jedoch nicht nur eine fachliche Heimat. Mein Mann Joachim hat nie an mir gezweifelt und mich immer unterstützt. Auch nach unserem Umzug nach Rechberghausen hat er mein Pendeln nach Kaiserslautern und das Brachliegen unseres Haushalts mitgetragen. Ohne meine Eltern Irma und Helmut Weber und meine Patentante Anita Schäfer wäre insbesondere nach der Geburt unserer Tochter Hannah diese Arbeit nie fertiggestellt worden. Sie haben Hannah liebevoll behütet, so dass ich lesen, schreiben und arbeiten konnte, ohne ein schlechtes Gewissen zu haben. Hannah hat von Anfang an gelernt, dass Forschung ein Leben bereichern kann und die Zeit mit ihrer erweiterten Familie und die vielen Reisen nach Kaiserslautern genossen. Gewidmet ist diese Arbeit meiner Mutter, die leider die Fertigstellung nicht mehr erleben durfte, und Hannah, die auch heute noch Geduld aufbringt, wenn ihre Mutter am Computer für ihre “Schule” arbeitet. Rechberghausen, im August 2004 Birgit Reinert ## Für meine Mutter und Hannah ###### Contents 1. 1 Introduction 1. 1.1 The History of Gröbner Bases 2. 1.2 Two Definitions of Gröbner Bases 3. 1.3 Applications of Gröbner Bases 4. 1.4 Generalizations of Gröbner Bases 5. 1.5 Gröbner Bases in Function Rings – A Guide for Introducing Reduction Relations to Algebraic Structures 6. 1.6 Applications of Gröbner Bases Generalized to Function Rings 7. 1.7 Organization of the Contents 2. 2 Basic Definitions 1. 2.1 Algebra 2. 2.2 The Notion of Reduction 3. 2.3 Gröbner Bases in Polynomial Rings 3. 3 Reduction Rings 1. 3.1 Reduction Rings 2. 3.2 Quotients of Reduction Rings 3. 3.3 Sums of Reduction Rings 4. 3.4 Modules over Reduction Rings 5. 3.5 Polynomial Rings over Reduction Rings 4. 4 Function Rings 1. 4.1 The General Setting 2. 4.2 Right Ideals and Right Standard Representations 1. 4.2.1 The Special Case of Function Rings over Fields 2. 4.2.2 Function Rings over Reduction Rings 3. 4.2.3 Function Rings over the Integers 3. 4.3 Right ${\cal F}$-Modules 4. 4.4 Ideals and Standard Representations 1. 4.4.1 The Special Case of Function Rings over Fields 2. 4.4.2 Function Rings over Reduction Rings 3. 4.4.3 Function Rings over the Integers 5. 4.5 Two-sided Modules 5. 5 Applications of Gröbner Bases 1. 5.1 Natural Applications 2. 5.2 Quotient Rings 3. 5.3 Elimination Theory 4. 5.4 Polynomial Mappings 5. 5.5 Systems of One-sided Linear Equations in Function Rings over the Integers 6. 6 Conclusions 0.4pt0.2pt ## Chapter 1 Introduction One of the amazing features of computers is the ability to do extensive computations impossible to be done by hand. This enables to overcome the boundaries of constructive algebra as proposed by mathematicians as Kronecker. He demanded that definitions of mathematical objects should be given in such a way that it is possible to decide in a finite number of steps whether a definition applies to an object. While in the beginning computers were used to do incredible numerical calculations, a new dimension was added when they were used to do symbolical mathematical manipulations substantial to many fields in mathematics and physics. These new possibilities led to open up whole new areas of mathematics and computer science. In the wake of these developments has come a new access to abstract algebra in a computational fashion – computer algebra. One important contribution to this field which is the subject of this work is the theory of Gröbner bases – the result of Buchberger’s algorithm for manipulating systems of polynomials. ### 1.1 The History of Gröbner Bases In 1965 Buchberger introduced the theory of Gröbner bases111Note that similar concepts appear in a paper of Hironaka where the notion of a complete set of polynomials is called a standard basis [Hir64]. for polynomial ideals in commutative polynomial rings over fields [Buc65, Buc70]. Let ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ be a polynomial ring over a computable field ${\mathbb{K}}$ and ${\mathfrak{i}}$ an ideal in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$. Then the quotient ${\mathbb{K}}[X_{1},\ldots,X_{n}]/{\mathfrak{i}}$ is a ${\mathbb{K}}$-algebra. If this quotient is zero-dimensional the algebra has a finite basis consisting of power products $X_{1}^{i_{1}}\ldots X_{n}^{i_{n}}$. This was the starting point for Buchberger’s PhD thesis. His advisor Wolfgang Gröbner wanted to compute the multiplication table and had suggested a procedure for zero- dimensional ideals, for which termination conditions were lacking. The result of Buchberger’s studies then was a terminating algorithm which turned a basis of an ideal into a special basis which allowed to solve Gröbner’s question of writing down an explicit multiplication for the multiplication table of the quotient in the zero-dimensional case and was even applicable to arbitrary polynomial ideals. Buchberger called these special bases of ideals Gröbner bases. ### 1.2 Two Definitions of Gröbner Bases In literature there are two main ways to define Gröbner bases in polynomial rings over fields. They both require an admissible222An ordering $\succeq$ on the set of terms is called an admissible term ordering if for every term $s,t,u$, $s\succeq 1$ holds, and $s\succeq t$ implies $s\circ u\succeq t\circ u$. An ordering fulfilling the latter condition is also said to be compatible with the respective multiplication $\circ$. ordering on the set of terms. With respect to such an ordering, given a polynomial $f$ the maximal term occurring in $f$ is called the head term denoted by ${\sf HT}(f)$. One way to characterize Gröbner bases in an algebraic fashion is to use the concept of term division: A term $X_{1}^{i_{1}}\ldots X_{n}^{i_{n}}$ is said to divide another term $X_{1}^{j_{1}}\ldots X_{n}^{j_{n}}$ if and only if $i_{l}\leq j_{l}$ for all $1\leq l\leq n$. Then a set $G$ of polynomials is called a Gröbner basis of the ideal $\mathfrak{i}$ it generates if and only if for every $f$ in $\mathfrak{i}$ there exists a polynomial $g\in G$ such that ${\sf HT}(g)$ divides ${\sf HT}(f)$. Another way to define Gröbner bases in polynomial rings is to establish a rewriting approach to the theory of polynomial ideals. Polynomials can be used as rules by using the largest monomial according to the admissible ordering as a left hand side of a rule. Then a term is reducible by a polynomial as a rule if the head term of the polynomial divides the term. A Gröbner basis $G$ then is a set of polynomials such that every polynomial in the polynomial ring has a unique normal form with respect to this reduction relation using the polynomials in $G$ as rules (especially the polynomials in the ideal generated by $G$ reduce to zero using $G$). Of course both definitions coincide for polynomial rings since the reduction relation defined by Buchberger can be compared to division of one polynomial by a set of finitely many polynomials. ### 1.3 Applications of Gröbner Bases The method of Gröbner bases allows to solve many problems related to polynomial ideals in a computational fashion. It was shown by Hilbert (compare Hilbert’s basis theorem) that every ideal in a polynomial ring has a finite generating set. However, an arbitrary finite generating set need not provide much insight into the nature of the ideal. Let $f_{1}=X_{1}^{2}+X_{2}$ and $f_{2}=X_{1}^{2}+X_{3}$ be two polynomials in the polynomial ring333${\mathbb{Q}}$ denotes the rational numbers. ${\mathbb{Q}}[X_{1},X_{2},X_{3}]$. Then ${\mathfrak{i}}=\\{f_{1}\ast g_{1}+f_{2}\ast g_{2}\mid g_{1},g_{2}\in{\mathbb{Q}}[X_{1},X_{2},X_{3}]\\}$ is the ideal they generate and it is not hard to see that the polynomial $X_{2}-X_{3}$ belongs to ${\mathfrak{i}}$ since $X_{2}-X_{3}=f_{1}-f_{2}$. But what can be said about the polynomial $f=X_{3}^{3}+X_{1}+X_{3}$? Does it belong to ${\mathfrak{i}}$ or not? The problem to decide whether a given polynomial lies in a given ideal is called the membership problem for ideals. In case the generating set is a Gröbner basis this problem becomes immediately decidable, as the membership problem then reduces to checking whether the polynomial reduces to zero using the elements of the Gröbner basis for reduction. In our example the set $\\{X_{1}^{2}+X_{3},X_{2}-X_{3}\\}$ is a generating set of ${\mathfrak{i}}$ which is in fact a Gröbner basis. Now returning to the polynomial $f=X_{3}^{3}+X_{1}+X_{3}$ we find that it cannot belong to ${\mathfrak{i}}$ since neither $X_{1}^{2}$ nor $X_{2}$ is a divisor of a term in $f$ and hence $f$ cannot be reduced to zero using the polynomials in the Gröbner basis as rules. The terms $X_{1}^{i_{1}}X_{2}^{i_{2}}X_{3}^{i_{3}}$ which are not reducible by the set $\\{X_{1}^{2}+X_{3},X_{2}-X_{3}\\}$ form a basis of the ${\mathbb{Q}}$-algebra ${\mathbb{Q}}[X_{1},X_{2},X_{3}]/{\mathfrak{i}}$. By inspecting the head terms $X_{1}^{2}$ and $X_{2}$ of the Gröbner basis we find that the (infinite) set $\\{X_{3}^{i},X_{1}X_{3}^{i}\mid i\in{\mathbb{N}}\\}$ is such a basis. Moreover, an ideal is zero-dimensional, i.e. this set is finite, if and only if for each variable $X_{i}$ the Gröbner basis contains a polynomial with head term $X_{i}^{k_{i}}$ for some $k_{i}\in{\mathbb{N}}^{+}$. Similarly the form of the Gröbner basis reveals whether the ideal is trivial: ${\mathfrak{i}}={\mathbb{K}}[X_{1},\ldots,X_{n}]$ if and only if every444Notice that if one Gröbner basis contains an element from ${\mathbb{K}}$ so will all the others. Gröbner basis contains an element from ${\mathbb{K}}$. Further applications of Gröbner bases come from areas as widespread as robotics, computer vision, computer-aided design, geometric theorem proving, Petrie nets and many more. More details can be found e.g. in Buchberger [Buc87], or the books of Becker and Weispfenning [BW92], Cox, Little and O’Shea [CLO92], and Adams and Loustaunau [AL94]. ### 1.4 Generalizations of Gröbner Bases In the last years, the method of Gröbner bases and its applications have been extended from commutative polynomial rings over fields to various types of algebras over fields and other rings. In general for such rings arbitrary finitely generated ideals will not have finite Gröbner bases. Nevertheless, there are interesting classes for which every finitely generated (left, right or even two-sided) ideal has a finite Gröbner basis which can be computed by appropriate variants of completion based algorithms. First successful generalizations were extensions to commutative polynomial rings over coefficient domains other than fields. It was shown by several authors including Buchberger, Kandri-Rody, Kapur, Narendran, Lauer, Stifter, and Weispfenning that Buchberger’s approach remains valid for polynomial rings over the integers, or even Euclidean rings, and over regular rings (see e.g. [Buc83, Buc85, KRK84, KRK88, KN85, Lau76, Sti87, Wei87b]). For regular rings Weispfenning has to deal with the situation that zero-divisors in the coefficient domain have to be considered. He uses a technique he calls Boolean closure to repair this problem and this technique can be regarded as a special saturating process555Saturation techniques are used in various fields to enrich a generating set of a structure in such a way, that the new set still describes the same structure but allows more insight. For example symmetrization in groups can be regarded as such a saturating process.. We will later on see how such saturating techniques become important ingredients of Gröbner basis methods in many algebraic structures. Since the development of computer algebra systems for commutative algebras made it possible to perform tedious calculations using computers, attempts to generalize such systems and especially Buchberger’s ideas to non-commutative algebras followed. Originating from special problems in physics, Lassner in [Las85] suggested how to extend existing computer algebra systems in order to additionally handle special classes of non-commutative algebras, e.g. Weyl algebras. He studied structures where the elements could be represented using the usual representations of polynomials in commutative variables and the non- commutative multiplication could be performed by a so-called “twisted product” which required only procedures involving commutative algebra operations and differentiation. Later on together with Apel he extended Buchberger’s algorithm to enveloping fields of Lie algebras [AL88]. Because these ideas use representations of the elements by commutative polynomials, Dickson’s Lemma666Dickson’s Lemma in the context of commutative terms is as follows: For every infinite sequence of terms $t_{s}$, $s\in{\mathbb{N}}$, there exists an index $k\in{\mathbb{N}}$ such that for every index $i>k$ there exists an index $j\leq k$ and a term $w$ such that $t_{i}=t_{j}w$. can be carried over. By this the existence and construction of finite Gröbner bases for finitely generated left ideals can be ensured using the same arguments as in the original approach. On the other hand, Mora gave a concept of Gröbner bases for a class of non- commutative algebras by saving an other property of the commutative polynomial ring – admissible orderings – while losing the validity of Dickson’s Lemma. The usual polynomial ring can be viewed as a monoid ring where the monoid is a finitely generated free commutative monoid. Mora studied the class where the free commutative monoid is substituted by a free monoid – the class of finitely generated free monoid rings (compare e.g. [Mor85, Mor94]). The ring operations are mainly performed in the coefficient domain while the terms are treated like words, i.e., the variables no longer commute with each other and multiplication is concatenation. The definitions of (one- and two-sided) ideals, reduction and Gröbner bases are carried over from the commutative case to establish a similar theory of Gröbner bases in “free non-commutative polynomial rings over fields”. But these rings are no longer Noetherian if they are generated by more than one variable. Mora presented a terminating completion procedure for finitely generated one-sided ideals and an enumeration procedure for finitely generated two-sided ideals with respect to some term ordering in free monoid rings. For the special instance of ideals generated by bases of the restricted form $\\{\ell_{i}-r_{i}\mid\ell_{i},r_{i}\mbox{ words},1\leq i\leq n\\}$, Mora’s procedure coincides with Knuth-Bendix completion for string rewriting systems and the one-sided cases can be related to prefix respectively suffix rewriting [MR98d, MR98c]. Hence many results known for finite string rewriting systems and their completion carry over to finitely generated ideals and the computation of their Gröbner bases. Especially the undecidability of the word problem yields non-termination for Mora’s general procedure (see also [Mor87]). Gröbner bases and Mora’s procedure have been generalized to path algebras (see [FCF93, Kel98]); free non-commutative polynomial rings are in fact a particular instance of path algebras. Another class of non-commutative rings where the elements can be represented by the usual polynomials and which allow the construction of finite Gröbner bases for arbitrary ideals are the solvable polynomial rings, a class intermediate between commutative and general non-commutative polynomial rings. They were studied by Kandri-Rody, Weispfenning and Kredel [KRW90, Kre93]. Solvable polynomial rings can be described by ordinary polynomial rings ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ provided with a “new” definition of multiplication which coincides with the ordinary multiplication except for the case that a variable $X_{j}$ is multiplied with a variable $X_{i}$ with lower index, i.e., $i<j$. In the latter case multiplication can be defined by equations of the form $X_{j}\star X_{i}=c_{ij}X_{i}X_{j}+p_{ij}$ where $c_{ij}$ lies in ${\mathbb{K}}^{}={\mathbb{K}}\backslash\\{0\\}$ and $p_{ij}$ is a polynomial “smaller” than $X_{i}X_{j}$ with respect to a fixed admissible term ordering on the polynomial ring. The more special case of twisted semi-group rings, where $c_{ij}=0$ is possible, has been studied in [Ape88, Mor89]. In [Wei87a] Weispfenning showed the existence of finite Gröbner bases for arbitrary finitely generated ideals in non-Noetherian skew polynomial rings over two variables $X,Y$ where a “new” multiplication $\star$ is introduced such that $X\star Y=XY$ and $Y\star X=X^{e}Y$ for some fixed $e$ in ${\mathbb{N}}^{+}$. Ore extensions have been successfully studied by Pesch in his PhD Thesis [Pes97] and his results on two-sided Gröbner bases are also presented in [Pes98]. Most of the results cited so far assume admissible well-founded orderings on the set of terms so that in fact the reduction relations can be defined by considering the head monomials mainly (compare the algebraic definition of Gröbner bases in Section 1.2). This is essential to characterize Gröbner bases in the respective ring with respect to the corresponding reduction relation777These reduction relations are based on divisibility of terms, namely the term to be reduced is divisible by the head term of the polynomial used as rule for the reduction step. in a finitary manner and to enable to decide whether a finite set is a Gröbner basis by checking whether the s-polynomials are reducible to zero888Note that we always assume that the reduction relation in the ring is effective.. There are rings combined with reduction relations where admissible well- founded orderings cannot be accomplished and, therefore, other concepts to characterize Gröbner bases have been developed. For example in case the ring contains zero-divisors a well-founded ordering on the ring is no longer compatible with the ring multiplication999When studying monoid rings over reduction rings it is possible that the ordering on the ring is not compatible with scalar multiplication as well as with multiplication with monomials or polynomials.. This phenomenon has been studied for the case of zero-divisors in the coefficient domain by Kapur and Madlener [KM89] and by Weispfenning for the special case of regular rings [Wei87b]. In his PhD thesis [Kre93], Kredel described problems occurring when dropping the axioms guaranteeing the existence of admissible orderings in the theory of solvable polynomial rings by allowing $c_{ij}=0$ in the defining equations above. He sketched the idea of using saturation techniques to repair some of them. Saturation enlarges the generating sets of ideals in order to ensure that enough head terms exist to do all necessary reduction steps and this process can often be related to additional special critical pairs. Similar ideas can be found in the PhD thesis of Apel [Ape88]. For special cases, e.g. for the Grassmann (exterior) algebras, positive results can be achieved (compare the paper of Stokes [Sto90]). Another important class of rings where reduction relations can be introduced and completion techniques can be applied to enumerate and sometimes compute Gröbner bases are monoid and group rings. They have been studied in detail by various authors, e.g. free group rings ([Ros93]), monoid and group rings ([MR93a, MR97a, Rei95, Rei96, MR98a]) (including finite and free monoids and finite, free, plain and polycyclic groups), and polycyclic group rings ([Lo98]). In this setting we again need saturation techniques to repair a severe defect due to the fact that in general we cannot expect the ordering on the set of terms (here of course now the monoid or group elements) to be both, well-founded and admissible. Let ${\cal F}$ be the free group generated by one element $a$. Then for the polynomial $a+1$ in ${\mathbb{Q}}[{\cal F}]$ we have $(a+1)\ast a^{-1}=1+a^{-1}$, i.e., after multiplication with the inverse element $a^{-1}$ the largest term of the new polynomial no longer results from the largest one of the original polynomial. Moreover, assuming our ordering is well-founded, it cannot be compatible with the group multiplication101010Assuming $a\succ 1$ compatibility with multiplication would imply $1\succ a^{-1}$ giving rise to an infinite descending chain $a^{-1}\succ a^{-2}\succ\ldots$ contradicting the well-foundedness of the ordering. On the other hand for $1\succ a$ compatibility with multiplication immediately gives us an infinite descending chain $a\succ a^{2}\succ\ldots$.. All approaches cited in this section can be basically divided into two main streams: One extension was to study structures which still allow to present their elements by ordinary “commutative” polynomials. The advantage of this generalization is that Dickson’s Lemma, which is essential in proving termination for Buchberger’s algorithm, carries over. The other idea of generalization was to view the polynomial ring as a special monoid ring and to try to extend Buchberger’s approach to other monoid and group rings. Since then in general Dickson’s Lemma no longer holds, other ways to prove termination, if possible, have to be established. Notice that solvable rings, skew-polynomial rings and arbitrary quotients of non-commutative polynomial rings cannot be interpreted as monoid rings. Hence to find a generalization which will subsume all results cited here, a more general setting is needed. In his habilitation thesis [Ape98], Apel provides one generalization which basically extends the first one of these two in such a way that Mora’s approach can be incorporated. He uses an abstraction of graded structures which needs admissible well-founded orderings. Hence he cannot deal with group rings and many cases of monoid rings where such orderings cannot exists. On the other hand he is much more interested in algebraic characterizations of Gröbner bases and the division algorithms associated to them. In order to characterize structures where the well-founded ordering is no longer admissible, we extend Gröbner basis techniques to an abstract setting called function rings. ### 1.5 Gröbner Bases in Function Rings – A Guide for Introducing Reduction Relations to Algebraic Structures The aim of this work is to give a general setting which comprises all generalizations mentioned above and which is a basis for studying further structures in the light of introducing reduction relations and Gröbner basis techniques. All structures mentioned so far can be viewed as rings of functions with finite support. For such rings we introduce the familiar concepts of polynomials, (right) ideals, standard representations, standard bases, reduction relations and Gröbner bases. A general characterization of Gröbner bases in an “algorithmic fashion” is provided. It is shown that in fact polynomial rings, solvable polynomial rings, free respectively finite monoid rings, and free, finite, plain, respectively polycyclic group rings are examples of our generalization where finite Gröbner bases can be computed. While most of the examples cited above are presented in the literature as rings over fields we will here also present the more general concept of function rings over reduction rings (compare [Mad86, Rei95, MR98b]) and the impotant special case of function rings over the integers. ### 1.6 Applications of Gröbner Bases Generalized to Function Rings For polynomial rings over fields many algebraic questions related to ideals can be solved using Gröbner bases and their associated reduction relations. Hence the question arises which of these applications can be extended to more general settings. While some questions e.g. concerning algebraic geometry are strongly connected to polynomial rings over fields, many other applications carry over. They include natural ones such as the membership problem for ideals, as well as special techniques such as elimination theory or the treatment of systems of linear equations. ### 1.7 Organization of the Contents Chapter 2 introduces some of the basic themes of this book. We need some basic notions from the theory of algebra as well as from the theory of rewriting systems. Furthermore, as the aim of this book is to provide a systematic study of Gröbner basis methods, a short introduction to the original case of Gröbner bases in polynomial rings over fields is presented. Chapter 3 concentrates on rings with reduction relations, which are studied with regard to the existence of Gröbner bases. They are called reduction rings in case they allow finite Gröbner bases for finitely generated ideals. Moreover, special ring constructions are presented, which in many cases preserve the existence of Gröbner bases. These constructions include quotients and sums of reduction rings as well as modules and polynomial rings over reduction rings. Many structures with reduction relations allowing Gröbner bases can already be found in this setting. For example knowing that the integers ${\mathbb{Z}}$ for certain reduction relations allow finite Gröbner bases, using the results of this chapter, we can conclude that the module ${\mathbb{Z}}^{k}$ as well as the polynomial rings ${\mathbb{Z}}[X_{1},\ldots,X_{n}]$ and ${\mathbb{Z}}^{k}[X_{1},\ldots,X_{n}]$ allow the computation of finite Gröbner bases. Chapter 4 is the heart of this book. It establishes a generalizing framework for structures enriched with reduction relations and studied with respect to the existence of Gröbner bases in the literature. Reduction relations are defined for the setting of function rings over fields and later on generalized to reduction rings. Definitions for terms such as variations of standard representations, standard bases and Gröbner bases are given and compared to the known terms from the theory of Gröbner bases over polynomial rings. It turns out that while completion procedures will still involve equivalents to s-polynomials or the more general concept of g- and m-polynomials for the ring case, these situations are no longer sufficient to characterize Gröbner bases. Saturation techniques, which enrich the bases by additional polynomials, are needed. Moreover, for function rings over reduction rings the characterizations no longer describe Gröbner bases but only weak111111Weak Gröbner bases are bases such that any polynomial in the ideal they generate can be reduced to zero. For fields this property already characterizes Gröbner bases as the Translation Lemma holds. In general this is not true and while weak Gröbner bases allows to solve the ideal membership problem they no longer guarantee the existence of unique normal forms for elements of the quotient. Gröbner bases, since the Translation Lemma121212The Translation Lemma establishes that if for two polynomials $f,g$ we have that $f-g$ reduces to zero, both polynomials reduce to the same normal form. no longer holds. Since the ring of integers viewed as a reduction ring is of special interest in the literature and allows more insight into the respective chosen reduction relations, this special case is studied. Chapter 5 outlines how some applications known for Gröbner bases in the literature carry over to function rings. These applications include natural ones such as the ideal membership problem, representation problems, the ideal inclusion problem, the ideal triviality problem, and many more. Another focus is on doing computations in quotient rings using Gröbner bases. The powerful elimination methods are also generalized. One of their applications to study polynomial mappings is outlined. Finally solutions for linear equations over function rings in terms of Gröbner bases are provided. ## Chapter 2 Basic Definitions After introducing the necessary definitions required from algebra we focus on the subject of this book — Gröbner bases. One way of characterizing Gröbner bases is in terms of algebraic simplification or reduction. The aim of this chapter is to introduce an abstract concept for the notion of reduction which is the basis of many syntactical methods for studying structures in mathematics or theoretical computer science in Section 2.2. It is the foundation for e.g. term rewriting and string rewriting and we introduce a reduction relation for polynomials in the commutative polynomial ring over a field in a similar fashion. Gröbner bases then arise naturally when doing completion in this setting in Section 2.3. ### 2.1 Algebra Mathematical theories are closely related with the study of two objects, namely sets and functions. Algebra can be regarded as the study of algebraic operations on sets, i.e., functions that take elements from a set to the set itself. Certain algebraic operations on sets combined with certain axioms are again the objects of independent theories. This chapter is a short introduction to some of the algebraic systems used later on: monoids, groups, rings, fields, ideals and modules. ###### Definition 2.1.1 A non-empty set of elements ${\cal M}$ together with a binary operation $\circ_{{\cal M}}$ is said to form a monoid, if for all $\alpha,\beta,\gamma$ in ${\cal M}$ 1. 1. ${\cal M}$ is closed under $\circ_{{\cal M}}$, i.e., $\alpha\circ_{{\cal M}}\beta\in{\cal M}$, 2. 2. the associative law holds for $\circ_{{\cal M}}$, i.e., $\alpha\circ_{{\cal M}}(\beta\circ_{{\cal M}}\gamma)=_{{\cal M}}(\alpha\circ_{{\cal M}}\beta)\circ_{{\cal M}}\gamma$, and 3. 3. there exists $1_{{\cal M}}\in{\cal M}$ such that $\alpha\circ_{{\cal M}}1_{{\cal M}}=_{{\cal M}}1_{{\cal M}}\circ_{{\cal M}}\alpha=_{{\cal M}}\alpha$. The element $1_{{\cal M}}$ is called identity. $\diamond$ For simplicity of notation we will henceforth drop the index ${\cal M}$ and write $\circ$ respectively $=$ if no confusion is likely to arise. Furthermore, we will often talk about a monoid without mentioning its binary operation explicitly. The monoid operation will often be called multiplication or addition. Since the algebraic operation is associative we can omit brackets, hence the product $\alpha_{1}\circ\ldots\circ\alpha_{n}$ is uniquely defined. ###### Example 2.1.2 Let $\Sigma=\\{a_{1},\ldots,a_{n}\\}$ be a set of letters. Then $\Sigma^{}$ denotes the set of words over this alphabet. For two words $u,v\in\Sigma^{}$ we define $u\circ v=uv$, i.e., the word which arises from concatenating the two words $u$ and $v$. Then $\Sigma^{}$ is a monoid with respect to this binary operation and its identity element is the empty word, i.e., the word containing no letters. This monoid is called the free monoid over the alphabet $\Sigma$. $\diamond$ For some $n$ in ${\mathbb{N}}$111In the following ${\mathbb{N}}$ denotes the set of natural numbers including zero and ${\mathbb{N}}^{+}={\mathbb{N}}\backslash\\{0\\}$. the product of $n$ times the same element $\alpha$ is called the n-th power of $\alpha$ and will be denoted by $\alpha^{n}$, where $\alpha^{0}=1$. ###### Definition 2.1.3 An element $\alpha$ of a monoid ${\cal M}$ is said to have infinite order in case for all $n,m\in{\mathbb{N}}$, $\alpha^{n}=\alpha^{m}$ implies $n=m$. We say that $\alpha$ has finite order in case the set $\\{\alpha^{n}\mid n\in{\mathbb{N}}^{+}\\}$ is finite and the cardinality of this set is then called the order of $\alpha$. $\diamond$ A subset of a monoid ${\cal M}$ which is again a monoid is called a submonoid of ${\cal M}$. Other special subsets of monoids are (one-sided) ideals. ###### Definition 2.1.4 For a subset $S$ of a monoid ${\cal M}$ we call 1. 1. ${\sf ideal}_{r}^{{\cal M}}(S)=\\{\sigma\circ\alpha\mid\sigma\in S,\alpha\in{\cal M}\\}$ the right ideal, 2. 2. ${\sf ideal}_{l}^{{\cal M}}(S)=\\{\alpha\circ\sigma\mid\sigma\in S,\alpha\in{\cal M}\\}$ the left ideal, and 3. 3. ${\sf ideal}^{{\cal M}}(S)=\\{\alpha\circ\sigma\circ\alpha^{\prime}\mid\sigma\in S,\alpha,\alpha^{\prime}\in{\cal M}\\}$ the ideal generated by $S$ in ${\cal M}$. $\diamond$ A monoid ${\cal M}$ is called commutative (Abelian) if we have $\alpha\circ\beta=\beta\circ\alpha$ for all elements $\alpha,\beta$ in ${\cal M}$. A natural example for a commutative monoid are the integers together with multiplication or addition. Another example which will be of interest later on is the set of terms. ###### Example 2.1.5 Let $X_{1},\ldots,X_{n}$ be a set of (ordered) variables. Then ${\cal T}=\\{X_{1}^{i_{1}}\ldots X_{n\phantom{1}}^{i_{n}}\mid i_{1},\ldots i_{n}\in{\mathbb{N}}\\}$ is called the set of terms over these variables. The multiplication $\circ$ is defined as $X_{1}^{i_{1}}\ldots X_{n\phantom{1}}^{i_{n}}\circ X_{1}^{j_{1}}\ldots X_{n\phantom{1}}^{j_{n}}=X_{1}^{i_{1}+j_{1}}\ldots X_{n\phantom{1}}^{i_{n}+j_{n}}$. The identity is the empty term $1_{\cal T}=X_{1}^{0}\ldots X_{n\phantom{1}}^{0}$. $\diamond$ A mapping $\phi$ from one monoid ${\cal M}_{1}$ to another monoid ${\cal M}_{2}$ is called a homomorphism, if $\phi(1_{{\cal M}_{1}})=1_{{\cal M}_{2}}$ and for all $\alpha,\beta$ in ${\cal M}_{1}$, $\phi(\alpha\circ_{{\cal M}_{1}}\beta)=\phi(\alpha)\circ_{{\cal M}_{2}}\phi(\beta)$. In case $\phi$ is surjective we call it an epimorphism, in case $\phi$ is injective a monomorphism and in case it is both an isomorphism. The fact that two structures $S_{1}$, $S_{2}$ are isomorphic will be denoted by $S_{1}\cong S_{2}$. A monoid is called left-cancellative (respectively right-cancellative) if for all $\alpha,\beta,\gamma$ in ${\cal M}$, $\gamma\circ\alpha=\gamma\circ\beta$ (respectively $\alpha\circ\gamma=\beta\circ\gamma$) implies $\alpha=\beta$. In case a monoid is both, left- and right-cancellative, it is called cancellative. In case $\alpha\circ\gamma=\beta$ we say that $\alpha$ is a left divisor of $\beta$ and $\gamma$ is called a right divisor of $b$. If $\gamma\circ\alpha\circ\delta=\beta$ then $\alpha$ is called a divisor of $\beta$. A special class of monoids fulfill that for all $\alpha,\beta$ in ${\cal M}$ there exist $\gamma,\delta$ in ${\cal M}$ such that $\alpha\circ\gamma=\beta$ and $\delta\circ\alpha=\beta$, i.e., right and left divisors always exist. These structures are called groups and they can be specified by extending the definition of monoids and we do so by adding one further axiom. ###### Definition 2.1.6 A monoid ${\cal M}$ together with its binary operation $\circ$ is said to form a group if additionally 1. 4. for every $\alpha\in{\cal M}$ there exists an element ${\sf inv}\/(\alpha)\in{\cal M}$ (called inverse of $\alpha$) such that $\alpha\circ{\sf inv}\/(\alpha)={\sf inv}\/(\alpha)\circ\alpha=1$. $\diamond$ Obviously, the integers form a group with respect to addition, but this is no longer true for multiplication. A subset of a group ${\cal G}$ which is again a group is called a subgroup of ${\cal M}$. A subgroup ${\cal H}$ of a group ${\cal G}$ is called normal if for each $\alpha$ in ${\cal G}$ we have $\alpha{\cal H}={\cal H}\alpha$ where $\alpha{\cal H}=\\{\alpha\circ\beta\mid\beta\in{\cal H}\\}$ and ${\cal H}\alpha=\\{\beta\circ\alpha\mid\beta\in{\cal H}\\}$. We end this section by briefly introducing some more algebraic structures that will be used throughout. ###### Definition 2.1.7 A nonempty set ${\sf R}$ is called an (associative) ring (with unit element) if there are two binary operations $+$ (addition) and $\star$ (multiplication) such that for all $\alpha,\beta,\gamma$ in ${\sf R}$ 1. 1. ${\sf R}$ together with $+$ is an Abelian group with zero element $0$ and inverse $-\alpha$, 2. 2. ${\sf R}$ is closed under $\star$, i.e., $\alpha\star\beta\in{\sf R}$, 3. 3. $\star$ is associative, i.e., $\alpha\star(\beta\star\gamma)=(\alpha\star\beta)\star\gamma$, 4. 4. the distributive laws hold, i.e., $\alpha\star(\beta+\gamma)=\alpha\star\beta+\alpha\star\gamma$ and $(\beta+\gamma)\star\alpha=\beta\star\alpha+\gamma\star\alpha$, 5. 5. there is an element $1\in{\sf R}$ (called unit) such that $1\star\alpha=\alpha\star 1=\alpha$. $\diamond$ A ring is called commutative (Abelian) if $\alpha\star\beta=\beta\star\alpha$ for all $\alpha,\beta$ in ${\sf R}$. The integers together with addition and multiplication are a well-known example of a ring. Other rings which will be of interest later on are monoid rings. ###### Example 2.1.8 Let ${\mathbb{Z}}$ be the ring of integers and ${\cal M}$ a monoid. Further let ${\mathbb{Z}}[{\cal M}]$ denote the set of all mappings $f:{\cal M}\longrightarrow{\mathbb{Z}}$ where the sets ${\sf supp}(f)=\\{\alpha\in{\cal M}\mid f(\alpha)\neq 0\\}$ are finite. We call ${\mathbb{Z}}[{\cal M}]$ the monoid ring of ${\cal M}$ over ${\mathbb{Z}}$. The sum of two elements $f$ and $g$ is denoted by $f+g$ where $(f+g)(\alpha)=f(\alpha)+g(\alpha)$. The product is denoted by $f\star g$ where $(f\star g)(\alpha)=\sum_{\beta\circ\gamma=\alpha}f(\beta)\star g(\gamma)$. Polynomial rings are a special case of monoid rings namely over the set of terms as defined in Example 2.1.5. A ring ${\sf R}$ is said to contain zero-divisors, if there exist not necessarily different elements $\alpha,\beta$ in ${\sf R}$ such that $\alpha\neq 0$ and $\beta\neq 0$, but $\alpha\star\beta=0$. Then $\alpha$ is called a left zero-divisor and $\beta$ is called a right zero-divisor. ###### Definition 2.1.9 A commutative ring is called a field if its non-zero elements form a group under multiplication. $\diamond$ Similar to our proceeding in group theory we will now look at subsets of a ring ${\sf R}$. For a subset $U\subseteq{\sf R}$ to be a subring of $R$ with the operations $+$ and $\star$ it is necessary and sufficient that 1. 1. $U$ is a subgroup of $({\sf R},+)$, i.e., for $a,b\in U$ we have $a-b\in U$, and 2. 2. for all $\alpha,\beta\in U$ we have $\alpha\star\beta\in U$. We will now take a closer look at special subrings that play a role similar to normal subgroups in group theory. ###### Definition 2.1.10 A nonempty subset $\mathfrak{i}$ of a ring ${\sf R}$ is called a right (left) ideal of ${\sf R}$, if 1. 1. for all $\alpha,\beta\in\mathfrak{i}$ we have $\alpha-\beta\in\mathfrak{i}$, and 2. 2. for every $\alpha\in\mathfrak{i}$ and $\rho\in{\sf R}$, the element $\alpha\star\rho$ (respectively $\rho\star\alpha$) lies in $\mathfrak{i}$. A subset that is both, a right and a left ideal, is called a (two-sided) ideal of ${\sf R}$. $\diamond$ For each ring the sets $\\{0\\}$ and ${\sf R}$ are trivial ideals. Similar to subgroups, ideals can be described in terms of generating sets. ###### Lemma 2.1.11 Let $F$ be a non-empty subset of ${\sf R}$. Then 1. 1. ${\sf ideal}^{{\sf R}}(F)=\\{\sum_{i=1}^{n}\rho_{i}\star\alpha_{i}\star\sigma_{i}\mid\alpha_{i}\in F,\rho_{i},\sigma_{i}\in{\sf R},n\in{\mathbb{N}}\\}$ is an ideal of ${\sf R}$, 2. 2. ${\sf ideal}_{r}^{{\sf R}}(F)=\\{\sum_{i=1}^{n}\alpha_{i}\star\rho_{i}\mid\alpha_{i}\in F,\rho_{i}\in{\sf R},n\in{\mathbb{N}}\\}$ is a right ideal of ${\sf R}$, and 3. 3. ${\sf ideal}_{l}^{{\sf R}}(F)=\\{\sum_{i=1}^{n}\rho_{i}\star\alpha_{i}\mid\alpha_{i}\in F,\rho_{i}\in{\sf R},n\in{\mathbb{N}}\\}$ is a left ideal of ${\sf R}$. $\diamond$ Notice that the empty sum $\sum_{i=1}^{0}\alpha_{i}$ is zero. We will simply write ${\sf ideal}(F)$, ${\sf ideal}_{r}(F)$ and ${\sf ideal}_{l}(F)$ if the context is clear. Many algebraic problems for rings are related to ideals and we will close this section by stating two of them222For more information on such problems in the special case of commutative polynomial rings see e.g. [Buc87].. The Ideal Membership Problem --- Given: \| An element $\alpha\in{\sf R}$ and a set of elements $F\subseteq{\sf R}$. Question: \| Is $\alpha$ in the ideal generated by $F$? ###### Definition 2.1.12 Two elements $\alpha,\beta\in{\sf R}$ are said to be congruent modulo ${\sf ideal}(F)$, denoted by $\alpha\equiv_{{\sf ideal}(F)}\beta$, if $\alpha=\beta+\rho$ for some $\rho\in{\sf ideal}(F)$, i.e., $\alpha-\beta\in{\sf ideal}(F)$. $\diamond$ The Congruence Problem --- Given: \| Two elements $\alpha,\beta\in{\sf R}$ and a set of elements $F\subseteq{\sf R}$. Question: \| Are $\alpha$ and $\beta$ congruent modulo the ideal generated by $F$? Note that both problems can similarly be specified for left and right ideals. We have seen that a non-empty subset of ${\sf R}$ is an ideal if it is closed under addition and closed under multiplication with arbitrary elements of ${\sf R}$. Modules now can be viewed as a natural generalization of the concept of ideals to arbitrary commutative groups. ###### Definition 2.1.13 Let ${\sf R}$ be a ring. A left ${\sf R}$-module $M$ is an additive commutative group with an additional operation $\cdot:{\sf R}\times M\longrightarrow M$, called scalar multiplication, such that for all $\alpha,\beta\in{\sf R}$ and $a,b\in M$, the following hold: 1. 1. $\alpha\cdot(a+b)=\alpha\cdot a+\alpha\cdot b$, 2. 2. $(\alpha+\beta)\cdot a=\alpha\cdot a+\beta\cdot a$, 3. 3. $(\alpha\star\beta)\cdot a=\alpha\cdot(\beta\cdot a)$, and 4. 4. $1\cdot a=a$. $\diamond$ We can define right ${\sf R}$-modules and (two-sided) ${\sf R}$-modules (also called ${\sf R}$-bimodules) in a similar fashion. Notice that a (left, right) ideal $\mathfrak{i}\subseteq{\sf R}$ forms a (left, right) ${\sf R}$-module with respect to the addition and multiplication in ${\sf R}$. This obviously holds for the trivial (left, right) ideals $\\{0\\}$ and ${\sf R}$ of ${\sf R}$. Another example of (left, right) ${\sf R}$-modules we will study are the finite direct products of the ring called free (left, right) ${\sf R}$-modules ${\sf R}^{k}$, $k\in{\sf R}$. An additive subset of a (left, right) ${\sf R}$-module is called a (left, right) submodule if it is closed under scalar multiplication with elements of ${\sf R}$. For a subset $F\subseteq M$ let $\langle F\rangle$ denote the submodule generated by $F$ in $M$. The Submodule Membership Problem --- Given: \| An element $a\in M$ and a set of elements $F\subseteq M$. Question: \| $a\in\langle F\rangle$? Similar to the congruence problem for ideals we can specify the congruence problem for submodules as follws: ###### Definition 2.1.14 Two elements $a,b\in{\sf R}$ are said to be congruent modulo the submodule $\langle F\rangle$ for some $F\subseteq M$, denoted by $a\equiv_{\langle F\rangle}b$, if $a-b\in\langle F\rangle$. $\diamond$ The Congruence Problem for submodules --- Given: \| Two elements $a,b\in{\sf R}$ and a set of elements $F\subseteq M$. Question: \| $a\equiv_{\langle F\rangle}b$? ### 2.2 The Notion of Reduction This section summarizes some important notations and definitions of reduction relations and basic properties related to them, as can be found more explicitly for example in the work of Huet or Book and Otto ([Hue80, Hue81, BO93]). Let ${\cal E}$ be a set of elements and $\longrightarrow$ a binary relation on ${\cal E}$ called reduction. For $a,b\in{\cal E}$ we will write $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}b$ in case $(a,b)\in\;\;\longrightarrow$. A pair $({\cal E},\longrightarrow)$ will be called a reduction system. Then we can expand the binary relation as follows: $\,\stackrel{{\scriptstyle 0}}{{\longrightarrow}}\\!\\!\mbox{}\,$ \| \| \| denotes the identity on ${\cal E}$, ---\|---\|---\|--- $\,\stackrel{{\scriptstyle}}{{\longleftarrow}}\\!\\!\mbox{}\,$$\,\stackrel{{\scriptstyle+}}{{\longleftrightarrow}}\\!\\!\mbox{}\,$ \| \| \| denotes the inverse relation for $\longrightarrow$, $\,\stackrel{{\scriptstyle n+1}}{{\longrightarrow}}\\!\\!\mbox{}\,$ \| $:=$ \| $\mbox{$\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}\,$}\circ\longrightarrow$ \| where $\circ$ denotes composition of relations and $n\in{\mathbb{N}}$, $\,\stackrel{{\scriptstyle\leq n}}{{\longrightarrow}}\\!\\!\mbox{}\,$ \| $:=$ \| $\;\\!\bigcup_{0\leq i\leq n}\\!\\!\mbox{$\,\stackrel{{\scriptstyle i}}{{\longrightarrow}}\\!\\!\mbox{}\,$}$, $\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}\,$ \| $:=$ \| $\;\\!\bigcup_{n>0}\\!\mbox{$\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}\,$}$ \| denotes the transitive closure of $\longrightarrow$, $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$ \| $:=$ \| $\mbox{$\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}\,$}\cup\mbox{$\,\stackrel{{\scriptstyle 0}}{{\longrightarrow}}\\!\\!\mbox{}\,$}$ \| denotes the reflexive transitive closure of $\longrightarrow$, $\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}\,$ \| $:=$ \| $\;\\!\longleftarrow\cup\longrightarrow$$\,\stackrel{{\scriptstyle+}}{{\longleftrightarrow}}\\!\\!\mbox{}\,$ \| denotes the symmetric closure of $\longrightarrow$, $\,\stackrel{{\scriptstyle+}}{{\longleftrightarrow}}\\!\\!\mbox{}\,$ \| \| \| denotes the symmetric transitive closure of $\longrightarrow$, $\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}\,$$\,\stackrel{{\scriptstyle+}}{{\longleftrightarrow}}\\!\\!\mbox{}\,$ \| \| \| denotes the reflexive symmetric transitive closure of $\longrightarrow$. A well-known decision problem related to a reduction system is the word problem. ###### Definition 2.2.1 The word problem for a reduction system $({\cal E},\longrightarrow)$ is to decide for $a,b$ in ${\cal E}$, whether $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}\,$}b$ holds. $\diamond$ Instances of this problem are well-known in the literature and undecidable in general. In the following we will outline sufficient conditions such that a reduction system $({\cal E},\longrightarrow)$ has solvable word problem. An element $a\in{\cal E}$ is said to be reducible (with respect to $\longrightarrow$) if there exists an element $b\in{\cal E}$ such that $a\longrightarrow b$. All elements $b\in{\cal E}$ such that $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}b$ are called successors of $a$ and in case $a\mbox{$\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}\,$}b$ they are called proper successors. An element which has no proper successors is called irreducible. In case $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}b$ and $b$ is irreducible, $b$ is called a normal form of $a$. Notice that for an element $a$ in ${\cal E}$ there can be no, one or many normal forms. ###### Definition 2.2.2 A reduction system $({\cal E},\longrightarrow)$ is said to be Noetherian (or terminating) in case there are no infinitely descending reduction chains $a_{0}\longrightarrow a_{1}\longrightarrow\ldots\;$, with $a_{i}\in{\cal E}$, $i\in{\mathbb{N}}$. $\diamond$ In case a reduction system $({\cal E},\longrightarrow)$ is Noetherian every element in ${\cal E}$ has at least one normal form. ###### Definition 2.2.3 A reduction system $({\cal E},\longrightarrow)$ is called confluent, if for all $a,a_{1},a_{2}\in{\cal E}$, $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{1}$ and $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{2}$ implies the existence of $a_{3}\in{\cal E}$ such that $a_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{3}$ and $a_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{3}$, and $a_{1}$, $a_{2}$ are called joinable. $\diamond$ In case a reduction system $({\cal E},\longrightarrow)$ is confluent every element has at most one normal form. We can combine these two properties to give sufficient conditions for the solvability of the word problem. ###### Definition 2.2.4 A reduction system $({\cal E},\longrightarrow)$ is said to be complete (or convergent) in case it is both, Noetherian and confluent. $\diamond$ Complete reduction systems with effective or computable333By effective or computable we mean that given an element we can always construct a successor in case one exists. reduction relations have solvable word problem, as every element has a unique normal form and two elements are equal if and only if their normal forms are equal. Of course we cannot always expect $({\cal E},\longrightarrow)$ to be complete. Even worse, both properties – termination and confluence – are undecidable in general. Nevertheless, there are weaker conditions which guarantee completeness. ###### Definition 2.2.5 A reduction system $({\cal E},\longrightarrow)$ is said to be locally confluent, if for all $a,a_{1},a_{2}\in{\cal E}$, $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{1}$ and $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{2}$ implies the existence of an element $a_{3}\in{\cal E}$ such that $a_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{3}$ and $a_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{3}$. $\diamond$ I.e. local confluence is a special instance of confluence, namely a localization of confluence to one-reduction-step successors of elements only. The next lemma gives an important connection between local confluence and confluence. ###### Lemma 2.2.6 (Newman) Let $({\cal E},\longrightarrow)$ be a Noetherian reduction system. Then $({\cal E},\longrightarrow)$ is confluent if and only if $({\cal E},\longrightarrow)$ is locally confluent. To prove Newman’s lemma we need the concept of Noetherian induction which is based on the following definition. ###### Definition 2.2.7 Let $({\cal E},\longrightarrow)$ be a reduction system. A predicate ${\cal P}$ on ${\cal E}$ is called $\longrightarrow$-complete, in case for every $a\in{\cal E}$ the following implication holds: if ${\cal P}(b)$ is true for all proper successors of $a$, then ${\cal P}(a)$ is true. $\diamond$ The Principle of Noetherian Induction: In case $({\cal E},\longrightarrow)$ is a Noetherian reduction system and ${\cal P}$ is a predicate that is $\longrightarrow$-complete, then for all $a\in{\cal E}$, ${\cal P}(a)$ is true. Proof of Newman’s lemma: Suppose, first, that the reduction system $({\cal E},\longrightarrow)$ is confluent. This immediately implies the local confluence of $({\cal E},\longrightarrow)$ as a special case. To show the converse, since $({\cal E},\longrightarrow)$ is Noetherian we can apply the principle of Noetherian induction to the following predicate: ${\cal P}(a)$ if and only if for all $a_{1},a_{2}\in{\cal E}$, $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{1}$ and $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{2}$ implies that $a_{1}$ and $a_{2}$ are joinable. All we have to do now is to show that ${\cal P}$ is $\longrightarrow$-complete. Let $a\in{\cal E}$ and let ${\cal P}(b)$ be true for all proper successors $b$ of $a$. We have to prove that ${\cal P}(a)$ is true. Suppose $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{1}$ and $a\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{2}$. In case $a=a_{1}$ or $a=a_{2}$ there is nothing to show. Therefore, let us assume $a\neq a_{1}$ and $a\neq a_{2}$, i.e., $a\longrightarrow\tilde{a}_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{1}$ and $a\longrightarrow\tilde{a}_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$}a_{2}$. Then we can deduce the following figure where $b_{0}$ exists, as $({\cal E},\longrightarrow)$ is locally confluent and $b_{1}$ and $b$ exist by our induction hypothesis since $a_{1}$, $b_{0}$ as well as $a_{2}$, $b_{1}$ are proper successors of $a$. Hence $a_{1}$ and $a_{2}$ must be joinable, i.e., the reduction system $({\cal E},\longrightarrow)$ is confluent. q.e.d. Therefore, if the reduction system is terminating, a check for confluence can be reduced to a check for local confluence. The concept of completion then is based on two steps: 1. 1. Check the system for local confluence. If it is locally confluent, then it is also complete. 2. 2. Add new relations arising from situations where the system is not locally confluent. For many reduction systems, e.g. string rewriting systems or term rewriting systems, the check for local confluence again can be localized, often to finite test sets of so-called critical pairs. The relations arising from such critical situations are either confluent or give rise to new relations which stay within the congruence described by the reduction system. Hence adding them in order to increase the descriptive power of the reduction system is correct. This can be done until a complete set is reached. If fair strategies are used in the test for local confluence, the limit system will be complete. We close this section by providing sufficient conditions to ensure a reduction system $({\cal E},\longrightarrow)$ to be Noetherian. ###### Definition 2.2.8 A binary relation $\succeq$ on a set $M$ is said to be a partial ordering, if for all $a,b,c$ in $M$: 1. 1. $\succeq$ is reflexive, i.e., $a\succeq a$, 2. 2. $\succeq$ is transitive, i.e., $a\succeq b$ and $b\succeq c$ imply $a\succeq c$, and 3. 3. $\succeq$ is anti-symmetrical, i.e., $a\succeq b$ and $b\succeq a$ imply $a=b$. $\diamond$ A partial ordering is called total, if for all $a,b\in M$ either $a\succeq b$ or $b\succeq a$ holds. Further a partial ordering $\succeq$ defines a transitive irreflexive ordering $\succ$, where $a\succ b$ if and only if $a\succeq b$ and $a\neq b$, which is often called a proper or strict ordering. We call a partial ordering $\succeq$ well-founded, if the corresponding strict ordering $\succ$ allows no infinite descending chains $a_{0}\succ a_{1}\succ\ldots\;$, with $a_{i}\in M$, $i\in{\mathbb{N}}$. Now we can give a sufficient condition for a reduction system to be terminating. ###### Lemma 2.2.9 Let $({\cal E},\longrightarrow)$ be a reduction system and suppose there exists a partial ordering $\succeq$ on ${\cal E}$ which is well-founded such that $\longrightarrow\;\;\subseteq\;\;\succ$. Then $({\cal E},\longrightarrow)$ is Noetherian. Proof : Suppose the reduction system $({\cal E},\longrightarrow)$ is not Noetherian. Then there is an infinite sequence $a_{0}\longrightarrow a_{1}\longrightarrow\ldots\;$, $a_{i}\in{\cal E}$, $i\in{\mathbb{N}}$. As $\longrightarrow\;\subseteq\;\succ$ this sequence gives us an infinite sequence $a_{0}\succ a_{1}\succ\ldots\;$, with $a_{i}\in{\cal E}$, $i\in{\mathbb{N}}$ contradicting our assumption that $\succeq$ is well-founded on ${\cal E}$. q.e.d. ### 2.3 Gröbner Bases in Polynomial Rings The main interest in this section is the study of ideals in polynomial rings over fields. Let ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ denote a polynomial ring over the (ordered) variables $X_{1},\ldots,X_{n}$ and the computable field ${\mathbb{K}}$. By ${\cal T}=\\{X_{1}^{i_{1}}\ldots X_{n\phantom{1}}^{i_{n}}\mid i_{1},\ldots i_{n}\in{\mathbb{N}}\\}$ we define the set of terms in this structure. A polynomial then is a formal sum $\sum_{i=1}^{n}\alpha_{i}\cdot t_{i}$ with non-zero coefficients $\alpha_{i}\in{\mathbb{K}}\backslash\\{0\\}$ and terms $t_{i}\in{\cal T}$. The products $\alpha\cdot t$ for $\alpha\in{\mathbb{K}}$, $t\in{\cal T}$ are called monomials and will often be denoted as $m=\alpha\cdot t$. We recall that a subset $F$ of ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ generates an ideal ${\sf ideal}(F)=\\{\sum_{i=1}^{k}f_{i}\ast g_{i}\mid k\in{\mathbb{N}},f_{i}\in F,g_{i}\in{\mathbb{K}}[X_{1},\ldots,X_{n}]\\}$ and $F$ is called a basis of this ideal. It was shown by Hilbert using non-constructive arguments that every ideal in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ in fact has a finite basis, but such a generating set need not allow algorithmic solutions for the membership or congruence problem related to the ideal as we have seen in the introduction. It was Buchberger who developed a special type of basis, namely the Gröbner basis, which allows algorithmic solutions for several algebraic problems concerning ideals. He introduced a reduction relation to ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ by transforming polynomials into “rules” and gave a terminating procedure to “complete” an ideal basis interpreted as a reduction system. This procedure is called Buchberger’s algorithm in the literature. We will give a sketch of his approach below. Let $\succeq$ be a total well-founded ordering on the set of terms ${\cal T}$, which is admissible, i.e., $t\succeq 1$, and $s\succ t$ implies $s\circ u\succ t\circ u$ for all $s,t,u$ in ${\cal T}$. The latter property is called compatibility with the multiplication $\circ$. In this context $\circ$ denotes the multiplication in ${\cal T}$, i.e., $X_{1}^{i_{1}}\ldots X_{n\phantom{1}}^{i_{n}}\circ X_{1}^{j_{1}}\ldots X_{n\phantom{1}}^{j_{n}}=X_{1}^{i_{1}+j_{1}}\ldots X_{n\phantom{1}}^{i_{n}+j_{n}}$. With respect to this multiplication we say that a term $s=X_{1}^{i_{1}}\ldots X_{n\phantom{1}}^{i_{n}}$ divides a term $t=X_{1}^{j_{1}}\ldots X_{n\phantom{1}}^{j_{n}}$, if for all $1\leq l\leq n$ we have $i_{l}\leq j_{l}$. The least common multiple ${\sf LCM}(s,t)$ of the terms $s$ and $t$ is the term $X_{1}^{\max\\{i_{1},j_{1}\\}}\ldots X_{n\phantom{1}}^{\max\\{i_{n},j_{n}\\}}$. Note that ${\cal T}$ can be interpreted as the free commutative monoid generated by $X_{1},\ldots,X_{n}$ with the same multiplication $\circ$ as defined above and identity $1=X_{1}^{0}\ldots X_{n\phantom{1}}^{0}$ (recall Example 2.1.5). We proceed to give an example for a total well-founded admissible ordering on the set of terms ${\cal T}$. ###### Example 2.3.1 A total degree ordering $\succ$ on ${\cal T}$ is specified as follows: $X_{1}^{i_{1}}\ldots X_{n\phantom{1}}^{i_{n}}\succ X_{1}^{j_{1}}\ldots X_{n\phantom{1}}^{j_{n}}$ if and only if $\sum_{s=1}^{n}i_{s}>\sum_{s=1}^{n}j_{s}$ or $\sum_{s=1}^{n}i_{s}=\sum_{s=1}^{n}j_{s}$ and there exists $k$ such that $i_{k}>j_{k}$ and $i_{s}=j_{s},1\leq s<k$. $\diamond$ Henceforth, let $\succeq$ denote a total admissible ordering on ${\cal T}$ which is of course well-founded. ###### Definition 2.3.2 Let $p=\sum_{i=1}^{k}\alpha_{i}\cdot t_{i}$ be a non-zero polynomial in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ such that $\alpha_{i}\in{\mathbb{K}}^{}={\mathbb{K}}\backslash\\{0\\}$, $t_{i}\in{\cal T}$ and $t_{1}\succ\ldots\succ t_{n}$. Then we let ${\sf HM}(p)=\alpha_{1}\cdot t_{1}$ denote the head monomial, ${\sf HT}(p)=t_{1}$ the head term and ${\sf HC}(p)=\alpha_{1}$ the head coefficient of $p$. ${\sf RED}(p)=p-{\sf HM}(p)$ stands for the reductum of $p$. We call $p$ monic in case ${\sf HC}(p)=1$. These definitions can be extended to sets $F$ of polynomials by setting ${\sf HT}(F)=\\{{\sf HT}(f)\mid f\in F\\}$, ${\sf HC}(F)=\\{{\sf HC}(f)\mid f\in F\\}$, respectively ${\sf HM}(F)=\\{{\sf HM}(f)\mid f\in F\\}$. $\diamond$ Using the notions of this definition we can recursively extend $\succeq$ from ${\cal T}$ to a partial well-founded admissible ordering $\geq$ on ${\mathbb{K}}[X_{1},\ldots,X_{n}]$. ###### Definition 2.3.3 Let $p,q$ be two polynomials in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$. Then we say $p$ is greater than $q$ with respect to a total well-founded admissible ordering $\succeq$ on ${\cal T}$, i.e., $p>q$, if 1. 1. ${\sf HT}(p)\succ{\sf HT}(q)$ or 2. 2. ${\sf HM}(p)={\sf HM}(q)$ and ${\sf RED}(p)>{\sf RED}(q)$. $\diamond$ Now one first specialization of right ideal bases in terms of the representations they allow can be given according to standard representations as introduced e.g. in [BW92] for polynomial rings over fields. ###### Definition 2.3.4 Let $F$ be a set of polynomials in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ and $g$ a non-zero polynomial in ${\sf ideal}(F)\subseteq{\mathbb{K}}[X_{1},\ldots,X_{n}]$. A representations of the form $\displaystyle g$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}=\alpha_{i}\cdot t_{i},\alpha_{i}\in{\mathbb{K}},t_{i}\in{\cal T},n\in{\mathbb{N}}$ (2.1) where additionally ${\sf HT}(g)\succeq{\sf HT}(f_{i}\star m_{i})$ holds for $1\leq i\leq n$ is called a standard representation of $g$ in terms of $F$. If every $g\in{\sf ideal}(F)\backslash\\{0\\}$ has such a representation in terms of $F$, then $F$ is called a standard basis of ${\sf ideal}(F)$. $\diamond$ What distinguishes an arbitrary representation from a standard representation is the fact that the former may contain polynomial multiples with head terms larger than the head term of the represented polynomial. For example let $f_{1}=X_{1}+X_{2}$, $f_{2}=X_{1}+X_{3}$ and $F=\\{f_{1},f_{2}\\}$ in ${\mathbb{Q}}[X_{1},X_{2}]$ with $X_{1}\succ X_{2}\succ X_{3}$. Then for the polynomial $g=X_{2}-X_{3}$ we have the representation $g=f_{1}+(-1)\cdot f_{2}$ which is no standard one as ${\sf HT}(g)=X_{2}\prec{\sf HT}(f_{1})={\sf HT}(f_{2})=X_{1}$. Obviously the larger head terms have to vanish in the sum. Therefore, in order to change an arbitrary representation into one fulfilling our additional condition (2.1) we have to deal with special sums of polynomials related to such situations. ###### Definition 2.3.5 Let $F$ be a set of polynomials in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ and $t$ an element in ${\cal T}$. Then we define the set of critical situations ${\cal C}(t,F)$ related to $t$ and $F$ to contain all tuples of the form $(t,f_{1},\ldots,f_{k},m_{1},\ldots,m_{k})$, $k\in{\mathbb{N}}$, $f_{1},\ldots,f_{k}\in F$444Notice that $f_{1},\ldots,f_{k}$ are not necessarily different polynomials from $F$., $m_{i}=\alpha_{i}\cdot t_{i}$, such that 1. 1. ${\sf HT}(f_{i}\star m_{i})=t$, $1\leq i\leq k$, and 2. 2. $\sum_{i=1}^{k}{\sf HM}(f_{i}\star m_{i})=0$. We set ${\cal C}(F)=\bigcup_{t\in{\cal T}}{\cal C}(t,F)$. $\diamond$ In our example the tuple $(X_{1},f_{1},f_{2},1,-1)$ is an elements of the critical set ${\cal C}(X_{1},F)$. We can characterize standard bases using these special sets. ###### Theorem 2.3.6 Let $F$ be a set of polynomials in ${\mathbb{K}}[X_{1},\ldots,X_{n}]\backslash\\{0\\}$. Then $F$ is a standard basis of ${\sf ideal}(F)$ if and only if for every tuple $(t,f_{1},\ldots,f_{k},m_{1},\ldots,m_{k})$ in ${\cal C}(F)$ as specified in Definition 2.3.5 the polynomial $\sum_{i=1}^{k}f_{i}\star m_{i}$ has a standard representation with respect to $F$. Proof : In case $F$ is a standard basis since these polynomials are all elements of ${\sf ideal}(F)$ they must have standard representations with respect to $F$. To prove the converse, it remains to show that every element in ${\sf ideal}(F)$ has a standard representation with respect to $F$. Hence, let $g=\sum_{j=1}^{m}f_{j}\star m_{j}$ be an arbitrary representation of a non- zero polynomial $g\in{\sf ideal}(F)$ such that $f_{j}\in F$, and $m_{j}=\alpha_{j}\cdot t_{j}$ with $\alpha_{j}\in{\mathbb{K}}$, $t_{j}\in{\cal T}$. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(f_{j}\star t_{j})\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $f_{j}\star t_{j}$ with head term $t$. Then $t\succeq{\sf HT}(g)$ and in case ${\sf HT}(g)=t$ this immediately implies that this representation is already a standard representation. Else we proceed by induction on the term $t$. Without loss of generality let $f_{1},\ldots,f_{K}$ be the polynomials in the corresponding representation such that $t={\sf HT}(f_{i}\star t_{i})$, $1\leq i\leq K$. Then the tuple $(t,f_{1},\ldots,f_{K},m_{1},\ldots,m_{K})$ is in ${\cal C}(F)$ and let $h=\sum_{i=1}^{K}f_{i}\star m_{i}$. We will now change our representation of $g$ in such a way that for the new representation of $g$ we have a smaller maximal term. Let us assume $h$ is not $0$555In case $h=0$, just substitute the empty sum for the representation of $h$ in the equations below.. By our assumption, $h$ has a standard representation with respect to $F$, say $\sum_{j=1}^{n}h_{j}\star n_{j}$, where $h_{j}\in F$, and $n_{j}=\beta_{j}\cdot s_{j}$ with $\beta_{j}\in{\mathbb{K}}$, $s_{j}\in{\cal T}$ and all terms occurring in the sum are bounded by $t\succ{\sf HT}(h)$ as $\sum_{i=1}^{K}{\sf HM}(f_{i}\star m_{i})=0$. This gives us: $\displaystyle g$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{K}f_{i}\star m_{i}+\sum_{i=K+1}^{m}f_{i}\star m_{i}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{n}h_{j}\star n_{j}+\sum_{i=K+1}^{m}f_{i}\star m_{i}$ which is a representation of $g$ where the maximal term is smaller than $t$. q.e.d. In fact for the case of polynomial rings over fields one can show that it is sufficient to consider critical sets for subsets of $F$ of size 2 and we can restrict the terms to the least common multiples of the head terms of the respective two polynomials. These sets then correspond to the concept of s-polynomials used to characterize Gröbner bases which will be introduced later on. Reviewing our example on page 2.3.4 we find that the set $F=\\{X_{1}+X_{2},X_{1}+X_{3}\\}$ is no standard basis as the polynomial $g=X_{2}-X_{3}$ has no standard representation although it is an elements of ${\sf ideal}(F)$. However the set $F\cup\\{g\\}$ then is a standard basis of ${\sf ideal}(F)$. In the literature standard representations in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ are closely related to reduction relations based on the divisibility of terms and standard bases are in fact Gröbner bases. Here we want to introduce Gröbner bases in terms of rewriting. Hence we continue by introducing the concept of reduction to ${\mathbb{K}}[X_{1},\ldots,X_{n}]$. We can split a non-zero polynomial $p$ into a rule ${\sf HM}(p)\longrightarrow-{\sf RED}(p)$ and we have ${\sf HM}(p)>-{\sf RED}(p)$. Therefore, a set of polynomials gives us a binary relation $\longrightarrow$ on ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ which induces a one-step reduction relation as follows. ###### Definition 2.3.7 Let $p,f$ be two polynomials in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$. We say $f$ reduces $p$ to $q$ at a monomial $m=\alpha\cdot t$ of $p$ in one step, denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{f}\,$}q$, if 1. (a) ${\sf HT}(f)\circ u=t$ for some $u\in{\cal T}$, i.e., ${\sf HT}(f)$ divides $t$, and 2. (b) $q=p-\alpha\cdot{\sf HC}(f)^{-1}\cdot f\ast u$. We write $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{f}\,$}$ if there is a polynomial $q$ as defined above and $p$ is then called reducible by $f$. Further, we can define $\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}\,$},\mbox{$\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}\,$}$, and $\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}\,$ as usual. Reduction by a set $F\subseteq{\mathbb{K}}[X_{1},\ldots,X_{n}]$ is denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}q$ and abbreviates $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{f}\,$}q$ for some $f\in F$, which is also written as $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{f\in F}\,$}q$. $\diamond$ Note that if $f$ reduces $p$ to $q$ at a monomial $m=\alpha\cdot t$ then $t$ is no longer among the terms of $q$. We call a set of polynomials $F\subseteq{\mathbb{K}}[X_{1},\ldots,X_{n}]$ interreduced, if no $f\in F$ is reducible by a polynomial in $F\backslash\\{f\\}$. In the classical case of polynomial rings over fields the existence of a standard representation for a polynomial immediately implies reducibility of the head monomial of the polynomial by any reduction relation based on divisibility of terms, hence by the reduction relation defined here. This is due to the fact that if a polynomial $g$ has a standard representation in terms of a set of polynomials $F$ for at least one polynomial $f$ in $F$ and some term $t$ in ${\cal T}$ we have ${\sf HT}(g)={\sf HT}(f\star t)={\sf HT}(f)\circ t$ and hence $g$ is reducible at the monomial ${\sf HM}(g)$ by $f$. Notice that this is no longer true for polynomial rings over the integers. Let $F=\\{3\cdot X^{2}+X,2\cdot X^{2}+X\\}$ be a subset of ${\mathbb{Z}}[X]$. Then the polynomial $g=(3\cdot X^{2}+X)-(2\cdot X^{2}+X)=X^{2}$ has a standard representation in terms of $F$ but neither $3\cdot X^{2}$ nor $2\cdot X^{2}$ are divisors of the monomial $X^{2}$ as neither $3$ nor $2$ devide $1$ in ${\mathbb{Z}}$. Notice that we have $\longrightarrow\;\subseteq\;\;>$ and indeed one can show that our reduction relation on ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ is Noetherian. Therefore, we can restrict ourselves to ensuring local confluence when describing a completion procedure to compute Gröbner bases later on. But first we have to provide a definition of Gröbner bases in the context of rewriting. ###### Definition 2.3.8 A set $G\subseteq{\mathbb{K}}[X_{1},\ldots,X_{n}]$ is said to be a Gröbner basis of the ideal it generates, if 1. 1. $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{G}\,$}=\;\;\equiv_{{\sf ideal}(G)}$, and 2. 2. $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{G}\,$ is confluent. $\diamond$ The first statement expresses that the reduction relation describes the ideal congruence. It holds for any basis of an ideal in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ and is hence normally omitted in the definitions provided in the literature. However, when generalizing the concept of Gröbner bases to other structures it is no longer guaranteed and hence we have included it in our definition. The second statement ensures the existence of unique normal forms. If we additionally require a Gröbner basis to be interreduced, such a basis is unique in case we assume that the polynomials are monic, i.e., their head coefficients are $1$. The following lemma gives some properties of the reduction relation, which are essential in giving a constructive description of a Gröbner basis not only in the setting of commutative polynomial rings over fields. ###### Lemma 2.3.9 Let $F$ be a set of polynomials and $p,q,h$ some polynomials in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$. Then the following statements hold: 1. 1. Let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}h$. Then there are polynomials $p^{\prime},q^{\prime}\in{\mathbb{K}}[X_{1},\ldots,X_{n}]$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}p^{\prime}$, $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}q^{\prime}$ and $h=p^{\prime}-q^{\prime}$. 2. 2. Let $0$ be a normal form of $p-q$ with respect to $F$. Then there exists a polynomial $g\in{\mathbb{K}}[X_{1},\ldots,X_{n}]$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}g$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}g$. 3. 3. $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}q\mbox{ if and only if }p-q\in{\sf ideal}(F)$. 4. 4. $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ implies $\alpha\cdot p\ast u\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ for all $\alpha\in{\mathbb{K}}$ and $u\in{\cal T}$. 5. 5. $\alpha\cdot p\ast u\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{p}\,$}0$ for all $\alpha\in{\mathbb{K}}^{}$ and $u\in{\cal T}$. The second statement of this lemma is often called the Translation Lemma in the literature. Statement 3 shows that Buchberger’s reduction relation always captures the ideal congruence. Statement 4 is connected to the important fact that reduction steps are preserved under multiplication with monomials. The set $F=\\{X_{1}+X_{2},X_{1}+X_{3}\\}$ of polynomials in ${\mathbb{Q}}[X_{1},X_{2},X_{3}]$ from page 2.3.4 is an example of an ideal basis which is not complete, i.e. the reduction relation is not complete666Note that we call a set of polynomials complete (confluent, etc.) if the reduction relation induced by these polynomials used as rules is complete (confluent, etc.).. This follows as the polynomial $X_{1}$ can be reduced by $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$ to $-X_{2}$ as well as to $-X_{3}$ and the latter two polynomials cannot be joined using $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$. Of course we cannot expect an arbitrary ideal basis to be complete. But Buchberger was able to show that in order to “complete” a given basis one only has to add finitely many special polynomials which arise from critical situations as described in the context of reduction systems in the previous section and Definition 2.3.5. The term $X_{1}$ in our example describes such a critical situation which is in fact the only one relevant for completing the set $F$. ###### Definition 2.3.10 The s-polynomial for two non-zero polynomials $p,q\in{\mathbb{K}}[X_{1},\ldots,X_{n}]$ is defined as ${\sf spol}(p,q)={\sf HC}(p)^{-1}\cdot p\ast u-{\sf HC}(q)^{-1}\cdot q\ast v,$ where ${\sf LCM}({\sf HT}(p),{\sf HT}(q))={\sf HT}(p)\circ u={\sf HT}(q)\circ v$ for some $u,v\in{\cal T}$. $\diamond$ An s-polynomial will be called non-trivial in case it is not zero and notice that for non-trivial s-polynomials we always have ${\sf HT}({\sf spol}(p,q))\prec{\sf LCM}({\sf HT}(p),{\sf HT}(q))$. The s-polynomial for $p$ and $q$ belongs to the set of critical situations ${\cal C}({\sf LCM}({\sf HT}(p),{\sf HT}(q)),\\{p,q\\})$. In our example we find ${\sf spol}(X_{1}+X_{2},X_{1}+X_{3})=X_{1}+X_{2}-(X_{1}+X_{3})=X_{2}-X_{3}$. Why are s-polynomials related to testing for local confluence? To answer this question we have to look at critical situations related to the reduction relation as defined in Definition 2.3.7. Given two polynomials $p,q\in{\mathbb{K}}[X_{1},\ldots,X_{n}]$ the smallest situation where both of them can be applied as rules is the least common multiple of their head terms. Let ${\sf LCM}({\sf HT}(p),{\sf HT}(q))={\sf HT}(p)\circ u={\sf HT}(q)\circ v=t$ for some $u,v\in{\cal T}$. This gives us the following situation: Then we get $p^{\prime}-q^{\prime}=t-{\sf HC}(q)^{-1}\cdot q\ast v-(t-{\sf HC}(p)^{-1}\cdot p\ast u)={\sf HC}(p)^{-1}\cdot p\ast u-{\sf HC}(q)^{-1}\cdot q\ast v={\sf spol}(p,q)$, i.e., the s-polynomial is derived from the two one- step successors by subtraction. Now by Lemma 2.3.9 we know that ${\sf spol}(p,q)\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ implies the existence of a common normal form for the polynomials $p^{\prime}$ and $q^{\prime}$. Since the reduction relation based on Definition 2.3.7 is terminating, the confluence test can hence be reduced to checking whether all s-polynomials reduce to zero. The following theorem now gives a constructive characterization of Gröbner bases based on these ideas. ###### Theorem 2.3.11 For a set of polynomials $F$ in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$, the following statements are equivalent: 1. 1. $F$ is a Gröbner basis. 2. 2. For all polynomials $g\in{\sf ideal}(F)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$. 3. 3. For all polynomials $f_{k},f_{l}\in F$ we have ${\sf spol}(f_{k},f_{l})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$. Proof : $1\Longrightarrow 2:$ Let $F$ be a Gröbner basis and $g\in{\sf ideal}(F)$. Then $g$ is congruent to $0$ modulo the ideal generated by $F$, i.e., $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$. Thus, as $0$ is irreducible and $G$ is confluent, we get $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$. $2\Longrightarrow 1:$ By Lemma 2.3.9 3 we know $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{G}\,$}=\;\;\equiv_{{\sf ideal}(G)}$. Hence it remains to show that reduction with respect to $F$ is confluent. Since our reduction is terminating it is sufficient to show local confluence. Thus, suppose there are three different polynomials $g,h_{1},h_{2}$ such that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}h_{1}$ and $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}h_{2}$. Then we know $h_{1}\equiv_{{\sf ideal}(F)}g\equiv_{{\sf ideal}(F)}h_{2}$ and hence $h_{1}-h_{2}\in{\sf ideal}(F)$. Now by lemma 2.3.9 (the translation lemma), $h_{1}-h_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ implies the existence of a polynomial $h\in{\mathbb{K}}[X_{1},\ldots,X_{n}]$ such that $h_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}h$ and $h_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}h$. Hence, $h_{1}$ and $h_{2}$ are joinable. $2\Longrightarrow 3:$ By definition 2.3.10 the s-polynomial for two non-zero polynomials $f_{k},f_{l}\in{\mathbb{K}}[X_{1},\ldots,X_{n}]$ is defined as ${\sf spol}(f_{k},f_{l})={\sf HC}(f_{k})^{-1}\cdot f_{k}\ast u-{\sf HC}(f_{l})^{-1}\cdot f_{l}\ast v,$ where ${\sf LCM}({\sf HT}(p),{\sf HT}(q))={\sf HT}(p)\circ u={\sf HT}(q)\circ v$ and, hence, ${\sf spol}(f_{k},f_{l})\in{\sf ideal}(F)$. Therefore, ${\sf spol}(f_{k},f_{l})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ follows immediately. $3\Longrightarrow 2:$ We have to show that every $g\in{\sf ideal}(F)\backslash\\{0\\}$ is $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$-reducible to zero. Remember that for $h\in{\sf ideal}(F)$, $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}h^{\prime}$ implies $h^{\prime}\in{\sf ideal}(F)$. As $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$ is Noetherian, thus it suffices to show that every $g\in{\sf ideal}(F)\backslash\\{0\\}$ is $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$-reducible. Let $g=\sum_{j=1}^{m}\alpha_{j}\cdot f_{j}\ast w_{j}$ be an arbitrary representation of $g$ with $\alpha_{j}\in{\mathbb{K}}^{}$, $f_{j}\in F$, and $w_{j}\in{\cal T}$. Depending on this representation of $g$ and a total well-founded admissible ordering $\succeq$ on ${\cal T}$ we define $t=\max\\{{\sf HT}(f_{j})\circ w_{j}\mid j\in\\{1,\ldots,m\\}\\}$ and $K$ is the number of polynomials $f_{j}\ast w_{j}$ containing $t$ as a term. Then $t\succeq{\sf HT}(g)$ and in case ${\sf HT}(g)=t$ this immediately implies that $g$ is $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$-reducible. Thus we will prove that $g$ has a representation where every occurring term is less or equal to ${\sf HT}(g)$, i.e., there exists a representation such that $t={\sf HT}(g)$777Such representations are often called standard representations in the literature (compare [BW92]).. This will be done by induction on $(t,K)$, where $(t^{\prime},K^{\prime})<(t,K)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $K^{\prime}<K)$888Note that this ordering is well-founded since $\succ$ is well-founded on ${\cal T}$ and $K\in{\mathbb{N}}$.. In case $t\succ{\sf HT}(g)$ there are two polynomials $f_{k},f_{l}$ in the corresponding representation999Not necessarily $f_{l}\neq f_{k}$. such that ${\sf HT}(f_{k})\circ w_{k}={\sf HT}(f_{l})\circ w_{l}=t$. By definition 2.3.10 we have an s-polynomial ${\sf spol}(f_{k},f_{l})={\sf HC}(f_{k})^{-1}\cdot f_{k}\ast z_{k}-{\sf HC}(f_{l})^{-1}\cdot f_{l}\ast z_{l}$ such that ${\sf HT}(f_{k})\circ z_{k}={\sf HT}(f_{l})\circ z_{l}={\sf LCM}({\sf HT}(f_{k}),{\sf HT}(f_{l}))$. Since ${\sf HT}(f_{k})\circ w_{k}={\sf HT}(f_{l})\circ w_{l}$ there exists an element $z\in{\cal T}$ such that $w_{k}=z_{k}\circ z$ and $w_{l}=z_{l}\circ z$. We will now change our representation of $g$ by using the additional information on this s-polynomial in such a way that for the new representation of $g$ we either have a smaller maximal term or the occurrences of the term $t$ are decreased by at least 1. Let us assume that ${\sf spol}(f_{k},f_{l})$ is not trivial101010In case ${\sf spol}(f_{k},f_{l})=0$, just substitute $0$ for the sum $\sum_{i=1}^{n}\delta_{i}\cdot h_{i}\ast v_{i}$ in the equations below.. Then the reduction sequence ${\sf spol}(f_{k},f_{l})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ results in a representation of the form ${\sf spol}(f_{k},f_{l})=\sum_{i=1}^{n}\delta_{i}\cdot h_{i}\ast v_{i}$, where $\delta_{i}\in{\mathbb{K}}^{},h_{i}\in F,v_{i}\in{\cal T}$. As the $h_{i}$ are due to the reduction of the s-polynomial, all terms occurring in the sum are bounded by the term ${\sf HT}({\sf spol}(f_{k},f_{l}))$. Moreover, since $\succeq$ is admissible on ${\cal T}$ this implies that all terms of the sum $\sum_{i=1}^{n}\delta_{i}\cdot h_{i}\ast v_{i}\ast z$ are bounded by ${\sf HT}({\sf spol}(f_{k},f_{l}))\circ z\prec t$, i.e., they are strictly bounded by $t$111111This can also be concluded by statement four of lemma 2.3.9 since ${\sf spol}(f_{k},f_{l})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ implies ${\sf spol}(f_{k},f_{l})\ast z\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ and ${\sf HT}({\sf spol}(f_{k},f_{l})\ast z)\prec t$.. We can now do the following transformations: $\displaystyle\alpha_{k}\cdot f_{k}\ast w_{k}+\alpha_{l}\cdot f_{l}\ast w_{l}$ (2.2) $\displaystyle=$ $\displaystyle\alpha_{k}\cdot f_{k}\ast w_{k}+\underbrace{\alpha^{\prime}_{l}\cdot\beta_{k}\cdot f_{k}\ast w_{k}-\alpha^{\prime}_{l}\cdot\beta_{k}\cdot f_{k}\ast w_{k}}_{=\,0}+\alpha^{\prime}_{l}\cdot\beta_{l}\cdot f_{l}\ast w_{l}$ $\displaystyle=$ $\displaystyle(\alpha_{k}+\alpha^{\prime}_{l}\cdot\beta_{k})\cdot f_{k}\ast w_{k}-\alpha^{\prime}_{l}\cdot\underbrace{(\beta_{k}\cdot f_{k}\ast w_{k}-\beta_{l}\cdot f_{l}\ast w_{l})}_{=\,{\sf spol}(f_{k},f_{l})\ast z}$ $\displaystyle=$ $\displaystyle(\alpha_{k}+\alpha^{\prime}_{l}\cdot\beta_{k})\cdot f_{k}\ast w_{k}-\alpha^{\prime}_{l}\cdot(\sum_{i=1}^{n}\delta_{i}\cdot h_{i}\ast(v_{i}\circ z))$ where, $\beta_{k}={\sf HC}(f_{k})^{-1}$, $\beta_{l}={\sf HC}(f_{l})^{-1}$, and $\alpha^{\prime}_{l}\cdot\beta_{l}=\alpha_{l}$. By substituting (2.2) in our representation of $g$ either $t$ disappears or $K$ is decreased. q.e.d. The second item of this theorem immediately implies the correctness of the algebraic definition of Gröbner bases, which is equivalent to Definition 2.3.8. ###### Definition 2.3.12 A set $G$ of polynomials in ${\mathbb{K}}[X_{1},\ldots,X_{n}]\backslash\\{0\\}$ is said to be a Gröbner basis, if ${\sf HT}({\sf ideal}(G))=\\{{\sf HT}(g)\ast t\mid g\in G,t\in{\cal T}\\}$. $\diamond$ ###### Remark 2.3.13 A closer inspection of the proof of $3\Longrightarrow 2$ given above reveals a concept which is essential in the proofs of similar theorems for specific function rings in the following chapters. The heart of this proof consists in transforming an arbitrary representation of an element $g$ belonging to the ideal generated by the set $F$ in such a way that we can deduce a top reduction sequence for $g$ to zero, i.e., a reduction sequence where the reductions only take place at the respective head term. Such a representation of $g$ then is a standard representation and Gröbner bases are standard bases. $\diamond$ As a consequence of Theorem 2.3.11 it is decidable whether a finite set of polynomials is a Gröbner basis. Moreover, this theorem gives rise to the following completion procedure for sets of polynomials. Procedure: Buchberger’s Algorithm Given: \| A finite set of polynomials $F\subseteq{\mathbb{K}}[X_{1},\ldots,X_{n}]$. ---\|--- Find: \| $\mbox{\sc Gb}(F)$, a Gröbner basis of $F$. $G$ := $F$; --- $B$ := $\\{(q_{1},q_{2})\mid q_{1},q_{2}\in G,q_{1}\neq q_{2}\\}$; while $B\neq\emptyset$ do \| $(q_{1},q_{2}):={\rm remove}(B)$; \| % Remove an element from the set $B$ \| $h:={\rm normalform}({\sf spol}(q_{1},q_{2}),\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{G}\,$})$ \| % Compute a normal form of ${\sf spol}(q_{1},q_{2})$ with respect to $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{G}\,$ \| if \| $h\neq 0$ \| \| then \| $B:=B\cup\\{(f,h)\mid f\in G\\}$; \| \| \| $G:=G\cup\\{h\\}$; \| endif endwhile $\mbox{\sc Gb}(F):=G$ Applying this procedure to our example $F=\\{X_{1}+X_{2},X_{1}+X_{3}\\}$ from page 2.3.4 gives us $h=X_{2}-X_{3}$ and $G=F\cup\\{h\\}$ is a Gröbner basis as all other critical situations are resolvable. Termination of the procedure can be shown by using a slightly different characterization of Gröbner bases (see Section 1.2): A subset $G$ of ${\sf ideal}^{{\mathbb{K}}[X_{1},\ldots,X_{n}]}(F)$ is a Gröbner basis of ${\sf ideal}^{{\mathbb{K}}[X_{1},\ldots,X_{n}]}(F)$ if and only if ${\sf HT}({\sf ideal}^{{\mathbb{K}}[X_{1},\ldots,X_{n}]}(F)\backslash\\{0\\})={\sf ideal}^{\cal T}({\sf HT}(G))$, i.e., the set of the head terms of the polynomials in the ideal generated by $F$ in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$ coincides with the ideal (in ${\cal T}$) generated by the head terms of the polynomials in $G$. Reviewing the procedure, we find that every polynomial added in the while loop has the property that its head term cannot be divided by the head terms of the polynomials already in $G$. By Dickson’s Lemma or Hilbert’s Basis Theorem, the head terms of the polynomials in $G$ will at some step form a basis for the set of head terms of the polynomials of the ideal generated by $F$ which itself is the ideal in ${\cal T}$ generated by the head terms of the polynomials in $G$. From this time on for every new polynomial $h$ computed by the algorithm the head term ${\sf HT}(h)$ must lie in this ideal. Therefore, its head term must be divisible by at least one of the head terms of the polynomials in $G$, i.e., ${\sf HT}(h)$ and hence $h$ cannot be in normal form with respect to $G$ unless it is zero. ## Chapter 3 Reduction Rings In this chapter we proceed to distinguish sufficient conditions, which allow to define a reduction relation for a ring in such a way that every finitely generated ideal in the ring has a finite Gröbner basis with respect to that reduction relation. Such rings will be called reduction rings. Often additional conditions can be given to ensure effectivity for the ring operations, the reduction relation and the computation of the Gröbner bases – the ring is then called an effective reduction ring. Naturally the question arises, when and how the property of being a reduction ring is preserved under various ring constructions. This can be studied from an existential as well as from a constructive point of view. One main goal of studying abstract reduction rings is to provide universal methods for constructing new reduction rings without having to generalize the whole setting individually for each new structure: e.g. knowing that the integers ${\mathbb{Z}}$ are a reduction ring and that the property lifts to polynomials in one variable, we find that ${\mathbb{Z}}[X]$ is again a reduction ring and we can immediately conclude that also ${\mathbb{Z}}[X_{1},\ldots,X_{n}]$ is a reduction ring. Similarly, as sums of reduction rings are again reduction rings, we can directly conclude that ${\mathbb{Z}}^{k}[X_{1},\ldots,X_{n}]$ or even $({\mathbb{Z}}[Y_{1},\ldots,Y_{m}])^{k}[X_{1},\ldots,X_{n}]$ are reduction rings. Moreover, since ${\mathbb{Z}}$ is an effective reduction ring it can be shown that these new reduction rings again are effective. Commutative effective reduction rings have been studied by Buchberger, Madlener, and Stifter in [Buc83, Mad86, Sti87]. On the other hand, many rings of interest are non-commutative, e.g. rings of matrices, the ring of quaternions, Bezout rings and various monoid rings, and since in many cases they can be regarded as reduction rings, they are again candidates for applying ring constructions. More interesting examples of non- commutative reduction rings have been studied by Pesch in [Pes97]. A general framework for reduction rings and ring constructions including the non-commutative case was presented at the Linz conference “33 years of Gröbner Bases” in [MR98b]. Here we extend this framework by giving more details and insight. Additionally, we add a section on modules over reduction rings, as this concept arises naturally as a generalization of ideals in rings. Of course there are also rings of interest, which can be enriched by a reduction relation, but will not allow finite Gröbner bases for all ideals. Monoid and group rings provide such a setting. For such structures still many of the properties studied here are of interest and can be shown in weaker forms, e.g. provided a monoid ring with a reduction relation we can define a reduction relation for the polynomial ring with one variable over the monoid ring. The chapter is organized as follows: In Section 3.1 we introduce axioms for specifying reduction relations in rings and give two concepts involving special forms of ideal bases – weak reduction rings and reduction rings. In Section 3.2 – 3.5 we study quotients, sums, modules, and polynomial rings of these structures. ### 3.1 Reduction Rings Let ${\sf R}$ be a ring with unit $1$ and a (not necessarily effective) reduction relation $\Longrightarrow_{B}\subseteq{\sf R}\times{\sf R}$ associated with subsets $B\subseteq{\sf R}$ satisfying the following axioms: 1. (A1) $\Longrightarrow_{B}\;=\;\bigcup_{\beta\in B}\Longrightarrow_{\beta}$, $\Longrightarrow_{B}$ is terminating for all finite subsets $B\subseteq{\sf R}$. 2. (A2) $\alpha\Longrightarrow_{\beta}\gamma$ implies $\alpha-\gamma\in{\sf ideal}^{{\sf R}}(\beta)$. 3. (A3) $\alpha\Longrightarrow_{\alpha}0$ for all $\alpha\in{\sf R}\backslash\\{0\\}$. Part one of Axiom (A1) states how a reduction relation using sets is defined in terms of a reduction relation using elements of ${\sf R}$ and is hence applicable to arbitrary sets $B\subseteq{\sf R}$. However, Axiom (A1) does not imply termination of reduction with respect to arbitrary sets: Just assume for example the ring ${\sf R}={\mathbb{Q}}[\\{X_{i}\mid i\in{\mathbb{N}}\\}]$, i.e., the polynomial ring with infinitely many indeterminates, and the reduction relation based on divisibility of head terms with respect to the length-lexicographical ordering induced by $X_{1}\succ X_{2}\succ\ldots$. Then although reduction when using a finite set of polynomials is terminating, this is no longer true for infinite sets. For example the infinite set $\\{X_{i}-X_{i+1}\mid i\in{\mathbb{N}}\\}$ gives rise to an infinite reduction sequence $X_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{X_{1}-X_{2}}\,$}X_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{X_{2}-X_{3}}\,$}X_{3}\ldots$. This phenomenon of course has many consequences. Readers familiar with Gröbner bases in polynomial rings know that when proving that a set of polynomials is a Gröbner basis if and only if all ideal elements reduce to zero using the set, this is shown by proving that every ideal element is reducible by some element in the set (compare Theorem 2.3.11). Unfortunately, this only implies reducibility to zero in case the reduction relation is terminating. Without this property other methods have to be applied. In order to ensure termination for arbitrary subsets of ${\sf R}$ it is possible to give a more restricted form of Axiom (A1): 1. (A1’) $\Longrightarrow_{B}\;=\;\bigcup_{\beta\in B}\Longrightarrow_{\beta}$, $\Longrightarrow_{B}$ is terminating for all subsets $B\subseteq{\sf R}$. Then of course reduction sequences are always terminating and many additional restrictions, which we have to add later, are no longer necessary. Still we prefer the more general formulation of the axiom since it allows to state more clearly why and where termination is needed and how it can be achieved. Axiom (A2) states how reduction steps are related to the ideal congruence, namely that one reduction step using an element $\beta\in{\sf R}$ is captured by the congruence generated by ${\sf ideal}^{{\sf R}}(\beta)$. We will later on see that this extends to the reflexive transitive symmetric closure $\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$ of any reduction relation $\Longrightarrow_{B}$ for arbitrary sets $B\subseteq{\sf R}$. Notice that in case ${\sf R}$ is commutative (A2) implies $\gamma=\alpha-\beta\cdot\rho$ for some $\rho$ in ${\sf R}$. In the non- commutative case using a single element $\beta$ for reduction $\alpha-\gamma\in{\sf ideal}^{{\sf R}}(\beta)$ only implies $\gamma=\alpha-\sum_{i=1}^{k}\rho_{i1}\cdot\beta\cdot\rho_{i2}$ for some $\rho_{i1},\rho_{i2}\in{\sf R}$, $1\leq i\leq k$, hence possibly involving $\beta$ more than once with different multipliers. This provides a large range of possibilities for defining reduction steps, e.g. by subtracting one or more appropriate multiples of $\beta$ from $\alpha$. Notice further that on the converse Axiom (A2) does not provide any information on how $\alpha$, $\gamma\in{\sf R}$ with $\alpha-\gamma\in{\sf ideal}^{{\sf R}}(\beta)$ are related with respect to the reduction relation $\Longrightarrow_{\\{\beta\\}}$. As a consequence many properties of specialized reduction relations as known from the literature, e.g. the useful Translation Lemma, cannot be shown to hold in this general setting. We can define one-sided (right or left) reduction relations in rings by refining Axiom (A2) as follows: 1. (A2r) $\alpha\Longrightarrow_{\beta}\gamma$ implies $\alpha-\gamma\in{\sf ideal}_{r}^{{\sf R}}(\beta)$, respectively 2. (A2l) $\alpha\Longrightarrow_{\beta}\gamma$ implies $\alpha-\gamma\in{\sf ideal}_{l}^{{\sf R}}(\beta)$. In these special cases again we always get $\gamma=\alpha-\beta\cdot\rho$ respectively $\gamma=\alpha-\rho\cdot\beta$ for some $\rho\in{\sf R}$. Remember that Axiom (A2) while not specific on the exact form of the reduction step ensures that reduction steps “stay” within the ideal congruence. Let us now study the situation for a set $B\subseteq{\sf R}$ and let $\equiv_{{\mathfrak{i}}}$ denote the congruence generated by the ideal ${\mathfrak{i}}={\sf ideal}(B)$, i.e., $\alpha\equiv_{{\mathfrak{i}}}\beta$ if and only if $\alpha-\beta\in{\mathfrak{i}}$. Then (A1)111We only need the first part of Axiom (A1), namely how $\Longrightarrow_{B}$ is defined, and hence we do not have to restrict ourselves to finite sets. and (A2) immediately imply $\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}\subseteq\;\;\equiv_{{\mathfrak{i}}}$. Hence, in case the reduction relation is effective one method for deciding the membership problem for a finitely generated ideal ${{\mathfrak{i}}}$ is to transform a finite generating set $B$ into a finite set $B^{\prime}$ such that $B^{\prime}$ still generates ${\mathfrak{i}}$ and $\Longrightarrow_{B^{\prime}}$ is confluent on ${\mathfrak{i}}$. Notice that $0$ has to be irreducible222$0$ cannot be reducible by itself since this would contradict the termination property in (A1). Similarly, $0\Longrightarrow_{\beta}0$ and $0\Longrightarrow_{\beta}\gamma$, both $\beta$ and $\gamma$ not equal $0$, give rise to infinite reduction sequences again contradicting (A1). for all $\Longrightarrow_{\alpha}$, $\alpha\in{\sf R}$. Therefore, $0$ has to be the normal form of the ideal elements. Hence the goal is to achieve $\alpha\in{{\mathfrak{i}}}$ if and only if $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}0$. In particular ${\mathfrak{i}}$ is one equivalence class of $\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$. The different definitions of reduction relations for rings existing in literature show that for deciding the membership problem of an ideal ${\mathfrak{i}}$ it is not necessary to enforce $\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}=\;\equiv_{{\mathfrak{i}}}$. For example the D-reduction notion given by Pan in [Pan85] does not have this property but is still sufficient to decide $\equiv_{{\mathfrak{i}}}$-equivalence of two elements because $\alpha\equiv_{{\mathfrak{i}}}\beta$ if and only if $\alpha-\beta\in{{\mathfrak{i}}}$. It may even happen that D-reduction is not only confluent on ${{\mathfrak{i}}}$ but confluent everywhere and still $\alpha\equiv_{{\mathfrak{i}}}\beta$ does not imply that the normal forms with respect to D-reduction are the same. This phenomenon is illustrated in the next example. ###### Example 3.1.1 Let us look at different ways of introducing reduction relations for the ring of integers ${\mathbb{Z}}$. For $\alpha,\beta,\gamma\in{\mathbb{Z}}$ we define: * • $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{\beta}\,$}\gamma$ if and only if $\alpha=\kappa\cdot\|\beta\|+\gamma$ where $0\leq\gamma<\|\beta\|$ and $\kappa\in{\mathbb{Z}}$ (division with remainder), * • $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm D}}_{\beta}\,$}0$ if and only if $\alpha=\kappa\cdot\beta$, i.e. $\beta$ is a proper divisor of $\alpha$ (D-reduction). Then for example we have $5\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{4}\,$}1$ but $5\mbox{$\,\,\,\,{\not\\!\\!\\!\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm D}}_{4}}\,$}$. It is easy to show that both reduction relations satisfy (A1) – (A3). Moreover, all elements in ${\mathbb{Z}}$ have unique normal forms. An element belongs to ${\sf ideal}(4)$ if and only if it is reducible to zero using $4$. For $\Longrightarrow$-reduction the normal forms are unique representatives of the quotient ${\mathbb{Z}}/{\sf ideal}(4)$. This is no longer true for $\Longrightarrow^{D}$-reduction, since e.g. $3\equiv_{{\sf ideal}(4)}7$ since $7=3+4$, but both are $\Longrightarrow^{D}$-irreducible. On the other hand, as $\Longrightarrow_{\alpha}^{D}$ is only applicable to multiples $\kappa\cdot\alpha$ and then reduces them to zero, $\Longrightarrow_{4}^{D}$ is confluent everywhere on ${\mathbb{Z}}$. $\diamond$ Since confluence of a reduction relation on the ideal is already sufficient to solve its membership problem, bases with this property called weak Gröbner bases have been studied in the literature. We proceed here by defining such weak Gröbner bases in our context. ###### Definition 3.1.2 A subset $B$ of ${\sf R}$ is called a weak Gröbner basis of the ideal ${\mathfrak{i}}={\sf ideal}(B)$ it generates, if $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$ is terminating and $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$ for all $\alpha\in{\mathfrak{i}}$. $\diamond$ Notice that in Theorem 2.3.11 this property was one way of characterizing Gröbner bases in ${\mathbb{K}}[X_{1},\ldots,X_{n}]$. We will later on see why in polynomial rings the terms weak Gröbner basis and Gröbner basis coincide. ###### Definition 3.1.3 A ring $({\sf R},\Longrightarrow)$ satisfying (A1) – (A3) is called a weak reduction ring if every finitely generated ideal in ${\sf R}$ has a finite weak Gröbner basis. $\diamond$ As stated before such a weak Gröbner basis is sufficient to decide the ideal membership problem in case the reduction relation is effective. However, if we want unique normal forms for all elements in ${\sf R}$ such that each congruence has one unique representative we need a stronger kind of ideal basis. ###### Definition 3.1.4 A subset $B$ of ${\sf R}$ is called a Gröbner basis of the ideal ${\mathfrak{i}}={\sf ideal}(B)$ it generates, if $\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}=\;\;\equiv_{{\mathfrak{i}}}$ and $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$ is complete333Notice that in the literature definitions of Gröbner bases normally only require that $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$ is “confluent”. This is due to the fact that in these cases $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$ is terminating. In our context, however for arbitrary sets $B\subseteq{\sf R}$ we have seen that $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$ need not be Noetherian. Hence we have to incorporate this additional requirement into our definition, which is done by demanding completeness. Hence here we have a point where the weaker form (A1) demands more care in defining the term “Gröbner basis”. In rings where the reduction relation using an arbitrary set of elements is always Noetherian, the weaker demand for (local) confluence is of course sufficient.. $\diamond$ Of course Gröbner bases are also weak Gröbner bases. This can be shown by induction on $k$, where for $\alpha\in{\sf ideal}(B)$ we have $\alpha\mbox{$\,\stackrel{{\scriptstyle k}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}0$. In case $k=1$ we immediately get that $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$ must hold as $0$ is irreducible. In case $k>1$ we find $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}\beta\mbox{$\,\stackrel{{\scriptstyle k-1}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}0$ and by our induction hypothesis $\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$ must hold. Now either $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\beta$ and we are done or $\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\alpha$. In the latter case the completeness of our reduction relation combined with the irreducibility of zero then must yield $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$ and we are done. The converse is not true. To see this let us review the definition of $\Longrightarrow^{D}$-reduction for ${\mathbb{Z}}$ as presented in Example 3.1.1. Then the set $\\{2\\}$ is a weak Gröbner basis of the ideal $2\cdot{\mathbb{Z}}=\\{2\cdot\alpha\mid\alpha\in{\mathbb{Z}}\\}$ as for every $\alpha\in(2\cdot{\mathbb{Z}})\backslash\\{0\\}$ we have $\alpha\Longrightarrow_{\\{2\\}}^{D}0$. On the other hand elements in ${\mathbb{Z}}\backslash(2\cdot{\mathbb{Z}})$ are irreducible and hence $3$ and $5$ are in normal form with respect to $\Longrightarrow_{\\{2\\}}^{D}$. Therefore, $3\mbox{$\,\,\,\,{\not\\!\\!\\!\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}^{{\rm D}}_{\\{2\\}}}\,$}5$ although $5\equiv_{2\cdot{\mathbb{Z}}}3$ as $5=3+1\cdot 2$. However, for many rings as e.g. polynomial rings over fields, weak Gröbner bases are also Gröbner bases. This is due to the fact that many rings with reduction relations studied in the literature fulfill a certain property for the reduction relation called the Translation Lemma (compare Lemma 2.3.9 (2)). Rephrased in our context the Translation Lemma states that for a set $F\subseteq{\sf R}$ and for all $\alpha,\beta\in{\sf R}$, $\alpha-\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{F}\,$}0$ implies the existence of $\gamma\in{\sf R}$ such that $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{F}\,$}\gamma$ and $\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{F}\,$}\gamma$. As mentioned before, the validity of this lemma for a reduction relation in a ring has consequences on the relation between weak Gröbner bases and Gröbner bases. ###### Theorem 3.1.5 Let ${\sf R}$ be a ring with a reduction relation $\Longrightarrow$ fulfilling (A1) – (A3). If additionally the Translation Lemma holds for the reduction relation $\Longrightarrow$ in ${\sf R}$, then weak Gröbner bases are also Gröbner bases. Proof : Let ${\sf R}$ be a ring where the Translation Lemma holds for the reduction relation $\Longrightarrow$. Further let $B$ be a weak Gröbner basis of the ideal ${\mathfrak{i}}={\sf ideal}(B)$. In order to prove that $B$ is in fact a Gröbner basis we have to show two properties: 1. 1. $\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}=\;\;\equiv_{{\mathfrak{i}}}$: The inclusion $\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}\subseteq\;\;\equiv_{{\mathfrak{i}}}$ follows by (A1) and (A2). To see the converse let $\alpha\equiv_{{\mathfrak{i}}}\beta$. Then $\alpha-\beta\in{\mathfrak{i}}$, and $\alpha-\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$, as $B$ is a weak Gröbner basis. But then the Translation Lemma yields that $\alpha$ and $\beta$ are joinable by $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$ and hence $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}\beta$. 2. 2. $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$ is complete: Since $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$ is terminating it suffices to show local confluence. Let $\alpha,\beta_{1},\beta_{2}\in{\sf R}$ such that $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\beta_{1}$ and $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\beta_{2}$. Then again $\beta_{1}-\beta_{2}\in{\mathfrak{i}}$, and $\beta_{1}-\beta_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$, since $B$ is a weak Gröbner basis. As before the Translation Lemma yields that $\beta_{1}$ and $\beta_{2}$ are joinable by $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$ and we are done. q.e.d. On the other hand, looking at proofs of variations of the Translation Lemma in the literature we find that in order to show this property for a ring with a reduction relation we need more information on the reduction step as is provided by the very general form of Axiom (A2). Hence in this general setting weak Gröbner bases and Gröbner bases have to be distinguished. Rings where finitely generated ideals have finite Gröbner bases are of particular interest. ###### Definition 3.1.6 A ring $({\sf R},\Longrightarrow)$ satisfying (A1) – (A3) is called a reduction ring if every finitely generated ideal in ${\sf R}$ has a finite Gröbner basis. $\diamond$ The connection between weak reduction rings and reduction rings follows from Theorem 3.1.5. ###### Corollary 3.1.7 Let $({\sf R},\Longrightarrow)$ be a weak reduction ring. If additionally the Translation Lemma holds, then $({\sf R},\Longrightarrow)$ is a reduction ring. To simplify notations sometimes we will identify $({\sf R},\Longrightarrow)$ with ${\sf R}$ in case $\Longrightarrow$ is known or irrelevant. The notion of one-sided weak reduction rings and one-sided reduction rings is straightforward444An example for a one-sided weak reduction ring which is not a one-sided reduction ring can be given using the two different reduction relations $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}\,$ and $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm D}}\,$ for the integers provided in Example 3.1.1. Then the free monoid ring ${\mathbb{Z}}[\\{a,b\\}]$ with prefix reduction induced by $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}\,$ is a one-sided reduction ring while for prefix reduction induced by $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm D}}\,$ we get a one-sided weak reduction ring.. Effective or computable weak reduction rings and effective or computable reduction rings can be defined similar to Buchberger’s commutative reduction rings (see [Buc83, Sti87]), in our case by demanding that the ring operations are computable, the reduction relation is effective, and, additionally, Gröbner bases can be computed. Procedures which compute Gröbner bases are normally completion procedures based on effective tests for local confluence to decide whether a finite set is a Gröbner basis and to enrich that set if not. But of course other procedures are also possible, e.g. when using division with remainders as reduction relation in ${\mathbb{Z}}$ the Euclidean algorithm can be used for computing Gröbner bases of ideals. Notice that Definition 3.1.6 does not imply that Noetherian rings satisfying the Axioms (A1), (A2) and (A3) are indeed reduction rings. This is due to the fact that while of course all ideals then have finite bases, the property of being a Gröbner basis strongly depends on the reduction ring which is of course itself strongly dependent on the reduction relation chosen for the ring. Hence the existence of finite ideal bases does not imply the existence of finite Gröbner bases as the following example shows: Given an arbitrary Noetherian ring ${\sf R}$ we can associate a (very simple) reduction relation to elements of ${\sf R}$ by defining for any $\alpha\in{\sf R}\backslash\\{0\\}$, $\alpha\Longrightarrow_{\beta}$ if and only if $\alpha=\beta$. Additionally we define $\alpha\Longrightarrow_{\alpha}0$. Then the Axioms (A1), (A2) and (A3) are fulfilled but although every ideal in the Noetherian ring ${\sf R}$ has a finite basis (in the sense of a generating set), infinite ideals will not have finite Gröbner bases, as for any ideal ${\mathfrak{i}}\subseteq{\sf R}$ in this setting the set ${\mathfrak{i}}\backslash\\{0\\}$ is the only possible Gröbner basis. Another interesting question concerns which changes to ideal bases preserve the property of being a Gröbner basis. Extensions of (weak) Gröbner bases by ideal elements are not critical555Extensions of (weak) Gröbner bases by elements not belonging to the ideal make no sense in our context as then the reduction relation no longer is a proper means for describing the original ideal congruence.. ###### Remark 3.1.8 If $B$ is a finite (weak) Gröbner basis of ${\mathfrak{i}}$ and $\alpha\in{\mathfrak{i}}$, then $B^{\prime}=B\cup\\{\alpha\\}$ is again a (weak) Gröbner basis of ${\mathfrak{i}}$: First of all we find $\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}\subseteq\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}\subseteq\;\;\equiv_{{\mathfrak{i}}}\;\;=\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}$. Moreover, since $B^{\prime}$ is again a finite set, $\Longrightarrow_{B^{\prime}}$ is terminating. Finally $\Longrightarrow_{B^{\prime}}$ inherits its confluence from $\Longrightarrow_{B}$ since $\beta\Longrightarrow_{\alpha}\gamma$ implies $\beta\equiv_{{\mathfrak{i}}}\gamma$, and hence $\beta$ and $\gamma$ have the same normal form with respect to $\Longrightarrow_{B}$. $\diamond$ Hence, if $B$ is a finite Gröbner basis of an ideal ${\mathfrak{i}}$ and $\beta\in B$ is reducible by $B\backslash\\{\beta\\}$ to $\alpha$, then $B\cup\\{\alpha\\}$ is again a Gröbner basis of ${\mathfrak{i}}$. The same is true for weak Gröbner bases. Removing elements from a set is critical as we might decrease the set of elements which are reducible with respect to the set. Hence if the set is a Gröbner basis, after removing elements the ideal elements might no longer reduce to zero using the remaining set. Reviewing the example presented in Section 1.3 we find that while the set $\\{X_{,}^{2}+X_{2},X_{1}^{2}+X_{3},X_{2}-X_{3}\\}$ is a Gröbner basis in ${\mathbb{Q}}[X_{1},X_{2},X_{3}]$ the subset $\\{X_{,}^{2}+X_{2},X_{1}^{2}+X_{3}\\}$, although it generates the same ideal, is none. In order to remove $\beta$ from a Gröbner basis $B$ without losing the Gröbner basis property it is important for the reduction relation $\Longrightarrow$ to satisfy an additional axiom: 1. (A4) $\alpha\Longrightarrow_{\beta}$ and $\beta\Longrightarrow_{\gamma}\delta$ imply $\alpha\Longrightarrow_{\gamma}$ or $\alpha\Longrightarrow_{\delta}$. It is not easy to give a simple example for a ring with a reduction relation fulfilling (A1) – (A3) but not (A4) as the reduction rings we have introduced so far all satisfy (A4)666An example using a right reduction relation in a monoid ring can be found in Example 3.6 in [MR98d]: Let $\Sigma=\\{a,b,c\\}$ and $T=\\{a^{2}\longrightarrow 1,b^{2}\longrightarrow 1,c^{2}\longrightarrow 1\\}$ be a monoid presentation of ${\cal M}$ with a length-lexicographical ordering induced by $a\succ b\succ c$. For $p,f\in{\mathbb{K}}[{\cal M}]$ a (right) reduction relation is defined by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm s}}_{f}\,$}q$ at a monomial $\alpha\cdot t$, if (a) ${\sf HT}(f\ast w)=t$ for some $w\in{\cal M}$, and (b) $q=p-\alpha\cdot{\sf HC}(f\ast w)^{-1}\cdot f\ast w$. Looking at $p=ba+b,q=bc+1$ and $r=ac+b\in{\mathbb{Q}}[{\cal G}]$ we get $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm s}}_{q}\,$}p-q\ast ca=-ca+b$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm s}}_{r}\,$}q-r\ast c=-a+1=q_{1}$, but $p\mbox{$\,\,\,\,{\not\\!\\!\\!\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm s}}_{\\{r,q_{1}\\}}}\,$}$. Trying to reduce $ba$ by $r$ or $q_{1}$ we get $r\ast a=\underline{aca}+ba,r\ast caba=ba+\underline{bcaba}$ and $q_{1}\ast aba=-ba+\underline{aba},q_{1}\ast ba=-\underline{aba}+ba$ all violating condition (a). Trying to reduce $b$ we get the same problem as $r\ast cab=b+\underline{bcab},q_{1}\ast ab=-b+\underline{a}$ and $q_{1}\ast b=-\underline{ab}+b$.. ###### Lemma 3.1.9 Let $({\sf R},\Longrightarrow)$ be a reduction ring satisfying (A4). Further let $B\subseteq{\sf R}$ be a (finite) Gröbner basis of a finitely generated ideal in ${\sf R}$ and $B^{\prime}\subseteq B$ such that for all $\beta\in B$, $\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}0$ holds. Then $B^{\prime}$ is a Gröbner basis of ${\sf ideal}^{{\sf R}}(B)$. In particular, for all $\alpha\in{\sf R}$, $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$ implies $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}0$. Proof : In this proof let $\alpha\\!\\!\Downarrow_{B}$ denote a normal form of $\alpha$ with respect to $\Longrightarrow_{B}$ and let ${\rm IRR}\/(\Longrightarrow_{B})$ denote the $\Longrightarrow_{B}$-irreducible elements in ${\sf R}$. Notice that by the Axioms (A1) and (A4) and our assumptions on $B^{\prime}$, all elements reducible by $B$ are also reducible by $B^{\prime}$: We show a more general claim by induction on $n$: If $\alpha,\beta\in{\sf R}$ such that $\alpha\Longrightarrow_{\beta}$ and $\beta\mbox{$\,\stackrel{{\scriptstyle n}}{{\Longrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}0$, then $\alpha\Longrightarrow_{B^{\prime}}$. The base case $n=1$ is a direct consequence of (A4), as $\alpha\Longrightarrow_{\beta}$ and $\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{\beta^{\prime}\in B^{\prime}}\,$}0$ immediately imply $\alpha\Longrightarrow_{\beta^{\prime}\in B^{\prime}}$. In the induction step we find $\beta\Longrightarrow_{\beta^{\prime}\in B^{\prime}}\delta\mbox{$\,\stackrel{{\scriptstyle n-1}}{{\Longrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}0$ and either $\alpha\Longrightarrow_{\beta^{\prime}\in B^{\prime}}$ or $\alpha\Longrightarrow_{\delta}$ and our induction hypothesis yields $\alpha\Longrightarrow_{B^{\prime}}$. Hence we can conclude ${\rm IRR}\/(\Longrightarrow_{B^{\prime}})\subseteq{\rm IRR}\/(\Longrightarrow_{B})$. We want to show that $B^{\prime}$ is a Gröbner basis of ${\sf ideal}^{{\sf R}}(B)$: Assuming $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\alpha\\!\\!\Downarrow_{B}$ but $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}\alpha\\!\\!\Downarrow_{B^{\prime}}\neq\alpha\\!\\!\Downarrow_{B}$, we find $\alpha\\!\\!\Downarrow_{B^{\prime}}\in{\sf ideal}^{{\sf R}}(B)$ and $\alpha\\!\\!\Downarrow_{B^{\prime}}\in{\rm IRR}\/(\Longrightarrow_{B^{\prime}})\subseteq{\rm IRR}\/(\Longrightarrow_{B})$, contradicting the confluence of $\Longrightarrow_{B}$. Hence, $\alpha\\!\\!\Downarrow_{B^{\prime}}=\alpha\\!\\!\Downarrow_{B}$, implying that $\Longrightarrow_{B^{\prime}}$ is also confluent, as $\alpha\\!\\!\Downarrow_{B}$ is unique. Now it remains to show that $\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}\subseteq\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}$ holds. This follows immediately, as for $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}\beta$ we have $\alpha\\!\\!\Downarrow_{B^{\prime}}=\alpha\\!\\!\Downarrow_{B}=\beta\\!\\!\Downarrow_{B}=\beta\\!\\!\Downarrow_{B^{\prime}}$ which implies $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}\beta$. q.e.d. This result carries over for weak Gröbner bases. ###### Corollary 3.1.10 Let $({\sf R},\Longrightarrow)$ be a weak reduction ring satisfying (A4). Further let $B\subseteq{\sf R}$ be a (finite) weak Gröbner basis of a finitely generated ideal in ${\sf R}$ and $B^{\prime}\subseteq B$ such that for all $\beta\in B$, $\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}0$ holds. Then $B^{\prime}$ is a weak Gröbner basis of ${\sf ideal}^{{\sf R}}(B)$. In particular, for all $\alpha\in{\sf R}$, $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$ implies $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}0$. Proof : As in the proof of Lemma 3.1.9 we can conclude ${\rm IRR}\/(\Longrightarrow_{B^{\prime}})\subseteq{\rm IRR}\/(\Longrightarrow_{B})$. Hence assuming that $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$ while $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}\alpha\\!\\!\Downarrow_{B^{\prime}}\neq 0$ would imply $\alpha\\!\\!\Downarrow_{B^{\prime}}\in{\rm IRR}\/(\Longrightarrow_{B})$. As $B^{\prime}\subseteq B$ this would give us a contradiction since then $\alpha\in{\sf ideal}^{{\sf R}}(B)$ would have two different normal forms at least one of them not equal to zero with respect to $B$ contradicting the fact that $B$ is supposed to be a weak Gröbner basis. q.e.d. Remark 3.1.8 and Lemma 3.1.9 are closely related to interreduction and reduced (weak) Gröbner bases. We call a (weak) Gröbner basis $B\subseteq{\sf R}$ reduced if no element $\beta\in B$ is reducible by $\Longrightarrow_{B\backslash\\{\beta\\}}$. The results of this section carry over to rings with appropriate one-sided reduction relations. In the remaining sections of this chapter we study the question which ring constructions preserve the property of being a (weak) reduction ring. ### 3.2 Quotients of Reduction Rings Let ${\sf R}$ be a ring with a reduction relation $\Longrightarrow$ fulfilling (A1) – (A3) and ${\mathfrak{i}}$ a finitely generated ideal in ${\sf R}$ with a finite Gröbner basis $B$. Then every element $\alpha\in{\sf R}$ has a unique normal form $\alpha\\!\\!\Downarrow_{B}$ with respect to $\Longrightarrow_{B}$. We choose the set of $\Longrightarrow_{B}$-irreducible elements of ${\sf R}$ as representatives for the elements in the quotient ${\sf R}/{\mathfrak{i}}$. Addition is defined by $\alpha+\beta:=(\alpha+\beta)\\!\\!\Downarrow_{B}$ and multiplication by $\alpha\cdot\beta:=(\alpha\cdot\beta)\\!\\!\Downarrow_{B}$. Then a natural reduction relation can be defined on the quotient ${\sf R}/{\mathfrak{i}}$ as follows: ###### Definition 3.2.1 Let $\alpha,\beta,\gamma\in{\sf R}/{\mathfrak{i}}$. We say $\beta$ reduces $\alpha$ to $\gamma$ in one step, denoted by $\alpha\longrightarrow_{\beta}\gamma$, if there exists $\gamma^{\prime}\in{\sf R}$ such that $\alpha\Longrightarrow_{\beta}\gamma^{\prime}$ and $(\gamma^{\prime})\\!\\!\Downarrow_{B}=\gamma$. $\diamond$ First we ensure that the Axioms (A1) – (A3) hold for the reduction relation in ${\sf R}/{\mathfrak{i}}$ based on Definition 3.2.1: $\longrightarrow_{S}\;=\bigcup_{s\in S}\longrightarrow_{s}$ is terminating for all finite $S\subseteq{\sf R}/{\mathfrak{i}}$ since otherwise $\Longrightarrow_{B\cup S}$ would not be terminating in ${\sf R}$ although $B\cup S$ is finite. Hence (A1) is satisfied. If $\alpha\longrightarrow_{\beta}\gamma$ for some $\alpha,\beta,\gamma\in{\sf R}/{\mathfrak{i}}$ we know $\alpha\Longrightarrow_{\beta}\gamma^{\prime}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\gamma$, i.e., $\alpha-\gamma\in{\sf ideal}^{{\sf R}}(\\{\beta\\}\cup B)$, and hence $\alpha-\gamma\in{\sf ideal}^{{\sf R}/{\mathfrak{i}}}(\beta)$. Therefore, (A2) is also fulfilled. Finally Axiom (A3) holds since $\alpha\Longrightarrow_{\alpha}0$ for all $\alpha\in{\sf R}\backslash\\{0\\}$ implies $\alpha\longrightarrow_{\alpha}0$. Moreover, in case (A4) holds in ${\sf R}$ this is also true for ${\sf R}/{\mathfrak{i}}$: For $\alpha,\beta,\gamma,\delta\in{\sf R}/{\mathfrak{i}}$ we have that $\alpha\longrightarrow_{\beta}$ and $\beta\longrightarrow_{\gamma}\delta$ imply $\alpha\Longrightarrow_{\beta}$ and $\beta\Longrightarrow_{\gamma}\delta^{\prime}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\delta$ and since $\alpha$ is $\Longrightarrow_{B}$-irreducible777Remember that in the proof of Lemma 3.1.9 we have shown that $\alpha\Longrightarrow_{\beta}$ and $\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B^{\prime}}\,$}0$ imply $\alpha\Longrightarrow_{B^{\prime}}$. This carries over to our situation in the form that $\alpha\Longrightarrow_{\beta}$ and $\beta\Longrightarrow_{\gamma}\delta^{\prime}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\delta$ implies $\alpha\Longrightarrow_{\\{\gamma,\delta^{\prime},\delta\\}\cup B}$ and using induction to $\alpha\Longrightarrow_{\\{\gamma,\delta\\}\cup B}$. this implies $\alpha\Longrightarrow_{\\{\gamma,\delta\\}}$ and hence $\alpha\longrightarrow_{\\{\gamma,\delta\\}}$. ###### Theorem 3.2.2 If $({\sf R},\Longrightarrow)$ is a reduction ring with (A4), then for every finitely generated ideal ${\mathfrak{i}}$ the quotient $({\sf R}/{\mathfrak{i}},\longrightarrow)$ again is a reduction ring with (A4). Proof : Since reduction in ${\sf R}/{\mathfrak{i}}$ as defined above inherits (A1) – (A4) from ${\sf R}$, it remains to show that every finitely generated ideal ${\mathfrak{j}}\subseteq{\sf R}/{\mathfrak{i}}$ has a finite Gröbner basis. Let ${\mathfrak{j}}_{{\sf R}}=\\{\alpha\in{\sf R}\mid\alpha\\!\\!\Downarrow_{B}\in{\mathfrak{j}}\\}$ be an ideal888${\mathfrak{j}}_{{\sf R}}$ is an ideal in ${\sf R}$ since 1. $0\in{\mathfrak{j}}_{{\sf R}}$ as $0\in{\mathfrak{j}}$. 2. $\alpha,\beta\in{\mathfrak{j}}_{{\sf R}}$ implies $\alpha\\!\\!\Downarrow_{B},\beta\\!\\!\Downarrow_{B}\in{\mathfrak{j}}$, hence $\alpha\\!\\!\Downarrow_{B}+\beta\\!\\!\Downarrow_{B}=(\alpha+\beta)\\!\\!\Downarrow_{B}\in{\mathfrak{j}}$ and $\alpha+\beta\in{\mathfrak{j}}_{{\sf R}}$. 3. $\alpha\in{\mathfrak{j}}_{{\sf R}}$ and $\gamma\in{\sf R}$ implies $\alpha\\!\\!\Downarrow_{B}\in{\mathfrak{j}}$ and $\gamma\cdot\alpha\\!\\!\Downarrow_{B}=(\gamma\cdot\alpha)\\!\\!\Downarrow_{B}\in{\mathfrak{j}}$, $\alpha\\!\\!\Downarrow_{B}\cdot\gamma=(\alpha\cdot\gamma)\\!\\!\Downarrow_{B}\in{\mathfrak{j}}$, hence $\gamma\cdot\alpha,\alpha\cdot\gamma\in{\mathfrak{j}}_{{\sf R}}$. in ${\sf R}$ corresponding to ${\mathfrak{j}}$. Then ${\mathfrak{j}}_{{\sf R}}$ is finitely generated as an ideal in ${\sf R}$ by its finite basis in ${\sf R}/{\mathfrak{i}}$ viewed as elements of ${\sf R}$ and the finite basis of ${\mathfrak{i}}$. Hence ${\mathfrak{j}}_{{\sf R}}$ has a finite Gröbner basis in ${\sf R}$, say $G_{{\sf R}}$. Then $G=\\{\alpha\\!\\!\Downarrow_{B}\mid\alpha\in G_{{\sf R}}\\}\backslash\\{0\\}$ is a finite Gröbner basis of ${\mathfrak{j}}$: If $\alpha\in{\mathfrak{j}}$ we have $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}0$ and ${\sf ideal}^{{\sf R}/{\mathfrak{i}}}(G)={\mathfrak{j}}$, as every element which is reducible with an element $\beta\in G_{{\sf R}}$ is also reducible with an element of $G\cup B$ because (A4) holds. Since $G\cup B$ is also a Gröbner basis of ${\mathfrak{j}}_{{\sf R}}$ and $\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}\subseteq\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{G\cup B}\,$}$, when restricted to elements in ${\sf R}/{\mathfrak{i}}$ we have ${\rm IRR}\/(\longrightarrow_{G})={\rm IRR}\/(\Longrightarrow_{G\cup B})$ and $\longrightarrow_{G}$ is confluent. Furthermore, since $\equiv_{\mathfrak{j}}\;=\;\equiv_{{\mathfrak{j}}_{{\sf R}}}$ when restricted to ${\sf R}/{\mathfrak{i}}$ we get $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{G}\,$}=\;\equiv_{\mathfrak{j}}$ on ${\sf R}/{\mathfrak{i}}$ implying that ${\sf R}/{\mathfrak{i}}$ is a reduction ring. q.e.d. In Example 3.1.1 we have seen how to associate the integers with a reduction relation $\Longrightarrow$ and in fact $({\mathbb{Z}},\Longrightarrow)$ is a reduction ring. Theorem 3.2.2 then states that for every $m\in{\mathbb{Z}}$ the quotient ${\mathbb{Z}}/{\sf ideal}(m)$ again is a reduction ring with respect to the reduction relation defined analogue to Definition 3.2.1. In particular reduction rings with zero divisors can be constructed in this way. Of course if we only assume that ${\sf R}$ is a weak reduction ring we no longer have unique normal forms for the elements in the quotient. Still comparing elements is possible as $\alpha=\beta$ in ${\sf R}/{\mathfrak{i}}$ if and only if $\alpha-\beta\in{\mathfrak{i}}$ if and only if $\alpha-\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$ for a weak Gröbner basis $B$ of ${\mathfrak{i}}$. Hence the elements in the quotient are no longer given by unique elements but by the respective sets of all representatives with respect to the weak Gröbner basis chosen for the ideal999Such an element $\alpha$ in the quotient can be represented by any element which is equivalent to it. When doing computations then of course to decide whether $\alpha=\beta$ in ${\sf R}/{\mathfrak{i}}$ one has to check if $\alpha-\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}0$ for a weak Gröbner basis $B$ of ${\mathfrak{i}}$.. ###### Corollary 3.2.3 If $({\sf R},\Longrightarrow)$ is a weak reduction ring with (A4), then for every finitely generated ideal ${\mathfrak{i}}$ the quotient $({\sf R}/{\mathfrak{i}},\longrightarrow)$ again is a weak reduction ring with (A4). Proof : It remains to show that every finitely generated ideal ${\mathfrak{j}}\subseteq{\sf R}/{\mathfrak{i}}$ has a finite weak Gröbner basis. Let $B$ be a finite weak Gröbner basis of ${\mathfrak{i}}$ in ${\sf R}$ and $B_{\mathfrak{j}}$ a finite generating set for the ideal ${\mathfrak{j}}$ in ${\sf R}/{\mathfrak{i}}$. Let ${\mathfrak{j}}_{{\sf R}}=\bigcup_{\alpha\in{\mathfrak{j}}}\\{\beta\in{\sf R}\mid\beta\Longleftrightarrow^{}_{B}\alpha\\}$, be an ideal in ${\sf R}$ corresponding to ${\mathfrak{j}}$. Then ${\mathfrak{j}}_{{\sf R}}$ is finitely generated by the set $B\cup\tilde{B}_{\mathfrak{j}}$ where for each element $\alpha\in B_{\mathfrak{j}}$ the set $\tilde{B}_{\mathfrak{j}}$ contains some $\tilde{\alpha}\in\\{\beta\in{\sf R}\mid\beta\Longleftrightarrow^{}_{B}\alpha\\}$. Moreover, ${\mathfrak{j}}_{{\sf R}}$ has a finite weak Gröbner basis, say $G_{{\sf R}}$. Then the set $G=\\{\alpha\\!\\!\Downarrow_{B}\mid\alpha\in G_{{\sf R}}\\}\backslash\\{0\\}$ containing for each $\alpha\in G_{{\sf R}}$ one not necessarily unique normal form $\alpha\\!\\!\Downarrow_{B}$ is a finite weak Gröbner basis of ${\mathfrak{j}}$: If $\alpha\in{\mathfrak{j}}$ we have $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}0$ and ${\sf ideal}^{{\sf R}/{\mathfrak{i}}}(G)={\mathfrak{j}}$, as every element in ${\mathfrak{j}}$ (i.e. in particular irreducible with respect to $B$) which is reducible with an element $\beta\in G_{{\sf R}}$ is also reducible with an element of $G$ because (A4) holds101010Since $\alpha\in{\mathfrak{j}}$ is irreducible by $B$, we have $\alpha\Longrightarrow_{\beta}\delta^{\prime}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{G_{{\sf R}}}\,$}\delta$ and $\beta\not\in B$. Then looking at the situation $\alpha\Longrightarrow_{\beta}$ and $\beta\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{G_{{\sf R}}}\,$}\beta\\!\\!\Downarrow_{B}$, (A4) yields $\alpha\Longrightarrow_{\beta\\!\\!\Downarrow_{B}}$.. q.e.d. Now if $({\sf R},\Longrightarrow)$ is an effective reduction ring, then $B$ can be computed and addition and multiplication in ${\sf R}/{\mathfrak{i}}$ as well as the reduction relation based on Definition 3.2.1 are computable operations. Moreover, Theorem 3.2.2 can be generalized: ###### Corollary 3.2.4 If $({\sf R},\Longrightarrow)$ is an effective reduction ring with (A4), then for every finitely generated ideal ${\mathfrak{i}}$ the quotient $({\sf R}/{\mathfrak{i}},\longrightarrow)$ again is an effective reduction ring with (A4). Proof : Given ${\sf R}$, $B$ and a finite generating set $F$ for an ideal ${\mathfrak{j}}$ in ${\sf R}/{\mathfrak{i}}$ we can compute a finite Gröbner basis for ${\mathfrak{j}}$ using the method for computing Gröbner bases in ${\sf R}$: Compute a Gröbner basis $G_{{\sf R}}$ of the ideal generated by $B\cup F$ in ${\sf R}$. Then the set $G=\\{\alpha\\!\\!\Downarrow_{B}\mid\alpha\in G_{{\sf R}}\\}$, where $\alpha\\!\\!\Downarrow_{B}$ is the normal form of $g$ with respect to $\Longrightarrow_{B}$ in ${\sf R}$ and hence an element of ${\sf R}/{\mathfrak{i}}$, is a Gröbner basis of ${\mathfrak{j}}$ in ${\sf R}/{\mathfrak{i}}$. q.e.d. The same is true for effective weak reduction rings. Finally the results carry over to the case of one-sided reduction rings with (A4) provided that the two-sided ideal has a finite right respectively left Gröbner basis. ### 3.3 Sums of Reduction Rings Let ${\sf R}_{1},{\sf R}_{2}$ be rings with reduction relations $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}\,$ respectively $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}\,$ fulfilling (A1) – (A3). Then ${\sf R}={\sf R}_{1}\times{\sf R}_{2}=\\{(\alpha_{1},\alpha_{2})\mid\alpha_{1}\in{\sf R}_{1},\alpha_{2}\in{\sf R}_{2}\\}$ is called the direct sum of ${\sf R}_{1}$ and ${\sf R}_{2}$. Addition and multiplication are defined component wise, the unit is $(1_{1},1_{2})$ where $1_{i}$ is the respective unit in ${\sf R}_{i}$. A natural reduction relation can be defined on ${\sf R}$ as follows: ###### Definition 3.3.1 Let $\alpha=(\alpha_{1},\alpha_{2})$, $\beta=(\beta_{1},\beta_{2})$, $\gamma=(\gamma_{1},\gamma_{2})\in{\sf R}$. We say that $\beta$ reduces $\alpha$ to $\gamma$ in one step, denoted by $\alpha\longrightarrow_{\beta}\gamma$, if either $(\alpha_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}_{\beta_{1}}\,$}\gamma_{1}$ and $\alpha_{2}=\gamma_{2})$ or $(\alpha_{1}=\gamma_{1}$ and $\alpha_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}_{\beta_{2}}\,$}\gamma_{2})$ or $(\alpha_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}_{\beta_{1}}\,$}\gamma_{1}$ and $\alpha_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}_{\beta_{2}}\,$}\gamma_{2})$. $\diamond$ Again we have to prove that the Axioms (A1) – (A3) hold for the reduction relation in ${\sf R}$: $\longrightarrow_{B}=\bigcup_{\beta\in B}\longrightarrow_{\beta}$ is terminating for finite sets $B\subseteq{\sf R}$ since this property is inherited from the termination of the respective reduction relations in ${\sf R}_{i}$. Hence (A1) holds. (A2) is satisfied since $\alpha\longrightarrow_{\beta}\gamma$ implies $\alpha-\gamma\in{\sf ideal}^{{\sf R}}(\beta)$. (A3) is true as $\alpha\longrightarrow_{\alpha}(0_{1},0_{2})$ holds for all $\alpha\in{\sf R}\backslash\\{(0_{1},0_{2})\\}$. Moreover, it is easy to see that if condition (A4) holds for $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}\,$ and $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}\,$ then this is inherited by $\longrightarrow$. ###### Theorem 3.3.2 If $({\sf R}_{1},\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}\,$})$, $({\sf R}_{2},\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}\,$})$ are reduction rings, then $({\sf R}={\sf R}_{1}\times{\sf R}_{2},\longrightarrow)$ is again a reduction ring. Proof : Since the reduction relation in ${\sf R}$ as defined above inherits (A1) – (A3) respectively (A4) from the reduction relations in the ${\sf R}_{i}$, it remains to show that every finitely generated ideal ${\mathfrak{i}}\subseteq{\sf R}$ has a finite Gröbner basis. To see this notice that the restrictions ${\mathfrak{i}}_{1}=\\{\alpha_{1}\mid(\alpha_{1},\alpha_{2})\in{\mathfrak{i}}\mbox{ for some }\alpha_{2}\in{\sf R}_{2}\\}$ and ${\mathfrak{i}}_{2}=\\{\alpha_{2}\mid(\alpha_{1},\alpha_{2})\in{\mathfrak{i}}\mbox{ for some }\alpha_{1}\in{\sf R}_{1}\\}$ are finitely generated ideals in ${\sf R}_{1}$ respectively ${\sf R}_{2}$ and hence have finite Gröbner bases $B_{1}$ respectively $B_{2}$. We claim that $B=\\{(\beta_{1},0_{2}),(0_{1},\beta_{2})\mid\beta_{1}\in B_{1},\beta_{2}\in B_{2}\\}$ is a finite Gröbner basis of ${\mathfrak{i}}$. Notice that ${\mathfrak{i}}={\mathfrak{i}}_{1}\times{\mathfrak{i}}_{2}$. Then ${\sf ideal}(B)={\mathfrak{i}}$ and $\alpha\in{\mathfrak{i}}$ implies $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{B}\,$}(0_{1},0_{2})$ due to the fact that for $\alpha=(\alpha_{1},\alpha_{2})$ we have $\alpha_{1}\in{\mathfrak{i}}_{1}$ and $\alpha_{2}\in{\mathfrak{i}}_{2}$ implying $\alpha_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}_{B_{1}}\,$}0_{1}$ and $\alpha_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}_{B_{2}}\,$}0_{2}$. Similarly $\longrightarrow_{B}$ is confluent because $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}_{B_{1}}\,$ and $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}_{B_{2}}\,$ are confluent. Finally $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}=\;\equiv_{\mathfrak{i}}$ since $(\alpha_{1},\alpha_{2})\equiv_{\mathfrak{i}}(\beta_{1},\beta_{2})$ implies $\alpha_{1}\equiv_{{\mathfrak{i}}_{1}}\beta_{1}$ respectively $\alpha_{2}\equiv_{{\mathfrak{i}}_{2}}\beta_{2}$ and hence $\alpha_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}^{{\rm 1}}_{B_{1}}\,$}\beta_{1}$ respectively $\alpha_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}^{{\rm 2}}_{B_{2}}\,$}\beta_{2}$. q.e.d. Special regular rings as introduced by Weispfenning in [Wei87b] provide examples of such sums of reduction rings, e.g. any direct sum of fields. ###### Corollary 3.3.3 If $({\sf R}_{1},\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}\,$})$, $({\sf R}_{2},\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}\,$})$ are weak reduction rings, then $({\sf R}={\sf R}_{1}\times{\sf R}_{2},\longrightarrow)$ is again a weak reduction ring. Proof : Reviewing the proof of Theorem 3.3.2 it remains to show that every finitely generated ideal ${\mathfrak{i}}\subseteq{\sf R}$ has a finite weak Gröbner basis. Again we look at the restrictions ${\mathfrak{i}}_{1}=\\{\alpha_{1}\mid(\alpha_{1},\alpha_{2})\in{\mathfrak{i}}\mbox{ for some }\alpha_{2}\in{\sf R}_{2}\\}$ and ${\mathfrak{i}}_{2}=\\{\alpha_{2}\mid(\alpha_{1},\alpha_{2})\in{\mathfrak{i}}\mbox{ for some }\alpha_{1}\in{\sf R}_{1}\\}$ which are finitely generated ideals in ${\sf R}_{1}$ respectively ${\sf R}_{2}$ and hence have finite weak Gröbner bases $B_{1}$ respectively $B_{2}$. We claim that $B=\\{(\beta_{1},0_{2}),(0_{1},\beta_{2})\mid\beta_{1}\in B_{1},\beta_{2}\in B_{2}\\}$ is a finite weak Gröbner basis of ${\mathfrak{i}}$. As before ${\mathfrak{i}}={\mathfrak{i}}_{1}\times{\mathfrak{i}}_{2}$ and ${\sf ideal}(B)={\mathfrak{i}}$. Then $\alpha\in{\mathfrak{i}}$ implies $\alpha\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{B}\,$}(0_{1},0_{2})$ due to the fact that for $\alpha=(\alpha_{1},\alpha_{2})$ we have $\alpha_{1}\in{\mathfrak{i}}_{1}$ and $\alpha_{2}\in{\mathfrak{i}}_{2}$ implying $\alpha_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}_{B_{1}}\,$}0_{1}$ and $\alpha_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}_{B_{2}}\,$}0_{2}$ as $B_{1}$ and $B_{2}$ are respective weak Gröbner bases, and we are done. q.e.d. Now if $({\sf R}_{1},\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}\,$})$, $({\sf R}_{2},\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}\,$})$ are effective reduction rings, then addition and multiplication in ${\sf R}$ as well as the reduction relation based on Definition 3.3.1 are computable operations. Moreover, Theorem 3.3.2 can be generalized: ###### Corollary 3.3.4 If $({\sf R}_{1},\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 1}}\,$})$, $({\sf R}_{2},\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}^{{\rm 2}}\,$})$ are effective reduction rings, then $({\sf R}={\sf R}_{1}\times{\sf R}_{2},\longrightarrow)$ is again an effective reduction ring. Proof : Given a finite generating set $F=\\{(\alpha_{i},\beta_{i})\mid 1\leq i\leq k,\alpha_{i}\in{\sf R}_{1},\beta_{i}\in{\sf R}_{2}\\}$ a Gröbner basis of the ideal generated by $F$ can be computed using the respective methods for Gröbner basis computation in ${\sf R}_{1}$ and ${\sf R}_{2}$: Compute $B_{1}$ a Gröbner basis of the ideal generated by $\\{\alpha_{1},\ldots,\alpha_{k}\\}$ in ${\sf R}_{1}$ and $B_{2}$ a Gröbner basis of the ideal generated by $\\{\beta_{1},\ldots,\beta_{k}\\}$ in ${\sf R}_{2}$. Then $B=\\{(\gamma_{1},0_{2}),(0_{1},\gamma_{2})\mid\gamma_{1}\in B_{1},\gamma_{2}\in B_{2}\\}$ is a finite Gröbner basis of the ideal generated by $F$ in ${\sf R}$. q.e.d. A similar result holds for effective weak reduction rings. Due to the “simple” multiplication used when defining direct sums, Theorem 3.3.2 and Corollary 3.3.4 extend directly to one-sided reduction rings. More complicated multiplications are possible and have to be treated individually. ### 3.4 Modules over Reduction Rings Another structure which can be studied by reduction techniques are modules and their submodules. Given a ring ${\sf R}$ with unit $1$ and a natural number $k$, let ${\sf R}^{k}=\\{{\bf a}=(\alpha_{1},\ldots,\alpha_{k})\mid\alpha_{i}\in{\sf R}\\}$ be the set of all vectors of length $k$ with coordinates in ${\sf R}$. Obviously ${\sf R}^{k}$ is an additive commutative group with respect to ordinary vector addition and we denote the zero by ${\bf 0}$. Moreover, ${\sf R}^{k}$ is an ${\sf R}$-module for scalar multiplication defined as $\alpha\ast(\alpha_{1},\ldots,\alpha_{k})=(\alpha\cdot\alpha_{1},\ldots,\alpha\cdot\alpha_{k})$ and $(\alpha_{1},\ldots,\alpha_{k})\ast\alpha=(\alpha_{1}\cdot\alpha,\ldots,\alpha_{k}\cdot\alpha)$. Additionally ${\sf R}^{k}$ is called free as it has a basis111111Here the term basis is used in the meaning of being a linearly independent set of generating vectors.. One such basis is the set of unit vectors ${\bf e}_{1}=(1,0,\ldots,0),{\bf e}_{2}=(0,1,0,\ldots,0),\ldots,{\bf e}_{k}=(0,\ldots,0,1)$. Using this basis the elements of ${\sf R}^{k}$ can be written uniquely as ${\bf a}=\sum_{i=1}^{k}\alpha_{i}\ast{\bf e}_{i}$ where ${\bf a}=(\alpha_{1},\ldots,\alpha_{k})$. ###### Definition 3.4.1 A subset of ${\sf R}^{k}$ which is again an ${\sf R}$-module is called a submodule of ${\sf R}^{k}$. $\diamond$ For example any ideal of ${\sf R}$ is an ${\sf R}$-module and even a submodule of the ${\sf R}$-module ${\sf R}^{1}$. Provided a set of vectors $S=\\{{\bf a}_{1},\ldots,{\bf a}_{n}\\}$ the set $\\{\sum_{i=1}^{n}\sum_{j=1}^{m_{i}}\beta_{ij}\ast{\bf a}_{i}\ast{\beta_{ij}}^{\prime}\mid\beta_{ij},{\beta_{ij}}^{\prime}\in{\sf R}\\}$ is a submodule of ${\sf R}^{k}$. This set is denoted as $\langle S\rangle$ and $S$ is called its generating set. Now similar to the case of modules over commutative polynomial rings, being Noetherian is inherited by ${\sf R}^{k}$ from ${\sf R}$. ###### Theorem 3.4.2 Let ${\sf R}$ be a Noetherian ring. Then every submodule in ${\sf R}^{k}$ is also finitely generated. Proof : Let ${\cal S}$ be a submodule of ${\sf R}^{k}$. We show our claim by induction on $k$. For $k=1$ we find that ${\cal S}$ is in fact an ideal in ${\sf R}$ and hence by our hypothesis must be finitely generated. For $k>1$ let us look at the set ${\mathfrak{i}}=\\{\beta_{1}\mid(\beta_{1},\ldots,\beta_{k})\in{\cal S}\\}$ which is again an ideal in ${\sf R}$ and hence finitely generated by some set $\\{\gamma_{1},\ldots,\gamma_{s}\mid\gamma_{i}\in{\sf R}\\}$. Choose121212In this step we need the Axiom of Choice and hence the construction is not constructive. $H=\\{{\bf c}_{1},\ldots,{\bf c}_{s}\\}\subseteq{\cal S}$ such that the first coordinate of ${\bf c}_{i}$ is $\gamma_{i}$. Similarly the set ${\cal M}=\\{(\beta_{2},\ldots,\beta_{k})\mid(0,\beta_{2},\ldots,\beta_{k})\in{\cal S}\\}$ is a submodule in ${\sf R}^{k-1}$ and therefore finitely generated by our induction hypothesis. Let $\\{(\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$ be such a finite generating set. Then ${\bf d}_{i}=(0,\delta_{2}^{i},\ldots,\delta^{i}_{k})\in{\cal S}$, $1\leq i\leq w$ and the set $G=\\{{\bf c}_{1},\ldots,{\bf c}_{s}\\}\cup\\{{\bf d}_{i}\mid 1\leq i\leq w\\}$ is a finite generating set for ${\cal S}$. To see this assume ${\bf t}=(\tau_{1},\ldots,\tau_{k})\in{\cal S}$. Then $\tau_{1}=\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\zeta_{ij}\cdot\gamma_{i}\cdot{\zeta_{ij}}^{\prime}$ for some $\zeta_{ij},{\zeta_{ij}}^{\prime}\in{\sf R}$ and ${\bf t}^{\prime}={\bf t}-\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\zeta_{ij}\ast{\bf c}_{i}\ast{\zeta_{ij}}^{\prime}\in{\cal S}$ with first coordinate $0$. Hence ${\bf t}^{\prime}=\sum_{i=1}^{w}\sum_{j=1}^{m_{i}}\eta_{ij}\ast{\bf d}_{i}\ast{\eta_{ij}}^{\prime}$ for some $\eta_{ij},{\eta_{ij}}^{\prime}\in{\sf R}$ giving rise to ${\bf t}={\bf t}^{\prime}+\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\zeta_{ij}\ast{\bf c}_{i}\ast{\zeta_{ij}}^{\prime}=\sum_{i=1}^{w}\sum_{j=1}^{m_{i}}\eta_{ij}\ast{\bf d}_{i}\ast{\eta_{ij}}^{\prime}+\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\zeta_{ij}\ast{\bf c}_{i}\ast{\zeta_{ij}}^{\prime}.$ q.e.d. We will now study submodules of modules using reduction relations. Let $\Longrightarrow$ be a reduction relation on ${\sf R}$ fulfilling (A1) – (A3). A natural reduction relation on ${\sf R}^{k}$ can be defined using the representations as polynomials with respect to the basis of unit vectors as follows: ###### Definition 3.4.3 Let ${\bf a}=\sum_{i=1}^{k}\alpha_{i}\ast{\bf e}_{i}$, ${\bf b}=\sum_{i=1}^{k}\beta_{i}\ast{\bf e}_{i}\in{\sf R}^{k}$. We say that ${\bf b}$ reduces ${\bf a}$ to ${\bf c}$ at $\alpha_{s}\ast{\bf e}_{s}$ in one step, denoted by ${\bf a}\longrightarrow_{\bf b}{\bf c}$, if (a) $\beta_{j}=0$ for $1\leq j<s$, * (b) $\alpha_{s}\Longrightarrow_{\beta_{s}}\gamma_{s}$ with $\alpha_{s}=\gamma_{s}+\sum_{i=1}^{n}\delta_{i}\cdot\beta_{s}\cdot{\delta_{i}}^{\prime}$, $\delta_{i},{\delta_{i}}^{\prime}\in{\sf R}$, and * (c) ${\bf c}={\bf a}-\sum_{i=1}^{n}\delta_{i}\ast{\bf b}\ast{\delta_{i}}^{\prime}=(\alpha_{1},\ldots,\alpha_{s-1},\gamma_{s},\alpha_{s+1}-\sum_{i=1}^{n}\delta_{i}\cdot\beta_{s+1}\cdot{\delta_{i}}^{\prime},\ldots,\alpha_{k}-\sum_{i=1}^{n}\delta_{i}\cdot\beta_{k}\cdot{\delta_{i}}^{\prime})$. $\diamond$ The Axioms (A1) – (A3) hold for this reduction relation on ${\sf R}^{k}$: $\longrightarrow_{B}=\bigcup_{{\bf b}\in B}\longrightarrow_{\bf b}$ is terminating for finite $B\subseteq{\sf R}^{k}$ since this property is inherited from the termination of the respective reduction relation $\Longrightarrow$ in ${\sf R}$. Hence (A1) holds. (A2) is satisfied now of course in the context of submodules since ${\bf a}\longrightarrow_{\bf b}{\bf c}$ implies ${\bf a}-{\bf c}\in\langle\\{{\bf b}\\}\rangle$. (A3) is true as ${\bf a}\longrightarrow_{\bf a}{\bf 0}$ holds for all ${\bf a}\in{\sf R}^{k}\backslash\\{{\bf 0}\\}$. Moreover, it is easy to see that if condition (A4) holds for $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}\,$ then this is inherited by $\longrightarrow$ as defined in Definition 3.4.3 for ${\sf R}^{k}$. First we show how the existence of weak Gröbner bases carries over for Noetherian ${\sf R}$. ###### Definition 3.4.4 A subset $B$ of ${\sf R}^{k}$ is called a weak Gröbner basis of the submodule ${\cal S}=\langle B\rangle$, if $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{B}\,$ is terminating and ${\bf a}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{B}\,$}{\bf 0}$ for all ${\bf a}\in{\cal S}$. $\diamond$ ###### Theorem 3.4.5 Let ${\sf R}$ be a Noetherian ring with reduction relation $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}\,$ fulfilling (A1) – (A3). If in ${\sf R}$ every ideal has a finite weak Gröbner basis, then the same holds for submodules in $({\sf R}^{k},\longrightarrow)$. Proof : Let ${\cal S}$ be a submodule of ${\sf R}^{k}$. We show our claim by induction on $k$. For $k=1$ we find that ${\cal S}$ is in fact an ideal131313At this point we could also proceed with a much weaker hypothesis, namely instead of requiring ${\sf R}$ to be Noetherian assuming that ${\cal S}$ is finitely generated. Then still the fact that ${\sf R}$ is supposed to be a weak reduction ring would imply the existence of a finite weak Gröbner basis for ${\cal S}$. in ${\sf R}$ and hence by our hypothesis must have a finite weak Gröbner basis. For $k>1$ let us look at the set ${\mathfrak{i}}=\\{\beta_{1}\mid(\beta_{1},\ldots,\beta_{k})\in{\cal S}\\}$ which is again an ideal141414Here it still would be sufficient to require that ${\cal S}$ is finitely generated as the first coordinates of a finite generating set for ${\cal S}$ then would generate ${\mathfrak{i}}$ hence implying that the ideal is finitely generated as well.. Hence ${\mathfrak{i}}$ must have a finite weak Gröbner basis $\\{\gamma_{1},\ldots,\gamma_{s}\mid\gamma_{i}\in{\sf R}\\}$. Choose $H=\\{{\bf c}_{1},\ldots,{\bf c}_{s}\\}\subseteq{\cal S}$ such that the first coordinate of ${\bf c}_{i}$ is $\gamma_{i}$. Similarly the set ${\cal M}=\\{(\beta_{2},\ldots,\beta_{k})\mid(0,\beta_{2},\ldots,\beta_{k})\in{\cal S}\\}$ is a submodule151515Now we really need that ${\sf R}^{k-1}$ is Noetherian. Assuming that ${\cal S}$ is finitely generated would not help to deduce that ${\cal M}$ is finitely generated. in ${\sf R}^{k-1}$ which by our induction hypothesis must have a finite weak Gröbner basis $\\{(\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$. Then the set $G=\\{{\bf c}_{1},\ldots,{\bf c}_{s}\\}\cup\\{{\bf d}_{i}=(0,\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$ is a weak Gröbner basis for ${\cal S}$. That $G$ is a generating set for ${\cal S}$ follows as in the proof of Theorem 3.4.2. It remains to show that $G$ is in fact a weak Gröbner basis, i.e., for every ${\bf t}=(\tau_{1},\ldots,\tau_{k})\in{\cal S}$ we have ${\bf t}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}{\bf 0}$. Since $\tau_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{\\{\gamma_{1},\ldots,\gamma_{s}\\}}\,$}0$ with $\tau_{1}=\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\zeta_{ij}\cdot\gamma_{i}\cdot{\zeta_{ij}}^{\prime}$, by the definition of $G$ we get ${\bf t}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{\\{{\bf c}_{1},\ldots,{\bf c}_{s}\\}}\,$}{\bf t}-\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\zeta_{ij}\ast{\bf c}_{i}\ast{\zeta_{ij}}^{\prime}={\bf t}^{\prime}$ where ${\bf t}^{\prime}=(0,{\tau_{2}}^{\prime},\ldots,{\tau_{k}}^{\prime})\in{\cal M}$. Hence, as $({\tau_{2}}^{\prime},\ldots,{\tau_{k}}^{\prime})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{\\{(\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}}\,$}{\bf 0}$, we get ${\bf t}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}{\bf 0}$ and are done. q.e.d. Now we turn our attention to Gröbner bases of submodules in ${\sf R}^{k}$. ###### Definition 3.4.6 A subset $B$ of ${\sf R}^{k}$ is called a Gröbner basis of the submodule ${\cal S}=\langle B\rangle$, if $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}=\;\;\equiv_{{\cal S}}$ and $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{B}\,$ is complete. $\diamond$ ###### Theorem 3.4.7 Let ${\sf R}$ be a Noetherian ring with reduction relation $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}\,$ fulfilling (A1) – (A3). If in ${\sf R}$ every ideal has a finite Gröbner basis, then the same holds for submodules in $({\sf R}^{k},\longrightarrow)$. Proof : The candidate for the Gröbner basis can be built similar to the set $G$ in the proof of Theorem 3.4.5 now of course using Gröbner bases in the construction instead of weak Gröbner bases: Let ${\cal S}$ be a submodule of ${\sf R}^{k}$. We show our claim by induction on $k$. For $k=1$ we find that ${\cal S}$ is in fact an ideal in ${\sf R}$ and hence by our hypothesis must have a finite Gröbner basis. For $k>1$ let us look at the set ${\mathfrak{i}}=\\{\beta_{1}\mid(\beta_{1},\ldots,\beta_{k})\in{\cal S}\\}$ which is again an ideal in ${\sf R}$. Hence ${\mathfrak{i}}$ must have a finite Gröbner basis $\\{\gamma_{1},\ldots,\gamma_{s}\mid\gamma_{i}\in{\sf R}\\}$ by our assumption. Choose $H=\\{{\bf c}_{1},\ldots,{\bf c}_{s}\\}\subseteq{\cal S}$ such that the first coordinate of ${\bf c}_{i}$ is $\gamma_{i}$. Similarly the set ${\cal M}=\\{(\beta_{2},\ldots,\beta_{k})\mid(0,\beta_{2},\ldots,\beta_{k})\in{\cal S}\\}$ is a submodule in ${\sf R}^{k-1}$ finitely generated as ${\sf R}^{k-1}$ is Noetherian. Hence by our induction hypothesis ${\cal M}$ then must have a finite Gröbner basis $\\{(\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$. Let $G=\\{{\bf c}_{1},\ldots,{\bf c}_{s}\\}\cup\\{{\bf d}_{i}=(0,\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$. Since $G$ generates ${\cal S}$ (see the proof of Theorem 3.4.5) it remains to show that it is a Gröbner basis. By the definition of the reduction relation in ${\sf R}^{k}$ we immediately find $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{G}\,$}\subseteq\;\;\equiv_{{\cal S}}$. To see the converse let ${\bf r}=(\rho_{1},\ldots,\rho_{k})\equiv_{{\cal S}}{\bf s}=(\sigma_{1},\ldots,\sigma_{k})$. Then as $\rho_{1}\equiv_{\\{\beta_{1}\mid{\bf b}=(\beta_{1},\ldots,\beta_{k})\in{\cal S}\\}}\sigma_{1}$ by the definition of $G$ we get $\rho_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{\\{\gamma_{1},\ldots,\gamma_{s}\\}}\,$}\sigma_{1}$. But this gives us ${\bf r}\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{H}\,$}{\bf r}+\sum_{i=1}^{s}\sum_{j=1}^{m_{i}}\chi_{ij}\ast{\bf c}_{i}\ast{\chi_{ij}}^{\prime}={\bf r}^{\prime}=(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})$ and we get $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})\equiv_{\cal S}(\sigma_{1},\ldots,\sigma_{k})$. Hence $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})-(\sigma_{1},\ldots,\sigma_{k})=(0,{\rho_{2}}^{\prime}-\sigma_{2},\ldots,{\rho_{k}}^{\prime}-\sigma_{k})\in{\cal S}$, implying $({\rho_{2}}^{\prime}-\sigma_{2},\ldots,{\rho_{k}}^{\prime}-\sigma_{k})\in{\cal M}$. Now we have to be more careful since we cannot conclude that $({\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime}),(\sigma_{2},\ldots,\sigma_{k})\in{\cal M}$. But we know $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})=(\sigma_{1},\ldots,\sigma_{k})+(0,{\rho_{2}}^{\prime}-\sigma_{2},\ldots,{\rho_{k}}^{\prime}-\sigma_{k})=(\sigma_{1},\ldots,\sigma_{k})+\sum_{i=1}^{w}\sum_{j=1}^{n_{i}}\eta_{ij}\ast{\bf d}_{i}\ast{\eta_{ij}}^{\prime}$ where $(0,{\rho_{2}}^{\prime}-\sigma_{2},\ldots,{\rho_{k}}^{\prime}-\sigma_{k})=\sum_{i=1}^{w}\sum_{j=1}^{n_{i}}\eta_{ij}\ast{\bf d}_{i}\ast{\eta_{ij}}^{\prime}$ for $\eta_{ij},{\eta_{ij}}^{\prime}\in{\sf R}$, i.e., $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})\equiv_{\langle{\bf d}_{1},\ldots,{\bf d}_{w}\rangle}(\sigma_{1},\ldots,\sigma_{k})$. Hence, as $\\{(\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$ is a Gröbner basis of ${\cal M}$ both vectors $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})$ and $(\sigma_{1},\ldots,\sigma_{k})$ must have a common normal form using $\\{{\bf d}_{i}=(0,\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$ for reduction161616The elements in this set cannot influence the first coordinate which is $\sigma_{1}$ for both vectors. and we are done. The same argument applies to show local confluence. Let us assume there are ${\bf r}$, ${\bf s}_{1}$, ${\bf s}_{2}\in{\sf R}^{k}$ such that ${\bf r}\longrightarrow_{G}{\bf s}_{1}$ and ${\bf r}\longrightarrow_{G}{\bf s}_{2}$. Then by the definition of $G$, the first coordinates $\sigma^{1}_{1}$ and $\sigma_{1}^{2}$ of ${\bf s}_{1}$ respectively ${\bf s}_{2}$ are joinable by $\\{\gamma_{1},\ldots,\gamma_{s}\\}$ to some element, say $\sigma$, giving rise to the elements ${\bf r}_{1}={\bf s}_{1}+\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}\chi_{ij}\ast{\bf c}_{i}\ast{\chi_{ij}}^{\prime}$ and ${\bf r}_{2}={\bf s}_{2}+\sum_{i=1}^{s}\sum_{j=1}^{m_{i}}\psi_{ij}\ast{\bf c}_{i}\ast{\psi_{ij}}^{\prime}$ with first coordinate $\sigma$. Again we know $(\sigma,{\rho^{1}_{2}},\ldots,{\rho^{1}_{k}})=(\sigma,{\rho^{2}_{2}},\ldots,{\rho^{2}_{k}})+(0,{\rho^{1}_{2}}-{\rho^{2}_{2}},\ldots,{\rho^{1}_{k}}-{\rho^{2}_{k}})$ with $({\rho^{1}_{2}}-{\rho^{2}_{2}},\ldots,{\rho^{1}_{k}}-{\rho^{2}_{k}})\in{\cal M}$. Hence $(\sigma,{\rho^{1}_{2}},\ldots,{\rho^{1}_{k}})=(\sigma,{\rho^{2}_{2}},\ldots,{\rho^{2}_{k}})+\sum_{i=1}^{w}\sum_{j=1}^{n_{i}}\eta_{ij}\ast{\bf d}_{i}\ast{\eta_{ij}}^{\prime}$ for $\eta_{ij},{\eta_{ij}}^{\prime}\in{\sf R}$, i.e., $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})\equiv_{\langle{\bf d}_{1},\ldots,{\bf d}_{w}\rangle}(\sigma_{1},\ldots,\sigma_{k})$. As again $\\{(\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$ is a Gröbner basis of ${\cal M}$ both vectors must have a common normal with respect to reduction using $\\{{\bf d}_{i}=(0,\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$. q.e.d. Let us close this section with a remark on why the additional property of being Noetherian is so important. In the proofs of Theorem 3.4.5 and 3.4.7 in the induction step the “projection” of ${\cal S}$ on ${\sf R}^{k-1}$ plays an essential role. If this projection is defined as ${\cal M}=\\{(\beta_{2},\ldots,\beta_{k})\mid(0,\beta_{2},\ldots,\beta_{k})\in{\cal S}\\}$ we have to show that this module is again finitely generated. In assuming Noetherian for ${\sf R}$ this then follows as ${\cal M}$ is a submodule of ${\sf R}^{k-1}$ which is again Noetherian. Assuming that ${\cal S}$ is finitely generated by some set $\\{{\bf a}_{1},\ldots,{\bf a}_{n}\\}$ does not improve the situation as in general we cannot extract a finite generating set for ${\cal M}$ from this set171717 Another idea might be to look at an other projection of ${\cal S}$: ${\cal M^{\prime}}=\\{(\beta_{2},\ldots,\beta_{k})\mid\mbox{ there exists }\beta_{1}\in{\sf R}\mbox{ such that }(\beta_{1},\beta_{2},\ldots,\beta_{k})\in{\cal S}\\}$. ${\cal M^{\prime}}$ then is again a module now finitely generated by $(\alpha^{1}_{2},\ldots,\alpha^{1}_{k}),\ldots,(\alpha^{n}_{2},\ldots,\alpha^{n}_{k})$. Unfortunately in this case having a Gröbner basis for this module is of no use as we can no longer lift this special basis to ${\sf R}^{k}$. The trick with adding $0$ as the first coordinate will no longer work as for some $(\gamma_{2},\ldots,\gamma_{k})\in{\cal M^{\prime}}$ we only know that there exists some $\gamma\in{\sf R}$ such that $(\gamma,\gamma_{2},\ldots,\gamma_{k})\in{\cal S}$ and we cannot enforce that $\gamma=0$. However, if we lift the set by adding appropriate elements $\gamma\in{\sf R}$ as first coordinates, then the resulting set does not lift the Gröbner basis properties for the reduction relation. Especially in the induction step the first coordinate of the vector being modified can no longer be expected to be left unchanged which is the case when using vectors with first coordinate $0$ for reduction.. The situation improves if we look at one- sided reduction rings ${\sf R}$ and demand that in ${\sf R}$ all (left respectively right) syzygy modules have finite bases. ${\sf R}^{k}$ is a right ${\sf R}$-module with scalar multiplication $(\alpha_{1},\ldots,\alpha_{k})\ast\alpha=(\alpha_{1}\cdot\alpha,\ldots,\alpha_{k}\cdot\alpha)$. Provided a finite subset $\\{\alpha_{1},\ldots,\alpha_{n}\\}\subseteq{\sf R}$ the set of solutions of the equation $\alpha_{1}\cdot X_{1}+\ldots+\alpha_{n}\cdot X_{n}=0$ is a submodule of the right ${\sf R}$-module ${\sf R}^{n}$. It is called the (first) module of syzygies of $\\{\alpha_{1},\ldots,\alpha_{n}\\}$ in the literature. We will see that these special modules can be used to characterize Gröbner bases of submodules in ${\sf R}^{k}$. A reduction relation can be defined similarly to Definition 3.4.3. ###### Definition 3.4.8 Let ${\bf a}=\sum_{i=1}^{k}{\bf e}_{i}\ast\alpha_{i}$, ${\bf b}=\sum_{i=1}^{k}{\bf e}_{i}\ast\beta_{i}\in{\sf R}^{k}$. We say that ${\bf b}$ right reduces ${\bf a}$ to ${\bf c}$ at the monomial ${\bf e}_{s}\ast\alpha_{s}$ in one step, denoted by ${\bf a}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{\bf b}\,$}{\bf c}$, if * (a) $\beta_{j}=0$ for $1\leq j<s$, * (b) $\alpha_{s}\Longrightarrow_{\beta_{s}}\gamma_{s}$ with $\alpha_{s}=\gamma_{s}+\beta_{s}\cdot{\delta}$, $\delta\in{\sf R}$, and * (c) ${\bf c}={\bf a}-{\bf b}\ast\delta=(\alpha_{1},\ldots,\alpha_{s-1},\gamma_{s},\alpha_{s+1}-\beta_{s+1}\cdot\delta,\ldots,\alpha_{k}-\beta_{k}\cdot\delta)$. $\diamond$ ###### Theorem 3.4.9 Let ${\sf R}$ be a ring with a right reduction relation $\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}\,$ fulfilling (A1) – (A3). Additionally let every right module of syzygies in ${\sf R}$ have a finite basis. If every finitely generated right ideal in ${\sf R}$ has a finite Gröbner basis, then the same holds for every finitely generated right submodule in $({\sf R}^{k},\longrightarrow)$. Proof : Again the candidate for the right Gröbner basis can be built similar to the set $G$ in the proofs of Theorem 3.4.5 and 3.4.7: Let ${\cal S}$ be a right submodule of ${\sf R}^{k}$ which is finitely generated by a set $\\{{\bf a}_{1},\ldots,{\bf a}_{n}\\}$. We show our claim by induction on $k$. For $k=1$ we find that ${\cal S}$ is in fact a finitely generated right ideal in ${\sf R}$ and hence by our hypothesis must have a finite right Gröbner basis. For $k>1$ let us look at the set ${\mathfrak{i}}=\\{\beta_{1}\mid(\beta_{1},\ldots,\beta_{k})\in{\cal S}\\}$ which is again a right ideal in ${\sf R}$ finitely generated by $\\{\alpha_{1}^{1},\ldots,\alpha_{1}^{n}\\}$ where ${\bf a}_{i}=(\alpha_{1}^{i},\ldots,\alpha_{k}^{i})$. Hence ${\mathfrak{i}}$ must have a finite right Gröbner basis $\\{\gamma_{1},\ldots,\gamma_{s}\mid\gamma_{i}\in{\sf R}\\}$ by our assumption. Choose $H=\\{{\bf c}_{1},\ldots,{\bf c}_{s}\\}\subseteq{\cal S}$ such that the first coordinate of ${\bf c}_{i}$ is $\gamma_{i}$. On the other hand the right syzygy module $\\{(\psi_{1},\ldots,\psi_{n})\mid\sum_{i=1}^{n}\alpha_{1}^{i}\cdot\psi_{i}=0,\psi_{i}\in{\sf R}\\}$ has a finite basis $B=\\{(\beta_{1}^{j},\ldots,\beta_{n}^{j})\mid 1\leq j\leq m\\}\subseteq{\sf R}^{n}$. Then the set $\\{\sum_{i=1}^{n}{\bf a}_{i}\ast\beta_{i}^{j}\mid 1\leq j\leq m\\}\cup\\{{\bf a}_{i}\mid\alpha_{1}^{i}=0,1\leq i\leq n\\}$ is a finite generating set for the submodule ${\cal M}=\\{(\beta_{2},\ldots,\beta_{k})\mid(0,\beta_{2},\ldots,\beta_{k})\in{\cal S}\\}$ of ${\sf R}^{k-1}$. To see this let $(0,\beta_{2},\ldots,\beta_{k})\in{\cal S}$. Then $(0,\beta_{2},\ldots,\beta_{k})=\sum_{i=1}^{n}{\bf a}_{i}\ast\zeta_{i}$, $\zeta_{i}\in{\sf R}$ implies $\sum_{i=1}^{n}\alpha_{1}^{i}\cdot\zeta_{i}=0$ and hence $(\zeta_{1},\ldots,\zeta_{n})$ lies in the right syzygy module and we are done. Hence by our induction hypothesis ${\cal M}$ then must have a finite right Gröbner basis $\\{(\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$. Let $G=\\{{\bf c}_{1},\ldots,{\bf c}_{s}\\}\cup\\{{\bf d}_{i}=(0,\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$. Since $G$ generates ${\cal S}$ it remains to show that it is a right Gröbner basis. By the definition of the reduction relation in ${\sf R}^{k}$ we immediately find $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{G}\,$}\subseteq\;\;\equiv_{{\cal S}}$. To see the converse let ${\bf r}=(\rho_{1},\ldots,\rho_{k})\equiv_{{\cal S}}{\bf s}=(\sigma_{1},\ldots,\sigma_{k})$. Then as $\rho_{1}\equiv_{\\{\alpha_{1}\mid{\bf a}=(\alpha_{1},\ldots,\alpha_{k})\in{\cal S}\\}}\sigma_{1}$ by the definition of $G$ we get $\rho_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{\\{\gamma_{1},\ldots,\gamma_{s}\\}}\,$}\sigma_{1}$. But this gives us ${\bf r}\mbox{$\,\stackrel{{\scriptstyle}}{{\Longleftrightarrow}}\\!\\!\mbox{}_{H}\,$}{\bf r}+\sum_{i=1}^{s}{\bf c}_{i}\ast\chi_{i}={\bf r}^{\prime}=(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})$, $\chi_{i}\in{\sf R}$, and we get $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})\equiv_{\cal S}(\sigma_{1},\ldots,\sigma_{k})$. Hence $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})-(\sigma_{1},\ldots,\sigma_{k})=(0,{\rho_{2}}^{\prime}-\sigma_{2},\ldots,{\rho_{k}}^{\prime}-\sigma_{k})\in{\cal S}$ implying $({\rho_{2}}^{\prime}-\sigma_{2},\ldots,{\rho_{k}}^{\prime}-\sigma_{k})\in{\cal M}$. Now we have to be more careful since we cannot conclude that $({\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime}),(\sigma_{2},\ldots,\sigma_{k})\in{\cal M}$. But we know $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})=(\sigma_{1},\ldots,\sigma_{k})+(0,{\rho_{2}}^{\prime}-\sigma_{2},\ldots,{\rho_{k}}^{\prime}-\sigma_{k})=(\sigma_{1},\ldots,\sigma_{k})+\sum_{i=1}^{w}{\bf d}_{i}\ast\eta_{i}$ where $(0,{\rho_{2}}^{\prime}-\sigma_{2},\ldots,{\rho_{k}}^{\prime}-\sigma_{k})=\sum_{i=1}^{w}{\bf d}_{i}\ast\eta_{i}$ for $\eta_{i}\in{\sf R}$, i.e., $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})\equiv_{\langle{\bf d}_{1},\ldots,{\bf d}_{w}\rangle}(\sigma_{1},\ldots,\sigma_{k})$. Hence, as $\\{(\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$ is a right Gröbner basis of ${\cal M}$ both vectors $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})$ and $(\sigma_{1},\ldots,\sigma_{k})$ must have a common normal form using $\\{{\bf d}_{i}=(0,\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$ for reduction181818The elements in this set cannot influence the first coordinate which is $\sigma_{1}$ for both vectors. and we are done. The same argument applies to show local confluence. Let us assume there are ${\bf r}$, ${\bf s}_{1}$, ${\bf s}_{2}\in{\sf R}^{k}$ such that ${\bf r}\longrightarrow_{G}{\bf s}_{1}$ and ${\bf r}\longrightarrow_{G}{\bf s}_{2}$. Then by the definition of $G$ the first coordinates $\sigma^{1}_{1}$ and $\sigma_{1}^{2}$ of ${\bf s}_{1}$ respectively ${\bf s}_{2}$ are joinable by $\\{\gamma_{1},\ldots,\gamma_{s}\\}$ to some element say $\sigma$ giving rise to elements ${\bf r}_{1}={\bf s}_{1}+\sum_{i=1}^{s}{\bf c}_{i}\ast\chi_{i}$ and ${\bf r}_{2}={\bf s}_{2}+\sum_{i=1}^{s}{\bf c}_{i}\ast\psi_{i}$ with first coordinate $\sigma$. Again we know $(\sigma,{\rho^{1}_{2}},\ldots,{\rho^{1}_{k}})=(\sigma,{\rho^{2}_{2}},\ldots,{\rho^{2}_{k}})+(0,{\rho^{1}_{2}}-{\rho^{2}_{2}},\ldots,{\rho^{1}_{k}}-{\rho^{2}_{k}})$ with $({\rho^{1}_{2}}-{\rho^{2}_{2}},\ldots,{\rho^{1}_{k}}-{\rho^{2}_{k}})\in{\cal M}$. Hence $(\sigma,{\rho^{1}_{2}},\ldots,{\rho^{1}_{k}})=(\sigma,{\rho^{2}_{2}},\ldots,{\rho^{2}_{k}})+\sum_{i=1}^{w}{\bf d}_{i}\ast\eta_{i}$ for $\eta_{i}\in{\sf R}$, i.e., $(\sigma_{1},{\rho_{2}}^{\prime},\ldots,{\rho_{k}}^{\prime})\equiv_{\langle{\bf d}_{1},\ldots,{\bf d}_{w}\rangle}(\sigma_{1},\ldots,\sigma_{k})$. As again $\\{(\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$ is a right Gröbner basis of ${\cal M}$ both vectors must have a common normal with respect to reduction using $\\{{\bf d}_{i}=(0,\delta_{2}^{i},\ldots,\delta^{i}_{k})\mid 1\leq i\leq w\\}$. q.e.d. The task of describing two-sided syzygy modules is much more complicated. We follow the ideas given by Apel in his habilitation [Ape98]. Let ${\cal R}$ be the free Abelian group with basis elements $\alpha\otimes\beta$ where $\alpha,\beta\in{\sf R}$. We define a new vector space ${\cal S}$ with formal sums as elements $\sum_{i=1}^{n}\gamma_{i}\cdot\alpha_{i}\otimes\beta_{i}\cdot\delta_{i}$ where $\gamma_{i},\delta_{i}\in{\sf R}$ and $\alpha_{i}\otimes\beta_{i}\in{\cal R}$. Let ${\cal U}$ be the subspace of ${\cal S}$ generated by the vectors $\alpha\otimes(\beta_{1}+\beta_{2})-\alpha\otimes\beta_{1}-\alpha\otimes\beta_{2}$ $(\alpha_{1}+\alpha_{2})\otimes\beta-\alpha_{1}\otimes\beta-\alpha_{2}\otimes\beta$ $\alpha\otimes(\gamma\cdot\beta)-\gamma\cdot(\alpha\otimes\beta)$ $(\gamma\cdot\alpha)\otimes\beta-\gamma\cdot(\alpha\otimes\beta)$ $\alpha\otimes(\beta\cdot\gamma)-(\alpha\otimes\beta)\cdot\gamma$ $(\alpha\cdot\gamma)\otimes\beta-(\alpha\otimes\beta)\cdot\gamma$ where $\alpha,\alpha_{i},\beta,\beta_{i},\gamma\in{\sf R}$. Then the quotient ${\cal S}/{\cal U}$ is called the tensor product denoted by ${\sf R}\otimes{\sf R}$. The sets we are interested in can be defined as follows: Let $R$ be some subset of ${\sf R}$. Syzygies of $R$ are solutions of the equations $\sum_{i=1}^{n}\sum_{j=1}^{n_{i}}\alpha_{i,j}\cdot\rho_{i}\cdot\beta_{i,j}=0,\alpha_{i,j},\beta_{i,j}\in{\sf R},\rho_{i}\in R$. The set containing all such solutions is called the syzygy module of $R$. We can now describe these sets using objects of the “polynomial” structure ${\cal S}[{\sf R}]$ which contains formal sums of the form $\sum_{i=1}^{n}\sum_{j=1}^{n_{i}}(\alpha_{i,j}\otimes\beta_{i,j})\cdot\gamma_{i}$, $\alpha_{i},\beta_{i},\gamma_{i}\in{\sf R}$. We can associate a mapping $\phi:{\cal S}[{\sf R}]\longrightarrow{\sf R}$ by $\sum_{i=1}^{n}\sum_{j=1}^{n_{i}}(\alpha_{i,j}\otimes\beta_{i,j})\cdot\gamma_{i}\mapsto\sum_{i=1}^{n}\sum_{j=1}^{n_{i}}\alpha_{i,j}\cdot\gamma_{i}\cdot\beta_{i,j}$. Then for the set $R$ we are interested in, the set of “solutions” is $\bigcup_{\rho_{1},\ldots,\rho_{k}\in R,k\in{\mathbb{N}}}S_{\rho_{1},\ldots,\rho_{k}}$ with ordered lists of not necessarily different elements from $R$ such that $S_{\rho_{1},\ldots,\rho_{k}}=\\{(\sum_{j=1}^{n_{1}}\alpha_{1,j}\otimes\beta_{1,j},\ldots,\sum_{j=1}^{n_{k}}\alpha_{k,j}\otimes\beta_{k,j})\mid\phi(\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}(\alpha_{i,j}\otimes\beta_{i,j})\cdot\rho_{i})=0,\alpha_{i,j},\beta_{i,j}\in{\sf R}\\}$. Then these sets $S_{\rho_{1},\ldots,\rho_{k}}$ are in fact modules 1. 1. $S_{\rho_{1},\ldots,\rho_{k}}$ is closed under scalar multiplication, i.e., $(\sum_{j=1}^{n_{1}}\alpha_{1,j}\otimes\beta_{1,j},\ldots,\sum_{j=1}^{n_{k}}\alpha_{k,j}\otimes\beta_{k,j})\in S_{\rho_{1},\ldots,\rho_{k}}$ and $\gamma\in{\sf R}$ implies $\gamma\cdot(\sum_{j=1}^{n_{1}}\alpha_{1,j}\otimes\beta_{1,j},\ldots,\sum_{j=1}^{n_{k}}\alpha_{k,j}\otimes\beta_{k,j})=(\gamma\cdot(\sum_{j=1}^{n_{1}}\alpha_{1,j}\otimes\beta_{1,j}),\ldots,\gamma\cdot(\sum_{j=1}^{n_{k}}\alpha_{k,j}\otimes\beta_{k,j}))\in S_{\rho_{1},\ldots,\rho_{k}}$: $\phi(\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}(\alpha_{i,j}\otimes\beta_{i,j})\cdot\rho_{i})=0$ implies $\phi(\gamma\cdot(\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}(\alpha_{i,j}\otimes\beta_{i,j})\cdot\rho_{i}))=0$ as $\gamma\cdot(\alpha_{i,j}\otimes\beta_{i,j})=(\gamma\cdot\alpha_{i,j})\otimes\beta_{i,j}$ and hence $\phi(\gamma\cdot(\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}(\alpha_{i,j}\otimes\beta_{i,j})\cdot\rho_{i}))=\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\gamma\cdot\alpha_{i,j}\cdot\rho_{i}\cdot\beta_{i,j}=0$. Multiplication from the right can be treated similarly. 2. 2. $S_{\rho_{1},\ldots,\rho_{k}}$ is closed under addition, i.e., $(\sum_{j=1}^{n_{1}}\alpha_{1,j}\otimes\beta_{1,j},\ldots,\sum_{j=1}^{n_{k}}\alpha_{k,j}\otimes\beta_{k,j})$, $(\sum_{j=1}^{\tilde{n_{1}}}\tilde{\alpha}_{1,j}\otimes\tilde{\beta}_{1,j},\ldots,\sum_{j=1}^{\tilde{n_{k}}}\tilde{\alpha}_{k,j}\otimes\tilde{\beta}_{k,j})\in S_{\rho_{1},\ldots,\rho_{k}}$ implies $(\sum_{j=1}^{n_{1}}\alpha_{1,j}\otimes\beta_{1,j}+\sum_{j=1}^{\tilde{n_{1}}}\tilde{\alpha}_{1,j}\otimes\tilde{\beta}_{1,j},\ldots,\sum_{j=1}^{n_{k}}\alpha_{k,j}\otimes\beta_{k,j}+\sum_{j=1}^{\tilde{n_{k}}}\tilde{\alpha}_{k,j}\otimes\tilde{\beta}_{k,j})\in S_{\rho_{1},\ldots,\rho_{k}}$: The question arises when such modules have useful bases for characterizing syzygy modules in non-commutative reduction rings. This would mean the existence of sets $B_{\rho_{1},\ldots,\rho_{k}}=\\{B_{i}\in({\sf R}\otimes{\sf R})^{k}\mid i\in I\\}$ such that for each $(\sum_{j=1}^{n_{1}}\alpha_{1,j}\otimes\beta_{1,j},\ldots,\sum_{j=1}^{n_{k}}\alpha_{k,j}\otimes\beta_{k,j})\in S_{\rho_{1},\ldots,\rho_{k}}$ there exist $\gamma_{ij},\delta_{ij}\in{\sf R}$ with $(\sum_{j=1}^{n_{1}}\alpha_{1,j}\otimes\beta_{1,j},\ldots,\sum_{j=1}^{n_{k}}\alpha_{k,j}\otimes\beta_{k,j})=\sum_{i\in I}\sum_{j=1}^{n_{i}}\gamma_{ij}\cdot B_{i}\cdot\delta_{ij}$. But even if this is possible it still remains the problem that we have to handle infinitely many sets of solutions associated to ordered subsets of a set admitting elements to occur more than once. This problem arises from the fact that in contrary to one-sided syzygy modules or syzygy modules in commutative structures the summands in the representations cannot be “collected” and “combined” in such a way that for a set $R$ the sums can be written as a $\sum_{\rho\in R}\alpha_{\rho}\cdot\rho\cdot\beta_{\rho}$. Let us close this section by illustrating the situation with two examples. ###### Example 3.4.10 Let $\Sigma=\\{a,b\\}$ and $\Sigma^{}$ the free monoid on the alphabet $\Sigma$. Further let ${\sf R}={\mathbb{Q}}[\Sigma^{}]$ the monoid ring over $\Sigma^{}$ and ${\mathbb{Q}}$. Let us look at the syzygy module of the set $\\{a,b\\}\subset{\sf R}$, i.e. the set of solutions of the equations $\sum_{j=1}^{n_{1}}\alpha_{1,j}\cdot a\cdot\beta_{1,j}+\sum_{j=1}^{n_{2}}\alpha_{2,j}\cdot b\cdot\beta_{2,j}=0,\alpha_{i,j},\beta_{i,j}\in{\sf R}$. Then we find $\\{(-1\otimes b,a\otimes 1),(-b\otimes 1,1\otimes a)\\}\subseteq S_{a,b}$ and this set is a finite basis for $S_{a,b}$. $\diamond$ ###### Example 3.4.11 Let ${\cal M}$ be the monoid presented by $(\\{a,b,c\\};\\{ab=a,ac=a,bc=b\\})$ and ${\sf R}={\mathbb{Q}}[{\cal M}]$ the monoid ring over ${\cal M}$ and ${\mathbb{Q}}$. Let us look at the syzygy module of the set $\\{a,b\\}\subset{\sf R}$. Then we find $\\{(1\otimes 1,-a\otimes c^{i}b^{j})\mid i,j\in{\mathbb{N}}\\}\subseteq S_{a,b}$ and hence $S_{a,b}$ has no finite basis. $\diamond$ Hence the task of two-sided syzygies is much more complicated than the one- sided case. This was also observed by Apel for graded structures where we have more structural information [Ape98]. ### 3.5 Polynomial Rings over Reduction Rings For a ring ${\sf R}$ with a reduction relation $\Longrightarrow$ fulfilling (A1) – (A3) we adopt the usual notations in ${\sf R}[X]$ the polynomial ring in one variable $X$ where multiplication is denoted by $\star$. Notice that for scalar multiplication with $\alpha\in{\sf R}$ we assume $\alpha\cdot X=X\cdot\alpha$ (see [Pes97] for other possibilities). We specify an ordering on the set of terms in one variable by defining that if $X^{i}$ divides $X^{j}$, i.e. $0\leq i\leq j$, then $X^{i}\preceq X^{j}$. Using this ordering, the head term ${\sf HT}(p)$, the head monomial ${\sf HM}(p)$ and the head coefficient ${\sf HC}(p)$ of a polynomial $p\in{\sf R}[X]$ are defined as usual, and ${\sf RED}(p)=p-{\sf HM}(p)$. We extend the function ${\sf HT}$ to sets of polynomials $F\subseteq{\sf R}[X]$ by ${\sf HT}(F)=\\{{\sf HT}(f)\mid f\in F\\}$. Let ${\mathfrak{i}}\subseteq{\sf R}[X]$ be a finitely generated ideal in ${\sf R}[X]$. It is easy to see that given a term $t$ the set $C(t,{\mathfrak{i}})=\\{{\sf HC}(f)\mid f\in{\mathfrak{i}},{\sf HT}(f)=t\\}\cup\\{0\\}$ is an ideal in ${\sf R}$. In order to guarantee that these ideals are also finitely generated we will assume that ${\sf R}$ is a Noetherian ring191919We run into similar problems as in the module case in Section 3.4 as we cannot conclude that the ideal $C(t,{\mathfrak{i}})$ is finitely generated from the fact that ${\mathfrak{i}}$ is.. Note that for any two terms $t$ and $s$ such that $t$ divides $s$ we have $C(t,{\mathfrak{i}})\subseteq C(s,{\mathfrak{i}})$. This follows, as for $s=t\star u$, $u\in\\{X^{i}\mid i\in{\mathbb{N}}\\}$, we find that ${\sf HC}(f)\in C(t,{\mathfrak{i}})$ implies ${\sf HC}(f\star u)={\sf HC}(f)\in C(s,{\mathfrak{i}})$ since $f\in{\mathfrak{i}}$ implies $f\star u\in{\mathfrak{i}}$. We additionally define a partial ordering on ${\sf R}$ by setting for $\alpha,\beta\in{\sf R}$, $\alpha>_{{\sf R}}\beta$ if and only if there exists a finite set $B\subseteq{\sf R}$ such that $\alpha\mbox{$\,\stackrel{{\scriptstyle+}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\beta$. Then we can define an ordering on ${\sf R}[X]$ as follows: For $f,g\in{\sf R}[X]$, $f>g$ if and only if either ${\sf HT}(f)\succ{\sf HT}(g)$ or $({\sf HT}(f)={\sf HT}(g)$ and ${\sf HC}(f)>_{{\sf R}}{\sf HC}(g))$ or $({\sf HM}(f)={\sf HM}(g)$ and ${\sf RED}(f)>{\sf RED}(g))$. Notice that this ordering in general is neither total nor Noetherian on ${\sf R}[X]$. ###### Definition 3.5.1 Let $p,f$ be two non-zero polynomials in ${\sf R}[X]$. We say $f$ reduces $p$ to $q$ at a monomial $\alpha\cdot X^{i}$ in one step, denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}q$, if 1. (a) ${\sf HT}(f)$ divides $X^{i}$, i.e. ${\sf HT}(f)\star X^{j}=X^{i}$ for some term $X^{j}$, 2. (b) $\alpha\Longrightarrow_{{\sf HC}(f)}\beta$, with $\alpha=\beta+\sum_{i=1}^{k}\gamma_{i}\cdot{\sf HC}(f)\cdot\delta_{i}$ for some $\beta,\gamma_{i},\delta_{i}\in{\sf R}$, $1\leq i\leq k$, and 3. (c) $q=p-\sum_{i=1}^{k}(\gamma_{i}\cdot f\cdot\delta_{i})\star X^{j}$. $\diamond$ Notice that if $f$ reduces $p$ to $q$ at a monomial $\alpha\cdot t$ the term $t$ can still occur in the resulting polynomial $q$. Hence termination of this reduction cannot be shown by arguments involving terms only as in the case of polynomial rings over fields. But when using a finite set of polynomials for reduction we know by (A1) that reducing $\alpha$ in ${\sf R}$ with respect to the finite set of head coefficients of the applicable polynomials must terminate and then either the monomial containing the term $t$ disappears or is irreducible. Hence the reduction relation as defined in Definition 3.5.1 is Noetherian when using finite sets of polynomials. Therefore it fulfills Axiom (A1). It is easy to see that (A2) and (A3) are also true and if the reduction relation $\Longrightarrow$ satisfies (A4) this is inherited by the reduction relation $\longrightarrow$ in ${\sf R}[X]$. ###### Theorem 3.5.2 If $({\sf R},\Longrightarrow)$ is a Noetherian reduction ring, then $({\sf R}[X],\longrightarrow)$ is a Noetherian reduction ring. Proof : By Hilbert’s basis theorem ${\sf R}[X]$ is Noetherian as ${\sf R}$ is Noetherian. We only have to prove that every ideal ${\mathfrak{i}}\neq\\{0\\}$ in ${\sf R}[X]$ has a finite Gröbner basis. A finite basis $G$ of ${\mathfrak{i}}$ will be defined in stages according to the degree of terms occurring as head terms among the polynomials in ${\mathfrak{i}}$ and then we will show that $G$ is in fact a Gröbner basis. Let $G_{0}$ be a finite Gröbner basis of the ideal $C(X^{0},{\mathfrak{i}})$ in ${\sf R}$, which must exist since ${\sf R}$ is supposed to be Noetherian and a reduction ring. Further, at stage $i>0$, if for each $X^{j}$ with $j<i$ we have $C(X^{j},{\mathfrak{i}})\subsetneqq C(X^{i},{\mathfrak{i}})$, include for each $\alpha$ in Gb$(C(X^{i},{\mathfrak{i}}))$ (a finite Gröbner basis of $C(X^{i},{\mathfrak{i}})$) a polynomial $p_{\alpha}$ from ${\mathfrak{i}}$ in $G_{i}$ such that ${\sf HM}(p)=\alpha\cdot X^{i}$. Notice that in this construction we use the axiom of choice, when choosing the $p_{\alpha}$ from the infinite set ${\mathfrak{i}}$, and hence the construction is non- constructive. At each stage only a finite number of polynomials can be added since the respective Gröbner bases Gb$(C(X^{i},{\mathfrak{i}}))$ are always finite, and at most one polynomial from ${\mathfrak{i}}$ is included for each element in Gb$(C(X^{i},{\mathfrak{i}}))$. If a polynomial with head term $X^{i}$ is included, then $C(X^{j},{\mathfrak{i}})\subsetneqq C(X^{i},{\mathfrak{i}})$ for every $j<i$. So if $X^{i}\in HT({\mathfrak{i}})$ is not included as a head term of a polynomial in $G_{i}$, then there is a term $X^{j}$ occurring as a head term in some set $G_{j}$, $j<i$, $C(X^{i},{\mathfrak{i}})=C(X^{j},{\mathfrak{i}})$ and $C(X^{j},G_{j})$ is a Gröbner basis for the ideal $C(X^{j},{\mathfrak{i}})=C(X^{i},{\mathfrak{i}})$ in ${\sf R}$. We claim that the set $G=\bigcup_{i\geq 0}G_{i}$ is a finite Gröbner basis of ${\mathfrak{i}}$. To show that $G$ is finite it suffices to prove that the set ${\sf HT}(G)$ is finite, since in every stage only finitely many polynomials all having new head terms are added. Assuming that ${\sf HT}(G)$ is infinite, there is a sequence $X^{n_{i}}$, $i\in{\mathbb{N}}$ of different terms such that $n_{i}<n_{i+1}$. But then by construction there is an ascending sequence of ideals in ${\sf R}$, namely $C(X^{n_{0}},{\mathfrak{i}})\subsetneqq C(X^{n_{1}},{\mathfrak{i}})\subsetneqq\ldots$ which contradicts the fact that ${\sf R}$ is supposed to be Noetherian. So after some step $m$ no more polynomials $p$ from ${\mathfrak{i}}$ can be found such that for ${\sf HT}(p)=X^{i}$ the set $C(X^{i},{\mathfrak{i}})$ is different from all $C(X^{j},{\mathfrak{i}})$, $j<i$. Notice that for all $p\in{\mathfrak{i}}$ we have $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}0$ and $G$ generates ${\mathfrak{i}}$. This follows immediately from the construction of $G$. Hence $G$ is at least a wesk Gröbner basis. To see that $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$ is confluent, let $p$ be a polynomial which has two distinct normal forms with respect to $G$, say $p_{1}$ and $p_{2}$. Let $t$ be the largest term on which $p_{1}$ and $p_{2}$ differ and let $\alpha_{1}$ and $\alpha_{2}$ be the respective coefficients of $t$ in $p_{1}$ and $p_{2}$. Since $p_{1}-p_{2}\in{\mathfrak{i}}$ this polynomial reduces to $0$ using $G$ and without loss of generality we can assume that these reductions always take place at the respective head terms of the polynomials in the reduction sequence. Let $s\in{\sf HT}(G)$ be the head term of the polynomial in $G$ which reduces ${\sf HT}(p_{1}-p_{2})$, i.e., $s$ divides $t$, $\alpha_{1}-\alpha_{2}\in C(s,{\mathfrak{i}})$, and hence $\alpha_{1}\equiv_{{\mathfrak{i}}}\alpha_{2}$. Therefore, not both $\alpha_{1}$ and $\alpha_{2}$ can be in normal form with respect to any Gröbner basis of $C(s,{\mathfrak{i}})$ and hence with respect to the set of head coefficients of polynomials in $G$ with head term $s$. So both, $\alpha_{1}\cdot t$ and $\alpha_{2}\cdot t$ cannot be in normal form with respect to $G$, which is a contradiction to the fact that $p_{1}$ and $p_{2}$ are supposed to be in normal form with respect to $G$. Finally we have to prove $\equiv_{{\mathfrak{i}}}\;=\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{G}\,$}$. Let $p\equiv_{{\mathfrak{i}}}q$ both be in normal form with respect to $G$. Then as before $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}0$ implies $p=q$. Hence we have shown that $G$ is in fact a finite Gröbner basis of ${\mathfrak{i}}$. q.e.d. This theorem of course can be applied to ${\sf R}[X]$ and a new variable $X_{2}$ and by iteration we immediately get the following: ###### Corollary 3.5.3 If $({\sf R},\Longrightarrow)$ is a Noetherian reduction ring, then ${\sf R}[X_{1},\ldots,X_{n}]$ is a Noetherian reduction ring with the respective extended reduction relation. Notice that other definitions of reduction relations in ${\sf R}[X_{1},\ldots,X_{n}]$ are known in the literature. These are usually based on divisibility of terms and admissible term orderings on the set of terms to distinguish the head terms. The proof of Theorem 3.5.2 can be generalized for these cases. Moreover, these results also hold for weak reduction rings. ###### Corollary 3.5.4 If $({\sf R},\Longrightarrow)$ is a Noetherian weak reduction ring, then ${\sf R}[X_{1},\ldots,X_{n}]$ is a Noetherian weak reduction ring with the respective extended reduction relation. Proof : This follows immediately by using weak Gröbner bases $G_{i}$ for the definition of $G$ in the proof of Theorem 3.5.2. As before the property that for all $p\in{\mathfrak{i}}$ we have $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}0$ and $G$ generates ${\mathfrak{i}}$ follows immediately from the construction of $G$. Hence the result holds for ${\sf R}[X_{1}]$ and can be extended to ${\sf R}[X_{1},\ldots,X_{n}]$. q.e.d. Now if $({\sf R},\Longrightarrow)$ is an effective reduction ring, then addition and multiplication in ${\sf R}[X]$ as well as reduction as defined in Definition 3.5.1 are computable operations. However, the proof of Theorem 3.5.2 does not specify how Gröbner bases for finitely generated ideals in ${\sf R}[X]$ can be constructed using Gröbner basis methods for ${\sf R}$. So we cannot conclude that for effective reduction rings the polynomial ring again will be effective. A more suitable characterization of Gröbner bases requiring ${\sf R}$ to fulfill additional conditions is needed. In order to provide completion procedures to compute Gröbner bases, various characterizations of Gröbner bases by finite test sets of special polynomials in certain commutative reduction rings (e.g. the integers and Euclidean domains) can be found in the literature (see e.g. [KN85, KRK84, Mor89]). A general approach to characterize commutative reduction rings allowing the computation of Gröbner bases using Buchberger’s approach was presented by Stifter in [Sti87]. Let us close this section by providing similar characterizations for polynomial rings over non-commutative reduction rings and outlining the arising problems. For simplicity we restrict ourselves to the case of ${\sf R}[X]$ but this is no general restriction. Given a generating set $F\subseteq{\sf R}[X]$ the key idea is to distinguish special elements of ${\sf ideal}(F)$ which have representations $\sum_{i=1}^{n}g_{i}\star f_{i}\star h_{i}$, $g_{i},h_{i}\in{\sf R}[X]$, $f_{i}\in F$ such that the head terms ${\sf HT}(g_{i}\star f_{i}\star h_{i})$ are all the same within the representation. Then on one hand the respective coefficients ${\sf HC}(g_{i}\star f_{i}\star h_{i})$ can add up to zero which in the commutative case means that the sum of the head coefficients is in an appropriate module generated by the coefficients ${\sf HC}(f_{i})$ — m(odule)-polynomials are related to these situations. If the result is not zero the sum of the coefficients ${\sf HC}(g_{i}\star f_{i}\star h_{i})$ as in the commutative case can be described in terms of a Gröbner basis of the coefficients ${\sf HC}(f_{i})$ — g(röbner)-polynomials are related to these situations. Zero divisors in the reduction ring occur as a special instance of m-polynomials where $F=\\{f\\}$ and $\alpha\star f\star\beta$, $\alpha,\beta\in{\sf R}$ are considered. In case ${\sf R}$ is a commutative or one-sided reduction ring the first problem is related to solving linear homogeneous equations in ${\sf R}$ and to the existence of finite bases of the respective modules. Let us become more precise and look into the definitions of m- and g-polynomials for the special case of rings with right reduction relations. ###### Definition 3.5.5 Let $P=\\{p_{1},\ldots,p_{k}\\}$ be a finite set of polynomials in ${\sf R}[X]$, $u_{1},\ldots,u_{k}$ terms in $\\{X^{j}\mid j\in{\mathbb{N}}\\}$ such that for the term $t=\max\\{{\sf HT}(p_{i})\mid 1\leq i\leq k\\}$ we have $t={\sf HT}(p_{i})\star u_{i}$ and $\gamma_{i}={\sf HC}(p_{i})$ for $1\leq i\leq k$. Let $G$ be a right Gröbner basis of the right ideal generated by $\\{\gamma_{i}\mid 1\leq i\leq k\\}$ in ${\sf R}$ and $\alpha=\sum_{i=1}^{k}\gamma_{i}\cdot\beta_{i}^{\alpha}$ for $\alpha\in G$, $\beta_{i}^{\alpha}\in{\sf R}$. Then we define the g-polynomials (Gröbner polynomials) corresponding to $P$ and $t$ by setting $g_{\alpha}=\sum_{i=1}^{k}p_{i}\star u_{i}\cdot\beta_{i}^{\alpha}$ where ${\sf HT}(p_{i})\star u_{i}=t$. Notice that ${\sf HM}(g_{\alpha})=\alpha\cdot t$. For the right module $M=\\{(\delta_{1},\ldots,\delta_{k})\mid\sum_{i=1}^{k}\gamma_{i}\cdot\delta_{i}=0\\}$, let the set $\\{B_{j}\mid j\in I_{M}\\}$ be a basis with $B_{j}=(\beta_{j,1},\ldots,\beta_{j,k})$ for $\beta_{j,l}\in{\sf R}$ and $1\leq l\leq k$. We define the m-polynomials (module polynomials) corresponding to $P$ and $t$ by setting $h_{j}=\sum_{i=1}^{k}p_{i}\star u_{i}\cdot\beta_{j,i}\mbox{ for each }j\in I_{M}$ where ${\sf HT}(p_{i})\star u_{i}=t$. Notice that ${\sf HT}(h_{j})\prec t$ for each $j\in I_{M}$. $\diamond$ Given a set of polynomials $F$ the corresponding m- and g-polynomials are those resulting for every subset $P\subseteq F$ according to this definition. In case we want effectiveness, we have to require that the bases in this definition are computable. Of course for commutative reduction rings the definition extends to characterize two-sided ideals. However, the whole situation becomes more complicated for non-commutative two-sided reduction rings, as the equations are no longer linear and we have to distinguish right and left multipliers simultaneously. Moreover the set of m-polynomials is a much more complicated structure. In some cases the problem for two-sided ideals can be translated into the one-sided case and hence solved via one- sided reduction techniques [KRW90]. But the general case is much more involved, see Definition 3.5.6 below. The g-polynomials corresponding to right Gröbner bases of right ideals in ${\sf R}$ can successfully be treated whenever finite right Gröbner bases exist. Here, if we want effectiveness, we have to require that a right Gröbner basis as well as representations for its elements in terms of the generating set are computable. Using m- and g-polynomials, right Gröbner bases can be characterized similar to the characterizations in terms of syzygies (a direct generalization of the approaches by Kapur and Narendran in [KN85] respectively Möller in [Mor89]): In case for the respective subsets $P\subseteq F$ the respective terms $t=\max\\{{\sf HT}(p)\mid p\in P\\}$ only give rise to finitely many m- and g-polynomials, these situations can be localized to finitely many terms. One can provide a completion procedure based on this characterization which will indeed compute a finite right Gröbner basis if ${\sf R}$ is Noetherian. In principal ideal rings, where the function ${\sf gcd}$ (greatest common divisor) is defined it is sufficient to consider subsets $P\subseteq F$ of size $2$ (compare [KN85]). Now let us look at two-sided ideals and two-sided reduction relations. ###### Definition 3.5.6 Let $P=\\{p_{1},\ldots,p_{k}\\}$ be a finite set of polynomials in ${\sf R}[X]$, $u_{1},\ldots,u_{k}$ terms in $\\{X^{j}\mid j\in{\mathbb{N}}\\}$ such that for the term $t=\max\\{{\sf HT}(p_{i})\mid 1\leq i\leq k\\}$ we have $t={\sf HT}(p_{i})\star u_{i}$ and $\gamma_{i}={\sf HC}(p_{i})$ for $1\leq i\leq k$. Let $G$ be a Gröbner basis of the ideal generated by $\\{\gamma_{i}\mid 1\leq i\leq k\\}$ in ${\sf R}$ and $\alpha=\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\beta_{i,j}^{\alpha}\cdot\gamma_{i}\cdot\delta_{i,j}^{\alpha}$ for $\alpha\in G$, $\beta_{i,j}^{\alpha},\delta_{i,j}^{\alpha}\in{\sf R}$, $1\leq i\leq k$,$1\leq j\leq n_{i}$. Then we define the g-polynomials (Gröbner polynomials) corresponding to $P$ and $t$ by setting $g_{\alpha}=\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\beta_{i,j}^{\alpha}\cdot p_{i}\star u_{i}\cdot\delta_{i,j}^{\alpha}$ where ${\sf HT}(p_{i})\star u_{i}=t$. Notice that ${\sf HM}(g_{\alpha})=\alpha\cdot t$. We define the m-polynomials (module polynomials) corresponding to $P$ and $t$ as $h=\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\beta_{i,j}\cdot p_{i}\star u_{i}\cdot\delta_{i,j}$ where $\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\beta_{i,j}\cdot\gamma_{i}\cdot\delta_{i,j}=0$. Notice that ${\sf HT}(h)\prec t$. $\diamond$ Given a set of polynomials $F$, the set of corresponding g- and m-polynomials contains those which are specified by Definition 3.5.6 for each subset $P\subseteq F$ fulfilling the respective conditions. For a set consisting of one polynomial the corresponding m-polynomials also reflect the multiplication of the polynomial with zero-divisors of the head coefficient, i.e., by a basis of the annihilator of the head coefficient. Notice that given a finite set of polynomials the corresponding sets of g- and m-polynomials in general can be infinite. We can use g- and m-polynomials to characterize finite weak Gröbner bases. Notice that this characterization does not require ${\sf R}$ to be Noetherian. In order to characterize Gröbner bases in this fashion the Translation Lemma must hold for the reduction ring. ###### Theorem 3.5.7 Let $F$ be a finite set of polynomials in ${\sf R}[X]\backslash\\{0\\}$. Then $F$ is a weak Gröbner basis of the ideal it generates if and only if all g-polynomials and all m-polynomials corresponding to $F$ as specified in Definition 3.5.6 reduce to zero. Proof : First let $F$ be a weak Gröbner basis. By Definition 3.5.6 the g- and m-polynomials are elements of the ideal generated by $F$ and hence reduce to zero using $F$. It remains to show that every $g\in{\sf ideal}(F)\backslash\\{0\\}$ reduces to zero by $F$. Remember that for $g\in{\sf ideal}(F)$, $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g^{\prime}$ implies $g^{\prime}\in{\sf ideal}(F)$. As $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$ is Noetherian202020To achieve this we have demanded that $F$ is finite., thus it suffices to show that every $g\in{\sf ideal}(F)\backslash\\{0\\}$ is $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$-reducible. Let $g=\sum_{i=1}^{m}\alpha_{i}\cdot f_{i}\star u_{i}\cdot\beta_{i}$ be an arbitrary representation of $g$ with $\alpha_{i},\beta_{i}\in{\sf R}$, $u_{i}\in\\{X^{j}\mid j\in{\mathbb{N}}\\}$, and $f_{i}\in F$ (not necessarily different polynomials). Depending on this representation of $g$ and the degree ordering $\succeq$ on $\\{X^{j}\mid j\in{\mathbb{N}}\\}$ we define the maximal occurring term of this representation of $g$ to be $t=\max\\{{\sf HT}(f_{i}\star u_{i})\mid 1\leq i\leq m\\}$ and $K$ is the number of polynomials $f_{i}\star u_{i}$ containing $t$ as a term. Then $t\succeq{\sf HT}(g)$. We will show that $G$ is reducible by induction on $(t,K)$, where $(t^{\prime},K^{\prime})<(t,K)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $K^{\prime}<K)$212121Note that this ordering is well- founded since $\succ$ is well-founded on $\\{X^{j}\mid j\in{\mathbb{N}}\\}$ and $K\in{\mathbb{N}}$.. Without loss of generality let the first $K$ multiples occurring in our representation of $g$ be those with head term $t$, i.e., for $\sum_{i=1}^{K}\alpha_{i}\cdot f_{i}\star u_{i}\cdot\beta_{i}$ we have ${\sf HT}(f_{i}\star u_{i})=t$ for $1\leq i\leq K$, and ${\sf HT}(\alpha_{i}\cdot f_{i}\star u_{i}\cdot\beta_{i})\prec t$ for $K<i\leq m$. In case $t\succ{\sf HT}(g)$ there is an m-polynomial corresponding to the set of polynomials $P=\\{f_{1},\ldots,f_{K}\\}$ and by our assumption this polynomial is reducible to zero using $F$ hence yielding the existence of a representation $\sum_{i=1}^{n}\gamma_{i}\cdot f_{i}\star v_{i}\cdot\delta_{i}$ with $t\succ\tilde{t}=\max\\{{\sf HT}(f_{i}\star v_{i})\mid i\in\\{1,\ldots n\\}\\}$. We can then change the original representation of $g$ by substituting this sum for $\sum_{i=1}^{K}\alpha_{i}\cdot f_{i}\star u_{i}\cdot\beta_{i}$ yielding a new representation with smaller maximal term than $t$. On the other hand, if $t={\sf HT}(g)$ then again we can assume that the first $K$ multiples have head term $t$. In this case there exists a g-polynomial corresponding to the set of polynomials $P=\\{f_{1},\ldots,f_{K}\\}$ and by our assumption this polynomial is reducible to zero using $F$. Now as the head monomial of the g-polynomial and the head monomial of $g$ are equal, then $g$ must be reducible by $F$ as well. q.e.d. In order to characterize infinite sets $F$ as weak Gröbner bases we have to be more careful since we can no longer assume that $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$ is terminating222222This can of course be achieved by requiring the stronger axiom (A1’) to hold for the reduction relation.. But inspecting the proof of the previous theorem closely we see that this is not necessary. Under the stronger assumption that the g-polynomial reduces to zero using reduction at head monomials only, i.e., we have a terminating reduction sequence using finitely many polynomials in $F$ only, we can conclude that the polynomials used to extinguish the term $t$ in the g-polynomial can equally be applied to extinguish the head monomial of $g$. Since there cannot be an infinite sequence of decreasing terms $t$ one can show that $g$ reduces to zero by iterating arguments involving g- and m-polynomials. ###### Corollary 3.5.8 Let $F$ be a set of polynomials in ${\sf R}[X]\backslash\\{0\\}$. Then $F$ is a weak Gröbner basis of the ideal it generates if and only if all g-polynomials and all m-polynomials corresponding to $F$ as specified in Definition 3.5.6 reduce to zero using reduction at head monomials only. ###### Corollary 3.5.9 Let $F$ be a set of polynomials in ${\sf R}[X]\backslash\\{0\\}$. Additionally let the Translation Lemma hold in ${\sf R}$. Then $F$ is a Gröbner basis of the ideal it generates if and only if all g-polynomials and all m-polynomials corresponding to $F$ as specified in Definition 3.5.6 reduce to zero using reduction at head monomials only. Still the problem remains that the set of m-polynomials does not have a nice characterization as an algebraic structure. Remember that in the one-sided case or the case of commutative reduction rings the m-polynomials for a finite set of polynomials $P$ correspond to submodules of ${\sf R}^{\|P\|}$, as they correspond to solutions of linear equations. When attempting to describe the setting for two-sided ideals in non-commutative reduction rings one runs into the same problems as in the previous section on modules. ## Chapter 4 Function Rings In the literature Gröbner bases and reduction relations have been introduced to various algebraic structures such as the classical commutative polynomial rings over fields, non-commutative polynomial rings over fields, commutative polynomial rings over reduction rings, skew polynomial rings, Lie algebras, monoid and group rings and many more. This chapter is intended to give a generalized setting subsuming these approaches and outlining a framework for introducing reduction relations and Gröbner bases to other structures fitting the appropriate requirements. An additional aim was to work out what conditions are necessary at what point in order to give more insight into the ideas behind algebraic characterizations such as specialized standard representations for ideal elements as well as into the idea of using rewriting techniques for achieving confluent reduction relations describing the ideal congruence. This chapter is organized as follows: Section 4.1 introduces the general structure we are looking into called function rings. Section 4.2 gives the algebraic characterization for the case of right ideals in form of right standard representations. To work out the difficulties involved by our notion of terms and coefficients separately, Section 4.2.1 first treats the easier case of function rings over fields while Section 4.2.2 then goes into the details when taking a reduction ring as introduced in Chapter 3 as coefficient domain. Since for function rings over general reduction rings only a feasible characterization of weak Gröbner bases is possible, we show that this situation can be improved when looking at the special case of function rings over the integers in Section 4.2.3. Section 4.3 is dedicated to the study of a generalization of the concept of right ideals – right modules. The remaining Sections 4.4 – 4.5 then treat the same concepts and problems now in the more complex setting of two-sided ideals. ### 4.1 The General Setting Let ${\cal T}$ be a set and let ${\sf R}=({\sf R},+,\cdot,0,1)$ be an associative ring with $1$. By ${\cal F}^{{\cal T}}_{{\sf R}}$ we will denote the set of all functions $f:{\cal T}\longrightarrow{\sf R}$ with finite support ${\sf supp}(f)=\\{t\mid t\in{\cal T},f(t)\neq 0\\}$. We will simply write ${\cal F}$ if the context is clear. By $o$ we will denote the function with empty support, i.e., ${\sf supp}(o)=\emptyset$. This function will be called the zero function. Two elements of ${\cal F}$ are equal if they are equal as functions, i.e., they have the same support and coincide in their respective values. We require the set ${\cal T}$ to be independent in the sense that a function $f$ has unique support. ${\cal F}$ can be viewed as a group with respect to a binary operation $\oplus:{\cal F}\times{\cal F}\longrightarrow{\cal F}$ called addition by associating to $f,g$ in ${\cal F}$ the function in ${\cal F}$, denoted by $f\oplus g$, which has support ${\sf supp}(f\oplus g)\subseteq{\sf supp}(f)\cup{\sf supp}(g)$ and values $(f\oplus g)(t)=f(t)+g(t)$ for $t\in{\sf supp}(f)\cup{\sf supp}(g)$. The zero function $o$ fulfills $o\oplus f=f\oplus o=f$, hence is neutral with respect to $\oplus$. For an element $f\in{\cal F}$ we define the element $-f$ with ${\sf supp}(-f)={\sf supp}(f)$ and for all $t\in{\sf supp}(f)$ the value of $(-f)(t)$ is the inverse of the element $f(t)$ with respect to $+$ in ${\sf R}$ denoted by $-f(t)$. Notice that since in ${\sf R}$ every element has such an inverse the inverse of an element in ${\cal F}\backslash\\{o\\}$ is always defined. Then $-f$ is the (left and right) inverse of $f$, since $f\oplus(-f)$ as well as $(-f)\oplus f$ equals $o$, i.e., has empty support. This follows as for all $t\in{\sf supp}(f)$ we have $(f\oplus(-f))(t)=f(t)+(-f)(t)=f(t)-f(t)=0=-f(t)+f(t)=(-f)(t)+f(t)=((-f)\oplus f)(t)$. We will write $f-g$ to abbreviate $f\oplus(-g)$ for $f,g$ in ${\cal F}$. If the context is clear we will also write $f+g$ instead of $f\oplus g$. Notice that $({\cal F},\oplus,o)$ is an Abelian group since $({\sf R},+,0)$ is Abelian. Sums of functions $f_{1},\ldots,f_{m}$ will be abbreviated by $f_{1}\oplus\ldots\oplus f_{m}=\sum_{i=1}^{m}f_{i}$ as usual. Now if ${\sf R}$ is a computable ring111A ring ${\sf R}$ is called computable, if the ring operations $+$ and $\cdot$ are computable, i.e. for $\alpha,\beta\in{\sf R}$ we can compute $\alpha+\beta$ and $\alpha\cdot\beta$., then $({\cal F},\oplus)$ is a computable group. In the next lemma we provide a syntactical representation for elements of the function ring. ###### Lemma 4.1.1 Every $f\in{\cal F}\backslash\\{o\\}$ has a finite representation of the form $f=\sum_{t\in{\sf supp}(f)}m_{t}$ where $m_{t}\in{\cal F}$ such that ${\sf supp}(m_{t})=\\{t\\}$ and $f(t)=m_{t}(t)$. The representation of $o$ is the empty sum. Proof : This can be shown by induction on $n=\|{\sf supp}(f)\|$. For $n=0$ we have the empty sum which is the zero function $o$ and are done. Hence let ${\sf supp}(f)=\\{t_{1},\ldots,t_{n}\\}$ and $n>0$. Furthermore let $f(t_{1})=\alpha\in{\sf R}$ and $m\in{\cal F}$ be the unique function with ${\sf supp}(m)=\\{t_{1}\\}$ and $m(t_{1})=\alpha$. Then there exists an inverse function $-m$ and a function $(-m)\oplus f\in{\cal F}$ such that $f=(m\oplus(-m))\oplus f=m\oplus((-m)\oplus f)$ and ${\sf supp}((-m)\oplus f)=\\{t_{2},\ldots t_{n}\\}$. Hence by our induction hypothesis ${\sf supp}((-m)\oplus f)$ has a representation $\sum_{t\in\\{t_{2},\ldots t_{n}\\}}m_{t}$ yielding $f=m\oplus((-m)\oplus f)=m\oplus\sum_{t\in\\{t_{2},\ldots t_{n}\\}}m_{t}=\sum_{t\in{\sf supp}(f)}m_{t}$ with $m_{t_{1}}=m$. q.e.d. This presentation is unique up to permutations. We will call such a representation of an element as a formal sum of special functions a polynomial representation or a polynomial to stress the similarity with the objects known as polynomials in other fields of mathematics. Polynomial representations in terms of these functions are unique up to permutations of the respective elements of their support. Since these special functions are of interest we define the following subsets of ${\cal F}$: ${\sf M}({\cal F})=\\{f\in{\cal F}\mid\|{\sf supp}(f)\|=1\\}$ will be called the set of monomial functions or monomials in ${\cal F}$. Monomials will often be denoted by $m_{t}$ where the suffix $t$ is the element of the support, i.e., ${\sf supp}(m_{t})=\\{t\\}$. A subset of this set, namely ${\sf T}({\cal F})=\\{m_{t}\in{\sf M}({\cal F})\mid m_{t}(t)=1\\}$ where $1$ denotes the unit in ${\sf R}$ will be called the set of term functions or terms of ${\cal F}$. Notice that this set can be viewed as an embedding of ${\cal T}$ in ${\cal F}$ via the mapping $t\longmapsto f$ with ${\sf supp}(f)=\\{t\\}$ and $f(t)=1$. Further we assume the existence of a second binary operation called multiplication $\star:{\cal F}\times{\cal F}\longrightarrow{\cal F}$ such that $({\cal F},\oplus,\star,o)$ is a ring. In particular we have $o\star f=f\star o=o$ for all $f$ in ${\cal F}$. This ring is called a function ring222Notice that in the literature the term function ring is usually restricted to those rings where the multiplication is defined pointwise as in Example 4.1.3. Here we want to allow more interpretations for $\star$.. In case $\star$ is a computable operation, ${\cal F}$ is a computable function ring. ###### Definition 4.1.2 An element ${\bf 1}^{r}_{{\cal F}}\in{\cal F}$ is called a right unit of ${\cal F}$ if for all $f\in{\cal F}$ we have $f\star{\bf 1}^{r}_{{\cal F}}=f$. Similarly ${\bf 1}^{\ell}_{{\cal F}}\in{\cal F}$ is called a left unit of ${\cal F}$ if for all $f\in{\cal F}$ we have ${\bf 1}^{\ell}_{{\cal F}}\star f=f$. An element ${\bf 1}_{{\cal F}}\in{\cal F}$ is called a unit if for all $f\in{\cal F}$ we have ${\bf 1}_{{\cal F}}\star f=f\star{\bf 1}_{{\cal F}}=f$. $\diamond$ In general ${\cal F}$ need not have a left or right unit. If ${\cal F}$ does not have a unit this can be achieved by enlarging the set ${\cal T}$ by a new element, say $\Lambda$, and associating to $\Lambda$ a function $f_{\Lambda}$ with support $\\{\Lambda\\}$ and $f_{\Lambda}(\Lambda)=1$. The definition of $\star$ must be extended such that for all $f\in{\cal F}$ we have $f\star f_{\Lambda}=f_{\Lambda}\star f=f$. Similarly we could add a left or right unit by requiring $f\star f_{\Lambda}^{r}=f$ respectively $f_{\Lambda}^{\ell}\star f=f$. When adding a new element $f_{\Lambda}$ as a unit to ${\cal F}$ we have $f_{\Lambda}\in{\sf T}({\cal F})\subseteq{\sf M}({\cal F})$. We will not specify our ring multiplication $\star$ further at the moment except for giving some examples. Our first example outlines the situation for multiplying two elements by multiplying the respective values of the support. This is the definition of multiplication normally associated to function rings in the mathematical literature. ###### Example 4.1.3 Let us specify our multiplication $\star$ by associating to $f,g$ in ${\cal F}$ the function in ${\cal F}$, denoted by $f\star g$, which has support ${\sf supp}(f\star g)\subseteq{\sf supp}(f)\cap{\sf supp}(g)$ and values $(f\star g)(t):=f(t)\cdot g(t)$ for $t\in{\sf supp}(f)\cap{\sf supp}(g)$. Notice that in this case ${\cal F}$ can only contain a (right, left) unit if ${\cal T}$ is finite, since otherwise a unit function would have infinite support and hence be no element of ${\cal F}$. But the set of special functions $u_{S}=\sum_{t\in S}u_{t}$ where $S\subseteq{\cal T}$ finite, ${\sf supp}(u_{t})=\\{t\\}$ and $u_{t}(t)=1$ is an approximation of a unit, since for every function $f$ in ${\cal F}$ and all functions $u_{S}$ with ${\sf supp}(f)\subseteq S$ we have $f\star u_{S}=u_{S}\star f=f$. However, if we want a real unit, adding a new symbol $\Lambda$ to ${\cal T}$ and $f_{\Lambda}$ with $f_{\Lambda}(\Lambda)=1$ to ${\cal F}$ together with an extension of the definition of $\star$ by $f_{\Lambda}\star f=f\star f_{\Lambda}=f$ for all $f\in{\cal F}$ will do the trick. $\diamond$ Remember that by Lemma 4.1.1 polynomials have representations of the form $f=\sum_{t\in{\sf supp}(f)}m_{t}$ and $g=\sum_{s\in{\sf supp}(g)}n_{s}$ yielding $f\star g=(\sum_{t\in{\sf supp}(f)}m_{t})\star(\sum_{s\in{\sf supp}(g)}n_{s})=\sum_{t\in{\sf supp}(f),s\in{\sf supp}(g)}m_{t}\star n_{s}$ since the multiplication $\star$ must satisfy the distributivity law of the ring axioms. Hence knowing the behaviour of the multiplication for monomials, i.e. $\star:{\sf M}({\cal F})\times{\sf M}({\cal F})\longrightarrow{\cal F}$, is enough to characterize the multiplication $\star$. For all examples from the literature mentioned in this work, we can even state that the multiplication can be defined by specifying $\star:{\cal T}\times{\cal T}\longrightarrow{\cal F}$, and then lifting it to ${\sf M}({\cal F})$ and ${\cal F}$. This is done by defining $m_{t}\star n_{s}=(m_{t}(t)\cdot n_{s}(s))\cdot(t\star s)$ and extending this to the formal sums of monomials333Notice that this lifting requires that when writing a monomial $m_{t}$ as $m_{t}(t)\cdot t$ we have $m_{t}(t)\cdot t=t\cdot m_{t}(t)$.. A well-known example for the special instance $\star:{\cal T}\times{\cal T}\longrightarrow{\cal T}$ are the polynomial rings from Section 2.3. ###### Example 4.1.4 For a set of variables $X_{1},\ldots,X_{n}$ let us define the set of commutative terms ${\cal T}=\\{X_{1}^{i_{1}}\ldots X_{n}^{i_{n}}\mid i_{1},\ldots i_{n}\in{\mathbb{N}}\\}$ and let ${\cal F}^{{\cal T}}_{{\mathbb{Q}}}$ be the set of all functions $f:{\cal T}\longrightarrow{\mathbb{Q}}$ with finite support, where ${\mathbb{Q}}$ are the rational numbers. Multiplication $\star:{\cal T}\times{\cal T}\longrightarrow{\cal T}$ is specified as $X_{1}^{i_{1}}\ldots X_{n}^{i_{n}}\star X_{1}^{j_{1}}\ldots X_{n}^{j_{n}}=X_{1}^{i_{1}+j_{1}}\ldots X_{n}^{i_{n}+j_{n}}$. Hence here we have an example where the set ${\cal T}$ is a monoid with unit element $X_{1}^{0}\ldots X_{n}^{0}$. Then ${\cal F}$ can be interpreted as the ordinary polynomial ring ${\mathbb{Q}}[X_{1},\ldots,X_{n}]$ with the usual multiplication $(\alpha\cdot t)\star(\beta\cdot s)=(\alpha\cdot\beta)\cdot(t\star s)$ where $\alpha,\beta\in{\mathbb{Q}},s,t\in{\cal T}$. $\diamond$ Notice that in this example the unit element is an element of the set ${\cal T}$ embedded in ${\cal F}$. This does not have to be the case as the next example shows. ###### Example 4.1.5 Let us fix a finite set ${\cal T}=\\{e_{11},e_{12},e_{21},e_{22}\\}$ and let ${\cal F}^{{\cal T}}_{{\mathbb{Q}}}$ be the set of all functions $f:{\cal T}\longrightarrow{\mathbb{Q}}$, where ${\mathbb{Q}}$ are the rational numbers. We specify the multiplication $\star$ on ${\cal F}^{{\cal T}}_{{\mathbb{Q}}}$ by the action on ${\cal T}$ as follows: $e_{ij}\star e_{kl}=o$ in case $j\neq k$ and $e_{ij}\star e_{jl}=e_{il}$ for $i,j,l,k\in\\{1,2\\}$. Then multiplication is not Abelian since $e_{11}\star e_{12}=e_{12}$ whereas $e_{12}\star e_{11}=o$. $({\cal F}^{{\cal T}}_{{\mathbb{Q}}},\oplus,\star,o)$ is a ring, in fact isomorphic to the ring of $2\times 2$ rational matrices444This interpretation can be extended to arbitrary rings of $n\times n$ matrices over a field ${\mathbb{K}}$ by setting ${\cal T}=\\{e_{ij}\mid 1\leq i,j\leq n\\}$, $e_{ij}\star e_{kl}=o$ in case $j\neq k$ and $e_{ij}\star e_{jl}=e_{il}$ else. The unit element then is $e_{11}+\ldots+e_{nn}$. It contains a unit element, namely $e_{11}+e_{22}$. $\diamond$ Notice that in this example the unit element is not an element of the set ${\cal T}$ embedded in ${\cal F}$. Moreover, the multiplication here arises from the situation $\star:{\cal T}\times{\cal T}\longrightarrow{\cal T}\cup\\{o\\}$. The next example even allows multiplications of terms to result in polynomials, i.e., $\star:{\cal T}\times{\cal T}\longrightarrow{\cal F}$. ###### Example 4.1.6 For a set of variables $X_{1},X_{2},X_{3}$ let us define the set of commutative terms ${\cal T}=\\{X_{1}^{i_{1}}X_{2}^{i_{2}}X_{3}^{i_{3}}\mid i_{1},i_{2},i_{3}\in{\mathbb{N}}\\}$ and let ${\cal F}^{{\cal T}}_{{\mathbb{Q}}}$ be the set of all functions $f:{\cal T}\longrightarrow{\mathbb{Q}}$ with finite support, where ${\mathbb{Q}}$ are the rational numbers. Multiplication $\star:{\cal T}\times{\cal T}\longrightarrow{\cal F}$ is lifted from the following multiplication of the variables: $X_{2}\star X_{1}=X_{2}+X_{3}$, $X_{3}\star X_{1}=X_{1}X_{3}$, $X_{3}\star X_{2}=X_{2}X_{3}$ and $X_{i}\star X_{j}=X_{i}X_{j}$ for $i<j$. Then ${\cal F}$ can be interpreted as a skew-polynomial ring ${\mathbb{Q}}[X_{1},X_{2},X_{3}]$ with unit element $X_{1}^{0}X_{2}^{0}X_{3}^{0}\in{\cal F}^{{\cal T}}_{{\mathbb{Q}}}$. $\diamond$ Finally, many examples for function rings will be taken from monoid rings and hence we close this subsection by giving an example of a monoid ring. ###### Example 4.1.7 Let ${\cal T}=\\{a^{i},b^{i},1\mid i\in{\mathbb{N}}^{+}\\}$, where $1$ is the empty word in $\\{a,b\\}^{}$, and let the multiplication $\star$ be defined by the following multiplication table: $1$ $a^{j}$ $b^{j}$ $1$ $1$ $a^{j}$ $b^{j}$ $a^{i}$ $a^{i}$ $a^{i+j}$ $a^{i\mbox{ {\tiny\sf monus} }j}b^{j\mbox{ {\tiny\sf monus} }i}$ $b^{i}$ $b^{i}$ $a^{j\mbox{ {\tiny\sf monus} }i}b^{i\mbox{ {\tiny\sf monus} }j}$ $b^{i+j}$ where $i,j\in{\mathbb{N}}^{+}$ and $i\mbox{ {\tiny\sf monus} }j=i-j$ if $i\geq j$ and $0$ else. In fact ${\cal T}$ is the free group on one generator which can be presented as a monoid by $(\\{a,b\\};\\{ab=ba=1\\})$. Let ${\cal F}^{{\cal T}}_{{\mathbb{Q}}}$ be the set of all functions $f:{\cal T}\longrightarrow{\mathbb{Q}}$ with finite support. Then ${\cal F}^{{\cal T}}_{{\mathbb{Q}}}$ is a ring and is known as a special case of the free group ring. Its unit element is $1\in{\cal F}^{{\cal T}}_{{\mathbb{Q}}}$. $\diamond$ For the special case that we have $\star:{\cal T}\times{\cal T}\longrightarrow{\cal T}$, and some subring ${\sf R}^{\prime}\subseteq{\sf R}$ we get that the function ring ${\cal F}^{{\cal T}}_{{\sf R}^{\prime}}$ is a subring of ${\cal F}^{{\cal T}}_{{\sf R}}$. This follows directly as then for $f,g\in{\cal F}^{{\cal T}}_{{\sf R}^{\prime}}$ we have $f+(-g),f\star g\in{\cal F}^{{\cal T}}_{{\sf R}^{\prime}}$. This is no longer true if $\star:{\cal T}\times{\cal T}\longrightarrow{\cal F}^{{\cal T}}_{{\sf R}}$. Let ${\sf R}={\mathbb{Q}}$, ${\sf R}^{\prime}={\mathbb{Z}}$ and ${\cal T}=\\{X_{1}^{i}X_{2}^{j}\mid i,j\in{\mathbb{N}}\\}$ with $\star$ induced by $X_{2}\star X_{1}=\frac{1}{2}\cdot X_{1}X_{2}$, $X_{1}\star X_{2}=X_{1}X_{2}$. Then for $X_{2},X_{1}\in{\cal F}^{{\cal T}}_{{\mathbb{Z}}}$ we get $X_{2}\star X_{1}=\frac{1}{2}\cdot X_{1}X_{2}\in{\cal F}^{{\cal T}}_{{\mathbb{Q}}}$. Similarly, if we have ${\cal T}^{\prime}\subseteq{\cal T}$ and $\star:{\cal T}^{\prime}\times{\cal T}^{\prime}\longrightarrow{\cal F}^{{\cal T}^{\prime}}_{{\sf R}}$, then ${\cal F}^{{\cal T}^{\prime}}_{{\sf R}}$ is a subring of ${\cal F}^{{\cal T}}_{{\sf R}}$. Again this follows as for $f,g\in{\cal F}^{{\cal T}^{\prime}}_{{\sf R}}$ we have $f+(-g),f\star g\in{\cal F}^{{\cal T}^{\prime}}_{{\sf R}}$. Let us review Example 4.1.6: There we have ${\cal T}=\\{X_{1}^{i_{1}}X_{2}^{i_{2}}X_{3}^{i_{3}}\mid i_{1},i_{2},i_{3}\in{\mathbb{N}}\\}$ and the multiplication $\star:{\cal T}\times{\cal T}\longrightarrow{\cal F}^{{\cal T}}_{{\mathbb{Q}}}$ is lifted from the following multiplication of the variables: $X_{2}\star X_{1}=X_{2}+X_{3}$, $X_{3}\star X_{1}=X_{1}X_{3}$, $X_{3}\star X_{2}=X_{2}X_{3}$ and $X_{i}\star X_{j}=X_{i}X_{j}$ for $i<j$. Then for ${\cal T}^{\prime}=\\{X_{2}^{i_{2}}X_{3}^{i_{3}}\mid i_{2},i_{3}\in{\mathbb{N}}\\}$ we have $\star:{\cal T}^{\prime}\times{\cal T}^{\prime}\longrightarrow{\cal F}^{{\cal T}^{\prime}}_{{\mathbb{Q}}}$ and hence ${\cal F}^{{\cal T}^{\prime}}_{{\mathbb{Q}}}$ is a subring of ${\cal F}^{{\cal T}}_{{\mathbb{Q}}}$. ### 4.2 Right Ideals and Right Standard Representations Since ${\cal F}$ is a ring, we can define right, left or two-sided ideals. In this section in a first step we will restrict our attention to one-sided ideals, in particular to right ideals since left ideals in general can be treated in a symmetrical manner. A subset $\mathfrak{i}\subseteq{\cal F}$ is called a right ideal, if 1. 1. $o\in\mathfrak{i}$, 2. 2. for $f,g\in\mathfrak{i}$ we have $f\oplus g\in\mathfrak{i}$, and 3. 3. for $f\in\mathfrak{i}$, $g\in{\cal F}$ we have $f\star g\in\mathfrak{i}$. Right ideals can also be specified in terms of generating sets. For $F\subseteq{\cal F}\backslash\\{o\\}$ let ${\sf ideal}_{r}(F)=\\{\sum_{i=1}^{n}f_{i}\star g_{i}\mid f_{i}\in F,g_{i}\in{\cal F},n\in{\mathbb{N}}\\}=\\{\sum_{i=1}^{n}f_{i}\star m_{i}\mid f_{i}\in F,m_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}\\}$. These generated sets are subsets of ${\cal F}$ since for $f,g\in{\cal F}$ $f\star g$ as well as $f\oplus g$ are again elements of ${\cal F}$, and it is easily checked that they are in fact right ideals: 1. 1. $o\in{\sf ideal}_{r}(F)$ since $o$ can be written as the empty sum. 2. 2. For two elements $\sum_{i=1}^{n}f_{i}\star g_{i}$ and $\sum_{j=1}^{m}f_{j}\star h_{j}$ in ${\sf ideal}_{r}(F)$, the resulting sum $\sum_{i=1}^{n}f_{i}\star g_{i}\oplus\sum_{j=1}^{m}f_{j}\star h_{j}$ is again an element in ${\sf ideal}_{r}(F)$. 3. 3. For an element $\sum_{i=1}^{n}f_{i}\star g_{i}$ in ${\sf ideal}_{r}(F)$ and a polynomial $h$ in ${\cal F}$, the product $(\sum_{i=1}^{n}f_{i}\star g_{i})\star h=\sum_{i=1}^{n}f_{i}\star(g_{i}\star h)$ is again an element in ${\sf ideal}_{r}(F)$. Given a right ideal $\mathfrak{i}\subseteq{\cal F}$ we call a set $F\subseteq{\cal F}\backslash\\{o\\}$ a basis or a generating set of $\mathfrak{i}$ if $\mathfrak{i}={\sf ideal}_{r}(F)$. Then every element $g\in{\sf ideal}_{r}(F)\backslash\\{o\\}$ has different representations of the form $g=\sum_{i=1}^{n}f_{i}\star h_{i},f_{i}\in F,h_{i}\in{\cal F},n\in{\mathbb{N}}.$ Of course the distributivity law in ${\cal F}$ then allows to convert any such representation into one of the form $g=\sum_{j=1}^{m}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}),m\in{\mathbb{N}}.$ As we have seen in Section 1.3, it is not obvious whether some polynomial belongs to an ideal. Let again $f_{1}=X_{1}^{2}+X_{2}$ and $f_{2}=X_{1}^{2}+X_{3}$ be two polynomials in the polynomial ring ${\mathbb{Q}}[X_{1},X_{2},X_{3}]$ and ${\mathfrak{i}}=\\{f_{1}\ast g_{1}+f_{2}\ast g_{2}\mid g_{1},g_{2}\in{\mathbb{Q}}[X_{1},X_{2},X_{3}]\\}$ the (right) ideal generated by them. It is not hard to see that the polynomial $X_{2}-X_{3}$ belongs to ${\mathfrak{i}}$ since $X_{2}-X_{3}=f_{1}-f_{2}$ is a representation of $X_{2}-X_{3}$ in terms of $f_{1}$ and $f_{2}$. The same is true for the polynomial $X_{2}^{2}-X_{2}X_{3}$ where now we have to use multiples of $f_{1}$ and $f_{2}$, namely $X_{2}^{2}-X_{2}X_{3}=f_{1}\star X_{2}-f_{2}\star X_{2}$. However, when looking at the polynomial $X_{3}^{3}+X_{1}+X_{3}$ we find that there is no obvious algorithm to find such appropriate multiples. The problem is that for an arbitrary generating set for an ideal we have to look at arbitrary polynomial multiples with no boundary. One first improvement for the situation can be achieved if we can represent ideal elements by special representations in terms of the given generating set. In polynomial rings such representations are studied as variations of the term standard representations in the literature (see also Section 2.3). They will also be introduced in this setting. Since standard representations are in general distinguished by conditions involving an ordering on the set of polynomials, we will start by introducing the notion of an ordering to ${\cal F}$. Let $\succeq$ be a total well-founded ordering on the set ${\cal T}$. This enables us to make our polynomial representations of functions unique by using the ordering $\succeq$ to arrange the elements of the support: $f=\sum_{i=1}^{k}m_{t_{i}},\mbox{ where }{\sf supp}(f)=\\{t_{1},\ldots,t_{k}\\},t_{1}\succ\ldots\succ t_{k}.$ Using the ordering $\succeq$ on ${\cal T}$ we are now able to give some notions for polynomials which are essential in introducing standard representations, standard bases and Gröbner bases in the classical approach. We call the monomial with the largest term according to $\succeq$ the head monomial of $f$ denoted by ${\sf HM}(f)$, consisting of the head term denoted by ${\sf HT}(f)$ and the head coefficient denoted by ${\sf HC}(f)=f({\sf HT}(f))$. $f-{\sf HM}(f)$ is called the reductum of $f$ denoted by ${\sf RED}(f)$. Note that ${\sf HM}(f)\in{\sf M}({\cal F})$, ${\sf HT}(f)\in{\cal T}$ and ${\sf HC}(f)\in{\sf R}$. These notions can be extended to sets of functions $F\subseteq{\cal F}\backslash\\{o\\}$ by setting ${\sf HM}(F)=\\{{\sf HM}(f)\mid f\in F\\}$, ${\sf HT}(F)=\\{{\sf HT}(f)\mid f\in F\\}$ and ${\sf HC}(F)=\\{{\sf HC}(f)\mid f\in F\\}$. Notice that for some polynomial $f=\sum_{i=1}^{k}m_{t_{i}}\in{\cal F}$, and some term $t\in{\cal T}$ we cannot conclude that for the terms occurring in the multiple $f\star t=\sum_{i=1}^{k}m_{t_{i}}\star t$ we have $t_{1}\star t\succ\ldots\succ t_{k}\star t$ (in case the multiplication of terms again results in terms) or ${\sf HT}(t_{1}\star t)\succ\ldots\succ{\sf HT}(t_{k}\star t)$ as the ordering need not be compatible with multiplication in ${\cal F}$. ###### Example 4.2.1 Let ${\cal T}=\\{x,1\\}$ and $\star$ induced by the following multiplication on ${\cal T}$: $x\star x=1\star 1=1$, $x\star 1=1\star x=x$. Then assuming $x\succ 1$, after multiplying both sides of the equation with $x$, we get $x\star x=1\prec 1\star x=x$. On the other hand, assuming the precedence $1\succ x$ similarly we get $x=1\star x\prec 1=x\star x$. Hence the ordering is not compatible with multiplication using elements in ${\cal T}$. $\diamond$ We will later on see that this lack of compatibility leads to additional requirements when defining standard representations, standard bases and Gröbner bases. Since the elements of ${\cal T}$ can be identified with the terms in ${\sf T}({\cal F})$, the ordering $\succeq$ can be extended as a total well-founded555An ordering $\succeq$ on a set ${\cal M}$ will be called well-founded if its strict part $\succ$ is well-founded, i.e., does not allow infinite descending chains of the form $m_{1}\succ m_{2}\succ\ldots$. ordering on ${\sf T}({\cal F})$. Additionally we can provide orderings on ${\sf M}({\cal F})$ and ${\cal F}$ as follows. ###### Definition 4.2.2 Let $\succeq$ be a total well-founded ordering on ${\cal T}$. Let $>_{{\sf R}}$ be a (not necessarily total) well-founded ordering on ${\sf R}$. We define an ordering on ${\sf M}({\cal F})$ by $m_{t_{1}}\succ m_{t_{2}}$ if $t_{1}\succ t_{2}$ or ($t_{1}=t_{2}$ and $m_{t_{1}}(t_{1})>_{{\sf R}}m_{t_{2}}(t_{2})$). For two elements $f,g$ in ${\cal F}$ we define $f\succ g$ iff ${\sf HM}(f)\succ{\sf HM}(g)$ or $({\sf HM}(f)={\sf HM}(g)$ and ${\sf RED}(f)\succ{\sf RED}(g))$. We further define $f\succ o$ for all $f\in{\cal F}\backslash\\{o\\}$. $\diamond$ Notice that the total well-founded ordering on ${\sf T}({\cal F})$ extends to a well-founded ordering on ${\sf M}({\cal F})$. For a field ${\mathbb{K}}$ we have the trivial ordering $>_{{\mathbb{K}}}$ where $\alpha>_{{\mathbb{K}}}0$ for all $\alpha\in{\mathbb{K}}\backslash{\\{0\\}}$ and no other elements are comparable. Then the resulting ordering on the respective function ring corresponds to the one given in Definition 2.3.3 for polynomial rings over fields. ###### Lemma 4.2.3 The ordering $\succ$ on ${\cal F}$ is well-founded. Proof : The proof of this lemma will use a method known as Cantor’s second diagonal argument (compare e.g. [BW92] Chapter 4). Let us assume that $\succ$ is not well-founded on ${\cal F}$. We will show that this gives us a contradiction to the fact that the ordering $\succeq$ on ${\sf M}({\cal F})$ inducing $\succ$ is well-founded. Hence, let us suppose $f_{0}\succ f_{1}\succ\ldots\succ f_{k}\succ\ldots\;$, $k\in{\mathbb{N}}$ is a strictly descending chain in ${\cal F}$. Then we can construct a sequence of sets of pairs $\\{\\{(m_{t_{k}},g_{kn})\mid n\in{\mathbb{N}}\\}\mid k\in{\mathbb{N}}\\}$ recursively as follows: For $k=0$ let $m_{t_{0}}=\min_{\succeq}\\{{\sf HM}(f_{i})\mid i\in{\mathbb{N}}\\}$ which is well-defined since $\succeq$ is well-founded on ${\sf M}({\cal F})$. Now let $j\in{\mathbb{N}}$ be the least index such that we have $m_{t_{0}}={\sf HM}(f_{j})$. Then $m_{t_{0}}={\sf HM}(f_{j+n})$ holds for all $n\in{\mathbb{N}}$ and we can set $g_{0n}=f_{j+n}-{\sf HM}(f_{j+n})$, i.e., $m_{t_{0}}\succ{\sf HM}(g_{0n})$ for all $n\in{\mathbb{N}}$. For $k+1$ we let $m_{t_{k+1}}=\min_{\succeq}\\{{\sf HM}(g_{ki})\mid i\in{\mathbb{N}}\\}$ and again let $j\in{\mathbb{N}}$ be the least index such that $m_{t_{k+1}}={\sf HM}(g_{kj})$ holds, i.e., $m_{t_{k+1}}={\sf HM}(g_{k(j+n)})$ for all $n\in{\mathbb{N}}$. Again we set $g_{(k+1)n}=g_{k(j+n)}-{\sf HM}(g_{k(j+n)})$. Then the following statements hold for every $k\in{\mathbb{N}}$: 1. 1. For all monomials $m$ occuring in the polynomials $g_{kn}$, $n\in{\mathbb{N}}$, we have $m_{t_{k}}\succ m$. 2. 2. $g_{k0}\succ g_{k1}\succ\ldots\;$ is a strictly descending chain in ${\cal F}$. Hence we get that $m_{t_{0}}\succ m_{t_{1}}\succ\ldots\;$ is a strictly descending chain in ${\sf M}({\cal F})$ contradicting the fact that $\succeq$ is supposed to be well-founded on this set. q.e.d. Characterizations of ideal bases in terms of special standard representations they allow are mainly provided for polynomial rings over fields in the literature (compare [BW92] and Section 2.3). Hence we will first take a closer look at possible generalizations of these concepts to function rings over fields. #### 4.2.1 The Special Case of Function Rings over Fields Let ${\cal F}_{{\mathbb{K}}}$ be a function ring over a field ${\mathbb{K}}$. Remember that for a set $F$ of polynomials in ${\cal F}_{{\mathbb{K}}}$ every polynomial $g\in{\sf ideal}_{r}(F)$ has a representation of the form $g=\sum_{i=1}^{n}f_{i}\star h_{i},f_{i}\in F,h_{i}\in{\cal F}_{{\mathbb{K}}},n\in{\mathbb{N}}.$ However, such an arbitrary representation can contain monomials larger than ${\sf HM}(g)$ which are cancelled in the sum. A first idea of standard representations in the literature now is to represent $g$ as a sum of polynomial multiples $f_{i}\star h_{i}$ such that no cancellation of monomials larger than ${\sf HM}(g)$ takes place, i.e. ${\sf HM}(g)\succeq{\sf HM}(f_{i}\star h_{i})$. Hence in a first step we look at the following analogon of a definition of standard representations (compare [BW92], page 218): ###### Definition 4.2.4 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $g$ a non- zero polynomial in ${\sf ideal}_{r}(F)$. A representation of the form $\displaystyle g$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}f_{i}\star h_{i},f_{i}\in F,h_{i}\in{\cal F}_{{\mathbb{K}}},n\in{\mathbb{N}}$ (4.1) where additionally ${\sf HT}(g)\succeq{\sf HT}(f_{i}\star h_{i})$ holds for $1\leq i\leq n$ is called a (general) right standard representation of $g$ in terms of $F$. If every $g\in{\sf ideal}_{r}(F)\backslash\\{o\\}$ has such a representation in terms of $F$, then $F$ is called a (general) right standard basis of ${\sf ideal}_{r}(F)$. $\diamond$ What distinguishes an arbitrary representation from a (general) right standard representation is the fact that the former may contain polynomial multiples $f_{i}\star h_{i}$ with head terms ${\sf HT}(f_{i}\star h_{i})$ larger than the head term of the represented polynomial $g$. Therefore, in order to change an arbitrary representation into one fulfilling our additional condition (4.1) we have to deal with special sums of polynomials. ###### Definition 4.2.5 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $t$ an element in ${\cal T}$. Then we define the critical set ${\cal C}_{gr}(t,F)$ to contain all tuples of the form $(t,f_{1},\ldots,f_{k},h_{1},\ldots,h_{k})$, $k\in{\mathbb{N}}$, $f_{1},\ldots,f_{k}\in F$666As in the case of commutative polynomials, $f_{1},\ldots,f_{k}$ are not necessarily different polynomials from $F$., $h_{1},\ldots,h_{k}\in{\cal F}_{{\mathbb{K}}}$ such that 1. 1. ${\sf HT}(f_{i}\star h_{i})=t$, $1\leq i\leq k$, and 2. 2. $\sum_{i=1}^{k}{\sf HM}(f_{i}\star h_{i})=o$. We set ${\cal C}_{gr}(F)=\bigcup_{t\in{\cal T}}{\cal C}_{gr}(t,F)$. $\diamond$ Notice that for the sums of polynomial multiples in this definition we get ${\sf HT}(\sum_{i=1}^{k}f_{i}\star h_{i})\prec t$. This definition is motivated by the definition of syzygies of polynomials in commutative polynomial rings over rings. However, it differs from the original definition insofar as we need not have ${\sf HT}(f\star h)={\sf HT}({\sf HT}(f)\star{\sf HT}(h))$, i.e., we cannot localize the definition to the head monomials of the polynomials in $F$. Still we can characterize (general) right standard bases using this concept. ###### Theorem 4.2.6 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a (general) right standard basis of ${\sf ideal}_{r}(F)$ if and only if for every tuple $(t,f_{1},\ldots,f_{k},h_{1},\ldots,h_{k})$ in ${\cal C}_{gr}(F)$ the polynomial $\sum_{i=1}^{k}f_{i}\star h_{i}$ (i.e., the element in ${\cal F}_{{\mathbb{K}}}$ corresponding to this sum) has a (general) right standard representation with respect to $F$. Proof : In case $F$ is a (general) right standard basis, since these polynomials are all elements of ${\sf ideal}_{r}(F)$, they must have (general) right standard representations with respect to $F$. To prove the converse, it remains to show that every element in ${\sf ideal}_{r}(F)$ has a (general) right standard representation with respect to $F$. Hence, let $g=\sum_{j=1}^{m}f_{j}\star h_{j}$ be an arbitrary representation of a non-zero polynomial $g\in{\sf ideal}_{r}(F)$ such that $f_{j}\in F$, $h_{j}\in{\cal F}_{{\mathbb{K}}}$, $m\in{\mathbb{N}}$. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(f_{j}\star h_{j})\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $f_{j}\star h_{j}$ with head term $t$. Then $t\succeq{\sf HT}(g)$ and in case ${\sf HT}(g)=t$ this immediately implies that this representation is already a (general) right standard one. Else we proceed by induction on $t$. Without loss of generality let $f_{1},\ldots,f_{K}$ be the polynomials in the corresponding representation such that $t={\sf HT}(f_{i}\star h_{i})$, $1\leq i\leq K$. Then the tuple $(t,f_{1},\ldots,f_{K},h_{1},\ldots,h_{K})$ is in ${\cal C}_{gr}(F)$ and let $h=\sum_{i=1}^{K}f_{i}\star h_{i}$. We will now change our representation of $g$ in such a way that for the new representation of $g$ we have a smaller maximal term. Let us assume $h$ is not $o$777In case $h=o$, just substitute the empty sum for the representation of $h$ in the equations below.. By our assumption, $h$ has a (general) right standard representation with respect to $F$, say $\sum_{j=1}^{n}p_{j}\star q_{j}$, where $p_{j}\in F$, $q_{j}\in{\cal F}_{{\mathbb{K}}}$, $n\in{\mathbb{N}}$ and all terms occurring in the sum are bounded by $t\succ{\sf HT}(h)$ as $\sum_{i=1}^{K}{\sf HM}(f_{i}\star h_{i})=o$. This gives us: $\displaystyle g$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{K}f_{i}\star h_{i}+\sum_{i=K+1}^{m}f_{i}\star h_{i}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{n}p_{j}\star q_{j}+\sum_{i=K+1}^{m}f_{i}\star h_{i}$ which is a representation of $g$ where the maximal term of the involved polynomial multiples is smaller than $t$. q.e.d. Remember that by the distributivity law in ${\cal F}_{{\mathbb{K}}}$ any representation of a polynomial $g$ of the form $g=\sum_{i=1}^{n}f_{i}\star h_{i},f_{i}\in F,h_{i}\in{\cal F}_{{\mathbb{K}}},n\in{\mathbb{N}}$ can be converted into one of the form $g=\sum_{j=1}^{m}f_{j}\star m_{j},f_{j}\in F,m_{j}\in{\sf M}({\cal F}_{{\mathbb{K}}}),m\in{\mathbb{N}}.$ Now for polynomial rings the conversion of a (general right) standard representation from a sum of polynomial multiples into a sum of monomial multiples again results in a standard representation. This is due to the fact that the orderings used for the polynomial rings are compatible with multiplication. Now let us look at a second analogon to this kind of standard representations in our setting. ###### Definition 4.2.7 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $g$ a non- zero polynomial in ${\sf ideal}_{r}(F)$. A representation of the form $\displaystyle g$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}}),n\in{\mathbb{N}}$ (4.2) where additionally ${\sf HT}(g)\succeq{\sf HT}(f_{i}\star m_{i})$ holds for $1\leq i\leq n$ is called a right standard representation of $g$ in terms of $F$. If every $g\in{\sf ideal}_{r}(F)\backslash\\{o\\}$ has such a representation in terms of $F$, then $F$ is called a right standard basis of ${\sf ideal}_{r}(F)$. $\diamond$ If our ordering $\succ$ on ${\cal F}_{{\mathbb{K}}}$ is compatible with $\star$ we can conclude that the conversion of a general right standard representation into a sum involving only monomial multiples again results in a right standard representation as defined in Definition 4.2.7. But since in general the ordering and the multiplication are not compatible (review Example 4.2.1) a polynomial multiple $f\star h$ can contain monomials $m,m^{\prime}\in{\sf M}(f\star m_{j})$ where $h=\sum_{j=1}^{n}m_{j}$ such that $m$ and $m^{\prime}$ are larger than ${\sf HM}(f\star h)$ and $m=m^{\prime}$. Hence just applying the distributivity to a sum of polynomial multiples no longer changes a standard representation as defined in Definition 4.2.4 into one as defined in Definition 4.2.7. Remember that this was true for polynomial rings over fields where both definitions are equivalent. Let us look at the monoid ring ${\mathbb{Q}}[{\cal M}]$ where ${\cal M}$ is the monoid presented by $(\\{a,b,c\\};ab=a)$. Moreover, let $\succ$ be the length-lexicographical ordering induced by the precedence $c\succ b\succ a$. Then for the polynomials $f=ca+1$, $h=b^{2}-b\in{\mathbb{Q}}[{\cal M}]$ we get ${\sf HT}(f\star b^{2})={\sf HT}(ca+b^{2})=ca$ and ${\sf HT}(f\star b)={\sf HT}(ca+b)=ca$. On the other hand ${\sf HT}(f\star h)={\sf HT}(ca+b^{2}-ca-b)={\sf HT}(b^{2}-b)=b^{2}$. Hence for the polynomial $g=b^{2}-b$ the polynomial multiple $f\star h$ is a general right standard representation as defined in Definition 4.2.4 while the sum of monomial multiples $f\star b^{2}-f\star b$ is no right standard representation as defined in Definition 4.2.7. We can even state that $g$ has no right standard representation in terms of the polynomial $f$. Now as our aim is to link standard representations of polynomials to reduction relations, a closer inspection of the concept of general right standard representations shows that a reduction relation related to them has to involve polynomial multiples for defining the reduction steps. Right standard representations can also be linked to special instances of such reduction relations but are traditionally linked to reduction relations involving monomial multiples. There is no example known from the literature where reduction relations involving polynomial multiples gain real advantages over reduction relations involving monomial multiples only888Examples where reduction relations involving polynomial multiples are studied for the original case of Gröbner bases in commutative polynomial rings can be found in [Tri78, Zac78].. Therefore we will restrict our attention to right standard representations as presented in Definition 4.2.7. Again, in order to change an arbitrary representation into one fulfilling our additional condition (4.2) of Definition 4.2.7 we have to deal with special sums of polynomials. ###### Definition 4.2.8 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $t$ an element in ${\cal T}$. Then we define the critical set ${\cal C}_{r}(t,F)$ to contain all tuples of the form $(t,f_{1},\ldots,f_{k},m_{1},\ldots,m_{k})$, $k\in{\mathbb{N}}$, $f_{1},\ldots,f_{k}\in F$999As in the case of commutative polynomials, $f_{1},\ldots,f_{k}$ are not necessarily different polynomials from $F$., $m_{1},\ldots,m_{k}\in{\sf M}({\cal F})$ such that 1. 1. ${\sf HT}(f_{i}\star m_{i})=t$, $1\leq i\leq k$, and 2. 2. $\sum_{i=1}^{k}{\sf HM}(f_{i}\star m_{i})=o$. We set ${\cal C}_{r}(F)=\bigcup_{t\in{\cal T}}{\cal C}_{r}(t,F)$. $\diamond$ As before, we can characterize right standard bases using this concept. ###### Theorem 4.2.9 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a right standard basis of ${\sf ideal}_{r}(F)$ if and only if for every tuple $(t,f_{1},\ldots,f_{k},m_{1},\ldots,m_{k})$ in ${\cal C}_{r}(F)$ the polynomial $\sum_{i=1}^{k}f_{i}\star m_{i}$ (i.e., the element in ${\cal F}$ corresponding to this sum) has a right standard representation with respect to $F$. Proof : In case $F$ is a right standard basis, since these polynomials are all elements of ${\sf ideal}_{r}(F)$, they must have right standard representations with respect to $F$. To prove the converse, it remains to show that every element in ${\sf ideal}_{r}(F)$ has a right standard representation with respect to $F$. Hence, let $g=\sum_{j=1}^{m}f_{j}\star m_{j}$ be an arbitrary representation of a non-zero polynomial $g\in{\sf ideal}_{r}(F)$ such that $f_{j}\in F$, $m_{j}\in{\sf M}({\cal F}_{{\mathbb{K}}})$, $m\in{\mathbb{N}}$. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(f_{j}\star m_{j})\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $f_{j}\star m_{j}$ with head term $t$. Then $t\succeq{\sf HT}(g)$ and in case ${\sf HT}(g)=t$ this immediately implies that this representation is already a right standard one. Else we proceed by induction on $t$. Without loss of generality let $f_{1},\ldots,f_{K}$ be the polynomials in the corresponding representation such that $t={\sf HT}(f_{i}\star m_{i})$, $1\leq i\leq K$. Then the tuple $(t,f_{1},\ldots,f_{K},m_{1},\ldots,m_{K})$ is in ${\cal C}_{r}(F)$ and let $h=\sum_{i=1}^{K}f_{i}\star m_{i}$. We will now change our representation of $g$ in such a way that for the new representation of $g$ we have a smaller maximal term. Let us assume $h$ is not $o$101010In case $h=o$, just substitute the empty sum for the representation of $h$ in the equations below.. By our assumption, $h$ has a right standard representation with respect to $F$, say $\sum_{j=1}^{n}h_{j}\star l_{j}$, where $h_{j}\in F$, $l_{j}\in{\sf M}({\cal F}_{{\mathbb{K}}})$, $n\in{\mathbb{N}}$ and all terms occurring in the sum are bounded by $t\succ{\sf HT}(h)$ as $\sum_{i=1}^{K}{\sf HM}(f_{i}\star m_{i})=o$. This gives us: $\displaystyle g$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{K}f_{i}\star m_{i}+\sum_{i=K+1}^{m}f_{i}\star m_{i}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{n}h_{j}\star l_{j}+\sum_{i=K+1}^{m}f_{i}\star m_{i}$ which is a representation of $g$ where the maximal term of the involved monomial multiples is smaller than $t$. q.e.d. For commutative polynomial rings over fields standard bases are in fact Gröbner bases. Remember that in algebraic terms a set $F$ is a Gröbner basis of the ideal ${\sf ideal}(F)$ it generates if and only if ${\sf HT}({\sf ideal}(F))=\\{t\star w\mid t\in{\sf HT}(F),w\mbox{ a term}\\}$ (compare Definition 2.3.12). The localization to the set of head terms only is possible as the ordering and multiplication are compatible, i.e. ${\sf HT}(f\star w)={\sf HT}(f)\star w$ for any $f\in F$ and any term $w$. Then of course if every $g\in{\sf ideal}(F)$ has a standard representation in terms of $F$ we immediately get that ${\sf HT}(g)={\sf HT}(f\star w)={\sf HT}(f)\star w$ for some $f\in F$ and some term $w$. Moreover, for any reduction relation based on divisibility of terms we get that $g$ is reducible at its head monomial by this polynomial $f$. This of course corresponds to the second definition of Gröbner bases in rewriting terms – a set $F$ is a Gröbner basis of the ideal it generates if and only if the reduction relation $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$ associated to the polynomials in $F$ is confluent111111The additional properties of capturing the ideal congruence and being terminating required by Definition 3.1.4 trivially hold for polynomial rings over fields. (compare Definition 2.3.8). Central in both definitions of Gröbner bases is the idea of “dividing” terms. Important in this context is the fact that divisors are smaller than the terms they divide with respect to term orderings and moreover the ordering on the terms is stable under multiplication with monomials. The algebraic definition states that every head term of a polynomial in ${\sf ideal}(G)$ has a head term of a polynomial in $G$ as a divisor121212 When generalizing this definition to our setting of function rings we have to be very careful as in reality this implies that every polynomial in the ideal is reducible to zero which is the definition of a weak Gröbner basis (compare Definition 3.1.2). Gröbner bases and weak Gröbner bases coincide in polynomial rings over fields due to the Translation Lemma (compare Lemma 2.3.9 (2)).. Similarly the reduction relation is based on divisibility of terms (compare Definition 2.3.7). The stability of the ordering under multiplication is important for the correctness of these characterizations of Gröbner bases since it allows finite localizations for the test sets to s-polynomials (Lemma 2.3.9 is central in this context). In our context now the ordering $\succ$ and the multiplication $\star$ on ${\cal F}_{{\mathbb{K}}}$ in general are not compatible. Hence, a possible algebraic definition of Gröbner bases and a definition of a reduction relation related to right standard representations must involve the whole polynomials and not only their head terms. ###### Definition 4.2.10 A subset $F$ of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ is called a weak right Gröbner basis of ${\sf ideal}_{r}(F)$ if ${\sf HT}({\sf ideal}_{r}(F)\backslash\\{o\\})={\sf HT}(\\{f\star m\mid f\in F,m\in{\sf M}({\cal F}_{{\mathbb{K}}})\\}\backslash\\{o\\})$. $\diamond$ Instead of considering multiples of head terms of the generating set $F$ we look at head terms of monomial multiples of polynomials in $F$. In the setting of function rings over fields, in order to localize the definitions of standard representations and weak Gröbner bases to head terms instead of head monomials and show their equivalence we have to view ${\cal F}$ as a vector space with scalars from ${\mathbb{K}}$. We define a natural left scalar multiplication $\cdot:{\mathbb{K}}\times{\cal F}\longrightarrow{\cal F}$ by associating to $\alpha\in{\mathbb{K}}$ and $f\in{\cal F}$ the function in ${\cal F}$, denoted by $\alpha\cdot f$, which has support ${\sf supp}(\alpha\cdot f)\subseteq{\sf supp}(f)$ and values $(\alpha\cdot f)(t)=\alpha\cdot f(t)$ for $t\in{\sf supp}(f)$. Notice that if $\alpha\neq 0$ we have ${\sf supp}(\alpha\cdot f)={\sf supp}(f)$. Similarly, we can define a natural right scalar multiplication $\cdot:{\cal F}\times{\mathbb{K}}\longrightarrow{\cal F}$ by associating to $\alpha\in{\mathbb{K}}$ and $f\in{\cal F}$ the function in ${\cal F}$, denoted by $f\cdot\alpha$, which has support ${\sf supp}(f\cdot\alpha)\subseteq{\sf supp}(f)$ and values $(f\cdot\alpha)(t)=f(t)\cdot\alpha$ for $t\in{\sf supp}(f)$. Since ${\mathbb{K}}$ is associative we have $\displaystyle((\alpha\cdot f)\cdot\beta)(t)$ $\displaystyle=$ $\displaystyle(\alpha\cdot f)(t)\cdot\beta$ $\displaystyle=$ $\displaystyle(\alpha\cdot f(t))\cdot\beta$ $\displaystyle=$ $\displaystyle\alpha\cdot(f(t)\cdot\beta)$ $\displaystyle=$ $\displaystyle\alpha\cdot((f\cdot\beta)(t))$ $\displaystyle=$ $\displaystyle(\alpha\cdot(f\cdot\beta))(t)$ and we will write $\alpha\cdot f\cdot\beta$. Monomials can be represented as $m=\alpha\cdot t$ where ${\sf supp}(m)=\\{t\\}$ and $m(t)=\alpha$. Additionally we have to state how scalar multiplication and ring multiplication are compatible. Remember that we have introduced the elements of our function rings as formal sums of monomials. We want to treat these objects similar to those occurring in the examples known from the literature. In particular we want to achieve that multiplication in ${\cal F}_{{\mathbb{K}}}$ can be specified by defining a multiplication on the terms and lifting it to the monomials. Hence we require the following equations $(\alpha\cdot f)\star g=\alpha\cdot(f\star g)$ and $f\star(g\cdot\alpha)=(f\star g)\cdot\alpha$ to hold131313Then of course since ${\mathbb{K}}$ is Abelian we have $(\alpha\cdot f)\star g=\alpha\cdot(f\star g)=f\star(\alpha\cdot g)=f\star(g\cdot\alpha)=(f\star g)\cdot\alpha$.. These equations are valid in the examples from the literature studied here. The condition of course then implies that multiplication in ${\cal F}_{{\mathbb{K}}}$ can be specified by knowing $\star:{\cal T}\times{\cal T}\longrightarrow{\cal F}_{{\mathbb{K}}}$. This follows as for $\alpha,\beta\in{\mathbb{K}}$ and $t,s\in{\cal T}$ we have $\displaystyle(\alpha\cdot t)\star(\beta\cdot s)$ $\displaystyle=$ $\displaystyle\alpha\cdot(t\star(\beta\cdot s))$ $\displaystyle=$ $\displaystyle\alpha\cdot(t\star(s\cdot\beta))$ $\displaystyle=$ $\displaystyle\alpha\cdot(t\star s)\cdot\beta$ $\displaystyle=$ $\displaystyle(\alpha\cdot\beta)\cdot(t\star s).$ If ${\cal F}$ contains a unit element ${\bf 1}$ the field can be embedded into ${\cal F}$ by $\alpha\longmapsto\alpha\cdot{\bf 1}$. Then for $\alpha\in{\mathbb{K}}$ and $f\in{\cal F}_{{\mathbb{K}}}$ the equations $\alpha\cdot f=(\alpha\cdot{\bf 1})\star f$ and $f\cdot\alpha=f\star(\alpha\cdot{\bf 1})$ hold. Moreover, as ${\mathbb{K}}$ is Abelian $\alpha\cdot f\cdot\beta=\alpha\cdot\beta\cdot f$ for any $\alpha,\beta\in{\mathbb{K}}$, $f\in{\cal F}_{{\mathbb{K}}}$. In the next lemma we show that in fact both characterizations of special bases, right standard bases and weak Gröbner bases, coincide as in the case of polynomial rings over fields. ###### Lemma 4.2.11 Let $F$ be a subset of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a right standard basis if and only if it is a weak right Gröbner basis. Proof : Let us first assume that $F$ is a right standard basis, i.e., every polynomial $g$ in ${\sf ideal}_{r}(F)$ has a right standard representation with respect to $F$. In case $g\neq o$ this implies the existence of a polynomial $f\in F$ and a monomial $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HT}(g)={\sf HT}(f\star m)$. Hence ${\sf HT}(g)\in{\sf HT}(\\{f\star m\mid m\in{\sf M}({\cal F}_{{\mathbb{K}}}),f\in F\\}\backslash\\{o\\})$. As the converse, namely ${\sf HT}(\\{f\star m\mid m\in{\sf M}({\cal F}_{{\mathbb{K}}}),f\in F\\}\backslash\\{o\\})\subseteq{\sf HT}({\sf ideal}_{r}(F)\backslash\\{o\\})$ trivially holds, $F$ then is a weak right Gröbner basis. Now suppose that $F$ is a weak right Gröbner basis and again let $g\in{\sf ideal}_{r}(F)$. We have to show that $g$ has a right standard representation with respect to $F$. This will be done by induction on ${\sf HT}(g)$. In case $g=o$ the empty sum is our required right standard representation. Hence let us assume $g\neq o$. Since then ${\sf HT}(g)\in{\sf HT}({\sf ideal}_{r}(F)\backslash\\{o\\})$ by the definition of weak right Gröbner bases we know there exists a polynomial $f\in F$ and a monomial $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HT}(g)={\sf HT}(f\star m)$. Then there exists a monomial $\tilde{m}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HM}(g)={\sf HM}(f\star\tilde{m})$, namely141414Notice that this step requires that we can view ${\cal F}_{{\mathbb{K}}}$ as a vector space. In order to get a similar result without introducing vector spaces we would have to use a different definition of weak right Gröbner bases. E.g. requiring that ${\sf HM}({\sf ideal}_{r}(F)\backslash\\{o\\})={\sf HM}(\\{f\star m\mid f\in F,m\in{\sf M}({\cal F}_{{\mathbb{K}}})\\}\backslash\\{o\\}\\})$ would be a possibility. However, then no localization of critical situations to head terms is possible, which is the advantage of having a field as coefficient domain. $\tilde{m}=({\sf HC}(g)\cdot{\sf HC}(f\star m)^{-1})\cdot m)$. Let $g_{1}=g-f\star\tilde{m}$. Then ${\sf HT}(g)\succ{\sf HT}(g_{1})$ implies the existence of a right standard representation for $g_{1}$ which can be added to the multiple $f\star\tilde{m}$ to give the desired right standard representation of $g$. q.e.d. Inspecting this proof closer we get the following corollary. ###### Corollary 4.2.12 Let a subset $F$ of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ be a weak right Gröbner basis. Then every $g\in{\sf ideal}_{r}(F)$ has a right standard representation in terms of $F$ of the form $g=\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}}),n\in{\mathbb{N}}$ such that ${\sf HM}(g)={\sf HM}(f_{1}\star m_{1})$ and ${\sf HT}(f_{1}\star m_{1})\succ{\sf HT}(f_{2}\star m_{2})\succ\ldots\succ{\sf HT}(f_{n}\star m_{n})$. Notice that we hence get stronger representations as specified in Definition 4.2.7 for the case that the set $F$ is a weak right Gröbner basis or a right standard basis. In the literature Gröbner bases are linked to reduction relations. These reduction relations in general then correspond to the respective standard representations as follows: if $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, then the monomial multiples involved in the respective reduction steps add up to a standard representation of $g$ in terms of $F$. One possible reduction relation related to right standard representations as defined in Definition 4.2.7 is called strong reduction151515Strong reduction has been studied extensively for monoid rings in [Rei95]. where a monomial $m_{1}$ is reducible by some polynomial $f$, if there exists some monomial $m_{2}$ such that $m_{1}={\sf HM}(f\star m_{2})$. Notice that such a reduction step eliminates the occurence of the term ${\sf HT}(m_{1})$ in the resulting reductum $m_{1}-f\star m_{2}$. When generalizing this reduction relation to function rings we can no longer localize the reduction step to checking whether ${\sf HM}(f)$ divides $m_{1}$, as now the whole polynomial is involved in the reduction step. We can no longer conclude that ${\sf HM}(f)$ divides $m_{1}$ but only that $m_{1}={\sf HM}(f\star m_{2})$. Our definition of weak right Gröbner bases using the condition ${\sf HT}({\sf ideal}_{r}(F)\backslash\\{o\\})$ $={\sf HT}(\\{f\star m\mid f\in F,m\in{\sf M}({\cal F}_{{\mathbb{K}}})\\}\backslash\\{o\\})$ in Definition 4.2.10 corresponds to this problem that in many cases orderings on ${\cal T}$ are not compatible with the multiplication $\star$. Let us review Example 4.2.1 where the ordering $\succeq$ induced by $x\succ 1$ on terms respectively monomials is well-founded but in general not compatible with multiplication, due to the algebraic structure of ${\cal T}$. There for the polynomial $f=x+1$ and the term $x$ we get ${\sf HM}(f\star x)=x$ while ${\sf HM}(f)\star x=1$. Behind this phenomenon lies the fact that the definition of “divisors” arising from the algebraic characterization of weak Gröbner bases in the context of function rings does not have the same properties as divisors in polynomial rings. One such important property is that divisors are smaller with respect to the ordering on terms and that this ordering is transitive. Hence if $t_{1}$ is a divisor of $t_{2}$ and $t_{2}$ is a divisor of $t_{3}$ then $t_{1}$ is also a divisor of $t_{3}$. This is the basis of localizations when checking for the Gröbner basis property in polynomial rings over fields (compare Lemma 2.3.9). Unfortunately this is no longer true for function rings in general. Now $m_{1}\in{\sf HM}({\sf ideal}_{r}(G))$ implies the existence of $m_{2}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HM}(f\star m_{2})=m_{1}$. Reviewing the previous example we see that for $f=x+1$, $m_{2}=x$ and $m_{1}={\sf HM}(f)=x$ we get ${\sf HM}(f\star m_{2})={\sf HM}((x+1)\star x)=x$, i.e. ${\sf HM}(f\star m_{2})$ divides $m_{1}$. On the other hand $m_{1}=x$ divides $1$ as $x\star x=1$. But ${\sf HM}({\sf HM}(f\star m_{2})\star x)=1$ while ${\sf HM}(f\star m_{2}\star x)=x$, i.e. the head monomial of the multiple involving the polynomial $f\star m_{2}$ does not divide $1$. Notice that even if we restrict the concept of right divisors to monomials only we do not get transitivity. We are interested when for some monomials $m_{1},m_{2},m_{3}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ the facts that $m_{1}$ divides $m_{2}$ and $m_{2}$ divides $m_{3}$ imply that $m_{1}$ divides $m_{3}$. Let $m,m^{\prime}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HM}(m_{1}\star m)=m_{2}$ and ${\sf HM}(m_{2}\star m^{\prime})=m_{3}$. Then $m_{3}={\sf HM}(m_{2}\star m^{\prime})={\sf HM}({\sf HM}(m_{1}\star m)\star m^{\prime})$. When does this equal ${\sf HM}(m_{1}\star m\star m^{\prime})$ or even ${\sf HM}(m_{1}\star{\sf HM}(m\star m^{\prime}))$? Obviously if we have $\star:{\sf M}({\cal F}_{{\mathbb{K}}})\times{\sf M}({\cal F}_{{\mathbb{K}}})\mapsto{\sf M}({\cal F}_{{\mathbb{K}}})$, which is true for the Examples 4.1.3, 4.1.4 and 4.1.5, this is true. However if multiplication of monomials results in polynomials we are in trouble. Let us look at the skew-polynomial ring ${\mathbb{Q}}[X_{1},X_{2},X_{3}]$, $X_{1}\succ X_{2}\succ X_{3}$, defined in Example 4.1.6, i.e. $X_{2}\star X_{1}=X_{2}+X_{3}$,$X_{3}\star X_{1}=X_{1}X_{3}$, $X_{3}\star X_{2}=X_{2}X_{3}$ and $X_{i}\star X_{j}=X_{i}X_{j}$ for $i<j$. Then from the fact that $X_{2}$ divides $X_{2}$ we get ${\sf HM}(X_{2}\star X_{1})=X_{2}$ and since again $X_{2}$ divides $X_{2}$, ${\sf HM}({\sf HM}(X_{2}\star X_{1})\star X_{1})={\sf HM}(X_{2}\star X_{1})=X_{2}$. But ${\sf HM}(X_{2}\star X_{1}\star X_{1})={\sf HM}(X_{1}X_{3}+X_{2}+X_{3})=X_{1}X_{3}$. Next we will show how using a restricted set of divisors only will enable some sort of transitivity. To establish a certain kind of compatibility for the ordering $\succeq$ and the multiplication $\star$, additional requirements can be added. One way to do this is by giving an additional ordering on ${\cal T}$ which is in some sense weaker than $\succeq$ but adds more information on compatibility with right multiplication. Examples from the literature, where this technique is successfully applied, include special monoid and group rings (see e.g. [Rei95, MR98a, MR98d]). There restrictions of the respective orderings on the monoid or group elements are of syntactical nature involving the presentation of the monoid or group (e.g. prefix orderings of various kinds for commutative monoids and groups, free groups and polycyclic groups). ###### Definition 4.2.13 We will call an ordering $\geq$ on ${\cal T}$ a right reductive restriction of the ordering $\succeq$ or simply right reductive, if the following hold: 1. 1. $t\geq s$ implies $t\succeq s$ for $t,s\in{\cal T}$. 2. 2. $\geq$ is a partial ordering on ${\cal T}$ which is compatible with multiplication $\star$ from the right in the following sense: if for $t,t_{1},t_{2},w\in{\cal T}$, $t_{2}\geq t_{1}$, $t_{1}\succ t$ and $t_{2}={\sf HT}(t_{1}\star w)$ hold, then $t_{2}\succ t\star w$. $\diamond$ Notice that if $\succeq$ is a partial well-founded ordering on ${\cal T}$ so is $\geq$. We can now distinguish special “divisors” of monomials: For $m_{1},m_{2}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ we call $m_{1}$ a stable left divisor of $m_{2}$ if and only if ${\sf HT}(m_{2})\geq{\sf HT}(m_{1})$ and there exists $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that $m_{2}={\sf HM}(m_{1}\star m)$. Then $m$ is called a stable right multiplier of $m_{1}$. If ${\cal T}$ contains a unit element161616I.e. ${\bf 1}\star t=t\star{\bf 1}=t$ for all $t\in{\cal T}$. ${\bf 1}$ and ${\bf 1}\preceq t$ for all terms $t\in{\cal T}$ this immediately171717As there are no terms smaller than ${\bf 1}$ the second condition of Definition 4.2.13 trivially holds. implies ${\bf 1}\leq t$ and hence ${\bf 1}$ is a stable divisor of any monomial $m$. It remains to show that stable division is also transitive. For three monomials $m_{1},m_{2},m_{3}\in{\sf M}({\cal F})$ let $m_{1}$ be a stable divisor of $m_{2}$ and $m_{2}$ a stable divisor of $m_{3}$. Then there exist monomials $m,m^{\prime}\in{\sf M}({\cal F})$ such that $m_{2}={\sf HM}(m_{1}\star m)$ with ${\sf HT}(m_{2})\geq{\sf HT}(m_{1})$ and $m_{3}={\sf HM}(m_{2}\star m^{\prime})$ with ${\sf HT}(m_{3})\geq{\sf HT}(m_{2})$. Let us have a look at the monomial ${\sf HM}({\sf HM}(m_{1}\star m)\star m^{\prime})$. Remember how on page 4.2.12 we have seen that the case $m_{1}\star m\in{\sf M}({\cal F})$ is not critical as then we immediately have that this monomial equals ${\sf HM}(m_{1}\star m\star m^{\prime})={\sf HM}(m_{1}\star{\sf HM}(m\star m^{\prime}))$. Hence let us assume that $m_{1}\star m\not\in{\sf M}({\cal F})$. Then for all terms $s\in{\sf T}(m_{1}\star m)\backslash{\sf HT}(m_{1}\star m)$ we know $s\prec{\sf HT}(m_{1}\star m)={\sf HT}(m_{2})$. Moreover ${\sf HT}(m_{3})\geq{\sf HT}(m_{2})$ and ${\sf HT}(m_{3})={\sf HT}({\sf HT}(m_{2})\star{\sf HT}(m^{\prime}))$ then implies ${\sf HT}(m_{3})\succ{\sf HT}(s\star{\sf HT}(m^{\prime}))$ and hence ${\sf HM}({\sf HM}(m_{1}\star m)\star m^{\prime})={\sf HM}(m_{1}\star m\star m^{\prime})$. In both cases now ${\sf HT}(m_{3})\geq{\sf HT}(m_{1})$. However, we cannot conclude that ${\sf HM}(m_{1}\star m\star m^{\prime})={\sf HM}(m_{1}\star{\sf HM}(m\star m^{\prime}))$. Still $m_{1}$ is a stable right divisor of $m_{3}$ as in case $m\star m^{\prime}$ is a polynomial there exists some monomial $\tilde{m}$ in this polynomial such that ${\sf HM}(m_{1}\star m\star m^{\prime})={\sf HM}(m_{1}\star\tilde{m})$. The intention of restricting the ordering is that now, if ${\sf HT}(m_{2})\geq{\sf HT}(m_{1})$ and $m_{2}=m_{1}\star m$, then for all terms $t$ with ${\sf HT}(m_{1})\succ t$ we then can conclude ${\sf HT}(m_{2})\succ{\sf HT}(t\star m)$, which will be used to localize the multiple ${\sf HT}(m_{1}\star m)$ to ${\sf HT}(m_{1})$ achieving an equivalent to the properties of “divisors” in the case of commutative polynomial rings. Under certain conditions reduction relations based on this divisibility property for terms will have the stability properties we desire. On the other hand, restricting the choice of divisors in this way will lead to reduction relations which in general no longer capture the respective right ideal congruences181818Prefix reduction for monoid rings is an example where the right ideal congruence is lost. See e.g. [MR98d] for more on this topic.. ###### Example 4.2.14 In Example 4.1.4 of a commutative polynomial ring we can state a reductive restriction of any term ordering by $t\geq s$ for two terms $t$ and $s$ if and only if $s$ divides $t$ as a term, i.e. for $t=X_{1}^{i_{1}}\ldots X_{n}^{i_{n}}$, $s=X_{1}^{j_{1}}\ldots X_{n}^{j_{n}}$ we have $j_{l}\leq i_{l}$, $1\leq l\leq n$. The same is true for skew-polynomial rings as defined by Kredel in his PhD thesis [Kre93]. The situation changes if for the defining equations of skew-polynomial rings, $X_{j}\star X_{i}=c_{ij}\cdot X_{i}X_{j}+p_{ij}$ where $i<j$, $p_{ij}\prec X_{i}X_{j}$, we allow $c_{ij}=0$. Then other restrictions of the ordinary term orderings have to be considered due to the possible vanishing of head terms. Let $X_{2}\star X_{1}=X_{1},X_{3}\star X_{1}=X_{1}X_{3},X_{3}\star X_{2}=X_{2}X_{3}$ and $\succ$ a term ordering with precedence $X_{3}\succ X_{2}\succ X_{1}$. Then, although $X_{2}\succ X_{1}$, as $X_{2}\star(X_{1}X_{2})=X_{1}X_{2}$ and $X_{1}\star(X_{1}X_{2})=X_{1}^{2}X_{2}\succ X_{1}X_{2}$, we get $X_{2}\star(X_{1}X_{2})\prec X_{1}\star(X_{1}X_{2})$. Hence, since $X_{2}$ is a divisor of $X_{1}X_{2}$ as a term, the classical restriction for polynomial rings no longer holds as $X_{2}$ is no stable divisor of $X_{1}X_{2}$. For these cases the restriction to $u<v$ if and only if $u$ is a prefix of $v$ as a word will work. Then we know that for the respective term $w$ with $u\star w=v$ multiplication is just concatenation of $u$ and $w$ as words and hence for all $t\prec u$ the result of $t\star w$ is again smaller than $u\star w$. $\diamond$ Let us continue with algebraic consequences related to the right reductive restriction of our ordering by distinguishing special standard representations. Notice that for standard representations in commutative polynomial rings we already have that ${\sf HT}(g)={\sf HT}(f_{i}\star m_{i})$ implies ${\sf HT}(g)={\sf HT}(f_{i})\star{\sf HT}(m_{i})$ and for all $t\prec{\sf HT}(f_{i})$ we have $t\star w\prec{\sf HT}(f_{i})\star w$ for any term $w$. In the setting of function rings an analogon to the latter property now can be achieved by restricting the monomial multiples in the representation to stable ones. Herefore we have different possibilities to incorporate these restrictions into the condition ${\sf HT}(g)\succeq{\sf HT}(f_{i}\star m_{i})$ of Definition 2.3.4 and Definition 4.2.7. The most general one is to require ${\sf HT}(g)={\sf HT}(f_{1}\star m_{1})={\sf HT}({\sf HT}(f_{1})\star m_{1})\geq{\sf HT}(f_{1})$ and ${\sf HT}(g)\succeq{\sf HT}(f_{i}\star m_{i})$ for all $2\leq i\leq n$. Then a representation of $g$ can contain further monomial multiples $f_{j}\star m_{j}$, $2\leq j\leq n$ with ${\sf HT}(g)={\sf HT}(f_{j}\star m_{j})$ not fullfilling the restriction on the first multiple of $f_{1}$. Hence when defining critical situations we have to look at the same set as in Definition 4.2.8. Another generalization is to demand ${\sf HT}(g)={\sf HT}(f_{1}\star m_{1})={\sf HT}({\sf HT}(f_{1})\star m_{1})\geq{\sf HT}(f_{1})$ and ${\sf HT}(g)\succeq{\sf HT}(f_{i}\star m_{i})={\sf HT}({\sf HT}(f_{i})\star m_{i})\geq{\sf HT}(f_{i})$ for all $2\leq i\leq n$. Then critical situations can be localized to stable multiplers. But we can also give a weaker analogon as follows: ###### Definition 4.2.15 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $g$ a non- zero polynomial in ${\sf ideal}_{r}(F)$. A representation of the form $g=\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}}),n\in{\mathbb{N}}$ such that ${\sf HT}(g)={\sf HT}(f_{i}\star m_{i})={\sf HT}({\sf HT}(f_{i})\star m_{i})\geq{\sf HT}(f_{i})$ for $1\leq i\leq k$, for some $k\geq 1$, and ${\sf HT}(g)\succ{\sf HT}(f_{i}\star m_{i})$ for $k<i\leq n$ is called a right reductive standard representation in terms of $F$. $\diamond$ Notice that we restrict the possible multipliers to stable ones if the monomial multiple has the same head term as $g$, i.e. contributes to the head term of $g$. For definitions sake we will let the empty sum be the right reductive standard representation of $o$. The idea behind right reductive standard representations is that for an appropriate definition of a reduction relation based now on stable divisors such representations will again allow a reduction step to take place at the head monomial. In case we have $\star:{\cal T}\times{\cal T}\longrightarrow{\cal T}$ we can rephrase the condition in Definition 4.2.15 to ${\sf HT}(g)={\sf HT}(f_{i}\star m_{i})={\sf HT}(f_{i})\star{\sf HT}(m_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq k$. ###### Definition 4.2.16 A set $F\subseteq{\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ is called a right reductive standard basis (with respect to the reductive ordering $\geq$) of ${\sf ideal}_{r}(F)$ if every polynomial $f\in{\sf ideal}_{r}(F)$ has a right reductive standard representation in terms of $F$. $\diamond$ Again, in order to change an arbitrary representation into one fulfilling our additional condition of Definition 4.2.15 we have to deal with special sums of polynomials. ###### Definition 4.2.17 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $t$ an element in ${\cal T}$. Then we define the critical set ${\cal C}_{rr}(t,F)$ to contain all tuples of the form $(t,f_{1},\ldots,f_{k},m_{1},\ldots,m_{k})$, $k\in{\mathbb{N}}$, $f_{1},\ldots,f_{k}\in F$191919As in the case of commutative polynomials, $f_{1},\ldots,f_{k}$ are not necessarily different polynomials from $F$., $m_{1},\ldots,m_{k}\in{\sf M}({\cal F})$ such that 1. 1. ${\sf HT}(f_{i}\star m_{i})={\sf HT}({\sf HT}(f_{i})\star m_{i})=t$, $1\leq i\leq k$, 2. 2. ${\sf HT}(f_{i}\star m_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq k$, and 3. 3. $\sum_{i=1}^{k}{\sf HM}(f_{i}\star m_{i})=o$. We set ${\cal C}_{rr}(F)=\bigcup_{t\in{\cal T}}{\cal C}_{rr}(t,F)$. $\diamond$ Unfortunately, in contrary to the characterization of right standard bases in Theorem 4.2.9 these critical situations will not be sufficient to characterize right reductive standard bases. To see this let us consider the following example: ###### Example 4.2.18 Let us recall the description of the free group ring in Example 4.1.7 with ${\cal T}=\\{a^{i},b^{i},1\mid i\in{\mathbb{N}}^{+}\\}$ and let $\succeq$ be the ordering induced by the length-lexicographical odering on ${\cal T}$ resulting from the precedence $a\succ b$. Then the set consisting of the polynomial $a+1$ does not give rise to non- trivial critical situations, but still is no right reductive standard basis as the polynomial $b+1\in{\sf ideal}_{r}(\\{a+1\\})$ has no right reductive standard representation with respect to $a+1$. $\diamond$ However, the failing situation $b+1=(a+1)\star b$ described in Example 4.2.18 describes the only kind of additional critical situations which have to be resolved in order to characterize right reductive standard bases. ###### Theorem 4.2.19 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a right reductive standard basis of ${\sf ideal}_{r}(F)$ if and only if 1. 1. for every $f\in F$ and every $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ the multiple $f\star m$ has a right reductive standard representation in terms of $F$, 2. 2. for every tuple $(t,f_{1},\ldots,f_{k},m_{1},\ldots,m_{k})$ in ${\cal C}_{rr}(F)$ the polynomial $\sum_{i=1}^{k}f_{i}\star m_{i}$ (i.e., the element in ${\cal F}$ corresponding to this sum) has a right reductive standard representation with respect to $F$. Proof : In case $F$ is a right reductive standard basis, since these polynomials are all elements of ${\sf ideal}_{r}(F)$, they must have right reductive standard representations with respect to $F$. To prove the converse, it remains to show that every element in ${\sf ideal}_{r}(F)$ has a right reductive standard representation with respect to $F$. Hence, let $g=\sum_{j=1}^{m}f_{j}\star m_{j}$ be an arbitrary representation of a non-zero polynomial $g\in{\sf ideal}_{r}(F)$ such that $f_{j}\in F$, and $m_{j}\in{\sf M}({\cal F}_{{\mathbb{K}}})$. By our first statement every such monomial multiple $f_{j}\star m_{j}$ has a right reductive standard representation in terms of $F$ and we can assume that all multiples are replaced by them. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(f_{j}\star m_{j})\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $f_{j}\star m_{j}$ with head term $t$. Then for each multiple $f_{j}\star m_{j}$ with ${\sf HT}(f_{j}\star m_{j})=t$ we know that ${\sf HT}(f_{j}\star m_{j})={\sf HT}({\sf HT}(f_{j})\star m_{j})\geq{\sf HT}(f_{j})$ holds. Then $t\succeq{\sf HT}(g)$ and in case ${\sf HT}(g)=t$ this immediately implies that this representation is already a right reductive standard one. Else we proceed by induction on $t$. Without loss of generality let $f_{1},\ldots,f_{K}$ be the polynomials in the corresponding representation such that $t={\sf HT}(f_{i}\star m_{i})$, $1\leq i\leq K$. Then the tuple $(t,f_{1},\ldots,f_{K},m_{1},\ldots,m_{K})$ is in ${\cal C}_{rr}(F)$ and let $h=\sum_{i=1}^{K}f_{i}\star m_{i}$. We will now change our representation of $g$ in such a way that for the new representation of $g$ we have a smaller maximal term. Let us assume $h$ is not $o$202020In case $h=o$, just substitute the empty sum for the representation of $h$ in the equations below.. By our assumption, $h$ has a right reductive standard representation with respect to $F$, say $\sum_{j=1}^{n}h_{j}\star l_{j}$, where $h_{j}\in F$, and $l_{j}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ and all terms occurring in the sum are bounded by $t\succ{\sf HT}(h)$ as $\sum_{i=1}^{K}{\sf HM}(f_{i}\star m_{i})=o$. This gives us: $\displaystyle g$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{K}f_{i}\star m_{i}+\sum_{i=K+1}^{m}f_{i}\star m_{i}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{n}h_{j}\star l_{j}+\sum_{i=K+1}^{m}f_{i}\star m_{i}$ which is a representation of $g$ where the maximal term is smaller than $t$. q.e.d. We can similarly refine Definition 4.2.10 with respect to a reductive restriction $\geq$ of the ordering $\succeq$. ###### Definition 4.2.20 A set $F\subseteq{\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ is called a weak right reductive Gröbner basis (with respect to the reductive ordering $\geq$) of ${\sf ideal}_{r}(F)$ if ${\sf HT}({\sf ideal}_{r}(F)\backslash\\{o\\})={\sf HT}(\\{f\star m\mid f\in F,m\in{\sf M}({\cal F}_{{\mathbb{K}}}),{\sf HT}(f\star m)={\sf HT}({\sf HT}(f)\star m)\geq{\sf HT}(f)\\}\backslash\\{o\\})$. $\diamond$ This definition now localizes the characterization of the Gröbner basis to the head terms of the generating set of polynomials. The next lemma states that in fact both characterizations of special bases, right reductive standard bases and weak right reductive Gröbner bases, coincide as in the case of polynomial rings over fields. ###### Lemma 4.2.21 Let $F$ be a subset of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a right reductive standard basis if and only if it is a weak right reductive Gröbner basis. Proof : Let us first assume that $F$ is a right reductive standard basis, i.e., every polynomial $g$ in ${\sf ideal}_{r}(F)$ has a right reductive standard representation with respect to $F$. In case $g\neq o$ this implies the existence of a polynomial $f\in F$ and a monomial $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HT}(g)={\sf HT}(f\star m)={\sf HT}({\sf HT}(f)\star m)\geq{\sf HT}(f)$. Hence ${\sf HT}(g)\in{\sf HT}(\\{f\star m\mid m\in{\sf M}({\cal F}_{{\mathbb{K}}}),f\in F,{\sf HT}(f\star m)={\sf HT}({\sf HT}(f)\star m)\geq{\sf HT}(f)\\}\backslash\\{o\\})$. As the converse, namely ${\sf HT}(\\{f\star m\mid m\in{\sf M}({\cal F}_{{\mathbb{K}}}),f\in F,{\sf HT}(f\star m)={\sf HT}({\sf HT}(f)\star m)\geq{\sf HT}(f)\\}\backslash\\{o\\})\subseteq{\sf HT}({\sf ideal}_{r}(F)\backslash\\{o\\})$ trivially holds, $F$ is then a weak right reductive Gröbner basis. Now suppose that $F$ is a weak right reductive Gröbner basis and again let $g\in{\sf ideal}_{r}(F)$. We have to show that $g$ has a right reductive standard representation with respect to $F$. This will be done by induction on ${\sf HT}(g)$. In case $g=o$ the empty sum is our required right reductive standard representation. Hence let us assume $g\neq o$. Since then ${\sf HT}(g)\in{\sf HT}({\sf ideal}_{r}(F)\backslash\\{o\\})$ by the definition of weak right reductive Gröbner bases we know there exists a polynomial $f\in F$ and a monomial $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HT}(g)={\sf HT}(f\star m)={\sf HT}({\sf HT}(f)\star m)\geq{\sf HT}(f)$. Then there exists a monomial $\tilde{m}\in{\sf M}({\cal F})$ such that ${\sf HM}(g)={\sf HM}(f\star\tilde{m})$, namely212121Notice that this step again requires that we can view ${\cal F}$ as a vector space. $\tilde{m}=({\sf HC}(g)\cdot{\sf HC}(f\star m)^{-1})\cdot m)$. Let $g_{1}=g-f\star\tilde{m}$. Then ${\sf HT}(g)\succ{\sf HT}(g_{1})$ implies the existence of a right reductive standard representation for $g_{1}$ which can be added to the multiple $f\star\tilde{m}$ to give the desired right reductive standard representation of $g$. q.e.d. An inspection of the proof shows that in fact we can require a stronger condition for the head terms of the monomial multiples involved in right reductive standard representations in terms of right reductive Gröbner bases. ###### Corollary 4.2.22 Let a subset $F$ of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ be a weak right reductive Gröbner basis. Then every $g\in{\sf ideal}_{r}(F)$ has a right reductive standard representation in terms of $F$ of the form $g=\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}$ such that ${\sf HT}(g)={\sf HT}(f_{1}\star m_{1})\succ{\sf HT}(f_{2}\star m_{2})\succ\ldots\succ{\sf HT}(f_{n}\star m_{n})$, and ${\sf HT}(f_{i}\star m_{i})={\sf HT}({\sf HT}(f_{i})\star m_{i})\geq{\sf HT}(f_{i})$ for all $1\leq i\leq n$. The importance of Gröbner bases in commutative polynomial rings stems from the fact that they can be characterized by special polynomials, the so-called s-polynomials, and that only finitely many such polynomials have to be checked in order to decide whether a set is a Gröbner basis. This test can be combined with adding ideal elements to the generating set leading to an algorithm which computes finite Gröbner bases by means of completion. These finite sets then can be used to solve many problems related to the ideals they generate. Given a field as coefficient domain the critical situations for function rings now lead to s-polynomials as in the original case and can be identified by studying term multiples of polynomials. Let $p$ and $q$ be two non-zero polynomials in ${\cal F}_{{\mathbb{K}}}$. We are interested in terms $t,u_{1},u_{2}$ such that ${\sf HT}(p\star u_{1})={\sf HT}({\sf HT}(p)\star u_{1})=t={\sf HT}(q\star u_{2})={\sf HT}({\sf HT}(q)\star u_{2})$ and ${\sf HT}(p)\leq t$, ${\sf HT}(q)\leq t$. Let ${\cal C}_{s}(p,q)$ (this is a specialization of Definition 4.2.17) be the critical set containing all such tuples $(t,u_{1},u_{2})$ (as a short hand for $(t,p,q,u_{1},u_{2})$). We call the polynomial ${\sf HC}(p\star u_{1})^{-1}\cdot p\star u_{1}-{\sf HC}(q\star u_{2})^{-1}\cdot q\star u_{2}={\sf spol}_{r}(p,q,t,u_{1},u_{2})$ the s-polynomial of $p$ and $q$ related to the tuple $(t,u_{1},u_{2})$. ###### Theorem 4.2.23 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a weak right reductive Gröbner basis of ${\sf ideal}_{r}(F)$ if and only if 1. 1. for all $f$ in $F$ and for $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ the multiple $f\star m$ has a right reductive standard representation in terms of $F$, and 2. 2. for all $p$ and $q$ in $F$ and every tuple $(t,u_{1},u_{2})$ in ${\cal C}_{s}(p,q)$ the respective s-polynomial ${\sf spol}_{r}(p,q,t,u_{1},u_{2})$ has a right reductive standard representation in terms of $F$. Proof : In case $F$ is a weak right reductive Gröbner basis it is also a right reductive standard basis, and since all multiples $f\star m$ and s-polynomials ${\sf spol}_{r}(p,q,t,u_{1},u_{2})$ stated above are elements of ${\sf ideal}_{r}(F)$, they must have right reductive standard representations in terms of $F$. The converse will be proven by showing that every element in ${\sf ideal}_{r}(F)$ has a right reductive standard representation in terms of $F$. Now, let $g=\sum_{j=1}^{m}f_{j}\star m_{j}$ be an arbitrary representation of a non-zero polynomial $g\in{\sf ideal}_{r}(F)$ such that $f_{j}\in F$, $m_{j}\in{\sf M}({\cal F})$, $m\in{\mathbb{N}}$. By our first assumption every multiple $f_{j}\star m_{j}$ in this sum has a right reductive representation. Hence without loss of generaltity we can assume that ${\sf HT}({\sf HT}(f_{j})\star m_{j})={\sf HT}(f_{j}\star m_{j})\geq{\sf HT}(f_{j})$ holds. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(f_{j}\star m_{j})\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $f_{j}\star m_{j}$ with head term $t$. Without loss of generality we can assume that the multiples with head term $t$ are just $f_{1}\star m_{1},\ldots,f_{K}\star m_{K}$. We proceed by induction on $(t,K)$, where $(t^{\prime},K^{\prime})<(t,K)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $K^{\prime}<K)$222222Note that this ordering is well- founded since $\succ$ is well-founded on ${\cal T}$ and $K\in{\mathbb{N}}$.. Obviously, $t\succeq{\sf HT}(g)$ holds. If $K=1$ this gives us $t={\sf HT}(g)$ and by our assumptions our representation is already of the required form. Hence let us assume $K>1$, then there are two not necessarily different polynomials $f_{1},f_{2}$ and corresponding monomials $m_{1}=\alpha_{1}\cdot w_{1}$, $m_{2}=\alpha_{2}\cdot w_{2}$ with $\alpha_{1},\alpha_{2}\in{\mathbb{K}}$, $w_{1},w_{2}\in{\cal T}$ in the corresponding representation such that $t={\sf HT}({\sf HT}(f_{1})\star w_{1})={\sf HT}(f_{1}\star w_{1})={\sf HT}(f_{2}\star w_{2})={\sf HT}({\sf HT}(f_{2})\star w_{2})$ and $t\geq{\sf HT}(f_{1})$, $t\geq{\sf HT}(f_{2})$. Then the tuple $(t,w_{1},w_{2})$ is in ${\cal C}_{s}(f_{1},f_{2})$ and we have an s-polynomial $h={\sf HC}(f_{1}\star w_{1})^{-1}\cdot f_{1}\star w_{1}-{\sf HC}(f_{2}\star w_{2})^{-1}\cdot f_{2}\star w_{2}$ corresponding to this tuple. We will now change our representation of $g$ by using the additional information on this s-polynomial in such a way that for the new representation of $g$ we either have a smaller maximal term or the occurrences of the term $t$ are decreased by at least 1. Let us assume the s-polynomial is not $o$232323In case $h=o$, just substitute the empty sum for the right reductive representation of $h$ in the equations below.. By our assumption, $h$ has a right reductive standard representation in terms of $F$, say $\sum_{i=1}^{n}h_{i}\star l_{i}$, where $h_{i}\in F$, and $l_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ and all terms occurring in the sum are bounded by $t\succ{\sf HT}(h)$. This gives us: $\displaystyle f_{1}\star m_{1}+f_{2}\star m_{2}$ (4.3) $\displaystyle=$ $\displaystyle\alpha_{1}\cdot f_{1}\star w_{1}+\alpha_{2}\cdot f_{2}\star w_{2}$ $\displaystyle=$ $\displaystyle\alpha_{1}\cdot f_{1}\star w_{1}+\underbrace{\alpha^{\prime}_{2}\cdot\beta_{1}\cdot f_{1}\star w_{1}-\alpha^{\prime}_{2}\cdot\beta_{1}\cdot f_{1}\star w_{1}}_{=\,0}+\underbrace{\alpha^{\prime}_{2}\cdot\beta_{2}}_{=\alpha_{2}}\cdot f_{2}\star w_{2}$ $\displaystyle=$ $\displaystyle(\alpha_{1}+\alpha^{\prime}_{2}\cdot\beta_{1})\cdot f_{1}\star w_{1}-\alpha^{\prime}_{2}\cdot\underbrace{(\beta_{1}\cdot f_{1}\star w_{1}-\beta_{2}\cdot f_{2}\star w_{2})}_{=\,h}$ $\displaystyle=$ $\displaystyle(\alpha_{1}+\alpha^{\prime}_{2}\cdot\beta_{1})\cdot f_{1}\star w_{1}-\alpha^{\prime}_{2}\cdot(\sum_{i=1}^{n}h_{i}\star l_{i})$ where $\beta_{1}={\sf HC}(f_{1}\star w_{1})^{-1}$, $\beta_{2}={\sf HC}(f_{2}\star w_{2})^{-1}$ and $\alpha^{\prime}_{2}\cdot\beta_{2}=\alpha_{2}$. By substituting (4.3) in our representation of $g$ it becomes smaller. q.e.d. Notice that both test sets in this characterization in general are not finite. Remember that in commutative polynomial rings over fields we can restrict these critical situations to one s-polynomial arising from the least common multiple of the head terms ${\sf HT}(p)$ and ${\sf HT}(q)$. Here we can introduce a similar concept of least common multiples, but now two terms can have no, one, finitely many and even infinitely many such multiples. Given two non-zero polynomials $p$ and $q$ in ${\cal F}_{{\mathbb{K}}}$ let $S(p,q)=\\{t\mid\mbox{ there exist }u_{1},u_{2}\in{\cal T}\mbox{ such that }{\sf HT}(p\star u_{1})={\sf HT}({\sf HT}(p)\star u_{1})=t={\sf HT}(q\star u_{2})={\sf HT}({\sf HT}(q)\star u_{2})\mbox{ and }{\sf HT}(p)\leq t,{\sf HT}(q)\leq t\\}$. A subset $LCM(p,q)$ of $S(p,q)$ is called a set of least common multiples for $p$ and $q$ if for any $t\in S(p,q)$ there exists $t^{\prime}\in LCM(p,q)$ such that $t^{\prime}\leq t$ and all other $s\in LCM(p,q)$ are not comparable with $t^{\prime}$ with respect to the reductive ordering $\leq$. For polynomial rings over fields a term $t$ is smaller than another term $s$ with respect to the reductive ordering if $t$ is a divisor of $s$ and $LCM(p,q)$ consists of the least common multiple of the head terms ${\sf HT}(p)$ and ${\sf HT}(q)$. But for function rings in general other situations are possible. Two polynomials do not have to give rise to any s-polynomial. Just take ${\cal T}$ to be the free monoid on $\\{a,b\\}$ and ${\mathbb{K}}={\mathbb{Q}}$. Then for the two polynomials $p=a+1$ and $q=b+1$ we have $S(p,q)=\emptyset$ as there are no terms $u_{1},u_{2}$ in ${\cal T}$ such that $a\star u_{1}=b\star u_{2}$. Next we give an example where the set $LCM(p,q)$ is finite but larger that one element. ###### Example 4.2.24 Let our set of terms ${\cal T}$ be presented as a monoid by $(\\{a,b,c,d_{1},d_{2},x_{1},x_{2}\\};\\{ax_{i}=cx_{i},bx_{i}=cx_{i},d_{j}x_{i}=x_{i}d_{j}\mid i,j\in\\{1,2\\}\\})$, $\succeq$ is the length-lexicographical ordering induced by the precedence $x_{2}\succ x_{1}\succ a\succ b\succ c\succ d_{1}\succ d_{2}$ and the reductive ordering $\geq$ is the prefix ordering. Then for the two polynomials $p=a+d_{1}$ and $q=b+d_{2}$ we get the respective sets $S(p,q)=\\{cx_{1}w,cx_{2}w\mid w\in{\cal T}\\}$ and $LCM(p,q)=\\{cx_{1},cx_{2}\\}$ with resulting s-polynomials ${\sf spol}_{r}(p,q,cx_{1},x_{1},x_{1})=x_{1}d_{1}-x_{1}d_{2}$ and ${\sf spol}_{r}(p,q,cx_{2},x_{2},x_{2})=x_{2}d_{1}-x_{2}d_{2}$. $\diamond$ It is also possible to have infinitely many least common multiples. ###### Example 4.2.25 Let our set of terms ${\cal T}$ be presented as a monoid by $(\\{a,b,c,d_{1},d_{2},x_{i}\mid i\in{\mathbb{N}}\\};\\{ax_{i}=cx_{i},bx_{i}=cx_{i},d_{j}x_{i}=x_{i}d_{j}\mid i\in{\mathbb{N}},j\in\\{1,2\\}\\})$, $\succeq$ is the length-lexicographical ordering induced by the precedence $\ldots\succ x_{n}\succ\ldots\succ x_{1}\succ a\succ b\succ c\succ d_{1}\succ d_{2}$ and the reductive ordering $\geq$ is the prefix ordering. Then for the two polynomials $p=a+d_{1}$ and $q=b+d_{2}$ we get the respective set $S(p,q)=\\{cx_{i}w\mid i\in{\mathbb{N}},w\in{\cal T}\\}$ and the infinite set $LCM(p,q)=\\{cx_{i}\mid i\in{\mathbb{N}}\\}$ with infinitely many resulting s-polynomials ${\sf spol}_{r}(p,q,cx_{i},x_{i},x_{i})=x_{i}d_{1}-x_{i}d_{2}$. $\diamond$ However, we have to show that we can restrict the set ${\cal C}_{s}(p,q)$ to those tuples corresponding to terms in $LCM(p,q)$. Remember that one problem which is related to the fact that the ordering $\succeq$ and the multiplication $\star$ in general are not compatible is that an important property fulfilled for representations of polynomials in commutative polynomial rings over fields no longer holds. This property in fact underlies Lemma 2.3.9 (4), which is essential in Buchberger’s characterization of Gröbner bases in polynomial rings: $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ implies $p\star m\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ for any monomial $m$. Notice that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}0$ implies that $p$ has a standard representation with respect to $F$, say $\sum_{i=1}^{n}f_{i}\star m_{i}$, and it is easy to see that then $\sum_{i=1}^{n}f_{i}\star m_{i}\star m$ is a standard representation of $p\star m$ with respect to $F$. This lemma is central in localizing all the critical situations related to two polynomials to the one s-polynomial resulting from the least common multiple of the respective head terms. Unfortunately, neither the lemma nor its implication for the existence of the respective standard representations holds in our more general setting. There, if $g\in{\sf ideal}_{r}(F)$ has a right reductive standard representation $g=\sum_{i=1}^{n}f_{i}\star m_{i}$, then the sum $\sum_{i=1}^{n}f_{i}\star m_{i}\star m$ in general is no right reductive standard representation not even a right standard representation of the multiple $g\star m$ for $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$. Even while $g\in{\sf ideal}_{r}(\\{g\\})$ has the trivial right reductive standard representation $g=g$, the multiple $g\star m$ is in general no right reductive standard representation of the function $g\star m$ for $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$. Recall the example on page 4.2.12 where for $g=x+1$ we have ${\sf HM}(g\star x)=x$ while ${\sf HM}(g)\star x=1$ as $x\star x=1$. Similarly, while $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{g}\,$}0$ must hold for any reduction relation, this no longer will imply $g\star m\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{g}\,$}0$. To see this let us review Example 4.2.18: For $g=a+1$ and $m=b$ we get the multiple $g\star m=(a+1)\star b=1+b$, but ${\sf HT}(g\star m)=b\neq 1={\sf HT}({\sf HT}(g)\star m)$. Moreover, $b+1$ is not reducible by $a+1$ for any reduction relation based on head monomial divisibility. In order to give localizations of the test sets from Theorem 4.2.23 it is important to study under which conditions the stability of right reductive standard representations with respect to multiplication by monomials can be restored. The next lemma provides a sufficient condition. ###### Lemma 4.2.26 Let $F\subseteq{\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ and $p$ a non-zero polynomial in ${\cal F}_{{\mathbb{K}}}$. Moreover, we assume that $p$ has a right reductive standard representation in terms of $F$ and $m$ is a monomial such that ${\sf HT}(p\star m)={\sf HT}({\sf HT}(p)\star m)\geq{\sf HT}(p)$. Then $p\star m$ again has a right reductive standard representation in terms of $F$. Proof : Let $p=\sum_{i=1}^{n}f_{i}\star m_{i}$ with $n\in{\mathbb{N}}$, $f_{i}\in F$, $m_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ be a right reductive standard representation of $p$ in terms of $F$, i.e., ${\sf HT}(p)={\sf HT}(f_{i}\star m_{i})={\sf HT}({\sf HT}(f_{i})\star m_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq k$ and ${\sf HT}(p)\succ{\sf HT}(f_{i}\star m_{i})$ for all $k+1\leq i\leq n$. Let us first analyze $f_{j}\star m_{j}\star m$ for $1\leq j\leq k$: Let ${\sf T}(f_{j}\star m_{j})=\\{s_{1},\ldots,s_{l}\\}$ with $s_{1}\succ s_{i}$, $2\leq i\leq l$, i.e. $s_{1}={\sf HT}(f_{j}\star m_{j})={\sf HT}({\sf HT}(f_{j})\star m_{j})={\sf HT}(p)$. Hence ${\sf HT}({\sf HT}(p)\star m)={\sf HT}(s_{1}\star m)\geq{\sf HT}(p)=s_{1}$ and as $s_{1}\succ s_{i}$, $2\leq i\leq l$, by Definition 4.2.13 we can conclude ${\sf HT}({\sf HT}(p)\star m)={\sf HT}(s_{1}\star m)\succ s_{i}\star m\succeq{\sf HT}(s_{i}\star m)$ for $2\leq i\leq l$. This implies ${\sf HT}({\sf HT}(f_{j}\star m_{j})\star m)={\sf HT}(f_{j}\star m_{j}\star m)$. Hence we get $\displaystyle{\sf HT}(p\star m)$ $\displaystyle=$ $\displaystyle{\sf HT}({\sf HT}(p)\star m)$ $\displaystyle=$ $\displaystyle{\sf HT}({\sf HT}(f_{j}\star m_{j})\star m),\mbox{ as }{\sf HT}(p)={\sf HT}(f_{j}\star m_{j})$ $\displaystyle=$ $\displaystyle{\sf HT}(f_{j}\star m_{j}\star m)$ and since ${\sf HT}(p\star m)\geq{\sf HT}(p)\geq{\sf HT}(f_{j})$ we can conclude ${\sf HT}(f_{j}\star m_{j}\star m)\geq{\sf HT}(f_{j})$. It remains to show that $f_{j}\star m_{j}\star m$ has a right reductive standard representation in terms of $F$. First we show that ${\sf HT}({\sf HT}(f_{j})\star m_{j}\star m)\geq{\sf HT}(f_{j})$: We know ${\sf HT}(f_{j})\star m_{j}\succeq{\sf HT}({\sf HT}(f_{j})\star m_{j})={\sf HT}(f_{j}\star m_{j})$ and hence ${\sf HT}({\sf HT}(f_{j})\star m_{j}\star m)={\sf HT}({\sf HT}(f_{j}\star m_{j})\star m)={\sf HT}(f_{j}\star m_{j}\star m)\geq{\sf HT}(f_{j})$. Now in case $m_{j}\star m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ we are done as then $f_{j}\star(m_{j}\star m)$ is a right reductive standard representation in terms of $F$. Hence let us assume $m_{j}\star m=\sum_{i=1}^{k}\tilde{m}_{i}$, $\tilde{m}_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}})$. Let ${\sf T}(f_{j})=\\{t_{1},\ldots,t_{s}\\}$ with $t_{1}\succ t_{p}$, $2\leq p\leq s$, i.e. $t_{1}={\sf HT}(f_{j})$. As ${\sf HT}({\sf HT}(f_{j})\star m_{j})\geq{\sf HT}(f_{j})\succ t_{p}$,$2\leq p\leq s$, again by Definition 4.2.13 we can conclude ${\sf HT}({\sf HT}(f_{j})\star m_{j})\succ t_{p}\star m_{j}\succeq{\sf HT}(t_{p}\star m_{j})$, and ${\sf HT}(f_{j})\star m_{j}\succ\sum_{p=2}^{s}t_{p}\star m_{1}$. Then for each $s_{i}$, $2\leq i\leq l$ there exists $t_{p}\in{\sf T}(f_{j})$ such that $s_{i}\in{\sf supp}(t_{p}\star m_{j})$. Since ${\sf HT}(p)\succ s_{i}$ and even242424${\sf HT}(p)\succ t_{p}\star m_{j}$ if ${\sf HT}(f_{j}\star m_{j})\not\in{\sf supp}(t_{p}\star m_{j})$. ${\sf HT}(p)\succeq t_{p}\star m_{j}$ we find that either ${\sf HT}(p\star m)\succeq{\sf HT}((t_{p}\star m_{j})\star m)={\sf HT}(t_{p}\star(m_{j}\star m))$ in case ${\sf HT}(t_{p}\star m_{j})={\sf HT}(f_{j}\star m_{j})$ or ${\sf HT}(p\star m)\succ{\sf HT}((t_{p}\star m_{j})\star m)={\sf HT}(t_{p}\star(m_{j}\star m))$. Hence we can conclude $f_{j}\star\tilde{m}_{i}\preceq{\sf HT}(p\star m)$, $1\leq i\leq l$ and for at least one $\tilde{m}_{i}$ we get ${\sf HT}(f_{j}\star\tilde{m}_{i})={\sf HT}(f_{j}\star m_{j}\star m)\geq{\sf HT}(f_{j})$. It remains to analyze the situation for the function $(\sum_{i=k+1}^{n}f_{i}\star m_{i})\star m$. Again we find that for all terms $s$ in the $f_{i}\star m_{i}$, $k+1\leq i\leq n$, we have ${\sf HT}(p)\succ s$ and we get ${\sf HT}(p\star m)\succ{\sf HT}(s\star m)$. Hence all polynomial multiples of the $f_{i}$ in the representation $\sum_{i=k+1}^{n}\sum_{j=1}^{k_{i}}f_{i}\star\tilde{m}^{i}_{j}$, where $m_{i}\star m=\sum_{j=1}^{k_{i}}\tilde{m}^{i}_{j}$, are bounded by ${\sf HT}(p\star m)$. q.e.d. Notice that these observations are no longer true in case we only require ${\sf HT}(p\star m)={\sf HT}({\sf HT}(p)\star m)\succeq{\sf HT}(p)$, as then ${\sf HT}(p)\succ s$ no longer implies that ${\sf HT}(p\star m)\succ{\sf HT}(s\star m)$ will hold. Of course this lemma now implies that if for two polynomials $p$ and $q$ in ${\cal F}_{{\mathbb{K}}}$ all s-polynomials related to the set $LCM(p,q)$ have right reductive standard representations so have all s-polynomials related to any tuple in ${\cal C}_{s}(p,q)$. So far we have characterized weak right reductive Gröbner bases as special right ideal bases providing right reductive standard representations for the right ideal elements. In the literature the existence of such representations is normally established by means of reduction relations. The special representations presented here can be related to a reduction relation based on the divisibility of terms as defined in the context of right reductive restrictions of our ordering following Definition 4.2.13. Let $\geq$ be such a right reductive restriction of the ordering $\succeq$. ###### Definition 4.2.27 Let $f,p$ be two non-zero polynomials in ${\cal F}_{{\mathbb{K}}}$. We say $f$ right reduces $p$ to $q$ at a monomial $\alpha\cdot t$ in one step, denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}q$, if there exists $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that 1. 1. $t\in{\sf supp}(p)$ and $p(t)=\alpha$, 2. 2. ${\sf HT}(f\star m)={\sf HT}({\sf HT}(f)\star m)=t\geq{\sf HT}(f)$, 3. 3. ${\sf HM}(f\star m)=\alpha\cdot t$, and 4. 4. $q=p-f\star m$. We write $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}$ if there is a polynomial $q$ as defined above and $p$ is then called right reducible by $f$. Further, we can define $\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$},\mbox{$\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$}$ and $\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$ as usual. Right reduction by a set $F\subseteq{\cal F}_{{\mathbb{K}}}$ is denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q$ and abbreviates $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}q$ for some $f\in F$. $\diamond$ Notice that if $f$ right reduces $p$ to $q$ at $\alpha\cdot t$ then $t\not\in{\sf supp}(q)$. If for some $w\in{\cal T}$ we have ${\sf HT}(f\star w)={\sf HT}({\sf HT}(f)\star w)=t\geq{\sf HT}(f)$ we can always reduce $\alpha\cdot t$ in $p$ by $f$ using the monomial $m=(\alpha\cdot{\sf HC}(f\star w)^{-1})\cdot w$. Other definitions of reduction relations are possible, e.g. substituting item 2 by the condition ${\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)$ (called right reduction in the context of monoid rings in [Rei95]; such a reduction relation would be connected to standard representations as defined in Definition 4.2.7) or by the condition ${\sf HT}(f\star m)=t$ (called strong reduction in the context of monoid rings in [Rei95] and for function rings on page 4.2.1). We have chosen this particular reduction relation as it provides the necessary information to apply Lemma 4.2.26 to give localizations for the test sets in Theorem 4.2.23 later on. Let us continue by studying some of the properties of our reduction relation. ###### Lemma 4.2.28 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. 1. 1. For $p,q\in{\cal F}_{{\mathbb{K}}}$, $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f\in F}\,$}q$ implies $p\succ q$, in particular ${\sf HT}(p)\succeq{\sf HT}(q)$. 2. 2. $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$ is Noetherian. Proof : 1. 1. Assuming that the reduction step takes place at a monomial $\alpha\cdot t$, by Definition 4.2.27 we know ${\sf HM}(f\star m)=\alpha\cdot t$ which yields $p\succ p-f\star m$ since ${\sf HM}(f\star m)\succ{\sf RED}(f\star m)$. 2. 2. This follows directly from 1. as the ordering $\succeq$ on ${\cal T}$ is well- founded (compare Theorem 4.2.3). q.e.d. The next lemma shows how reduction sequences and right reductive standard representations are related. ###### Lemma 4.2.29 Let $F\subseteq{\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ and $p\in{\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$ implies that $p$ has a right reductive standard representation in terms of $F$. Proof : This follows directly by adding up the polynomials used in the reduction steps occurring in the reduction sequence $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, say $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{1}}\,$}p_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{2}}\,$}\ldots\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{n}}\,$}o$. If the reduction steps take place at the respective head monomials only, we can additionally state that $p=\sum_{i=1}^{n}f_{i}\star m_{i}$, ${\sf HT}(f_{i}\star m_{i})={\sf HT}({\sf HT}(f_{i})\star m_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq n$, and even ${\sf HT}(f_{1}\star m_{1})\succ{\sf HT}(f_{2}\star m_{2})\succ\ldots{\sf HT}(f_{n}\star m_{n})$. q.e.d. If $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q$, then $p$ has a right reductive standard representation in terms of $F\cup\\{q\\}$, respectively $p-q$ has a right reductive standard representation in terms of $F$. On the other hand, if a polynomial $g$ has a right reductive standard representation in terms of some set $F$ it is reducible by a polynomial in $F$. To see this let $g=\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}}),n\in{\mathbb{N}}$ be a right reductive standard representation of $g$ in terms of $F$. Then ${\sf HT}(g)={\sf HT}(f_{1}\star m_{1})={\sf HT}({\sf HT}(f_{1})\star m_{1})\geq{\sf HT}(f_{1})$ and by Definition 4.2.27 this implies that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{1}}\,$}g-\alpha\cdot f_{1}\star m_{1}=g^{\prime}$ where $\alpha\in{\mathbb{K}}$ such that $\alpha\cdot{\sf HC}(f_{1}\star m_{1})={\sf HC}(g)$. So far we have given an algebraic characterization of weak right reductive Gröbner bases in Definition 4.2.20 and a characterization of them as right reductive standard bases in Lemma 4.2.21. Another characterization known from the literature is that for a Gröbner basis in a polynomial ring every element of the ideal it generates reduces to zero using the Gröbner basis. Reviewing Definition 3.1.2 we find that this is in fact only the definition of a weak Gröbner basis. However in polynomial rings over fields and many other structures in the literature the definitions of weak Gröbner bases and Gröbner bases coincide as the Translation Lemma holds (see Lemma 2.3.9 (2)). This is also true for function rings over fields. The first part of the following lemma is only needed for the proof of the second part which is an analogon of the Translation Lemma for function rings over fields. ###### Lemma 4.2.30 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $p,q,h$ polynomials in ${\cal F}_{{\mathbb{K}}}$. 1. 1. Let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}h$. Then there exist $p^{\prime},q^{\prime}\in{\cal F}_{{\mathbb{K}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}p^{\prime}$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q^{\prime}$ and $h=p^{\prime}-q^{\prime}$. 2. 2. Let $o$ be a normal form of $p-q$ with respect to $F$. Then there exists $g\in{\cal F}_{{\mathbb{K}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g$. Proof : 1. 1. Let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}h$ at the monomial $\alpha\cdot t$, i.e., $h=p-q-f\star m$ for some $f\in F$,$m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)=t\geq{\sf HT}(f)$ and ${\sf HM}(f\star m)=\alpha\cdot t$, i.e., $\alpha$ is the coefficient of $t$ in $p-q$. We have to distinguish three cases: 1. (a) $t\in{\sf supp}(p)$ and $t\in{\sf supp}(q)$: Then we can eliminate the occurrence of $t$ in the respective polynomials by right reduction and get $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}p-\alpha_{1}\cdot f\star m=p^{\prime}$, $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}q-\alpha_{2}\cdot f\star m=q^{\prime}$, where $\alpha_{1}\cdot{\sf HC}(f\star m)$ and $\alpha_{2}\cdot{\sf HC}(f\star m)$ are the coefficients of $t$ in $p$ respectively $q$. Moreover, $\alpha_{1}\cdot{\sf HC}(f\star m)-\alpha_{2}\cdot{\sf HC}(f\star m)=\alpha$ and hence $\alpha_{1}-\alpha_{2}=1$, as ${\sf HC}(f\star m)=\alpha$. This gives us $p^{\prime}-q^{\prime}=p-\alpha_{1}\cdot f\star m-q+\alpha_{2}\cdot f\star m=p-q-(\alpha_{1}-\alpha_{2})\cdot f\star m=p-q-f\star m=h$. 2. (b) $t\in{\sf supp}(p)$ and $t\not\in{\sf supp}(q)$: Then we can eliminate the term $t$ in the polynomial $p$ by right reduction and get $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}p-f\star m=p^{\prime}$, $q=q^{\prime}$, and, therefore, $p^{\prime}-q^{\prime}=p-f\star m-q=h$. 3. (c) $t\in{\sf supp}(q)$ and $t\not\in{\sf supp}(p)$: Then we can eliminate the term $t$ in the polynomial $q$ by right reduction and get $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}q+f\star m=q^{\prime}$, $p=p^{\prime}$, and, therefore, $p^{\prime}-q^{\prime}=p-(q+f\star m)=h$. 2. 2. We show our claim by induction on $k$, where $p-q\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. In the base case $k=0$ there is nothing to show as then $p=q$. Hence, let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}h\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Then by 1. there are polynomials $p^{\prime},q^{\prime}\in{\cal F}_{{\mathbb{K}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}p^{\prime}$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q^{\prime}$ and $h=p^{\prime}-q^{\prime}$. Now the induction hypothesis for $p^{\prime}-q^{\prime}\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$ yields the existence of a polynomial $g\in{\cal F}_{{\mathbb{K}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g$. q.e.d. The essential part of the proof is that right reducibility is connected to stable divisors of terms. We will later see that for function rings over arbitrary reduction rings, when the coefficient is also involved in the reduction step, this lemma no longer holds. ###### Definition 4.2.31 A subset $G$ of ${\cal F}_{{\mathbb{K}}}$ is called a right Gröbner basis (with respect to the reduction relation $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$) of the right ideal ${\mathfrak{i}}={\sf ideal}_{r}(G)$ it generates, if $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{G}\,$}=\;\;\equiv_{{\mathfrak{i}}}$ and $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{G}\,$ is confluent. Recall the free group ring in Example 4.2.18. There the polynomial $b+1$ lies in the right ideal generated by the polynomial $a+1$. Unlike in the case of polynomial rings over fields where for any set of polynomials $F$ we have $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}=\;\;\equiv_{{\sf ideal}(F)}$, here we have $b+1\equiv_{{\sf ideal}_{r}(\\{a+1\\})}o$ but $b+1\mbox{$\,\,\,\,{\not\\!\\!\\!\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{a+1}}\,$}o$. Hence the first condition of Definition 4.2.31 now becomes necessary while it can be omitted in the definition of Gröbner bases for ordinary polynomial rings. Now by Lemma 4.2.30 and Theorem 3.1.5 weak right reductive Gröbner bases are right Gröbner bases and can be characterized as follows: ###### Corollary 4.2.32 Let $G$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. $G$ is a right Gröbner basis if and only if for every $g\in{\sf ideal}_{r}(G)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{G}\,$}o$. Finally we can characterize right Gröbner bases similar to Theorem 2.3.11. ###### Theorem 4.2.33 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a right Gröbner basis if and only if 1. 1. for all $f$ in $F$ and for all $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ we have $f\star m\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, and 2. 2. for all $p$ and $q$ in $F$ and every tuple $(t,u_{1},u_{2})$ in ${\cal C}_{s}(p,q)$ and the respective s-polynomial ${\sf spol}_{r}(p,q,t,u_{1},u_{2})$ we have ${\sf spol}_{r}(p,q,t,u_{1},u_{2})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. However, the importance of Gröbner bases in the classical case stems from the fact that we only have to check a finite set of s-polynomials for $F$ in order to decide, whether $F$ is a Gröbner basis. Hence, we are interested in localizing the test sets in Theorem 4.2.33 – if possible to finite ones. ###### Definition 4.2.34 A set of polynomials $F\subseteq{\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ is called weakly saturated, if for every monomial $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ and every polynomial $f$ in $F$ we have $f\star m\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. $\diamond$ Then for a weakly saturated set $F$ and any monomial $m\in{\sf M}({\cal F}_{{\cal T}})$, $f\in F$, the multiple $f\star m$ has a right reductive standard representation in terms of $F$. Notice that since the coefficient domain is a field and ${\cal F}$ a vector space we can even restrict ourselves to multiples with elements of ${\cal T}$. However, for reduction rings as coefficient domains, we will need monomial multiples and hence we give the more general definition. For the free group ring in Example 4.2.18 the set $\\{a+1,b+1\\}$ is weakly saturated. ###### Definition 4.2.35 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{0\\}$. A set ${\sf SAT}(F)\subseteq\\{f\star m\mid f\in F,m\in{\sf M}({\cal F}_{{\mathbb{K}}})\\}$ is called a stable saturator for $F$ if for any $f\in F$, $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ there exist $s\in{\sf SAT}(F)$, $m^{\prime}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that $f\star m=s\star m^{\prime}$ and ${\sf HT}(f\star m)={\sf HT}({\sf HT}(s)\star m^{\prime})\geq{\sf HT}(s)$. Notice that a stable saturator need not be weakly saturated. Let $s\in{\sf SAT}(F)\subseteq\\{f\star m\mid f\in F,m\in{\sf M}({\cal F}_{{\mathbb{K}}})\\}$ and $m^{\prime}\in{\sf M}({\cal F}_{{\mathbb{K}}})$. For ${\sf SAT}(F)$ to be weakly saturated then $s\star m^{\prime}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{{\sf SAT}(F)}\,$}o$ must hold. We know that $s=f\star m$ for some $f\in F,m\in{\sf M}({\cal F}_{{\mathbb{K}}})$. In case $m\star m^{\prime}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ we are done. But this is no longer true if the monomial multiple results in a polynomial. Let our set of terms consist of words on the alphabet $\\{a,b,c\\}$ with multiplication $\star$ deduced form the equations $a\star b=a,b\star a=b^{2}-b,a\star a=o$. As ordering on ${\cal T}$ we take the length lexicographical ordering with precedence $a\succ b\succ c$ and as reductive restriction the prefix ordering. For the polynomial $f=ca+1$ we get a stable saturator ${\sf SAT}(\\{f\\})=\\{ca+1,ca+b,ca+b^{2},b^{3}+ca,a\\}$. Then the polynomial multiple $(f\star b)\star a=f\star(b\star a)=f\star(b^{2}-b)=ca+b^{2}-(ca+b)=b^{2}-b$ is not reducible by ${\sf SAT}(\\{f\\})$ while $f\star b=ca+b\in{\sf SAT}(\\{f\\})$. ###### Corollary 4.2.36 Let ${\sf SAT}(F)$ be a stable saturator of a set $F\subseteq{\cal F}_{{\mathbb{K}}}$. Then for any $f\in F$, $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ there exists $s\in{\sf SAT}(F)$ such that $f\star m\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{s}\,$}o$. ###### Lemma 4.2.37 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{0\\}$. If for all $s$ in a stable saturator ${\sf SAT}(F)$ we have $s\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, then for every $m$ in ${\sf M}({\cal F}_{{\mathbb{K}}})$ and every polynomial $f$ in $F$ the right multiple $f\star m$ has a right reductive standard representation in terms of $F$. Proof : This follows immediately from Lemma 4.2.29 and Lemma 4.2.26. q.e.d. ###### Definition 4.2.38 Let $p$ and $q$ be two non-zero polynomials in ${\cal F}_{{\mathbb{K}}}$. Then a subset $C\subseteq\\{{\sf spol}_{r}(p,q,t,u_{1},u_{2})\mid(t,u_{1},u_{2})\in{\cal C}_{s}(p,q)\\}$ is called a stable localization for the critical situations if for every s-polynomial ${\sf spol}_{r}(p,q,t,u_{1},u_{2})$ related to a tuple $(t,u_{1},u_{2})$ in ${\cal C}_{s}(p,q)$ there exists a polynomial $h\in C$ and a monomial $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that 1. 1. ${\sf HT}(h)\leq{\sf HT}({\sf spol}_{r}(p,q,t,u_{1},u_{2}))$, 2. 2. ${\sf HT}(h\star m)={\sf HT}({\sf HT}(h)\star m)={\sf HT}({\sf spol}_{r}(p,q,t,u_{1},u_{2}))$, 3. 3. ${\sf spol}_{r}(p,q,t,u_{1},u_{2})=h\star m$. $\diamond$ The set $LCM(p,q)$ (see page 4.2.23) allows a stable localization as follows: $C=\\{{\sf spol}_{r}(p,q,t,u_{1},u_{2})\mid t\in LCM(p,q),(t,u_{1},u_{2})\in{\cal C}_{s}(p,q)\\}$. ###### Corollary 4.2.39 Let $C\subseteq\\{{\sf spol}_{r}(p,q,t,u_{1},u_{2})\mid(t,u_{1},u_{2})\in{\cal C}_{s}(p,q)\\}$ be a stable localization for two polynomials $p,q\in{\cal F}_{{\mathbb{K}}}$. Then for any s-polynomial ${\sf spol}_{r}(p,q,t,u_{1},u_{2})$ there exists $h\in C$ such that ${\sf spol}_{r}(p,q,t,u_{1},u_{2})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{h}\,$}o$. ###### Lemma 4.2.40 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{0\\}$. If for all $h$ in a stable localization $C\subseteq\\{{\sf spol}_{r}(p,q,t,u_{1},u_{2})\mid(t,u_{1},u_{2})\in{\cal C}_{s}(p,q)\\}$, we have $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, then for every $(t,u_{1},u_{2})$ in ${\cal C}_{s}(p,q)$ the s-polynomial ${\sf spol}_{r}(p,q,t,u_{1},u_{2})$ has a right reductive standard representation in terms of $F$. Proof : This follows immediately from Lemma 4.2.29 and Lemma 4.2.26. q.e.d. So far we have seen that basically the theory for right Gröbner bases and the refined notion of right reductive standard bases (for right ideals of course) carries over similar from the case of polynomial rings over fields. Now Lemma 4.2.26 and Lemma 4.2.29 allow a localization of the test situations from Theorem 4.2.33. ###### Theorem 4.2.41 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{0\\}$. Then $F$ is a right Gröbner basis if and only if 1. 1. for all $s$ in a stable saturator ${\sf SAT}(F)$ we have $s\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, and 2. 2. for all $p$ and $q$ in $F$, and every polynomial $h$ in a stable localization $C\subseteq\\{{\sf spol}_{r}(p,q,t,u_{1},u_{2})\mid(t,u_{1},u_{2})\in{\cal C}_{s}(p,q)\\}$, we have $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Proof : In case $F$ is a right Gröbner basis by Lemma 4.2.32 all elements of ${\sf ideal}_{r}(F)$ must right reduce to zero by $F$. Since the polynomials in question all belong to the right ideal generated by $F$ we are done. The converse will be proven by showing that every element in ${\sf ideal}_{r}(F)$ has a right reductive representation in terms of $F$. Now, let $g=\sum_{j=1}^{m}f_{j}\star m_{j}$ be an arbitrary representation of a non- zero polynomial $g\in{\sf ideal}_{r}(F)$ such that $f_{j}\in F$, and $m_{j}\in{\sf M}({\cal F}_{{\mathbb{K}}})$. By our first assumption for every multiple $f_{j}\star m_{j}$ in this sum we have some $s\in{\sf SAT}(F)$, $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that $f_{j}\star m_{j}=s\star m$ and ${\sf HT}(f_{j}\star m_{j})={\sf HT}(s\star m)={\sf HT}({\sf HT}(s)\star m)\geq{\sf HT}(s)$. Since we have $s\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, by Lemma 4.2.26 we can conclude that each $f_{j}\star m_{j}$ has a right reductive standard representation in terms of $F$. Therefore, we can assume that ${\sf HT}({\sf HT}(f_{j})\star m_{j})={\sf HT}(f_{j}\star m_{j})\geq{\sf HT}(f_{j})$ holds. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(f_{j}\star m_{j})\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $f_{j}\star m_{j}$ with head term $t$. Without loss of generality we can assume that the polynomial multiples with head term $t$ are just $f_{1}\star m_{1},\ldots,f_{K}\star m_{K}$. We proceed by induction on $(t,K)$, where $(t^{\prime},K^{\prime})<(t,K)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $K^{\prime}<K)$252525Note that this ordering is well-founded since $\succ$ is well-founded on ${\cal T}$ and $K\in{\mathbb{N}}$.. Obviously, $t\succeq{\sf HT}(g)$ must hold. If $K=1$ this gives us $t={\sf HT}(g)$ and by our assumption our representation is already of the required form. Hence let us assume $K>1$, then for the two not necessarily different polynomials $f_{1},f_{2}$ and corresponding monomials $m_{1}=\alpha_{1}\cdot w_{1}$, $m_{2}=\alpha_{2}\cdot w_{2}$, $\alpha_{1},\alpha_{2}\in{\mathbb{K}}$, $w_{1},w_{2}\in{\cal T}$, in the corresponding representation we have $t={\sf HT}({\sf HT}(f_{1})\star w_{1})={\sf HT}(f_{1}\star w_{1})={\sf HT}(f_{2}\star w_{2})={\sf HT}({\sf HT}(f_{2})\star w_{2})$ and $t\geq{\sf HT}(f_{1})$, $t\geq{\sf HT}(f_{2})$. Then the tuple $(t,w_{1},w_{2})$ is in ${\cal C}_{s}(f_{1},f_{2})$ and we have a polynomial $h$ in a stable localization $C\subseteq\\{{\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2})\mid(t,w_{1},w_{2})\in{\cal C}_{s}(f_{1},f_{2})\\}$ and $m\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2})={\sf HC}(f_{1}\star w_{1})^{-1}\cdot f_{1}\star w_{1}-{\sf HC}(f_{2}\star w_{2})^{-1}\cdot f_{2}\star w_{2}=h\star m$ and ${\sf HT}({\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2}))={\sf HT}(h\star m)={\sf HT}({\sf HT}(h)\star m)\geq{\sf HT}(h)$. We will now change our representation of $g$ by using the additional information on this situation in such a way that for the new representation of $g$ we either have a smaller maximal term or the occurrences of the term $t$ are decreased by at least 1. Let us assume the s-polynomial is not $o$262626In case $h=o$, just substitute the empty sum for the right reductive representation of $h$ in the equations below.. By our assumption, $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$ and by Lemma 4.2.29 $h$ then has a right reductive standard representation in terms of $F$. Then by Lemma 4.2.26 the multiple $h\star m$ again has a right reductive standard representation in terms of $F$, say $\sum_{i=1}^{n}h_{i}\star l_{i}$, where $h_{i}\in F$, and $l_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ and all terms occurring in this sum are bounded by $t\succ{\sf HT}(h\star m)$. This gives us: $\displaystyle\alpha_{1}\cdot f_{1}\star w_{1}+\alpha_{2}\cdot f_{2}\star w_{2}$ (4.4) $\displaystyle=$ $\displaystyle\alpha_{1}\cdot f_{1}\star w_{1}+\underbrace{\alpha^{\prime}_{2}\cdot\beta_{1}\cdot f_{1}\star w_{1}-\alpha^{\prime}_{2}\cdot\beta_{1}\cdot f_{1}\star w_{1}}_{=\,0}+\underbrace{\alpha^{\prime}_{2}\cdot\beta_{2}}_{=\alpha_{2}}\cdot f_{2}\star w_{2}$ $\displaystyle=$ $\displaystyle(\alpha_{1}+\alpha^{\prime}_{2}\cdot\beta_{1})\cdot f_{1}\star w_{1}-\alpha^{\prime}_{2}\cdot\underbrace{(\beta_{1}\cdot f_{1}\star w_{1}-\beta_{2}\cdot f_{2}\star w_{2})}_{=\,h\star m}$ $\displaystyle=$ $\displaystyle(\alpha_{1}+\alpha^{\prime}_{2}\cdot\beta_{1})\cdot f_{1}\star w_{1}-\alpha^{\prime}_{2}\cdot(\sum_{i=1}^{n}h_{i}\star l_{i})$ where $\beta_{1}={\sf HC}(f_{1}\star w_{1})^{-1}$, $\beta_{2}={\sf HC}(f_{2}\star w_{2})^{-1}$ and $\alpha^{\prime}_{2}\cdot\beta_{2}=\alpha_{2}$. By substituting (4.4) our representation of $g$ becomes smaller. q.e.d. Obviously we now have criteria for when a set is a right Gröbner basis. As in the case of completion procedures such as the Knuth-Bendix procedure or the Buchberger algorithm, elements from these test sets which do not reduce to zero can be added to the set being tested, to gradually describe a not necessarily finite right Gröbner basis. Of course in order to get a computable completion procedure certain assumptions on the test sets have to be made, e.g. they should themselves be recursively enumerable, and normal forms with respect to finite sets have to be computable. Then provided such enumeration procedures for stable saturators and critical situations, an enumeration procedure for a respective right Gröbner basis has to ensure that all necessary candidates are enumerated and tested for reducibility to $o$. If this is not the case they are added to the right Gröbner basis, have to be added to the enumeration of the stable saturator candidates and the new arising critical situations have to be added to the respective enumeration process. We close this subsection by outlining how different structures known to allow finite Gröbner bases can be interpreted as function rings. Using the respective interpretations the terminology can be adapted at once to the respective structures and in general the resulting characterizations of Gröbner bases coincide with the results known from literature. ##### Polynomial Rings A commutative polynomial ring ${\mathbb{K}}[x_{1},\ldots,x_{n}]$ is a function ring according to the following interpretation: * • ${\cal T}$ is the set of terms $\\{x_{1}^{i_{1}}\ldots x_{n}^{i_{n}}\mid i_{1},\ldots,i_{n}\in{\mathbb{N}}\\}$. * • $\succ$ can be any admissible term ordering on ${\cal T}$. For the reductive ordering $\geq$ we have $t\geq s$ if $s$ divides $t$ as as term272727Apel has studied another possible reductive ordering $\geq$ where we have $t\geq s$ if $s$ is a prefix of $t$. This ordering gives rise to Janet bases.. * • Multiplication $\star$ is specified by the action on terms, i.e. $\star:{\cal T}\times{\cal T}\longrightarrow{\cal T}$ where $x_{1}^{i_{1}}\ldots x_{n}^{i_{n}}\star x_{1}^{j_{1}}\ldots x_{n}^{j_{n}}=x_{1}^{i_{1}+j_{1}}\ldots x_{n}^{i_{n}+j_{n}}$. We do not need the concept of weak saturation. A stable localization of ${\cal C}_{s}(p,q)$ is already provided by the tuple corresponding to the least common multiple of the terms ${\sf HT}(p)$ and ${\sf HT}(q)$. Since this structure is Abelian, one-sided and two-sided ideals coincide. Buchberger’s Algorithm provides an effictive procedure to compute finite Gröbner bases. ##### Solvable Polynomial Rings According to [KRW90, Kre93], a solvable polynomial ring ${\mathbb{K}}\\{x_{1},\ldots,x_{n};p_{ij};c_{ij}\\}$ with $1\leq j<i\leq n$, $p_{ij}\in{\mathbb{K}}[x_{1},\ldots,x_{n}]$, $c_{ij}\in{\mathbb{K}}^{}$ is a function ring according to the following interpretation: • ${\cal T}$ is the set of terms $\\{x_{1}^{i_{1}}\ldots x_{n}^{i_{n}}\mid i_{1},\ldots,i_{n}\in{\mathbb{N}}\\}$. * • $\succ$ can be any admissible term ordering on ${\cal T}$ for which $x_{j}x_{i}\succ p_{ij}$, $j<i$, must hold. For the reductive ordering $\geq$ we have $t\geq s$ if $s$ divides $t$ as as term. * • Multiplication $\star$ is specified by lifting the following action on the variables: $x_{i}\star x_{j}=x_{i}x_{j}$ if $i\leq j$ and $x_{i}\star x_{j}=c_{ij}\cdot x_{j}x_{i}+p_{ij}$ if $i>j$. We do not need the concept of weak saturation except in case we also allow $c_{ij}=0$. Then appropriate term multiples which “delete” head terms have to be taken into account. This critical set can be described in a finitary manner. For the reductive ordering $\geq$ then we can chose $t\geq s$ if $s$ is a prefix of $t$ (compare Example 4.2.14). The set ${\cal C}_{s}(p,q)$ again contains as a stable localization the tuple corresponding to the least common multiple of the terms ${\sf HT}(p)$ and ${\sf HT}(q)$. This structure is no longer Abelian, but finite Gröbner bases can be computed for one- and two-sided ideals (see [KRW90, Kre93]). ##### Non-commutative Polynomial Rings A non-commutative polynomial ring ${\mathbb{K}}[\\{x_{1},\ldots,x_{n}\\}^{}]$ is a function ring according to the following interpretation: • ${\cal T}$ is the set of words on $\\{x_{1},\ldots,x_{n}\\}$. * • $\succ$ can be any admissible ordering on ${\cal T}$. For the reductive ordering $\geq$ we can chose $t\geq s$ if $s$ is a subword of $t$. * • Multiplication $\star$ is specified by the action on words which is just concatenation. We do not need the concept of weak saturation. A stable localization of ${\cal C}_{s}(p,q)$ is already provided by the tuples corresponding to word overlaps resulting from the equations $u_{1}{\sf HT}(p)v_{1}={\sf HT}(q)$, $u_{2}{\sf HT}(q)v_{2}={\sf HT}(p)$, $u_{3}{\sf HT}(p)={\sf HT}(q)v_{3}$ respectively $u_{4}{\sf HT}(q)={\sf HT}(p)v_{4}$ with the restriction that $\|u_{3}\|<\|{\sf HT}(q)\|$ and $\|u_{4}\|<\|{\sf HT}(p)\|$, $u_{i},v_{i}\in{\cal T}$. This structure is not Abelian. For the case of one-sided ideals finite Gröbner bases can be computed (see e.g. [Mor94]). The case of two-sided ideals only allows an enumerating procedure. This is not surprising as the word problem for monoids can be reduced to the problem of computing the respective Gröbner bases (see e.g. [Mor87, MR98d]). ##### Monoid and Group Rings A monoid or group ring ${\mathbb{K}}[{\cal M}]$ is a function ring according to the following interpretation: * • ${\cal T}$ is the monoid or group ${\cal M}$. In the cases studied by us as well as in [Ros93, Lo96], it is assumed that the elements of the monoid or group have a certain form. This presentation is essential in the approach. We will assume that the given monoid or group is presented by a convergent semi- Thue system. * • $\succ$ will be the completion ordering induced from the presentation of ${\cal M}$ to ${\cal M}$ and hence to ${\cal T}$. The reductive ordering $\geq$ depends on the choice of the presentation. * • Multiplication $\star$ is specified by lifting the monoid or group operation. The concept of weak saturation and the choice of stable localizations of ${\cal C}_{s}(p,q)$ again depend on the choice of the presentation. We will close this section by listing monoids and groups which allow finite Gröbner bases for the respective monoid or group ring and pointers to the literature where the appropriate solutions can be found. Structure \| Ideals \| Quote ---\|---\|--- Finite monoid \| one- and two-sided \| [Rei96, MR97b] Free monoid \| one-sided \| [Mor94, MR97b] Finite group \| one- and two-sided \| [Rei95, MR97b] Free group \| one-sided \| [MR93a, Ros93, Rei95, MR97b] Plain group \| one-sided \| [MR93a, Rei95, MR97b] Context-free group \| one-sided \| [Rei95, MR97b] Nilpotent group \| one- and two-sided \| [Rei95, MR97a] Polycyclic group \| one- and two-sided \| [Lo96, Rei96] #### 4.2.2 Function Rings over Reduction Rings The situation becomes more complicated for a function ring ${\cal F}_{{\sf R}}$ where ${\sf R}$ is not a field. We will abbreviate ${\cal F}_{{\sf R}}$ by ${\cal F}$. Notice that similar to the previous section it is possible to study generalizations of standard representations for function rings over reduction rings with respect to the orderings $\succeq$ and $\geq$ on ${\cal T}$. General right standard representations as defined in Definition 4.2.4, as well as the corresponding critical situations from Definition 4.2.5 and the characterization of general right standard bases as in Theorem 4.2.6 carry over to our function ring ${\cal F}$. The same is true for right standard representations as defined in Definition 4.2.7, the corresponding critical situations from Definition 4.2.8 and the characterization of right standard bases as in Theorem 4.2.6. However, these standard representations can no longer be linked to weak right Gröbner bases as defined in Definition 4.2.10. This is of course obvious as for function rings over fields we have a characterization of such Gröbner bases by head terms which is no longer possible for function rings over reduction rings. This is already the case for polynomial rings over the integers. For example take the polynomial $3\cdot X$ in ${\mathbb{Q}}[X]$. Then obviously for $F_{1}=\\{3\cdot X\\}$ and $F_{2}=\\{X\\}$ we get that ${\sf HT}({\sf ideal}_{r}(F_{1})\backslash\\{0\\})={\sf HT}(\\{3\cdot X\star X^{i}\mid i\in{\mathbb{N}}\\})={\sf HT}(\\{X\star X^{i}\mid i\in{\mathbb{N}}\\})={\sf HT}({\sf ideal}_{r}(F_{2})\backslash\\{0\\})$ while of course $F_{1}$ is no right Gröbner basis of ${\sf ideal}_{r}(F_{2})$ and $F_{2}$ is no right Gröbner basis of ${\sf ideal}_{r}(F_{1})$. One possible generalizing of Definition 4.2.10 is as follows: $F$ is a weak right Gröbner basis of ${\sf ideal}_{r}(F)$ if ${\sf HM}({\sf ideal}_{r}(F)\backslash\\{0\\})={\sf HM}(\\{f\star m\mid f\in F,m\in{\sf M}({\cal F})\\})$. But this does not solve the problem as there is no equivalent to Lemma 4.2.11 to link these right Gröbner bases to the respective standard bases. The reason for this is that the definitions of standard representations as provided by Definition 4.2.4 and 4.2.7 are no longer related to reduction relations corresponding to Gröbner bases. Of course it is possible to study other generalizations of these definitions, e.g. involving the ordering on the coefficients, but we take a different approach. Our studies of standard representations for function rings over fields revealed that in fact we can identify stronger conditions for such representations in terms of weak right Gröbner bases (review e.g. Corollary 4.2.12 and 4.2.22). These special represenations arise from reduction sequences. Hence we will proceed by studying such standard representations which can be directly related to reduction relations in our function ring. Similar to function rings over fields we need to view ${\cal F}$ as a vector space now over ${\sf R}$, a reduction ring as described in Section 3.1. In general ${\sf R}$ is not Abelian and hence we have to distinguish right and left scalar multiplication as defined on page 4.2.10. However, since ${\sf R}$ is associative as in the case of fields we can write $\alpha\cdot f\cdot\beta$. Notice that for $f,g$ in ${\cal F}$ and $\alpha,\beta\in{\sf R}$ we have 1. 1. $\alpha\cdot(f\oplus g)=\alpha\cdot f\oplus\alpha\cdot g$ 2. 2. $\alpha\cdot(\beta\cdot f)=(\alpha\cdot\beta)\cdot f$ 3. 3. $(\alpha+\beta)\cdot f=\alpha\cdot f\oplus\beta\cdot f$, i.e., ${\cal F}$ is a left ${\sf R}$-module. Similarly we have 1. 1. $(f\oplus g)\cdot\alpha=f\cdot\alpha\oplus g\cdot\alpha$ 2. 2. $(f\cdot\alpha)\cdot\beta=f\cdot(\alpha\cdot\beta)$ 3. 3. $f\cdot(\alpha+\beta)=f\cdot\alpha\oplus f\cdot\beta$, i.e., ${\cal F}$ is a right ${\sf R}$-module as well. Moreover, as $(\alpha\cdot f)\cdot\beta=\alpha\cdot(f\cdot\beta)$ for all $f\in{\cal F}$, $\alpha,\beta\in{\sf R}$, ${\cal F}$ is an ${\sf R}$-${\sf R}$ bimodule. In order to state how scalar multiplication and ring multiplication are compatible, we again require $(\alpha\cdot f)\star g=\alpha\cdot(f\star g)$ and $f\star(g\cdot\alpha)=(f\star g)\cdot\alpha$ to hold. This is true for all examples we know from the literature. If we additionally require that for $\alpha,\beta\in{\sf R}$ and $t,s\in{\cal T}$ we have $(\alpha\cdot t)\star(\beta\cdot s)=(\alpha\cdot\beta)\cdot(t\star s)$, then the multiplication in ${\cal F}$ can be specified by knowing $\star:{\cal T}\times{\cal T}\longrightarrow{\cal F}$. If ${\cal F}$ contains a unit ${\bf 1}$, ${\sf R}$ can be embedded into ${\cal F}$ via the mapping $\alpha\longmapsto\alpha\cdot{\bf 1}$. Then for $\alpha\in{\sf R}$ and $f\in{\cal F}$ the equations $\alpha\cdot f=(\alpha\cdot{\bf 1})\star f$ and $f\cdot\alpha=f\star(\alpha\cdot{\bf 1})$ hold. Since for $\alpha\in{\sf R}$ and $t\in{\cal T}$ we have $\alpha\cdot t=t\cdot\alpha$ this implies $(\alpha\cdot t)\star(\beta\cdot s)=(\alpha\cdot\beta)\cdot(t\star s)$282828$(\alpha\cdot t)\star(\beta\cdot s)=(\alpha\cdot t)\star((\beta\cdot{\bf 1})\star s)=((\alpha\cdot t)\star(\beta\cdot{\bf 1}))\star s=(\alpha\cdot(t\star(\beta\cdot{\bf 1}))\star s=(\alpha\cdot(t\cdot\beta))\star s=(\alpha\cdot(\beta\cdot t))\star s=(\alpha\cdot\beta)\cdot(t\star s)$. . Moreover, if ${\sf R}$ is Abelian, we get $\alpha\cdot(f\star g)=f\star(\alpha\cdot g)$ and ${\cal F}$ is an algebra. Remember that we want to study standard representations directly related to reduction relations on ${\cal F}$. Since we have a function ring over a reduction ring such a reduction relation originates from the reduction relation on the reduction ring ${\sf R}$. Here we want to distinguish two such reduction relations on ${\cal F}$. One possible generalization in the spirit of these ideas for function rings over reduction rings is as follows: ###### Definition 4.2.42 Let $F$ be a set of polynomials in ${\cal F}$ and $g$ a non-zero polynomial in ${\sf ideal}_{r}(F)$. A representation of the form $g=\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}$ such that ${\sf HT}(g)={\sf HT}({\sf HT}(f_{1})\star m_{1})={\sf HT}(f_{1}\star m_{1})\geq{\sf HT}(f_{1})$ and ${\sf HT}(g)\succ{\sf HT}(f_{i}\star m_{i})$ for all $2\leq i\leq n$ is called a right reductive standard representation in terms of $F$. A set $F\subseteq{\cal F}\backslash\\{o\\}$ is called a right reductive standard basis of ${\sf ideal}_{r}(F)$ if every polynomial $f\in{\sf ideal}_{r}(F)$ has a right reductive standard representation in terms of $F$. $\diamond$ Notice that that this definition differs from Definition 4.2.15 insofar as we demand ${\sf HT}(g)\succ{\sf HT}(f_{i}\star m_{i})$ for all $2\leq i\leq n$. In fact we use those special standard representations which arise in the case of function rings for $g\in{\sf ideal}_{r}(F)$ when $F$ already is a right reductive standard basis (compare Corollary 4.2.22). This definition is directly related to the reduction relation presented in Definition 4.2.27 for ${\cal F}_{{\mathbb{K}}}$ generalized to ${\cal F}$. A possible definition of reduction can be given in the following fashion where we require that the reduction step eliminates the respective monomial it is applied to. ###### Definition 4.2.43 Let $f,p$ be two non-zero polynomials in ${\cal F}$. We say $f$ right reduces $p$ to $q$ eliminating the monomial $\alpha\cdot t$ in one step, denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}_{f}\,$}q$, if there exists $m\in{\sf M}({\cal F})$ such that 1. 1. $t\in{\sf supp}(p)$ and $p(t)=\alpha$, 2. 2. ${\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)=t\geq{\sf HT}(f)$, 3. 3. ${\sf HM}(f\star m)=\alpha\cdot t$, such that $\alpha\Longrightarrow_{{\sf HC}(f\star m)}0$, and 4. 4. $q=p-f\star m$. We write $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}_{f}\,$}$ if there is a polynomial $q$ as defined above and $p$ is then called right reducible by $f$. Further, we can define $\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}\,$},\mbox{$\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}\,$}$ and $\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}\,$ as usual. Right reduction by a set $F\subseteq{\cal F}$ is denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}_{F}\,$}q$ and abbreviates $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}_{f}\,$}q$ for some $f\in F$. $\diamond$ This reduction relation is related to a special instance292929Compare Pan’s reduction relation for the integers as defined in Example 3.1.1. of the reduction relation $\Longrightarrow$. Notice that by Axiom (A2) $\alpha\Longrightarrow_{{\sf HC}(f\star m)}0$ implies $\alpha\in{\sf ideal}_{r}^{{\sf R}}({\sf HC}(f\star m))$ and hence $\alpha={\sf HC}(f\star m)\cdot\beta$ for some $\beta\in{\sf R}$. Notice that in contrary to ${\cal F}_{{\mathbb{K}}}$ now for $g,f\in{\cal F}$ and $m\in{\sf M}({\cal F})$ the situation ${\sf HT}(g)={\sf HT}(f\star m)={\sf HT}({\sf HT}(f)\star m)\geq{\sf HT}(f)$ alone no longer implies that ${\sf HM}(g)$ is right reducible by $f$. This is due to the fact that we can no longer modify the coefficients involved in the reduction step in the appropriate manner since reduction rings in general will not contain inverse elements. Let us continue by studying our reduction relation. ###### Lemma 4.2.44 Let $F$ be a set of polynomials in ${\cal F}\backslash\\{o\\}$. 1. 1. For $p,q\in{\cal F}$ $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}_{F}\,$}q$ implies $p\succ q$, in particular ${\sf HT}(p)\succeq{\sf HT}(q)$. 2. 2. $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}_{F}\,$ is Noetherian. Proof : 1. 1. Assuming that the reduction step takes place at a monomial $\alpha\cdot t$, by Definition 4.2.43 we know ${\sf HM}(f\star m)=\alpha\cdot t$ which yields $p\succ p-f\star m$ since ${\sf HM}(f\star m)\succ{\sf RED}(f\star m)$. 2. 2. This follows from 1. q.e.d. The Translation Lemma no longer holds for this reduction relation. This is already so for polynomial rings over the integers. ###### Example 4.2.45 Let ${\mathbb{Z}}[X]$ be the polynomial ring in one indeterminant over ${\mathbb{Z}}$. Moreover, let $\Longrightarrow$ be the reduction relation on ${\mathbb{Z}}$ where for $\alpha,\beta\in{\mathbb{Z}}$, $\alpha\Longrightarrow_{\beta}$ if and only if there exists $\gamma\in{\mathbb{Z}}$ such that $\alpha=\beta\cdot\gamma$ (compare Example 3.1.1). Let $p=2\cdot x$, $q=-3\cdot X$ and $f=5\cdot X$. Then $p-q=5\cdot X\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}_{f}\,$}0$ while $p\mbox{$\,\,\,\,{\not\\!\\!\\!\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}_{f}}\,$}$ and $q\mbox{$\,\,\,\,{\not\\!\\!\\!\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}_{f}}\,$}$. $\diamond$ The reduction relation $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r,e}}\,$ in polynomial rings over the integers is known as Pan’s reduction in the literature. The generalization of Gröbner bases then are weak Gröbner bases as by completion one can achieve that all ideal elements reduce to zero. Next we present a proper algebraic characterization of weak right Gröbner bases related to right reductive standard representations and the reduction relation defined in Definition 4.2.43. Notice that it differs from Definition 4.2.20 for function rings over fields insofar as we now have to look at the head monomials of the right ideal instead of the head terms only. ###### Definition 4.2.46 A set $F\subseteq{\cal F}\backslash\\{o\\}$ is called a weak right reductive Gröbner basis of ${\sf ideal}_{r}(F)$ if ${\sf HM}({\sf ideal}_{r}(F)\backslash\\{o\\})={\sf HM}(\\{f\star m\mid f\in F,m\in{\sf M}({\cal F}),{\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)\geq{\sf HT}(f)\\}\backslash\\{o\\})$. $\diamond$ Similar to Lemma 4.2.21 right reductive standard bases and weak right reductive Gröbner bases coincide. ###### Lemma 4.2.47 Let $F$ be a subset of ${\cal F}\backslash\\{o\\}$. Then $F$ is a right reductive standard basis if and only if it is a weak right reductive Gröbner basis. Proof : Let us first assume that $F$ is a right reductive standard basis, i.e., every polynomial $g$ in ${\sf ideal}_{r}(F)$ has a right reductive standard representation with respect to $F$. In case $g\neq o$ this implies the existence of a polynomial $f\in F$ and a monomial $m\in{\sf M}({\cal F})$ such that ${\sf HT}(g)={\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)\geq{\sf HT}(f)$ and ${\sf HM}(g)={\sf HM}(f\star m)$303030Notice that if we had generalized the original Definition 4.2.15 this would not holds.. Hence ${\sf HM}(g)\in{\sf HM}(\\{f\star m\mid m\in{\sf M}({\cal F}),f\in F,{\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)\geq{\sf HT}(f)\\}\backslash\\{o\\})$. As the converse, namely ${\sf HM}(\\{f\star m\mid m\in{\sf M}({\cal F}),f\in F,{\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)\geq{\sf HT}(f)\\}\backslash\\{o\\})\subseteq{\sf HM}({\sf ideal}_{r}(F)\backslash\\{o\\})$ trivially holds, $F$ is a weak right reductive Gröbner basis. Now suppose that $F$ is a weak right reductive Gröbner basis and again let $g\in{\sf ideal}_{r}(F)$. We have to show that $g$ has a right reductive standard representation with respect to $F$. This will be done by induction on ${\sf HT}(g)$. In case $g=o$ the empty sum is our required right reductive standard representation. Hence let us assume $g\neq o$. Since then ${\sf HM}(g)\in{\sf HM}({\sf ideal}_{r}(F)\backslash\\{o\\})$ by the definition of weak right reductive Gröbner bases we know there exists a polynomial $f\in F$ and a monomial $m\in{\sf M}({\cal F})$ such that ${\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)\geq{\sf HT}(f)$ and ${\sf HM}(g)={\sf HM}(f\star m)$. Let $g_{1}=g-f\star m$. Then ${\sf HT}(g)\succ{\sf HT}(g_{1})$ implies the existence of a right reductive standard representation for $g_{1}$ which can be added to the multiple $f\star m$ to give the desired right reductive standard representation of $g$. q.e.d. ###### Corollary 4.2.48 Let $F$ a subset of ${\cal F}\backslash\\{o\\}$ be a weak right reductive Gröbner basis. Then every $g\in{\sf ideal}_{r}(F)$ has a right reductive standard representation in terms of $F$ of the form $g=\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}$ such that ${\sf HT}(g)={\sf HT}({\sf HT}(f_{1})\star m_{1})={\sf HT}(f_{1}\star m_{1})\geq{\sf HT}(f_{1})$ and ${\sf HT}(f_{1}\star m_{1})\succ{\sf HT}(f_{2}\star m_{2})\succ\ldots\succ{\sf HT}(f_{n}\star m_{n})$. Proof : This follows from inspecting the proof of Lemma 4.2.47. q.e.d. Another consequence of Lemma 4.2.47 is the characterization of weak right reductive Gröbner bases in rewriting terms. ###### Lemma 4.2.49 A subset $F$ of ${\cal F}\backslash\\{o\\}$ is a weak right reductive Gröbner basis if for all $g\in{\sf ideal}_{r}(F)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Now to find some analogon to s-polynomials in ${\cal F}$ we again study what polynomial multiples occur when changing arbitrary representations of right ideal elements into right reductive standard representations. Given a generating set $F\subseteq{\cal F}$ of a right ideal in ${\cal F}$ the key idea in order to characterize weak right Gröbner bases is to distinguish special elements of ${\sf ideal}_{r}(F)$ which have representations $\sum_{i=1}^{n}f_{i}\star m_{i}$, $f_{i}\in F$, $m_{i}\in{\sf M}({\cal F})$ such that the head terms ${\sf HT}(f_{i}\star m_{i})$ are all the same within the representation. Then on one hand the respective coefficients ${\sf HC}(f_{i}\star m_{i})$ can add up to zero which means that the sum of the head coefficients is in an appropriate module in ${\sf R}$ — m-polynomials are related to these situations (see also Definition 4.2.8). If the result is not zero the sum of the coefficients ${\sf HC}(f_{i}\star m_{i})$ can be described in terms of a (weak) right Gröbner basis in ${\sf R}$ — g-polynomials are related to these situations. Zero divisors in the reduction ring eliminating the head monomial of a polynomial occur as a special instance of m-polynomials where $F=\\{f\\}$ and $f\cdot\alpha$, $\alpha\in{\sf R}$ are considered. The first problem is related to solving linear homogeneous equations in ${\sf R}$ and to the existence of possibly finite bases of the respective modules. In case we want effectiveness, we have to require that these bases are computable. The g-polynomials can successfully be treated when possibly finite (weak) right Gröbner bases exist for finitely generated right ideals in ${\sf R}$. Here, in case we want effectiveness, we have to require that the (weak) right Gröbner bases as well as representations for their elements in terms of the generating set are computable. Using m- and g-polynomials, weak right Gröbner bases can again be characterized as in Section 3.5. The definition of m- and g-polynomials is inspired by Definition 3.5.5. One main difference however is that in function rings multiples of one polynomial by different terms can result in the same head terms for the multiples while the multiples themselves are different. These multiples have to be treated as different ones contributing to the same overlap although they arise from the same polynomial. Hence when looking at sets of polynomials we now have to assume that we have multisets which can contain polynomials more than once. Additionally, while in Definition 3.5.5 we can restrict our attention to overlaps equal to the maximal head term of the polynomials involved now we have to introduce the overlapping term as an additional variable. ###### Definition 4.2.50 Let $P=\\{p_{1},\ldots,p_{k}\\}$ be a multiset of not necessarily different polynomials in ${\cal F}$ and $t$ an element in ${\cal T}$ such that there are $w_{1},\ldots,w_{k}\in{\cal T}$ with ${\sf HT}(p_{i}\star w_{i})={\sf HT}({\sf HT}(p_{i})\star w_{i})=t\geq{\sf HT}(p_{i})$, for all $1\leq i\leq k$. Further let $\gamma_{i}={\sf HC}(p_{i}\star w_{i})$ for $1\leq i\leq k$. Let $G$ be a (weak) right Gröbner basis of $\\{\gamma_{1},\ldots,\gamma_{k}\\}$ in ${\sf R}$ with respect to $\Longrightarrow$. Additionally let $\alpha=\sum_{i=1}^{k}\gamma_{i}\cdot\beta_{i}^{\alpha}$ for $\alpha\in G$, $\beta^{\alpha}_{i}\in{\sf R}$, $1\leq i\leq k$. Then we define the g-polynomials (Gröbner polynomials) corresponding to $p_{1},\ldots,p_{k}$ and $t$ by setting $g_{\alpha}=\sum_{i=1}^{k}p_{i}\star w_{i}\cdot\beta^{\alpha}_{i}.$ Notice that ${\sf HM}(g_{\alpha})=\alpha\cdot t$. For the right module $M=\\{(\delta_{1},\ldots,\delta_{k})\mid\sum_{i=1}^{k}\gamma_{i}\cdot\delta_{i}=0\\}$, let the set $\\{B_{j}\mid j\in I_{M}\\}$ be a basis with $B_{j}=(\beta_{j,1},\ldots,\beta_{j,k})$ for $\beta_{j,l}\in{\sf R}$ and $1\leq l\leq k$. Then we define the m-polynomials (module polynomials) corresponding to $P$ and $t$ by setting $h_{j}=\sum_{i=1}^{k}p_{i}\star w_{i}\cdot\beta_{j,i}\mbox{ for each }j\in I_{M}.$ Notice that ${\sf HT}(h_{j})\prec t$ for each $j\in I_{M}$. $\diamond$ Given a set of polynomials $F$, the set of corresponding g- and m-polynomials contains those which are specified by Definition 4.2.50 for each term $t\in{\cal T}$ fulfilling the respective conditions. For a set consisting of one polynomial the corresponding m-polynomials reflect the multiplication of the polynomial with zero-divisors of the head monomial, i.e., by a basis of the annihilator of the head monomial. Notice that given a finite set of polynomials the corresponding sets of g- and m-polynomials in general can be infinite. As in Theorem 4.2.23 we can use g- and m-polynomials instead of s-polynomials to characterize special bases in function rings. As before we also have to take into account right multiples of the generating set as Example 4.2.18 does not require a field as coefficient domain. ###### Theorem 4.2.51 Let $F$ be a set of polynomials in ${\cal F}\backslash\\{o\\}$. Then $F$ is a weak right Gröbner basis of ${\sf ideal}_{r}(F)$ if and only if 1. 1. for all $f$ in $F$ and for all $m$ in ${\sf M}({\cal F})$, $f\star m$ has a right reductive standard representation in terms of $F$, and 2. 2. all g- and m-polynomials corresponding to $F$ as specified in Definition 4.2.50 have right reductive standard representations in terms of $F$. Proof : In case $F$ is a weak right Gröbner basis it is also a right reductive standard basis, and since the multiples $f\star m$ and the respective g- and m-polynomials are all elements of ${\sf ideal}_{r}(F)$ they must have right reductive standard representations. The converse will be proven by showing that every element in ${\sf ideal}_{r}(F)$ has a right reductive standard representation in terms of $F$. Let $g\in{\sf ideal}_{r}(F)$ have a representation in terms of $F$ of the following form: $g=\sum_{j=1}^{m}f_{j}\star(w_{j}\cdot\alpha_{j})$ such that $f_{j}\in F$, $w_{j}\in{\cal T}$ and $\alpha_{j}\in{\sf R}$. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(f_{j}\star(w_{j}\cdot\alpha_{j}))\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $f_{j}\star(w_{j}\cdot\alpha_{j})$ with head term $t$. We show our claim by induction on $(t,K)$, where $(t^{\prime},K^{\prime})<(t,K)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $K^{\prime}<K)$. Since by our first assumption every multiple $f_{j}\star(w_{j}\cdot\alpha_{j})$ in this sum has a right reductive standard representation in terms of $F$, we can assume that ${\sf HT}({\sf HT}(f_{j})\star w_{j})={\sf HT}(f_{j}\star w_{j})\geq{\sf HT}(f_{j})$ holds. Moreover, without loss of generality we can assume that the polynomial multiples with head term $t$ are just $f_{1}\star w_{1},\ldots,f_{K}\star w_{K}$. Notice that these assumptions on the representation of $g$ neither change $t$ nor $K$. Obviously, $t\succeq{\sf HT}(g)$ must hold. If $K=1$ this gives us $t={\sf HT}(g)$ and by our assumptions our representation is already a right reductive one and we are done. Hence let us assume $K>1$. First let $\sum_{j=1}^{K}{\sf HM}(f_{j}\star(w_{j}\cdot\alpha_{j}))=o$. Then by Definition 4.2.50 there exists a tuple $(\alpha_{1},\ldots,\alpha_{K})\in M$, as $\sum_{j=1}^{K}{\sf HC}(f_{j}\star w_{j})\cdot\alpha_{j}=0$. Hence there are $\delta_{1},\ldots,\delta_{K}\in{\sf R}$ such that $\sum_{i=1}^{l}A_{i}\cdot\delta_{i}=(\alpha_{1},\ldots,\alpha_{K})$ for some $l\in{\mathbb{N}}$, $A_{i}=(\alpha_{i,1},\ldots,\alpha_{i,K})\in\\{A_{j}\mid j\in I_{M}\\}$, and $\alpha_{j}=\sum_{i=1}^{l}\alpha_{i,j}\cdot\delta_{i}$, $1\leq j\leq K$. By our assumption there are module polynomials $h_{i}=\sum_{j=1}^{K}f_{j}\star w_{j}\cdot\alpha_{i,j}$,$1\leq i\leq l$, all having right reductive standard representations in terms of $F$. Then since $\displaystyle\sum_{j=1}^{K}f_{j}\star(w_{j}\cdot\alpha_{j})$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{K}f_{j}\star w_{j}\cdot(\sum_{i=1}^{l}\alpha_{i,j}\cdot\delta_{i})$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{K}\sum_{i=1}^{l}(f_{j}\star w_{j}\cdot\alpha_{i,j})\cdot\delta_{i}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{l}(\sum_{j=1}^{K}f_{j}\star w_{j}\cdot\alpha_{i,j})\cdot\delta_{i}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{l}h_{i}\cdot\delta_{i}$ we can change the representation of $g$ to $\sum_{i=1}^{l}h_{i}\cdot\delta_{i}+\sum_{j=K+1}^{m}f_{j}\star(w_{j}\cdot\alpha_{j})$ and replace each $h_{i}$ by its right reductive standard representation in terms of $F$. Remember that for all $h_{i}$, $1\leq i\leq l$ we have ${\sf HT}(h_{i})\prec t$. Hence, for this new representation we now have maximal term smaller than $t$ and by our induction hypothesis we have a right reductive standard representation for $g$ in terms of $F$ and are done. It remains to study the case where $\sum_{j=1}^{K}{\sf HM}(f_{j}\star(w_{j}\cdot\alpha_{j}))\neq 0$. Then we have ${\sf HT}(f_{1}\star(w_{1}\cdot\alpha_{1})+\dots+f_{K}\star(w_{K}\cdot\alpha_{K}))=t={\sf HT}(g)$, ${\sf HC}(g)={\sf HC}(f_{1}\star(w_{1}\cdot\alpha_{1})+\dots+f_{K}\star(w_{K}\cdot\alpha_{K}))\in{\sf ideal}_{r}(\\{{\sf HC}(f_{1}\star w_{1}),\ldots,{\sf HC}(f_{K}\star w_{K})\\})$ and ${\sf HM}(f_{1}\star(w_{1}\cdot\alpha_{1})+\dots+f_{K}\star(w_{K}\cdot\alpha_{K}))={\sf HM}(g)$. Hence ${\sf HC}(g)=\alpha\cdot\delta$ with $\delta\in{\sf R}$ and $\alpha\in G$313131Remember that we assume the reduction relation $\Longrightarrow$ on ${\sf R}$ based on division, see the remark after Definition 4.2.43., $G$ being a (weak) right Gröbner basis of ${\sf ideal}_{r}(\\{{\sf HC}(f_{1}\star w_{1}),\ldots,{\sf HC}(f_{K}\star w_{K})\\})$ (compare Definition 4.2.50). Let $g_{\alpha}$ be the respective g-polynomial corresponding to $\alpha$. Then the polynomial $g^{\prime}=g-g_{\alpha}\cdot\delta$ lies in ${\sf ideal}_{r}(F)$. Since the multiple323232Note that right reductive standard representations are stable under multiplication with coefficients which are no zero-divisors of the head coefficient. $g_{\alpha}\cdot\delta$ has a right reductive standard representation in terms of $F$, say $\sum_{j=1}^{l}f_{j}\star m_{j}$, for the situation $\sum_{j=1}^{K}f_{j}\star(w_{j}\cdot\alpha_{j})-f_{1}\star m_{1}$ all polynomial multiples involved in this sum have head term $t$ and their head monomials add up to $o$. Therefore, this situation again corresponds to an m-polynomial of $F$. Hence we can apply our results from above and get that the polynomial $g^{\prime}$ has a smaller representation than $g$, especially the maximal term $t^{\prime}$ is smaller. Moreover, we can assume that $g^{\prime}$ has a right reductive standard representation in terms of $F$, say $g^{\prime}=\sum_{i=1}^{n}f_{i}\star\tilde{m}_{i}$. Then $g=\sum_{i=1}^{n}f_{i}\star\tilde{m}_{i}+g_{\alpha}\cdot\delta=\sum_{i=1}^{n}f_{i}\star\tilde{m}_{i}+\sum_{j=1}^{l}f_{j}\star m_{j}$ is a right reductive standard representation of $g$ in terms of $F$ and we are done. q.e.d. Since in general we will have infinitely many g- and m-polynomials related to $F$, it is important to look for possible localizations of these situations. We are looking for concepts similar to those of weak saturation and stable localizations in the previous section. Remember that Lemma 4.2.26 is central there. It describes when the existence of a right reductive standard representation for some polynomial implies the existence of a right reductive standard representation for a multiple of the polynomial. Unfortunately we cannot establish an analogon to this lemma for right reductive standard representations in ${\cal F}$ as defined in Definition 4.2.42. ###### Example 4.2.52 Let ${\cal F}$ be a function ring over the integers with ${\cal T}=\\{X_{1},\ldots,X_{7}\\}$ and multiplication $\star:{\cal T}\times{\cal T}\mapsto{\cal F}$ defined by the following equations: $X_{1}\star X_{2}=X_{4}$, $X_{4}\star X_{3}=X_{5}$, $X_{2}\star X_{3}=X_{6}+X_{7}$, $X_{1}\star X_{6}=3\cdot X_{5}$, $X_{1}\star X_{7}=-2\cdot X_{5}$ and else $X_{i}\star X_{j}=o$. Additionally let $X_{5}>X_{4}>X_{1}\succ X_{2}\succ X_{3}\succ X_{6}\succ X_{7}$. Then for $p=X_{4}$, $f=X_{1}$ and $m=X_{3}$ we find that 1. 1. $p$ has a right reductive standard representation in terms of $\\{f\\}$, namely $p=f\star X_{2}$. 2. 2. ${\sf HT}(p\star m)={\sf HT}({\sf HT}(p)\star m)\geq{\sf HT}(p)$ as $X_{5}=X_{4}\star X_{3}>X_{4}$ and for all $X_{i}\prec X_{4}$ we have $X_{i}\star X_{3}\prec X_{5}$. 3. 3. $p\star m=X_{5}$ has no right reductive standard representation in terms of $\\{f\\}$ as only $X_{1}\star X_{j}\neq o$ for $j=\\{2,6,7\\}$, namely $X_{1}\star X_{2}=X_{4}$, $X_{1}\star X_{6}=3\cdot X_{5}$, $X_{1}\star X_{7}=-2\cdot X_{5}$, and $X_{1}\star(X_{j}\cdot\alpha)\neq X_{5}$ for all $j\in\\{2,6,7\\}$, $\alpha\in{\mathbb{Z}}$. Notice that these problems are due to the fact that while $(X_{1}\star X_{2})\star X_{3}=X_{1}\star(X_{2}\star X_{3})=X_{5}$, $X_{1}\star(X_{2}\star X_{3})=X_{1}\star(X_{6}+X_{7})=X_{1}\star X_{6}+X_{1}\star X_{7}$ does not give us a right reductive standard representation in terms of $X_{1}$ as ${\sf HT}(X_{1}\star X_{6})=X_{5}$ and ${\sf HT}(X_{1}\star X_{7})=X_{5}$ (compare Definition 4.2.42). This was the crucial point in the proof of Lemma 4.2.26 and it is only fulfilled for the weaker form of right reductive standard representations in ${\cal F}_{{\mathbb{K}}}$ as defined in Definition 4.2.15. $\diamond$ As this example shows an analogon to Lemma 4.2.26 does not hold in our general case. Note that the trouble arises from the fact that we allow multiplication of two terms to result in a polynomial. If we restrict ourselves to multiplications where multiples of monomials are again monomials, the proof of Lemma 4.2.26 carries over and we can look for appropriate localizations. However, the reduction relation defined in Definition 4.2.43 is only one way of defining a reduction relation in ${\cal F}$ and we stated that the main motivation behind it is to link the reduction relation with special standard representations as it is done in the case of ${\cal F}_{{\mathbb{K}}}$. The question now arises whether this motivation is as appropriate for ${\cal F}$ as it was for ${\cal F}_{{\mathbb{K}}}$. In ${\cal F}_{{\mathbb{K}}}$ any reduction relation based on stable divisibility of terms can be linked to right reductive standard representations as defined in Definition 4.2.15 and hence the approach is very powerful. It turns out that for different reduction relations in ${\cal F}$ based on stable right divisibility this is no longer so. Let us look at another familiar way of generalizing a reduction relation for ${\cal F}$ from one defined in the reduction ring. From now on we require a (not necessarily Noetherian) partial ordering on ${\sf R}$: for $\alpha,\beta\in{\sf R}$, $\alpha>_{{\sf R}}\beta$ if and only if there exists a finite set $B\subseteq{\sf R}$ such that $\alpha\mbox{$\,\stackrel{{\scriptstyle+}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\beta$. This ordering ensures that reduction in ${\cal F}$ is terminating when using a finite set of polynomials. ###### Definition 4.2.53 Let $f,p$ be two non-zero polynomials in ${\cal F}$. We say $f$ right reduces $p$ to $q$ at a monomial $\alpha\cdot t$ in one step, denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}q$, if there exists $m\in{\sf M}({\cal F})$ such that 1. 1. $t\in{\sf supp}(p)$ and $p(t)=\alpha$, 2. 2. ${\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)=t\geq{\sf HT}(f)$, 3. 3. $\alpha\Longrightarrow_{{\sf HC}(f\star m)}\beta$, with $\alpha={\sf HC}(f\star m)+\beta$ for some $\beta\in{\sf R}$, and 4. 4. $q=p-f\star m$. We write $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}$ if there is a polynomial $q$ as defined above and $p$ is then called right reducible by $f$. Further, we can define $\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$},\mbox{$\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$}$ and $\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$ as usual. Right reduction by a set $F\subseteq{\cal F}\backslash\\{o\\}$ is denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q$ and abbreviates $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}q$ for some $f\in F$. $\diamond$ Notice that in specifying this reduction relation we use a special instance of $\alpha\Longrightarrow_{{\sf HC}(f\star m)}\beta$, namely the case that $\alpha={\sf HC}(f\star m)+\beta$ for some $\beta\in{\sf R}$. Moreover, for this reduction relation we can still have $t\in{\sf supp}(q)$. Hence other arguments than used in the proof of Lemma 4.2.44 have to be provided to show termination. It turns out that for infinite subsets of polynomials $F$ in ${\cal F}$ the reduction relation $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$ need not terminate. ###### Example 4.2.54 Let ${\sf R}={\mathbb{Q}}[\\{X_{i}\mid i\in{\mathbb{N}}\\}]$ with $X_{1}\succ X_{2}\succ\ldots$ be the polynomial ring over the rationals with infinitely many indeterminates. We associate this ring with the reduction relation based on divisibility of terms. Let ${\cal F}={\sf R}[Y]$ be our function ring. Elements of ${\cal F}$ are polynomials in $Y^{i}$, $i\in{\mathbb{N}}$ with coefficients in ${\sf R}$. Then for $p=X_{1}\cdot Y$ and the infinite set $F=\\{f_{i}=(X_{i}-X_{i+1})\cdot Y\mid i\in{\mathbb{N}}\\}$ we get the infinite reduction sequence $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{1}}\,$}X_{2}\cdot Y\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{2}}\,$}X_{3}\cdot Y\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{3}}\,$}\ldots$ $\diamond$ However, if we restrict ourselves to finite sets of polynomials the reduction relation is Noetherian. ###### Lemma 4.2.55 Let $F$ be a finite set of polynomials in ${\cal F}\backslash\\{o\\}$. 1. 1. For $p,q\in{\cal F}$ $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q$ implies $p\succ q$, in particular ${\sf HT}(p)\succeq{\sf HT}(q)$. 2. 2. $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$ is Noetherian. Proof : 1. 1. Assuming that the reduction step takes place at a monomial $\alpha\cdot t$, by Definition 4.2.53 we know ${\sf HM}(\alpha\cdot t-f\star m)=\beta\cdot t$ which yields $p\succ p-f\star m$ since $\alpha>_{{\sf R}}\beta$. 2. 2. This follows from 1. and Axiom (A1) as long as only finite sets of polynomials are involved. Since we have ${\sf HT}(f\star m)={\sf HT}({\sf HT}(f)\star m)\geq{\sf HT}(f)$ we get ${\sf HC}(f\star m)={\sf HC}(f)\cdot{\sf HC}({\sf HT}(f)\star m)$. Then $\alpha\Longrightarrow_{{\sf HC}(f\star m)}\beta$ implies $\alpha\Longrightarrow_{{\sf HC}(f)}$. Hence an infinite reduction sequence would give rise to an infinite reduction sequence in ${\sf R}$ with respect to the finite set of head coefficients $\\{{\sf HC}(f)\mid f\in F\\}$ contradicting our assumption. q.e.d. Now if we try to link the reduction relation in Definition 4.2.53 to special standard representations, we find that this is no longer as natural as in the cases studied before, where for ${\cal F}_{{\mathbb{K}}}$ we linked the reduction relation from Definition 4.2.27 to the right reductive standard representations in Definition 4.2.15 respectively for ${\cal F}$ the right reduction relation from Definition 4.2.43 to right reductive standard representations as defined in Definition 4.2.42. Hence we claim that for generalizing Gröbner bases to ${\cal F}$, the rewriting approach is more suitable. Hence we use the following definition of weak right Gröbner bases in terms of our reduction relation. ###### Definition 4.2.56 A set $F\subseteq{\cal F}\backslash\\{o\\}$ is called a weak right Gröbner basis (with respect to $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$) of ${\sf ideal}_{r}(F)$ if for all $g\in{\sf ideal}_{r}(F)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. $\diamond$ Every reduction sequence $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$ gives rise to a special representation of $g$ in terms of $F$ which could be taken as a new definition of standard representations. ###### Corollary 4.2.57 Let $F$ be a set of polynomials in ${\cal F}$ and $g$ a non-zero polynomial in ${\sf ideal}_{r}(F)$ such that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Then $g$ has a representation of the form $g=\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}$ such that ${\sf HT}(g)={\sf HT}({\sf HT}(f_{i})\star m_{i})={\sf HT}(f_{i}\star m_{i})\geq{\sf HT}(f_{i})$ for $1\leq i\leq k$, and ${\sf HT}(g)\succ{\sf HT}(f_{i}\star m_{i})$ for all $k+1\leq i\leq n$. Proof : We show our claim by induction on $n$ where $g\mbox{$\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. If $n=0$ we are done. Else let $g\mbox{$\,\stackrel{{\scriptstyle 1}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g_{1}\mbox{$\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. In case the reduction step takes place at the head monomial, there exists a polynomial $f\in F$ and a monomial $m\in{\sf M}({\cal F})$ such that ${\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)={\sf HT}(g)\geq{\sf HT}(f)$ and ${\sf HC}(g)\Longrightarrow_{{\sf HC}(f\star m)}\beta$ with ${\sf HC}(g)={\sf HC}(f\star m)+\beta$ for some $\beta\in{\sf R}$. Moreover the induction hypothesis then is applied to $g_{1}=g-f\star m\cdot\beta$. If the reduction step takes place at a monomial with term smaller ${\sf HT}(g)$ for the respective monomial multiple $f\star m$ we immediately get ${\sf HT}(g)\succ{\sf HT}(f\star m)$ and we can apply our induction hypothesis to the resulting polynomial $g_{1}$. In both cases we can arrange the monomial multiples $f\star m$ arising from the reduction steps in such a way that gives us th desired representation. q.e.d. Notice that on the other hand the existence of such a representation for a polynomial does not imply reducibility. For example take the polynomial ring ${\mathbb{Z}}[X]$ with Pan’s reduction. Then with respect to the polynomials $F=\\{2\cdot X,3\cdot X\\}$ the polynomial $g=5\cdot X$ has a representation $5\cdot X=2\cdot X+3\cdot X$ of the desired form but is neither reducible by $2\cdot X$ nor $3\cdot X$. This is of course a consequence of the fact that $\\{2,3\\}$ is no Gröbner basis in ${\mathbb{Z}}$ with respect to Pan’s reduction. In fact Corollary 4.2.57 provides additional information for the head coefficient of $g$, namely ${\sf HC}(g)=\sum_{i=1}^{k}{\sf HC}(f_{i})\cdot{\sf HC}(m_{i})$ and this is a standard representation of ${\sf HC}(g)$ in terms of $\\{{\sf HC}(f_{i})\mid 1\leq i\leq k\\}$ in the reduction ring ${\sf R}$. We can characterize weak right Gröbner bases similar to Theorem 4.2.51. Of course the g-polynomials in Definition 4.2.50 depend on the reduction relation $\Longrightarrow$ in ${\sf R}$ which now is defined according to Definition 4.2.53. Notice that the characterization will only hold for finite sets as the proof requires the reduction relation to be Noetherian. Additionally we need that the reduction ring fulfills Axiom (A4), i.e., for $\alpha,\beta,\gamma,\delta\in{\sf R}$, $\alpha\Longrightarrow_{\beta}$ and $\beta\Longrightarrow_{\gamma}\delta$ imply $\alpha\Longrightarrow_{\gamma}$ or $\alpha\Longrightarrow_{\delta}$333333Notice that (A4) is no basis for localizing test sets, as this would require that $\alpha\Longrightarrow_{\beta}$ and $\beta\Longrightarrow_{\gamma}\delta$ imply $\alpha\Longrightarrow_{\gamma}$. Hence even if the reduction relation in ${\cal F}$ satisfies (A4), this does not substitute Lemma 4.2.26 or its variants.. ###### Theorem 4.2.58 Let $F$ be a finite set of polynomials in ${\cal F}\backslash\\{o\\}$ where the reduction ring satisfies (A4). Then $F$ is a weak right Gröbner basis of ${\sf ideal}_{r}(F)$ if and only if 1. 1. for all $f$ in $F$ and for all $m$ in ${\sf M}({\cal F})$ we have $f\star m\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, and 2. 2. all g- and m-polynomials corresponding to $F$ as specified in Definition 4.2.50 reduce to $o$ using $F$. Proof : In case $F$ is a weak right Gröbner basis, since the multiples $f\star m$ and the respective g- and m-polynomials are all elements of ${\sf ideal}_{r}(F)$ they must reduce to zero using $F$. The converse will be proven by showing that every element in ${\sf ideal}_{r}(F)$ is reducible by $F$. Then as $g\in{\sf ideal}_{r}(F)$ and $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g^{\prime}$ implies $g^{\prime}\in{\sf ideal}_{r}(F)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Notice that this only holds in case the reduction relation $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$ is Noetherian. This follows as by our assumption $F$ is finite (Lemma 4.2.55). Let $g\in{\sf ideal}_{r}(F)$ have a representation in terms of $F$ of the following form: $g=\sum_{j=1}^{m}f_{j}\star(w_{j}\cdot\alpha_{j})$ such that $f_{j}\in F$, $w_{j}\in{\cal T}$, $\alpha_{j}\in{\sf R}$. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(f_{j}\star(w_{j}\cdot\alpha_{j}))\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $f_{j}\star(w_{j}\cdot\alpha_{j})$ with head term $t$. We show our claim by induction on $(t,K)$, where $(t^{\prime},K^{\prime})<(t,K)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $K^{\prime}<K)$. Since by our first assumption every multiple $f_{j}\star(w_{j}\cdot\alpha_{j})$ in this sum reduces to zero using $F$ and hence has a right representation as defined in Corollary 4.2.57, we can assume that ${\sf HT}({\sf HT}(f_{j})\star w_{j})={\sf HT}(f_{j}\star w_{j})\geq{\sf HT}(f_{j})$ holds. Moreover, without loss of generality we can assume that the polynomial multiples with head term $t$ are just $f_{1}\star(w_{1}\cdot\alpha_{1}),\ldots,f_{K}\star(w_{K}\cdot\alpha_{K})$. Notice that these assumptions neither change $t$ nor $K$ for our representation of $g$. Obviously, $t\succeq{\sf HT}(g)$ must hold. If $K=1$ this gives us $t={\sf HT}(g)$ and even ${\sf HM}(g)={\sf HM}(f_{1}\star(w_{1}\cdot\alpha_{1}))$, implying that $g$ is right reducible at ${\sf HM}(g)$ by $f_{1}$. Hence let us assume $K>1$. First let $\sum_{j=1}^{K}{\sf HM}(f_{j}\star(w_{j}\cdot\alpha_{j}))=o$. Then by Definition 4.2.50 we know $(\alpha_{1},\ldots,\alpha_{K})\in M$, as $\sum_{j=1}^{K}{\sf HC}(f_{j}\star w_{j})\cdot\alpha_{j}=0$. Hence there are $\delta_{1},\ldots,\delta_{K}\in{\sf R}$ such that $\sum_{i=1}^{l}A_{i}\cdot\delta_{i}=(\alpha_{1},\ldots,\alpha_{K})$ for some $l\in{\mathbb{N}}$, $A_{i}=(\alpha_{i,1},\ldots,\alpha_{i,K})\in\\{A_{j}\mid j\in I_{M}\\}$, and $\alpha_{j}=\sum_{i=1}^{l}\alpha_{i,j}\cdot\delta_{i}$, $1\leq j\leq K$. By our assumption there are module polynomials $h_{i}=\sum_{j=1}^{K}f_{j}\star w_{j}\cdot\alpha_{i,j}$,$1\leq i\leq l$, all having representations in terms of $F$ as defined in Corollary 4.2.57. Then since $\displaystyle\sum_{j=1}^{K}f_{j}\star(w_{j}\cdot\alpha_{j})$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{K}f_{j}\star w_{j}\cdot(\sum_{i=1}^{l}\alpha_{i,j}\cdot\delta_{i})$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{K}\sum_{i=1}^{l}(f_{j}\star w_{j}\cdot\alpha_{i,j})\cdot\delta_{i}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{l}(\sum_{j=1}^{K}f_{j}\star w_{j}\cdot\alpha_{i,j})\cdot\delta_{i}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{l}h_{i}\cdot\delta_{i}$ we can change the representation of $g$ to $\sum_{i=1}^{l}h_{i}\cdot\delta_{i}+\sum_{j=K+1}^{m}f_{j}\star(w_{j}\cdot\alpha_{j})$ and replace each $h_{i}$ by its respective representation in terms of $F$. Remember that for all $h_{i}$, $1\leq i\leq l$ we have ${\sf HT}(h_{i})\prec t$. Hence, for this new representation we now have maximal term smaller than $t$ and by our induction hypothesis $g$ is reducible by $F$ and we are done. It remains to study the case where $\sum_{j=1}^{K}{\sf HM}(f_{j}\star(w_{j}\cdot\alpha_{j}))\neq 0$. Then we have ${\sf HT}(f_{1}\star(w_{1}\cdot\alpha_{1})+\dots+f_{K}\star(w_{K}\cdot\alpha_{K}))=t={\sf HT}(g)$, ${\sf HC}(g)={\sf HC}(f_{1}\star(w_{1}\cdot\alpha_{1})+\dots+f_{K}\star(w_{K}\cdot\alpha_{K}))\in{\sf ideal}_{r}(\\{{\sf HC}(f_{1}\star w_{1}),\ldots,{\sf HC}(f_{K}\star w_{K})\\})$ and even ${\sf HM}(f_{1}\star(w_{1}\cdot\alpha_{1})+\dots+f_{K}\star(w_{K}\cdot\alpha_{K}))={\sf HM}(g)$. Hence ${\sf HC}(g)$ is $\Longrightarrow$-reducible by some $\alpha$, $\alpha\in G$, a (weak) right Gröbner basis of ${\sf ideal}_{r}(\\{{\sf HC}(f_{1}\star w_{1}),\ldots,{\sf HC}(f_{K}\star w_{K})\\})$ in ${\sf R}$ with respect to the reduction relation $\Longrightarrow$. Let $g_{\alpha}$ be the respective g-polynomial corresponding to $\alpha$ and $t$. Then we know that $g_{\alpha}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Moreover, we know that the head monomial of $g_{\alpha}$ is reducible by some polynomial $f\in F$ and we assume ${\sf HT}(g_{\alpha})={\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)\geq{\sf HT}(f)$ and ${\sf HC}(g_{\alpha})\Longrightarrow_{{\sf HC}(f\star m)}$. Then, as ${\sf HC}(g)$ is $\Longrightarrow$-reducible by ${\sf HC}(g_{\alpha})$, ${\sf HC}(g_{\alpha})$ is $\Longrightarrow$-reducible and (A4) holds, the head monomial of $g$ is also reducible by some $f^{\prime}\in F$ and we are done. q.e.d. Of course this theorem is also true for infinite $F$ if we can show that for the respective function ring the reduction relation is terminating. Now the question arises when the critical situations in this characterization can be localized to subsets of the respective sets as in Theorem 4.2.41. Reviewing the Proof of Theorem 4.2.41 we find that Lemma 4.2.26 is central as it describes when multiples of polynomials which have a right reductive standard representation in terms of some set $F$ again have such a representation. As we have seen above, this will not hold for function rings over reduction rings in general. Now one way to introduce localizations would be to restrict the attention to those ${\cal F}$ satisfying Lemma 4.2.26. Then appropriate adaptions of Definition 4.2.34, 4.2.35 and 4.2.38 would allow a localization of the critical situations. However, we have stated that it is not natural to link right reduction as defined in Definition 4.2.43 to special standard representations. Hence, to give localizations of Theorem 4.2.58 another property for ${\cal F}$ is sufficient: ###### Definition 4.2.59 A set $C\subset S\subseteq{\cal F}$ is called a stable localization of $S$ if for every $g\in S$ there exists $f\in C$ such that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}o$. $\diamond$ In case ${\cal F}$ and $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$ allow such stable localizations, we can rephrase Theorem 4.2.58 as follows: ###### Theorem 4.2.60 Let $F$ be a finite set of polynomials in ${\cal F}\backslash\\{o\\}$ where the reduction ring satisfies (A4). Then $F$ is a weak right Gröbner basis of ${\sf ideal}_{r}(F)$ if and only if 1. 1. for all $s$ in a stable localization of $\\{f\star m\mid f\in{\cal F},m\in{\sf M}({\cal F})\\}$ we have $s\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, and 2. 2. for all $h$ in a stable localization of the g- and m-polynomials corresponding to $F$ as specified in Definition 4.2.50 we have $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. We have stated that for arbitrary reduction relations in ${\cal F}$ it is not natural to link them to special standard representations. Still, when proving Theorem 4.2.60, we will find that in order to change the representation of an arbitrary right ideal element, Definition 4.2.59 is not enough to ensure reducibility. However, we can substitute the critical situation using an analogon of Lemma 4.2.26, which, while not related to reducibility, in this case will still be sufficient to make the representation smaller. ###### Lemma 4.2.61 Let $F$ be a subset of polynomials in ${\cal F}\backslash\\{o\\}$ and $f$, $p$ non-zero polynomials in ${\cal F}$. If $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}o$ and $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, then $p$ has a standard representation of the form $p=\sum_{i=1}^{n}f_{i}\star l_{i},f_{i}\in F,l_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}$ such that ${\sf HT}(p)={\sf HT}({\sf HT}(f_{i})\star l_{i})={\sf HT}(f_{i}\star l_{i})\geq{\sf HT}(f_{i})$ for $1\leq i\leq k$ and ${\sf HT}(p)\succ{\sf HT}(f_{i}\star l_{i})$ for all $k+1\leq i\leq n$ (compare Definition 4.2.15). Proof : If $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}o$ then $p=f\star m$ with $m\in{\sf M}({\cal F})$ and ${\sf HT}(p)={\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)\geq{\sf HT}(f)$. Similarly $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$ implies $f=\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}$ such that ${\sf HT}(f)={\sf HT}({\sf HT}(f_{i})\star m_{i})={\sf HT}(f_{i}\star m_{i})\geq{\sf HT}(f_{1})$, $1\leq i\leq k$, and ${\sf HT}(f)\succ{\sf HT}(f_{i}\star m_{i})$ for all $k+1\leq i\leq n$ (compare Corollary 4.2.57). Let us first analyze $f_{i}\star m_{i}\star m$ with ${\sf HT}(f_{i}\star m_{i})={\sf HT}(f)$, $1\leq i\leq k$. Let ${\sf T}(f_{i}\star m_{i})=\\{s_{1}^{i},\ldots,s_{k_{i}}^{i}\\}$ with $s_{1}^{i}\succ s_{j}^{i}$, $2\leq j\leq k_{i}$, i.e., $s_{1}^{i}={\sf HT}(f_{i}\star m_{i})={\sf HT}({\sf HT}(f_{i})\star m_{i})={\sf HT}(f)$. Hence ${\sf HT}(f)\star m=s_{1}^{i}\star m\geq{\sf HT}(f)=s_{1}^{i}$ and as $s_{1}^{i}\succ s_{j}^{i}$, $2\leq j\leq k_{i}$, by Definition 4.2.13 we can conclude that ${\sf HT}({\sf HT}(f)\star m)={\sf HT}(s_{1}^{i}\star m)\succ s_{j}^{i}\star m\succeq{\sf HT}(s_{j}^{i}\star m)$ for $2\leq j\leq k_{i}$. This implies ${\sf HT}({\sf HT}(f_{i}\star m_{i})\star m)={\sf HT}(f_{i}\star m_{i}\star m)$. Hence we get $\displaystyle{\sf HT}(f\star m)$ $\displaystyle=$ $\displaystyle{\sf HT}({\sf HT}(f)\star m)$ $\displaystyle=$ $\displaystyle{\sf HT}({\sf HT}(f_{i}\star m_{i})\star m),\mbox{ as }{\sf HT}(f)={\sf HT}(f_{i}\star m_{i})$ $\displaystyle=$ $\displaystyle{\sf HT}(f_{i}\star m_{i}\star m)$ and since ${\sf HT}(f\star m)\geq{\sf HT}(f)\geq{\sf HT}(f_{i})$ we can conclude ${\sf HT}(f_{i}\star m_{i}\star m)\geq{\sf HT}(f_{i})$. It remains to show that the $f_{i}\star m_{i}\star m$ have representations of the desired form in terms of $F$. First we show that ${\sf HT}({\sf HT}(f_{i})\star m_{i}\star m)\geq{\sf HT}(f_{i})$. We know ${\sf HT}(f_{i})\star m_{i}\succeq{\sf HT}({\sf HT}(f_{i})\star m_{i})={\sf HT}(f_{i}\star m_{i})$343434Notice that ${\sf HT}(f_{i})\star m_{i}$ can be a polynomial and hence we cannot conclude ${\sf HT}(f_{i})\star m_{i}={\sf HT}({\sf HT}(f_{i})\star m_{i})$. and hence ${\sf HT}({\sf HT}(f_{i})\star m_{i}\star m)={\sf HT}({\sf HT}(f_{i}\star m_{i})\star m)={\sf HT}(f_{i}\star m_{i}\star m)\geq{\sf HT}(f_{i})$. Then in case $m_{i}\star m\in{\sf M}({\cal F})$ we are done as then $f_{i}\star(m_{i}\star m)$ is a representation of the desired form. Hence let us assume $m_{i}\star m=\sum_{r=1}^{k_{i}}\tilde{m}^{i}_{r}$, $\tilde{m}^{i}_{r}\in{\sf M}({\cal F})$. Let ${\sf T}(f_{i})=\\{t^{i}_{1},\ldots,t^{i}_{w_{i}}\\}$ with $t^{i}_{1}\succ t^{i}_{l}$, $2\leq l\leq w_{i}$, i.e., $t^{i}_{1}={\sf HT}(f_{i})$. As ${\sf HT}({\sf HT}(f_{i})\star m_{i})\geq{\sf HT}(f_{i})\succ t_{l}^{i}$, $2\leq l\leq w_{i}$, again by Definition 4.2.13 we can conclude that ${\sf HT}({\sf HT}(f_{i})\star m_{i})\succ t^{i}_{l}\star m_{i}\succeq{\sf HT}(t^{i}_{l}\star m_{i})$, $2\leq l\leq w_{i}$, and ${\sf HT}(f_{i})\star m_{i}\succ\sum_{l=2}^{w_{i}}t_{l}^{i}\star m_{i}$. Then for each $s_{j}^{i}$, $2\leq j\leq k_{i}$, there exists $t_{l}^{i}\in{\sf T}(f_{i})$ such that $s\in{\sf supp}(t_{l}^{i}\star m_{i})$. Since ${\sf HT}(f)\succ s_{j}^{i}$ and even ${\sf HT}(f)\succ t_{l}^{i}\star m_{i}$ we find that either ${\sf HT}(f\star m)\succeq{\sf HT}((t_{l}^{i}\star m_{i})\star m)={\sf HT}(t_{l}^{i}\star(m_{i}\star m))$ in case ${\sf HT}(t_{l}^{i}\star m_{i})={\sf HT}(f_{i}\star m_{i})$ or ${\sf HT}(f\star m)\succ(t_{l}^{i}\star m_{i})\star m=t_{l}^{i}\star(m_{i}\star m)$. Hence we can conclude $f_{i}\star\tilde{m}^{i}_{r}\preceq{\sf HT}(f\star m)$, $1\leq r\leq k_{i}$ and for at least one $\tilde{m}^{i}_{r}$ we get ${\sf HT}(f_{i}\star\tilde{m}^{i}_{r})={\sf HT}(f_{i}\star m_{i}\star m)\geq{\sf HT}(f_{i})$. It remains to analyze the situation for the function $(\sum_{i=k+1}^{n}f_{i}\star m_{i})\star m$. Again we find that for all terms $s$ in the $f_{i}\star m_{i}$, $k+1\leq i\leq n$, we have ${\sf HT}(f)\succ s$ and we get ${\sf HT}(f\star m)\succ{\sf HT}(s\star m)$. Hence all polynomial multiples of the $f_{i}$ in the representation $\sum_{i=k+1}^{n}\sum_{j=1}^{k_{i}}f_{i}\star\tilde{m}^{i}_{j}$, where $m_{i}\star m=\sum_{j=1}^{k_{i}}\tilde{m}^{i}_{j}$, are bounded by ${\sf HT}(f\star m)$. q.e.d. Now we are able to prove Theorem 4.2.60. Proof of Theorem 4.2.60: The proof is basically the same as for Theorem 4.2.58. Due to Lemma 4.2.61 we can substitute the multiples $f_{j}\star m_{j}$ by appropriate representations without changing $(t,K)$. Hence, we only have to ensure that despite testing less polynomials we are able to apply our induction hypothesis. Taking the notations from the proof of Theorem 4.2.58, let us first check the situation for m-polynomials. Let $\sum_{j=1}^{K}{\sf HM}(f_{j}\star(w_{j}\cdot\alpha_{j}))=o$. Then by Definition 4.2.50 we know $(\alpha_{1},\ldots,\alpha_{K})\in M$, as $\sum_{j=1}^{K}{\sf HC}(f_{j}\star w_{j})\cdot\alpha_{j}=0$. Hence there are $\delta_{1},\ldots,\delta_{K}\in{\sf R}$ such that $\sum_{i=1}^{l}A_{i}\cdot\delta_{i}=(\alpha_{1},\ldots,\alpha_{K})$ for some $l\in{\mathbb{N}}$, $A_{i}=(\alpha_{i,1},\ldots,\alpha_{i,K})\in\\{A_{j}\mid j\in I_{M}\\}$, and $\alpha_{j}=\sum_{i=1}^{l}\alpha_{i,j}\cdot\delta_{i}$, $1\leq j\leq K$. There are module polynomials $h_{i}=\sum_{j=1}^{K}f_{j}\star w_{j}\cdot\alpha_{i,j}$,$1\leq i\leq l$ and by our assumption there are polynomials $h_{i}^{\prime}$ in the stable localization such that $h_{i}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{h_{i}^{\prime}}\,$}o$. Moreover, $h_{i}^{\prime}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Then by Lemma 4.2.61 the m-polynomials $h_{i}$ all have representations bounded by $t$. Again we get $\displaystyle\sum_{j=1}^{K}f_{j}\star(w_{j}\cdot\alpha_{j})$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{K}f_{j}\star w_{j}\cdot(\sum_{i=1}^{l}\alpha_{i,j}\cdot\delta_{i})$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{K}\sum_{i=1}^{l}(f_{j}\star w_{j}\cdot\alpha_{i,j})\cdot\delta_{i}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{l}(\sum_{j=1}^{K}f_{j}\star w_{j}\cdot\alpha_{i,j})\cdot\delta_{i}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{l}h_{i}\cdot\delta_{i}$ and we can change the representation of $g$ to $\sum_{i=1}^{l}h_{i}\cdot\delta_{i}+\sum_{j=K+1}^{m}f_{j}\star(w_{j}\cdot\alpha_{j})$ and replace each $h_{i}$ by the respective special standard representation in terms of $F$. Remember that for all $h_{i}$, $1\leq i\leq l$ we have ${\sf HT}(h_{i})\prec t$. Hence, for this new representation we now have maximal term smaller than $t$ and by our induction hypothesis $g$ is reducible by $F$ and we are done. It remains to study the case where $\sum_{j=1}^{K}{\sf HM}(f_{j}\star(w_{j}\cdot\alpha_{j}))\neq 0$. Then we have ${\sf HT}(f_{1}\star(w_{1}\cdot\alpha_{1})+\dots+f_{K}\star(w_{K}\cdot\alpha_{K}))=t={\sf HT}(g)$, ${\sf HC}(g)={\sf HC}(f_{1}\star(w_{1}\cdot\alpha_{1})+\dots+f_{K}\star(w_{K}\cdot\alpha_{K}))\in{\sf ideal}_{r}(\\{{\sf HC}(f_{1}\star w_{1}),\ldots,{\sf HC}(f_{K}\star w_{K})\\})$ and even ${\sf HM}(f_{1}\star(w_{1}\cdot\alpha_{1})+\dots+f_{K}\star(w_{K}\cdot\alpha_{K}))={\sf HM}(g)$. Hence ${\sf HC}(g)$ is $\Longrightarrow$-reducible by some $\alpha$, $\alpha\in G$, $G$ being a (weak) right Gröbner basis of ${\sf ideal}_{r}(\\{{\sf HC}(f_{1}\star w_{1}),\ldots,{\sf HC}(f_{K}\star w_{K})\\})$ in ${\sf R}$ with respect to the reduction relation $\Longrightarrow$. Let $g_{\alpha}$ be the respective g-polynomial corresponding to $\alpha$ and $t$. Then we know that $g_{\alpha}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{g_{\alpha}^{\prime}}\,$}o$ for some $g_{\alpha}^{\prime}$ in the stable localization and $g_{\alpha}^{\prime}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Moreover, we know that the head monomial of $g_{\alpha}^{\prime}$ is reducible by some polynomial $f\in F$ and we assume ${\sf HT}(g_{\alpha})={\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)\geq{\sf HT}(f)$ and ${\sf HC}(g_{\alpha})\Longrightarrow_{{\sf HC}(f\star m)}$. Then, as ${\sf HC}(g)$ is $\Longrightarrow$-reducible by ${\sf HC}(g_{\alpha})$, ${\sf HC}(g_{\alpha})$ is $\Longrightarrow$-reducible by ${\sf HC}(g_{\alpha}^{\prime})$, ${\sf HC}(g_{\alpha}^{\prime})$ is $\Longrightarrow$-reducible to zero and (A4) holds, the head monomial of $g$ is also reducible by some $f^{\prime}\in F$ and we are done. q.e.d. Again, if for infinite $F$ we can assure that the reduction relation is Noetherian, the proof still holds. #### 4.2.3 Function Rings over the Integers In the previous section we have seen that for the reduction relations for ${\cal F}$ as defined in Definition 4.2.43 and 4.2.53 the Translation Lemma no longer holds. This is due to the fact that the first definition is based on divisibility in ${\sf R}$ and hence too weak and the second definition is based on the abstract notion of the reduction relation $\Longrightarrow$ and hence there is not enough information on the reduction step involving the coefficient. When studying special reduction rings where we have more information on the specific reduction relation $\Longrightarrow$ the situation often can be improved. Here we want to go into the details for the case that ${\sf R}$ is the ring of the integers ${\mathbb{Z}}$. Remember that there are various ways of defining a reduction relation for the integers. In Example 3.1.1 two possibilities are presented. Here we want to use the second one based on division with remainders in order to introduce a reduction relation to ${\cal F}_{{\mathbb{Z}}}$. We follow the ideas presented in [MR93b] for characterizing prefix Gröbner bases in monoid rings ${\mathbb{Z}}[{\cal M}]$ where ${\cal M}$ is presented by a finite convergent string rewriting system. In order to use elements of ${\cal F}_{{\mathbb{Z}}}$ as rules for a reduction relation we need an ordering on ${\mathbb{Z}}$. We specify a total well–founded ordering on ${\mathbb{Z}}$ as follows353535If not stated otherwise $<$ is the usual ordering on ${\mathbb{Z}}$, i.e. $\ldots<-3<-2<-1<0<1<2<3\ldots$.: $\alpha<_{Z}\beta\mbox{ iff }\left\\{\begin{array}[]{l}\alpha\geq 0\mbox{ and }\beta<0\\\ \alpha\geq 0,\beta>0\mbox{ and }\alpha<\beta\\\ \alpha<0,\beta<0\mbox{ and }\alpha>\beta\end{array}\right.$ and $\alpha\leq_{Z}\beta$ iff $\alpha=\beta$ or $\alpha<_{Z}\beta$. Hence we get $0\leq_{Z}1\leq_{Z}2\leq_{Z}3\leq_{Z}\ldots\leq_{Z}-1\leq_{Z}-2\leq_{Z}-3\leq_{Z}\ldots$. Then we can make the following important observation: Let $\gamma\in{\mathbb{N}}$. We call the positive numbers $0,\ldots,\gamma-1$ the remainders of $\gamma$. Then for each $\delta\in{\mathbb{Z}}$ there are unique $\alpha,\beta\in{\mathbb{Z}}$ such that $\delta=\alpha\cdot\gamma+\beta$ and $\beta$ is a remainder of $\gamma$. We get $\beta<\gamma$ and in case $\delta>0$ and $\alpha\not=0$ even $\gamma\leq\delta$. Further $\gamma$ does not divide $\beta_{1}-\beta_{2}$, if $\beta_{1},\beta_{2}$ are different remainders of $\gamma$. As we will later on only use polynomials with head coefficients in ${\mathbb{N}}$ for reduction, we will mainly require the part of the ordering on ${\mathbb{N}}$ which then coincides with the natural ordering on this set. Then we will drop the suffix363636In the literature other orderings on the integers are used by Buchberger and Stifter [Sti87] and Kapur and Kandri-Rody [KRK88]. They then have to consider s- and t-polynomials as critical situations.. This ordering $<_{Z}$ can be used to induce an ordering on ${\cal F}_{{\mathbb{Z}}}$ as follows: for two elements $f,g$ in ${\cal F}$ we define $f\succ g$ iff ${\sf HT}(f)\succ{\sf HT}(g)$ or ($({\sf HT}(f)={\sf HT}(g)$ and ${\sf HC}(f)>_{Z}{\sf HC}(g)$) or ($({\sf HM}(f)={\sf HM}(g)$ and ${\sf RED}(f)\succ{\sf RED}(g)$). The reduction relation presented in Definition 4.2.53 now can be adapted to this special case: Let $\Longrightarrow$ be our reduction relation on ${\mathbb{Z}}$ where $\alpha\Longrightarrow_{\gamma}\beta$, if $\gamma>0$ and for some $\delta\in{\mathbb{Z}}$ we have $\alpha=\gamma\cdot\delta+\beta$ with $0\leq\beta<\gamma$, i.e. $\beta$ is the remainder of $\alpha$ modulo $\gamma$. ###### Definition 4.2.62 Let $p$, $f$ be two non-zero polynomials in ${\cal F}_{{\mathbb{Z}}}$. We say $f$ right reduces $p$ to $q$ at a monomial $\alpha\cdot t$ in one step, i.e. $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}q$, if there exists $s\in{\sf T}({\cal F}_{{\mathbb{Z}}})$ such that 1. 1. $t\in{\sf supp}(p)$ and $p(t)=\alpha$, 2. 2. ${\sf HT}({\sf HT}(f)\star s)={\sf HT}(f\star s)=t\geq{\sf HT}(f)$, 3. 3. $\alpha\geq_{{\mathbb{Z}}}{\sf HC}(f\star m)>0$ and $\alpha\Longrightarrow_{{\sf HC}(f\star s)}\delta$ where $\alpha={\sf HC}(f\star s)\cdot\beta+\delta$ with $\beta,\delta\in{\mathbb{Z}}$, $0\leq\delta<{\sf HC}(f\star s)$, and 4. 4. $q=p-f\star m$ where $m=\beta\cdot s$. We write $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}$ if there is a polynomial $q$ as defined above and $p$ is then called right reducible by $f$. Further, we can define $\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$},\mbox{$\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$}$ and $\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$ as usual. Right reduction by a set $F\subseteq{\cal F}\backslash\\{o\\}$ is denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q$ and abbreviates $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}q$ for some $f\in F$. $\diamond$ As before, for this reduction relation we can still have $t\in{\sf supp}(q)$. Hence other arguments than those used in the proof of Lemma 4.2.44 have to be used to show termination. The important part now is that if we still have $t\in{\sf supp}(q)$ then its coefficient will be smaller according to our ordering $<_{{\mathbb{Z}}}$ chosen for ${\mathbb{Z}}$ and since this ordering is well-founded we are done. Notice that in contrary to Lemma 4.2.55 we do not have to restrict ourselves to finite sets of polynomials in order to ensure termination. The additional information we have on the coefficients before and after the reduction step now enables us to prove an analogon of the Translation Lemma for function rings over the integers. The first and second part of the lemma are only needed to prove the essential third part. ###### Lemma 4.2.63 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{Z}}}$ and $p,q,h$ polynomials in ${\cal F}_{{\mathbb{Z}}}$. 1. 1. Let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}h$ such that the reduction step takes place at the monomial $\alpha\cdot t$ and we additionally have $t\not\in{\sf supp}(h)$. Then there exist $p^{\prime},q^{\prime}\in{\cal F}_{{\mathbb{Z}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}p^{\prime}$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q^{\prime}$ and $h=p^{\prime}-q^{\prime}$. 2. 2. Let $o$ be the unique normal form of $p$ with respect to $F$ and $t={\sf HT}(p)$. Then there exists a polynomial $f\in F$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}p^{\prime}$ and $t\not\in{\sf supp}(p^{\prime})$. 3. 3. Let $o$ be the unique normal form of $p-q$ with respect to $F$. Then there exists $g\in{\cal F}_{{\mathbb{Z}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g$. Proof : 1. 1. Let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}h$ at the monomial $\alpha\cdot t$, i.e., $h=p-q-f\star m$ for some $m=\beta\cdot s\in{\sf M}({\cal F}_{{\mathbb{Z}}})$ such that ${\sf HT}({\sf HT}(f)\star s)={\sf HT}(f\star s)=t\geq{\sf HT}(f)$ and ${\sf HC}(f\star s)>0$. Remember that $\alpha$ is the coefficient of $t$ in $p-q$. Then as $t\not\in{\sf supp}(h)$ we know $\alpha={\sf HC}(f\star m)$. Let $\alpha_{1}$ respectively $\alpha_{2}$ be the coefficients of $t$ in $p$ respectively $q$ and $\alpha_{1}={\sf HC}(f\star m)\cdot\beta_{1}+\gamma_{1}$ respectively $\alpha_{2}={\sf HC}(f\star m)\cdot\beta_{2}+\gamma_{2}$ for some $\beta_{1},\beta_{2},\gamma_{1},\gamma_{2}\in{\mathbb{Z}}$ where $0\leq\gamma_{1},\gamma_{2}<{\sf HC}(f\star s)\leq{\sf HC}(f\star m)$. Then $\alpha={\sf HC}(f\star m)=\alpha_{1}-\alpha_{2}={\sf HC}(f\star m)\cdot(\beta_{1}-\beta_{2})+(\gamma_{1}-\gamma_{2})$, and as $\gamma_{1}-\gamma_{2}$ is no multiple of ${\sf HC}(f\star m)$ we have $\gamma_{1}-\gamma_{2}=0$ and hence $\beta_{1}-\beta_{2}=1$. We have to distinguish two cases: 1. (a) $\beta_{1}\neq 0$ and $\beta_{2}\neq 0$: Then $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}p-f\star m\cdot\beta_{1}=p^{\prime}$, $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q-f\star m\cdot\beta_{2}=q^{\prime}$ and $p^{\prime}-q^{\prime}=p-f\star m\cdot\beta_{1}-q+f\star m\cdot\beta_{2}=p-q-f\star m=h$. 2. (b) $\beta_{1}=0$ and $\beta_{2}=-1$ (the case $\beta_{2}=0$ and $\beta_{1}=1$ being symmetric): Then $p^{\prime}=p$, $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q-f\star m\cdot\beta_{2}=q+f\star m\cdot\beta=q^{\prime}$ and $p^{\prime}-q^{\prime}=p-q-f\star m=h$. 2. 2. Since $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, ${\sf HM}(p)=\alpha\cdot t$ must be $F$-reducible. Let $f_{1},\ldots,f_{k}\in F$ be all polynomials in $F$ such that $\alpha\cdot t$ is reducible by them. Let $m_{1},\ldots m_{k}$ be the respective monomials involved in possible reduction steps. Moreover, let $\gamma=\min_{1\leq i\leq k}\\{{\sf HC}(f_{i}\star m_{i})\\}$ and without loss of generality ${\sf HM}(f\star m)=\gamma\cdot t$ for some $f\in F$, ${\sf HT}({\sf HT}(f)\star m)={\sf HT}(f\star m)\geq{\sf HT}(f)$. We claim that for $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{1}}\,$}p-f\star m=p^{\prime}$ we have $t\not\in{\sf supp}(p^{\prime})$. Suppose ${\sf HT}(p^{\prime})=t$. Then by our definition of reduction we must have $0<{\sf HC}(p^{\prime})<{\sf HC}(f\star m)$. But then $p^{\prime}$ would no longer be $F$-reducible contradicting our assumption that $o$ is the unique normal form of $p$. 3. 3. Since $o$ is the unique normal form of $p-q$ by 2. there exists a reduction sequence $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{i_{1}}}\,$}h_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{i_{2}}}\,$}\ldots\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f_{i_{k}}}\,$}o$ such that for the head terms we get ${\sf HT}(p-q)\succ{\sf HT}(h_{1})\succ\ldots$. We show our claim by induction on $k$, where $p-q\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$ is such a reduction sequence. In the base case $k=0$ there is nothing to show as then $p=q$. Hence, let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}h\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Then by 1. there are polynomials $p^{\prime},q^{\prime}\in{\cal F}_{{\mathbb{Z}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}p^{\prime}$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}q^{\prime}$ and $h=p^{\prime}-q^{\prime}$. Now the induction hypothesis for $p^{\prime}-q^{\prime}\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$ yields the existence of a polynomial $g\in{\cal F}_{{\mathbb{Z}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g$. q.e.d. Hence weak Gröbner bases are in fact Gröbner bases and can be characterized as follows: ###### Definition 4.2.64 A set $F\subseteq{\cal F}_{{\mathbb{Z}}}\backslash\\{o\\}$ is called a (weak) right Gröbner basis of ${\sf ideal}_{r}(F)$ if for all $g\in{\sf ideal}_{r}(F)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. $\diamond$ ###### Corollary 4.2.65 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{Z}}}$ and $g$ a non- zero polynomial in ${\sf ideal}_{r}(F)$ such that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Then $g$ has a representation of the form $g=\sum_{i=1}^{n}f_{i}\star m_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}_{{\mathbb{Z}}}),n\in{\mathbb{N}}$ such that ${\sf HT}(g)={\sf HT}({\sf HT}(f_{i})\star m_{i})={\sf HT}(f_{i}\star m_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq k$, and ${\sf HT}(g)\succ{\sf HT}(f_{i}\star m_{i})={\sf HT}({\sf HT}(f_{i})\star m_{i})$ for all $k+1\leq i\leq n$. Proof : We show our claim by induction on $n$ where $g\mbox{$\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. If $n=0$ we are done. Else let $g\mbox{$\,\stackrel{{\scriptstyle 1}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g_{1}\mbox{$\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. In case the reduction step takes place at the head monomial, there exists a polynomial $f\in F$ and a monomial $m=\beta\cdot s\in{\sf M}({\cal F})$ such that ${\sf HT}({\sf HT}(f)\star s)={\sf HT}(f\star s)={\sf HT}(g)\geq{\sf HT}(f)$ and ${\sf HC}(g)\Longrightarrow_{{\sf HC}(f\star s)}\delta$ with ${\sf HC}(g)={\sf HC}(f\star s)\cdot\beta+\delta$ for some $\beta,\delta\in{\mathbb{Z}}$, $0\leq\delta<{\sf HC}(f\star s)$. Moreover the induction hypothesis then is applied to $g_{1}=g-f\star m$. If the reduction step takes place at a monomial with term smaller ${\sf HT}(g)$ for the respective monomial multiple $f\star m$ we immediately get ${\sf HT}(g)\succ{\sf HT}(f\star m)$ and we can apply our induction hypothesis to the resulting polynomial $g_{1}$. In both cases we can arrange the monomial multiples $f\star m$ arising from the reduction steps in such a way that gives us the desired representation. q.e.d. We can even state that ${\sf HC}(g)\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}_{\\{{\sf HC}(f_{i}\star m_{i})\mid 1\leq i\leq k\\}}\,$}0$. Now right Gröbner bases can be characterized using the concept of s-polynomials combined with the technique of saturation which is necessary in order to describe the whole right ideal congruence by the reduction relation. ###### Definition 4.2.66 Let $p_{1},p_{2}$ be two polynomials in ${\cal F}_{{\mathbb{Z}}}$. If there are respective terms $t,u_{1},u_{2}\in{\cal T}$ such that ${\sf HT}({\sf HT}(p_{i})\star u_{i})={\sf HT}(p_{i}\star u_{i})=t\geq{\sf HT}(p_{i})$ let $HC(p_{i}\star u_{i})=\gamma_{i}$. Assuming $\gamma_{1}\geq\gamma_{2}>0$373737Notice that $\gamma_{i}>0$ can always be achieved by studying the situation for $-p_{i}$ in case we have $HC(p_{i}\star u_{i})<0$., there are $\beta,\delta\in{\mathbb{Z}}$ such that $\gamma_{1}=\gamma_{2}\cdot\beta+\delta$ and $0\leq\delta<\gamma_{2}$ and we get the following s-polynomial ${\sf spol}_{r}(p_{1},p_{2},t,u_{1},u_{2})=p_{2}\star u_{2}\cdot\beta- p_{1}\star u_{1}.$ The set ${\sf SPOL}(\\{p_{1},p_{2}\\})$ then is the set of all such s-polynomials corresponding to $p_{1}$ and $p_{2}$. $\diamond$ These sets can be infinite383838This is due to the fact that in general we cannot always find finite locations for $t$. One well-studied field are monoid rings.. ###### Theorem 4.2.67 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{Z}}}\backslash\\{o\\}$. Then $F$ is a right Gröbner basis of ${\sf ideal}_{r}(F)$ if and only if 1. 1. for all $f$ in $F$ and for all $m$ in ${\sf M}({\cal F}_{{\mathbb{Z}}})$ we have $f\star m\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, and 2. 2. all s-polynomials corresponding to $F$ as specified in Definition 4.2.66 reduce to $o$ using $F$. Proof : In case $F$ is a right Gröbner basis, since the multiples $f\star m$ and the respective s-polynomials are all elements of ${\sf ideal}_{r}(F)$ they must reduce to zero using $F$. The converse will be proven by showing that every element in ${\sf ideal}_{r}(F)$ is reducible by $F$. Then as $g\in{\sf ideal}_{r}(F)$ and $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}g^{\prime}$ implies $g^{\prime}\in{\sf ideal}_{r}(F)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. Notice that this is sufficient as the reduction relation $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$ is Noetherian. Let $g\in{\sf ideal}_{r}(F)$ have a representation in terms of $F$ of the following form: $g=\sum_{j=1}^{m}f_{j}\star w_{j}\cdot\alpha_{j}$ such that $f_{j}\in F$, $w_{j}\in{\cal T}$ and $\alpha_{j}\in{\mathbb{Z}}$. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(f_{j}\star w_{j})\mid 1\leq j\leq m\\}$, $K$ as the number of polynomials $f_{j}\star w_{j}$ with head term $t$, and $M=\\{\\{{\sf HC}(f_{i}\star w_{i})\mid{\sf HT}(f_{j}\star w_{j})=t\\}\\}$ a multiset in ${\mathbb{Z}}$. We show our claim by induction on $(t,M)$, where $(t^{\prime},M^{\prime})<(t,M)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $M^{\prime}\ll M)$393939We define $M^{\prime}\ll M$ if $M$ can be transformed into $M^{\prime}$ by substituting elements in $M$ with sets of smaller elements (with respect to our ordering on the integers.. Since by our first assumption every multiple $f_{j}\star w_{j}$ in this sum reduces to zero using $F$ and hence has a representation as specified in Corollary 4.2.65, we can assume that ${\sf HT}({\sf HT}(f_{j})\star w_{j})={\sf HT}(f_{j}\star w_{j})\geq{\sf HT}(f_{j})$ holds. Moreover, without loss of generality we can assume that the polynomial multiples with head term $t$ are just $f_{1}\star w_{1},\ldots,f_{K}\star w_{K}$ and additionally we can assume ${\sf HC}(f_{j}\star w_{j})>0$404040This can easily be achieved by adding $-f$ to $F$ for all $f\in F$ and using $(-f_{j})\star w_{j}$ in case ${\sf HC}(f_{j}\star w_{j})<0$.. Obviously, $t\succeq{\sf HT}(g)$ must hold. If $K=1$ this gives us $t={\sf HT}(g)$ and even ${\sf HM}(g)={\sf HM}(f_{1}\star w_{1}\cdot\alpha_{1})$, implying that $g$ is right reducible at ${\sf HM}(g)$ by $f_{1}$. Hence let us assume $K>1$. Without loss of generality we can assume that ${\sf HC}(f_{1}\star w_{1})\geq{\sf HC}(f_{2}\star w_{2})>0$ and there are $\alpha,\beta\in{\mathbb{Z}}$ such that ${\sf HC}(f_{2}\star w_{2})\cdot\alpha+\beta={\sf HC}(f_{1}\star w_{1})$ and ${\sf HC}(f_{2}\star w_{2})>\beta\geq 0$. Since $t={\sf HT}(f_{1}\star w_{1})={\sf HT}(f_{2}\star w_{2})$ by Definition 4.2.66 we have an s-polynomial ${\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2})=f_{2}\star w_{2}\cdot\alpha-f_{1}\star w_{1}$. If ${\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2})\neq o$414141In case ${\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2})=o$ the proof is similar. We just have to substitute $o$ in the equations below which immediately gives us a smaller representation of $g$. then ${\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$ implies ${\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2})=\sum_{i=1}^{k}\delta_{i}\cdot h_{i}\star v_{i}$, $\delta_{i}\in{\mathbb{Z}}$, $h_{i}\in F$, $v_{i}\in{\cal T}$ where this sum is a representation in the sense of Corollary 4.2.65 with terms bounded by ${\sf HT}({\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2}))\leq t$. This gives us $\displaystyle g$ $\displaystyle=$ $\displaystyle f_{1}\star w_{1}\cdot\alpha_{1}+f_{2}\star w_{2}\cdot\alpha_{2}+\sum_{j=3}^{m}f_{j}\star w_{j}\cdot\alpha_{j}$ $\displaystyle=$ $\displaystyle f_{1}\star w_{1}\cdot\alpha_{1}+\underbrace{f_{2}\star w_{2}\cdot\alpha_{1}\cdot\alpha- f_{2}\star w_{2}\cdot\alpha_{1}\cdot\alpha}_{=o}+f_{2}\star w_{2}\cdot\alpha_{2}+\sum_{j=3}^{m}f_{j}\star w_{j}\cdot\alpha_{j}$ $\displaystyle=$ $\displaystyle f_{2}\star w_{2}\cdot(\alpha_{1}\cdot\alpha+\alpha_{2})-\underbrace{(f_{2}\star w_{2}\cdot\alpha-f_{1}\star w_{1}}_{={\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2})}\cdot\alpha_{1}+\sum_{j=3}^{m}f_{j}\star w_{j}\cdot\alpha_{j}$ $\displaystyle=$ $\displaystyle f_{2}\star w_{2}\cdot(\alpha_{1}\cdot\alpha+\alpha_{2})-(\sum_{i=1}^{k}\delta_{i}\cdot h_{i}\star v_{i})\cdot\alpha_{1}+\sum_{j=3}^{m}f_{j}\star w_{j}\cdot\alpha_{j}$ and depending on this new representation of $g$ we define $t^{\prime}=\max_{\succeq}\\{{\sf HT}(f_{j}\star w_{j}),{\sf HT}(h_{j}\star v_{j})\mid f_{j},h_{j}\mbox{ appearing in the new representation }\\}$, and $M^{\prime}=\\{\\{{\sf HC}(f_{i}\star w_{i}),{\sf HC}(h_{j}\star v_{j})\mid{\sf HT}(f_{j}\star w_{j})={\sf HT}(h_{j}\star v_{j})=t^{\prime}\\}\\}$ and we either get $t^{\prime}\prec t$ and have a smaller representation for $g$ or in case $t^{\prime}=t$ we have to distinguish two cases 1. 1. $\alpha_{1}\cdot\alpha+\alpha_{2}=0$. Then $M^{\prime}=M-\\{\\{{\sf HC}(f_{1}\star w_{1}),{\sf HC}(f_{2}\star w_{2})\\}\\}\cup\\{\\{{\sf HC}(h_{j}\star v_{j})\mid{\sf HT}(h_{j}\star v_{j})=t\\}\\}$. As those polynomials $h_{j}$ with ${\sf HT}(h_{j}\star v_{j})=t$ are used to right reduce the monomial $\beta\cdot t={\sf HM}({\sf spol}_{r}(f_{1},f_{2},t,w_{1},w_{2}))$ we know that for them we have $0<{\sf HC}(h_{j}\star v_{j})\leq\beta<{\sf HC}(f_{2}\star w_{2})\leq{\sf HC}(f_{1}\star w_{1})$. Hence $M^{\prime}\ll M$ and we have a smaller representation for $g$. 2. 2. $\alpha_{1}\cdot\alpha+\alpha_{2}\neq 0$. Then $M^{\prime}=(M-\\{\\{{\sf HC}(f_{1}\star w_{1})\\}\\})\cup\\{\\{{\sf HC}(h_{j}\star v_{j})\mid{\sf HT}(h_{j}\star v_{j})=t\\}\\}$. Again $M^{\prime}\ll M$ and we have a smaller representation for $g$. Notice that the case $t^{\prime}=t$ and $M^{\prime}\ll M$ cannot occur infinitely often but has to result in either $t^{\prime}<t$ or will lead to $t^{\prime}=t$ and $K=1$ and hence to reducibility by $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$. q.e.d. Now the question arises when the critical situations in this characterization can be localized to subsets of the respective sets as in Theorem 4.2.41. Reviewing the Proof of Theorem 4.2.41 we find that Lemma 4.2.26 is central as it describes when multiples of polynomials which have a right reductive standard representation in terms of some set $F$ again have such a representation. As we have seen before, this will not hold for function rings over reduction rings in general. As in Section 4.2.2, to give localizations of Theorem 4.2.67 the concept of stable subsets is sufficient: ###### Definition 4.2.68 A set $C\subset S\subseteq{\cal F}_{{\mathbb{Z}}}$ is called a stable localization of $S$ if for every $g\in S$ there exists $f\in C$ such that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}o$. $\diamond$ Stable localizations for the sets of s-polynomials again arise from the appropriate sets of least common multiples as presented on page 4.2.23. In case ${\cal F}_{{\mathbb{Z}}}$ and $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}\,$ allow such stable localizations, we can rephrase Theorem 4.2.67 as follows: ###### Theorem 4.2.69 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{Z}}}\backslash\\{o\\}$. Then $F$ is a right Gröbner basis of ${\sf ideal}_{r}(F)$ if and only if 1. 1. for all $s$ in a stable localization of $\\{f\star m\mid f\in{\cal F}_{{\mathbb{Z}}},m\in{\sf M}({\cal F}_{{\mathbb{Z}}})\\}$ we have $s\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, and 2. 2. for all $h$ in a stable localization of the s-polynomials corresponding to $F$ as specified in Definition 4.2.66 we have $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$. When proving Theorem 4.2.69, we can substitute the critical situation using an analogon of Lemma 4.2.26, which will be sufficient to make the representation used in the proof smaller. It is a direct consequence of Lemma 4.2.61. ###### Corollary 4.2.70 Let $F\subseteq{\cal F}_{{\mathbb{Z}}}\backslash\\{o\\}$ and $f$, $p$ non-zero polynomials in ${\cal F}_{{\mathbb{Z}}}$. If $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{f}\,$}o$ and $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$, then $p$ has a representation of the form $p=\sum_{i=1}^{n}f_{i}\star l_{i},f_{i}\in F,l_{i}\in{\sf M}({\cal F}_{{\mathbb{Z}}}),n\in{\mathbb{N}}$ such that ${\sf HT}(p)={\sf HT}({\sf HT}(f_{i})\star l_{i})={\sf HT}(f_{i}\star l_{i})\geq{\sf HT}(f_{i})$ for $1\leq i\leq k$ and ${\sf HT}(p)\succ{\sf HT}(f_{i}\star l_{i})$ for all $k+1\leq i\leq n$ (compare Definition 4.2.15). Proof Theorem 4.2.69: The proof is basically the same as for Theorem 4.2.67. Due to Corollary 4.2.70 we can substitute the multiples $f_{j}\star w_{j}$ by appropriate representations. Hence, we only have to ensure that despite testing less polynomials we are able to apply our induction hypothesis. Taking the notations from the proof of Theorem 4.2.67, let us check the situation for $K>1$. Without loss of generality we can assume that ${\sf HC}(f_{1}\star w_{1})\geq{\sf HC}(f_{2}\star w_{2})>0$ and there are $\alpha,\beta\in{\mathbb{Z}}$ such that ${\sf HC}(f_{2}\star w_{2})\cdot\alpha+\beta={\sf HC}(f_{1}\star w_{1})$ and ${\sf HC}(f_{2}\star w_{2})>\beta\geq 0$. Since $t={\sf HT}(f_{1}\star w_{1})={\sf HT}(f_{2}\star w_{2})$ by Definition 4.2.66 we have an s-polynomial $h\in{\sf SPOL}(f_{1},f_{2})$ and $m\in{\sf M}({\cal F}_{{\mathbb{Z}}})$ such that $h\star m=\alpha\cdot f_{2}\star w_{2}-f_{1}\star w_{1}$. If $h\neq o$424242In case $h=o$ the proof is similar. We just have to substitute $o$ in the equations below which immediately gives us a smaller representation of $g$. then by Corollary 4.2.70 $f_{2}\star w_{2}\cdot\alpha-f_{1}\star w_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{h}\,$}o$ and $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{F}\,$}o$ imply $f_{2}\star w_{2}\cdot\alpha-f_{1}\star w_{1}=\sum_{i=1}^{k}h_{i}\star v_{i}\cdot\delta_{i}$, $\delta_{i}\in{\mathbb{Z}}$, $h_{i}\in F$, $v_{i}\in{\cal T}$ where this sum is a representation in the sense of Corollary 4.2.65 with terms bounded by ${\sf HT}(h\star m)\leq t$. As in the proof of Theorem 4.2.67 we now can use this bounded representation to get a smaller representation of $g$ and are done. q.e.d. We close this subsection by outlining how different structures known to allow finite Gröbner bases can be interpreted as function rings. Using the respective interpretations the terminology can be adapted at once to the respective structures and in general the resulting characterizations of Gröbner bases coincide with the results known from literature. ##### Polynomial Rings A commutative polynomial ring ${\mathbb{Z}}[x_{1},\ldots,x_{n}]$ is a function ring according to the following interpretation: • ${\cal T}$ is the set of terms $\\{x_{1}^{i_{1}}\ldots x_{n}^{i_{n}}\mid i_{1},\ldots,i_{n}\in{\mathbb{N}}\\}$. * • $\succ$ can be any admissible term ordering on ${\cal T}$. For the reductive ordering $\geq$ we have $t\geq s$ if $s$ divides $t$ as as term. * • Multiplication $\star$ is specified by the action on terms, i.e. $\star:{\cal T}\times{\cal T}\longrightarrow{\cal T}$ where $x_{1}^{i_{1}}\ldots x_{n}^{i_{n}}\star x_{1}^{j_{1}}\ldots x_{n}^{j_{n}}=x_{1}^{i_{1}+j_{1}}\ldots x_{n}^{i_{n}+j_{n}}$. We do not need the concept of weak saturation. Since the integers are an instance of euclidean domains, similar reductions to those given by Kandri-Rodi and Kapur in [KRK88] arise. A stable localization of ${\cal C}_{s}(p,q)$ is already provided by the tuple corresponding to the least common multiple of the terms ${\sf HT}(p)$ and ${\sf HT}(q)$. In contrast to the s- and t-polynomials studied by Kandri-Rodi and Kapur, we restrict ourselves to s-polynomials as described in Definition 4.2.66. Since this structure is Abelian, one-sided and two-sided ideals coincide. Buchberger’s Algorithm provides an effictive procedure to compute finite Gröbner bases. ##### Non-commutative Polynomial Rings A non-commutative polynomial ring ${\mathbb{Z}}[\\{x_{1},\ldots,x_{n}\\}^{}]$ is a function ring according to the following interpretation: • ${\cal T}$ is the set of words on $\\{x_{1},\ldots,x_{n}\\}$. * • $\succ$ can be any admissible ordering on ${\cal T}$. For the reductive ordering $\geq$ we can chose $t\geq s$ if $s$ is a subword of $t$. * • Multiplication $\star$ is specified by the action on words which is just concatenation. We do not need the concept of weak saturation. A stable localization of ${\cal C}_{s}(p,q)$ is already provided by the tuples corresponding to word overlaps resulting from the equations $u_{1}{\sf HT}(p)v_{1}={\sf HT}(q)$, $u_{2}{\sf HT}(q)v_{2}={\sf HT}(p)$, $u_{3}{\sf HT}(p)={\sf HT}(q)v_{3}$ respectively $u_{4}{\sf HT}(q)={\sf HT}(p)v_{4}$ with the restriction that $\|u_{3}\|<\|{\sf HT}(q)\|$ and $\|u_{4}\|<\|{\sf HT}(p)\|$, $u_{i},v_{i}\in{\cal T}$. The coefficients arise as described in Definition 4.2.66. This structure is not Abelian. For the case of one-sided ideals finite Gröbner bases can be computed. The case of two-sided ideals only allows an enumerating procedure. This is not surprising as the word problem for monoids can be reduced to the problem of computing the respective Gröbner bases (see e.g. [Mor87, MR98d]). ##### Monoid and Group Rings A monoid or group ring ${\mathbb{Z}}[{\cal M}]$ is a function ring according to the following interpretation: * • ${\cal T}$ is the monoid or group ${\cal M}$. In the cases studied by us as well as in [Ros93, Lo96], it is assumed that the elements of the monoid or group have a certain form. This presentation is essential in the approach. We will assume that the given monoid or group is presented by a convergent semi- Thue system. * • $\succ$ will be the completion ordering induced from the presentation of ${\cal M}$ to ${\cal M}$ and hence to ${\cal T}$. The reductive ordering $\geq$ depends on the choice of the presentation. * • Multiplication $\star$ is specified by lifting the monoid or group operation. The concept of weak saturation and the choice of stable localizations of ${\cal C}_{s}(p,q)$ again depend on the choice of the presentation. More on this topic can be found in [Rei95]. ### 4.3 Right ${\cal F}$-Modules The concept of modules arises naturally as a generalization of the concept of an ideal in a ring: Remember that an ideal of a ring is an additive subgroup of the ring which is additionally closed under multiplication with ring elements. Extending this idea to arbitrary additive groups then gives us the concept of modules. In this section we turn our attention to right modules, but left modules can be defined similarly and all results carry over (with the respective modifications of the terms “right” and “left”). Let ${\cal F}$ be a function ring with unit ${\bf 1}$. ###### Example 4.3.1 Let us provide some examples for right ${\cal F}$-modules. 1. 1. Any right ideal in ${\cal F}$ is of course a right ${\cal F}$-module. 2. 2. The set ${\cal M}=\\{{\bf 0}\\}$ with right scalar multiplication ${\bf 0}\star f={\bf 0}$ is a right ${\cal F}$-module called the trivial right ${\cal F}$-module. 3. 3. Given a function ring ${\cal F}$ and a natural number $k$, let ${\cal F}^{k}=\\{(f_{1},\ldots,f_{k})\mid f_{i}\in{\cal F}\\}$ be the set of all vectors of length $k$ with coordinates in ${\cal F}$. Obviously ${\cal F}^{k}$ is an additive commutative group with respect to ordinary vector addition. Moreover, ${\cal F}^{k}$ is a right ${\cal F}$-module with right scalar multiplication $\star:{\cal F}^{k}\times{\cal F}\longrightarrow{\cal F}^{k}$ defined by $(f_{1},\ldots,f_{k})\star f=(f_{1}\star f,\ldots,f_{k}\star f)$. $\diamond$ ###### Definition 4.3.2 A subset of a right ${\cal F}$-module ${\cal M}$ which is again a right ${\cal F}$-module is called a right submodule of ${\cal M}$. $\diamond$ For example any right ideal of ${\cal F}$ is a right submodule of the right ${\cal F}$-module ${\cal F}^{1}$. Provided a set of vectors $S\subset{\cal M}$ the set $\\{\sum_{i=1}^{s}{\bf m}_{i}\star{g_{i}}\mid s\in{\mathbb{N}},g_{i}\in{\cal F},{\bf m}_{i}\in S\\}$ is a right submodule of ${\cal M}$. This set is denoted as $\langle S\rangle_{r}$ and $S$ is called its generating set. If $\langle S\rangle_{r}={\cal M}$ then $S$ is a generating set of the right module itself. If $S$ is finite then ${\cal M}$ is said to be finitely generated. A generating set is called linearly independent or a basis if for all $s\in{\mathbb{N}}$, pairwise different ${\bf m}_{1},\ldots,{\bf m}_{s}\in S$ and $g_{1},\ldots,g_{s}\in{\cal F}$, $\sum_{i=1}^{s}{\bf m}_{i}\star g_{i}={\bf 0}$ implies $g_{1}=\ldots=g_{s}=o$. A right ${\cal F}$-module is called free if it has a basis. The right ${\cal F}$-module ${\cal F}^{k}$ is free and one such basis is the set of unit vectors ${\bf e}_{1}=({\bf 1},o,\ldots,o),{\bf e}_{2}=(o,{\bf 1},o,\ldots,o),\ldots,{\bf e}_{k}=(o,\ldots,o,{\bf 1})$. Using this basis the elements of ${\cal F}^{k}$ can be written uniquely as ${\bf f}=\sum_{i=1}^{k}{\bf e}_{i}\star f_{i}$ where ${\bf f}=(f_{1},\ldots,f_{k})$. Moreover, ${\cal F}^{k}$ has special properties similar to the special case of ${\mathbb{K}}[x_{1},\ldots,x_{n}]$ and we will continue to state some of them. ###### Theorem 4.3.3 Let ${\cal F}$ be right Noetherian. Then every right submodule of ${\cal F}^{k}$ is finitely generated. Proof : Let ${\cal S}$ be a right submodule of ${\cal F}^{k}$. We show our claim by induction on $k$. For $k=1$ we find that ${\cal S}$ is in fact a right ideal in ${\cal F}$ and hence by our hypothesis finitely generated. For $k>1$ let us look at the set $I=\\{f_{1}\mid(f_{1},\ldots,f_{k})\in{\cal S}\\}$. Then again $I$ is a right ideal in ${\cal F}$ and hence finitely generated. Let $\\{g_{1},\ldots,g_{s}\mid g_{i}\in{\cal F}\\}$ be a generating set of $I$. Choose ${\bf g}_{1},\ldots,{\bf g}_{s}\in{\cal S}$ such that the first coordinate of ${\bf g}_{i}$ is $g_{i}$. Similarly, the set $\\{(f_{2},\ldots,f_{k})\mid(o,f_{2},\ldots,f_{k})\in{\cal S}\\}$ is a submodule of ${\cal F}^{k-1}$ and hence finitely generated by some set $\\{(n_{2}^{i},\ldots,n_{k}^{i}),1\leq i\leq w\\}$. Then the set $\\{{\bf g}_{1},\ldots,{\bf g}_{s}\\}\cup\\{{\bf n}_{i}=(o,n_{2}^{i},\ldots,n_{k}^{i})\mid 1\leq i\leq w\\}$ is a generating set for ${\cal S}$. To see this assume ${\bf m}=(m_{1},\ldots,m_{k})\in{\cal S}$. Then $m_{1}=\sum_{i=1}^{s}g_{i}\star h_{i}$ for some $h_{i}\in{\cal F}$ and ${\bf m^{\prime}}={\bf m}-\sum_{i=1}^{s}{\bf g}_{i}\star h_{i}\in{\cal S}$ with first coordinate $o$. Hence ${\bf m^{\prime}}=\sum_{i=1}^{w}{\bf n}_{i}\star l_{i}$ for some $l_{i}\in{\cal F}$ giving rise to ${\bf m}={\bf m^{\prime}}+\sum_{i=1}^{s}{\bf g}_{i}\star h_{i}=\sum_{i=1}^{w}{\bf n}_{i}\star l_{i}+\sum_{i=1}^{s}{\bf g}_{i}\star h_{i}.$ q.e.d. ${\cal F}^{k}$ is called right Noetherian if and only if all its right submodules are finitely generated. If ${\cal F}$ is a right reduction ring, results on the existence of right Gröbner bases for the right submodules carry over from modifications of the proofs in Section 4.3. A natural reduction relation using the right reduction relation in ${\cal F}$ denoted by $\Longrightarrow$ can be defined using the representation as (module) polynomials with respect to the basis of unit vectors as follows: ###### Definition 4.3.4 Let ${\bf f}=\sum_{i=1}^{k}{\bf e}_{i}\star f_{i}$, ${\bf p}=\sum_{i=1}^{k}{\bf e}_{i}\star p_{i}\in{\cal F}^{k}$. We say that ${\bf f}$ reduces ${\bf p}$ to ${\bf q}$ at ${\bf e}_{s}\star p_{s}$ in one step, denoted by ${\bf p}\longrightarrow_{\bf f}{\bf q}$, if 1. 1. $p_{j}=o$ for $1\leq j<s$, 2. 2. $p_{s}\Longrightarrow_{f_{s}}q_{s}$, 3. 3. ${\bf q}$ \| = \| ${\bf p}-\sum_{i=1}^{n}{\bf f}\cdot d_{i}$ ---\|---\|--- \| = \| $(0,\ldots,0,q_{s},p_{s+1}-\sum_{i=1}^{n}f_{s+1}\cdot d,\ldots,p_{k}-\sum_{i=1}^{n}f_{k}\cdot d)$. $\diamond$ Notice that item 2 of this definition is dependant on the definition of the reduction relation $\Longrightarrow$ in ${\cal F}$. If we assume that the reduction relation is the one specified in Definition 4.2.43 we get $p_{s}=q_{s}+f_{s}\cdot d$, $d\in{\sf M}({\cal F})$, but there are other possibilities. Reviewing the introduction of right modules to reduction rings we could substitute 2. by $p_{s}=q_{s}+f_{s}\cdot d$, $d\in{\cal F}$ as well (compare Definition 3.4.8). To show that our reduction relation is terminating we have to extend the ordering from ${\cal F}$ to ${\cal F}^{k}$. For two elements ${\bf p}=(p_{1},\ldots,p_{k})$, ${\bf q}=(q_{1},\ldots,q_{k})\in{\cal F}^{k}$ we define ${\bf p}\succ{\bf q}$ if and only if there exists $1\leq s\leq k$ such that $p_{i}=q_{i}$, $1\leq i<s$, and $p_{s}\succ q_{s}$. ###### Lemma 4.3.5 Let $F$ be a finite set of module polynomials in ${\cal F}^{k}$. 1. 1. For ${\bf p},{\bf q}\in{\cal F}^{k}$ ${\bf p}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}{\bf q}$ implies ${\bf p}\succ{\bf q}$. 2. 2. $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$ is Noetherian in case $\Longrightarrow_{F_{i}}$ is for $1\leq i\leq k$ and $F_{i}=\\{f_{i}\mid f=(f_{1},\ldots,f_{k})\in F\\}$.434343Notice that $F_{i}\subseteq{\cal F}$.. Proof : 1. 1. Assuming that the reduction step takes place at ${\bf e}_{s}\star p_{s}$, by Definition 4.3.4 we know $p_{s}\Longrightarrow_{f_{s}}q_{s}$ and $p_{s}>q_{s}$ implying ${\bf p}\succ{\bf q}$. 2. 2. This follows from 1. and Axiom (A1). q.e.d. ###### Definition 4.3.6 A subset $B$ of ${\cal F}^{k}$ is called a right Gröbner basis of the right submodule ${\cal S}=\langle B\rangle_{r}$, if $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{B}\,$}=\;\;\equiv_{{\cal S}}$ and $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{B}\,$ is convergent. $\diamond$ For any reduction relation in ${\cal F}$ fulfilling the Axioms (A1)–(A3), the following theorem holds. ###### Theorem 4.3.7 If in $({\cal F},\mbox{$\,\stackrel{{\scriptstyle}}{{\Longrightarrow}}\\!\\!\mbox{}\,$})$ every finitely generated right ideal has a finite right Gröbner basis, then the same holds for finitely generated right submodules in $({\cal F}^{k},\longrightarrow)$. Proof : Let ${\cal S}=\langle\\{{\bf s}_{1},\ldots,{\bf s}_{n}\\}\rangle$ be a finitely generated right submodule of ${\cal F}^{k}$. We show our claim by induction on $k$. For $k=1$ we find that ${\cal S}$ is in fact a finitely generated right ideal in ${\cal F}$ and hence by our hypothesis must have a finite right Gröbner basis. For $k>1$ let us look at the set $I=\\{f_{1}\mid(f_{1},\ldots,f_{k})\in{\cal S}\\}$ which is in fact the right ideal generated by $\\{s_{1}^{i}\mid{\bf s}_{i}=(s_{1}^{i},\ldots,s_{k}^{i}),1\leq i\leq n\\}$. Hence $I$ must have a finite right Gröbner basis $H=\\{g_{1},\ldots,g_{s}\mid g_{i}\in{\cal F}\\}$. Choose ${\bf g}_{1},\ldots,{\bf g}_{s}\in{\cal S}$ such that the first coordinate of ${\bf g}_{i}$ is $g_{i}$. Similarly the set ${\cal S}^{\prime}=\\{(f_{2},\ldots,f_{k})\mid(o,f_{2},\ldots,f_{k})\in{\cal S}\\}$ is a submodule in ${\cal F}^{k-1}$ which by our induction hypothesis then must have a finite right Gröbner basis $\\{(\tilde{g}_{2}^{i},\ldots,\tilde{g}_{k}^{i}),1\leq i\leq w\\}$. Then the set $G=\\{{\bf g}_{1},\ldots,{\bf g}_{s}\\}\cup\\{{\bf\tilde{g}}_{i}=(o,\tilde{g}_{2}^{i},\ldots,\tilde{g}_{k}^{i})\mid 1\leq i\leq w\\}$ is a right Gröbner basis for ${\cal S}$. As shown in the proof of Theorem 4.3.3, $G$ is a generating set for ${\cal S}$. It remains to show that $G$ is in fact a right Gröbner basis. First we have to show $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{G}\,$}=\;\;\equiv_{{\cal S}}$. By the definition of the reduction relation in ${\cal F}^{k}$ we immediately find $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{G}\,$}\subseteq\;\;\equiv_{{\cal S}}$. To see the converse let ${\bf p}=(p_{1},\ldots,p_{k})\equiv_{{\cal S}}{\bf q}=(q_{1},\ldots,q_{k})$. Then $p_{1}\equiv_{\langle\\{s^{1}_{i}\mid{\bf s}_{i}=(s_{1}^{i},\ldots,s_{k}^{i}),1\leq i\leq n\\}\rangle_{r}}q_{1}$ and hence by the definition of $G$ we get $p_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{\\{g_{1}^{i}\mid{\bf g}_{i}=(g_{1}^{i},\ldots,g_{k}^{i}),1\leq i\leq s\\}}\,$}q_{1}$. But this gives us ${\bf p}\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{H}\,$}{\bf p}+\sum_{i=1}^{s}{\bf g}_{i}\star r_{i}={\bf p}^{\prime}=(q_{1},{p_{2}}^{\prime},\ldots,{p_{k}}^{\prime})$, $r_{i}\in{\cal F}$, and we get $(q_{1},{p_{2}}^{\prime},\ldots,{p_{k}}^{\prime})\equiv_{{\cal S}}(q_{1},q_{2},\ldots,q_{k})$ and hence $(q_{1},{p_{2}}^{\prime},\ldots,{p_{k}}^{\prime})-(q_{1},q_{2},\ldots,q_{k})=(o,{p_{2}}^{\prime}-q_{2},\ldots,{p_{k}}^{\prime}-q_{k})\in{\cal S}$ implying $({p_{2}}^{\prime}-q_{2},\ldots,{p_{k}}^{\prime}-q_{k})\in{\cal S}^{\prime}$ and $(o,{p_{2}}^{\prime}-q_{2},\ldots,{p_{k}}^{\prime}-q_{k})=\sum_{i=1}^{w}{\bf\tilde{g}}_{i}\star{\eta_{i}}$ for $\eta_{i}\in{\cal F}$. Hence $(q_{1},{p_{2}}^{\prime},\ldots,{p_{k}}^{\prime})$ and $(q_{1},q_{2},\ldots,q_{k})=(q_{1},{p_{2}}^{\prime},\ldots,{p_{k}}^{\prime})-(o,{p_{2}}^{\prime}-q_{2},\ldots,{p_{k}}^{\prime}-q_{k})=(q_{1},{p_{2}}^{\prime},\ldots,{p_{k}}^{\prime})-\sum_{i=1}^{w}{\bf\tilde{g}}_{i}\star{\eta_{i}}$ must be joinable by $\\{{\bf\tilde{g}}_{i}\mid 1\leq i\leq w\\}$ as the restriction of this set without the first coordinate is a right Gröbner basis of ${\cal S}^{\prime}$. Since the reduction relation using the finite set $G$ is terminating we only have to show local confluence. Let us assume there are ${\bf p}$, ${\bf q}_{1}$, ${\bf q}_{2}\in{\cal F}^{k}$ such that ${\bf p}\longrightarrow_{G}{\bf q}_{1}$ and ${\bf p}\longrightarrow_{G}{\bf q}_{2}$. Then by the definition of $G$ the first coordinates $q^{1}_{1}$ and $q^{2}_{1}$ are joinable to some element say $s$ by $H=\\{g_{1},\ldots,g_{s}\\}$ giving rise to the elements ${\bf p}_{1}={\bf q}_{1}+\sum_{i=1}^{s}{\bf g}_{i}\star h_{i}$ and ${\bf p}_{2}={\bf q}_{2}+\sum_{i=1}^{s}{\bf g}_{i}\star\tilde{h}_{i}$ with first coordinate $s$. As before, ${\bf p}_{1}={\bf p}_{2}+\sum_{i=1}^{w}{\bf\tilde{g}}_{i}\star{\eta_{i}}$ and hence ${\bf p}_{1}$ and ${\bf p}_{2}$ must be joinable by $\\{{\bf\tilde{g}}_{i}\mid 1\leq i\leq w\\}$. q.e.d. Now given a right submodule ${\cal S}$ of ${\cal M}$, we can define ${\cal M}/{\cal S}=\\{{\bf f}+{\cal S}\mid{\bf f}\in{\cal M}\\}$. Then with addition defined as $({\bf f}+{\cal S})+({\bf g}+{\cal S})=({\bf f}+{\bf g})+{\cal S}$ the set ${\cal M}/{\cal S}$ is an Abelian group and can be turned into a right ${\cal F}$-module by the action $({\bf f}+{\cal S})\star g={\bf f}\star g+{\cal S}$. ${\cal M}/{\cal S}$ is called the right quotient module of ${\cal M}$ by ${\cal S}$. As usual this quotient can be related to homomorphisms. The results carry over from commutative module theory as can be found in [AL94]. Recall that for two right ${\cal F}$-modules ${\cal M}$ and ${\cal N}$, a function $\phi:{\cal M}\longrightarrow{\cal N}$ is a right ${\cal F}$-module homomorphism if $\phi({\bf f}+{\bf g})=\phi({\bf f})+\phi({\bf g})\mbox{ for all }{\bf f,g}\in{\cal M}$ and $\phi({\bf f})\star g=\phi({\bf f}\star g)\mbox{ for all }{\bf f}\in{\cal M},g\in{\cal F}.$ The homomorphism is called an isomorphism if $\phi$ is one to one and we then write ${\cal M}\cong{\cal N}$. Let ${\cal S}={\rm ker}(\phi)=\\{{\bf f}\in{\cal M}\mid\phi({\bf f})={\bf 0}\\}$. Then ${\cal S}$ is a right submodule of ${\cal M}$ and $\phi({\cal M})$ is a right submodule of ${\cal N}$. Since all are Abelian groups we know ${\cal M}/{\cal S}\cong\phi({\cal M})$ under the mapping ${\cal M}/{\cal S}\longrightarrow\phi({\cal M})$ with ${\bf f}+{\cal S}\mapsto\phi({\bf f})$ which is in fact an isomorphism. All right submodules of the quotient ${\cal M}/{\cal S}$ are of the form ${\cal L}/{\cal S}$ where ${\cal L}$ is a right submodule of ${\cal M}$ containing ${\cal S}$. We can even show that every finitely generated right ${\cal F}$-module is of a special form. ###### Lemma 4.3.8 Every finitely generated right ${\cal F}$-module ${\cal M}$ is isomorphic to ${\cal F}^{k}/{\cal N}$ for some $k\in{\mathbb{N}}$ and some right submodule ${\cal N}$ of ${\cal F}^{k}$. Proof : Let ${\cal M}$ be a finitely generated right ${\cal F}$-module with generating set ${\bf f}_{1},\ldots{\bf f}_{k}\in{\cal M}$. Consider the mapping $\phi:{\cal F}^{k}\longrightarrow{\cal M}$ defined by $\phi(g_{1},\ldots,g_{k})=\sum_{i=1}^{k}{\bf f}_{i}\star g_{i}$. Then $\phi$ is an ${\cal F}$-module homomorphism with image ${\cal M}$. Let ${\cal N}$ be the kernel of $\phi$, then the First Isomorphism Theorem for modules yields our claim. Note that $\phi$ is uniquely defined by specifying the image of each unit vector ${\bf e}_{1},\ldots,{\bf e}_{k}$, namely by $\phi({\bf e}_{i})={\bf f}_{i}$. q.e.d. Now, there are two ways to give a finitely generated right ${\cal F}$-module ${\cal M}\subset{\cal F}^{k}$. One is to be given explicit ${\bf f}_{1},\ldots{\bf f}_{t}\in{\cal F}^{k}$ such that ${\cal M}=\langle\\{{\bf f}_{1},\ldots{\bf f}_{s}\\}\rangle_{r}$. The other way is to give a right submodule ${\cal N}=\langle\\{{\bf g}_{1},\ldots{\bf g}_{s}\\}\rangle_{r}$ for explicit ${\bf g}_{1},\ldots{\bf g}_{s}\in{\cal F}^{k}$ such that ${\cal M}\cong{\cal F}^{k}/{\cal N}$. This is called a presentation of ${\cal M}$. Presentations are chosen when studying right ideals of ${\cal F}$ as right ${\cal F}$-modules. To see how this is done let $\mathfrak{i}$ be the right ideal generated by $\\{f_{1},\ldots,f_{k}\\}$ in ${\cal F}$. Let us consider the right ${\cal F}$-module homomorphism defined as a mapping $\phi:{\cal F}^{k}\longrightarrow\mathfrak{i}$ with $\phi(g_{1},\ldots,g_{k})=\sum_{i=1}^{k}f_{i}\star g_{i}$. Then $\mathfrak{i}\cong{\cal F}^{k}/{\rm ker}(\phi)$ as ${\cal F}$-modules. ${\rm ker}(\phi)$ is called the right syzygy of $\\{f_{1},\ldots,f_{k}\\}$ denoted by ${\rm Syz}(f_{1},\ldots,f_{k})$. In fact ${\rm Syz}(f_{1},\ldots,f_{k})$ is the set of all solutions of the linear equation $f_{1}X_{1}+\ldots+f_{k}X_{k}=o$ in ${\cal F}$. Syzygies play an important role in Gröbner basis theory for ordinary polynomial rings. ### 4.4 Ideals and Standard Representations A subset $\mathfrak{i}\subseteq{\cal F}$ is called a (two-sided) ideal, if 1. 1. $o\in\mathfrak{i}$, 2. 2. for $f,g\in\mathfrak{i}$ we have $f\oplus g\in\mathfrak{i}$, and 3. 3. for $f\in\mathfrak{i}$, $g,h\in{\cal F}$ we have $g\star f\star h\in\mathfrak{i}$. Ideals can also be specified in terms of a generating set. For $F\subseteq{\cal F}\backslash\\{o\\}$ let ${\sf ideal}(F)=\\{\sum_{i=1}^{n}g_{i}\star f_{i}\star h_{i}\mid f_{i}\in F,g_{i},h_{i}\in{\cal F},n\in{\mathbb{N}}\\}=\\{\sum_{i=1}^{m}m_{i}\star f_{i}\star l_{i}\mid f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}\\}$. These generated sets are in fact subsets of ${\cal F}$ since for $f,g\in{\cal F}$ we have that $f\star g$ as well as $f\oplus g$ are again elements of ${\cal F}$, and it is easily checked that they are in fact ideals: 1. 1. $o\in{\sf ideal}(F)$ since $o$ can be written as the empty sum. 2. 2. For two elements $\sum_{i=1}^{n}g_{i}\star f_{i}\star h_{i}$ and $\sum_{i=1}^{m}\tilde{g}_{i}\star\tilde{f}_{i}\star\tilde{h}_{i}$ in ${\sf ideal}(F)$, the sum $\sum_{i=1}^{n}g_{i}\star f_{i}\star h_{i}\oplus\sum_{i=1}^{m}\tilde{g}_{i}\star\tilde{f}_{i}\star\tilde{h}_{i}$ is again an element in ${\sf ideal}(F)$. 3. 3. For an element $\sum_{i=1}^{n}g_{i}\star f_{i}\star h_{i}$ in ${\sf ideal}_{r}(F)$ and two polynomials $g,h$ in ${\cal F}$, the product $g\star(\sum_{i=1}^{n}g_{i}\star f_{i}\star h_{i})\star h=\sum_{i=1}^{n}(g\star g_{i})\star f_{i}\star(h_{i}\star h)$ is again an element in ${\sf ideal}(F)$. Given an ideal $\mathfrak{i}\subseteq{\cal F}$ we call a set $F\subseteq{\cal F}\backslash\\{o\\}$ a basis of $\mathfrak{i}$ if $\mathfrak{i}={\sf ideal}(F)$. Then every element $g\in{\sf ideal}(F)\backslash\\{o\\}$ can have different representations of the form $g=\sum_{i=1}^{n}g_{i}\star f_{i}\star h_{i},f_{i}\in F,g_{i},h_{i}\in{\cal F},n\in{\mathbb{N}}.$ Notice that the $f_{i}$ occurring in this sum are not necessarily different. The distributivity law in ${\cal F}$ allows to convert such a representation into one of the form $g=\sum_{j=1}^{m}m_{i}\star f_{i}\star l_{i},f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}.$ Again special representations can be distinguished in order to characterize special ideal bases. An ordering on ${\cal F}$ is used to define appropriate standard representations. As in the case of right ideals we will first look at generalizations of standard representations for the case of function rings over fields. #### 4.4.1 The Special Case of Function Rings over Fields Let ${\cal F}_{{\mathbb{K}}}$ be a function ring over a field ${\mathbb{K}}$. We first look at an analogon to Definition 4.2.7 ###### Definition 4.4.1 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $g$ a non- zero polynomial in ${\sf ideal}(F)$. A representations of the form $g=\sum_{i=1}^{n}m_{i}\star f_{i}\star l_{i},f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}}),n\in{\mathbb{N}}$ where additionally ${\sf HT}(g)\succeq{\sf HT}(m_{i}\star f_{i}\star l_{i})$ holds for $1\leq i\leq n$ is called a standard representation of $g$ in terms of $F$. If every $g\in{\sf ideal}(F)\backslash\\{o\\}$ has such a representation in terms of $F$, then $F$ is called a standard basis of ${\sf ideal}(F)$. $\diamond$ Notice that since we assume $f\cdot\alpha=\alpha\cdot f$, we can also substitute the monomials $l_{i}$ by terms $w_{i}\in{\cal T}$, i.e. study representations of the form $g=\sum_{i=1}^{n}m_{i}\star f_{i}\star w_{i},f_{i}\in F,m_{i}\in{\sf M}({\cal F}),w_{i}\in{\cal T},n\in{\mathbb{N}}.$ We will use this additional information in some proofs later on. As with right standard representations, in order to change an arbitrary representation of an ideal element into a standard representation we have to deal with special sums of polynomials. We get the following analogon to Definition 4.2.8. ###### Definition 4.4.2 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $t$ an element in ${\cal T}$. Then we define a set ${\cal C}(F,t)$ to contain all tuples of the form $(t,f_{1},\ldots,f_{k},m_{1},\ldots,m_{k},l_{1},\ldots,l_{k})$, $k\in{\mathbb{N}}$, $f_{1},\ldots,f_{k}\in F$, $m_{1},\ldots,m_{k},l_{1},\ldots,l_{k}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that 1. 1. ${\sf HT}(m_{i}\star f_{i}\star l_{i})=t$, $1\leq i\leq k$, and 2. 2. $\sum_{i=1}^{k}{\sf HM}(m_{i}\star f_{i}\star l_{i})=0$. We set ${\cal C}(F)=\bigcup_{t\in{\cal T}}{\cal C}(F,t)$. $\diamond$ Notice that this definition is motivated by the definition of syzygies of head monomials in commutative polynomial rings over rings. We can characterize standard bases using this concept (compare Theorem 4.2.9). ###### Theorem 4.4.3 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a standard basis of ${\sf ideal}(F)$ if and only if for every tuple $(t,f_{1},\ldots,f_{k},m_{1},\ldots,m_{k},l_{1},\ldots,l_{k})$ in ${\cal C}(F)$ the polynomial $\sum_{i=1}^{k}m_{i}\star f_{i}\star l_{i}$ (i.e. the element in ${\cal F}_{{\mathbb{K}}}$ corresponding to this sum) has a standard representation with respect to $F$. Proof : In case $F$ is a standard basis since the polynomials related to the tuples are all elements of ${\sf ideal}(F)$ they must have standard representations with respect to $F$. To prove the converse, it remains to show that every element in ${\sf ideal}(F)$ has a standard representation with respect to $F$. Hence, let $g=\sum_{j=1}^{m}m_{j}\star f_{j}\star l_{j}$ be an arbitrary representation of a non-zero polynomial $g\in{\sf ideal}(F)$ such that $f_{j}\in F$, $m_{j},l_{j}\in{\sf M}({\cal F}_{{\mathbb{K}}})$, $m\in{\mathbb{N}}$. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(m_{j}\star f_{j}\star l_{j})\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $m_{j}\star f_{j}\star l_{j}$ with head term $t$. Then $t\succeq{\sf HT}(g)$ and in case ${\sf HT}(g)=t$ this immediately implies that this representation is already a standard one. Else we proceed by induction on $t$. Without loss of generality let $f_{1},\ldots,f_{K}$ be the polynomials in the corresponding representation such that $t={\sf HT}(m_{j}\star f_{j}\star l_{j})$, $1\leq j\leq K$. Then the tuple $(t,f_{1},\ldots,f_{K},m_{1},\ldots,m_{K},l_{1},\ldots,l_{K})$ is in ${\cal C}(F)$ and let $h=\sum_{j=1}^{K}m_{j}\star f_{j}\star l_{j}$. We will now change our representation of $g$ in such a way that for the new representation of $g$ we have a smaller maximal term. Let us assume $h$ is not $o$444444In case $h=o$, just substitute the empty sum for the representation of $h$ in the equations below.. By our assumption, $h$ has a standard representation with respect to $F$, say $\sum_{i=1}^{n}\tilde{m}_{i}\star\tilde{f}_{i}\star\tilde{l}_{i}$, where $\tilde{f}_{i}\in F$, and $\tilde{m}_{i},\tilde{l}_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ and all terms occurring in the sum are bounded by $t\succ{\sf HT}(h)$. This gives us: $\displaystyle g$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{K}m_{j}\star f_{j}\star l_{j}+\sum_{j=K+1}^{m}m_{j}\star f_{j}\star l_{j}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\tilde{m}_{i}\star\tilde{f}_{i}\star\tilde{l}_{i}+\sum_{j=K+1}^{m}m_{j}\star f_{j}\star l_{j}$ which is a representation of $g$ where the maximal term of the involved monomial multiples is decreased. q.e.d. Weak Gröbner bases can be defined as in Definition 4.2.10. Since the ordering $\succeq$ and the multiplication $\star$ in general are not compatible, instead of considering multiples of head terms of the generating set $F$ we look at head terms of monomial multiples of polynomials in $F$. ###### Definition 4.4.4 A subset $F$ of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ is called a weak Gröbner basis of ${\sf ideal}(F)$ if ${\sf HT}({\sf ideal}(F)\backslash\\{o\\})={\sf HT}(\\{m\star f\star l\mid f\in F,m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})\\}\backslash\\{o\\})$. $\diamond$ In the next lemma we show that in fact both characterizations of special bases, standard bases and weak Gröbner bases, coincide as in the case of polynomial rings over fields (compare Lemma 4.2.11). ###### Lemma 4.4.5 Let $F$ be a subset of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a standard basis if and only if it is a weak Gröbner basis. Proof : Let us first assume that $F$ is a standard basis, i.e., every polynomial $g$ in ${\sf ideal}(F)$ has a standard representation with respect to $F$. In case $g\neq o$ this implies the existence of a polynomial $f\in F$ and monomials $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HT}(g)={\sf HT}(m\star f\star l)$. Hence ${\sf HT}(g)\in{\sf HT}(\\{m\star f\star l\mid m,l\in{\sf M}({\cal F}_{{\mathbb{K}}}),f\in F\\}\backslash\\{o\\})$. As the converse, namely ${\sf HT}(\\{m\star f\star l\mid m,l\in{\sf M}({\cal F}_{{\mathbb{K}}}),f\in F\\}\backslash\\{o\\})\subseteq{\sf HT}({\sf ideal}(F)\backslash\\{o\\})$ trivially holds, $F$ then is a weak Gröbner basis. Now suppose that $F$ is a weak Gröbner basis and again let $g\in{\sf ideal}(F)$. We have to show that $g$ has a standard representation with respect to $F$. This will be done by induction on ${\sf HT}(g)$. In case $g=o$ the empty sum is our required standard representation. Hence let us assume $g\neq o$. Since then ${\sf HT}(g)\in{\sf HT}({\sf ideal}(F)\backslash\\{o\\})$ by the definition of weak Gröbner bases we know there exists a polynomial $f\in F$ and monomials $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HT}(g)={\sf HT}(m\star f\star l)$. Then there exists a monomial $\tilde{m}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HM}(g)={\sf HM}(\tilde{m}\star f\star l)$, namely454545Notice that this step requires that we can view ${\cal F}_{{\mathbb{K}}}$ as a vector space. In order to get a similar result without introducing vector spaces we would have to use a different definition of weak Gröbner bases. E.g. requiring that ${\sf HM}({\sf ideal}(F)\backslash\\{o\\})={\sf HM}(\\{m\star f\star l\mid f\in F,m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})\\}\backslash\\{o\\}\\})$ would be a possibility. However, then no localization of critical situations to head terms is possible, which is the advantage of having a field as coefficient domain. $\tilde{m}=({\sf HC}(g)\cdot{\sf HC}(m\star f\star l)^{-1})\cdot m)$. Let $g_{1}=g-\tilde{m}\star f\star l$. Then ${\sf HT}(g)\succ{\sf HT}(g_{1})$ implies the existence of a standard representation for $g_{1}$ which can be added to the multiple $\tilde{m}\star f\star l$ to give the desired standard representation of $g$. q.e.d. Inspecting this proof closer we get the following corollary (compare Corollary 4.2.12). ###### Corollary 4.4.6 Let a subset $F$ of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ be a weak Gröbner basis. Then every $g\in{\sf ideal}(F)$ has a standard representation in terms of $F$ of the form $g=\sum_{i=1}^{n}m_{i}\star f_{i}\star l_{i},f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}}),n\in{\mathbb{N}}$ such that ${\sf HM}(g)={\sf HM}(m_{1}\star f_{1}\star l_{1})$ and ${\sf HT}(m_{1}\star f_{1}\star l_{1})\succ{\sf HT}(m_{2}\star f_{2}\star l_{2})\succ\ldots\succ{\sf HT}(m_{n}\star f_{n}\star l_{n})$. Notice that we hence get stronger representations as specified in Definition 4.4.1 for the case that the set $F$ is a weak Gröbner basis or a standard basis. In order to proceed as before in the case of one-sided ideals we have to extend our restriction of the ordering $\succeq$ on ${\cal F}$ to cope with two-sided multiplication similar to Definition 4.2.13. ###### Definition 4.4.7 We will call an ordering $\geq$ on ${\cal T}$ a reductive restriction of the ordering $\succeq$ or simply reductive, if the following hold: 1. 1. $t\geq s$ implies $t\succeq s$ for $t,s\in{\cal T}$. 2. 2. $\geq$ is a partial well-founded ordering on ${\cal T}$ which is compatible with multiplication $\star$ in the following sense: if for $t,t_{1},t_{2},w_{1},w_{2}\in{\cal T}$ $t_{2}\geq t_{1}$, $t_{1}\succ t$ and $t_{2}={\sf HT}(w_{1}\star t_{1}\star w_{2})$ hold, then $t_{2}\succ{\sf HT}(w_{1}\star t\star w_{2})$. $\diamond$ Again we can distiguish special “divisors” of monomials: For $m_{1},m_{2}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ we call $m_{1}$ a (stable) divisor of $m_{2}$ if and only if ${\sf HT}(m_{2})\geq{\sf HT}(m_{1})$ and there exist $l_{1},l_{2}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that $m_{2}={\sf HM}(l_{1}\star m_{1}\star l_{2})$. We then call $l_{1},l_{2}$ stable multipliers of $m_{1}$. The intention is that for all terms $t$ with ${\sf HT}(m_{1})\succ t$ we then can conclude ${\sf HT}(m_{2})\succ{\sf HT}(l_{1}\star t\star l_{2})$. Reduction relations based on this divisibility of terms will again have the stability properties we desire. In the commutative polynomial ring we can state a reductive restriction of any term ordering by $t\geq s$ for two terms $t$ and $s$ if and only if $s$ divides $t$ as a term. In the non-commutative polynomial ring we can state a reductive restriction of any term ordering by $t\geq s$ for two terms $t$ and $s$ if and only if $s$ is a subword of $t$. Let us continue with an algebraic consequence related to this reductive ordering by distinguishing special standard representations as we have done in Definition 4.2.15. ###### Definition 4.4.8 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $g$ a non- zero polynomial in ${\sf ideal}(F)$. A representation of the form $g=\sum_{i=1}^{n}m_{i}\star f_{i}\star l_{i},f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}}),n\in{\mathbb{N}}$ such that ${\sf HT}(g)={\sf HT}(m_{i}\star f_{i}\star l_{i})={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq k$ for some $k\geq 1$, and ${\sf HT}(g)\succ{\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})$ for $k<i\leq n$ is called a reductive standard representation in terms of $F$. $\diamond$ Again the empty sum is taken as reductive standard representation of $o$. In case we have $\star:{\cal T}\times{\cal T}\longrightarrow{\cal T}$ the condition can be rephrased as ${\sf HT}(g)=m_{i}\star f_{i}\star l_{i}={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq k$. ###### Definition 4.4.9 A set $F\subseteq{\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ is called a reductive standard basis (with respect to the reductive ordering $\geq$) of ${\sf ideal}(F)$ if every polynomial $f\in{\sf ideal}(F)$ has a reductive standard representation in terms of $F$. $\diamond$ Again, in order to change an arbitrary representation into one fulfilling our additional condition of Definition 4.4.8 we have to deal with special sums of polynomials. ###### Definition 4.4.10 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $t$ an element in ${\cal T}$. Then we define the critical set ${\cal C}_{r}(t,F)$ to contain all tuples of the form $(t,f_{1},\ldots,f_{k},m_{1},\ldots,m_{k},l_{1},\ldots,l_{k})$, $k\in{\mathbb{N}}$, $f_{1},\ldots,f_{k}\in F$464646As in the case of commutative polynomials, $f_{1},\ldots,f_{k}$ are not necessarily different polynomials from $F$., $m_{1},\ldots,m_{k},l_{1},\ldots,l_{k}\in{\sf M}({\cal F})$ such that 1. 1. ${\sf HT}(m_{i}\star f_{i}\star l_{i})={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})=t$, $1\leq i\leq k$, 2. 2. ${\sf HT}(m_{i}\star f_{i}\star l_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq k$, and 3. 3. $\sum_{i=1}^{k}{\sf HM}(m_{i}\star f_{i}\star l_{i})=o$. We set ${\cal C}_{r}(F)=\bigcup_{t\in{\cal T}}{\cal C}_{r}(t,F)$. $\diamond$ Unfortunately, as in the case of right reductive standard bases, these critical situations will not be sufficient to characterize reductive standard bases (compare again Example 4.2.18). But we can give an analogon to Theorem 4.2.19. ###### Theorem 4.4.11 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a reductive standard basis of ${\sf ideal}(F)$ if and only if 1. 1. for every $f\in F$ and every $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ the multiple $m\star f\star l$ has a reductive standard representation in terms of $F$, 2. 2. for every tuple $(t,f_{1},\ldots,f_{k},m_{1},\ldots,m_{k},l_{1},\ldots,l_{k})$ in ${\cal C}_{r}(F)$ the polynomial $\sum_{i=1}^{k}m_{i}\star f_{i}\star l_{i}$ (i.e., the element in ${\cal F}$ corresponding to this sum) has a reductive standard representation with respect to $F$. Proof : In case $F$ is a reductive standard basis, since these polynomials are all elements of ${\sf ideal}(F)$, they must have reductive standard representations with respect to $F$. To prove the converse, it remains to show that every element in ${\sf ideal}(F)$ has a reductive standard representation with respect to $F$. Hence, let $g=\sum_{j=1}^{m}m_{j}\star f_{j}\star l_{j}$ be an arbitrary representation of a non-zero polynomial $g\in{\sf ideal}(F)$ such that $f_{j}\in F$, $m_{j},l_{j}\in{\sf M}({\cal F}_{{\mathbb{K}}})$, $m\in{\mathbb{N}}$. By our first statement every such monomial multiple $m_{j}\star f_{j}\star l_{j}$ has a reductive standard representation in terms of $F$ and we can assume that all multiples are replaced by them. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(m_{j}\star f_{j}\star l_{j})\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $m_{j}\star f_{j}\star l_{j}$ with head term $t$. Then for each monomial multiple $m_{j}\star f_{j}\star l_{j}$ with ${\sf HT}(m_{j}\star f_{j}\star l_{j})=t$ we know that ${\sf HT}(m_{j}\star f_{j}\star l_{j})={\sf HT}(m_{j}\star{\sf HT}(f_{j})\star l_{j})\geq{\sf HT}(f_{j})$ holds. Then $t\succeq{\sf HT}(g)$ and in case ${\sf HT}(g)=t$ this immediately implies that this representation is already a reductive standard one. Else we proceed by induction on $t$. Without loss of generality let $f_{1},\ldots,f_{K}$ be the polynomials in the corresponding representation such that $t={\sf HT}(m_{i}\star f_{i}\star l_{i})$, $1\leq i\leq K$. Then the tuple $(t,f_{1},\ldots,f_{K},m_{1},\ldots,m_{K},l_{1},\ldots,l_{K})$ is in ${\cal C}_{r}(F)$ and let $h=\sum_{i=1}^{K}m_{i}\star f_{i}\star l_{i}$. We will now change our representation of $g$ in such a way that for the new representation of $g$ we have a smaller maximal term. Let us assume $h$ is not $o$474747In case $h=o$, just substitute the empty sum for the representation of $h$ in the equations below.. By our assumption, $h$ has a reductive standard representation with respect to $F$, say $\sum_{j=1}^{n}\tilde{m}_{j}\star h_{j}\star\tilde{l}_{j}$, where $h_{j}\in F$, and $\tilde{m}_{j},\tilde{l}_{j}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ and all terms occurring in the sum are bounded by $t\succ{\sf HT}(h)$ as $\sum_{i=1}^{K}{\sf HM}(m_{i}\star f_{i}\star l_{i})=o$. This gives us: $\displaystyle g$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{K}m_{i}\star f_{i}\star l_{i}+\sum_{i=K+1}^{m}m_{i}\star f_{i}\star l_{i}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{n}\tilde{m}_{j}\star h_{j}\star\tilde{l}_{j}+\sum_{i=K+1}^{m}m_{i}\star f_{i}\star l_{i}$ which is a representation of $g$ where the maximal term is smaller than $t$. q.e.d. An algebraic characterization of weak Gröbner bases again can be given by a property of head monomials based on stable divisors of terms (compare Definition 4.2.20). ###### Definition 4.4.12 A set $F\subseteq{\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ is called a weak reductive Gröbner basis of ${\sf ideal}(F)$ (with respect to the reductive ordering $\geq$) if ${\sf HT}({\sf ideal}(F)\backslash\\{o\\})={\sf HT}(\\{m\star f\star l\mid f\in F,m,l\in{\sf M}({\cal F}_{{\mathbb{K}}}),{\sf HT}(m\star f\star l)={\sf HT}(m\star{\sf HT}(f)\star l)\geq{\sf HT}(f)\\}\backslash\\{o\\})$. $\diamond$ We will later on see that an analogon of the Translation Lemma holds for the reduction relation related to reductive standard representations. Hence weak reductive Gröbner bases and Gröbner bases coincide. This is again due to the fact that the coefficient domain is a field and will not carry over for reduction rings as coefficient domains. The next lemma states that in fact both characterizations of special bases provided so far coincide. ###### Lemma 4.4.13 Let $F$ be a subset of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a reductive standard basis if and only if it is a weak reductive Gröbner basis. Proof : Let us first assume that $F$ is a reductive standard basis, i.e., every polynomial $g$ in ${\sf ideal}(F)$ has a reductive standard representation with respect to $F$. In case $g\neq o$ this implies the existence of a polynomial $f\in F$ and monomials $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HT}(g)={\sf HT}(m\star f\star l)={\sf HT}(m\star{\sf HT}(f)\star l)\geq{\sf HT}(f)$. Hence ${\sf HT}(g)\in{\sf HT}(\\{m\star f\star l\mid m,l\in{\sf M}({\cal F}_{{\mathbb{K}}}),f\in F,{\sf HT}(m\star f\star l)={\sf HT}(m\star{\sf HT}(f)\star l)\geq{\sf HT}(f)\\}\backslash\\{o\\})$. As the converse, namely ${\sf HT}(\\{m\star f\star l\mid m,l\in{\sf M}({\cal F}_{{\mathbb{K}}}),f\in F,{\sf HT}(m\star f\star l)={\sf HT}(m\star{\sf HT}(f)\star l)\geq{\sf HT}(f)\\}\backslash\\{o\\})\subseteq{\sf HT}({\sf ideal}(F)\backslash\\{o\\})$ trivially holds, $F$ is a weak reductive Gröbner basis. Now suppose that $F$ is a weak reductive Gröbner basis and again let $g\in{\sf ideal}(F)$. We have to show that $g$ has a reductive standard representation with respect to $F$. This will be done by induction on ${\sf HT}(g)$. In case $g=o$ the empty sum is our required reductive standard representation. Hence let us assume $g\neq o$. Since then ${\sf HT}(g)\in{\sf HT}({\sf ideal}(F)\backslash\\{o\\})$ by the definition of weak reductive Gröbner bases we know there exists a polynomial $f\in F$ and monomials $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HT}(m\star f\star l)={\sf HT}(m\star{\sf HT}(f)\star l)\geq{\sf HT}(f)$ and there exists $\alpha\in{\mathbb{K}}$ such that ${\sf HC}(g)={\sf HC}(m\star f\star l)\cdot\alpha$, i.e., ${\sf HM}(g)={\sf HM}(m\star f\star l\cdot\alpha)$. Let $g_{1}=g-m\star f\star l\cdot\alpha$. Then ${\sf HT}(g)\succ{\sf HT}(g_{1})$ implies the existence of a reductive standard representation for $g_{1}$ which can be added to the multiple $m\star f\star l\cdot\alpha$ to give the desired reductive standard representation of $g$. q.e.d. A close inspection of this proof reveals that in fact we can provide a stronger condition for standard representations in terms of weak reductive Gröbner bases. ###### Corollary 4.4.14 Let a subset $F$ of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ be a weak reductive Gröbner basis. Every $g\in{\sf ideal}(F)$ has a reductive standard representation in terms of $F$ of the form $g=\sum_{i=1}^{n}m_{i}\star f_{i}\star l_{i},f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}}),n\in{\mathbb{N}}$ such that ${\sf HT}(g)={\sf HT}(m_{1}\star f_{1}\star l_{1})\succ{\sf HT}(m_{2}\star f_{2}\star l_{2})\succ\ldots\succ{\sf HT}(m_{n}\star f_{n}\star l_{n})$ and ${\sf HT}(m_{i}\star f_{i}\star l_{i})={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})\geq{\sf HT}(f_{i})$ for all $1\leq i\leq n$. The importance of Gröbner bases in commutative polynomial rings stems from the fact that they can be characterized by special polynomials, the so-called s-polynomials. This characterization can be combined with a reduction relation to an algorithm which computes finite Gröbner bases. We provide a first characterization for our function ring over the field ${\mathbb{K}}$. Here critical situations lead to s-polynomials as in the original case and can be identified by studying term multiples of polynomials. Let $p$ and $q$ be two non-zero polynomials in ${\cal F}_{{\mathbb{K}}}$. We are interested in terms $t,u_{1},u_{2},v_{1},v_{2}$ such that ${\sf HT}(u_{1}\star p\star v_{1})={\sf HT}(u_{1}\star{\sf HT}(p)\star v_{1})=t={\sf HT}(u_{2}\star q\star v_{2})={\sf HT}(u_{2}\star{\sf HT}(q)\star v_{2})$ and ${\sf HT}(p)\leq t$, ${\sf HT}(q)\leq t$. Let ${\cal C}_{s}(p,q)$ (this is a specialization of Definition 4.4.2) be the set containing all such tuples $(t,u_{1},u_{2},v_{1},v_{2})$ (as a short hand for $(t,p,q,u_{1},u_{2},v_{1},v_{2})$. We call the polynomial ${\sf HC}(u_{1}\star p\star v_{1})^{-1}\cdot u_{1}\star p\star v_{1}-{\sf HC}(u_{2}\star q\star v_{2})^{-1}\cdot u_{2}\star q\star v_{2}={\sf spol}{}(p,q,t,u_{1},u_{2},v_{1},v_{2})$ the s-polynomial of $p$ and $q$ related to the tuple $(t,u_{1},u_{2},v_{1},v_{2})$. Again these critical situations are not sufficient to characterize weak Gröbner bases (compare Example 4.2.18) and additionally we have to test monomial multiples of polynomials now from both sides. ###### Theorem 4.4.15 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a weak Gröbner basis of ${\sf ideal}(F)$ if and only if 1. 1. for all $f$ in $F$ and for all $m,l$ in ${\sf M}({\cal F}_{{\mathbb{K}}})$ the multiple $m\star f\star l$ has a reductive standard representation in terms of $F$, and 2. 2. for all $p$ and $q$ in $F$ and every tuple $(t,u_{1},u_{2},v_{1},v_{2})$ in ${\cal C}_{s}(p,q)$ the respective s-polynomial ${\sf spol}{}(p,q,t,u_{1},u_{2},v_{1},v_{2})$ has a reductive standard representation in terms of $F$. Proof : In case $F$ is a weak Gröbner basis it is also a reductive standard basis, and since the multiples $m\star f\star l$ as well as the respective s-polynomials are all elements of ${\sf ideal}(F)$ they must have reductive standard representations in terms of $F$. The converse will be proven by showing that every element in ${\sf ideal}(F)$ has a reductive standard representation in terms of $F$. Now, let $g=\sum_{j=1}^{m}\alpha_{j}\cdot v_{j}\star f_{j}\star w_{j}$ be an arbitrary representation of a non-zero polynomial $g\in{\sf ideal}(F)$ such that $\alpha_{j}\in{\mathbb{K}}^{},f_{j}\in F$, and $v_{j},w_{j}\in{\cal T}$. Since by our first assumption every multiple $v_{j}\star f_{j}\star w_{j}$ in this sum has a reductive standard representation we can assume that ${\sf HT}(v_{j}\star{\sf HT}(f_{j})\star w_{j})={\sf HT}(v_{j}\star f_{j}\star w_{j})\geq{\sf HT}(f_{j})$ holds. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(v_{j}\star f_{j}\star w_{j})\mid 1\leq j\leq m\\}$ and $K$ as the number of polynomials $v_{j}\star f_{j}\star w_{j}$ with head term $t$. Without loss of generality we can assume that the polynomial multiples with head term $t$ are just $v_{1}\star f_{1}\star w_{1},\ldots,v_{K}\star f_{K}\star w_{K}$. We proceed by induction on $(t,K)$, where $(t^{\prime},K^{\prime})<(t,K)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $K^{\prime}<K)$484848Note that this ordering is well-founded since $\succ$ is well-founded on ${\cal T}$ and $K\in{\mathbb{N}}$.. Obviously, $t\succeq{\sf HT}(g)$ must hold. If $K=1$ this gives us $t={\sf HT}(g)$ and by our assumptions our representation is already of the required form. Hence let us assume $K>1$. Then for the two polynomials $f_{1},f_{2}$ in the corresponding representation494949Not necessarily $f_{2}\neq f_{1}$. such that $t={\sf HT}(v_{1}\star{\sf HT}(f_{1})\star w_{1})={\sf HT}(v_{1}\star f_{1}\star w_{1})={\sf HT}(v_{2}\star f_{2}\star w_{2})={\sf HT}(v_{2}\star{\sf HT}(f_{2})\star w_{2})$ and $t\geq{\sf HT}(f_{1})$, $t\geq{\sf HT}(f_{2})$. Then the tuple $(t,v_{1},v_{2},w_{1},w_{2})$ is in ${\cal C}_{s}(f_{1},f_{2})$ and we have an s-polynomial $h={\sf HC}(v_{1}\star f_{1}\star w_{1})^{-1}\cdot v_{1}\star f_{1}\star w_{1}-{\sf HC}(v_{2}\star f_{2}\star w_{2})^{-1}\cdot v_{2}\star f_{2}\star w_{2}$ corresponding to this tuple. We will now change our representation of $g$ by using the additional information on this s-polynomial in such a way that for the new representation of $g$ we either have a smaller maximal term or the occurrences of the term $t$ are decreased by at least 1. Let us assume the s-polynomial is not $o$505050In case $h=o$, just substitute the empty sum for the reductive representation of $h$ in the equations below.. By our assumption, $h$ has a reductive standard representation in terms of $F$, say $\sum_{i=1}^{n}\tilde{\alpha}_{i}\cdot\tilde{v}_{i}\star\tilde{f}_{i}\star\tilde{w}_{i}$, where $\tilde{\alpha}_{i}\in{\mathbb{K}}^{},\tilde{f}_{i}\in F$, and $\tilde{v}_{i},\tilde{w}_{i}\in{\cal T}$ and all terms occurring in this sum are bounded by $t\succ{\sf HT}(h)$. This gives us: $\displaystyle\alpha_{1}\cdot v_{1}\star f_{1}\star w_{1}+\alpha_{2}\cdot v_{2}\star f_{2}\star w_{2}$ (4.6) $\displaystyle=$ $\displaystyle\alpha_{1}\cdot v_{1}\star f_{1}\star w_{1}+\underbrace{\alpha^{\prime}_{2}\cdot\beta_{1}\cdot v_{1}\star f_{1}\star w_{1}-\alpha^{\prime}_{2}\cdot\beta_{1}\cdot v_{1}\star f_{1}\star w_{1}}_{=\,0}$ $\displaystyle+\underbrace{\alpha^{\prime}_{2}\cdot\beta_{2}}_{\alpha_{2}}\cdot v_{2}\star f_{2}\star w_{2}$ $\displaystyle=$ $\displaystyle(\alpha_{1}+\alpha^{\prime}_{2}\cdot\beta_{1})\cdot v_{1}\star f_{1}\star w_{1}-\alpha^{\prime}_{2}\cdot\underbrace{(\beta_{1}\cdot v_{1}\star f_{1}\star w_{1}-\beta_{2}\cdot v_{2}\star f_{2}\star w_{2})}_{=\,h}$ $\displaystyle=$ $\displaystyle(\alpha_{1}+\alpha^{\prime}_{2}\cdot\beta_{1})\cdot v_{1}\star f_{1}\star w_{1}-\alpha^{\prime}_{2}\cdot(\sum_{i=1}^{n}\tilde{\alpha}_{i}\cdot\tilde{v}_{i}\star\tilde{f}_{i}\star\tilde{w}_{i})$ where $\beta_{1}={\sf HC}(v_{1}\star f_{1}\star w_{1})^{-1}$, $\beta_{2}={\sf HC}(v_{2}\star f_{2}\star w_{2})^{-1}$ and $\alpha^{\prime}_{2}\cdot\beta_{2}=\alpha_{2}$. Substituting (4.6) in the representation of $g$ gives rise to a smaller one. q.e.d. Notice that both test sets in this characterization in general cannot be described in a finitary manner, i.e., provide no finite test for the property of being a Gröbner basis. A problem which is related to the fact that the ordering $\succeq$ and the multiplication $\star$ in general are not compatible is that an important property fulfilled for representations of polynomials in commutative polynomial rings no longer holds: As in the case of right ideals the existence of a standard representation for some polynomial $f\in{\cal F}_{{\mathbb{K}}}$ no longer implies the existence of one for a multiple $m\star f\star l$ where $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$. However there are restrictions where this implication will hold (compare Lemma 4.2.26). ###### Lemma 4.4.16 Let $F$ be a subset of ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ and $p$ a non-zero polynomial in ${\cal F}_{{\mathbb{K}}}$. If $p$ has a reductive standard representation with respect to $F$ and $m,l$ are monomials such that ${\sf HT}(m\star p\star l)={\sf HT}(m\star{\sf HT}(p)\star l)\geq{\sf HT}(p)$, then the multiple $m\star p\star l$ again has a reductive standard representation with respect to $F$. Proof : Let $p=\sum_{i=1}^{n}m_{i}\star f_{i}\star l_{i}$ with $n\in{\mathbb{N}}$, $f_{i}\in F$, $m_{i},l_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ be a reductive standard representation of $p$ in terms of $F$, i.e., ${\sf HT}(p)={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})={\sf HT}(m_{i}\star f_{i}\star l_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq k$ and ${\sf HT}(p)\succeq{\sf HT}(m_{i}\star f_{i}\star l_{i})$ for all $k+1\leq i\leq n$. Let us first analyze the multiple $m\star m_{j}\star f_{j}\star l_{j}\star l$. Let ${\sf T}(m_{j}\star f_{j}\star l_{j})=\\{s_{1},\ldots,s_{k}\\}$ with $s_{1}\succ s_{i}$, $2\leq i\leq l$, i.e. $s_{1}={\sf HT}(m_{j}\star f_{j}\star l_{j})={\sf HT}(m_{j}\star{\sf HT}(f_{j})\star l_{j})={\sf HT}(p)$. Hence ${\sf HT}(m\star{\sf HT}(p)\star l)={\sf HT}(m\star s_{1}\star l)\geq{\sf HT}(p)=s_{1}$ and as $s_{1}\succ s_{i}$, $2\leq i\leq l$, by Definition 4.4.7 we can conclude ${\sf HT}(m\star{\sf HT}(p)\star l)={\sf HT}(m\star s_{1}\star l)\succ m\star s_{i}\star l\succeq{\sf HT}(m\star s_{i}\star l)$ for $2\leq i\leq l$. This implies ${\sf HT}(m\star{\sf HT}(m_{j}\star f_{j}\star l_{j})\star l)={\sf HT}(m\star m_{j}\star f_{j}\star l_{j}\star l)$. Hence we get $\displaystyle{\sf HT}(p\star m)$ $\displaystyle=$ $\displaystyle{\sf HT}(m\star{\sf HT}(p)\star l)$ $\displaystyle=$ $\displaystyle{\sf HT}(m\star{\sf HT}(m_{j}\star f_{j}\star l_{j})\star l),\mbox{ as }{\sf HT}(p)={\sf HT}(m_{j}\star f_{j}\star l_{j})$ $\displaystyle=$ $\displaystyle{\sf HT}(m\star m_{j}\star f_{j}\star l_{j}\star l)$ and since ${\sf HT}(m\star p\star l)\geq{\sf HT}(p)\geq{\sf HT}(f_{j})$ we can conclude ${\sf HT}(m\star m_{j}\star f_{j}\star l_{j}\star l)\geq{\sf HT}(f_{j})$. It remains to show that $m\star m_{j}\star f_{j}\star l_{j}\star l$ has a reductive standard representation in terms of $F$. First we show that ${\sf HT}(m\star m_{j}\star{\sf HT}(f_{j})\star l_{j}\star l)\geq{\sf HT}(f_{j})$. We know $m_{j}\star{\sf HT}(f_{j})\star l_{j}\succeq{\sf HT}(m_{j}\star{\sf HT}(f_{j})\star l_{j})={\sf HT}(m_{j}\star f_{j}\star l_{j})$515151Notice that $m_{j}\star{\sf HT}(f_{j})\star l_{j}$ can be a polynomial and hence we cannot conclude $m_{j}\star{\sf HT}(f_{j})\star l_{j}={\sf HT}(m_{j}\star{\sf HT}(f_{j})\star l_{j})$. and hence ${\sf HT}(m\star m_{j}\star{\sf HT}(f_{j})\star l_{j}\star l)={\sf HT}(m\star{\sf HT}(m_{j}\star f_{j}\star l_{j})\star l)={\sf HT}(m\star m_{j}\star f_{j}\star l_{j}\star l)\geq{\sf HT}(f_{j})$. Now in case $m\star m_{j},l_{j}\star l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ we are done as then $(m_{j}\star m)\star f_{j}\star(l_{j}\star l)$ is a reductive standard representation in terms of $F$. Hence let us assume $m\star m_{j}=\sum_{i=1}^{k_{1}}\tilde{m}_{i}$, $l_{j}\star l=\sum_{i^{\prime}=1}^{k_{1}^{\prime}}\tilde{l}_{i^{\prime}}$, $\tilde{m}_{i},\tilde{l}_{i^{\prime}}\in{\sf M}({\cal F}_{{\mathbb{K}}})$. Let ${\sf T}(f_{j})=\\{t_{1},\ldots,t_{w}\\}$ with $t_{1}\succ t_{i}$, $2\leq i\leq w$, i.e. $t_{1}={\sf HT}(f_{j})$. As ${\sf HT}(m_{j}\star{\sf HT}(f_{j})\star l_{j})\geq{\sf HT}(f_{j})\succ t_{p}$, $2\leq p\leq w$, again by Definition 4.4.7 we can conclude ${\sf HT}(m_{j}\star{\sf HT}(f_{j})\star l_{j})\succ m_{j}\star t_{p}\star l_{j}\succeq{\sf HT}(m_{j}\star t_{p}\star l_{j})$, and $m_{j}\star{\sf HT}(f_{j})\star l_{j}\succ\sum_{p=2}^{w}m_{j}\star t_{p}\star l_{j}$. Then for each $s_{i}$, $2\leq i\leq l$ there exists $t_{q}\in{\sf T}(f_{1})$ such that $s_{i}\in{\sf supp}(m_{j}\star t_{q}\star l_{j})$. Since ${\sf HT}(p)\succ s_{i}$ and even ${\sf HT}(p)\succeq m_{j}\star t_{q}\star l_{j}$ we find that either ${\sf HT}(m\star p\star l)\succeq{\sf HT}(m\star(m_{j}\star t_{q}\star l_{j})\star l)={\sf HT}((m\star m_{j})\star t_{q}\star(l_{j}\star l))$ in case ${\sf HT}(m_{j}\star t_{q}\star l_{j})={\sf HT}(m_{j}\star f_{j}\star l_{j})$ or ${\sf HT}(m\star p\star l)\succ{\sf HT}(m\star(m_{j}\star t_{q}\star l_{j})\star l)={\sf HT}((m\star m_{j})\star t_{q}\star(l_{j}\star l))$. Hence we can conlude $\tilde{m}_{i}\star f_{j}\star\tilde{l}_{i^{\prime}}\preceq{\sf HT}(m\star p\star l)$, $1\leq i\leq k_{1}$, $1\leq i^{\prime}\leq k_{1}^{\prime}$ and for at least one such multiple we get ${\sf HT}(\tilde{m}_{i}\star f_{1}\star\tilde{l}_{i^{\prime}})={\sf HT}(m\star m_{j}\star f_{j}\star l_{j}\star l)\geq{\sf HT}(f_{j})$. It remains to analyze the situation for the function $(\sum_{i=k+1}^{n}m\star(m_{i}\star f_{i}\star l_{i})\star l$. Again we find that for all terms $s$ in the $m_{i}\star f_{i}\star l_{i}$, $k+1\leq i\leq n$, we have ${\sf HT}(p)\succ s$ and we get ${\sf HT}(m\star p\star l)\succeq{\sf HT}(m\star s\star l)$. Hence all polynomial multiples of the $f_{i}$ in the representation $\sum_{i=k+1}^{n}((\sum_{j=1}^{k_{i}}\tilde{m}^{i}_{j})\star f_{i}\star(\sum_{j=1}^{k^{\prime}_{i}}\tilde{l}^{i}_{j}))$, where $m\star m_{i}=\sum_{j=1}^{k_{i}}\tilde{m}^{i}_{j}$, $l_{i}\star l=\sum_{j=1}^{k^{\prime}_{i}}\tilde{l}^{i}_{j}$, are bounded by ${\sf HT}(m\star p\star l)$. q.e.d. Notice that this lemma no longer holds in case we only require ${\sf HT}(m\star{\sf HT}(p)\star l)={\sf HT}(m\star p\star l)\succeq{\sf HT}(p)$, as then ${\sf HT}(p)\succ s$ no longer implies ${\sf HT}(m\star p\star l)\succ{\sf HT}(m\star s\star l)$. Our standard representations from Definition 4.4.8 are closely related to a reduction relation based on the divisibility of terms as defined in the context of reductive restrictions of orderings on page 4.4.7. ###### Definition 4.4.17 Let $f,p$ be two non-zero polynomials in ${\cal F}_{{\mathbb{K}}}$. We say $f$ reduces $p$ to $q$ at a monomial $\alpha\cdot t$ in one step, denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}q$, if there exist $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that 1. 1. $t\in{\sf supp}(p)$ and $p(t)=\alpha$, 2. 2. ${\sf HT}(m\star{\sf HT}(f)\star l)={\sf HT}(m\star f\star l)=t\geq{\sf HT}(f)$, 3. 3. ${\sf HM}(m\star f\star l)=\alpha\cdot t$, and 4. 4. $q=p-m\star f\star l$. We write $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}$ if there is a polynomial $q$ as defined above and $p$ is then called reducible by $f$. Further, we can define $\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$},\mbox{$\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}\,$}$ and $\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}\,$ as usual. Reduction by a set $F\subseteq{\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ is denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q$ and abbreviates $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}q$ for some $f\in F$. $\diamond$ Due to the fact that the coefficients lie in a field, again if for some terms $w_{1},w_{2}\in{\cal T}$ we have ${\sf HT}(w_{1}\star f\star w_{2})={\sf HT}(w_{1}\star{\sf HT}(f)\star w_{2})=t\geq{\sf HT}(f)$ this implies reducibility at the monomial $\alpha\cdot t$. ###### Lemma 4.4.18 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. 1. 1. For $p,q\in{\cal F}_{{\mathbb{K}}}$ we have that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q$ implies $p\succ q$, in particular ${\sf HT}(p)\succeq{\sf HT}(q)$. 2. 2. $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$ is Noetherian. Proof : 1. 1. Assuming that the reduction step takes place at a monomial $\alpha\cdot t$, by Definition 4.4.17 we know ${\sf HM}(m_{1}\star f\star m_{2})=\alpha\cdot t$ which yields $p\succ p-m_{1}\star f\star m_{2}$ since ${\sf HM}(m_{1}\star f\star m_{2})\succ{\sf RED}(m_{1}\star f\star m_{2})$. 2. 2. This follows directly from 1. as the ordering $\succeq$ on ${\cal T}$ is well- founded (compare Lemma 4.2.3). q.e.d. The next lemma shows how reduction sequences and reductive standard representations are related. ###### Lemma 4.4.19 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $p$ a non- zero polynomial in ${\cal F}_{{\mathbb{K}}}$. Then $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$ implies that $p$ has a reductive standard representation in terms of $F$. Proof : This follows directly by adding up the polynomials used in the reduction steps occurring in the reduction sequence $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. q.e.d. If $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q$, then $p$ has a reductive standard representation in terms of $F\cup\\{q\\}$, especially $p-q$ has one in terms of $F$. As stated before an analogon to the Translation Lemma holds. ###### Lemma 4.4.20 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}$ and $p,q,h$ polynomials in ${\cal F}_{{\mathbb{K}}}$. 1. 1. Let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}h$. Then there exist $p^{\prime},q^{\prime}\in{\cal F}_{{\mathbb{K}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}p^{\prime}$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q^{\prime}$ and $h=p^{\prime}-q^{\prime}$. 2. 2. Let $o$ be a normal form of $p-q$ with respect to $F$. Then there exists $g\in{\cal F}_{{\mathbb{K}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g$. Proof : 1. 1. Let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}h$ at the monomial $\alpha\cdot t$, i.e., $h=p-q-m\star f\star l$ for some $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf HT}(m\star{\sf HT}(f)\star l)={\sf HT}(m\star f\star l)=t\geq{\sf HT}(f)$ and ${\sf HM}(m\star f\star l)=\alpha\cdot t$. We have to distinguish three cases: 1. (a) $t\in{\sf supp}(p)$ and $t\in{\sf supp}(q)$: Then we can eliminate the occurence of $t$ in the respective polynomials by reduction and get $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}p-\alpha_{1}\cdot(m\star f\star l)=p^{\prime}$, $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}q-\alpha_{2}\cdot(m\star f\star l)=q^{\prime}$, where $\alpha_{1}\cdot{\sf HC}(m\star f\star l)$ and $\alpha_{2}\cdot{\sf HC}(m\star f\star l)$ are the coefficients of $t$ in $p$ respectively $q$. Moreover, $\alpha_{1}\cdot{\sf HC}(m\star f\star l)-\alpha_{2}\cdot{\sf HC}(m\star f\star l)=\alpha$ and hence $\alpha_{1}-\alpha_{2}=1$, as ${\sf HC}(m\star f\star l)=\alpha$. This gives us $p^{\prime}-q^{\prime}=p-\alpha_{1}\cdot(m\star f\star l)-q+\alpha_{2}\cdot(m\star f\star l)=p-q-(\alpha_{1}-\alpha_{2})\cdot(m\star f\star l)=p-q-m\star f\star l=h$. 2. (b) $t\in{\sf supp}(p)$ and $t\not\in{\sf supp}(q)$: Then we can eliminate the term $t$ in the polynomial $p$ by right reduction and get $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}p-m\star f\star l=p^{\prime}$, $q=q^{\prime}$, and, therefore, $p^{\prime}-q^{\prime}=p-m\star f\star l-q=h$. 3. (c) $t\in{\sf supp}(q)$ and $t\not\in{\sf supp}(p)$: Then we can eliminate the term $t$ in the polynomial $q$ by right reduction and get $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}q+m\star f\star l=q^{\prime}$, $p=p^{\prime}$, and, therefore, $p^{\prime}-q^{\prime}=p-(q+m\star f\star l)=h$. 2. 2. We show our claim by induction on $k$, where $p-q\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. In the base case $k=0$ there is nothing to show as then $p=q$. Hence, let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}h\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. Then by 1. there are polynomials $p^{\prime},q^{\prime}\in{\cal F}_{{\mathbb{K}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}p^{\prime}$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q^{\prime}$ and $h=p^{\prime}-q^{\prime}$. Now the induction hypothesis for $p^{\prime}-q^{\prime}\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$ yields the existence of a polynomial $g\in{\cal F}_{{\mathbb{K}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g$. q.e.d. The essential part of the proof is that reducibility as defined in Definition 4.4.17 is connected to stable divisors of terms and not to coefficients. We will later see that for function rings over reduction rings, when the coefficient is also involved in the reduction step, this lemma no longer holds. Next we state the definition of Gröbner bases based on the reduction relation. ###### Definition 4.4.21 A subset $G$ of ${\cal F}_{{\mathbb{K}}}$ is called a Gröbner basis (with respect to the reduction relation $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$) of the ideal ${\mathfrak{i}}={\sf ideal}(G)$, if $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{G}\,$}=\;\;\equiv_{{\mathfrak{i}}}$ and $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$ is confluent. Remember the free group ring in Example 4.2.18 where the polynomial $b+\lambda$ lies in the ideal generated by the polynomial $a+\lambda$. Then of course $b+\lambda$ also lies in the ideal generated by $a+\lambda$. Unlike in the case of polynomial rings over fields where for any set of polynomials $F$ we have $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}^{{\rm b}}_{F}\,$}=\;\;\equiv_{{\sf ideal}(F)}$, here we have $b+\lambda\equiv_{{\sf ideal}(\\{a+\lambda\\})}0$ but $b+\lambda\mbox{$\,\,\,\,{\not\\!\\!\\!\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{a+\lambda}}\,$}0$. Hence the first condition of Definition 4.4.21 is again neccessary. Now by Lemma 4.4.20 and Theorem 3.1.5 weak Gröbner bases are Gröbner bases and can be characterized as follows: ###### Corollary 4.4.22 Let $G$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. $G$ is a (weak) Gröbner basis of ${\sf ideal}(G)$ if and only if for every $g\in{\sf ideal}(G)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$. Finally we can characterize Gröbner bases similar to Theorem 2.3.11. ###### Theorem 4.4.23 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$. Then $F$ is a Gröbner basis of ${\sf ideal}(G)$ if and only if 1. 1. for all $f$ in $F$ and for all $m,l$ in ${\sf M}({\cal F}_{{\mathbb{K}}})$ we have $m\star f\star l\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, and 2. 2. for all $p$ and $q$ in $F$ and every tuple $(t,u_{1},u_{2},v_{1},v_{2})$ in ${\cal C}(p,q)$ and the respective s-polynomial ${\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})$ we have ${\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. We will later on prove a stronger version of this theorem. The importance of Gröbner bases in the classical case stems from the fact that we only have to check a finite set of s-polynomials for $F$ in order to decide, whether $F$ is a Gröbner basis. Hence, we are interested in localizing the test sets in Theorem 4.4.23 – if possible to finite ones. ###### Definition 4.4.24 A set of polynomials $F\subseteq{\cal F}_{{\mathbb{K}}}\backslash\\{o\\}$ is called weakly saturated, if for all monomials $m,l$ in ${\sf M}({\cal F}_{{\mathbb{K}}})$ and every polynomial $f\in F$ we have $m\star f\star l\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. $\diamond$ This of course implies that for a weakly saturated set $F$ and any $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$, $f\in F$ the multiple $m\star f\star l$ has a reductive standard representation in terms of $F$. Notice that since the coefficient domain is a field we could restrict ourselves to multiples with elements of ${\cal T}$. However, as we will later on allow reduction rings as coefficient domains, we present this more general definition. ###### Definition 4.4.25 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{0\\}$. A set ${\sf SAT}(F)\subseteq\\{m\star f\star l\mid f\in F,m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})\\}$ is called a stable saturator for $F$ if for any $f\in F$, $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ there exist $s\in{\sf SAT}(F)$, $m^{\prime},l^{\prime}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that $m\star f\star l=m^{\prime}\star s\star l^{\prime}$, ${\sf HT}(m\star f\star l)={\sf HT}(m^{\prime}\star{\sf HT}(s)\star l^{\prime})\geq{\sf HT}(s)$. ###### Corollary 4.4.26 Let ${\sf SAT}(F)$ be a stable saturator of a set $F\subseteq{\cal F}_{{\mathbb{K}}}$. Then for any $f\in F$, $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ there exists $s\in{\sf SAT}(F)$ such that $m\star f\star l\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{s}\,$}o$. ###### Lemma 4.4.27 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{0\\}$. If for all $s\in{\sf SAT}(F)$ we have $s\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, then for every $m$, $l$ in ${\sf M}({\cal F}_{{\mathbb{K}}})$ and every polynomial $f$ in $F$ the multiple $m\star f\star l$ has a reductive standard representation in terms of $F$. Proof : This follows immediately from Lemma 4.4.16 and Lemma 4.4.19. q.e.d. ###### Definition 4.4.28 Let $p$ and $q$ be two non-zero polynomials in ${\cal F}_{{\mathbb{K}}}$. Then a subset $C\subseteq\\{{\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})\mid(t,u_{1},u_{2},v_{1},v_{2})\in{\cal C}_{s}(p,q)\\}$ is called a stable localization for the critical situations if for every s-polynomial ${\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})$ related to a tuple $(t,u_{1},u_{2},v_{1},v_{2})$ in ${\cal C}_{s}(p,q)$ there exists a polynomial $h\in C$ and monomials $\alpha\cdot w_{1},1\cdot w_{2}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that 1. 1. ${\sf HT}(h)\leq{\sf HT}({\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2}))$, 2. 2. ${\sf HT}(w_{1}\star h\star w_{2})={\sf HT}(w_{1}\star{\sf HT}(h)\star w_{2})={\sf HT}({\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2}))$, 3. 3. ${\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})=(\alpha\cdot w_{1})\star h\star w_{2}$. $\diamond$ The idea behind this definition is to reduce the number of s-polynomials, which have to be considered when checking for the Gröbner basis property. ###### Corollary 4.4.29 Let $C\subseteq\\{{\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})\mid(t,u_{1},u_{2},v_{1},v_{2})\in{\cal C}_{s}(p,q)\\}$ be a stable localization for two polynomials $p,q\in{\cal F}_{{\mathbb{K}}}$. Then for any s-polynomial ${\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})$ there exists $h\in C$ such that ${\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{h}\,$}o$. ###### Lemma 4.4.30 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{0\\}$. If for all $h$ in a stable localization $C\subseteq\\{{\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})\mid(t,u_{1},u_{2},v_{1},v_{2})\in{\cal C}_{s}(p,q)\\}$, we have $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, then for every $(t,u_{1},u_{2},v_{1},v_{2})$ in ${\cal C}_{s}(p,q)$ the s-polynomial ${\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})$ has a reductive standard representation in terms of $F$. Proof : This follows immediately from Lemma 4.4.16 and Lemma 4.4.19. q.e.d. ###### Theorem 4.4.31 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{K}}}\backslash\\{0\\}$. Then $F$ is a Gröbner basis if and only if 1. 1. for all $s$ in ${\sf SAT}(F)$ we have $s\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, and 2. 2. for all $p$ and $q$ in $F$, and every polynomial $h$ in a stable localization $C\subseteq\\{{\sf spol}(p,q,t,u_{1},u_{2},v_{1},v_{2})\mid(t,u_{1},u_{2},v_{1},v_{2})\in{\cal C}(p,q)\\}$, we have $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. Proof : In case $F$ is a Gröbner basis by Lemma 4.4.22 all elements of ${\sf ideal}(F)$ must reduce to zero by $F$. Since the polynomials in the saturator and the respective localizations of the s-polynomials all belong to the ideal generated by $F$ we are done. The converse will be proven by showing that every element in ${\sf ideal}(F)$ has a reductive standard representation in terms of $F$. Now, let $g=\sum_{j=1}^{n}(\alpha_{j}\cdot w_{j})\star f_{j}\star z_{j}$ be an arbitrary representation of a non-zero polynomial $g\in{\sf ideal}(F)$ such that $\alpha_{j}\in{\mathbb{K}}^{},f_{j}\in F$, and $w_{j},z_{j}\in{\cal T}$. By the definition of the stable saturator for every multiple $w_{j}\star f_{j}\star z_{j}$ in this sum we have some $s\in{\sf SAT}(F)$, $m,l\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that $w_{j}\star f_{j}\star z_{j}=m\star s\star l$ and ${\sf HT}(w_{j}\star f_{j}\star z_{j})={\sf HT}(m\star s\star l)={\sf HT}(m\star{\sf HT}(s)\star l)\geq{\sf HT}(s)$. Since we have $s\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, by Lemma 4.4.16 we can conclude that each $w_{j}\star f_{j}\star z_{j}$ has a reductive standard representation in terms of $F$. Therefore, we can assume that ${\sf HT}(w_{j}\star{\sf HT}(f_{j})\star z_{j})={\sf HT}(w_{j}\star f_{j}\star z_{j})\geq{\sf HT}(f_{j})$ holds. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(w_{j}\star f_{j}\star z_{j})\mid 1\leq j\leq n\\}$ and $K$ as the number of polynomials $w_{j}\star f_{j}\star z_{j}$ with head term $t$. Without loss of generality we can assume that the polynomial multiples with head term $t$ are just $(\alpha_{1}\cdot w_{1})\star f_{1}\star z_{1},\ldots,(\alpha_{K}\cdot w_{K})\star f_{K}\star z_{K}$. We proceed by induction on $(t,K)$, where $(t^{\prime},K^{\prime})<(t,K)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $K^{\prime}<K)$525252Note that this ordering is well-founded since $\succ$ is well-founded on ${\cal T}$ and $K\in{\mathbb{N}}$.. Obviously, $t\succeq{\sf HT}(g)$ must hold. If $K=1$ this gives us $t={\sf HT}(g)$ and by our assumption our representation is already of the required form. Hence let us assume $K>1$, then for the two not necessarily different polynomials $f_{1},f_{2}$ in the corresponding representation we have $t={\sf HT}(w_{1}\star{\sf HT}(f_{1})\star z_{1})={\sf HT}(w_{1}\star f_{1}\star z_{1})={\sf HT}(w_{2}\star f_{2}\star z_{2})={\sf HT}(w_{2}\star{\sf HT}(f_{2})\star z_{2})$ and $t\geq{\sf HT}(f_{1})$, $t\geq{\sf HT}(f_{2})$. Then the tuple $(t,w_{1},w_{2},z_{1},z_{2})$ is in ${\cal C}(f_{1},f_{2})$ and we have a polynomial $h$ in a stable localization $C\subseteq\\{{\sf spol}(f_{1},f_{2},t,w_{1},w_{2},z_{1},z_{2})\mid(t,w_{1},w_{2},z_{1},z_{2})\in{\cal C}(f_{1},f_{2})\\}$ and $\alpha\cdot w,1\cdot z\in{\sf M}({\cal F}_{{\mathbb{K}}})$ such that ${\sf spol}(f_{1},f_{2},t,w_{1},w_{2},z_{1},z_{2})={\sf HC}(w_{1}\star f_{1}\star z_{1})^{-1}\cdot w_{1}\star f_{1}\star z_{1}-{\sf HC}(w_{2}\star f_{2}\star z_{2})^{-1}\cdot w_{2}\star f_{2}\star z_{2}=(\alpha\cdot w)\star h\star z$ and ${\sf HT}({\sf spol}(f_{1},f_{2},t,w_{1},w_{2},z_{1},z_{2})={\sf HT}(w\star h\star z)={\sf HT}(w\star{\sf HT}(h)\star z)\geq{\sf HT}(h)$. We will now change our representation of $g$ by using the additional information on this situation in such a way that for the new representation of $g$ we either have a smaller maximal term or the occurrences of the term $t$ are decreased by at least 1. Let us assume the s-polynomial is not $o$535353In case $h=o$, just substitute the empty sum for the right reductive representation of $h$ in the equations below.. By our assumption, $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$ and by Lemma 4.4.19 $h$ has a reductive standard representation in terms of $F$. Then by Lemma 4.4.16 the multiple $(\alpha\cdot w)\star h\star z$ again has a right reductive standard representation in terms of $F$, say $\sum_{i=1}^{n}m_{i}\star h_{i}\star l_{i}$, where $h_{i}\in F$, and $m_{i},l_{i}\in{\sf M}({\cal F}_{{\mathbb{K}}})$ and all terms occurring in this sum are bounded by $t\succ{\sf HT}((\alpha\cdot w)\star h\star z)$. This gives us: $\displaystyle(\alpha_{1}\cdot w_{1})\star f_{1}\star z_{1}+(\alpha_{2}\cdot w_{2})\star f_{2}\star z_{2}$ (4.7) $\displaystyle=$ $\displaystyle(\alpha_{1}\cdot w_{1})\star f_{1}\star z_{1}+\underbrace{(\alpha^{\prime}_{2}\cdot\beta_{1}\cdot w_{1})\star f_{1}\star z_{1}-(\alpha^{\prime}_{2}\cdot\beta_{1}\cdot w_{1})\star f_{1}\star z_{1}}_{=\,0}$ $\displaystyle+\underbrace{(\alpha^{\prime}_{2}\cdot\beta_{2}}_{=\alpha_{2}}\cdot w_{2})\star f_{2}\star z_{2}$ $\displaystyle=$ $\displaystyle((\alpha_{1}+\alpha^{\prime}_{2}\cdot\beta_{1})\cdot w_{1})\star f_{1}\star z_{1}-\alpha^{\prime}_{2}\cdot\underbrace{((\beta_{1}\cdot w_{1})\star f_{1}\star z_{1}-(\beta_{2}\cdot w_{2})\star f_{2}\star z_{2})}_{=\,(\alpha\cdot w)\star h\star z}$ $\displaystyle=$ $\displaystyle((\alpha_{1}+\alpha^{\prime}_{2}\cdot\beta_{1})\cdot w_{1})\star f_{1}\star z_{1}-\alpha^{\prime}_{2}\cdot(\sum_{i=1}^{n}m_{i}\star h_{i}\star l_{i})$ where $\beta_{1}={\sf HC}(w_{1}\star f_{1}\star z_{1})^{-1}$, $\beta_{2}={\sf HC}(w_{2}\star f_{2}\star z_{2})^{-1}$ and $\alpha^{\prime}_{2}\cdot\beta_{2}=\alpha_{2}$. By substituting (4.7) in our representation of $g$ the representation becomes smaller. q.e.d. Obviously this theorem states a criterion for when a set is a Gröbner basis. As in the case of completion procedures such as the Knuth-Bendix procedure or Buchberger’s algorithm, elements from these test sets which do not reduce to zero can be added to the set being tested, to gradually describe a not necessarily finite Gröbner basis. Of course in order to get a computable completion procedure certain assumptions on the test sets have to be made, e.g. they should themselves be recursively enumerable, and normal forms with respect to finite sets have to be computable. The examples from page 4.2.1 can also be studied with respect to two-sided ideals. For polynomial rings, skew- polynomial rings commutative monoid rings and commutative respectively poly- cyclic group rings finite Gröbner bases can be computed in the respective setting. #### 4.4.2 Function Rings over Reduction Rings The situation becomes more complicated if ${\sf R}$ is not a field. Let ${\sf R}$ be a non-commutative ring with a reduction relation $\Longrightarrow_{B}$ associated with subsets $B\subseteq{\sf R}$ as described in Section 3.1. When following the path of linking special standard representations and reduction relations we get the same results as in Section 4.2.2, i.e., such representations naturally arise from the respective reduction relations. Hence we proceed by studying a special reduction relation which subsumes the two reduction relations presented for one-sided ideals in function rings over reduction rings. As before for our ordering $>_{{\sf R}}$ on ${\sf R}$ we require: for $\alpha,\beta\in{\sf R}$, $\alpha>_{{\sf R}}\beta$ if and only if there exists a finite set $B\subseteq{\sf R}$ such that $\alpha\mbox{$\,\stackrel{{\scriptstyle+}}{{\Longrightarrow}}\\!\\!\mbox{}_{B}\,$}\beta$. This ordering will ensure that the reduction relation on ${\cal F}$ is terminating. The reduction relation on ${\sf R}$ can be used to define various reduction relations on the function ring. Here we want to present a reduction relation which in some sense is based on the “divisibility” of the term to be reduced by the head term of the polynomial used for reduction. ###### Definition 4.4.32 Let $f,p$ be two non-zero polynomials in ${\cal F}$. We say $f$ reduces $p$ to $q$ at a monomial $\alpha\cdot t$ in one step, denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}q$, if there exist monomials $m,l\in{\sf M}({\cal F})$ such that 1. 1. $t\in{\sf supp}(p)$ and $p(t)=\alpha$, 2. 2. ${\sf HT}(m\star{\sf HT}(f)\star l)={\sf HT}(m\star f\star l)=t\geq{\sf HT}(f)$, 3. 3. $\alpha\Longrightarrow_{{\sf HC}(m\star f\star l)}\beta$, with545454Remember that by Axiom (A2) for reduction rings $\alpha\Longrightarrow_{\gamma}\beta$ implies $\alpha-\beta\in{\sf ideal}(\gamma)$ and hence $\alpha=\sum_{i=1}^{k}\gamma_{i}\cdot\gamma\cdot\delta_{i}+\beta$, $\gamma_{i},\delta_{i}\in{\sf R}$. $\alpha=\sum_{i=1}^{k}\gamma_{i}\cdot{\sf HC}(m\star f\star l)\cdot\delta_{i}+\beta$ for some $\beta,\gamma_{i},\delta_{i}\in{\sf R}$, $1\leq i\leq k$, and 4. 4. $q=p-\sum_{i=1}^{k}\gamma_{i}\cdot m\star f\star l\cdot\delta_{i}$. We write $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}$ if there is a polynomial $q$ as defined above and $p$ is then called reducible by $f$. Further, we can define $\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$},\mbox{$\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}\,$}$ and $\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}\,$ as usual. Reduction by a set $F\subseteq{\cal F}\backslash\\{o\\}$ is denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q$ and abbreviates $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}q$ for some $f\in F$. $\diamond$ By specializing item 3. of this definition to $3.~{}\alpha\Longrightarrow_{{\sf HC}(m\star f\star l)}\mbox{ such that }\alpha={\sf HC}(m\star f\star l)$ we get an analogon to Definition 4.2.43. Similarly, specializing 3. to $3.~{}\alpha\Longrightarrow_{{\sf HC}(m\star f\star l)}\beta\mbox{ such that }{\sf HC}(m\star f\star l)+\beta$ gives us an analogon to Definition 4.2.53. Reviewing Example 4.2.54 we find that the reduction relation is not terminating when using infinite sets of polynomials for reduction. But for finite sets we get the following analogon of Lemma 4.2.55. ###### Lemma 4.4.33 Let $F$ be a finite set of polynomials in ${\cal F}\backslash\\{o\\}$. 1. 1. For $p,q\in{\cal F}$, $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q$ implies $p\succ q$, in particular ${\sf HT}(p)\succeq{\sf HT}(q)$. 2. 2. $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$ is Noetherian. Proof : 1. 1. Assuming that the reduction step takes place at a monomial $\alpha\cdot t$, by Definition 4.4.32 we know ${\sf HM}(p-\sum_{i=1}^{k}\gamma_{i}\cdot m_{1}\star f\star m_{2}\cdot\delta_{i})=\beta\cdot t$ which yields $p\succ p-\sum_{i=1}^{k}\gamma_{i}\cdot m_{1}\star f\star m_{2}\cdot\delta_{i}$ since $\alpha>_{{\sf R}}\beta$. 2. 2. This follows from 1. and Axiom (A1) as long as only finite sets of polynomials are involved. q.e.d. As for the one-sided case a Translation Lemma does not hold for this reduction relation. Hence we have to distinguish between weak Gröbner bases and Gröbner bases. ###### Definition 4.4.34 A set $F\subseteq{\cal F}\backslash\\{o\\}$ is called a weak Gröbner basis of ${\sf ideal}(F)$ if for all $g\in{\sf ideal}(F)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. $\diamond$ Now as for one-sided weak Gröbner bases, weak Gröbner bases allow special representations of the polynomials in the ideal they generate. ###### Corollary 4.4.35 Let $F$ be a set of polynomials in ${\cal F}$ and $g$ a non-zero polynomial in ${\sf ideal}(F)$ such that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. Then $g$ has a representation of the form $g=\sum_{i=1}^{n}m_{i}\star f_{i}\star l_{i},f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}$ such that ${\sf HT}(g)={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})={\sf HT}(m_{i}\star f_{i}\star l_{i})\geq{\sf HT}(f_{i})$ for $1\leq i\leq k$, and ${\sf HT}(g)\succ{\sf HT}(m_{i}\star f_{i}\star l_{i})$ for all $k+1\leq i\leq n$. Proof : We show our claim by induction on $n$ where $g\mbox{$\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. If $n=0$ we are done. Else let $g\mbox{$\,\stackrel{{\scriptstyle 1}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g_{1}\mbox{$\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. In case the reduction step takes place at the head monomial, there exists a polynomial $f\in F$ and monomial $m,l\in{\sf M}({\cal F})$ such that ${\sf HT}(m\star{\sf HT}(f)\star l)={\sf HT}(m\star f\star l)={\sf HT}(g)\geq{\sf HT}(f)$ and ${\sf HC}(g)\Longrightarrow_{{\sf HC}(m\star f\star l)}\beta$ with ${\sf HC}(g)\Longrightarrow_{{\sf HC}(m\star f\star l)}\beta$ with ${\sf HC}(g)=\beta+\sum_{i=1}^{k}\gamma_{i}\cdot{\sf HC}(m\star f\star l)\cdot\delta_{i}$ for some $\gamma_{i},\delta_{i}\in{\sf R}$, $1\leq i\leq k$. Moreover the induction hypothesis then is applied to $g_{1}=g-\sum_{i=1}^{k}\gamma_{i}\cdot m\star f\star l\cdot\delta_{i}$. If the reduction step takes place at a monomial with term smaller ${\sf HT}(g)$ for the respective monomial multiple $m\star f\star l$ we immediately get ${\sf HT}(g)\succ{\sf HT}(m\star f\star l)$ and we can apply our induction hypothesis to the resulting polynomial $g_{1}$. In both cases we can arrange the monomial multiples $m\star f\star l$ arising from the reduction steps in such a way that gives us th desired representation. q.e.d. As in Theorem 4.4.15 we can characterize weak Gröbner bases using g- and m-polynomials instead of s-polynomials. ###### Definition 4.4.36 Let $P=\\{p_{1},\ldots,p_{k}\\}$ be a multiset of (not necessarily different) polynomials in ${\cal F}$ and $t$ an element in ${\cal T}$ such that there are $u_{1},\ldots,u_{k},v_{1},\ldots,v_{k}\in{\cal T}$ with ${\sf HT}(u_{i}\star p_{i}\star v_{i})={\sf HT}(u_{i}\star{\sf HT}(p_{i})\star v_{i})=t$, for all $1\leq i\leq k$. Further let $\gamma_{i}={\sf HC}(u_{i}\star p_{i}\star v_{i})$ for $1\leq i\leq k$. Let $G$ be a (weak) Gröbner basis of $\\{\gamma_{1},\ldots,\gamma_{k}\\}$ with respect to $\Longrightarrow$ in ${\sf R}$ and $\alpha=\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\beta_{i,j}\cdot\gamma_{i}\cdot\delta_{i,j}$ for $\alpha\in G$, $\beta_{i,j},\delta_{i,j}\in{\sf R}$, $1\leq i\leq k$, and $1\leq j\leq n_{i}$. Then we define the g-polynomials (Gröbner polynomials) corresponding to $p_{1},\ldots,p_{k}$ and $t$ by setting $g_{\alpha}=\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\beta_{i,j}\cdot u_{i}\star p_{i}\star v_{i}\cdot\delta_{i,j}.$ Notice that ${\sf HM}(g_{\alpha})=\alpha\cdot t$. We define the m-polynomials (module polynomials) corresponding to $P$ and $t$ as those $h=\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\beta_{i,j}\cdot u_{i}\star p_{i}\star v_{i}\cdot\delta_{i,j}$ where $\sum_{i=1}^{k}\sum_{j=1}^{n_{i}}\beta_{i,j}\cdot\gamma_{i}\cdot\delta_{i,j}=0$. Notice that ${\sf HT}(h)\prec t$. $\diamond$ Notice that while we allow the multiplication of two terms to have influence on the coefficients of the result555555Skew-polynomial rings are a classical example, see Section 4.2.1. we require that $t\cdot\alpha=\alpha\cdot t$. Given a set of polynomials $F$, the set of corresponding g- and m-polynomials is again defined for all possible multisets of polynomials in $F$ and appropriate terms $t$ as specified by Definition 4.4.36. Notice that given a finite set of polynomials the corresponding sets of g- and m-polynomials in general will be infinite. We can use g- and m-polynomials to characterize special bases in function rings over reduction rings satisfying Axiom (A4) in case we add an additional condition as before. ###### Theorem 4.4.37 Let $F$ be a finite set of polynomials in ${\cal F}\backslash\\{o\\}$ where the reduction ring satisfies (A4). Then $F$ is a weak Gröbner basis if and only if 1. 1. for all $f$ in $F$ and for all $m,l$ in ${\sf M}({\cal F})$ we have $m\star f\star l\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, and 2. 2. all g- and all m-polynomials corresponding to $F$ as specified in Definition 4.4.36 reduce to zero using $F$. Proof : In case $F$ is a weak Gröbner basis, since the multiples $m\star f\star l$ as well as the respective g- and m-polynomials are all elements of ${\sf ideal}(F)$ they must reduce to zero using $F$. The converse will be proven by showing that every element in ${\sf ideal}(F)$ is reducible by $F$. Then as $g\in{\sf ideal}(F)$ and $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g^{\prime}$ implies $g^{\prime}\in{\sf ideal}(F)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. Notice that this only holds in case the reduction relation $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$ is Noetherian. This follows as by our assumption $F$ is finite. Let $g\in{\sf ideal}(F)$ have a representation in terms of $F$ of the following form: $g=\sum_{j=1}^{n}m_{j}\star f_{j}\star l_{j}$, $f_{j}\in F$ and $m_{j},l_{j}\in{\sf M}({\cal F})$. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(m_{j}\star f_{j}\star l_{j})\mid 1\leq j\leq n\\}$ and $K$ as the number of polynomials $m_{j}\star f_{j}\star l_{j}$ with head term $t$. We show our claim by induction on $(t,K)$, where $(t^{\prime},K^{\prime})<(t,K)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $K^{\prime}<K)$. Since by our first assumption every multiple $m_{j}\star f_{j}\star l_{j}$ in this sum reduces to zero using $F$ and hence has a representation as described in Corollary 4.4.35 we can assume that ${\sf HT}(m_{j}\star{\sf HT}(f_{j})\star l_{j})={\sf HT}(m_{j}\star f_{j}\star l_{j})\geq{\sf HT}(f_{j})$ holds. Without loss of generality we can assume that the polynomial multiples with head term $t$ are just $m_{1}\star f_{1}\star l_{1},\ldots,m_{K}\star f_{K}\star l_{K}$. Obviously, $t\succeq{\sf HT}(g)={\sf HT}(m_{1}\star{\sf HT}(f_{1})\star l_{1})\geq{\sf HT}(f_{1})$ must hold. If $K=1$ this gives us $t={\sf HT}(g)$ and even ${\sf HM}(g)={\sf HM}(m_{1}\star f_{1}\star l_{1})$, implying that $g$ is reducible at ${\sf HM}(g)$ by $f_{1}$. Hence let us assume $K>1$. First let $\sum_{j=1}^{K}{\sf HM}(m_{j}\star f_{j}\star l_{j})=o$. Then there is a m-polynomial $h$, corresponding to the polynomials $f_{1},\ldots,f_{K}$ and the term $t$ such that $\sum_{j=1}^{K}l_{j}\star f_{j}\star m_{j}=h$. We will now change our representation of $g$ by using the additional information on this m-polynomial in such a way that for the new representation of $g$ we have a smaller maximal term. Let us assume the m-polynomial is not $o$565656In case $h=o$, just substitute the empty sum for the reductive representation of $h$ in the equations below.. By our assumption, $h$ is reducible to zero using $F$ and hence has a representation with respect to $F$ as described in Corollary 4.4.35, say $\sum_{i=1}^{n}\tilde{m}_{i}\star\tilde{f}_{i}\star\tilde{l}_{i}$, where $\tilde{f}_{i}\in F$, $\tilde{m}_{i},\tilde{l}_{i}\in{\sf M}({\cal F})$ and all terms occurring in the sum are bounded by $t\succ{\sf HT}(h)$. Hence replacing the sum $\sum_{j=1}^{K}m_{j}\star f_{j}\star l_{j}$ by $\sum_{i=1}^{n}\tilde{m}_{i}\star\tilde{f}_{i}\star\tilde{l}_{i}$ gives us a smaller representation of $g$. Hence let us assume $\sum_{j=1}^{K}{\sf HM}(m_{j}\star f_{j}\star l_{j})\neq 0$. Then we have ${\sf HT}(m_{1}\star f_{1}\star l_{1}+\ldots+m_{K}\star f_{K}\star l_{K})=t={\sf HT}(g)$, ${\sf HC}(g)={\sf HC}(m_{1}\star f_{1}\star l_{1}+\ldots+m_{K}\star f_{K}\star l_{K})\in{\sf ideal}_{r}(\\{{\sf HC}(m_{1}\star f_{1}\star l_{1}),\ldots,{\sf HC}(m_{K}\star f_{K}\star l_{K})\\})$ and even ${\sf HM}(m_{1}\star f_{1}\star l_{1}+\ldots+m_{K}\star f_{K}\star l_{K})={\sf HM}(g)$. Hence ${\sf HC}(g)$ is $\Longrightarrow$-reducible by $\alpha$, $\alpha\in G$, $G$¡a (weak) right Gröbner basis of ${\sf ideal}_{r}(\\{{\sf HC}(m_{1}\star f_{1}\star l_{1}),\ldots,{\sf HC}(m_{K}\star f_{K}\star l_{K})\\})$ in ${\sf R}$ with respect to the reduction relation $\Longrightarrow$. Let $g_{\alpha}$ be the respective g-polynomial corresponding to $\alpha$. Then we know that $g_{\alpha}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. Moreover, we know that the head monomial of $g_{\alpha}$ is reducible by some polynomial $f\in F$ and we assume ${\sf HT}(g_{\alpha})={\sf HT}(m\star{\sf HT}(f)\star l)={\sf HT}(m\star f\star l)\geq{\sf HT}(f)$ and ${\sf HC}(g_{\alpha})\Longrightarrow_{{\sf HC}(m\star f\star l)}$. Then, as ${\sf HC}(g)$ is $\Longrightarrow$-reducible by ${\sf HC}(g_{\alpha})$, ${\sf HC}(g_{\alpha})$ is $\Longrightarrow$-reducible to zero and (A4) holds, the head monomial of $g$ is also reducible by some $f^{\prime}\in F$ and we are done. q.e.d. Of course this theorem is still true for infinite $F$ if we can show that for the respective function ring the reduction relation is terminating. Now the question arises when the critical situations in this characterization can be localized to subsets of the respective sets. Reviewing the Proof of Theorem 4.4.31 we find that Lemma 4.4.16 is central as it describes when multiples of polynomials which have a reductive standard representation in terms of some set $F$ again have such a representation. As before, this does not hold for function rings over reduction rings in general. We have stated that it is not natural to link right reduction as defined in Definition 4.4.32 to special standard representations. Hence, to give localizations of Theorem 4.4.37 another property for ${\cal F}$ is sufficient: ###### Definition 4.4.38 A set $C\subset S\subseteq{\cal F}$ is called a stable localization of $S$ if for every $g\in S$ there exists $f\in C$ such that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}o$. $\diamond$ In case ${\cal F}$ and $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$ allow such stable localizations, we can rephrase Theorem 4.4.37 as follows: ###### Theorem 4.4.39 Let $F$ be a finite set of polynomials in ${\cal F}\backslash\\{o\\}$ where the reduction ring satisfies (A4). Then $F$ is a weak Gröbner basis of ${\sf ideal}(F)$ if and only if 1. 1. for all $s$ in a stable localization of $\\{m\star f\star l\mid f\in{\cal F},m,l\in{\sf M}({\cal F})\\}$ we have $s\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, and 2. 2. for all $h$ in a stable localization of the g- and m-polynomials corresponding to $F$ as specified in Definition 4.4.36 we have $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. We have stated that for arbitrary reduction relations in ${\cal F}$ it is not natural to link them to special standard representations. Still, when proving Theorem 4.4.39, we will find that in order to change the representation of an arbitrary ideal element, Definition 4.4.38 is not enough to ensure reducibility. However, we can substitute the critical situation using an analogon of Lemma 4.4.16, which while not related to reducibility in this case will still be sufficient to make the representation smaller. ###### Lemma 4.4.40 Let $F\subseteq{\cal F}\backslash\\{o\\}$ and $f$, $p$ non-zero polynomials in ${\cal F}$. If $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}o$ and $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, then $p$ has a standard representation of the form $p=\sum_{i=1}^{n}m_{i}\star f_{i}\star l_{i},f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}$ such that ${\sf HT}(p)={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})={\sf HT}(m_{i}\star f_{i}\star l_{i})\geq{\sf HT}(f_{i})$ for $1\leq i\leq k$ and ${\sf HT}(p)\succ{\sf HT}(m_{i}\star f_{i}\star l_{i})$ for all $k+1\leq i\leq n$. Proof : If $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}o$ then $p=\sum_{j=1}^{s}\gamma_{j}\cdot m^{\prime}\star f\star l^{\prime}\cdot\delta_{j}$ with $m^{\prime},l^{\prime}\in{\sf M}({\cal F})$, $\gamma_{j},\delta_{j}\in{\sf R}$, and ${\sf HT}(p)={\sf HT}(m\star{\sf HT}(f)\star l)={\sf HT}(m\star f\star l)\geq{\sf HT}(f)$. Similarly $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$ implies575757Notice that in this representation we write the products of the form $\gamma\cdot m$ respectively $l\cdot\delta$ arising in the reduction steps as simple monomials. $f=\sum_{i=1}^{n}m_{i}\star f_{i}\star l_{i},f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}),n\in{\mathbb{N}}$ such that ${\sf HT}(f)={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})={\sf HT}(m_{i}\star f_{i}\star l_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq k$, and ${\sf HT}(f)\succ{\sf HT}(m_{i}\star f_{i}\star l_{i})$ for all $k+1\leq i\leq n$. We show our claim for all multiples with $\gamma_{j}\cdot m^{\prime}$ and $l^{\prime}\cdot\delta_{j}$ for $1\leq j\leq s$. Let $m=\gamma_{j}\star m^{\prime}$ and $l=l^{\prime}\cdot\delta_{j}$ and let us analyze $m\star m_{i}\star f_{i}\star l_{i}\star l$ with ${\sf HT}(m_{i}\star f_{i}\star l_{i})={\sf HT}(f)$, $1\leq i\leq k$. Let ${\sf T}(m_{i}\star f_{i}\star l_{i})=\\{s_{1}^{i},\ldots,s_{w_{i}}^{i}\\}$ with $s_{1}^{i}\succ s_{j}^{i}$, $2\leq j\leq w_{i}$, i.e. $s_{1}^{i}={\sf HT}(m_{i}\star f_{i}\star l_{i})={\sf HT}(p)$. Hence $m\star{\sf HT}(p)\star l=m\star s_{1}^{i}\star l\geq{\sf HT}(p)=s_{1}^{i}$ and as $s_{1}^{i}\succ s_{j}^{i}$, $2\leq j\leq w_{i}$, by Definition 4.4.7 we can conclude ${\sf HT}(m\star{\sf HT}(p)\star l)={\sf HT}(m\star s_{1}^{i}\star l)\succ m\star s_{j}^{i}\star l\succeq{\sf HT}(m\star s_{j}^{i}\star l)$ for $2\leq j\leq w_{i}$. This implies ${\sf HT}(m\star{\sf HT}(m_{i}\star f_{i}\star l_{i})\star l)={\sf HT}(m\star m_{i}\star f_{i}\star l_{i}\star l)$ Hence we get $\displaystyle{\sf HT}(m\star f\star l)$ $\displaystyle=$ $\displaystyle{\sf HT}(m\star{\sf HT}(f)\star l)$ $\displaystyle=$ $\displaystyle{\sf HT}(m\star{\sf HT}(m_{i}\star f_{i}\star l_{i})\star l),\mbox{ as }{\sf HT}(p)={\sf HT}(m_{i}\star f_{i}\star l_{i})$ $\displaystyle=$ $\displaystyle{\sf HT}(m\star m_{i}\star f_{i}\star l_{i}\star l)$ and since ${\sf HT}(m\star f\star l)\geq{\sf HT}(f)\geq{\sf HT}(f_{i})$ we can conclude ${\sf HT}(m\star m_{i}\star f_{i}\star l_{i}\star l)\geq{\sf HT}(f_{i})$. It remains to show that $m\star(m_{i}\star f_{i}\star l_{i})\star l=(m\star m_{i})\star f_{i}\star(l_{i}\star l)$ has representations of the desired form in terms of $F$. First we show that ${\sf HT}((m\star m_{i}\star{\sf HT}(f_{i})\star l_{i}\star l)\geq{\sf HT}(f_{i})$. We know $m_{i}\star{\sf HT}(f_{i})\star l_{i}\succeq{\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})={\sf HT}(m_{i}\star f_{i}\star l_{i})$ and hence ${\sf HT}(m\star m_{i}\star{\sf HT}(f_{i})\star l_{i}\star l)={\sf HT}(m\star{\sf HT}(m_{i}\star f_{i}\star l_{i})\star l)={\sf HT}(m\star m_{i}\star f_{i}\star l_{i}\star l)\geq{\sf HT}(f_{i})$. Now in case $m\star m_{i},l_{i}\star l\in{\sf M}({\cal F})$ we are done as then $(m\star m_{i})\star f_{i}\star(l_{i}\star l)$ is a representation of the desired form. Hence let us assume $m\star m_{i}=\sum_{j=1}^{k_{i}}\tilde{m}^{i}_{j}$,$l_{i}\star l=\sum_{j^{\prime}=1}^{k^{\prime}_{i}}\tilde{l}^{i}_{j}$ $\tilde{m}^{i}_{j},\tilde{l}^{i}_{j^{\prime}}\in{\sf M}({\cal F})$. Let ${\sf T}(f_{i})=\\{t^{i}_{1},\ldots,t^{i}_{w}\\}$ with $t^{i}_{1}\succ t^{i}_{j}$, $2\leq j\leq w$, i.e. $t^{i}_{1}={\sf HT}(f_{i})$. As ${\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})\geq{\sf HT}(f_{i})\succ t_{j}$, $2\leq j\leq w$, again by Definition 4.4.7 we can conclude that ${\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})\succ m_{i}\star t^{i}_{j}\star l_{i}\succeq{\sf HT}(m_{i}\star t^{i}_{j}\star l_{i})$, $2\leq j\leq l$, and $m_{i}\star{\sf HT}(f_{i})\star l_{i}\succ\sum_{j=2}^{w}m_{i}\star t^{i}_{j}\star l_{i}$. Then for each $s^{i}_{j}$, $2\leq j\leq w_{i}$, there exists $t^{i}_{j^{\prime}}\in{\sf T}(f_{i})$ such that $s^{i}_{j}\in{\sf supp}(m_{i}\star t^{i}_{j^{\prime}}\star l_{i})$. Since ${\sf HT}(f)\succ s^{i}_{j}$ and even ${\sf HT}(f)\succ m_{i}\star t^{i}_{j^{\prime}}\star l_{i}$ we find that either ${\sf HT}(m\star f\star l)\succeq{\sf HT}(m\star(m_{i}\star t^{i}_{j^{\prime}}\star l_{i})\star l)={\sf HT}((m\star m_{i})\star t^{i}_{j^{\prime}}\star(l_{i}\star l))$ in case ${\sf HT}(m_{i}\star t^{i}_{j^{\prime}}\star l_{i})={\sf HT}(m_{i}\star f_{1}\star l_{i})$ or ${\sf HT}(m\star f\star l)\succ m\star(m_{i}\star t^{i}_{j^{\prime}}\star l_{i})\star l=(m\star m_{i})\star t^{i}_{j^{\prime}}\star(l_{i}\star l)$. Hence we can conclude $\tilde{m}^{i}_{j}\star f_{i}\star\tilde{l}^{i}_{j^{\prime}}\preceq{\sf HT}(m\star f\star l)$, $1\leq j\leq k_{i}$, $1\leq j^{\prime}\leq K_{i}$ and for at least one $\tilde{m}^{i}_{j}$, $\tilde{l}^{i}_{j^{\prime}}$ we get ${\sf HT}(\tilde{m}^{i}_{j}\star f_{i}\star\tilde{l}^{i}_{j^{\prime}})={\sf HT}(m\star m_{i}\star f_{i}\star l_{i}\star l)\geq{\sf HT}(f_{i})$. It remains to analyze the situation for the functions $(\sum_{i=k+1}^{n}m\star(m_{i}\star f_{i}\star l_{i})\star l$. Again we find that for all terms $s$ in the $m_{i}\star f_{i}\star l_{i}$, $k+1\leq i\leq n$, we have ${\sf HT}(f)\succeq s$ and we get ${\sf HT}(m\star f\star l)\succeq{\sf HT}(m\star s\star l)$. Hence all polynomial multiples of the $f_{i}$ in the representation $\sum_{i=k+1}^{n}(\sum_{j=1}^{k_{i}}\tilde{m}^{i}_{j})\star f_{i}\star(\sum_{j=1}^{K_{i}}\tilde{l}^{i}_{j^{\prime}})$, where $m\star m_{i}=\sum_{j=1}^{k_{i}}\tilde{m}^{i}_{j}$, $l_{i}\star l=\sum_{j=1}^{K_{i}}\tilde{l}^{i}_{j^{\prime}}$, are bounded by ${\sf HT}(m\star f\star l)$. q.e.d. Proof Theorem 4.4.39: The proof is basically the same as for Theorem 4.4.37. Due to Lemma 4.4.40 we can substitute the multiples $m_{j}\star f_{j}\star l_{j}$ by appropriate representations without changing $(t,K)$. Hence, we only have to ensure that despite testing less polynomials we are able to apply our induction hypothesis. Taking the notations from the proof of Theorem 4.4.37, let us first check the situation for m-polynomials. Let $\sum_{j=1}^{K}{\sf HM}(m_{j}\star f_{j}\star l_{j})=o$. Then by Definition 4.4.36 there exists a module polynomial $h=\sum_{j=1}^{K}m_{j}\star f_{j}\star l_{j}$ and by our assumption there is a polynomial $h^{\prime}$ in the stable localization such that $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{h^{\prime}}\,$}o$. Moreover, $h^{\prime}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. Then by Lemma 4.4.40 the m-polynomial $h$ has a standard representations bounded by $t$. Hence we can change the representation of $g$ by substituting $h$ by its representation giving us a smaller representation and by our induction hypothesis $g$ is reducible by $F$ and we are done. It remains to study the case where $\sum_{j=1}^{K}{\sf HM}(m_{j}\star f_{j}\star l_{j})\neq 0$. Then we have ${\sf HT}(\sum_{j=1}^{K}m_{j}\star f_{j}\star l_{j})=t={\sf HT}(g)$, ${\sf HC}(g)={\sf HC}(\sum_{j=1}^{K}m_{j}\star f_{j}\star l_{j})\in{\sf ideal}(\\{{\sf HC}(m_{1}\star f_{1}\star l_{1}),\ldots,{\sf HC}(m_{K}\star f_{K}\star l_{K})\\})$ and even ${\sf HM}(\sum_{j=1}^{K}m_{j}\star f_{j}\star l_{j})={\sf HM}(g)$. Hence ${\sf HC}(g)$ is $\Longrightarrow$-reducible by $\alpha$, $\alpha\in G$, $G$ a (weak) Gröbner basis of ${\sf ideal}(\\{{\sf HC}(m_{1}\star f_{1}\star l_{1}),\ldots,{\sf HC}(m_{K}\star f_{K}\star l_{K})\\})$ in ${\sf R}$ with respect to the reduction relation $\Longrightarrow$. Let $g_{\alpha}$ be the respective g-polynomial corresponding to $\alpha$. Then we know that $g_{\alpha}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{g_{\alpha}^{\prime}}\,$}o$ for some $g_{\alpha}^{\prime}$ in the stable localization and $g_{\alpha}^{\prime}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. Moreover, we know that the head monomial of $g_{\alpha}^{\prime}$ is reducible by some polynomial $f\in F$ and we assume ${\sf HT}(g_{\alpha})={\sf HT}(m\star{\sf HT}(f)\star l)={\sf HT}(m\star f\star l)\geq{\sf HT}(f)$ and ${\sf HC}(g_{\alpha})\Longrightarrow_{{\sf HC}(m\star f\star l)}$. Then, as ${\sf HC}(g)$ is $\Longrightarrow$-reducible by ${\sf HC}(g_{\alpha})$, ${\sf HC}(g_{\alpha})$ is $\Longrightarrow$-reducible by ${\sf HC}(g_{\alpha}^{\prime})$, ${\sf HC}(g_{\alpha}^{\prime})$ is $\Longrightarrow$-reducible to zero and (A4) holds, the head monomial of $g$ is also reducible by some $f^{\prime}\in F$ and we are done. q.e.d. Again, if for infinite $F$ we can assure that the reduction relation is Noetherian, the proof still holds. #### 4.4.3 Function Rings over the Integers In the previous section we have seen that for the reduction relation for ${\cal F}$ based on the abstract notion of the reduction relation $\Longrightarrow_{{\sf R}}$ there is not enough information on the reduction step involving the coefficient and hence we cannot prove an analogon of the Translation Lemma. As in the case of studying one-sided ideals, when studying special reduction rings where we have more information on the specific reduction relation $\Longrightarrow_{{\sf R}}$ the situation often can be improved. Again we go into the details for the case that ${\sf R}$ is the ring of the integers ${\mathbb{Z}}$. The reduction relation presented in Definition 4.4.32 then can be reformulated for this special case as follows: ###### Definition 4.4.41 Let $p$, $f$ be two non-zero polynomials in ${\cal F}_{{\mathbb{Z}}}$. We say $f$ reduces $p$ to $q$ at $\alpha\cdot t$ in one step, i.e. $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}q$, if there exist $u,v\in{\sf T}({\cal F}_{{\mathbb{Z}}})$ such that 1. 1. $t\in{\sf supp}(p)$ and $p(t)=\alpha$, 2. 2. ${\sf HT}(u\star{\sf HT}(f)\star v)={\sf HT}(u\star f\star v)=t\geq{\sf HT}(f)$, 3. 3. $\alpha\geq_{{\mathbb{Z}}}{\sf HC}(u\star f\star v)>0$ and $\alpha\Longrightarrow_{{\sf HC}(u\star f\star v)}\delta$ where $\alpha={\sf HC}(u\star f\star v)\cdot\beta+\delta$ with $\beta,\delta\in{\mathbb{Z}}$, $0\leq\delta<{\sf HC}(u\star f\star v)$, and 4. 4. $q=p-u\star f\star v\cdot\beta$. We write $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}$ if there is a polynomial $q$ as defined above and $p$ is then called reducible by $f$. Further, we can define $\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$},\mbox{$\,\stackrel{{\scriptstyle+}}{{\longrightarrow}}\\!\\!\mbox{}\,$}$ and $\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}\,$ as usual. Reduction by a set $F\subseteq{\cal F}\backslash\\{o\\}$ is denoted by $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q$ and abbreviates $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}q$ for some $f\in F$. $\diamond$ As before, for this reduction relation we can still have $t\in{\sf supp}(q)$. The important part in showing termination now is that if we still have $t\in{\sf supp}(q)$ then its coefficient will be smaller according to our ordering chosen for ${\mathbb{Z}}$ (compare Section 4.2.3) and since this ordering is well-founded we are done. Due to the additional information on the coefficents, again we do not have to restrict ourselves to finite sets of polynomials in order to ensure termination. ###### Corollary 4.4.42 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{Z}}}\backslash\\{o\\}$. 1. 1. For $p,q\in{\cal F}_{{\mathbb{Z}}}$, $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q$ implies $p\succ q$, in particular ${\sf HT}(p)\succeq{\sf HT}(q)$. 2. 2. $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$ is Noetherian. Similarly, the additional information we have on the coefficients before and after the reduction step now enables us to prove an analogon of the Translation Lemma for function rings over the integers. The first and second part of the lemma are only needed to prove the essential third part. ###### Lemma 4.4.43 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{Z}}}$ and $p,q,h$ polynomials in ${\cal F}_{{\mathbb{Z}}}$. 1. 1. Let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}h$ such that the reduction step takes place at the monomial $\alpha\cdot t$ and we additionally have $t\not\in{\sf supp}(h)$. Then there exist $p^{\prime},q^{\prime}\in{\cal F}_{{\mathbb{Z}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}p^{\prime}$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q^{\prime}$ and $h=p^{\prime}-q^{\prime}$. 2. 2. Let $o$ be the unique normal form of $p$ with respect to $F$ and $t={\sf HT}(p)$. Then there exists a polynomial $f\in F$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}p^{\prime}$ and $t\not\in{\sf supp}(p^{\prime})$. 3. 3. Let $o$ be the unique normal form of $p-q$ with respect to $F$. Then there exists $g\in{\cal F}_{{\mathbb{Z}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g$. Proof : 1. 1. Let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}h$ at the monomial $\alpha\cdot t$, i.e., $h=p-q-u\star f\star v\cdot\beta$ for some $u,v\in{\sf T}({\cal F}_{{\mathbb{Z}}}),\beta\in{\mathbb{Z}}$ such that ${\sf HT}(u\star{\sf HT}(f)\star v)={\sf HT}(u\star f\star v)=t\geq{\sf HT}(f)$ and ${\sf HC}(u\star f\star v)>0$. Remember that $\alpha$ is the coefficient of $t$ in $p-q$. Then as $t\not\in{\sf supp}(h)$ we know $\alpha={\sf HC}(u\star f\star v)\cdot\beta$. Let $\alpha_{1}$ respectively $\alpha_{2}$ be the coefficients of $t$ in $p$ respectively $q$ and $\alpha_{1}=({\sf HC}(u\star f\star v)\cdot\beta)\cdot\beta_{1}+\gamma_{1}$ respectively $\alpha_{2}=({\sf HC}(u\star f\star v)\cdot\beta)\cdot\beta_{2}+\gamma_{2}$ for some $\beta_{1},\beta_{2},\gamma_{1},\gamma_{2}\in{\mathbb{Z}}$ where $0\leq\gamma_{1},\gamma_{2}<{\sf HC}(u\star f\star v)\cdot\beta$. Then $\alpha={\sf HC}(u\star f\star v)\cdot\beta=\alpha_{1}-\alpha_{2}=({\sf HC}(u\star f\star v)\cdot\beta)\cdot(\beta_{1}-\beta_{2})+(\gamma_{1}-\gamma_{2})$, and as $\gamma_{1}-\gamma_{2}$ is no multiple of ${\sf HC}(u\star f\star v)\cdot\beta$ we have $\gamma_{1}-\gamma_{2}=0$ and hence $\beta_{1}-\beta_{2}=1$. We have to distinguish two cases: 1. (a) $\beta_{1}\neq 0$ and $\beta_{2}\neq 0$: Then $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}p-u\star f\star v\cdot\beta\cdot\beta_{1}=p^{\prime}$, $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q-u\star f\star v\cdot\beta\cdot\beta_{2}=q^{\prime}$ and $p^{\prime}-q^{\prime}=p-u\star f\star v\cdot\beta\cdot\beta_{1}-q+u\star f\star v\cdot\beta\cdot\beta_{2}=p-q-u\star f\star v\cdot\beta\cdot\beta=h$. 2. (b) $\beta_{1}=0$ and $\beta_{2}=-1$ (the case $\beta_{2}=0$ and $\beta_{1}=1$ being symmetric): Then $p^{\prime}=p$, $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q-u\star f\star v\cdot\beta\cdot\beta_{2}=q+u\star f\star v\cdot\beta=q^{\prime}$ and $p^{\prime}-q^{\prime}=p-q-u\star f\star v\cdot\beta=h$. 2. 2. Since $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, ${\sf HM}(p)=\alpha\cdot t$ must be $F$-reducible. Let $f_{i}\in F$, $i\in I$ be a series of all not necessarily different polynomials in $F$ such that $\alpha\cdot t$ is reducible by them involving terms $u_{i},v_{i}$. Then ${\sf HC}(u_{i}\star f_{i}\star v_{i})>0$. Moreover, let $\gamma=\min_{\leq}\\{{\sf HC}(u_{i}\star f_{i}\star v_{i})\mid i\in I\\}$ and without loss of generality ${\sf HM}(u\star f\star v)=\gamma\cdot t$ for some $f\in F$, ${\sf HT}(u\star{\sf HT}(f)\star v)={\sf HT}(u\star f\star v)\geq{\sf HT}(f)$. We claim that for $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}p-\beta\cdot u\star f\star v=p^{\prime}$ where $\alpha=\beta\cdot\gamma+\delta$, $\beta,\delta\in{\mathbb{Z}}$, $0\leq\delta<\gamma$, we have $t\not\in{\sf supp}(p^{\prime})$. Suppose ${\sf HT}(p^{\prime})=t$. Then by our definition of reduction we must have $0<{\sf HC}(p^{\prime})<{\sf HC}(u\star f\star v)$. But then $p^{\prime}$ would no longer be $F$-reducible contradicting our assumption that $o$ is the unique normal form of $p$. 3. 3. Since $o$ is the unique normal form of $p-q$ by 2. there exists a reduction sequence $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f_{i_{1}}}\,$}h_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f_{i_{2}}}\,$}h_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f_{i_{3}}}\,$}\ldots\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f_{i_{k}}}\,$}o$ such that ${\sf HT}(p-q)\succ{\sf HT}(h_{1})\succ{\sf HT}(h_{2})\succ\ldots$. We show our claim by induction on $k$, where $p-q\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$ is such a reduction sequence. In the base case $k=0$ there is nothing to show as then $p=q$. Hence, let $p-q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}h\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. Then by 1. there are polynomials $p^{\prime},q^{\prime}\in{\cal F}_{{\mathbb{Z}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}p^{\prime}$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}q^{\prime}$ and $h=p^{\prime}-q^{\prime}$. Now the induction hypothesis for $p^{\prime}-q^{\prime}\mbox{$\,\stackrel{{\scriptstyle k}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$ yields the existence of a polynomial $g\in{\cal F}_{{\mathbb{Z}}}$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g$ and $q\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g$. q.e.d. Hence weak Gröbner bases are in fact Gröbner bases and can hence be characterized as follows (compare Definition 4.2.10): ###### Definition 4.4.44 A set $F\subseteq{\cal F}_{{\mathbb{Z}}}\backslash\\{o\\}$ is called a (weak) Gröbner basis of ${\sf ideal}(F)$ if for all $g\in{\sf ideal}(F)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. $\diamond$ ###### Corollary 4.4.45 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{Z}}}$ and $g$ a non- zero polynomial in ${\sf ideal}(F)$ such that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. Then $g$ has a representation of the form $g=\sum_{i=1}^{n}m_{i}\star f_{i}\star l_{i},f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}_{{\mathbb{Z}}}),n\in{\mathbb{N}}$ such that ${\sf HT}(g)={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})={\sf HT}(m_{i}\star f_{i}\star l_{i})\geq{\sf HT}(f_{i})$, $1\leq i\leq k$, and ${\sf HT}(g)\succ{\sf HT}(m_{i}\star f_{i}\star l_{i})={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})$ for all $k+1\leq i\leq n$. In case $o$ is the unique normal form of $g$ with respect to $F$ we even can find a representation where additionally ${\sf HT}(m_{1}\star f_{1}\star l_{1})\succ{\sf HT}(m_{2}\star f_{2}\star l_{2})\succ\ldots\succ{\sf HT}(m_{n}\star f_{n}\star l_{n})$. Proof : We show our claim by induction on $n$ where $g\mbox{$\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. If $n=0$ we are done. Else let $g\mbox{$\,\stackrel{{\scriptstyle 1}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g_{1}\mbox{$\,\stackrel{{\scriptstyle n}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. In case the reduction step takes place at the head monomial, there exists a polynomial $f\in F$ and $u,v\in{\sf T}({\cal F}_{{\mathbb{Z}}}),\beta\in{\mathbb{Z}}$ such that ${\sf HT}(u\star{\sf HT}(f)\star v)={\sf HT}(u\star f\star v)={\sf HT}(g)\geq{\sf HT}(f)$ and ${\sf HC}(g)\Longrightarrow_{{\sf HC}(u\star f\star v)}\delta$ with ${\sf HC}(g)={\sf HC}(u\star f\star v)\cdot\beta+\delta$ for some $\beta,\delta\in{\mathbb{Z}}$, $0\leq\delta<{\sf HC}(u\star f\star v)$. Moreover the induction hypothesis then is applied to $g_{1}=g-u\star f\star v\cdot\beta$. If the reduction step takes place at a monomial with term smaller ${\sf HT}(g)$ for the respective monomial multiple $u\star f\star v\cdot\beta$ we immediately get ${\sf HT}(g)\succ u\star f\star v\cdot\beta$ and we can apply our induction hypothesis to the resulting polynomial $g_{1}$. In both cases we can arrange the monomial multiples $u\star f\star v\cdot\beta$ arising from the reduction steps in such a way that gives us the desired representation. q.e.d. Now Gröbner bases can be characterized using the concept of s-polynomials combined with the technique of saturation which is neccessary in order to describe the whole ideal congruence by the reduction relation. ###### Definition 4.4.46 Let $p_{1},p_{2}$ be polynomials in ${\cal F}_{{\mathbb{Z}}}$. If there are respective terms $t,u_{1},u_{2},v_{1},v_{2}\in{\cal T}$ such that ${\sf HT}(u_{i}\star{\sf HT}(p_{i})\star v_{i})={\sf HT}(u_{i}\star p_{i}\star v_{i})=t\geq{\sf HT}(p_{i})$ let $HC(u_{i}\star p_{i}\star v_{i})=\gamma_{i}$. Assuming $\gamma_{1}\geq\gamma_{2}>0$585858Notice that $\gamma_{i}>0$ can always be achieved by studying the situation for $-p_{i}$ in case we have $HC(u_{i}\star p_{i}\star v_{i})<0$., there are $\beta,\delta\in{\mathbb{Z}}$ such that $\gamma_{1}=\gamma_{2}\cdot\beta+\delta$ and $0\leq\delta<\gamma_{2}$ and we get the following s-polynomial ${\sf spol}(p_{1},p_{2},t,u_{1},u_{2},v_{1},v_{2})=u_{2}\star p_{2}\star v_{2}\cdot\beta-u_{1}\star p_{1}\star v_{1}.$ The set ${\sf SPOL}(\\{p_{1},p_{2}\\})$ then is the set of all such s-polynomials corresponding to $p_{1}$ and $p_{2}$. $\diamond$ Again these sets in general are not finite. ###### Theorem 4.4.47 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{Z}}}\backslash\\{o\\}$. Then $F$ is a Gröbner basis if and only if 1. 1. for all $f$ in $F$ and for all $m,l$ in ${\sf M}({\cal F}_{{\mathbb{Z}}})$ we have $m\star f\star l\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, and 2. 2. all s-polynomials corresponding to $F$ as specified in Definition 4.4.46 reduce to $o$ using $F$. Proof : In case $F$ is a Gröbner basis, since these polynomials are all elements of ${\sf ideal}(F)$ they must reduce to zero using $F$. The converse will be proven by showing that every element in ${\sf ideal}(F)$ is reducible by $F$. Then as $g\in{\sf ideal}(F)$ and $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}g^{\prime}$ implies $g^{\prime}\in{\sf ideal}(F)$ we have $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. Notice that this is sufficient as the reduction relation $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$ is Noetherian. Let $g\in{\sf ideal}(F)$ have a representation in terms of $F$ of the following form: $g=\sum_{j=1}^{n}v_{j}\star f_{j}\star w_{j}\cdot\alpha_{j}$ such that $f_{j}\in F$, $v_{j},w_{j}\in{\cal T}$ and $\alpha_{j}\in{\mathbb{Z}}$. Depending on this representation of $g$ and the well-founded total ordering $\succeq$ on ${\cal T}$ we define $t=\max_{\succeq}\\{{\sf HT}(v_{j}\star f_{j}\star w_{j})\mid 1\leq j\leq m\\}$, $K$ as the number of polynomials $f_{j}\star w_{j}$ with head term $t$, and $M=\\{\\{{\sf HC}(v_{j}\star f_{j}\star w_{j})\mid{\sf HT}(v_{j}\star f_{j}\star w_{j})=t\\}\\}$ a multiset in ${\mathbb{Z}}$. We show our claim by induction on $(t,M)$, where $(t^{\prime},M^{\prime})<(t,M)$ if and only if $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $M^{\prime}\ll M)$. Since by our first assumption every multiple $v_{j}\star f_{j}\star w_{j}$ in this sum reduces to zero using $F$ and hence has a representation as specified in Corollary 4.4.45, we can assume that ${\sf HT}(v_{j}\star{\sf HT}(f_{j})\star w_{j})={\sf HT}(v_{j}\star f_{j}\star w_{j})\geq{\sf HT}(f_{j})$ holds. Moreover, without loss of generality we can assume that the polynomial multiples with head term $t$ are just $v_{1}\star f_{1}\star w_{1},\ldots,v_{K}\star f_{K}\star w_{K}$ and additionally we can assume ${\sf HC}(v_{j}\star f_{j}\star w_{j})>0$595959This can easily be achieved by adding $-f$ to $F$ for all $f\in F$ and using $v_{j}\star(-f_{j})\star w_{j}\cdot(-\alpha_{j})$ in case ${\sf HC}(v_{j}\star f_{j}\star w_{j})<0$.. Obviously, $t\succeq{\sf HT}(g)$ must hold. If $K=1$ this gives us $t={\sf HT}(g)$ and even ${\sf HM}(g)={\sf HM}(v_{1}\star f_{1}\star w_{1}\cdot\alpha_{1})$, implying that $g$ is right reducible at ${\sf HM}(g)$ by $f_{1}$. Hence let us assume $K>1$. Without loss of generality we can assume that ${\sf HC}(v_{1}\star f_{1}\star w_{1})\geq{\sf HC}(v_{2}\star f_{2}\star w_{2})>0$ and there are $\alpha,\beta\in{\mathbb{Z}}$ such that ${\sf HC}(v_{2}\star f_{2}\star w_{2})\cdot\alpha+\beta={\sf HC}(v_{1}\star f_{1}\star w_{1})$ and ${\sf HC}(v_{2}\star f_{2}\star w_{2})>\beta\geq 0$. Since $t={\sf HT}(v_{1}\star f_{1}\star w_{1})={\sf HT}(v_{2}\star f_{2}\star w_{2})$ by Definition 4.4.46 we have an s-polynomial ${\sf spol}(f_{1},f_{2},t,v_{1},v_{2},w_{1},w_{2})=v_{2}\star f_{2}\star w_{2}\cdot\alpha-v_{1}\star f_{1}\star w_{1}$. If ${\sf spol}(f_{1},f_{2},t,v_{1},v_{2},w_{1},w_{2})\neq o$606060In case ${\sf spol}(f_{1},f_{2},t,v_{1},v_{2},w_{1},w_{2})=o$ the proof is similar. We just have to subsitute $o$ in the equations below which immediately gives us a smaller representation of $g$. then ${\sf spol}(f_{1},f_{2},t,v_{1},v_{2},w_{1},w_{2})\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$ implies ${\sf spol}(f_{1},f_{2},t,v_{1},v_{2},w_{1},w_{2})=\sum_{i=1}^{k}m_{i}\star h_{i}\star l_{i}$, $h_{i}\in F$, $m_{i},l_{i}\in{\sf M}({\cal F}_{{\mathbb{Z}}})$ where this sum is a representation in the sense of Corollary 4.4.45 with terms bounded by ${\sf HT}({\sf spol}(f_{1},f_{2},t,v_{1},v_{2},w_{1},w_{2}))\leq t$. This gives us $\displaystyle v_{1}\star f_{1}\star w_{1}\cdot\alpha_{1}+v_{2}\star f_{2}\star w_{2}\cdot\alpha_{2}$ $\displaystyle=$ $\displaystyle v_{1}\star f_{1}\star w_{1}\cdot\alpha_{1}+\underbrace{v_{2}\star f_{2}\star w_{2}\cdot\alpha_{1}\cdot\alpha-v_{2}\star f_{2}\star w_{2}\cdot\alpha_{1}\cdot\alpha}_{=o}+v_{2}\star f_{2}\star w_{2}\cdot\alpha_{2}$ $\displaystyle=$ $\displaystyle v_{2}\star f_{2}\star w_{2}\cdot(\alpha_{1}\cdot\alpha+\alpha_{2})-\underbrace{(v_{2}\star f_{2}\star w_{2}\cdot\alpha-v_{1}\star f_{1}\star w_{1})}_{={\sf spol}(f_{1},f_{2},t,v_{1},v_{2},w_{1},w_{2})}\cdot\alpha_{1}$ $\displaystyle=$ $\displaystyle v_{2}\star f_{2}\star w_{2}\cdot(\alpha_{1}\cdot\alpha+\alpha_{2})-(\sum_{i=1}^{k}m_{i}\star h_{i}\star l_{i})\cdot\alpha_{1}$ and substituting this in the representation of $g$ we get a new representation with $t^{\prime}=\max_{\succeq}\\{{\sf HT}(v_{j}\star f_{j}\star w_{j}),{\sf HT}(m_{j}\star h_{j}\star l_{j})\mid f_{j},h_{j}\mbox{ appearing in the new representation }\\}$, and $M^{\prime}=\\{\\{{\sf HC}(v_{j}\star f_{j}\star w_{j}),{\sf HC}(m_{j}\star h_{j}\star l_{j})\mid{\sf HT}(v_{j}\star f_{j}\star w_{j})={\sf HT}(m_{j}\star h_{j}\star l_{j})=t^{\prime}\\}\\}$ and either $t^{\prime}\prec t$ and we have a smaller representation for $g$ or in case $t^{\prime}=t$ we have to distinguish two cases: 1. 1. $\alpha_{1}\cdot\alpha+\alpha_{2}=0$. Then $M^{\prime}=(M-\\{\\{{\sf HC}(v_{1}\star f_{1}\star w_{1}),{\sf HC}(v_{2}\star f_{2}\star w_{2})\\}\\})\cup\\{\\{{\sf HC}(m_{j}\star h_{j}\star l_{j})\mid{\sf HT}(m_{j}\star h_{j}\star l_{j})=t\\}\\}$. As those polynomials $h_{j}$ with ${\sf HT}(m_{j}\star h_{j}\star l_{j})=t$ are used to reduce the monomial $\beta\cdot t={\sf HM}({\sf spol}(f_{1},f_{2},t,v_{1},v_{2},w_{1},w_{2}))$ we know that for them we have $0<{\sf HC}(m_{j}\star h_{j}\star l_{j})\leq\beta<{\sf HC}(v_{2}\star f_{2}\star w_{2})\leq{\sf HC}(v_{1}\star f_{1}\star w_{1})$ and hence $M^{\prime}\ll M$ and we have a smaller representation for $g$. 2. 2. $\alpha_{1}\cdot\alpha+\alpha_{2}\neq 0$. Then $M^{\prime}=(M-\\{\\{{\sf HC}(v_{1}\star f_{1}\star w_{1})\\}\\})\cup\\{\\{{\sf HC}(m_{j}\star h_{j}\star l_{j})\mid{\sf HT}(m_{j}\star h_{j}\star l_{j})=t\\}\\}$. Again $M^{\prime}\ll M$ and we have a smaller representation for $g$. Notice that the case $t^{\prime}=t$ and $M^{\prime}\ll M$ cannot occur infinitely often but has to result in either $t^{\prime}<t$ or will lead to $t^{\prime}=t$ and $K=1$ and hence to reducibility by $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$. q.e.d. Now the question arises when the critical situations in this characterization can be localized to subsets of the respective sets as in Theorem 4.4.31. Reviewing the Proof of Theorem 4.4.31 we find that Lemma 4.4.16 is central as it describes when multiples of polynomials which have a reductive standard representation in terms of some set $F$ again have such a representation. As we have seen before, this will not hold for function rings over reduction rings in general. As in Section 4.4.2, to give localizations of Theorem 4.4.47 the concept of stable subsets is sufficient: ###### Definition 4.4.48 A set $C\subset S\subseteq{\cal F}_{{\mathbb{Z}}}$ is called a stable localization of $S$ if for every $g\in S$ there exists $f\in C$ such that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}o$. $\diamond$ In case ${\cal F}_{{\mathbb{Z}}}$ and $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}\,$ allow such stable localizations, we can rephrase Theorem 4.4.47 as follows: ###### Theorem 4.4.49 Let $F$ be a set of polynomials in ${\cal F}_{{\mathbb{Z}}}\backslash\\{o\\}$. Then $F$ is a Gröbner basis of ${\sf ideal}(F)$ if and only if 1. 1. for all $s$ in a stable localization of $\\{m\star f\star l\mid f\in{\cal F}_{{\mathbb{Z}}},m,l\in{\sf M}({\cal F}_{{\mathbb{Z}}})\\}$ we have $s\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, and 2. 2. for all $h$ in a stable localization of the s-polynomials corresponding to $F$ as specified in Definition 4.4.46 we have $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$. When proving Theorem 4.4.49, we can substitute the critical situation using an analogon of Lemma 4.4.16, which will be sufficient to make the representation used in the proof smaller. It is a direct consequence of Lemma 4.4.40. ###### Corollary 4.4.50 Let $F\subseteq{\cal F}_{{\mathbb{Z}}}\backslash\\{o\\}$ and $f$, $p$ non-zero polynomials in ${\cal F}_{{\mathbb{Z}}}$. If $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{f}\,$}o$ and $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$, then $p$ has a representation of the form $p=\sum_{i=1}^{n}m_{i}\star f_{i}\star l_{i},f_{i}\in F,m_{i},l_{i}\in{\sf M}({\cal F}_{{\mathbb{Z}}}),n\in{\mathbb{N}}$ such that ${\sf HT}(p)={\sf HT}(m_{i}\star{\sf HT}(f_{i})\star l_{i})={\sf HT}(m_{i}\star f_{i}\star l_{i})\geq{\sf HT}(f_{i})$ for $1\leq i\leq k$ and ${\sf HT}(p)\succ{\sf HT}(m_{i}\star f_{i}\star l_{i})$ for all $k+1\leq i\leq n$. Proof Theorem 4.4.49: The proof is basically the same as for Theorem 4.4.47. Due to Corollary 4.4.50 we can substitute the multiples $v_{j}\star f_{j}\star w_{j}$ by appropriate representations. Hence, we only have to ensure that despite testing less polynomials we are able to apply our induction hypothesis. Taking the notations from the proof of Theorem 4.4.47, let us check the situation for $K>1$. Without loss of generality we can assume that ${\sf HC}(v_{1}\star f_{1}\star w_{1})\geq{\sf HC}(v_{2}\star f_{2}\star w_{2})>0$ and there are $\alpha,\beta\in{\mathbb{Z}}$ such that ${\sf HC}(v_{2}\star f_{2}\star w_{2})\cdot\alpha+\beta={\sf HC}(v_{1}\star f_{1}\star w_{1})$ and ${\sf HC}(v_{2}\star f_{2}\star w_{2})>\beta\geq 0$. Since $t={\sf HT}(v_{1}\star f_{1}\star w_{1})={\sf HT}(v_{2}\star f_{2}\star w_{2})$ by Definition 4.4.46 we have an s-polynomial $h$ in the stable localization of ${\sf SPOL}(f_{1},f_{2})$ such that $v_{2}\star f_{2}\star w_{2}\cdot\alpha- v_{1}\star f_{1}\star w_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{h}\,$}o$. If $h\neq o$616161In case $h=o$ the proof is similar. We just have to subsitute $o$ in the equations below which immediately gives us a smaller representation of $g$. then by Corollary 4.4.50 $h\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$}o$ implies $v_{2}\star f_{2}\star w_{2}\cdot\alpha-v_{1}\star f_{1}\star w_{1}=\sum_{i=1}^{k}m_{i}\star h_{i}\star l_{i}$, $h_{i}\in F$, $m_{i},l_{i}\in{\sf M}({\cal F}_{{\mathbb{Z}}})$ where this sum is a representation in the sense of Corollary 4.4.45 with terms bounded by ${\sf HT}(m\star h\star l)\leq t$. This gives us $\displaystyle v_{1}\star f_{1}\star w_{1}\cdot\alpha_{1}+v_{2}\star f_{2}\star w_{2}\cdot\alpha_{2}$ $\displaystyle=$ $\displaystyle v_{1}\star f_{1}\star w_{1}\cdot\alpha_{1}+\underbrace{v_{2}\star f_{2}\star w_{2}\cdot\alpha_{1}\cdot\alpha-v_{2}\star f_{2}\star w_{2}\cdot\alpha_{1}\cdot\alpha}_{=o}+v_{2}\star f_{2}\star w_{2}\cdot\alpha_{2}$ $\displaystyle=$ $\displaystyle v_{2}\star f_{2}\star w_{2}\cdot(\alpha_{1}\cdot\alpha+\alpha_{2})-(v_{2}\star f_{2}\star w_{2}\cdot\alpha-v_{1}\star f_{1}\star w_{1})\cdot\alpha_{1}$ $\displaystyle=$ $\displaystyle v_{2}\star f_{2}\star w_{2}\cdot(\alpha_{1}\cdot\alpha+\alpha_{2})-(\sum_{i=1}^{k}m_{i}\star h_{i}\star l_{i})\cdot\alpha_{1}$ and substituting this in the representation of $g$ we get a new representation with $t^{\prime}=\max_{\succeq}\\{{\sf HT}(v_{j}\star f_{j}\star w_{j}),{\sf HT}(m_{j}\star h_{j}\star l_{j})\mid f_{j},h_{j}\mbox{ appearing in the new representation }\\}$, and $M^{\prime}=\\{\\{{\sf HC}(v_{j}\star f_{j}\star w_{j}),{\sf HC}(m_{j}\star h_{j}\star l_{j})\mid{\sf HT}(v_{j}\star f_{j}\star w_{j})={\sf HT}(m_{j}\star h_{j}\star l_{j})=t^{\prime}\\}\\}$ and either $t^{\prime}\prec t$ or $(t^{\prime}=t$ and $M^{\prime}\ll M)$ and in both cases we have a smaller representation for $g$. Notice that the case $t^{\prime}=t$ and $M^{\prime}\ll M$ cannot occur infinitely often but has to result in either $t^{\prime}<t$ or will lead to $t^{\prime}=t$ and $K=1$ and hence to reducibility by $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{F}\,$. q.e.d. ### 4.5 Two-sided Modules Given a function ring ${\cal F}$ with unit ${\bf 1}$ and a natural number $k$, let ${\cal F}^{k}=\\{(f_{1},\ldots,f_{k})\mid f_{i}\in{\cal F}\\}$ be the set of all vectors of length $k$ with coordinates in ${\cal F}$. Obviously ${\cal F}^{k}$ is an additive commutative group with respect to ordinary vector addition. Moreover, ${\cal F}^{k}$ is such an ${\cal F}$-module with respect to the scalar multiplication $f\star(f_{1},\ldots,f_{k})=(f\star f_{1},\ldots,f\star f_{k})$ and $(f_{1},\ldots,f_{k})\star f=(f_{1}\star f,\ldots,f_{k}\star f)$. Additionally ${\cal F}^{k}$ is called free as it has a basis626262Here the term basis is used in the meaning of being a linearly independent set of generating vectors.. One such basis is the set of unit vectors ${\bf e}_{1}=({\bf 1},o,\ldots,o),{\bf e}_{2}=(o,{\bf 1},o,\ldots,o),\ldots,{\bf e}_{k}=(o,\ldots,o,{\bf 1})$. Using this basis the elements of ${\cal F}^{k}$ can be written uniquely as ${\bf f}=\sum_{i=1}^{k}f_{i}\star{\bf e}_{i}$ where ${\bf f}=(f_{1},\ldots,f_{k})$. ###### Definition 4.5.1 A subset of ${\cal F}^{k}$ which is again an ${\cal F}$-module is called a submodule of ${\cal F}^{k}$. As before any ideal of ${\cal F}$ is an ${\cal F}$-module and even a submodule of the ${\cal F}$-module ${\cal F}^{1}$. Provided a set of vectors $S=\\{{\bf f}_{1},\ldots,{\bf f}_{s}\\}$ the set $\\{\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}g_{ij}\star{\bf f}_{i}\star{h_{ij}}\mid g_{ij},{h_{ij}}\in{\cal F}\\}$ is a submodule of ${\cal F}^{k}$. This set is denoted as $\langle S\rangle$ and $S$ is called a generating set. ###### Theorem 4.5.2 Let ${\cal F}$ be Noetherian. Then every submodule of ${\cal F}^{k}$ is finitely generated. Proof : Let ${\cal S}$ be a submodule of ${\cal F}^{k}$. Again we show our claim by induction on $k$. For $k=1$ we find that ${\cal S}$ is in fact an ideal in ${\cal F}$ and hence by our hypothesis finitely generated. For $k>1$ let us look at the set $I=\\{f_{1}\mid(f_{1},\ldots,f_{k})\in{\cal S}\\}$. Then again $I$ is an ideal in ${\cal F}$ and hence finitely generated. Let $\\{g_{1},\ldots,g_{s}\mid g_{i}\in{\cal F}\\}$ be a generating set of $I$. Choose ${\bf g}_{1},\ldots,{\bf g}_{s}\in{\cal S}$ such that the first coordinate of ${\bf g}_{i}$ is $g_{i}$. Note that the set $\\{(f_{2},\ldots,f_{k})\mid(o,f_{2},\ldots,f_{k})\in{\cal S}\\}$ is a submodule of ${\cal F}^{k-1}$ and hence finitely generated by some set $\\{(n_{2}^{i},\ldots,n_{k}^{i}),1\leq i\leq w\\}$. Then the set $\\{{\bf g}_{1},\ldots,{\bf g}_{s}\\}\cup\\{{\bf n}_{i}=(o,n_{2}^{i},\ldots,n_{k}^{i})\mid 1\leq i\leq w\\}$ is a generating set for ${\cal S}$. To see this assume ${\bf m}=(m_{1},\ldots,m_{k})\in{\cal S}$. Then $m_{1}=\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}h_{ij}\star g_{i}\star{h_{ij}}^{\prime}$ for some $h_{ij},{h_{ij}}^{\prime}\in{\cal F}$ and ${\bf m^{\prime}}={\bf m}-\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}h_{ij}\star{\bf g}_{i}\star{h_{ij}}^{\prime}\in{\cal S}$ with first coordinate $o$. Hence ${\bf m^{\prime}}=\sum_{i=1}^{w}\sum_{j=1}^{m_{i}}l_{ij}\star{\bf n}_{i}\star{l_{ij}}^{\prime}$ for some $l_{ij},{l_{ij}}^{\prime}\in{\cal F}$ giving rise to ${\bf m}={\bf m^{\prime}}+\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}h_{ij}\star{\bf g}_{i}\star{h_{ij}}^{\prime}=\sum_{i=1}^{w}\sum_{j=1}^{m_{i}}l_{ij}\star{\bf n}_{i}\star{l_{ij}}^{\prime}+\sum_{i=1}^{s}\sum_{j=1}^{n_{i}}h_{ij}\star{\bf g}_{i}\star{h_{ij}}^{\prime}.$ q.e.d. ${\cal F}^{k}$ is called Noetherian if and only if all its submodules are finitely generated. If ${\cal F}$ is a reduction ring Section 4.5 outlines how the existence of Gröbner bases for submodules can be shown. Now given a submodule ${\cal S}$ of ${\cal F}^{k}$, we can define ${\cal F}^{k}/{\cal S}=\\{{\bf f}+{\cal S}\mid{\bf f}\in{\cal F}^{k}\\}$. Then with addition defined as $({\bf f}+{\cal S})+({\bf g}+{\cal S})=({\bf f}+{\bf g})+{\cal S}$ the set ${\cal F}^{k}/{\cal S}$ is an abelian group and can be turned into an ${\cal F}$-module by the action $g\star({\bf f}+{\cal S})\star h=g\star{\bf f}\star h+{\cal S}$ for $g,h\in{\cal F}$. ${\cal F}^{k}/{\cal S}$ is called the quotient module of ${\cal F}^{k}$ by ${\cal S}$. As usual this quotient can be related to homomorphisms. The results carry over from commutative module theory as can be found in [AL94]. Recall that for two ${\cal F}$-modules ${\cal M}$ and ${\cal N}$, a function $\phi:{\cal M}\longrightarrow{\cal N}$ is an ${\cal F}$-module homomorphism if $\phi({\bf f}+{\bf g})=\phi({\bf f})+\phi({\bf g})\mbox{ for all }{\bf f,g}\in{\cal M}$ and $\phi(g\star{\bf f}\star h)=g\star\phi({\bf f})\star h\mbox{ for all }{\bf f}\in{\cal M},g,h\in{\cal F}.$ The homomorphism is called an isomorphism if $\phi$ is one to one and we write ${\cal M}\cong{\cal N}$. Let ${\cal S}={\rm ker}(\phi)=\\{{\bf f}\in{\cal M}\mid\phi({\bf f})={\bf 0}\\}$. Then ${\cal S}$ is a submodule of ${\cal M}$ and $\phi({\cal M})$ is a submodule of ${\cal N}$. Since all are abelian groups we know ${\cal M}/{\cal S}\cong\phi({\cal M})$ under the mapping ${\cal M}/{\cal S}\longrightarrow\phi({\cal M})$ with ${\bf f}+{\cal S}\mapsto\phi({\bf f})$ which is in fact an isomorphism. All submodules of the quotient ${\cal M}/{\cal S}$ are of the form ${\cal L}/{\cal S}$ where ${\cal L}$ is a submodule of ${\cal M}$ containing ${\cal S}$. Unfortunately, contrary to the one-sided case we can no longer show that every finitely generated ${\cal F}$-module ${\cal M}$ is isomorphic to some quotient of ${\cal F}^{k}$. Let ${\cal M}$ be a finitely generated ${\cal F}$-module with generating set ${\bf f}_{1},\ldots{\bf f}_{k}\in{\cal M}$. Consider the mapping $\phi:{\cal F}^{k}\longrightarrow{\cal M}$ defined by $\phi(g_{1},\ldots,g_{k})=\sum_{i=1}^{k}g_{i}\star{\bf f}_{i}$ for ${\cal M}$. The image of the ${\cal F}$-module homomorphis is no longer ${\cal M}$. ## Chapter 5 Applications of Gröbner Bases In this chapter we outline how the concept of Gröbner bases can be used to describe algebraic questions and when solutions can be achieved. We will describe the problems in the following manner Problem Given: A description of the algebraic setting of the problem. Problem: A description of the problem itself. Proceeding: A description of how the problem can be analyzed using Gröbner bases. In a first step we do not require finiteness or computability of the operations, especially of a Gröbner basis. Since an ideal itself is always a Gröbner basis itself, the assumption “Let G be a respective Gröbner basis” always holds and means a Gröbner basis of the ideal generated by $G$. In case a Gröbner basis is computable (though not necessarily finite) and the normal form computation for a polynomial with respect to a finite set is effective, our so-called proceedings give rise to procedures which can then be used to treat the problem in a constructive manner. If additionally the Gröbner basis computation terminates, these procedures terminate as well and the instance of the problem is decidable. In case Gröbner basis computation always terminates for a chosen setting the whole problem is decidable in this setting. Of course “termination” here is meant in a theoretical sense while as we know practical “termination” is already often not achievable for the Gröbner basis computation in the ordinary polynomial ring due to complexity issues although finite Gröbner bases always exist. The terminology extends to one-sided ideals and we note those problems, where the one-sided case also makes sense. We will also note when weak Gröbner bases are sufficient for the solution of a problem. ### 5.1 Natural Applications The most obvious problem related to Gröbner bases is the ideal membership problem. Characterizing Gröbner bases with respect to a reduction relation uses the important fact that an element belonging to the ideal will reduce to zero using the Gröbner basis. Ideal Membership Problem Given: A set $F\subseteq{\cal F}$ and an element $f\in{\cal F}$. Problem: $f\in{\sf ideal}(F)$? Proceeding: 1. Let $G$ be a Gröbner basis of ${\sf ideal}(F)$. 2. If $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$, then $f\in{\sf ideal}(F)$. Hence Gröbner bases give a semi-answer to this question in case they are computable and the normal form computation is effective. To give a negative answer the Gröbner basis computation must either terminate or one must explicitly prove, e.g. using properties of the enumerated Gröbner basis, that the element will never reduce to zero. These results carry over to one-sided ideals using the appropriate one-sided Gröbner bases. Moreover, weak Gröbner bases are sufficient to solve the problem. A normal form computation always gives rise to a special representation in terms of the polynomials used for reduction and in case the normal form is zero such representations are special standard representations. We give two instances of this problem. Representation Problem 1 Given: A Gröbner basis $G\subseteq{\cal F}$ and an element $f\in{\sf ideal}(G)$. Problem: Give a representation of $f$ in terms of $G$. Proceeding: Reducing $f$ to $o$ using $G$ yields such a representation. In case the normal form computation is effective, we can collect the polynomials and multiples used in the reduction process and combine them to the desired representation. Notice that since we know that the element is in the ideal, it is enough to additionally require that the Gröbner basis is recursively enumerable as a set. The result carries over to one-sided ideals using the appropriate one-sided Gröbner bases. Again, weak Gröbner bases are sufficient to solve the problem. Often the ideal is not presented in terms of a Gröbner basis. Then additional information is necessary which in the computational case is related to collecting the history of polynomials created during completion. Notice that the proceedings in this case require some equivalent to Lemma 4.4.16 to hold and hence the problem is restricted to function rings over fields. Representation Problem 2 Given: A set $F\subseteq{\cal F}_{{\mathbb{K}}}$ and an element $f\in{\sf ideal}(F)$. Problem: Give a representation of $f$ in terms of $F$. Proceeding: 1. Let $G$ be a Gröbner basis of ${\sf ideal}(F)$. 2. Let $g=\sum_{i=1}^{k_{g}}m_{i}\star f_{i}\star\tilde{m}_{i}$ be representations of the elements $g\in G$ in terms of $F$. 3. Let $f=\sum_{j=1}^{k}n_{i}\star g_{i}\star\tilde{n}_{i}$ be a representation of $f$ in terms of $G$. 4. The sums in 2. and 3. yield a representation of $f$ in terms of $F$. In case the Gröbner basis is computable by a completion procedure the procedure has to keep track of the history of polynomials by storing their representations in terms of $F$. If the completion stops we can reduce $f$ to zero and substitute the representations of the polynomials used by their “history representation”. If the Gröbner basis is only recursively enumerable both processes have to be interwoven and to continue until the normal form computation for $f$ reaches $o$. The result carries over to one-sided ideals using the appropriate one-sided Gröbner bases. Moreover, weak Gröbner bases are sufficient to solve the problem. Other problems are related to the comparison of ideals. For example given two ideals one can ask whether one is included in the other. Ideal Inclusion Problem Given: Two sets $F_{1},F_{2}\subseteq{\cal F}$. Problem: ${\sf ideal}(F_{1})\subseteq{\sf ideal}(F_{2})$? Proceeding: 1. Let $G$ be a Gröbner basis of ${\sf ideal}(F_{2})$ . 2. If $F_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$, then ${\sf ideal}(F_{1})\subseteq{\sf ideal}(F_{2})$. In case the Gröbner basis is computable and the normal form computation is effective this yields a semi-decision procedure for the problem. If additionally the Gröbner basis computation terminates for $F_{1}$ or we can prove that some element of the set $F_{1}$ does not belong to ${\sf ideal}(F_{2})$, e.g. by deriving knowledge from the enumerated Gröbner basis, we can also give a negative answer. The result carries over to one-sided ideals using the appropriate one-sided Gröbner bases. Weak Gröbner bases are sufficient to solve the problem. Applying the inclusion problem in both directions we get a characterization for equality of ideals. Ideal Equality Problem Given: Two sets $F_{1},F_{2}\subseteq{\cal F}$. Problem: ${\sf ideal}(F_{1})={\sf ideal}(F_{2})$? Proceeding: 1. Let $G_{1}$, $G_{2}$ be Gröbner bases of ${\sf ideal}(F_{1})$ respectively ${\sf ideal}(F_{2})$. 2. If $F_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G_{2}}\,$}o$, then ${\sf ideal}(F_{1})\subseteq{\sf ideal}(F_{2})$. 3. If $F_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G_{1}}\,$}o$, then ${\sf ideal}(F_{2})\subseteq{\sf ideal}(F_{1})$. 4. If 2. and 3. both hold, then ${\sf ideal}(F_{1})={\sf ideal}(F_{2})$. Again, Gröbner bases at least give a semi-answer in case they are computable and the normal form procedure is effective. We can confirm whether two generating sets are bases of one ideal. Of course, in case the computed Gröbner bases are finite, we can also give a negative answer. However, if the Gröbner bases are not finite, a negative answer is only possible, if we can prove either $F_{1}\not\subseteq{\sf ideal}(F_{2})$ or $F_{2}\not\subseteq{\sf ideal}(F_{1})$. The result carries over to one-sided ideals using the appropriate one-sided Gröbner bases. Again, weak Gröbner bases are sufficient to solve the problem. In case ${\cal F}$ contains a unit say ${\bf 1}$, we can ask whether an ideal is equal to the trivial ideal in ${\cal F}$ generated by the unit. Ideal Triviality Problem 1 Given: A set $F\subseteq{\cal F}$. Problem: ${\sf ideal}(F)={\sf ideal}(\\{{\bf 1}\\})$? Proceeding: 1. Let $G$ be a respective Gröbner basis. 2. If ${\bf 1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$, then ${\sf ideal}(F)={\sf ideal}(\\{{\bf 1}\\})$. Again Gröbner bases give a semi-answer in case they can be computed. If the Gröbner basis is additionally finite or we can prove that ${\bf 1}\not\in{\sf ideal}(F)$, then we can also confirm ${\sf ideal}(F)\neq{\sf ideal}(\\{{\bf 1}\\})$. Since ${\sf ideal}(\\{1\\})={\cal F}$ one can also rephrase the question for rings without a unit. Ideal Triviality Problem 2 Given: A set $F\subseteq{\cal F}$. Problem: ${\sf ideal}(F)={\cal F}$? Proceeding: 1. Let $G$ be a Gröbner basis of ${\sf ideal}(F)$. 2. If for every $t\in{\cal T}$, $t\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$, then ${\sf ideal}(F)={\cal F}$. Of course now we have the problem that the test set ${\cal T}$ in general will not be finite. Hence a Gröbner basis can give a semi-answer in case we can restrict this test set to a finite subset. If the Gröbner basis is additionally finite or we can prove that $t\mbox{$\,\,\,\,{\not\\!\\!\\!\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}}\,$}o$ for some $t$ in the finite sub test set of ${\cal T}$, then we can also confirm ${\sf ideal}(F)\neq{\cal F}$. Both of these result carry over to one-sided ideals using the appropriate one- sided Gröbner bases. As before, weak Gröbner bases are sufficient to solve the problem. Ideal Union Problem Given: Two sets $F_{1},F_{2}\subseteq{\cal F}$ and an element $f\in{\cal F}$. Problem: $f\in{\sf ideal}(F_{1})\cup{\sf ideal}(F_{2})$? Proceeding: 1. Let $G_{1}$, $G_{2}$ be Gröbner bases of ${\sf ideal}(F_{1})$ respectively ${\sf ideal}(F_{2})$. 2. If $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G_{1}}\,$}o$, then $f\in{\sf ideal}(F_{1})\cup{\sf ideal}(F_{2})$. 3. If $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G_{2}}\,$}o$, then $f\in{\sf ideal}(F_{1})\cup{\sf ideal}(F_{2})$. Notice that ${\sf ideal}(F_{1})\cup{\sf ideal}(F_{2})\neq{\sf ideal}(F_{1}\cup F_{2})$. Moreover $G_{1}\cup G_{2}$ is neither a Gröbner basis of ${\sf ideal}(F_{1})\cup{\sf ideal}(F_{2})$, which in general is no ideal itself, nor of ${\sf ideal}(F_{1}\cup F_{2})$. Again, weak Gröbner bases are sufficient to solve the problem. The ideal generated by the set $F_{1}\cup F_{2}$ is called the sum of the two ideals. ###### Definition 5.1.1 For two ideals $\mathfrak{i},\mathfrak{j}\subseteq{\cal F}$ the sum is defined as the set $\mathfrak{i}+\mathfrak{j}=\\{f\oplus g\mid f\in\mathfrak{i},g\in\mathfrak{j}\\}.$ As in the case of commutative polynomials one can show the following theorem. ###### Theorem 5.1.2 For two ideals $\mathfrak{i},\mathfrak{j}\subseteq{\cal F}$ the sum $\mathfrak{i}+\mathfrak{j}$ is again an ideal. In fact, it is the smallest ideal containing both, $\mathfrak{i}$ and $\mathfrak{j}$. If $F$ and $G$ are the respective generating sets for $\mathfrak{i}$ and $\mathfrak{j}$, then $F\cup G$ is a generating set for $\mathfrak{i}+\mathfrak{j}$. Proof : First we check that the sum is indeed an ideal: 1. 1. as $o\oplus o=o$ we get $o\in\mathfrak{i}+\mathfrak{j}$, 2. 2. for $h_{1},h_{2}\in\mathfrak{i}+\mathfrak{j}$ we have that there are $f_{1},f_{2}\in\mathfrak{i}$ and $g_{1},g_{2}\in\mathfrak{j}$ such that $h_{1}=f_{1}\oplus g_{1}$ and $h_{2}=f_{2}\oplus g_{2}$. Then $h_{1}\oplus h_{2}=(f_{1}\oplus g_{1})\oplus(f_{2}\oplus g_{2})=(f_{1}\oplus f_{2})\oplus(g_{1}\oplus g_{2})\in\mathfrak{i}+\mathfrak{j}$, and 3. 3. for $h_{1}\in\mathfrak{i}+\mathfrak{j}$, $h_{2}\in{\cal F}$ we have that there are $f\in\mathfrak{i}$ and $g\in\mathfrak{j}$ such that $h_{1}=f\oplus g$. Then $h_{1}\star h_{2}=(f\oplus g)\star h_{2}=f\star h_{2}\oplus g\star h_{2}\in\mathfrak{i}+\mathfrak{j}$ as well as $h_{2}\star h_{1}=h_{2}\star(f\oplus g)=h_{2}\star f\oplus h_{2}\star g\in\mathfrak{i}+\mathfrak{j}$. Since any ideal containing $\mathfrak{i}$ and $\mathfrak{j}$ contains $\mathfrak{i}+\mathfrak{j}$, this is the smallest ideal containing them. It is easy to see that $F\cup G$ is a generating set for the sum. q.e.d. Of course $F\cup G$ in general will not be a Gröbner basis. This becomes immediately clear when looking at the following corollary. ###### Corollary 5.1.3 For $F\subseteq{\cal F}$ we have ${\sf ideal}(F)=\bigcup_{f\in F}{\sf ideal}(f).$ But we have already seen that for function rings a polynomial in general is no Gröbner basis of the ideal or one-sided ideal it generates. Ideal Sum Problem Given: Two sets $F_{1},F_{2}\subseteq{\cal F}$ and an element $f\in{\cal F}$. Problem: $f\in{\sf ideal}(F_{1})+{\sf ideal}(F_{2})$? Proceeding: 1. Let $G$ be a Gröbner basis of ${\sf ideal}(F_{1}\cup F_{2})$. 2. If $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$, then $f\in{\sf ideal}(F_{1})+{\sf ideal}(F_{2})$. Both of these result carry over to one-sided ideals using the appropriate one- sided Gröbner bases. As before, weak Gröbner bases are sufficient to solve the problem. Similar to sums for commutative function rings we can define products of ideals. ###### Definition 5.1.4 For two ideals $\mathfrak{i},\mathfrak{j}$ in a commutative function ring ${\cal F}$ the product is defined as the set $\langle\mathfrak{i}\star\mathfrak{j}\rangle={\sf ideal}(\\{f_{i}\star g_{i}\mid f_{i}\in\mathfrak{i},g_{i}\in\mathfrak{j}\\}).$ $\diamond$ ###### Theorem 5.1.5 For two ideals $\mathfrak{i},\mathfrak{j}$ in a commutative function ring ${\cal F}$ the product $\langle\mathfrak{i}\star\mathfrak{j}\rangle$ is again an ideal. If $F$ and $G$ are the respective generating sets for $\mathfrak{i}$ and $\mathfrak{j}$, then $F\star G=\\{f\star g\mid f\in F,g\in G\\}$ is a generating set for $\mathfrak{i}\star\mathfrak{j}$. Proof : First we check that the product is indeed an ideal: 1. 1. as $o\in\mathfrak{i}$ and $o\in\mathfrak{j}$ we get $o\in\mathfrak{i}\star\mathfrak{j}$, 2. 2. for $f,g\in\mathfrak{i}\star\mathfrak{j}$ we have $f\oplus g\in\mathfrak{i}\star\mathfrak{j}$ by our definition, and 3. 3. for $f\in\mathfrak{i}\star\mathfrak{j}$, $h\in{\cal F}$ we have that there are $f_{i}\in\mathfrak{i}$ and $g_{i}\in\mathfrak{j}$ such that $f=\sum_{i=1}^{k}f_{i}\star g_{i}$ and then $f\star h=(\sum_{i=1}^{k}f_{i}\star g_{i})\star h=\sum_{i=1}^{k}f_{i}\star(g_{i}\star h)\in\mathfrak{i}\star\mathfrak{j}$. It is obvious that ${\sf ideal}(F\star G)\subseteq\langle\mathfrak{i}\star\mathfrak{j}\rangle$ as $F\star G\subseteq\mathfrak{i}\star\mathfrak{j}$. On the other hand every polynomial in $\langle\mathfrak{i}\star\mathfrak{j}\rangle$ can be written as a sum of products $\tilde{f}\star\tilde{g}$ where $\tilde{f}=\sum_{i=1}^{n}h_{i}\star f_{i}\in\mathfrak{i}$, $f_{i}\in F$, $h_{i}\in{\cal F}$ and $\tilde{g}=\sum_{j=1}^{m}g_{j}\star\tilde{h}_{j}$, $g_{j}\in G$, $\tilde{h}_{j}\in{\cal F}$. Hence every such product $\tilde{f}\star\tilde{g}$ is again of the desired form. q.e.d. Ideal Product Problem Given: Two subsets $F_{1},F_{2}$ of a commutative function ring ${\cal F}$ and an element $f\in{\cal F}$. Problem: $f\in\langle{\sf ideal}(F_{1})\star{\sf ideal}(F_{2})\rangle$? Proceeding: 1. Let $G$ be a Gröbner basis of ${\sf ideal}(F_{1}\star F_{2})$. 2. If $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$, then $f\in\langle{\sf ideal}(F_{1})\star{\sf ideal}(F_{2})\rangle$. Again, weak Gröbner bases are sufficient to solve the problem. We close this section by showing how Gröbner bases can help to detect the existence of inverse elements in ${\cal F}$ in case ${\cal F}$ has a unit say ${\bf 1}$. ###### Definition 5.1.6 Let ${\cal F}$ be a function ring with unit ${\bf 1}$ and $f\in{\cal F}$. An element $g\in{\cal F}$ is called a right inverse of $f$ in ${\cal F}$ if $f\star g={\bf 1}$. Similarly $g$ is called a left inverse of $f$ in ${\cal F}$ if $g\star f={\bf 1}$. $\diamond$ Inverse Element Problem Given: An element $f\in{\cal F}$. Problem: Does $f$ have a right or left inverse in ${\cal F}$? Proceeding: 1. Let $G_{r}$ be a respective right Gröbner basis of ${\sf ideal}_{r}(f)$. 2. If ${\bf 1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{G_{r}}\,$}o$, then $f$ has a right inverse. 1’. Let $G_{\ell}$ be a respective left Gröbner basis of ${\sf ideal}_{\ell}(f)$. 2’. If ${\bf 1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{G_{\ell}}\,$}o$, then $f$ has a left inverse. To see that this is correct we give the following argument for the right inverse case: It is clear that $f$ has a right inverse in ${\cal F}$ if and only if ${\sf ideal}_{r}(\\{f\\})={\cal F}$ since $f\star g-{\bf 1}=o$ for some $g\in{\cal F}$ if and only if ${\bf 1}\in{\sf ideal}_{r}(\\{f\\})$. So, in order to decide whether $f$ has a right inverse in ${\cal F}$ one has to distinguish the following two cases provided we have a right Gröbner basis $G_{r}$ of ${\sf ideal}_{r}(\\{f\\})$: If ${\bf 1}\mbox{$\,\,\,\,{\not\\!\\!\\!\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{G_{r}}}\,$}o$ then $f$ has no right inverse. If ${\bf 1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{G_{r}}\,$}o$ then we know ${\bf 1}\in{\sf ideal}_{r}(\\{f\\})$, i.e. there exist $h\in{\cal F}$ such that ${\bf 1}=f\star h$ and hence $h$ is a right inverse of $f$ in ${\cal F}$. A symmetric argument holds for the case of left inverses. Of course in case ${\cal F}$ is commutative, left inverses and right inverses coincide in case they exist and we can use the fact that $f\star g-{\bf 1}=g\star f-{\bf 1}=o$ if and only if ${\bf 1}\in{\sf ideal}(\\{f\\})$. Again, weak Gröbner bases are sufficient to solve the problem. It is also possible to ask for the existence of left and right inverses for elements of the quotient rings described in the next section. ### 5.2 Quotient Rings Let $F$ be a subset of ${\cal F}$ generating an ideal $\mathfrak{i}={\sf ideal}(F)$. The canonical homomorphism from ${\cal F}$ onto ${\cal F}/\mathfrak{i}$ is defined as $f\longmapsto[f]_{\mathfrak{i}}$ with $[f]_{\mathfrak{i}}=f+\mathfrak{i}$ denoting the congruence class of $f$ modulo $\mathfrak{i}$. The ring operations are given by $[f]_{\mathfrak{i}}+[g]_{\mathfrak{i}}=[f+g]_{\mathfrak{i}},$ $[f]_{\mathfrak{i}}\ast[g]_{\mathfrak{i}}=[f\star g]_{\mathfrak{i}}.$ A natural question now is whether two elements of ${\cal F}$ are in fact in the same congruence class modulo $\mathfrak{i}$. Congruence Problem Given: A set $F\subseteq{\cal F}$ and two elements $f,g\in{\cal F}$. Problem: $f=g$ in ${\cal F}/{\sf ideal}(F)$? Proceeding: 1. Let $G$ be a Gröbner basis of ${\sf ideal}(F)$. 2. If $f-g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$, then $f=g$ in ${\cal F}/{\sf ideal}(F)$. Hence if $G$ is a Gröbner basis for which normal form computation is effective, the congruence problem is solvable. Usually one element of the congruence class is identified as its representative and since normal forms with respect to Gröbner bases are unique, they can be chosen as such representatives. Notice that for weak Gröbner bases unique representations for the quotient can no longer be determined by reduction (review Example 3.1.1). Unique Representatives Problem Given: A set $F\subseteq{\cal F}$ and an element $f\in{\cal F}$. Problem: Determine a unique representative for $f$ in ${\cal F}/{\sf ideal}(F)$. Proceeding: 1. Let $G$ be a respective Gröbner basis. 2. The normal form of $f$ with respect to $G$ is a unique representative. Provided a Gröbner basis of $\mathfrak{i}$ together with an effective normal form algorithm we can specify unique representatives by $[f]_{\mathfrak{i}}:={\rm normal\\_form}(f,G),$ and define addition and multiplication in the quotient by $[f]_{\mathfrak{i}}+[g]_{\mathfrak{i}}:={\rm normal\\_form}(f+g,G),$ $[f]_{\mathfrak{i}}\ast[g]_{\mathfrak{i}}:={\rm normal\\_form}(f\star g,G).$ Similar to the case of polynomial rings for a function ring over a field ${\mathbb{K}}$ we can show that this structure is a ${\mathbb{K}}$-vector space with a special basis. ###### Lemma 5.2.1 For any ideal $\mathfrak{i}\subseteq{\cal F}_{{\mathbb{K}}}$ the following hold: 1. 1. ${\cal F}_{{\mathbb{K}}}/\mathfrak{i}$ is a ${\mathbb{K}}$-vector space. 2. 2. The set $B=\\{[t]_{\mathfrak{i}}\mid t\in{\cal T}\\}$ is a vector space basis and we can chose $[t]_{\mathfrak{i}}={\rm monic}({\rm normal\\_form}(t,G))$ for $G$ being a Gröbner basis of $\mathfrak{i}$. Proof : 1. 1. We have to show that the following properties hold for $V={\cal F}_{{\mathbb{K}}}/\mathfrak{i}$: 1. (a) There exists a mapping $K\times V\longrightarrow V$, $(\alpha,[f]_{\mathfrak{i}})\longmapsto\alpha\cdot[f]_{\mathfrak{i}}$ called multiplication with scalars. 2. (b) $(\alpha\cdot\beta)\cdot[f]_{\mathfrak{i}}=\alpha\cdot(\beta\cdot[f]_{\mathfrak{i}})$ for all $\alpha,\beta\in{\mathbb{K}}$, $[f]_{\mathfrak{i}}\in V$. 3. (c) $\alpha\cdot([f]_{\mathfrak{i}}+[g]_{\mathfrak{i}})=\alpha\cdot[f]_{\mathfrak{i}}+\alpha\cdot[g]_{\mathfrak{i}}$ for all $\alpha\in{\mathbb{K}}$, $[f]_{\mathfrak{i}},[g]_{\mathfrak{i}}\in V$. 4. (d) $(\alpha+\beta)\cdot[f]_{\mathfrak{i}}=\alpha\cdot[f]_{\mathfrak{i}}+\beta\cdot[f]_{\mathfrak{i}}$ for all $\alpha,\beta\in{\mathbb{K}}$, $[f]_{\mathfrak{i}}\in V$. 5. (e) ${\bf 1}\cdot[f]_{\mathfrak{i}}=[f]_{\mathfrak{i}}$ for all $[f]_{\mathfrak{i}}\in V$. It is easy to show that this follows from the natural definition $\alpha\cdot[f]_{\mathfrak{i}}:=[\alpha\cdot f]_{\mathfrak{i}}$ for $\alpha\in{\mathbb{K}}$, $[f]_{\mathfrak{i}}\in V$. 2. 2. It follows immediately that $B$ generates the quotient ${\cal F}_{{\mathbb{K}}}/\mathfrak{i}$. So it remains to show that this basis is free in the sense that $o$ cannot be represented as a non-trivial linear combination of elements in $B$. Let $G$ be a Gröbner basis of $\mathfrak{i}$. Then we can choose the elements of $B$ as the normal forms of the elements in ${\cal T}$ with respect to $G$. Since for a polynomial in normal form all its terms are also in normal form we can conclude that these normal forms are elements of ${\sf M}({\cal F}_{{\mathbb{K}}})$ and since ${\mathbb{K}}$ is a field we can make them monic. This leaves us with a basis $\\{\tilde{t}={\rm monic}({\rm normal\\_form}(t,G))\mid t\in{\cal T}\\}$ . Now let us assume that $B$ is not free, i.e. there exists $k\in{\mathbb{N}}$ minimal with $\alpha_{i}\in{\mathbb{K}}\backslash\\{0\\}$ and $[t_{i}]_{\mathfrak{i}}\in B$, $1\leq i\leq k$ such that $\sum_{i}^{k}\alpha_{i}\cdot[t_{i}]_{\mathfrak{i}}=o$. Since then we also get ${\rm normal\\_form}(\sum_{i}^{k}\alpha_{i}\cdot\tilde{t}_{i},G)=o$ and all $\tilde{t_{i}}$ are different and in normal form, all $\alpha_{i}$ must equal $0$ contradicting our assumption. q.e.d. If we can compute normal forms for the quotient elements, we can give a multiplication table for the quotient in terms of the vector space basis by $[t_{i}]_{\mathfrak{i}}\ast[t_{j}]_{\mathfrak{i}}=[t_{i}\star t_{j}]_{\mathfrak{i}}={\rm normal\\_form}(t_{i}\circ t_{j},G).$ Notice that for a function ring over a reduction ring the set $B=\\{[t]_{\mathfrak{i}}\mid t\in{\cal T}\\}$ also is a generating set where we can chose $[t]_{\mathfrak{i}}={\rm normal\\_form}(t,G)$. But we can no longer choose the representatives to be a subset of ${\cal T}$. This is due to the fact that if a monomial $\alpha\cdot t$ is reducible by some polynomial $g$ this does not imply that some other monomial $\beta\cdot t$ or even the term $t$ is reducible by $g$. For example let ${\sf R}={\mathbb{Z}}$, ${\cal T}=\\{a,\lambda\\}$ and $a\star a=2\cdot a$, $\lambda\star\lambda=\lambda$, $a\star\lambda=\lambda\star a=a$. Then $2\cdot a$ is reducible by $a$ while of course $a$ isn’t. In case ${\cal F}_{{\mathbb{K}}}$ contains a unit say ${\bf 1}$ we can ask whether an element of ${\cal F}_{{\mathbb{K}}}/\mathfrak{i}$ is invertible. ###### Definition 5.2.2 Let $f\in{\cal F}_{{\mathbb{K}}}$. An element $g\in{\cal F}_{{\mathbb{K}}}$ is called a right inverse of $f$ in ${\cal F}_{{\mathbb{K}}}/\mathfrak{i}$ if $f\star g={\bf 1}\mbox{ mod }\mathfrak{i}$. Similarly $g$ is called a left inverse of $f$ in ${\cal F}_{{\mathbb{K}}}/\mathfrak{i}$ if $g\star f={\bf 1}\mbox{ mod }\mathfrak{i}$. $\diamond$ In case ${\cal F}_{{\mathbb{K}}}$ is commutative, right and left inverses coincide if they exist and we can tackle the problem by using the fact that $f$ has an inverse in $\mathfrak{i}$ if and only if $f\star g-{\bf 1}\in\mathfrak{i}$ if and only if ${\bf 1}\in\mathfrak{i}+{\sf ideal}(\\{f\\})$. Hence, if we have a Gröbner basis $G$ of the ideal $\mathfrak{i}+{\sf ideal}(\\{f\\})$ the existence of an inverse of $f$ is equivalent to ${\bf 1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$. Even, weak Gröbner bases are sufficient to solve the problem. For the non-commutative case we introduce a new non-commuting tag variable $z$ by lifting the multiplication $z\star z=z$, $z\star t=zt$ and $t\star z=tz$ for $t\in{\cal T}$ and extending ${\cal T}$ to $z{\cal T}=\\{z^{i}t_{1}zt_{2}z\ldots zt_{k}z^{j}\mid k\in{\mathbb{N}},i,j\in\\{0,1\\},t_{i}\in{\cal T}\\}$. The order on this enlarged set of terms is induced by combining a syllable ordering with respect to $z$ with the original ordering on ${\cal T}$. By ${\cal F}_{{\mathbb{K}}}^{z{\cal T}}$ we denote the function ring over $z{\cal T}$. This technique of using a tag variable now allows to study the right ideal generated by $f$ in ${\cal F}_{{\mathbb{K}}}/\mathfrak{i}$, where $\mathfrak{i}={\sf ideal}(F)$ for some set $F\subseteq{\cal F}_{{\mathbb{K}}}$, by studying the ideal generated by $F\cup\\{z\star f\\}$ in ${\cal F}_{{\mathbb{K}}}^{z{\cal T}}$ because of the following fact: ###### Lemma 5.2.3 Let $F\subseteq{\cal F}_{{\mathbb{K}}}$ and $f\in{\cal F}_{{\mathbb{K}}}$. Then ${\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F\cup\\{z\star f\\})$ has a Gröbner basis of the form $G\cup\\{z\star p_{i}\mid i\in I,p_{i}\in{\cal F}_{{\mathbb{K}}}\\}$ with $G\subseteq{\cal F}_{{\mathbb{K}}}$. In fact the set $\\{p_{i}\mid i\in I\\}$ then is a right Gröbner basis of ${\sf ideal}_{r}^{{\cal F}_{{\mathbb{K}}}/\mathfrak{i}}(\\{f\\})$. Proof : Let $G\subseteq{\cal F}_{{\mathbb{K}}}$ be a Gröbner basis of ${\sf ideal}^{{\cal F}_{{\mathbb{K}}}}(F)$. Then obviously ${\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F\cup\\{z\star f\\})={\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(G\cup\\{z\star f\\})$. Theorem 4.4.31 specifies a criterion to check whether a set is a Gröbner basis and gives rise to test sets for a completion procedure. Notice that due to the ordering on $z{\cal T}$ which uses the tag variable to induce syllables, we can state the following important result: > If for a polynomial $q\in{\cal F}_{{\mathbb{K}}}$ the multiple $z\star q$ > has a standard representation, then so has every multiple $u\star(z\star > q)\star z\star v$ for $u,v\in z{\cal T}$. Moreover, since $G$ is already a Gröbner basis, no critical situation for polynomials in $G$ have to be considered. Then a completion of $G\cup\\{z\star f\\}$ can be obtained as follows: In a first step only three kinds of critical situations have to be considered: 1. 1. s-polynomials of the form $zu\star g\star v-z\star f\star w$ where $u,v,w\in{\cal T}$ such that ${\sf HT}(zu\star g\star v)={\sf HT}(z\star f\star w)$, 2. 2. s-polynomials of the form $z\star f\star u-z\star f\star v$ where $u,v\in{\cal T}$ such that ${\sf HT}(z\star f\star u)={\sf HT}(z\star f\star v)$, and 3. 3. polynomials of the form $z\star f\star u$ where $u\in{\cal T}$ such that ${\sf HT}(f\star u)\neq{\sf HT}(f)\star u$. Since normal forms of polynomials of the form $z\star p$, $p\in{\cal F}_{{\mathbb{K}}}$, with respect to subsets of ${\cal F}_{{\mathbb{K}}}\cup z\star{\cal F}_{{\mathbb{K}}}$ are again elements of $z\star{\cal F}_{{\mathbb{K}}}\cup\\{o\\}$, we can assume that from then on we are completing a set $G\cup\\{z\star q_{i}\mid q_{i}\in{\cal F}_{{\mathbb{K}}}\\}$ and again three kinds of critical situations have to be considered: 1. 1. s-polynomials of the form $zu\star g\star v-z\star q_{i}\star w$ where $u,v,w\in{\cal T}$ such that ${\sf HT}(zu\star g\star v)={\sf HT}(z\star q_{i}\star w)$, 2. 2. s-polynomials of the form $z\star q_{i}\star u-z\star q_{j}\star v$ where $u,v\in{\cal T}$ such that ${\sf HT}(z\star q_{i}\star u)={\sf HT}(z\star q_{j}\star v)$, and 3. 3. polynomials of the form $z\star p_{i}\star u$ where $u\in{\cal T}$ such that ${\sf HT}(p_{i}\star u)\neq{\sf HT}(p_{i})\star u$. Normal forms again are elements of $z\star{\cal F}_{{\mathbb{K}}}\cup\\{o\\}$. Hence a Gröbner basis of the form $G\cup\\{z\star p_{i}\mid i\in I,p_{i}\in{\cal F}_{{\mathbb{K}}}\\}$ with $G\subseteq{\cal F}_{{\mathbb{K}}}$ must exist. It remains to show that the set $\\{p_{i}\mid i\in I\\}$ is in fact a right Gröbner basis of ${\sf ideal}_{r}^{{\cal F}_{{\mathbb{K}}}/\mathfrak{i}}(\\{f\\})$. This follows immediately if we recall the history of the polynomials $p_{i}$. In the first step they arise as a normal form with respect to $G\cup\\{z\star f\\}$ of a polynomial either of the form $zu\star g\star v-z\star f\star w$, $z\star f\star u-z\star f\star v$ or $z\star f\star u$, hence belonging to ${\sf ideal}_{r}^{{\cal F}_{{\mathbb{K}}}/\mathfrak{i}}(\\{f\\})$. In the iteration step, the new $p_{n}$ arises as a normal form with respect to $G\cup\\{z\star p_{i}\mid i\in I_{old}\\}$ of a polynomial either of the form $zu\star g\star v-z\star p_{i}\star w$, $z\star p_{i}\star u-z\star p_{j}\star v$ or $z\star p_{i}\star u$, hence belonging to ${\sf ideal}_{r}^{{\cal F}_{{\mathbb{K}}}/\mathfrak{i}}(\\{p_{i}\mid i\in I_{old}\\})={\sf ideal}_{r}^{{\cal F}_{{\mathbb{K}}}/\mathfrak{i}}(\\{f\\})$. q.e.d. Since we require ${\cal F}_{{\mathbb{K}}}$ to have a unit (otherwise looking for inverse elements makes no sense), ${\cal F}_{{\mathbb{K}}}^{z{\cal T}}$ then will contain $z$. Inverse Element Problem Given: An element $f\in{\cal F}_{{\mathbb{K}}}$ and a generating set $F$ for $\mathfrak{i}$. Problem: Does $f$ have a right or left inverse in ${\cal F}_{{\mathbb{K}}}/\mathfrak{i}$? Proceeding: 1. Let $G$ be a Gröbner basis of ${\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F\cup\\{z\star f\\})$. 2. If $z\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$, then $f$ has a right inverse. 1’. Let $G$ be a Gröbner basis of ${\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F\cup\\{f\star z\\})$. 2’. If $z\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$, then $f$ has a left inverse. To see that this is correct we give the following argument for the case of right inverses: It is clear that $f$ has a right inverse in ${\cal F}_{{\mathbb{K}}}/\mathfrak{i}$ if and only if $f\star g-{\bf 1}\in\mathfrak{i}$ for some $g\in{\cal F}_{{\mathbb{K}}}$. On the other hand we get $f\star g-{\bf 1}\in\mathfrak{i}$ if and only if $z\star f\star g-z\in{\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F)\cap z\star{\cal F}_{{\mathbb{K}}}$: $f\star g-{\bf 1}\in\mathfrak{i}$ immediately implies $z\star(f\star g-{\bf 1})\in{\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F)\cap z\star{\cal F}_{{\mathbb{K}}}$ as $\mathfrak{i}\subseteq{\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F)$, $z\in z{\cal T}\subseteq{\cal F}_{{\mathbb{K}}}^{z{\cal T}}$ and $z\star(f\star g-{\bf 1})\in z\star{\cal F}_{{\mathbb{K}}}$. On the other hand, if $z\star f\star g-z\in{\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F)\cap z\star{\cal F}_{{\mathbb{K}}}\subseteq{\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F)$, then we have a representation $z\star f\star g-z=\sum_{i=1}^{k}h_{i}\star f_{i}\star\tilde{h}_{i}$, $h_{i},\tilde{h}_{i}\in{\cal F}_{{\mathbb{K}}}^{z{\cal T}}$, $f_{i}\in F\subseteq{\cal F}_{{\mathbb{K}}}$. For a polynomial $p\in{\cal F}_{{\mathbb{K}}}^{z{\cal T}}$ and some element $\alpha\in{\mathbb{K}}$ let $p[z=\alpha]$ be the polynomial which arises from $p$ by substituting $\alpha$ for the variable $z$. Then by substituting $z={\bf 1}$ we get $f\star g-{\bf 1}=\sum_{i=1}^{k}h_{i}[z={\bf 1}]\star f_{i}\star\tilde{h}_{i}[z={\bf 1}]$ with $h_{i}[z={\bf 1}],\tilde{h}_{i}[z={\bf 1}]\in{\cal F}_{{\mathbb{K}}}$ and are done. Now, in order to decide whether $f$ has a right inverse in $\mathfrak{i}$ one has to distinguish the following two cases provided we have a Gröbner basis $G$ of ${\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F\cup\\{z\star f\\})$: If $z\mbox{$\,\,\,\,{\not\\!\\!\\!\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}}\,$}o$ then there exists no $g\in{\cal F}_{{\mathbb{K}}}$ such that $f\star g-{\bf 1}\in\mathfrak{i}$ and hence $f$ has no right inverse. If $z\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$ then we know $z\in{\sf ideal}^{{\cal F}_{{\mathbb{K}}}^{z{\cal T}}}(F\cup\\{z\star f\\})$, and even $z\in{\sf ideal}_{r}^{{\cal F}_{{\mathbb{K}}}/\mathfrak{i}}(\\{z\star f\\})$ Hence there exist $m_{i},\tilde{m}_{i},n_{j}\in{\sf M}({\cal F}_{{\mathbb{K}}}^{z{\cal T}})$, $f_{i}\in F$ such that $z=\sum_{i=1}^{k}m_{i}\star f_{i}\star\tilde{m}_{i}+\sum_{j=1}^{l}z\star f\star n_{j}.$ Now substituting $z={\bf 1}$ gives us that for $h=\sum_{j=1}^{l}n_{j}$ we have $f\star h={\bf 1}(\mbox{ mod }\mathfrak{i})$ and we are done. As before, weak Gröbner bases are sufficient to solve the problem. ### 5.3 Elimination Theory In ordinary polynomial rings special term orderings called elimination orderings can be used to produce Gröbner bases with useful properties. Many problems, e.g. the ideal intersection problem or the subalgebra problem, can be solved using tag variables. The elimination orderings are then used to separate the ordinary variables from these additional tag variables. Something similar can be achieved for function rings. Let $Z=\\{z_{i}\mid i\in I\\}$ be a set of new tag variables commuting with terms. The multiplication $\star$ can be extended by $z_{i}\star z_{j}=z_{i}z_{j}$, $z\star t=zt$ and $t\star z=zt$ for $z,z_{i},z_{j}\in Z$ and $t\in{\cal T}$. The ordering $\succeq$ is lifted to $Z^{}{\cal T}=\\{wt\mid w\in Z^{},t\in{\cal T}\\}$ by $w_{1}t_{1}\succeq w_{2}t_{2}$ if and only if $w_{1}\geq_{\rm lex}w_{2}$ or $(w_{1}=w_{2}$ and $t_{1}\succeq t_{2}$) for all $w_{1},w_{2}\in Z^{}$, $t_{1},t_{2}\in{\cal T}$. Moreover, we require $w\succ t$ for all $w\in Z^{}$, $t\in{\cal T}$. This ordering is called an elimination ordering. Up to now we have studied ideals in ${\cal F}^{{\cal T}}$. Now we can view ${\cal F}^{{\cal T}}$ as a subring of ${\cal F}^{Z^{}{\cal T}}$ and study ideals in both rings. For a generating set $F\subset{\cal F}^{{\cal T}}$ we have ${\sf ideal}^{{\cal F}^{{\cal T}}}(F)\subseteq{\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(F)$. This follows immediately since for every $f=\sum_{i=1}^{k}m_{i}\star f_{i}\star\tilde{m}_{i}$, $m_{i},\tilde{m}_{i}\in{\sf M}({\cal F}^{{\cal T}})$ this immediately implies $m_{i},\tilde{m}_{i}\in{\sf M}({\cal F}^{Z^{}{\cal T}})$. ###### Lemma 5.3.1 Let $G$ be a weak Gröbner basis of an ideal in ${\cal F}^{Z^{}{\cal T}}$ with respect to an elimination ordering. Then the following hold: 1. 1. ${\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G)\cap{\cal F}^{{\cal T}}={\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})$. 2. 2. $G\cap{\cal F}^{{\cal T}}$ is a weak Gröbner basis for ${\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})$ with respect to $\succeq$. 3. 3. If $G$ is a Gröbner basis, then $G\cap{\cal F}^{{\cal T}}$ is a Gröbner basis for ${\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})$ with respect to $\succeq$. Proof : 1. 1. * • ${\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G)\cap{\cal F}^{{\cal T}}\subseteq{\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})$: Let $f\in{\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G)\cap{\cal F}^{{\cal T}}$. By the elimination ordering property for $w\in Z^{}$ and $t\in{\cal T}$ we have that $wt\succ w\succ t$ holds and we get that ${\sf HT}(f)\in{\cal T}$ if and only if $f\in{\cal F}^{{\cal T}}$. Since $f\in{\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G)$ we know that $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$ and as all monomials in $f$ are also in ${\cal F}^{{\cal T}}$ for each $g\in G$ used in this reduction sequence we know ${\sf HT}(g)\in{\cal T}$ and hence $g\in{\cal F}^{{\cal T}}$. Moreover, the reduction sequence gives us a representation $f=\sum_{i=1}^{k}m_{i}\star f_{i}\star\tilde{m}_{i}$ with $f_{i}\in G\cap{\cal F}^{{\cal T}}$ and $m_{i},\tilde{m}_{i}\in{\sf M}({\cal F}^{{\cal T}})$, implying $f\in{\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})$. • ${\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})\subseteq{\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G)\cap{\cal F}^{{\cal T}}$: Let $f\in{\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})$. Then $f=\sum_{i=1}^{k}m_{i}\star f_{i}\star\tilde{m}_{i}$ with $f_{i}\in G\cap{\cal F}^{{\cal T}}$ and $m_{i},\tilde{m}_{i}\in{\sf M}({\cal F}^{{\cal T}})$. Hence $f\in{\sf ideal}^{{\cal F}^{{\cal T}}}(G)\subseteq{\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G)$ and $f\in{\cal F}^{{\cal T}}$ imply $f\in{\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G)\cap{\cal F}^{{\cal T}}$. 2. 2. We show this by proving that for every $f\in{\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})$ we have $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G\cap{\cal F}^{{\cal T}}}\,$}o$. Since $G$ is a weak Gröbner basis of ${\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G)$ and ${\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})\subseteq{\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G\cap{\cal F}^{{\cal T}})\subseteq{\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G)$ we get $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$. On the other hand, as every monomial in $f$ is an element of ${\cal F}^{{\cal T}}$, only elements of $G\cap{\cal F}^{{\cal T}}$ are applicable for reduction. 3. 3. Let $G$ be a Gröbner basis with respect to some reduction relation $\longrightarrow$. To show that $G\cap{\cal F}^{{\cal T}}$ is a Gröbner basis of ${\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})$ we proceed in two steps: 1. (a) $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{G\cap{\cal F}^{{\cal T}}}\,$}=\;\;\equiv_{{\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})}$: $\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{G\cap{\cal F}^{{\cal T}}}\,$}\subseteq\;\;\equiv_{{\sf ideal}^{{\cal F}^{{\cal T}}}(G\cap{\cal F}^{{\cal T}})}$ trivially holds as because of Axiom (A2) reduction steps stay within the ideal congruence. To see the converse let $f\equiv_{{\sf ideal}(G\cap{\cal F}^{{\cal T}})}g$ for $f,g\in{\cal F}^{{\cal T}}$. Then, as $G$ is a Gröbner basis and also $f\equiv_{{\sf ideal}^{{\cal F}^{Z^{}{\cal T}}}(G)}g$ holds, we know $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longleftrightarrow}}\\!\\!\mbox{}_{G}\,$}g$ and as ${\sf HT}(f),{\sf HT}(g)\in{\cal F}^{{\cal T}}$, only elements from $G\cap{\cal F}^{{\cal T}}$ can be involved and we are done. 2. (b) $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G\cap{\cal F}^{{\cal T}}}\,$ is confluent: Let $g,g_{1},g_{2}\in{\cal F}^{{\cal T}}$ such that $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G\cap{\cal F}^{{\cal T}}}\,$}g_{1}$ and $g\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G\cap{\cal F}^{{\cal T}}}\,$}g_{2}$. Then, as $\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$ is confluent we know that there exists $f\in{\cal F}^{Z^{}{\cal T}}$ such that $g_{1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}f$ and $g_{2}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}f$. Now since ${\sf HT}(g)\in{\cal F}^{{\cal T}}$ we can conclude that $g_{1},g_{2},f\in{\cal F}^{{\cal T}}$ and hence all polynomials used for reduction in the reduction sequences lie in $G\cap{\cal F}^{{\cal T}}$ proving our claim. q.e.d. Given an ideal $\mathfrak{i}\subseteq{\cal F}^{Z^{}{\cal T}}$ the set $\mathfrak{i}\cap{\cal F}^{{\cal T}}$ is again an ideal, now in ${\cal F}^{{\cal T}}$. This follows as 1. 1. $o\in\mathfrak{i}\cap{\cal F}^{{\cal T}}$ since $o\in\mathfrak{i}$ and $o\in{\cal F}^{{\cal T}}$. 2. 2. For $f,g\in\mathfrak{i}\cap{\cal F}^{{\cal T}}$ we have $f+g\in\mathfrak{i}$ as $f,g\in\mathfrak{i}$ and $f+g\in{\cal F}^{{\cal T}}$ as $f,g\in{\cal F}^{{\cal T}}$ yielding $f+g\in\mathfrak{i}\cap{\cal F}^{{\cal T}}$. 3. 3. For $f\in\mathfrak{i}\cap{\cal F}^{{\cal T}}$ and $h\in{\cal F}^{{\cal T}}$ we have that $f\star h,h\star f\in\mathfrak{i}$ as $f\in\mathfrak{i}$ and $f\star h,h\star f\in{\cal F}^{{\cal T}}$ as $f,h\in{\cal F}^{{\cal T}}$ yielding $f\star h,h\star f\in\mathfrak{i}\cap{\cal F}^{{\cal T}}$. The ideal $\mathfrak{i}\cap{\cal F}^{{\cal T}}$ is called the elimination ideal of $\mathfrak{i}$ with respect to $Z$ since the occurrences of the tag variables $Z$ are eliminated. ###### Definition 5.3.2 For an ideal $\mathfrak{i}$ in ${\cal F}$ the set $\surd\mathfrak{i}=\\{f\in{\cal F}\mid\mbox{ there exists }m\in{\mathbb{N}}\mbox{ with }f^{m}\in\mathfrak{i}\\}$ is called the radical of $\mathfrak{i}$. $\diamond$ Obviously we always have $\mathfrak{i}\subseteq\surd\mathfrak{i}$. Moreover, if ${\cal F}$ is commutative the radical of an ideal is again an ideal. This follows as 1. 1. $o\in\surd\mathfrak{i}$ since $o\in\mathfrak{i}$, 2. 2. For $f,g\in\surd\mathfrak{i}$ we know $f^{m},g^{n}\in\mathfrak{i}$ for some $m,n\in{\mathbb{N}}$. Now $f+g\in\surd\mathfrak{i}$ if we can show that $(f+g)^{q}\in\mathfrak{i}$ for some $q\in{\mathbb{N}}$. Remember that for $q=m+n-1$ every term in the binomial expansion of $(f+g)^{q}$ has a factor of the form $f^{i}\star g^{j}$ with $i+j=m+n-1$. As either $i\geq m$ or $j\geq n$ we find $f^{i}\star g^{j}\in\mathfrak{i}$ yielding $(f+g)^{q}\in\mathfrak{i}$ and hence $f+g\in\surd\mathfrak{i}$. Notice that commutativity is essential in this setting. 3. 3. For $f\in\surd\mathfrak{i}$ we know $f^{m}\in\mathfrak{i}$ for some $m\in{\mathbb{N}}$. Hence for $h\in{\cal F}^{{\cal T}}$ we get $(f\star h)^{m}=f^{m}\star h^{m}\in\mathfrak{i}$ yielding $f\star h\in\surd\mathfrak{i}$. Again commutativity is essential in the proof. Unfortunately this no longer holds for non-commutative function rings. For example take ${\cal T}=\\{a,b\\}^{}$ with concatenation as multiplication. Then for $\mathfrak{i}={\sf ideal}(\\{a^{2}\\})=\\{\sum_{i=1}^{n}\alpha_{i}\cdot u_{i}a^{2}v_{i}\mid n\in{\mathbb{N}},\alpha_{i}\in{\mathbb{Q}},u_{i},v_{i}\in{\cal T}\\}$ we get $a\in\surd\mathfrak{i}$. But for $b\in{\cal F}$ there exists no $m\in{\mathbb{N}}$ such that $(ab)^{m}\in\mathfrak{i}$ and hence $\surd\mathfrak{i}$ is no ideal. In the commutative polynomial ring the question whether some polynomial $f$ lies in the radical of some ideal generated by a set $F$ can be answered by introducing a tag variable $z$ and computing a Gröbner basis of the ideal generated by the set $F\cup\\{fz-{\bf 1}\\}$. It can be shown that if a commutative function ring ${\cal F}$ contains a unit ${\bf 1}$ we get a similar result. ###### Theorem 5.3.3 Let $F\subseteq{\cal F}$ and $f\in{\cal F}$ where ${\cal F}$ is a commutative function ring containing a unit ${\bf 1}$. Then $f\in\surd{\sf ideal}^{{\cal F}^{{\cal T}}}(F)$ if and only if ${\bf 1}\in{\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(F\cup\\{z\star f-{\bf 1}\\})$ for some new tag variable $z$. Proof : If $f\in\surd{\sf ideal}^{{\cal F}^{{\cal T}}}(F)$, then $f^{m}\in{\sf ideal}^{{\cal F}^{{\cal T}}}(F)\subseteq{\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(F\cup\\{z\star f-{\bf 1}\\})$ for some $m\in{\mathbb{N}}$. But we also have that $z\star f-{\bf 1}\in{\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(F\cup\\{z\star f-{\bf 1}\\})$. Remember that for the tag variable we have $t\star z=zt$ for all $t\in{\cal T}$ and hence $f\star z=z\star f$ yielding $\displaystyle{\bf 1}$ $\displaystyle=$ $\displaystyle z^{m}\star f^{m}-(z^{m}\star f^{m}-{\bf 1})$ $\displaystyle=$ $\displaystyle\underbrace{z^{m}\star f^{m}}_{\in{\sf ideal}^{{\cal F}^{{\cal T}}}(F)}-\underbrace{(z\star f-{\bf 1})\star(\sum_{i=0}^{m-1}z^{i}\star f^{i})}_{{\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(F\cup\\{z\star f-{\bf 1}\\})}$ and hence ${\bf 1}\in{\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(F\cup\\{z\star f-{\bf 1}\\})$ and we are done. On the other hand, ${\bf 1}\in{\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(F\cup\\{z\star f-{\bf 1}\\})$ implies ${\bf 1}=\sum_{i=1}^{k}m_{i}\star f_{i}\star\tilde{m}_{i}+\sum_{j=1}^{l}n_{j}\star(z\star f-{\bf 1})\star\tilde{n}_{j}$ with $m_{i},\tilde{m}_{i},n_{j},\tilde{n}_{j}\in{\sf M}({\cal F}^{\\{z\\}^{}{\cal T}})$. Moreover, since for the tag variable we have $z\star t=t\star z=zt$ for all $t\in{\cal T}$ all terms occurring in $\sum_{i=1}^{k}g_{i}\star f_{i}\star h_{i}$ are of the form $z^{j}t$ for some $t\in{\cal T}$, $j\in{\mathbb{N}}$. Now, since $z\star f-{\bf 1}\in{\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(F\cup\\{z\star f-{\bf 1}\\})$, we have $z^{j}t\star f^{j}=t\star z^{j}\star f^{j}=t$ as well as $f^{j}\star z^{j}\star t=z^{j}\star f^{j}\star t=t$. Hence, the occurrences of $z$ in a term $z^{j}t$ with $t\in{\cal T}$ can be “cancelled” by multiplication with $f^{m}$, $m\geq j$. Therefore, by choosing $m\in{\mathbb{N}}$ sufficiently large to cancel all occurrences of $z$ in the terms of $\sum_{i=1}^{k}m_{i}\star f_{i}\star\tilde{m}_{i}$, multiplying the equation with $f^{m}$ from both sides yields $f^{2m}=\sum_{i=1}^{k}(f^{m}\star m_{i})\star f_{i}\star(\tilde{m}_{i}\star f^{m})$ and $f^{m}\star m_{i},\tilde{m}_{i}\star f^{m}\in{\cal F}^{{\cal T}}$. Hence $f^{2m}\in{\sf ideal}^{{\cal F}^{{\cal T}}}(F)$ and therefore $f\in\surd{\sf ideal}^{{\cal F}^{{\cal T}}}(F)$. q.e.d. This theorem now enables us to describe the membership problem for radicals of ideals in terms of Gröbner bases. Radical Membership Problem Given: A set $F\subseteq{\cal F}$ and an element $f\in{\cal F}$, ${\cal F}$ containing a unit ${\bf 1}$. Problem: $f\in\surd{\sf ideal}(F)$? Proceeding: 1. Let $G$ be a respective Gröbner basis of ${\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(F\cup\\{z\star f-{\bf 1}\\})$ for some new tag variable $z$. 2. If ${\bf 1}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}o$, then $f\in\surd{\sf ideal}(F)$. If additionally the function ring is commutative, remember that then $\surd\mathfrak{i}$ is an ideal and we then describe the equality problem for radicals of ideals. Notice that weak Gröbner bases are sufficient to solve the problem. Radical Equality Problem Given: Two sets $F_{1},F_{2}\subseteq{\cal F}$, ${\cal F}$ commutative containing a unit. Problem: $\surd{\sf ideal}(F_{1})=\surd{\sf ideal}(F_{2})$? Proceeding: 1. If for all $f\in F_{1}$ we have $f\in\surd{\sf ideal}(F_{2})$, then $\surd{\sf ideal}(F_{1})\subseteq\surd{\sf ideal}(F_{2})$. 2. If for all $f\in F_{2}$ we have $f\in\surd{\sf ideal}(F_{1})$, then $\surd{\sf ideal}(F_{2})\subseteq\surd{\sf ideal}(F_{1})$. 3. If 1. and 2. both hold, then $\surd{\sf ideal}(F_{1})=\surd{\sf ideal}(F_{2})$. Correctness can be shown as follows: Let us assume that for all $f\in F_{1}$ we have $f\in\surd{\sf ideal}(F_{2})$. Then, as ${\cal F}$ is commutative ${\sf ideal}(F_{1})\subseteq\surd{\sf ideal}(F_{2})$ holds. Now let $f\in\surd{\sf ideal}(F_{1})$. Then for some $m\in{\mathbb{N}}$ we have $f^{m}\in{\sf ideal}(F_{1})\subseteq\surd{\sf ideal}(F_{2})$ and hence $\surd{\sf ideal}(F_{1})\subseteq\surd{\sf ideal}(F_{2})$. If ${\cal F}$ is not commutative, ${\sf ideal}(F_{1})\subseteq\surd{\sf ideal}(F_{2})$ need not hold. Remember the function ring with ${\cal T}=\\{a,b\\}^{}$. Take $F_{1}=\\{a\\}$ and $F_{2}=\\{a^{2}\\}$. Then $a\in\surd{\sf ideal}(F_{2})$ since $a^{2}\in{\sf ideal}(F_{2})$. But while $ab\in{\sf ideal}(F_{1})$ we have $ab\not\in\surd{\sf ideal}(F_{2})$. Radicals of one-sided ideals can be defined as well and Theorem 5.3.3 is also valid in this setting and can be used to state the radical membership problem for one-sided ideals. Another problem which can be handled using tag variables and elimination orderings in the commutative polynomial ring is that of ideal intersections. Something similar can be done for function rings containing a unit. ###### Theorem 5.3.4 Let $\mathfrak{i}$ and $\mathfrak{j}$ be two ideals in ${\cal F}$ and $z$ a new tag variable. Then $\mathfrak{i}\cap\mathfrak{j}={\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(z\star\mathfrak{i}\cup(z-{\bf 1})\star\mathfrak{j})\cap{\cal F}$ where $z\star\mathfrak{i}=\\{z\star f\mid f\in\mathfrak{i}\\}$ and $(z-{\bf 1})\star\mathfrak{j}=\\{(z-{\bf 1})\star f\mid f\in\mathfrak{j}\\}$. Proof : Every polynomial $f\in\mathfrak{i}\cap\mathfrak{j}$ can be written as $f=z\star f-(z-{\bf 1})\star f$ and hence $f\in{\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(z\star\mathfrak{i}\cup(z-{\bf 1})\star\mathfrak{j})\cap{\cal F}$. On the other hand, $f\in{\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(z\star\mathfrak{i}\cup(z-{\bf 1})\star\mathfrak{j})\cap{\cal F}$ implies $f=\sum_{i=1}^{k}m_{i}\star z\star f_{i}\star\tilde{m}_{i}+\sum_{j=1}^{l}n_{j}\star(z-{\bf 1})\star\tilde{f}_{j}\star\tilde{n}_{j}$ with $f_{i}\in\mathfrak{i}$, $\tilde{f}_{j}\in\mathfrak{j}$ and $m_{i},\tilde{m}_{i},n_{j},\tilde{n}_{j}\in{\sf M}({\cal F}^{\\{z\\}^{}{\cal T}})$. Since $f\in{\cal F}^{{\cal T}}$, substituting $z={\bf 1}$ gives us $f\in\mathfrak{i}$ and $z=0$ gives us $f\in\mathfrak{j}$ and hence $f\in\mathfrak{i}\cap\mathfrak{j}$. q.e.d. Moreover, combining this result with Lemma 5.3.1 gives us the means to characterize a Gröbner basis of the intersection ideal. Intersection Problem Given: Two sets $F_{1},F_{2}\subseteq{\cal F}$. Problem: Determine a basis of ${\sf ideal}(F_{1})\cap{\sf ideal}(F_{2})$. Proceeding: 1. Let $G$ be a Gröbner basis of ${\sf ideal}^{{\cal F}^{\\{z\\}^{}{\cal T}}}(z\star\mathfrak{i}\cup(z-{\bf 1})\star\mathfrak{j})$ with respect to an elimination ordering with $z>{\cal T}$. 2. Then $G\cap{\cal F}$ is a Gröbner basis of ${\sf ideal}(F_{1})\cap{\sf ideal}(F_{2})$. These ideas extend to one-sided ideals as well. Again, weak Gröbner bases are sufficient to solve the problem. Of course Theorem 5.3.4 can be generalized to intersections of more than two ideals. The techniques can also be applied to treat quotients of ideals in case ${\cal F}_{{\mathbb{K}}}$ is commutative. ###### Definition 5.3.5 For two ideals $\mathfrak{i}$ and $\mathfrak{j}$ in a commutative function ring ${\cal F}_{{\mathbb{K}}}$ we define the quotient to be the set $\mathfrak{i}/\mathfrak{j}=\\{g\mid g\in{\cal F}_{{\mathbb{K}}}\mbox{ with }g\star\mathfrak{j}\subseteq\mathfrak{i}\\}$ where $g\star\mathfrak{j}=\\{g\star f\mid f\in\mathfrak{j}\\}$. $\diamond$ ###### Lemma 5.3.6 Let ${\cal F}_{{\mathbb{K}}}$ be a commutative function ring. Let $\mathfrak{i}$ and $\mathfrak{j}={\sf ideal}(F)$ be two ideals in ${\cal F}_{{\mathbb{K}}}$. Then $\mathfrak{i}/\mathfrak{j}=\bigcap_{f\in F}(\mathfrak{i}/{\sf ideal}(\\{f\\}).$ Proof : First let $g\in\mathfrak{i}/\mathfrak{j}$. Then $g\star\mathfrak{j}\subseteq\mathfrak{i}$. Since $\mathfrak{j}={\sf ideal}(F)$ we get $g\star f\in\mathfrak{i}$ for all $f\in F$. As ${\cal F}_{{\mathbb{K}}}$ is commutative we can conclude $g\star{\sf ideal}(\\{f\\})\subseteq\mathfrak{i}$ for all $f\in F$ and hence $g\in\mathfrak{i}/{\sf ideal}(\\{f\\})$ for all $f\in F$ yielding $g\in\bigcap_{f\in F}(\mathfrak{i}/{\sf ideal}(\\{f\\})$. On the other hand, $g\in\bigcap_{f\in F}(\mathfrak{i}/{\sf ideal}(\\{f\\})$ implies $g\in\mathfrak{i}/{\sf ideal}(\\{f\\})$ for all $f\in F$ and hence $g\star{\sf ideal}(\\{f\\})\subseteq\mathfrak{i}$ for all $f\in F$. Since $\mathfrak{j}={\sf ideal}(F)$ then $g\star\mathfrak{j}\subseteq\mathfrak{i}$ and hence $g\in\mathfrak{i}/\mathfrak{j}$. q.e.d. Hence we can describe quotients of ideals in terms of quotients of the special form $\mathfrak{i}/{\sf ideal}(\\{f\\})$. These special quotients now can be described using ideal intersection in case ${\cal F}_{{\mathbb{K}}}$ contains a unit element ${\bf 1}$. ###### Lemma 5.3.7 Let ${\cal F}_{{\mathbb{K}}}$ be a commutative function ring. Let $\mathfrak{i}$ be an ideal and $f\neq o$ a polynomial in ${\cal F}_{{\mathbb{K}}}$. Then $\mathfrak{i}/{\sf ideal}(\\{f\\})=(\mathfrak{i}\cap{\sf ideal}(\\{f\\}))\star f^{-1}$ where $f^{-1}$ is an element in ${\cal F}_{{\mathbb{K}}}$ such that $f\star f^{-1}={\bf 1}$. Proof : First let $g\in\mathfrak{i}/{\sf ideal}(\\{f\\})$. Then $g\star{\sf ideal}(\\{f\\})\subseteq\mathfrak{i}$ and $g\star f\in\mathfrak{i}$, even $g\star f\in\mathfrak{i}\cap{\sf ideal}(\\{f\\})$. Hence $g\in(\mathfrak{i}\cap{\sf ideal}(\\{f\\}))\star f^{-1}$. On the other hand let $g\in(\mathfrak{i}\cap{\sf ideal}(\\{f\\}))\star f^{-1}$. Then $g\star f\in\mathfrak{i}\cap{\sf ideal}(\\{f\\})\subseteq\mathfrak{i}$. Since ${\cal F}_{{\mathbb{K}}}$ is commutative, this implies $g\star{\sf ideal}(\\{f\\})\subseteq\mathfrak{i}$ and hence $g\in\mathfrak{i}/{\sf ideal}(\\{f\\})$. q.e.d. Hence we can study the quotient of $\mathfrak{i}$ and $\mathfrak{j}={\sf ideal}(F)$ by studying $(\mathfrak{i}\cap{\sf ideal}(\\{f\\}))\star f^{-1}$ for all $f\in F$. ### 5.4 Polynomial Mappings In this section we are interested in ${\mathbb{K}}$-algebra homomorphisms between the non-commutative polynomial ring ${\mathbb{K}}[Z^{}]$ where $Z=\\{z_{1},\ldots,z_{n}\\}$, and ${\cal F}_{{\mathbb{K}}}^{{\cal T}}$. Let $\phi:{\mathbb{K}}[Z^{}]\longrightarrow{\cal F}_{{\mathbb{K}}}^{{\cal T}}$ be a ring homomorphism which is determined by a linear mapping $\phi:z_{i}\longmapsto f_{i}$ with $f_{i}\in{\cal F}_{{\mathbb{K}}}^{{\cal T}}$, $1\leq i\leq n$. Then for a non-commutative polynomial $g\in{\mathbb{K}}[Z^{}]$ with $g=\sum_{j=1}^{m}\alpha_{j}\cdot w_{j}$, $w_{j}\in Z^{}$ we get $\phi(g)=\sum_{j=1}^{m}\alpha_{j}\cdot\phi(w_{j})$ where $\phi(w_{j})=w_{j}[z_{1}\longmapsto f_{1},\ldots,z_{n}\longmapsto f_{n}]$. The kernel of such a mapping is defined as ${\sf ker}(\phi)=\\{g\in{\mathbb{K}}[Z^{}]\mid\phi(g)=o\\}$ and the image is defined as ${\sf im}(\phi)=\\{f\in{\cal F}_{{\mathbb{K}}}^{{\cal T}}\mid\mbox{ there exists }g\in{\mathbb{K}}[Z^{}]\mbox{ such that }\phi(g)=f\\}.$ Note that ${\sf im}(\phi)$ is a subalgebra of ${\cal F}_{{\mathbb{K}}}^{{\cal T}}$. ###### Lemma 5.4.1 Let $\phi:{\mathbb{K}}[Z^{}]\longrightarrow{\cal F}_{{\mathbb{K}}}^{{\cal T}}$ be a ring homomorphism. Then ${\mathbb{K}}[Z^{}]/{\sf ker}(\phi)\cong{\sf im}(\phi)$. Proof : To see this inspect the mapping $\psi:{\mathbb{K}}[Z^{}]/{\sf ker}(\phi)\longrightarrow{\sf im}(\phi)$ defined by $g+{\sf ker}(\phi)\mapsto\phi(g)$. Then $\psi$ is an isomorphism. 1. 1. $\psi(g+{\sf ker}(\phi))=o$ for $g\in{\sf ker}(\phi)$ by the definition of ${\sf ker}(\phi)$. 2. 2. $\psi((g_{1}+{\sf ker}(\phi))+(g_{2}+{\sf ker}(\phi)))=\phi(g_{1}+g_{2})=\psi(g_{1}+{\sf ker}(\phi))+\psi(g_{2}+{\sf ker}(\phi))$. 3. 3. $\psi((g_{1}+{\sf ker}(\phi))\star(g_{2}+{\sf ker}(\phi)))=\phi(g_{1}\star g_{2})=\psi(g_{1}+{\sf ker}(\phi))\star\psi(g_{2}+{\sf ker}(\phi))$, as for $g\in{\mathbb{K}}[Z^{}]$ and $h\in{\sf ker}(\phi)$ we have $\psi(g\star h)=\psi(h\star g)=o$. 4. 4. $\psi$ is onto as its image is the image of $\phi$ and by the definition of the latter for each $f\in{\sf im}(\phi)={\sf im}(\psi)$ there exists $g\in{\mathbb{K}}[Z^{}]$ such that $\phi(g)=f$. Since for all $h\in{\sf ker}(\phi)$ we have $\phi(h)=o$ then $\psi(g+{\sf ker}(\phi))=\psi(g)=\phi(g)$. 5. 5. Assume that for $g_{1},g_{2}\in{\mathbb{K}}[Z^{}]$ we have $\psi(g_{1}+{\sf ker}(\phi))=\psi(g_{2}+{\sf ker}(\phi))$. Then $\phi(g_{1})=\phi(g_{2})$ and this immediately implies that $g_{1}-g_{2}\in{\sf ker}(\phi)$ and hence $\psi$ is also a monomorphism. q.e.d. Now the theory of elimination described in the previous section can be used to provide a Gröbner basis for ${\sf ker}(\phi)$. Remember that the tag variables commute with the elements on ${\cal T}$. Again we use the function ring ${\cal F}_{{\mathbb{K}}}^{Z^{}{\cal T}}$ and the fact that ${\mathbb{K}}[Z^{}]\subseteq{\cal F}_{{\mathbb{K}}}^{Z^{}{\cal T}}$ by mapping the polynomials to the respective functions in ${\cal F}_{{\mathbb{K}}}^{Z^{}}\subseteq{\cal F}_{{\mathbb{K}}}^{Z^{}{\cal T}}$. ###### Theorem 5.4.2 Let ${\mathfrak{i}}={\sf ideal}(\\{z_{1}-f_{1},\ldots,z_{n}-f_{n}\\})\subseteq{\cal F}_{{\mathbb{K}}}^{Z^{}{\cal T}}$. Then ${\sf ker}(\phi)={\mathfrak{i}}\cap{\mathbb{K}}[Z^{}]$. Proof : Let $g\in{\mathfrak{i}}\cap{\mathbb{K}}[Z^{}]$ . Then $g=\sum_{j=1}^{n}h_{j}\star s_{j}\star h_{j}^{\prime}$ with $s_{j}\in\\{z_{1}-f_{1},\ldots,z_{n}-f_{n}\\}$, $h_{j},h_{j}^{\prime}\in{\cal F}_{{\mathbb{K}}}^{Z^{}{\cal T}}$. As $\phi(z_{j}-f_{j})=o$ for all $1\leq j\leq n$ we get $\phi(g)=o$ and hence $g\in{\sf ker}(\phi)$. To see the converse let $g\in{\sf ker}(\phi)$. Then $g\in{\mathbb{K}}[Z^{}]$ and hence $g=\sum_{j=1}^{m}\alpha_{j}\cdot w_{j}$ where $w_{j}\in Z^{}$, $1\leq j\leq m$. On the other hand we know $\phi(g)=o$. Then $\displaystyle g$ $\displaystyle=$ $\displaystyle g-\phi(g)$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{m}\alpha_{j}\cdot w_{j}-\sum_{j=1}^{m}\alpha_{j}\cdot\phi(w_{j})$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{m}\alpha_{j}\cdot(w_{j}-\phi(w_{j}))$ It remains to show that $w-\phi(w)\in{\mathfrak{i}}$ for all $w\in Z^{}$ as this implies $g\in{\mathfrak{i}}\cap{\mathbb{K}}[Z^{}]$. This will be done by induction on $k=\|w\|$. For $k=1$ we get $w=z_{i}$ for some $1\leq i\leq n$ and $w-\phi(w)=z_{i}-f_{i}\in{\mathfrak{i}}$. In the induction step let $w\equiv a_{1}\ldots a_{k}$, $a_{i}\in Z$. Then we get $\displaystyle a_{1}(a_{2}\ldots a_{k}-\phi(a_{2}\ldots a_{k}))+(a_{1}-\phi(a_{1}))\phi(a_{2}\ldots a_{k})$ $\displaystyle=$ $\displaystyle a_{1}a_{2}\ldots a_{k}-a_{1}\phi(a_{2}\ldots a_{k})+a_{1}\phi(a_{2}\ldots a_{k})-\phi(a_{1})\phi(a_{2}\ldots a_{k})$ $\displaystyle=$ $\displaystyle a_{1}a_{2}\ldots a_{k}-\phi(a_{1}\ldots a_{k})$ Then, as $\|a_{2}\ldots a_{k}\|=k-1$ the induction hypothesis yields $a_{2}\ldots a_{k}-\phi(a_{2}\ldots a_{k})\in{\mathfrak{i}}$ and as of course $a_{1}-\phi(a_{1})\in{\mathfrak{i}}$ we find that $a_{1}a_{2}\ldots a_{k}-\phi(a_{1}\ldots a_{k})\in{\mathfrak{i}}$. q.e.d. Now if $G$ is a (weak) Gröbner basis of ${\mathfrak{i}}$ in ${\cal F}_{{\mathbb{K}}}^{Z^{}{\cal T}}$ with respect to an elimination ordering where the elements in $Z^{}$ are made smaller than those in ${\cal T}$, then $G\cap{\mathbb{K}}[Z^{}]$ is a (weak) Gröbner basis of the kernel of $\phi$. Hence, in case finite such bases exist or bases allowing to solve the membership problem, they can be used to treat the following question. Kernel of a Polynomial Mapping Given: A set $F=\\{z_{1}-f_{1},\ldots,z_{n}-f_{n}\\}\subseteq{\cal F}_{{\mathbb{K}}}^{Z^{}{\cal T}}$ encoding a mapping $\phi:{\mathbb{K}}[Z^{}]\longrightarrow{\cal F}_{{\mathbb{K}}}^{{\cal T}}$ and an element $f\in{\mathbb{K}}[Z^{}]$. Problem: $f\in{\sf ker}(\phi)$? Proceeding: 1. Let $G$ be a (weak) Gröbner basis of ${\sf ideal}(\\{z_{1}-f_{1},\ldots,z_{n}-f_{n}\\})$ with respect to an elimination ordering. 2. Let $G^{\prime}=G\cap{\mathbb{K}}[Z^{}]$. 3. If $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G^{\prime}}\,$}o$, then $f\in{\sf ker}(\phi)$. A similar question can be asked for the image of a polynomial mapping. Image of a Polynomial Mapping Given: A set $F=\\{z_{1}-f_{1},\ldots,z_{n}-f_{n}\\}\subseteq{\cal F}_{{\mathbb{K}}}^{Z^{}{\cal T}}$ encoding a mapping $\phi:{\mathbb{K}}[Z^{}]\longrightarrow{\cal F}_{{\mathbb{K}}}^{{\cal T}}$ and an element $f\in{\cal F}_{{\mathbb{K}}}^{{\cal T}}$. Problem: $f\in{\sf im}(\phi)$? Proceeding: 1. Let $G$ be a Gröbner basis of ${\sf ideal}(\\{z_{1}-f_{1},\ldots,z_{n}-f_{n}\\})$ with respect to an elimination ordering. 2. If $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}h$, with $h\in{\mathbb{K}}[Z^{}]$, then $f\in{\sf im}(\phi)$. The basis for this solution is the following theorem. ###### Theorem 5.4.3 Let ${\mathfrak{i}}={\sf ideal}(\\{z_{1}-f_{1},\ldots,z_{n}-f_{n}\\})\subseteq{\cal F}_{{\mathbb{K}}}^{Z^{}{\cal T}}$ and let $G$ be a Gröbner basis of ${\mathfrak{i}}$ with respect to an elimination ordering where the elements in $Z^{}$ are smaller than those in ${\cal T}$. Then $f\in{\cal F}_{{\mathbb{K}}}^{{\cal T}}$ lies in the image of $\phi$ if and only if there exists $h\in{\mathbb{K}}[Z^{}]$ such that $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}h$. Moreover, $f=\phi(h)$. Proof : Let $f\in{\sf im}(\phi)$, i.e., $f\in{\cal F}_{{\mathbb{K}}}^{{\cal T}}$. Then $f=\phi(g)$ for some $g\in{\mathbb{K}}[Z^{}]$. Moreover, $f-g=\phi(g)-g$, and similar to the proof of Theorem 5.4.2 we can show $f-g\in{\mathfrak{i}}$. Hence, $f$ and $g$ must reduce to the same normal form $h$ with respect to $G$. As $g\in{\mathbb{K}}[Z^{}]$ this implies $h\in{\mathbb{K}}[Z^{}]$ and we are done. To see the converse, for $f\in{\cal F}_{{\mathbb{K}}}^{{\cal T}}$ let $f\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}h$ with $h\in{\mathbb{K}}[Z^{}]$. Then $f-h\in{\mathfrak{i}}$ and hence $f-h=\sum_{j=1}^{k}g_{j}\star s_{j}\star g_{j}^{\prime}$ with $s_{j}\in\\{z_{1}-f_{1},\ldots,z_{n}-f_{n}\\}$, $g_{j},g_{j}^{\prime}\in{\cal F}_{{\mathbb{K}}}^{{\cal T}}$. As $\phi(s_{j})=o$ we get $f-\phi(h)=o$ and hence $f=\phi(h)$ is in the image of $\phi$. q.e.d. Obviously the question of whether an element lies in the image of $\phi$ then can be answered in case we can compute a unique normal form of the element with respect to the Gröbner basis of ${\mathfrak{i}}={\sf ideal}(\\{z_{1}-f_{1},\ldots,z_{n}-f_{n}\\})$. Another question is whether the mapping $\phi:{\mathbb{K}}[Z^{}]\longrightarrow{\cal F}_{{\mathbb{K}}}^{{\cal T}}$ is onto. This is the case if for every $t\in{\cal T}$ we have $t\in{\sf im}(\phi)$. A simpler solution can be found in case ${\cal T}\subseteq\Sigma^{}$ for some finite set of letters $\Sigma=\\{a_{1},\ldots,a_{k}\\}$ and additionally ${\cal T}$ is subword closed as a subset of $\Sigma^{}$. ###### Theorem 5.4.4 Let ${\mathfrak{i}}={\sf ideal}(\\{z_{1}-f_{1},\ldots,z_{n}-f_{n}\\})\subseteq{\cal F}_{{\mathbb{K}}}^{Z^{}{\cal T}}$ and let $G$ be a Gröbner basis of ${\mathfrak{i}}$ with respect to an elimination ordering where the elements in $Z^{}$ are smaller than those in ${\cal T}$. Then $f\in{\cal F}_{{\mathbb{K}}}^{{\cal T}}$ is onto if and only if for each $a_{j}\in\Sigma$, we have $a_{j}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}h_{j}$ where $h_{j}\in{\mathbb{K}}[Z^{}]$. Moreover, $a_{j}=\phi(h_{j})$. Proof : Remember that $\phi$ is onto if and only if $a_{j}\in{\sf im}(\phi)$ for $1\leq j\leq k$. Let us first assume that $\phi$ is onto, i.e., $a_{1},\ldots,a_{k}\in{\sf im}(\phi)$. Then by Theorem 5.4.3 there exist $h_{j}\in{\mathbb{K}}[Z^{}]$ such that $a_{j}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}h_{j}$, $1\leq j\leq k$. To see the converse, again, by Theorem 5.4.3 the existence of $h_{j}\in{\mathbb{K}}[Z^{}]$ such that $a_{j}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}h_{j}$, $1\leq j\leq k$ now implies $a_{1},\ldots,a_{k}\in{\sf im}(\phi)$ and we are done. q.e.d. ### 5.5 Systems of One-sided Linear Equations in Function Rings over the Integers Let ${\cal F}_{{\mathbb{Z}}}$ be the function ring over the integers ${\mathbb{Z}}$ as specified in Section 4.2.3. Additionally we require that multiplying terms by terms results in terms, i.e., $\star:{\cal T}\times{\cal T}\longrightarrow{\cal T}$. Then a reduction relation can be defined for ${\cal F}_{{\mathbb{Z}}}$ as follows: ###### Definition 5.5.1 Let $p$, $f$ be two non-zero polynomials in ${\cal F}_{{\mathbb{Z}}}$. We say $f$ reduces $p$ to $q$ at $\alpha\cdot t$ in one step, i.e. $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{g}\,$}q$, if 1. (a) $t={\sf HT}(f\star u)={\sf HT}(f)\star u$ for some $u\in{\cal T}$. 2. (b) ${\sf HC}(f)>0$ and $\alpha={\sf HC}(f)\cdot\beta+\delta$ with $\beta,\delta\in{\mathbb{Z}}$, $\beta\neq 0$, and $0\leq\delta<{\sf HC}(f)$. 3. (c) $q=p-f\star(\beta\cdot u)$. The definition of s-polynomials can be derived from Definition 4.2.66. ###### Definition 5.5.2 Let $p_{1},p_{2}$ be two polynomials in ${\cal F}_{{\mathbb{Z}}}$. If there are respective terms $t,u_{1},u_{2}\in{\cal T}$ such that ${\sf HT}(p_{i})\star u_{i}={\sf HT}(p_{i}\star u_{i})=t\geq{\sf HT}(p_{i})$ let $HC(p_{i})=\gamma_{i}$. Assuming $\gamma_{1}\geq\gamma_{2}>0$111Notice that $\gamma_{i}>0$ can always be achieved by studying the situation for $-p_{i}$ in case we have $HC(p_{i})<0$., there are $\beta,\delta\in{\mathbb{Z}}$ such that $\gamma_{1}=\gamma_{2}\cdot\beta+\delta$ and $0\leq\delta<\gamma_{2}$ and we get the following s-polynomial ${\sf spol}(p_{1},p_{2},t,u_{1},u_{2})=\beta\cdot p_{2}\star u_{2}-p_{1}\star u_{1}.$ The set ${\sf SPOL}(\\{p_{1},p_{2}\\})$ then is the set of all such s-polynomials corresponding to $p_{1}$ and $p_{2}$. $\diamond$ Notice that two polynomials can give rise to infinitely many s-polynomials. A subset $C$ of these possible s-polynomials ${\sf SPOL}(p_{1},p_{2})$ is called a stable localization if for any possible s-polynomial $p\in{\sf SPOL}(p_{1},p_{2})$ there exists a special s-polynomial $h\in C$ such that $p\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{h}\,$}o$. In the following let $f_{1},\ldots,f_{m}\in{\cal F}_{{\mathbb{Z}}}$. We describe a generating set of solutions for the linear one-sided inhomogeneous equation $f_{1}\star X_{1}+\ldots+f_{m}\star X_{m}=f_{0}$ in the variables $X_{1},\ldots,X_{m}$ provided a finite computable right Gröbner basis of the right ideal generated by $\\{f_{1},\ldots,f_{m}\\}$ in ${\cal F}_{{\mathbb{Z}}}$ exists. In order to find a generating set of solutions we have to find one solution of $\displaystyle f_{1}\star X_{1}+\ldots+f_{m}\star X_{m}$ $\displaystyle=$ $\displaystyle f_{0}$ (5.1) and if possible a finite set of generators for the solutions of the homogeneous equation $\displaystyle f_{1}\star X_{1}+\ldots+f_{m}\star X_{m}$ $\displaystyle=$ $\displaystyle o.$ (5.2) We proceed as follows assuming that we have a finite right Gröbner basis of the right ideal generated by $\\{f_{1},\ldots,f_{m}\\}$: 1. 1. Let $G=\\{g_{1},\ldots,g_{n}\\}$ be a right Gröbner basis of the right ideal generated by $\\{f_{1},\ldots,f_{m}\\}$ in ${\cal F}_{{\mathbb{Z}}}$, and ${\bf f}=(f_{1},\ldots,f_{m})$, ${\bf g}=(g_{1},\ldots,g_{n})$ the corresponding vectors. There are two linear mappings given by matrices $P\in{\sf M}_{m\times n}({\cal F}_{{\mathbb{Z}}})$, $Q\in{\sf M}_{n\times m}({\cal F}_{{\mathbb{Z}}})$ such that ${\bf f}\cdot P={\bf g}$ and ${\bf g}\cdot Q={\bf f}$. 2. 2. Equation 5.1 is solvable if and only if $f_{0}\in{\sf ideal}_{r}(\\{f_{1},\ldots,f_{m}\\})$. This is equivalent to $f_{0}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{G}\,$}0$ and the reduction sequence gives rise to a representation $f_{0}=\sum_{i=1}^{n}g_{i}\star h_{i}={\bf g}\cdot{\bf h}$ where ${\bf h}=(h_{1},\ldots,h_{n})$. Then, as ${\bf f}\cdot P={\bf g}$, we get ${\bf g}\cdot{\bf h}=({\bf f}\cdot P)\cdot{\bf h}$ and $P\cdot{\bf h}$ is such a solution of equation 5.1. 3. 3. Let $\\{{\bf z}_{1},\ldots,{\bf z}_{r}\\}$ be a generating set for the solutions of the homogeneous equation $\displaystyle g_{1}\star X_{1}+\ldots+g_{n}\star X_{n}$ $\displaystyle=$ $\displaystyle 0$ (5.3) and let $I_{m}$ be the $m\times m$ identity matrix. Further let ${\bf w}_{1},\ldots,{\bf w}_{m}$ be the columns of the matrix $P\cdot Q-I_{m}$. Since ${\bf f}\cdot(P\cdot Q-I_{m})={\bf f}\cdot P\cdot Q-{\bf f}\cdot I_{m}={\bf g}\cdot Q-{\bf f}=0$ these are solutions of equation 5.2. We can even show that the set $\\{P\cdot{\bf z}_{1},\ldots,P\cdot{\bf z}_{r},{\bf w}_{1},\ldots,{\bf w}_{m}\\}$ generates all solutions of equation 5.2: Let ${\bf q}=(q_{1},\ldots,q_{m})$ be an arbitrary solution of equation 5.2. Then $Q\cdot{\bf q}$ is a solution of equation 5.3 as ${\bf f}={\bf g}\cdot Q$. Hence there are $h_{1},\ldots,h_{r}\in{\cal F}_{{\mathbb{Z}}}$ such that $Q\cdot{\bf q}={\bf z}_{1}\cdot h_{1}+\ldots{\bf z}_{r}\cdot h_{r}$. Further we find ${\bf q}=P\cdot Q\cdot{\bf q}-(P\cdot Q-I_{m})\cdot{\bf q}=P\cdot{\bf z}_{1}\cdot h_{1}+\ldots P\cdot{\bf z}_{r}\cdot h_{r}+{\bf w}_{1}\cdot q_{1}+\ldots+{\bf w}_{m}\cdot q_{m}$ and hence ${\bf q}$ is a right linear combination of elements in $\\{P\cdot{\bf z}_{1},\ldots,P\cdot{\bf z}_{r},{\bf w}_{1},\ldots,{\bf w}_{m}\\}$. Now the important part is to find a generating set for the solutions of the homogeneous equation 5.3. In commutative polynomial rings is was sufficient to look at special vectors arising from those situations causing s-polynomials. These situations are again important in our setting: For every $g_{i},g_{j}\in G$ not necessarily different such that the stable localization $C_{i,j}\subseteq{\sf SPOL}(g_{i},g_{j})$ for the s-polynomials is not empty and additionally we require these sets to be finite, we compute vectors ${\bf a}^{\ell}_{ij}$, $1\leq\ell\leq\|C\|$ as follows: Let $t={\sf HT}(g_{i}\star u)={\sf HT}(g_{i})\star u={\sf HT}(g_{j})\star v={\sf HT}(g_{j}\star v)$, $t\geq{\sf HT}(g_{i})$, $t\geq{\sf HT}(g_{j})$, be the overlapping term corresponding to $h_{\ell}\in C_{i,j}$. Further let ${\sf HC}(g_{i})\geq{\sf HC}(g_{j})>0$ and ${\sf HC}(g_{i})=\alpha\cdot{\sf HC}(g_{j})+\beta$ for some $\alpha,\beta\in{\mathbb{Z}}$, $0\leq\beta<{\sf HC}(g_{j})$. Then $h_{\ell}=g_{i}\star u-g_{j}\star(\alpha\cdot v)=\sum_{l=1}^{n}g_{l}\star h_{l},$ where the polynomials $h_{l}\in{\cal F}_{{\mathbb{Z}}}$ are due to the reduction sequence $h_{\ell}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm r}}_{G}\,$}0$. Then ${\bf a}_{ij}^{\ell}=(a_{1},\ldots,a_{n})$, where $\displaystyle a_{i}$ $\displaystyle=$ $\displaystyle h_{i}-u,$ $\displaystyle a_{j}$ $\displaystyle=$ $\displaystyle h_{j}+\alpha\cdot v,$ $\displaystyle a_{l}$ $\displaystyle=$ $\displaystyle h_{l},$ $l\neq i,j$, is a solution of 5.3 as $\sum_{l=1}^{n}g_{l}\star h_{l}-g_{i}\star u+g_{j}\star\alpha\cdot v=0$. If all sets ${\sf SPOL}(g_{i},g_{j})$ are empty for $g_{i},g_{j}\in G$, in the case of ordinary Gröbner bases in polynomial rings one could conclude that the homogeneous equation 5.3 had no solution. This is no longer true for arbitrary function rings. ###### Example 5.5.3 Let ${\mathbb{Z}}[{\cal M}]$ be a monoid ring where ${\cal M}$ is presented by the complete string rewriting system $\Sigma=\\{a,b\\}$, $T=\\{ab\longrightarrow\lambda\\}$. Then for the homogeneous equation $(a+1)\star X_{1}+(b+1)\star X_{2}=0$ we find that the set $\\{a+1,b+1\\}$ is a prefix Gröbner basis of the right ideal it generates. Moreover neither of the head terms of the polynomials in this basis is prefix of the other and hence no s-polynomials with respect to prefix reduction exist. Still the equation can be solved: $(b,-1)$ is a solution since $(a+1)\star b-(b+1)=b+1-(b+1)=0$. Hence inspecting s-polynomials is not sufficient to describe all solutions. This phenomenon is due to the fact that as seen before in most function rings s-polynomials are not sufficient for a Gröbner basis test. Additionally the concept of saturation has to be incorporated. In Example 5.5.3 we know that $(a+1)\star b=1+b$, i.e. $b+1\in{\sf SAT}(a+1)$. Of course $(a+1)\star b\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{b+1}\,$}0$ and hence $(a+1)\star b=b+1$ gives rise to a solution $(b,-1)$ as required above. More general we can express these additional solutions as follows: For every $g_{i}\in G$ with ${\sf SAT}(g_{i})$ a stable saturator for $\\{g_{i}\\}$ and again we additionally require it to be finte, we define vectors ${\bf b}_{i,\ell}=(b_{1},\ldots,b_{n})$ $1\leq\ell\leq\|{\sf SAT}(g_{i})\|$ as follows: For $g_{i}\star w_{\ell}\in{\sf SAT}(g_{i})$ we know $g_{i}\star w_{\ell}=\sum_{l=1}^{n}g_{l}\star h_{l}$ as $G$ is a Gröbner basis. Then ${\bf b}_{i,\ell}=(b_{1},\ldots,b_{n})$, where $\displaystyle b_{i}$ $\displaystyle=$ $\displaystyle h_{i}-w_{\ell},$ $\displaystyle b_{l}$ $\displaystyle=$ $\displaystyle h_{l},$ $l\neq i$, is a solution of equation 5.3 as $\sum_{l=1}^{n}g_{l}\star h_{l}-g_{i}\star w_{\ell}=0$. ###### Lemma 5.5.4 Let $\\{g_{1},\ldots,g_{n}\\}$ be a finite right Gröbner basis. For $g_{i},g_{j}$ let $C_{i,j}$ be a stable localization of ${\sf SPOL}(g_{i},g_{j})$. The finitely many vectors ${\bf a}^{\ell_{1}}_{i,j},{\bf b}_{i,\ell_{2}}$, $1\leq i,j\leq n$, $1\leq\ell_{1}\leq\|C_{i,j}\|$, $1\leq\ell_{2}\leq\|{\sf SAT}(g_{i})\|$ form a right generating set for all solutions of equation 5.3. Proof : Let ${\bf p}=(p_{1},\ldots,p_{n})$ be an arbitrary (non-trivial) solution of equation 5.3, i.e., $\sum_{i=1}^{n}g_{i}\star p_{i}=0$. Let $T_{p}=\max\\{{\sf HT}(g_{i}\star t_{j}^{p_{i}})\mid 1\leq i\leq n,p_{i}=\sum_{j=1}^{n_{i}}\alpha_{j}^{p_{i}}\cdot t_{j}^{p_{i}}\\}$, $K_{p}$ the number of multiples $g_{i}\star t_{j}^{p_{i}}$ with $T_{p}={\sf HT}(g_{i}\star t_{j}^{p_{i}})\neq{\sf HT}(g_{i})\star t_{j}^{p_{i}}$, and $M_{p}=\\{\\{{\sf HC}(g_{i})\mid{\sf HT}(g_{i}\star t_{j}^{p_{i}})=T_{p}\\}\\}$ a multiset in ${\mathbb{Z}}$. A solution ${\bf q}$ is called smaller than ${\bf p}$ if either $T_{q}\prec T_{p}$ or ($T_{q}=T_{p}$ and $K_{q}<K_{p}$) or ($T_{q}=T_{p}$ and $K_{q}=K_{p}$ and $M_{q}\ll M_{p}$). We will prove our claim by induction on $T_{p}$, $K_{p}$ and $M_{p}$ and have to distinguish two cases: 1. 1. If there is $1\leq i\leq n$, $1\leq j\leq n_{i}$ such that $T_{p}={\sf HT}(g_{i}\star t_{j}^{p_{i}})\neq{\sf HT}(g_{i})\star t_{j}^{p_{i}}$, then there exists $s_{\ell}\in{\sf SAT}(g_{i})$ such that $g_{i}\star t_{j}^{p_{i}}=s_{\ell}\star v$ for some $v\in{\cal T}$, ${\sf HT}(s_{\ell}\star v)={\sf HT}(s_{\ell})\star v$ and $s_{\ell}=g_{i}\star w_{\ell}$, $w_{\ell}\in{\cal T}$. Then we can set ${\bf q}={\bf p}+\alpha_{j}^{p_{i}}\cdot{\bf b}_{i,\ell}\star v$ with $\displaystyle q_{i}$ $\displaystyle=$ $\displaystyle p_{i}+\alpha_{j}^{p_{i}}\cdot(h_{i}-w_{\ell})\star v$ $\displaystyle q_{l}$ $\displaystyle=$ $\displaystyle p_{l}+\alpha_{j}^{p_{i}}\cdot h_{l}\star v\mbox{ for }l\neq i$ which is again a solution of equation 5.3. It remains to show that it is a smaller one. To see this we have to examine the multiples $g_{l}\star t_{j}^{q_{l}}$ for all $1\leq l\leq n$, $1\leq j\leq m_{l}$ where $q_{l}=\sum_{j=1}^{m_{l}}\alpha_{j}^{q_{l}}\cdot t_{j}^{q_{l}}$. Remember that ${\sf HT}(s_{\ell})\leq{\sf HT}(s_{\ell}\star v)={\sf HT}(s_{\ell})\star v=T_{p}$. Moreover, for all terms $w_{j}^{h_{l}}$ in $h_{l}=\sum_{j=1}^{m_{l}}\beta_{j}^{h_{l}}\cdot w_{j}^{h_{l}}$ we know $w_{j}^{h_{l}}\preceq{\sf HT}(s_{\ell})$, as the $h_{l}$ arise from the reduction sequence $g_{i}\star w_{\ell}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}^{{\rm p}}_{G}\,$}0$, and hence ${\sf HT}(w_{j}^{h_{l}}\star v)\preceq{\sf HT}(s_{\ell}\star v)=T_{p}$. 1. (a) For $l=i$ we get $g_{i}\star q_{i}=g_{i}\star(p_{i}+\alpha_{j}^{p_{i}}\cdot(h_{i}-w_{\ell})\star v)=g_{i}\star p_{i}+\alpha_{j}^{p_{i}}\cdot g_{i}\star h_{i}\star v-\alpha_{j}^{p_{i}}\cdot g_{i}\star w_{\ell}\star v$ and as ${\sf HT}(g_{i}\star t_{j}^{p_{i}})={\sf HT}(g_{i}\star w_{\ell}\star v)$ and the resulting monomials add up to zero we get $\max\\{{\sf HT}(g_{i}\star w_{j}^{h_{i}})\mid 1\leq j\leq m_{i}\\}\leq T_{p}$. 2. (b) For $l\neq i$ we get $g_{l}\star q_{l}=g_{l}\star(p_{l}+\alpha_{j}^{p_{i}}\cdot h_{l}\star v)=g_{l}\star p_{l}+\alpha_{j}^{p_{i}}\cdot g_{l}\star h_{l}\star v$ and $\max\\{{\sf HT}(g_{i}\star w_{j}^{h_{l}})\mid 1\leq j\leq m_{l}\\}\preceq T_{p}$ as well as $\max\\{{\sf HT}(g_{i}\star w_{j}^{h_{l}})\mid 1\leq j\leq m_{l}\\}\preceq T_{p}$. Hence while still in one of the cases we must have $T_{q}=T_{p}$, the element $g_{i}\star t_{j}^{p_{i}}$ is replaced by the sum $\sum_{l=1}^{n}g_{l}\star h_{l}\star v$ where the $h_{l}$ arise from the reduction sequence $s_{\ell}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}0$. Let $h_{l}=\sum_{j=1}^{k_{l}}\alpha_{j}^{h_{l}}\cdot t_{j}^{h_{l}}$. Since $s_{\ell}$ is stable, for all elements $g_{l}\star t_{j}^{h_{l}}$ involved in the reduction of the head term of $s_{\ell}$ we know ${\sf HT}(g_{l}\star t_{j}^{h_{l}}\star v)={\sf HT}(g_{l})\star t_{j}^{h_{l}}\star v=T_{p}$ and no other elements result in this term. Hence $K_{q}<K_{p}$ and ${\bf q}$ is smaller than ${\bf p}$. 2. 2. Let us now assume there are $1\leq i_{1},i_{2}\leq n$, $1\leq j_{1}\leq n_{i_{1}}$, $1\leq j_{2}\leq n_{i_{2}}$ such that ${\sf HT}(g_{i_{1}}\star t_{j_{1}}^{p_{i_{1}}})={\sf HT}(g_{i_{1}})\star t_{j_{1}}^{p_{i_{1}}}=T_{p}={\sf HT}(g_{i_{2}})\star t_{j_{2}}^{p_{i_{2}}}={\sf HT}(g_{i_{2}}\star t_{j_{2}}^{p_{i_{2}}})$. Moreover, we assume ${\sf HC}(g_{i_{1}})\geq{\sf HC}(g_{i_{2}})>0$ and ${\sf HC}(g_{i_{1}})=\alpha\cdot{\sf HC}(g_{i_{2}})+\beta$, $\alpha,\beta\in{\mathbb{Z}}$, $0\leq\beta<{\sf HC}(g_{i_{2}})$. Let $h_{\ell_{2}}\in C_{i_{1},i_{2}}$ such that for the corresponding s-polynomial $p=g_{i_{1}}\star t_{j_{1}}^{p_{i_{1}}}-\alpha\cdot g_{i_{2}}\star t_{j_{2}}^{p_{i_{2}}}$ we have $p=h_{\ell_{2}}\star v$ and $h_{\ell_{2}}=g_{i_{1}}\star u_{1}-g_{i_{2}}\star(\alpha\cdot u_{2})$. Since we have a vector ${\bf a}_{{i_{1}},{i_{2}}}^{\ell_{2}}$ corresponding to $h_{\ell_{2}}$, we can define a new solution ${\bf q}={\bf p}+\alpha_{j_{1}}^{p_{i_{1}}}\cdot{\bf a}_{{i_{1}},{i_{2}}}\star v$ with $\displaystyle q_{i_{1}}$ $\displaystyle=$ $\displaystyle p_{i_{1}}+\alpha_{j_{1}}^{p_{i_{1}}}\cdot(h_{i_{1}}-u_{1})\star v$ $\displaystyle q_{i_{2}}$ $\displaystyle=$ $\displaystyle p_{i_{2}}+\alpha_{j_{1}}^{p_{i_{1}}}\cdot(h_{i_{2}}+\alpha\cdot u_{2})\star v$ $\displaystyle q_{l}$ $\displaystyle=$ $\displaystyle p_{l}+\alpha_{j_{1}}^{p_{i_{1}}}\cdot h_{l}\star v\mbox{ for }l\neq i,j.$ It remains to show that this solution indeed is smaller. To do this we examine the multiples $g_{l}\star t_{j}^{q_{l}}$ for all $1\leq l\leq n$, $1\leq j\leq m_{l}$ where $q_{l}=\sum_{j=1}^{m_{l}}\alpha_{j}^{q_{l}}\cdot t_{j}^{q_{l}}$. Let $h_{l}=\sum_{j=1}^{k_{l}}\alpha_{j}^{h_{l}}\cdot t_{j}^{h_{l}}$. Since the elements $g_{l}\star t_{j}^{h_{l}}$ arise from the reduction sequence $h_{\ell_{2}}\mbox{$\,\stackrel{{\scriptstyle}}{{\longrightarrow}}\\!\\!\mbox{}_{G}\,$}0$ and the s-polynomial is stable we have additional information on how these elements affect the size of the new solution ${\bf q}$. Since ${\sf HT}(g_{l}\star t_{j}^{h_{l}})={\sf HT}(g_{l})\star t_{j}^{h_{l}}\leq{\sf HT}(h_{\ell_{2}})$ we can conclude ${\sf HT}(g_{l}\star t_{j}^{q_{l}})\leq{\sf HT}(h_{\ell_{2}})\star v\preceq T_{p}$ and we get the following boundaries: 1. (a) For $l\neq i_{1},i_{2}$ we get $g_{l}\star q_{l}=g_{l}\star p_{l}+\alpha_{j_{1}}^{p_{i_{1}}}\cdot g_{l}\star h_{l}\star v$. This implies $\max\\{{\sf HT}(g_{l}\star t_{j}^{q_{l}})\mid 1\leq j\leq m_{l}\\}\preceq T_{p}$. 2. (b) For $l=i_{1}$ we get $g_{i_{1}}\star q_{i_{1}}=g_{i_{1}}\star p_{i_{1}}+\alpha_{j_{1}}^{p_{i_{1}}}\cdot g_{i_{1}}\star h_{i_{1}}\star v-\alpha_{j_{1}}^{p_{i_{1}}}\cdot g_{i_{1}}\star u_{1}\star v$. Since $\alpha_{j_{1}}^{p_{i_{1}}}\cdot{\sf HM}(g_{i_{1}})\star t_{j_{1}}^{p_{i_{1}}}=\alpha_{j_{1}}^{p_{i_{1}}}\cdot{\sf HM}(g_{i_{1}})\star u_{1}\star v$ we get $\max\\{\\{{\sf HT}(g_{i_{1}}\star t_{j}^{q_{i_{1}}})\mid 1\leq j\leq m_{i_{1}}\\}\backslash\\{{\sf HT}(g_{i_{1}})\star t_{j_{1}}^{p_{i_{1}}},{\sf HT}(g_{i_{1}})\star u_{1}\star v\\}\\}\preceq T_{p}$. 3. (c) For $l=i_{2}$ we get $g_{i_{2}}\star q_{i_{2}}=g_{i_{2}}\star p_{i_{2}}+\alpha_{j_{1}}^{p_{i_{1}}}\cdot g_{i_{2}}\star h_{i_{2}}\star v+\alpha_{j_{1}}^{p_{i_{1}}}\cdot g_{i_{2}}\star\alpha\star u_{2}\star v$. Again $\max\\{{\sf HT}(g_{i_{1}}\star t_{j}^{q_{i_{1}}})\mid 1\leq j\leq m_{i_{1}}\\}\preceq T_{p}$. Now in case $\beta=0$ we know that the equations are strict as then ${\sf HT}(h_{\ell_{2}})\star v\prec T_{p}$ holds. Then either $T_{q}\prec T_{p}$ or $(T_{q}=T_{p}$ and $K_{q}<K_{p})$. If $\beta\neq 0$ we have to be more carefull and have to show that then $M_{q}\ll M_{p}$. For the elements $g_{l}\star t_{j}^{h_{l}}$ arising from reducing the head of the s-polynomial we know that $g_{l}\star t_{j}^{h_{l}}\star v$ again has the same head coefficient as $g_{l}\star t_{j}^{h_{l}}$. Now as ${\sf HC}(h_{\ell_{2}})=\beta$, by the definition of our reduction relation we know that only $g_{l}$ with ${\sf HC}(g_{l})\leq\beta$ are applicable. Hence while two elements ${\sf HC}(g_{i_{1}}),{\sf HC}(g_{i_{2}})$ are removed from the multiset $M_{p}$ only ones less equal to $\beta<{\sf HC}(g_{i_{2}})\leq{\sf HC}(g_{i_{1}})$ are added and hence the multiset becomes smaller. Hence we find that in all cases above either $T_{q}\prec T_{p}$ or ( $T_{q}=T_{p}$ and $K_{q}<K_{p}$) or ($T_{q}=T_{p}$, $K_{q}=K_{p}$ and $M_{q}\ll M_{p}$). Therefore, in all cases, we can reach a smaller solution and since our ordering on solutions is well-founded, or claim holds. q.e.d. ###### Corollary 5.5.5 Let $\\{g_{1},\ldots,g_{n}\\}$ be a finite right Gröbner basis. For not necessarily finite localizations $C_{i,j}\subseteq{\sf SPOL}(g_{i},g_{j})$ and ${\sf SAT}(g_{i})$ the not necessarily finite set of vectors ${\bf a}^{\ell_{1}}_{i,j},{\bf b}_{i,\ell_{2}}$, $1\leq i,j\leq n$, $h_{\ell_{1}}\in C_{i,j}$, $s_{\ell_{2}}\in{\sf SAT}(g_{i})$ forms a right generating set for all solutions of equation 5.3. The approach extends to systems of linear equations by using Gröbner bases in right modules. A study of the situation for one-sided equations in integer monid and group rings can be found in [Rei00]. ## Chapter 6 Conclusions The aim of this work was to give a guide for introducing reduction relations and Gröbner basis theory to algebraic structures. We chose function rings as they allow a representation of their elements by formal sums. This gives a natural link to those algebraic structures known in the literature where the Gröbner basis method works. At the same time function rings provide enough flexibility to subsume these algebraic structures. In the general setting of function rings we introduced the algebraic terms which are vital in Gröbner basis theory: head monomials, head terms, standard representations, standard bases, reduction relations and of course (weak) Gröbner bases. Incorporating the technique of saturation we could give characterizations of Gröbner bases in terms of critical situations similar to the original approach. We have established the theory first for right ideals in function rings over fields as this is the easiest setting. This has been generalized to function rings over reduction rings - a very general setting. Then in order to show how more knowledge on the reduction relation can be used to get deeper results on characterizing Gröbner bases, we have studied the special reduction ring ${\mathbb{Z}}$, which is of interest in the literature. The same approach has been applied to two-sided ideals in function rings with of course weaker results but still providing characterizations of Gröbner bases. Important algebraic structures where the Gröbner basis method has been successfully applied in the literature have been outlined in the setting of function rings. It has also been shown how special applications from Gröbner basis theory in polynomial rings can be lifted to function rings. What remains to be done is to find out if this approach can be extended to function rings allowing infinite formal sums as elements. Such an extension would allow to subsume the work of Mora et. al. on power series which resulted in the tangent cone algorithm. These rings are covered by graded structures as defined by Apel in his habilitation ([Ape98]), by monomial structures as defined by Pesch in his PhD Thesis ([Pes97]) and by Mora in “The Eigth variation” (on Gröbner bases). However, these approaches require admissible orderings and hence do not cover general monoid rings. ## Bibliography [AL88] J. Apel and W. Lassner. An extension of Buchberger’s algorithm and calculations in enveloping fields of Lie algebras. Journal of Symbolic Computation, 6:361–370, 1988. * [AL94] W. Adams and P. Loustaunau. An Introduction to Gröbner Bases. Graduate Studies in Mathematics, AMS, Providence, 1994. * [Ape88] J. Apel. Gröbnerbasen in nichtkommutativen Algebren und ihre Anwendung. PhD thesis, Universität Leipzig, 1988. * [Ape98] J. Apel. Zu Berechenbarkeitsfragen der Idealtheorie. Habilitationsschrift, Universität Leipzig, 1998. * [BO93] R. Book and F. Otto. String Rewriting Systems. Springer, 1993. * [Buc65] B. Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. PhD thesis, Universität Innsbruck, 1965. * [Buc70] B. Buchberger. Ein algorithmisches kriterium für die lösbarkeit eines algebraischen gleichungssystems. Aequ. Math. 4/3, 1970. * [Buc83] B. Buchberger. A critical-pair completion algorithm for finitely generated ideals in rings. In Proc. Logic and Machines: Decision Problems and Complexity, LNCS 171, pages 137–161. Springer, 1983. * [Buc85] B. Buchberger. Gröbner bases: An algorithmic method in polynomial ideal theory. In N. K. Bose, editor, Multidimensional Systems Theory, chapter 6, pages 184–232. Reidel, Dordrecht, 1985. * [Buc87] B. Buchberger. Applications of Gröbner bases in non-linear computational geometry. In R. Janßen, editor, Trends in Computer Algebra, LNCS 296, pages 52–80. Springer, 1987. * [BW92] T. Becker and V. Weispfenning. Gröbner Bases - A Computational Approach to Commutative Algebra. Springer, 1992. * [CLO92] D. Cox, J. Little, and D. O’Shea. Ideals, Varieties, and Algorithms. Springer, 1992. * [FCF93] D. Farkas and E. Green C. Feustel. Synergy in the theories of Gröbner bases and path algebras. Canadian Journal of Mathematics, 45(4):727–739, 1993. * [Hir64] H. Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero. Annals of Mathematics, 79:109–326, 1964. * [Hue80] G. Huet. Confluent reductions: Abstract properties and applications to term rewriting systems. Journal of the ACM, 27(4):797–821, 1980. * [Hue81] G. Huet. A complete proof of correctness of the Knuth-Bendix completion algorithm. Journal of Computer and System Science, 23(1):11–21, 1981. * [Kel98] B. J. Keller. Alternatives in implementing noncommutative Gröbner basis systems. In Proceedings of the Workshop on Symbolic Rewriting Techniques, Monte Verita, 1995, pages 127–180. Birkhäuser, 1998. * [KM89] D. Kapur and K. Madlener. A completion procedure for computing a canonical basis for a k-subalgebra. In Computers and Mathematics, pages 1–11. Springer, 1989. * [KN85] D. Kapur and P. Narendran. A finite thue system with decidable word problem and without equivalent finite canonical system. Theoretical Computer Science, 35:337–344, 1985. * [Kre93] H. Kredel. Solvable Polynomial Rings. Verlag Shaker, Aachen, 1993. * [KRK84] A. Kandri-Rody and D. Kapur. An algorithm for computing the Gröbner basis of a polynomial ideal over an Euclidean ring. Technical report, General Electric Company Corporate Research and Development, Schenectady, NY 12345, December 1984. * [KRK88] A. Kandri-Rody and D. Kapur. Computing a Gröbner basis of a polynomial ideal over an Euclidean domain. Journal of Symbolic Computation, 6:37–57, 1988. * [KRW90] A. Kandri-Rody and V. Weispfenning. Non-commutative Gröbner bases in algebras of solvable type. Journal of Symbolic Computation, 9:1–26, 1990. * [Las85] W. Lassner. Symbol representations of noncommutative algebras. In EUROCAL’85, LNCS 204, pages 99–115. Springer, 1985. * [Lau76] M. Lauer. Kanonische Repräsentanten für die Restklassen nach einem Polynomideal. Master’s thesis, Universität Kaiserslautern, 1976. * [Lo96] E. Lo. A Polycyclic Quotient Algorithm. PhD thesis, Rutgers, New Brunswick, New Jersey, 1996. * [Lo98] E. Lo. A polycyclic quotient algorithm. Journal of Symbolic Computation, 25(1), 1998. * [Mad86] K. Madlener. Existence and construction of Gröbner bases for ideals in reduction rings. Universität Kaiserslautern, 1986. * [Mor85] F. Mora. Gröbner bases for non-commutative polynomial rings. In Proc. AAECC-3, LNCS 229, pages 353–362. Springer, 1985. * [Mor87] T. Mora. Gröbner bases and the word problem. Genova, 1987. * [Mor89] T. Mora. Gröbner bases for non-commutative algebras. In Proc. of ISSAC’88, LNCS 358, pages 150–161, 1989. * [Mor94] T. Mora. An introduction to commutative and non-commutative Gröbner bases. Theoretical Computer Science, 134:131–173, 1994. * [MR93a] K. Madlener and B. Reinert. Computing Gröbner bases in monoid and group rings. In M. Bronstein, editor, Proc. ISSAC’93, pages 254–263. ACM, 1993\. * [MR93b] K. Madlener and B. Reinert. On Gröbner bases in monoid and group rings. SEKI Report SR-93-08, Universität Kaiserslautern, 1993. * [MR97a] K. Madlener and B. Reinert. A generalization of Gröbner bases algorithms to nilpotent group rings. Applicable Algebra in Engineering, Communication and Computing, 8(2):103–123, 1997. * [MR97b] K. Madlener and B. Reinert. String rewriting and Gröbner bases – a general approach to monoid and group rings. Reports on Computer Algebra 16, Centre of Computer Algebra, Universität Kaiserslautern, Universität Kaiserslautern, 1997. Online available at http://www.mathematik.uni-kl.de/~zca. * [MR98a] K. Madlener and B. Reinert. A generalization of Gröbner basis algorithms to polycyclic group rings. Journal of Symbolic Computation, 25(1):23–45, 1998. * [MR98b] K. Madlener and B. Reinert. Gröbner bases in non-commutative reduction rings. In B. Buchberger and F. Winkler, editors, Gröbner Bases and Applications (Proc. of the Conference 33 Years of Gröbner Bases), volume 251 of London Mathematical Society Lecture Notes Series, pages 408–420. Cambridge University Press, 1998. * [MR98c] K. Madlener and B. Reinert. Relating rewriting techniques on monoids and rings: Congruences on monoids and ideals in monoid rings. Theoretical Computer Science, 208:3–31, 1998. * [MR98d] K. Madlener and B. Reinert. String rewriting and Gröbner bases – a general approach to monoid and group rings. In M. Bronstein, J. Grabmeier, and V. Weispfenning, editors, Proceedings of the Workshop on Symbolic Rewriting Techniques, Monte Verita, 1995, volume 15 of Progress in Computer Science and Applied Logic, pages 127–180. Birkhäuser, 1998. * [Pan85] L. Pan. On the Gröbner bases of ideals in polynomial rings over a prinicipal ideal domain. Technical report, Department of Mathematics, University of California, Santa Barbara, 1985. * [Pes97] M. Pesch. Gröbner Bases in Skew Polynomial Rings. PhD thesis, Universität Passau, 1997. * [Pes98] M. Pesch. Two-sided gröbner bases in iterated ore extensions. In M. Bronstein, J. Grabmeier, and V. Weispfenning, editors, Proceedings of the Workshop on Symbolic Rewriting Techniques, Monte Verita, 1995, volume 15 of Progress in Computer Science and Applied Logic. Birkhäuser, 1998. * [Rei95] B. Reinert. On Gröbner Bases in Monoid and Group Rings. PhD thesis, Universität Kaiserslautern, 1995. * [Rei96] B. Reinert. Introducing reduction to polycyclic group rings - a comparison of methods. Reports on Computer Algebra 9, Centre of Computer Algebra, Universität Kaiserslautern, 1996. Online available at http://www.mathematik.uni-kl.de/~zca. * [Rei00] B. Reinert. Solving systems of linear one-sided equations in integer monoid and group rings. In C. Traverso, editor, Proc. ISSAC’00, pages 281–287. ACM, 2000\. * [Ros93] A. Rosenmann. An algorithm for constructing Gröbner and free Schreier bases in free group algebras. Journal of Symbolic Computation, 16:523–549, 1993. * [Sti87] S. Stifter. A generalization of reduction rings. Journal of Symbolic Computation, 4:351–364, 1987. * [Sto90] T. Stokes. Gröbner bases in exterior algebras. Journal of Automated Reasoning, 6:233–250, 1990. * [Tri78] W. Trinks. Über B. Buchbergers Verfahren, Systeme algebraischer Gleichungen zu lösen. Journal of Number Theory, 10(4):475–488, 1978. * [Wei87a] V. Weispfenning. Finite Gröbner bases in non-noetherian skew polynomial rings. In P. Wang, editor, Proc. ISSAC’92, pages 329–334. ACM, 1987. * [Wei87b] V. Weispfenning. Gröbner basis for polynomial ideals over commutative regular rings. In Proc. EUROCAL’87, LNCS 378, pages 336–347. Springer, 1987. * [Zac78] G. Zacharias. Generalized Gröbner bases in commutative polynomial rings. Master’s thesis, M. I. T., Dept. of Comp. Sci., 1978.	arxiv-papers	2009-03-13T19:24:15	2024-09-04T02:49:01.135167	{ "license": "Public Domain", "authors": "Birgit Reinert", "submitter": "Claus-Peter Wirth", "url": "https://arxiv.org/abs/0903.2462" }
0903.2612	This paper has been withdrawn by the authors, In our work: 0903.2612, we calculate the production rate of single top-Higgs boson in the TC2 model which is a modified version of the original top- technicolor model. The similar process was discussed in arXiv:hep- ph/9905347v2. The TC2 model, as we discussed in the introduction part remedies some shortcomings and loophole of the old version. The top-Higgs in the TC2 model is a mixture of the top-Higgs of the toptechnicolor model and that of the ETC model, thus a parameter $\epsilon$ is introduced to denote the mixture. Moreover, we vary the mass range of the top-Higgs within 300 to 800 GeV while in arXiv:hep-ph/9905347v2, the mass range was taken as 200 to 400 GeV. In the work, our numerical results show that the production rate of single top-Higgs in the TC2 model is very close to that in the toptecnicolor model within the mass range of 200 to 400 GeV. This manifests that change from the original toptechnicolor model to the new TC2 version does not much affect the production rate of the top-Higgs even though the two top-Higgs in the two models are different. Beyond the 400 GeV, even the TC2 model predicts a negligible production rate at LHC. Since the phenomenological change is indeed not obvious, there is not much new to report. Even though the two models are somehow different, we believe that the result is not worth publishing. Therefore we decide to withdraw our manuscript.	arxiv-papers	2009-03-15T07:28:48	2024-09-04T02:49:01.171228	{ "license": "Public Domain", "authors": "Qing-Peng Qiao, Xue-Qian Li, Xue-Lei Wang", "submitter": "Qing-Peng Qiao", "url": "https://arxiv.org/abs/0903.2612" }
0903.2821	# Balls are maximizers of the Riesz-type functionals with supermodular integrands Hichem Hajaiej Justus-Liebig-Universität Giessen Mathematisches Institut Arnd Str 2, 35392 Giessen Germany hichem.hajaiej@gmail.com ###### Abstract. For a large class of supermodular integrands, we establish conditions under which balls are the unique (up to translations) maximizers of the Riesz-type functionals with constraints. ## 1\. Introduction Over the last decades, one field of intense research activity has been the study of extremals of integral functionals. The Riesz-type kind has attracted growing attention and played a crucial role in the resolution of Choquard’s conjecture in a breakthrough paper by E. H. Lieb [1]. The determination of cases of equality in the Riesz-rearrangement inequality has also received a large amount of interest from mathematicians due to its connection with many other functional inequalities and its several applications to physics [2, 3, 4]. Variational problems for steady axisymmetric vortex-rings in which kinetic energy is maximized subject to prescribed impulse involves Riesz-type functionals with constraints. In [5], G. R. Burton has proved the existence of maximizers in an extended constraint set, he has also showed that the maximizer is Schwarz symmetric (up to translations). His method hinges on a resolution of an optimization of a Riesz-type functional under constraint [5, Proposition 8]. The purpose of this paper is to answer the more general question: When do maximizers of the Riesz-type functional inherit the symmetry and monotonicity properties of the integrand involved in it? The method of G. R. Burton [5] cannot apply to solve the above problem. In this paper, we develop a self-contained approach. Let us give here a foretaste of our ideas. First, we recall that: A Riesz-type functional is a functional of the form: $R(f,g)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi\left(f(x),g(y)\right)\,r(x,y)\,dx\,dy.$ In this paper, we will consider $r(x,y)=j\left(\|x-y\|\right)$. We are interested in the following maximization problem: (P1) $\sup\limits_{(f,g)\in C}J(f,g)$ where (1.1) $J(f,g)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi\left(f(x),g(y)\right)\,j\left(\|x-y\|\right)\,dx\,dy.$ and (1.2) $C=(f,g):\begin{cases}f:&\mathbb{R}^{n}\rightarrow\mathbb{R};0\leq f\leq k_{1}\mbox{ and }\int_{\mathbb{R}^{n}}f\leq\ell_{1}\\\ g:&\mathbb{R}^{n}\rightarrow\mathbb{R};0\leq g\leq k_{2}\mbox{ and }\int_{\mathbb{R}^{n}}g\leq\ell_{2}\end{cases}$ $\ell_{1},k_{1},\ell_{2},k_{2}$ are positive numbers. For supermodular operators $\Psi$ and nonincreasing functions $j$, we know that $J(f,g)\leq J(f^{},g^{})$ [4, Theorem 1], where $u^{}$ denotes the Schwarz symmetrization of $u$. Hence the problem reduces to: (P2) $\sup\limits_{(f^{},g^{})\in C}J\left(f^{},g^{}\right)$ For continuous integrands $\Psi$ having the N-Luzin property (for any subset N having Lebesgue measure zero, $\Psi$(N) has the same property), lemma 2.6 enables us to assert that (P2) is equivalent to an optimization of a Hardy- Littlewood type functionals where balls are maximizers. We will then extend this study to supermodular non-continuous bounded functions $\Psi$ thanks to the decomposition of these functions into $\tilde{\Psi}\left(\varphi_{1}(s_{1}),\varphi_{2}(s_{2})\right)$ in the spirit of [4, 6]. The approximation of unbounded supermodular functions by bounded ones inheriting the monotonicity properties will enable us to prove that balls are maximizers in the general case. Main Result: Let $\Psi:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ be a H-Borel function satisfying: ($\Psi$1) $\Psi$ vanishes at hyperplanes; ($\Psi$2) $\Psi(b,d)-\Psi(b,c)-\Psi(a,d)+\Psi(a,c)\geq 0$ for all $0\leq a<b$ and $0\leq c<d$; ($\Psi$3)(i) $\Psi(tx,b_{2})-t\Psi(x,b_{2})-\Psi(tx,b_{1})+t\Psi(x,b_{1})\leq 0$ for all $x\geq 0$, $0\leq b_{1}<b_{2}$ and $0<t<1$; ($\Psi$3)(ii) $\Psi(a_{2},ty)-t\Psi(a_{2},y)-\Psi(a_{1},ty)+t\Psi(a_{1},y)\leq 0$ for all $y\geq 0$, $0\leq a_{1}<a_{2}$ and $0<t<1$; (j1) $j$ is nonincreasing. Suppose in addition that $\Psi$ is continuous with respect to each variable and has the N-Luzin property, then for all $(f_{1},f_{2})\in C$ $J\left(f_{1},f_{2}\right)\leq J\left(k_{1}\mathrm{1}_{B_{1}},k_{2}\mathrm{1}_{B_{2}}\right)$ where $B_{1}$ and $B_{2}$ are centered in the origin, $\mathrm{1}_{B}$ is the characteristic function of $B$, and $\mu(B_{1})=\ell_{1}/k_{1}$, $\mu(B_{2})=\ell_{2}/k_{2}$. Moreover, if ($\Psi$2) and ($\Psi$3) hold with strict inequality, $j$ is strictly decreasing and $J\left(f_{1},f_{2}\right)<\infty$ for any $(f_{1},f_{2})\in C$, then (P1) is attained by exactly two couples $\left(k_{1}\mathrm{1}_{B_{1}},k_{2}\mathrm{1}_{B_{2}}\right)$ and $(h_{1},h_{2})$ where $h_{1}$ and $h_{2}$ are translates by the same vector of $k_{1}\mathrm{1}_{B_{1}}$ and $k_{2}\mathrm{1}_{B_{2}}$ (respectively). ## 2\. Notations and preliminaries ###### Definition 2.1. If $A\subset\mathbb{R}^{n}$ is a measurable set of finite Lebesgue measures $\mu$, we define $A^{}$, the symmetric rearrangement of the set $A$ to be the open ball centered at the origin whose volume is that of $A$, thus $A^{}=\left\\{x\in\mathbb{R}^{n}:\|x\|<r\right\\}$ with $V_{n}r^{n}=\mu(A)$, $V_{n}$ is a constant. For a nonnegative measurable function $u$ on $\mathbb{R}^{n}$, we require $u$ to vanish at infinity in the sense that all its positive level sets $\\{x\in\mathbb{R}^{n}:u(x)>t\\}$ having finite measure for $t>0$. The set of these functions is denoted by $F_{n}$. The symmetric decreasing rearrangement $u^{}$ of $u$ is the unique upper semicontinuous, nonincreasing radial function that is equimeasurable with $u$. Explicitly, $u^{}(x)=\int\limits_{0}^{\infty}{\mathbf{1}}^{}_{\\{u>t\\}}(x)\,dt$ where ${\mathbf{1}}^{}_{A}={\mathbf{1}}_{A^{}}$. We say that $u$ is Schwarz symmetric if $u\equiv u^{}$. ###### Definition 2.2. A reflexion $\sigma$ on $\mathbb{R}^{n}$is an isometry with the properties: (i) $\sigma^{2}_{x}=\sigma_{x}\circ\sigma_{x}=x$ for all $x\in\mathbb{R}^{n}$; * (ii) the fixed point set of $H_{0}$ of $\sigma$ separates $\mathbb{R}^{n}$ into two half spaces $H_{+}$ and $H_{-}$ that are interchanged by $\sigma$; * (iii) $\|x-x^{\prime}\|<\|x-\sigma_{x^{\prime}}\|$ for all $x,x^{\prime}\in H_{+}$. $H_{+}$ is the half space containing the origin. The two point rearrangement or polarization of a real valued function $u$ with respect to a reflection $\sigma$ is defined by: (2.1) $u^{\sigma_{x}}=\begin{cases}\max\\{u(x),u(\sigma_{x})\\},x\in H_{+}\cup H_{0},\\\ \min\\{u(x),u(\sigma_{x})\\},x\in H_{-}.\end{cases}$ ###### Lemma 2.3. Let $j:[0,\infty)\rightarrow\mathbb{R}$ be a nonincreasing function then $\nu(x)=\int_{{\mathbb{R}}^{n}}j\left(\|x-y\|\right)h(y)\,dy$ is radial and radially decreasing for any Schwarz symmetric function $h$. If in addition $j$ is strictly radially decreasing then $\nu$ also inherits this property. Proof: we will use [7, Lemma 2.8]: $u=u^{}\Leftrightarrow u=u^{\sigma}$ for all $\sigma$. It is sufficient to prove that $u(x)\geq u(\sigma_{x})$ for all $x\in\mathbb{R}^{n}$, all $\sigma$. $\displaystyle u(x)$ $\displaystyle=$ $\displaystyle\int_{H^{+}}j\left(\|x-y\|\right)h(y)+j\left(\|x-\sigma_{y}\|\right)h(\sigma_{y})\,dy$ $\displaystyle u(\sigma_{x})$ $\displaystyle=$ $\displaystyle\int_{H^{+}}j\left(\|\sigma_{x}-y\|\right)h(y)+j\left(\|\sigma_{x}-\sigma_{y}\|\right)h(\sigma_{y})\,dy$ $\displaystyle u(x)-u(\sigma_{x})$ $\displaystyle=$ $\displaystyle\int_{H_{+}}j\left(\|x-y\|\right)[h(y)-h(\sigma_{y})]-j\left(\sigma_{x}-y\right)[h(y)-h(\sigma_{y})]\,dy$ $\displaystyle=$ $\displaystyle\int_{H_{+}}\left(j\left(\|x-y\|\right)-j(\sigma_{x}-y)\right)\left(h(y)-h(\sigma_{y})\right)\,dy$ By (iii) $\|x-y\|<\|\sigma_{x}-y\|$, it follows that $j\left(\|x-y\|\right)\geq j\left(\|\sigma_{x}-y\|\right)$. On the other hand $h$ is Schwarz symmetric, hence $h(y)\geq h(\sigma_{y})$ for all $y\in H_{+}$, the conclusion follows. ###### Definition 2.4. Let $\Psi:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$: (a) $\Psi$ is supermodular if $(\Psi 2)$ holds. * (b) We say that $\Psi$ vanishes at hyperplanes if $\Psi(s_{1},0)=\Psi(0,s_{2})=0$ for all $s_{1},s_{2}\geq 0$. An important property of functions satisfying (c) is that the composition $(x,y)\mapsto\Psi\left(f(x),g(y)\right)$ is measurable on $\mathbb{R}_{+}$ for every $f,g\in F_{n}$. Hence $j\left(\|x-y\|\right)\Psi\left(f(x),g(y)\right)$ is measurable on $\mathbb{R}_{+}\times\mathbb{R}_{+}$. In the spirit of [4] and [6], we obtain: ###### Lemma 2.5. Assume that $\Psi:\mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ is a supermodular bounded function vanishing at hyperplanes. Then there exist two bounded nondecreasing functions $\varphi_{1}$ and $\varphi_{2}$ on $\mathbb{R}_{+}$ with $\varphi_{i}(0)=0$ and a Lipschitz continuous function $\tilde{\Psi}$ on $\mathbb{R}_{+}^{2}$ such that $\Psi(u,v)=\tilde{\Psi}\left(\varphi(u),\varphi(v)\right)$. Proof: First, we will prove the following: If $\varphi$ is a nondecreasing real-valued function defined on an interval $I$, then for every $f$ on I satisfying $\|f(u)-f(v)\|<c\left(\varphi(v)-\varphi(u)\right)$ where $u<v\in I$, $c$ is a constant, there exists a Lipschitz continuous function $\tilde{f}:\mathbb{R}\rightarrow[\inf{f},\sup{f}]$ such that $f(x)=\tilde{f}\circ\varphi(x)$ (2.0). If $f$ is nondecreasing then $\tilde{f}$ is nondecreasing also. The result is obvious for $t=\varphi(v)$ and $s=\varphi(u)<t$ since we have $\|\tilde{f}(t)-\tilde{f}(s)\|=\left\|f\left(\varphi(v)\right)-f\left(\varphi(u)\right)\right\|\leq c\left(\varphi(v)\right)-\left(\varphi(u)\right)=c(t-s).$ Now $\tilde{f}$ has a unique extension to the closure of the image and the complement consists of a countable number of disjoint bounded intervals, it is sufficient to interpolate $\tilde{f}$ linearly between the values, that were assigned to end-points. By construction $f=\tilde{f}\circ\varphi$ and $\tilde{f}(\mathbb{R})=[\inf f,\sup f]$ the extension we have made by linear interpolation preserves of course the modulus of continuity of $\tilde{f}$: $\|\tilde{f}(t)-\tilde{f}(s)\|\leq c(t-s)$ for all $t>s$. If $f$ is nondecreasing, it is easy to check that this property is inherited by $\tilde{f}$. Now we can prove our lemma: First note that the fact that $\Psi$ is supermodular and vanishes at hyperplanes imply that it is nondecreasing with respect to each variable and it is nonnegative. Now set $\varphi_{1}(u)=\lim\limits_{u\rightarrow+\infty}\Psi(u,v)$. $\varphi_{1}$ is well-defined on $\mathbb{R}_{+}$ since $\Psi$ is bounded and nondecreasing in the second variable. By the supermodularity of $\Psi$, it follows that $\Psi(u+h_{1},v+h_{2})-\Psi(u,v+h_{2})-\Psi(u+h_{1},v)+\Psi(u,v)\geq 0$ for any $u,v,h_{1}$ and $h_{2}\geq 0$. Letting $h_{2}$ tend to infinity, we obtain $\varphi_{1}(u+h_{1})-\varphi(u)\geq\Psi(u+h_{1},v)-\Psi(u,v)\geq 0$ for all $u,v,h_{1}\geq 0$. For a fixed $v$, the last inequality enables us to apply (2.0) to $\Psi(\cdot,v)$. Hence, there exists $\Psi^{1}$ such that: $\Psi(u,v)=\Psi^{1}(\varphi_{1}(u),v)$. A moment’s consideration shows that $\Psi^{1}$ inherits all the properties of $\Psi$. Now set $\varphi_{2}(v)=\lim\limits_{u\rightarrow+\infty}\Psi(u,v)$, a similar argument ensures us that there exists $\tilde{\Psi}$ such that $\Psi^{1}\left(\varphi_{1}(u),u\right)=\tilde{\Psi}^{1}\left(\varphi_{1}(u),\varphi_{2}(v)\right)$. $\tilde{\Psi}$ has the same monotonicity property as $\Psi^{1}$ and consequently as $\Psi$. Note that $\varphi_{1}(0)=\varphi_{2}(0)=0$ and the monotonicity properties of $\Psi$ imply that $\varphi_{1}$ and $\varphi_{2}$ are nondecreasing. ###### Lemma 2.6. Let $l,k>0$, $D=\\{h:\mathbb{R}^{n}\rightarrow\mathbb{R}:0\leq h(x)\leq k\mbox{ and }\int_{\mathbb{R}^{n}}h(x)\,dx\leq l\\}$. Suppose that $\Gamma:\mathbb{R}_{+}\rightarrow\mathbb{R}$ is a function satisfying: 1. (1) $\Gamma(0)=0$, 2. (2) $\Gamma(tx)\leq t\Gamma(x)$ for all $x\geq 0$ and $0<t<1$. 3. Assume also that 4. (3) $u:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a Schwarz symmetric function. Then for every $\nu\in D:\int_{\mathbb{R}^{n}}u(x)\Gamma\left(\nu(x)\right)\,dx\leq\int_{\mathbb{R}^{n}}u(x)\Gamma\left(k{\mathbf{1}}_{B}(x)\right)\,dx$ where $B$ is the ball centered at the origin with $\mu(B)=\ell/k$. Proof: (2) implies that $\displaystyle\int_{\mathbb{R}^{n}}u(x)\Gamma\left(\nu(x)\right)\,dx\leq\int_{\mathbb{R}^{n}}u(x)\Gamma(k)\frac{\nu(x)}{k}\,dx$ $\displaystyle=$ $\displaystyle\Gamma(k)\left[\int_{B}u(x)\left[\frac{\nu(x)}{k}-1+1\right]\,dx+\int_{\mathbb{R}^{n}-B}\frac{u(x)\nu(x)}{k}\,dx\right]$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{n}}u(x)\Gamma\left(k{\mathbf{1}}_{B(x)}\right)\,dx$ $\displaystyle\quad+\Gamma(k)\left[\int_{B}u(x)\left[\frac{\nu(x)}{k}-1\right]\,dx+\int_{\mathbb{R}^{n}-B}\frac{u(x)\nu(x)}{k}\,dx\right]\,dx.$ Using (3), it follows that the above integrals are $\leq\int_{\mathbb{R}^{n}}u(x)\Gamma\left(k{\mathbf{1}}_{B(x)}\right)\,dx+\Gamma(k)u(r)\left[\int_{\mathbb{R}^{n}}\frac{\nu(x)}{k}\,dx-\mu(B)\right]$ where $\mu(B)=V_{r}r^{n}$ (see definition 2.1). Thus $\int_{\mathbb{R}^{n}}u(x)\Gamma\left(\nu(x)\right)\,dx\leq\int_{\mathbb{R}^{n}}u(x)\Gamma\left(k{\mathbf{1}}_{B(x)}\right)\,dx$, since $\int_{\mathbb{R}^{n}}\frac{\nu(x)}{k}\,dx\leq\mu(B)=\ell/k$. If additionally $\int_{\mathbb{R}^{n}}u(x)\Gamma\left(\nu(x)\right)\,dx<\infty$ for any $\nu\in D$, (2) holds with strict inequality and $u$ is strictly decreasing, we can prove that for every $\nu\in D$: $\int_{\mathbb{R}^{n}}u(x)\Gamma\left(\nu(x)\right)\,dx<\int_{\mathbb{R}^{n}}u(x)\Gamma\left(k{\mathbf{1}}_{B(x)}\right)\,dx$. ## 3\. Proof of the result For the convenience of the reader, the proof will be divided in three parts. First part: We suppose that: $\Psi(\cdot,s_{2})$ is absolutely continuous for every $s_{2}\geq 0$, and $\Psi(s_{1},\cdot)$ is absolutely continuous for every $s_{1}\geq 0$. First note that $(\Psi 1)$ and $(\Psi 2)$ imply that $\Psi$ is a non- decreasing function with respect to each variable and it is nonnegative. Let $(f_{1},f_{2})\in C$, $(\Psi 2)$ and (j1) imply that $\displaystyle J\left(f_{1},f_{2}\right)\leq J\left(f_{1}^{},f_{2}^{}\right)$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\Psi\left(f_{1}^{}(x),f_{2}^{}(y)\right)j\left(\|x-y\|\right)\,dx\,dy$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\left(\int_{0}^{f_{2}^{}(y)}F\left(f_{1}^{}(x),s\right)\,ds\right)j\left(\|x-y\|\right)\,dx\,dy$ where $\Psi(s_{1},s_{2})=\int_{0}^{s_{2}}F\left(s_{1},u\right)\,du$. Applying Tonelli’s theorem (see (3.0)), we obtain: $J\left(f_{1}^{},f_{2}^{}\right)=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}j\left(\|x-y\|\right){\mathbf{1}}_{\\{y\in\mathbb{R}^{n}:f_{2}^{}(y)\geq s\\}}F\left(f_{1}^{}(x),s\right)\,dy\,dx\,ds.$ Setting $u(x,s)=\int_{\mathbb{R}^{n}}{\mathbf{1}}_{\\{y\in\mathbb{R}^{n}:f_{2}^{}(y)\geq s\\}}j\left(\|x-y\|\right)\,dy$, it follows from lemma 2.3 that $u$ is radial and radially decreasing with respect to $x$ for every fixed $s$. $J\left(f_{1}^{},f_{2}^{}\right)=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}u(x,s)F\left(f_{1}^{}(x),s\right)\,dx\,ds.$ Now for a fixed $s_{1}\geq 0$, $\Psi(s_{1},x_{2})-\Psi(s_{1},x_{1})=\int_{x_{1}}^{x_{2}}F(s_{1},t)\,dt\geq 0$ for $x_{2}\geq x_{1}$; from which we deduce that $F(s_{1},t)$ is nonnegative for almost every $t\geq 0$. (3.0) On the other hand, $0=\Psi(0,s_{2})=\int_{0}^{s_{2}}F(0,u)\,du$. By the nonnegativity of $F$, we conclude that $F(0,s)=0$ for almost every $s\geq 0$. Moreover ($\Psi$3) says that: $\Psi(tx,d)-t\Psi(x,d)-\Psi(tx,c)+t\Psi(x,c)\leq 0$ for every $x\geq 0$, $d\geq c\geq 0$. Integrating this inequality, we have $\int_{c}^{d}F(tx,u)-tF(x,u)\,du\geq 0$ for every $x\geq 0$; $d\geq c\geq 0$. Hence $F(tx,u)\leq tF(x,u)$ for all $x\geq 0$, $t\in]0,1[$ and almost every $u\geq 0$. This shows that for almost every $s\geq 0$, the function $u(x,s)F\left(f_{1}^{}(x),s\right)$ satisfies all the hypotheses of lemma 2.6, consequently: For almost every $s\geq 0$ $\int_{\mathbb{R}^{n}}u(x,s)F\left(f_{1}^{}(x),s\right)\,dx\leq\int_{\mathbb{R}^{n}}u(x,s)F\left(k_{1}{\mathbf{1}}_{B_{1}}(x),s\right)\,dx$ and (3.1) $J\left(f_{1}^{},f_{2}^{}\right)\leq J\left(k_{1}{\mathbf{1}}_{B_{1}},f_{2}^{}\right).$ Using the same argument, we easily conclude that (3.2) $J\left(k_{1}{\mathbf{1}}_{B_{1}},f_{2}^{}\right)\leq J\left(k_{1}{\mathbf{1}}_{B_{1}},k_{2}{\mathbf{1}}_{B_{2}}\right).$ By [4, Theorem 2] we know that: (3.3) $J\left(f_{1},f_{2}\right)\leq J\left(f_{1}^{},f_{2}^{}\right).$ Combining these three inequalities, we obtain: $J\left(f_{1},f_{2}\right)\leq J\left(f_{1}^{},f_{2}^{}\right)\leq J\left(k_{1}{\mathbf{1}}_{B_{1}},f_{2}^{}\right)\leq J\left(k_{1}{\mathbf{1}}_{B_{1}},k_{2}{\mathbf{1}}_{B_{2}}\right).$ If in addition, we have strict inequality in ($\Psi$2) and ($\Psi$3), $j$ is strictly decreasing and $J\left(f_{1},f_{2}\right)<\infty$ for any $f_{1},f_{2}\in C$ then [4, Theorem 2] asserts that equality occurs in (3.3) if and only if there exists $x_{0}\in\mathbb{R}^{n}$ such that $f_{1}=f_{1}^{}(\cdot-x_{0})$ and $f_{2}=f_{2}^{}(\cdot-x_{0})$. On the other hand, by lemma 2.6, equality occurs in (3.1) if and only if $f_{1}^{}=k_{1}{\mathbf{1}}_{B_{1}}$. Similarly equality holds in (3.2) if and only if $f_{2}^{}=k_{2}{\mathbf{1}}_{B_{2}}$. Conclusion: we have proved that for any absolutely continuous function $\Psi$ satisfying ($\Psi$1), ($\Psi$2), ($\Psi$3) with a kernel $j$ satisfying (j1) $\left(k_{1}{\mathbf{1}}_{B_{1}},k_{2}{\mathbf{1}}_{B_{2}}\right)$ is a maximizer of $J$ under the constraint $C$. If additionally ($\Psi$2), ($\Psi$3) hold with strict inequality $j$ is strictly decreasing and $J\left(f_{1},f_{2}\right)<\infty$ for all $\left(f_{1},f_{2}\right)\in C$ then $\left(k_{1}{\mathbf{1}}_{B_{1}},k_{2}{\mathbf{1}}_{B_{2}}\right)$ is the unique maximizer of (P1) (up to a translation). Remark 1: $\Psi$ is a nondecreasing function with respect to each variable, it is then of bounded variations. The absolute continuity is then equivalent to its continuity and the fact that it satisfies the N-Luzin property. Remark 2: We can remove condition ($\Psi$1) from our theorem by modifying ($\Psi$3) and adding an integrability assumption in a same way as [8, Proposition 3.2]. Part 2: $\Psi$ is bounded. Applying lemma 2.5, we know that there exist $\varphi_{1},\varphi_{2}$ such that $\Psi(s_{1},s_{2})=\tilde{\Psi}\left(\varphi_{1}(s_{1}),\varphi_{2}(s_{2})\right)$, where $\tilde{\Psi}$ is Lipschitz continuous with respect to each variable, there exist a function $\tilde{F}$ defined on $\mathbb{R}_{+}$ such that $\tilde{\Psi}(s_{1},s_{2})=\int_{0}^{s_{2}}\tilde{F}(s_{1},u)\,du$. $\displaystyle J\left(f_{1},f_{2}\right)$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\tilde{\Psi}\left(\varphi_{1}\left(f_{1}^{}(x)\right),\varphi_{2}\left(f_{2}^{}(x)\right)\right)j\left(\|x-y\|\right)\,dx\,dy$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}\left(\int_{\mathbb{R}^{n}}\nu(x,s)\tilde{F}\left(\varphi_{1}\left(f_{1}^{}(x)\right),s\right)\right)\,dx\,ds$ where $\nu(x,s)=\int_{\mathbb{R}^{n}}{\mathbf{1}}_{\\{y\in\mathbb{R}^{n}:\varphi_{2}\left(f_{2}^{}(y)\right)\geq s\\}}j\left(\|x-y\|\right)\,dy$. The function ${\mathbf{1}}_{\\{y\in\mathbb{R}^{n}:\varphi_{2}\left(f_{2}^{}(y)\right)\geq s\\}}$ is Schwarz-symmetric for every $s$ since $\varphi_{2}$ is nondecreasing. We can then apply Part 1 and the result follows. Remark 3: Here we cannot obtain a uniqueness result since $\varphi_{1}$ and $\varphi_{2}$ do not inherit the strict monotonicity properties of $\Psi$. Part 3: $\Psi$ is not bounded. For $L>0$, set $\Psi^{L}\left(s_{1},s_{2}\right)=\Psi\left(\min(s_{1},L),\min(s_{2},L)\right)$. It is easy to check that $\Psi^{L}$ inherits all the properties of $\Psi$ stated in our result. Moreover Part 2 applies to $\Psi^{L}$ since it is a bounded function. Noticing that $\Psi^{L}\rightarrow\Psi$, the monotone convergence theorem enables us to conclude. ## References * [1] E. H. Lieb. Existence and uniqueness of the minimizing solution of the Choquard’s nonlinear equation. Studies in Applied Mathematics, 57:93–105, 1977. * [2] A. Burchard. Cases of equality in Riesz rearrangement inequality. Ann. Math. (2), 143:499–527, 1996. * [3] E. H. Lieb and M. Loss. Analysis, volume 14 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2001. * [4] A. Burchard and H. Hajaiej. Rearrangement inequalities for functionals with monotone integrands. J. Funct. Anal., 233(2):561–582, 2006. * [5] G. R. Burton. Vortex-rings of prescribed impulse. Math. Proc. Cambridge Philos. Soc., 134(3):515–528, 2003. * [6] A. Sklar. Functions de répartition à $n$ dimensions et leurs marges. Inst. Statist. Univ. Paris, 8:229–231, 1959. * [7] A. Burchard and M. Schmuckenschläger. Comparison theorems for exist times. Geom. Funct. Anal., 11:651–692, 2001. * [8] C. Draghici and H. Hajaiej. Uniqueness and characterization of maximizers of integral functionals with constraints. Preprint.	arxiv-papers	2009-03-16T18:16:49	2024-09-04T02:49:01.177070	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hichem Hajaiej", "submitter": "Hichem Hajaiej", "url": "https://arxiv.org/abs/0903.2821" }
0903.2826	# Quantitative stability estimate for an optimization problem under constraints H. Hajaiej ###### Abstract. A class of functionals maximized by characteristic functions of balls is identified by a mass transportation argument. A variational approach to the study of standing waves for the nonlinear Schrödinger equation leads to the minimization of functionals like $\frac{1}{2}\int_{\mathbb{R}^{n}}\|\nabla u(x)\|^{2}\,dx-\int_{\mathbb{R}^{n}}F(\|x\|,u(x))\,dx\,,$ (1) over all $u\in H^{1}(\mathbb{R}^{n})$, $u\geq 0$, such that $\int_{\mathbb{R}^{n}}u^{2}=1$, see [3] (here $n\in\mathbb{N},n\geq 1$). The function $F$ describes the index of refraction of the media in which the wave propagates. A typical example is $F(r,s)=p(r)s^{2}+q(r)s^{d}\,,\quad 2<d<2+\frac{4}{n}\,,$ where $p$ and $q$ are positive decreasing functions, and the constraint on $d$ has to be assumed so to avoid non-existence issues due to unbalanced scalings. The two terms of the energy (1) are in competition. Indeed, if we try to maximize $\int_{\mathbb{R}^{n}}F(\|x\|,u(x))dx$ under the additional constraint that $u\leq a$, then the unique maximizer is given by the function $a\,1_{rB}$, having infinite Dirichlet integral (here $B$ is the Euclidean unit ball and $r>0$ is such that $\int_{\mathbb{R}^{n}}(a\,1_{rB})^{2}=1$). In this note we identify a simple sufficient condition on an integrand $F$ ensuring that $\int_{\mathbb{R}^{n}}F(\|x\|,u(x))dx$ presents this behavior. More precisely, we are going to consider integrands $F:\mathbb{R}^{+}\times\mathbb{R}^{+}\to\mathbb{R}^{+}$ (here $\mathbb{R}^{+}:=[0,\infty)$), such that 1. (H1) for every $s\in\mathbb{R}^{+}$, $F(\cdot,s)$ is decreasing; for a.e. $r\in\mathbb{R}^{+}$, $F(r,\cdot)$ is continuous on $\mathbb{R}^{+}$; 2. (H2) there exist $\alpha\in L^{1}(\mathbb{R}^{+},r^{n-1}dr)$ and a locally bounded function $\beta:\mathbb{R}\to\mathbb{R}^{+}$ such that, for a.e. $r\in\mathbb{R}^{+}$ and every $s\in\mathbb{R}^{+}$, $F(r,s)\leq\alpha(r)\beta(s)$. Given $a>0$ and $p\geq 1$, we consider the convex subset of $L^{p}(\mathbb{R}^{n})$ $X:=\left\\{u\in L^{p}(\mathbb{R}^{n}):0\leq u\leq a\,,\int_{\mathbb{R}^{n}}u^{p}\leq 1\right\\}\,,$ and define a functional ${\mathcal{F}}$ on $X$ by setting $\displaystyle{\mathcal{F}}(u)=\int_{\mathbb{R}^{n}}F(\|x\|,u(x))dx\,,\quad\forall u\in X\,.$ Note that, thanks to (H1) and (H2), $x\in\mathbb{R}^{n}\mapsto F(\|x\|,u(x))$ is measurable and ${\mathcal{F}}(u)\in\mathbb{R}^{+}$ for every $u\in X$. We are going to prove the following theorem: ###### Theorem 1. Let $a>0$, $p\geq 1$, and let $F$ be such that (H1) and (H2) hold true. Assume that there exists $t>0$ such that the ball $E=\\{x\in\mathbb{R}^{n}:F(\|x\|,a)>t\\}$ satisfies $a^{p}\|E\|=1$, and that, for a.e. $r\in\mathbb{R}^{+}$ and for every $\lambda\in[0,1]$, $F(r,\lambda a)\leq\lambda^{p}F(r,a)\,.$ (2) Then the function $w=a\,1_{E}$ is a maximum of ${\mathcal{F}}$ on $X$. Moreover, if $F(\cdot,a)$ is strictly decreasing, then $w=a\,1_{E}$ is the unique maximizer of ${\mathcal{F}}$ on $X$. The proof of Theorem 1 is based on a basic result in mass transportation theory, namely the Brenier Theorem [1] (see also [4]): given two Radon measures $\mu_{1},\mu_{2}$ on $\mathbb{R}^{n}$, both absolutely continuous with respect to the Lebesgue measure and such that $\mu_{1}(\mathbb{R}^{n})=\mu_{2}(\mathbb{R}^{n})$, there exists a convex function $\varphi:\mathbb{R}^{n}\to[0,\infty]$ and a Borel measurable map $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that $T(x)=\nabla\varphi(x)$ at a.e. $x\in\mathbb{R}^{n}$ and $T$ pushes forward $\mu_{1}$ into $\mu_{2}$, i.e. $\int_{\mathbb{R}^{n}}H(y)d\mu_{2}(y)=\int_{\mathbb{R}^{n}}H(T(x))d\mu_{2}(x)\,,$ (3) for every Borel function $H:\mathbb{R}^{n}\to[0,\infty]$. The mass transportation approach to Theorem 1 allows also to deduce a quantitative stability estimate on the maximality of $w=a1_{E}$, see Corollary 2 below. We pass now to prove Theorem 1. ###### Proof of Theorem 1. By (2), as $F(r,\cdot)$ is continuous for a.e. $r\in\mathbb{R}^{+}$, we deduce that $F(r,0)=0$ for a.e. $r\in\mathbb{R}^{+}$. We let $S^{n-1}=\\{x\in\mathbb{R}^{n}:\|x\|=1\\}$, and denote by $\sigma$ the $(n-1)$-dimensional Hausdorff measure restricted to $S^{n-1}$. Step one: Let us fix $u\in X$ and construct an auxiliary function $v=a\,1_{G}$ by letting $G:=\left\\{x\in\mathbb{R}^{n}:\|x\|<\kappa\left(\frac{x}{\|x\|}\right)\right\\}\,,$ where we have introduced $\kappa:S^{n-1}\to\mathbb{R}^{+}$, $\kappa(\nu):=\left(\frac{n}{a^{p}}\int_{0}^{\infty}u(r\nu)^{p}r^{n-1}dr\right)^{1/n}\,,\quad\nu\in S^{n-1}\,.$ (4) Note that $v(r\nu)=a\,1_{[0,\kappa(\nu)]}(r)$, and that the value of $\kappa(\nu)$ has been chosen so that the measures $1_{\mathbb{R}^{+}}(r)u(r\nu)^{p}\,r^{n-1}\,dr\,\quad\mbox{and}\quad 1_{\mathbb{R}^{+}}(r)v(r\nu)^{p}\,r^{n-1}\,dr\,,$ have the same total mass on $\mathbb{R}$. For every $\nu\in S^{n-1}$, let $T_{\nu}$ denote the map given by Brenier theorem. By construction $T_{\nu}$ is increasing on $\mathbb{R}$, moreover, thanks to (3) we have $\int_{\mathbb{R}^{+}}H(r)v(\nu r)^{p}r^{n-1}dr=\int_{\mathbb{R}^{+}}H(T_{\nu}(r))u(\nu r)^{p}r^{n-1}dr\,,$ (5) for every Borel function $H:\mathbb{R}\to[0,\infty]$: in particular $T_{\nu}(r)\in[0,k(\nu)]$ for a.e. $r\in\mathbb{R}$. Note also that, as $0\leq u\leq a$, we clearly have $T_{\nu}(r)\leq r\,,\quad\mbox{for a.e. $r\in\mathbb{R}^{+}$.}$ (6) We are going to prove that ${\mathcal{F}}(u)\leq{\mathcal{F}}(v)$. By (2) we have that $\displaystyle{\mathcal{F}}(u)=\int d\sigma(\nu)\int_{\mathbb{R}^{+}}F(r,u(r\nu))r^{n-1}dr\leq\int d\sigma(\nu)\int_{\mathbb{R}^{+}}\frac{F(r,a)}{a^{p}}u(r\nu)^{p}r^{n-1}dr\,,$ (7) while at the same time, thanks to (5) $\displaystyle{\mathcal{F}}(v)$ $\displaystyle=$ $\displaystyle\int d\sigma(\nu)\int_{0}^{\kappa(\nu)}F(r,a)r^{n-1}dr=\int d\sigma(\nu)\int_{\mathbb{R}^{+}}\frac{F(r,a)}{a^{p}}v(r\nu)^{p}r^{n-1}dr$ $\displaystyle=$ $\displaystyle\int d\sigma(\nu)\int_{\mathbb{R}^{+}}\frac{F(T_{\nu}(r),a)}{a^{p}}u(r\nu)^{p}r^{n-1}dr\,,$ By (H1) and (6) it follows immediately that ${\mathcal{F}}(u)\leq{\mathcal{F}}(v)$. Step two: We are going to prove that ${\mathcal{F}}(v)\leq{\mathcal{F}}(w)$. We start by noticing that $\|E\|=\|G\|$. Indeed by (4) $\|G\|=\int\frac{\kappa(\nu)^{n}}{n}\,d\sigma(\nu)=\frac{1}{a^{p}}\int_{\mathbb{R}^{n}}u^{p}=\|E\|\,.$ In particular $\|E\setminus G\|=\|G\setminus E\|$, and, without loss of generality, $\|E\setminus G\|>0$. Consider the Brenier map $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ between $1_{E\setminus G}(x)dx$ and $1_{G\setminus E}(y)dy$. By (3), $\int_{E\setminus G}H(y)dy=\int_{G\setminus E}H(T(x))dx\,,$ (8) for every Borel function $H:\mathbb{R}^{n}\to[0,\infty]$. On choosing $H(y)=F(y,a)$ we find $\displaystyle\int_{E\setminus G}F(\|x\|,a)dx=\int_{G\setminus E}F(\|T(x)\|,a)dx\,,$ (9) while, on taking $H(y)=1_{E\setminus G}(y)$, we prove that $T(x)\in E\setminus G$ for a.e. $x\in G\setminus E$. As $E$ is a ball, this last remark implies that $\displaystyle\|T(x)\|\leq\|x\|\,,\quad\mbox{for a.e. $x\in G\setminus E$\,.}$ (10) On combining (10) with (9) we get $\displaystyle{\mathcal{F}}(w)$ $\displaystyle=$ $\displaystyle\int_{G\cap E}F(\|x\|,a)dx+\int_{E\setminus G}F(\|x\|,a)dx$ (11) $\displaystyle=$ $\displaystyle\int_{G\cap E}F(\|x\|,a)dx+\int_{G\setminus E}F(\|T(x)\|,a)dx$ $\displaystyle\geq$ $\displaystyle\int_{G\cap E}F(\|x\|,a)dx+\int_{G\setminus E}F(\|x\|,a)dx={\mathcal{F}}(v)\,,$ and the conclusion follows. Let us now assume that for every $s\in\mathbb{R}^{+}$ the function $F(\cdot,a)$ is strictly decreasing, and consider a function $u\in X$ that maximizes ${\mathcal{F}}$ on $X$, i.e. such that ${\mathcal{F}}(u)={\mathcal{F}}(w)$. We want to show that $u=w$ a.e. on $\mathbb{R}^{n}$. Let us prove that $G=E$ up to null sets. Indeed, let $R$ denote the radius of the ball $E$. If $\|G\setminus E\|>0$, then we can consider $T$ and repeat the above argument. Since $F(\cdot,a)$ is strictly decreasing and equality holds in (11), we find that $\|T(x)\|=\|x\|$ for a.e. $x\in G\setminus E$. Thus $\|T(x)\|\geq R$ for a.e. $x\in\mathbb{R}$; but $T(x)\in E\setminus G$ for a.e. $x\in G\setminus E$, therefore it must be $\|G\setminus E\|=0$, a contradiction. As $G=E$ up to null sets, we have $\kappa(\nu)=R$ for every $\nu\in S^{n-1}$. The equality sign in (7) implies that, for $\sigma$-a.e. $\nu\in S^{n-1}$, $T_{\nu}(r)=r$ for a.e. $r\in\\{t:u(\nu t)>0\\}$. As $0\leq T_{\nu}\leq\kappa(\nu)=R$, by (5) and (6) we deduce that $\\{t:u(\nu t)>0\\}\subset[0,R]$ for $\sigma$-a.e. $\nu\in S^{n-1}$. On applying (5) to $H=1_{\\{t:u(\nu t)>0\\}}$ we deduce $u(\nu r)=a$ on $\\{t:u(\nu t)>0\\}$, therefore that $u(\nu r)=a\,1_{[0,R]}(r)$. In particular $u=w$ a.e. on $\mathbb{R}^{n}$. ∎ We come now to a quantitative stability estimate: ###### Corollary 2. Under the assumptions of Theorem 1, let us assume the existence of $\lambda>0$ such that, whenever $0<r_{1}<r_{2}$, $F(r_{1},a)\geq F(r_{2},a)+\lambda(r_{2}-r_{1})\,.$ (12) Then, for every $u\in X$ we have that $\int_{\mathbb{R}^{n}}\|u-w\|^{p}\leq C(n,p,a)\sqrt{\frac{{\mathcal{F}}(w)-{\mathcal{F}}(u)}{\lambda}}\,.$ (13) where $C(n,p,a)$ is a constant depending only on $n$, $p$ and $a$. ###### Proof. Let $\delta:={\mathcal{F}}(w)-{\mathcal{F}}(u)$. Thanks to (12), from (7) and (11) we find that $\displaystyle\delta$ $\displaystyle\geq$ $\displaystyle\lambda\int_{G\setminus E}(\|x\|-\|T(x)\|)dx\,,$ (14) $\displaystyle\delta$ $\displaystyle\geq$ $\displaystyle\lambda\int d\sigma(\nu)\int_{0}^{\infty}(r-T_{\nu}(r))\frac{u(r\nu)^{p}}{a^{p}}r^{n-1}dr\,.$ (15) We now consider (14) and (15) separately: Step one: Let $\varepsilon\in(0,R)$, then $(R+\varepsilon)^{n}\leq R^{n}+c(n)R^{n-1}\varepsilon$. Thus $\displaystyle\frac{1}{a^{p}}\int_{\mathbb{R}^{n}}\|w-v\|^{p}$ $\displaystyle=$ $\displaystyle\|E\Delta G\|=2\|G\setminus E\|\leq 2\\{\|G\setminus B_{R+\varepsilon}\|+\|B_{R+\varepsilon}\setminus B_{R}\|\\}$ $\displaystyle\leq$ $\displaystyle C(n)\left\\{\|G\setminus B_{R+\varepsilon}\|+\varepsilon R^{n-1}\right\\}\,.$ If $x\in G\setminus B_{R+\varepsilon}$ then $\|x\|\geq R+\varepsilon\geq\|T(x)\|+\varepsilon$. By (14) we have $\|G\setminus B_{R+\varepsilon}\|\leq(\delta/\varepsilon\lambda)$, therefore we come to $\frac{1}{a^{p}}\int_{\mathbb{R}^{n}}\|w-v\|^{p}\leq C(n)\left\\{\frac{\delta}{\lambda\varepsilon}+\varepsilon R^{n-1}\right\\}\,.$ We minimize over $\varepsilon\in[0,R]$ and find $\frac{1}{a^{p}}\int_{\mathbb{R}^{n}}\|w-v\|^{p}\leq C(n)\max\left\\{\sqrt{\frac{\delta R^{n-1}}{\lambda}},\frac{\delta}{\lambda R}\right\\}\,.$ (16) Step two: We start by noticing that $\displaystyle\int_{\mathbb{R}^{n}}\|u-v\|^{p}$ $\displaystyle=$ $\displaystyle\int_{G}(a-u)^{p}+\int_{\mathbb{R}^{n}\setminus G}u^{p}\leq\int_{G}(a^{p}-u^{p})+\int_{\mathbb{R}^{n}\setminus G}u^{p}=2\int_{\mathbb{R}^{n}\setminus G}u^{p}$ $\displaystyle=$ $\displaystyle 2\int\tau_{2}(\nu)d\sigma(\nu)\,,$ where, for every $\nu\in S^{n-1}$, we have set $\tau_{1}(\nu):=\int_{0}^{\kappa(\nu)}u(r\nu)^{p}r^{n-1}dr\,,\quad\tau_{2}(\nu):=\int_{\kappa(\nu)}^{\infty}u(r\nu)^{p}r^{n-1}dr\,.$ Since $a^{p}\kappa(\nu)^{n}/n=\tau_{1}(\nu)+\tau_{2}(\nu)$, we have that $\displaystyle a^{p}\frac{T_{\nu}(r)^{n}}{n}$ $\displaystyle\leq$ $\displaystyle\tau_{1}(\nu)+\int_{\kappa(\nu)}^{r}u(t\nu)t^{n-1}dt\leq\tau_{1}(\nu)+a^{p}\frac{r^{n}}{n}-a^{p}\frac{\kappa(\nu)^{n}}{n}\,,$ for every $r\geq\kappa(\nu)$, i.e., $T_{\nu}(r)\leq\left(r^{n}-\frac{n\tau_{2}(\nu)}{a^{p}}\right)^{1/n}\,,\quad\forall r\geq\kappa(\nu)\,.$ Then by (15) we deduce that $\displaystyle\frac{\delta}{\lambda}$ $\displaystyle\geq$ $\displaystyle\int d\sigma(\nu)\int_{\kappa(\nu)}^{\infty}\left[r-\left(r^{n}-\frac{n\tau_{2}(\nu)}{a^{p}}\right)^{1/n}\right]u(r\nu)^{p}r^{n-1}dr$ $\displaystyle\geq$ $\displaystyle\int d\sigma(\nu)\int_{\kappa(\nu)}^{\infty}\frac{\tau_{2}(\nu)}{a^{p}}\left(r^{n}-\frac{n\tau_{2}(\nu)}{a^{p}}\right)^{(1/n)-1}u(r\nu)^{p}r^{n-1}dr$ $\displaystyle\geq$ $\displaystyle\int\frac{\tau_{2}(\nu)^{2}}{a^{p}}\left(\kappa(\nu)^{n}-\frac{n\tau_{2}(\nu)}{a^{p}}\right)^{(1/n)-1}d\sigma(\nu)$ $\displaystyle=$ $\displaystyle\frac{c(n)}{a^{p/n}}\int\tau_{2}(\nu)^{2}\tau_{1}(\nu)^{(1/n)-1}d\sigma(\nu)\,.$ By Hölder inequality $\displaystyle\int_{\mathbb{R}^{n}}\|u-v\|^{p}\leq 2\int\tau_{2}(\nu)d\sigma(\nu)\leq C(n)\sqrt{a^{p/n}\frac{\delta}{\lambda}}\sqrt{\int\tau_{1}(\nu)^{1-(1/n)}d\sigma(\nu)}\,.$ By Jensen inequality for concave functions, $\displaystyle\int\tau_{1}(\nu)^{1-(1/n)}d\sigma(\nu)$ $\displaystyle\leq$ $\displaystyle C(n)\left(\int\tau_{1}(\nu)d\sigma(\nu)\right)^{1-(1/n)}$ $\displaystyle\leq$ $\displaystyle C(n)\left(\int a^{p}\frac{\kappa(\nu)^{n}}{n}d\sigma(\nu)\right)^{1-(1/n)}=C(n)\,,$ and we come to conclude that $\int_{\mathbb{R}^{n}}\|u-v\|^{p}\leq C(n)\sqrt{a^{p/n}\frac{\delta}{\lambda}}$ (17) Step three: As $\int\|w-u\|^{p}\leq 2^{p}$, (13) follows trivially whenever $\delta\geq\leq$. Let us now assume that $\delta\leq\lambda$, then (13) is easily deduced from (16) and (17). ∎ ## References * [1] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (4) (1991) 375–417. * [2] C. Draghici & H. Hajaiej, Uniqueness and characterization of maximizers of integral functionals with constraints, preprint. * [3] H. Hajaiej & C. A. Stuart, On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation. Adv. Nonlinear Stud. 4 (2004), no. 4, 469–501. * [4] R.J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (2) (1995) 309–323. * [5] R.J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1) (1997) 153–179.	arxiv-papers	2009-03-16T18:28:03	2024-09-04T02:49:01.182688	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hichem Hajaiej", "submitter": "Hichem Hajaiej", "url": "https://arxiv.org/abs/0903.2826" }

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.